Stability, observer design and control of networks using Lyapunov methods

Transcription

1 Stablty, observer desgn and control of networks usng Lyapunov methods von Lars Naujok Dssertaton zur Erlangung des Grades enes Doktors der Naturwssenschaften - Dr. rer. nat. - Vorgelegt m Fachberech 3 (Mathematk & Informatk) der Unverstät Bremen m März 2012

2 Tag der Enrechung: Gutachter: Prof. Dr. Sergey Dashkovsky, Unverstät Bremen 2. Gutachter: Prof. Dr. Lars Grüne, Unverstät Bayreuth

3 Acknowledgement Frst of all I am deeply grateful to my advsor Sergey Dashkovsky, who supported me wthn the three years workng on ths thess by many dscussons, new research deas and takng tme for the contnuous mprovement of the qualty of my work. I lke to thank all the members of the workgroup Mathematcal Modelng of Complex Systems, whch s a research group assocated to the Center for Industral Mathematcs (ZeTeM) at the Unversty of Bremen, for the useful dscussons and semnars. Especally, I apprecate the cooperaton wth my colleague Mchael Kosmykov, who read parts of my thess and I enjoyed the nce atmosphere sharng an offce wth hm. Also, I lke to thank my colleague Andr Mronchenko for the useful dscussons and the cooperaton n the paper about mpulsve systems. There were helpful remarks and dscussons wth Lars Grüne, Hamd Reza Karm, Danel Lberzon, Andy Teel and Faban Wrth, whch nspred me for parts of ths thess. The results presented n ths thess, were developed durng my work as a research assstant at the Unversty of Bremen wthn the framework of the Collaboratve Research Center (CRC) 637 Autonomous Cooperatng Logstc Processes, subproject A5 Dynamcs of Autonomous Systems, supported by the German Research Foundaton (DFG). Especally, I lke to thank Mchael Görges and Thomas Jagalsk for the cooperaton wthn the subproject A5. Last but not least, I wll not forget the patence and support, whch I got from my famly, especally from my parents, Anke und Peter, and my grlfrend Yvonne. Thank you very much. Bremen and Hamburg, March the 1st, 2012 Lars Naujok

4 Abstract We nvestgate dfferent aspects of the analyss and control of nterconnected systems. Dfferent tools, based on Lyapunov methods, are provded to analyze such systems n vew of stablty, to desgn observers and to control systems subject to stablzaton. All the dfferent tools presented n ths work can be used for many applcatons and extend the analyss toolbox of networks. Consderng systems wth nputs, the stablty property nput-to-state dynamcal stablty (ISDS) has some advantages over nput-to-state stablty (ISS). We ntroduce the ISDS property for nterconnected systems and provde an ISDS small-gan theorem wth a constructon of an ISDS-Lyapunov functon and the rate and the gans of the ISDS estmaton for the whole system. Ths result s appled to observer desgn for sngle and nterconnected systems. Observers are used n many applcatons where the measurement of the state s not possble or dsturbed due to physcal reasons or the measurement s uneconomcal. By the help of error Lyapunov functons we desgn observers, whch have a so-called quas ISS or quas-isds property to guarantee that the dynamcs of the estmaton error of the systems state has the ISS or ISDS property, respectvely. Ths s appled to quantzed feedback stablzaton. In many applcatons, there occur tme-delays and/or nstantaneous jumps of the systems state. At frst, we provde tools to check whether a network of tme-delay systems has the ISS property usng ISS-Lyapunov-Razumkhn functons and ISS-Lyapunov-Krasovsk functonals. Then, these approaches are also used for nterconnected mpulsve systems wth tme-delays usng exponental Lyapunov-Razumkhn functons and exponental Lyapunov- Krasovsk functonals. We derve condtons to assure ISS of an mpulsve network wth tme-delays. Controllng a system n a desred and optmal way under gven constrants s a challengng task. One approach to handle such problems s model predctve control (MPC). In ths thess, we ntroduce the ISDS property for MPC of sngle and nterconnected systems. We provde condtons to assure the ISDS property of systems usng MPC, where the prevous result of ths thess, the ISDS small-gan theorem, s appled. Furthermore, we nvestgate the ISS property for MPC of tme-delay systems usng the Lyapunov-Krasovsk approach. We prove theorems, whch guarantee ISS for sngle and nterconnected systems usng MPC.

5 Contents Introducton 7 1 Prelmnares Input-to-state stablty Interconnected systems Input-to-state dynamcal stablty (ISDS) ISDS for sngle systems ISDS for nterconnected systems Examples Observer and quantzed output feedback stablzaton Quas-ISDS observer for sngle systems Quas-ISS and quas-isds observer for nterconnected systems Applcatons Dynamc quantzers ISS for tme-delay systems ISS for sngle tme-delay systems ISS for nterconnected tme-delay systems Lyapunov-Razumkhn approach Lyapunov-Krasovsk approach Applcatons n logstcs A certan scenaro ISS for mpulsve systems wth tme-delays Sngle mpulsve systems wth tme-delays The Lyapunov-Razumkhn methodology The Lyapunov-Krasovsk methodology Networks of mpulsve systems wth tme-delays The Lyapunov-Razumkhn approach The Lyapunov-Krasovsk approach Example

6 6 6 Model predctve control ISDS and MPC Sngle systems Interconnected systems ISS and MPC of tme-delay systems Sngle systems Interconnected systems Summary and Outlook ISDS Observer and quantzed output feedback stablzaton ISSforTDS ISS for mpulsve systems wth tme-delays MPC Bblography 115

7 Introducton In ths thess, we provde tools to analyze, to observe and to control networks wth regard to stablty based on Lyapunov methods. A network conssts of an arbtrary number of nterconnected subsystems. We consder such networks, whch can be modeled usng ordnary dfferental equatons of the form ẋ (t) =f (x 1 (t),...,x n (t),u(t)), =1,...,n, (1) whch can be seen as one sngle system of the form ẋ(t) =f(x(t),u(t)), (2) where the tme t s contnuous, x(t) =(x 1 (t),...,x n (t)) T R N, wth x (t) R N,N= N, denotes the state of the system and u R m s a measurable and essentally bounded nput functon of the system. For example, the dynamcs of a logstc network, such as a producton network, can be descrbed by a system of the form (1) [48, 15, 12, 13, 103]. In ths work, we nvestgate nterconnected systems n vew of stablty. We consder the noton of nput-to-state stablty (ISS), ntroduced n 1989 by Sontag, [114]. ISS means, roughly speakng, that the norm of the soluton of a system s bounded for all tmes by x(t; x 0,u) max {β( x 0,t),γ ISS ( u )}, (3) where x(t; x 0,u) denotes the soluton of a system wth ntal value x 0, where denotes the Eucldean norm and s the essental supremum norm. The functon β : R + R + R + ncreases n the frst argument and tends to zero, f the second argument tends to nfnty. The functon γ ISS : R + R + s strctly ncreasng wth γ ISS (0) = 0, called a K -functon. In contrast, nstablty of a system can lead to nfnte states. For example, n case of a logstc system the state can be the work n progress or the number of unsatsfed orders. Instablty, by means of an unbounded growth of a state, for example, may cause hgh nventory costs or loss of customers, f orders wll not be satsfed. Hence, for many applcatons t s necessary to analyze networks n vew of stablty and to provde tools to check whether a system s stable to avod such negatve outcomes descrbed above. Durng the last decades, several stablty concepts, such as exponental stablty, asymptotc stablty, global stablty and ISS were establshed, see [115, 64, 120], for example. Based on ISS, several related stablty propertes were nvestgated: nput-to-output stablty (IOS) [56], ntegral ISS (ISS) [116] and nput-to-state dynamcal stablty (ISDS) [35]. The ISS 7

8 8 property and ts varants became mportant durng the recent years for the stablty analyss of dynamcal systems wth dsturbances and they were appled n network control, engneerng, bologcal or economcal systems, for example. Survey papers about ISS and related stablty propertes can be found n [118, 11]. Furthermore, the stablty analyss of sngle systems can be performed n dfferent frameworks such as passvty, dsspatvty [108], and ts varatons [1, 36, 98, 57]. It can be a challengng task to check the stablty of a gven system or to desgn a stable system. Lyapunov functons are a helpful tool to nvestgate the stablty of a system, snce the exstence of a Lyapunov functon s suffcent for stablty, see [115, 64], for example. Moreover, the necessty of the exstence of a Lyapunov functon for stablty for some stablty propertes was proved [115, 64]. In [119, 74], t was shown that the ISS property for a system of the form (2) s equvalent to the exstence of an ISS-Lyapunov functon, whch s a locally Lpschtz contnuous functon V : R N R + that has the propertes ψ 1 ( x ) V (x) ψ 2 ( x ), x R N, V (x) χ ( u ) V (x) f(x, u) α (V (x)) for almost all x and all u, where ψ 1,ψ 2,χ K, α s a postve defnte functon and denotes the gradent of V. Based on a Lyapunov functon, we provde tools to check stablty, to desgn observers and to control networks. To ths end, we consder nterconnected systems of the form (1). The noton of ISS s a useful property for the nvestgaton of nterconnected systems n vew of stablty, because t can handle nternal and external nputs of a subsystem. The ISS estmaton of a subsystem s the followng: x (t; x 0,u) ( ) ) } max {β x 0,t, max j γiss j ( x j [0,t],γ ISS ( u ), (4) where [0,t] denotes the supremum norm over the nterval [0,t], γj ISS,γISS : R + R + are K -functons and are called (nonlnear) gans. Investgatng a whole system n vew of stablty, t turns out that a network must not possess the ISS property even f all subsystems are ISS. A method to check the stablty propertes of networks s the so-called small-gan condton. It s based on the gans and the nterconnecton structure of the system. For n =2coupled systems an ISS small-gan theorem was proved n [56] and ts Lyapunov verson n [55], where an explct constructon of the ISS-Lyapunov functon for the whole system was shown. For an arbtrary number of nterconnected systems, an ISS small-gan theorem was proved n [25, 98] and ts Lyapunov verson n [28]. For a local varant of ISS, namely LISS, a Lyapunov formulaton of the small-gan theorem can be found n [27]. Consderng ISS, a small-gan theorem can be found n [49] for two coupled systems and n [50] for n coupled systems. Another approach, usng the cycle small-gan condton, whch s equvalent to the maxmum formulaton of the small-gan condton n matrx form, was used n [57, 77] to establsh ISS of nterconnectons. A small-gan theorem consderng a

9 9 mxed formulaton of ISS subsystems n summaton and maxmum formulaton was proved n [19, 66]. General nonlnear systems were consdered n [58] and [59], where small-gan theorems were proved, usng vector Lyapunov functons. Applyng the mentoned tools to check whether a system has the ISS property one can derve the estmaton (3) or (4) of the norm of the soluton of a system. A stablty property equvalent to ISS, whch has some advantages over ISS, s the followng: Input-to-state dynamcal stablty The defnton of ISDS s motvated by the observaton that the ISS estmaton takes the supremum norm of the nput functon u nto account, despte ths the nput can change and especally can tend to zero. The ISDS estmaton takes essentally only the recent values of the nput u nto account and past values wll be forgotten by tme. Ths s known as the so-called memory fadng effect. The ISDS estmaton s of the form x(t; x 0,u) max{μ(η( x 0 ),t), ess sup μ(γ ISDS ( u(τ) ),t τ)}, τ [0,t] where the functon μ : R + R + R + ncreases n the frst argument, tends to zero, f the second argument tends to nfnty and has the property μ(r, t + s) =μ(μ(r, t),s), r, t, s 0. The beneft for logstc systems, for example producton networks, s the followng: consder the number of unprocessed parts wthn the system as the state, whch have to be stored n a warehouse. By the ISS estmaton, whch gves an upper bound for the trajectory of the state of the system, we can calculate the sze or the capacty of the warehouse to guarantee stablty. The costs for warehouses ncrease by ncreasng the sze or dmenson of the warehouse. Consder the case that the nflux of parts nto the system s large at the begnnng of the process,.e., the number of unprocessed parts n a system s relatvely large, and the nflux tends to zero or close to zero by tme. If the system has the ISS property, the number of unprocessed parts tends also to zero or close to zero by tme, whch means that the warehouse becomes almost empty by tme. Therefore, t s not necessary to provde a huge warehouse to satsfy the upper bound of parts calculated by the ISS estmaton. Takng recent values of the nput nto account by the ISDS estmaton, we can calculate tghter estmatons n contrast to ISS. The sze of the warehouse can be smaller, whch avods hgh costs caused by the over-dmensoned warehouse. Another advantage over ISS s that the ISDS property s equvalent to the exstence of an ISDS-Lyapunov functon, where μ, η and γ ISDS can be drectly taken from the defnton of an ISDS-Lyapunov functon. Consderng ISS-Lyapunov functons and the ISS property, the functons of the accordng defntons are dfferent, n general. There exst no results for the applcaton of ISDS and ts Lyapunov functon characterzaton to networks. Ths work flls ths gap and an ISDS small-gan theorem s proved, whch assures that a network consstng of ISDS subsystems s agan ISDS under a small-gan condton. An explct constructon of the Lyapunov functon and the correspondng gans of the whole system s gven. Ths result was publshed n [20] and presented at the CDC 2009, [26].

10 10 The advantages of the ISDS property wll be transfered to observer desgn: Observer and quantzed output feedback stablzaton In many applcatons, measurements are used to get knowledge about the systems state. To analyze such systems, we consder systems wth an output of the form ẋ = f(x, u), y = h(x), (5) where y R P s the output. In vew of producton networks, t can happen that the measurement of the state of a system s uneconomc or mpossble due to physcal crcumstances or dsturbed by perturbatons, for example. For these cases, observers are used to estmate the state. An observer for the state of the system (5) s of the form ˆξ = F (ȳ, ˆξ,u), ˆx = H(ȳ, ˆξ,u), (6) where ˆξ R L s the observer state, ˆx R N s the estmate of the system state x and ȳ R P s the measurement of y that may be dsturbed by d: ȳ = y + d. The state estmaton error s gven by x =ˆx x. Here, we transfer the dea of ISDS to the desgn of an observer: the challenge s that the observer of a general system or network should be desgned n such a way that the norm of the trajectory of the state estmaton error has the ISS property or ISDS property, respectvely. Frst approaches n the observer desgn usng the (quas-)iss property wth respect to the state estmaton error were performed n [110]. Motvated by the advantages of ISDS over ISS, we ntroduce the noton of quas-isds observers wth respect to the state estmaton error of a system. We show that a quas-isds observer can be desgned, provded that there exsts an error Lyapunov functon (see [88, 60]). The desgn of the observer s the same as for quas-iss observers, based on the works [112, 60, 72, 61, 110], for example, but t has the advantage that the estmaton of the error dynamcs takes only recent dsturbances nto account (see above for the ISDS property). Namely, f the perturbaton of the measurement tends to zero, then the estmaton of the norm of the error dynamcs tends to zero, whch s not the case usng the quas-iss property. The approach of the quas-iss/isds observer desgn s used here for nterconnected systems. We desgn quas-iss/isds observers for each subsystem and for the whole system, provded that error Lyapunov functons of the subsystems exst and a small-gan condton s satsfed. We apply the presented approach to stablzaton of sngle and nterconnected systems based on quantzed output feedback. The problem of output feedback stablzaton was nvestgated n [62, 63, 60, 71, 72, 110], for example. The queston, how to stablze a system,

11 11 plays an mportant role n the analyss of control systems. In ths work, we use quantzed output feedback stablzaton accordng to the results n [7, 70, 72, 110]. A quantzer s a devce, whch converts a real-valued sgnal nto a pecewse constant sgnal,.e., t maps R P nto a fnte and dscrete subset of R P. It may affect the process output or may also affect the control nput. We show that under suffcent condtons a quantzed output feedback law can be desgned usng quas-iss/isds observer, whch guarantee that a sngle system, subsystems of a network or the whole system are stable,.e., the norm of the trajectores of the systems are bounded. Furthermore, we nvestgate dynamc quantzers, where the quantzers can be adapted by a so-called zoomng varable. Ths leads to a feedback law, whch provdes asymptotc stablty of a sngle system, subsystems of a network or the whole system. The results were partally presented at the CDC 2010, [22]. Another type of systems s the followng: Tme-delay systems In many applcatons from areas such as bology, economcs, mechancs, physcs, socal scences and logstcs [5, 65], there occur tme-delays. For example, delays appear by consderng transportaton of materal, communcaton and computatonal delays n control loops, populaton dynamcs and prce fluctuatons [94]. A tme-delay systems (TDS) s gven n the form ẋ(t) =f(x t,u(t)), x 0 (τ) =ξ(τ), τ [ θ, 0], and t s also called a retarded functonal dfferental equaton. θ s the maxmum nvolved delay and the functon x t C ( [ θ, 0] ; R N) s gven by x t (τ) :=x(t + τ), τ [ θ, 0], where C ( [t 1,t 2 ];R N) denotes the Banach space of contnuous functons defned on [t 1,t 2 ] equpped wth the supremum norm. ξ C ( [ θ, 0] ; R N) s the ntal functon of the system. The tool of a Lyapunov functon for the stablty analyss of systems wthout tme-delays can not be drectly appled to TDS. Consderng sngle TDS, a natural generalzaton of a Lyapunov functon s a Lyapunov-Krasovsk functonal [44]. It was shown n [87] that the exstence of an ISS-Lyapunov-Krasovsk functonal s suffcent for the ISS property of a TDS. In contrast to functonals, the usage of a functon s more smpler for an analyss. Ths motvates the ntroducton of the Lyapunov-Razumkhn methodology for TDS. In [121], the suffcency of the exstence of an ISS-Lyapunov-Razumkhn functon for the ISS property of a sngle TDS was shown. In both methodologes, the necessty s not proved yet. The ISS property for nterconnected systems of TDS has not been nvestgated so far. In Chapter 4, we provde tools to analyze networks n vew of ISS and LISS, based on the Lyapunov-Razumkhn and Lyapunov-Krasovsk approaches, whch were presented at the MTNS 2010, [21]. The results are appled to a scenaro of a logstc network to demonstrate

12 12 the relevance of the stablty analyss n applcatons. Further applcatons of the ISS property for logstc networks can be found n [15, 16, 103, 12, 13], for example. The Lyapunov-Razumkhn and Lyapunov-Krasovsk approaches wll be used for mpulsve systems wth tme-delays: Impulsve systems Besdes tme-delays, also sudden changes or jumps, called mpulses of the state of a system occur n applcatons, such as loadng processes of vehcles n logstc networks, for example. Such systems are called mpulsve systems and they are closely related to hybrd systems, see [43, 100, 34, 66], for example. They combne contnuous and dscontnuous behavors of a system: ẋ(t) =f(x(t),u(t)), t t k,k N, x(t) =g(x (t),u (t)), t= t k,k N, where t R + and t k are the mpulse tmes. The ISS property for hybrd systems was nvestgated n [8] and for nterconnectons of hybrd subsystems n [66]. The ISS and ISS propertes for mpulsve systems were studed n [45] for the delayfree case and n [10] for non-autonomous tme-delay systems. Suffcent condtons, whch assure ISS and ISS of an mpulsve system, were derved usng exponental ISS-Lyapunov(- Razumkhn) functons and a so-called dwell-tme condton. In [45], the average dwell-tme condton, ntroduced n [46] for swtched systems, was used, whereas n [10] a fxed dwelltme condton was utlzed. The average dwell-tme condton takes the average of mpulses over an nterval nto account, whereas the fxed dwell-tme condton consders the (mnmal or maxmal) nterval between two mpulses. In mpulsve systems, also tme-delays can occur. For the stablty analyss for such knds of a system, we provde a Lyapunov-Krasovsk type and a Lyapunov-Razumkhn type ISS theorem for sngle mpulsve tme-delay systems usng the average dwell-tme condton. In contrast to the Razumkhn-type theorem from [10], we consder autonomous tme-delay systems and the average dwell-tme condton. Our theorem allows to verfy the ISS property for larger classes of mpulse tme sequences, however, we have used an addtonal techncal condton on the Lyapunov gan n our proofs. Networks of mpulsve systems wthout tme-delays and the ISS property were nvestgated n [66], where a small-gan theorem was proved under the average dwell-tme condton for networks. However, tme-delays were not consdered n the mentoned work. Consderng networks of mpulsve systems wth tme-delays, we prove that under a smallgan condton wth lnear gans and the dwell-tme condton accordng to [45, 66] the whole system has the ISS property. We use exponental Lyapunov-Razumkhn and exponental Lyapunov-Krasovsk functon(al)s. The results regardng mpulsve systems wth tme-delays were partally presented at the NOLCOS 2010, [18], and publshed n [17].

13 13 The analyss of networks wth tme-delays n vew of ISS motvates the nvestgaton of ISS for model predctve control (MPC) of tme-delay networks. Furthermore, the advantages of the ISDS property over ISS wll be used for MPC of networks: Model predctve control Model predctve control (MPC), also known as recedng horzon control, s an approach for an optmal control of systems under constrants. For example, MPC can be used to control a system n a optmal way (optmal accordng to small effort to acheve the goal, for example) such that the soluton of the system follows a certan trajectory or that the soluton s steered to an equlbrum pont, where certan constrants have to be fulflled. By the ncreasng applcaton of automaton processes n ndustry, MPC became more and more popular durng the last decades. It has many applcatons n the chemcal, ol or automotve and aerospace ndustry, for example, see the survey papers [89, 90]. MPC transforms the control problem nto an optmzaton problem: at samplng tmes t = kδ, k N, Δ > 0, the trajectory of a system wll be predcted untl a predcton horzon. A cost functon J wll be mnmzed wth respect to a control u and the soluton of ths optmzaton problem wll be mplemented untl the next samplng tme. Then, the predcton horzon s moved and the procedure starts agan. By the choce of the cost functon one has many degrees of freedom for the defnton and the achevement of the goals. The MPC procedure results n an optmal control to reach the goals and to satsfy possble constrants. There could be constrants to the state space, the control space or the termnal regon of the state of the system. More detals about MPC can be found n [78, 9, 38], for example. However, the stablty of MPC s not guaranteed n general, see [91], for example. Therefore, t s desred to derve condtons to assure stablty. An overvew of exstng results regardng (asymptotc) stablty and optmalty of MPC can be found n [81] and recent results regardng (asymptotc) stablty, optmalty and algorthms of MPC can be found n [92, 84, 68, 38], for example. The ISS property for MPC was nvestgated n [80, 79, 73, 69] for sngle nonlnear dscretetme systems wth dsturbances. There, suffcent condtons to guarantee ISS for MPC were establshed. Interconnectons and the ISS property for MPC were analyzed n [93]. The approach n these papers s that the cost functon s a Lyapunov functon, whch mples ISS. In ths thess, we want to combne the ISDS property and MPC, whch s not done yet. Consderng sngle nonlnear contnuous-tme systems, we show that the cost functon of the used MPC scheme s an ISDS-Lyapunov functon, whch mples ISDS of the system. For nterconnectons, we apply the ISDS small-gan theorem, whch s a result of ths thess, showng that the cost functon of the th subsystem of the nterconnecton s an ISDS-Lyapunov functon for the th subsystem. We establsh the ISDS property for MPC of sngle and nterconnected nonlnear systems. Consderng tme-delay systems, the asymptotc stablty for MPC was nvestgated for

14 14 sngle nonlnear contnuous-tme TDS n [29, 96, 95]. Besdes asymptotc stablty, the determnaton of the termnal cost, the termnal regon and the computaton of locally stablzng controller were performed n these papers, usng Lyapunov-Razumkhn and Lyapunov- Krasovsk arguments. The ISS property for MPC of TDS has not been nvestgated so far. Here, we want to ntroduce the ISS property for MPC of nonlnear sngle and nterconnected TDS. We show that the cost functon for a sngle system s an ISS-Lyapunov-Krasovsk functonal and apply the ISS-Lyapunov-Krasovsk approach of the chapter regardng TDS. Usng the ISS Lyapunov-Krasovsk small-gan theorem for nterconnectons, whch s a result of ths thess, we derve condtons to assure ISS for MPC of networks wth TDS. The tools presented n ths work enrch the toolbox for the analyss and control of nterconnected systems. They can be used n many applcatons from dfferent areas, such as logstcs and economcs, bology, mechancs and physcs, or socal scences, for example. Organzaton of the thess Chapter 1 contans all necessary notons for the analyss and for the man results of ths work. The ISDS property s nvestgated n Chapter 2, where we prove an ISDS small-gan theorem. Chapter 3 s devoted to the quas-iss/isds observer desgn for sngle systems and networks and the applcaton to quantzed output feedback stablzaton. Tme-delay systems are consdered n Chapter 4, where ISS small-gan theorems usng the Lyapunov-Razumkhn and Lyapunov-Krasovsk approach are proved. They are appled to a scenaro of a logstc network n Secton 4.3. Proceedng wth mpulsve systems wth tme-delays, ISS theorems are gven n Chapter 5. The tools wthn the framework of ISS/ISDS for model predctve control can be found n Chapter 6. Fnally, Chapter 7 summarzes the work wth an overvew of all results combned wth open questons and outlooks of possble future research actvtes.

15 Chapter 1 Prelmnares In ths chapter, all notatons and defntons are gven, whch are necessary for the followng chapters. More precsely, the defnton of ISS and ts Lyapunov functon characterzaton for sngle systems are ncluded. Consderng nterconnected systems, we recall the man theorems regardng ISS of networks. By x T we denote the transposton of a vector x R n, n N, furthermore R + := [0, ) and R n + denotes the postve orthant {x R n : x 0}, where we use the standard partal order for x, y R n gven by x y x y,=1,...,n, x y : x <y and x>y x >y,=1,...,n. For a nonempty ndex set I {1,...,n},n N, we denote by #I the number of elements of I and y I := (y ) I for y R n +. A projecton P I from R n + nto R #I + maps y to y I. By B(x, r) we denote the open ball wth respect to the Eucldean norm around x of radus r. denotes the Eucldean norm n R n. The essental supremum norm of a (Lebesgue-) measurable functon f : R R n s the smallest number K such that the set {x : f(x) >K} has (Lebesgue-) measure zero and t s denoted by f. x denotes the maxmum norm of x R n and V s the gradent of a functon V : R n R +. We denote the set of essentally bounded (Lebesgue-) measurable functons u from R to R m by L (R, R m ):={u : R R m measurable K>0: u(t) K, for almost all (f.a.a.) t}, where f.a.a. means for all t except the set {t : u(t) >K}, whch has measure zero. For t 1,t 2 R, t 1 <t 2, let C ( [t 1,t 2 ];R N) denote the Banach space of contnuous functons defned on [t 1,t 2 ] wth values n R N and equpped wth the norm φ [t1,t 2 ] := sup t 1 s t 2 φ(s) and takes values n R N. Let θ R +. The functon x t C ( [ θ, 0] ; R N) s gven by x t (τ) := x(t+τ), τ [ θ, 0]. PC ( [t 1,t 2 ];R N) denotes the Banach space of pecewse rght-contnuous functons defned on [t 1,t 2 ] equpped wth the norm [t1,t 2 ] and takes values n RN. 15

16 16 For a functon v : R + R m we defne ts restrcton to the nterval [s 1,s 2 ] by { v(t) f t [s1,s 2 ], v [s1,s 2 ](t) := 0 otherwse, t, s 1,s 2 R +. We defne the followng classes of functons: Defnton P := {f : R n R + f(0)=0, f(x) > 0, x 0}, K := {γ : R + R + γ s contnuous, γ(0)=0and strctly ncreasng}, K := {γ K γ s unbounded}, { } L := γ : R + R + γ s contnuous and decreasng wth lm γ(t) =0, t KL := {β : R + R + R + β s contnuous, β(,t) K, β(r, ) L, t, r 0}. We wll call functons of class P postve defnte. Note that, f γ K, then there exsts the nverse functon γ 1 : R + R + wth γ 1 K, see [98], Lemma To ntroduce nterconnected systems, we consder nonlnear systems descrbed by ordnary dfferental equatons of the form ẋ(t) =f(x(t),u(t)), (1.1) where t R + s the (contnuous) tme, ẋ denotes the dervatve of x R N, the nput u L (R +, R m ) and f : R N+m R N, N,m N. We assume that the ntal value x(t 0 )=x 0 s gven and wthout loss of generalty we consder t 0 =0. Systems of the form (1.1) are examples of dynamcal systems accordng to [115, 64, 48]. For the exstence and unqueness of a soluton of a system of the form (1.1), we need the noton of a locally Lpschtz contnuous functon. Defnton Let f : D R N R N be a functon. () f satsfes a Lpschtz condton n D, f there exsts a L 0 such that t holds x 1,x 2 D : f(x 1 ) f(x 2 ) L x 1 x 2. L s called Lpschtz constant and f s called Lpschtz contnuous. () f s called locally Lpschtz contnuous n D, f for each x D there exsts a neghborhood U(x) such that the restrcton f D U satsfes a Lpschtz condton n D U. Snce we are dealng wth locally Lpschtz contnuous functons, we recall the followng theorem. Theorem (Theorem of Rademacher). Let f : R N R N be a functon, whch satsfes a Lpschtz condton n R N. Then, f s dfferentable n R N almost everywhere (a.e.) (whch means everywhere except for the set wth (Lebesgue-)measure zero).

17 Chapter 1. Prelmnares 17 A proof can be found n [30], page 216, for example. To have exstence and unqueness of a soluton of (1.1) we use the followng theorem: Theorem (Carathéodory condtons). Consder a system of the form (1.1). Let the functon f be contnuous and for each R>0 there exsts a constant L R > 0 such that t holds f (x 1,u) f (x 2,u) L R x 1 x 2 for all x 1,x 2 R N and u L (R, R m ) wth x 1, x 2, u R. Then, for each x 0 R N and u L (R, R m ) there exsts a maxmal (open) nterval I wth 0 I and a unque absolute contnuous functon ξ(t), whch satsfes ξ(t) =x 0 + t 0 f (x(τ),u(τ)) dτ, t I. The proof can be found n [117], Appendx C. We denote the unque functon ξ from Theorem by x(t; x 0,u) or x(t) n short and call t soluton of the system (1.1) wth ntal value x 0 R N and u L (R +, R m ). For the exstence and unqueness of solutons of systems of the form (1.1), we assume n the rest of the thess that the functon f : R N R m R N satsfes the condtons n Theorem 1.0.4,.e., f s contnuous and locally Lpschtz n x unformly n u. 1.1 Input-to-state stablty It s desrable to have knowledge about the systems behavor. For example, n applcatons t s needed to know that the trajectory of a system wth bounded external nput remans n a ball around the orgn for all tmes whatever the nput s. Ths leads to the noton of (L)ISS, ntroduced by Sontag [114]: Defnton (Input-to-state stablty). The System (1.1) s called locally nput-to-state stable (LISS), f there exst ρ>0,ρ u > 0, β KLand γ ISS K such that for all x 0 ρ, u ρ u and all t R + t holds x(t; x 0,u) max {β( x 0,t),γ ISS ( u )}. (1.2) γ ISS s called gan. Ifρ = ρ u =, then system (1.1) s called nput-to-state stable (ISS). (L)ISS establshes an estmaton of the norm of the trajectory of a system. On the one hand, ths estmaton takes the ntal value nto account by the term β( x 0,t), whch tends to zero f t tends to nfnty. On the other hand, t takes the supremum norm of the nput nto account by the term γ ISS ( u ). Note that we get an equvalent defnton of LISS or ISS, respectvely, f we replace (1.2) by x(t; x 0,u) β( x 0,t)+γ ISS ( u ), (1.3)

18 Input-to-state stablty where β and γ ISS n (1.2) and (1.3) are dfferent n general. It s known for ISS systems that f lm sup t u(t) =0then also lm t x(t) =0holds, see [114, 118], for example. However, wth t, (1.2) provdes only a constant postve bound for u 0. The relatonshp between ISS and other stablty concepts was shown n [120]. One of these concepts s the 0-global asymptotc stablty (0-GAS) property, whch we use n the followng chapters and s defned as follows (see [120]): Defnton The system (1.1) wth u 0 s called 0-global asymptotcally stable (0- GAS), f there exsts β KLsuch that for all x 0 and for all t R + t holds x(t; x 0, 0) β( x 0,t), where 0 denotes the nput functon dentcally equal to zero on R +. It s not always an easy task to fnd the functons β and γ ISS to verfy the ISS property of a system. As for systems wthout nputs, Lyapunov functons are a helpful tool to check whether a system of the form (1.1) possesses the ISS property. Defnton A locally Lpschtz contnuous functon V : D R +, wth D R N open, s called a local ISS-Lyapunov functon of the system (1.1), f there exst ρ>0,ρ u > 0, ψ 1,ψ 2 K, γ ISS Kand α P such that B(0,ρ) D and ψ 1 ( x ) V (x) ψ 2 ( x ), x D, (1.4) V (x) γ ISS ( u ) V (x) f(x, u) α (V (x)) (1.5) for almost all x B(0,ρ)\{0} and all u ρ u.ifρ = ρ u =, then the functon V s called an ISS-Lyapunov functon of the system (1.1). γ ISS s called (L)ISS-Lyapunov gan. The functon V can be nterpreted as the energy of the system. The condton (1.4) states that V s postve defnte and radally bounded by two K -functons. The meanng of the condton (1.5) s that outsde of the regon {x : V (x) < γ ISS ( u )} the energy of the system s decreasng. In partcular, for every gven external nput wth fnte norm, the energy of the system s bounded, whch mples, by (1.4) that the trajectory of system also remans bounded for all tmes t>0. However, there s no general method to fnd a Lyapunov functon for arbtrary nonlnear systems. The equvalence of ISS and the exstence of an ISS-Lyapunov functon was shown n [119, 74]: Theorem The system (1.1) possesses the ISS property f and only f there exsts an ISS-Lyapunov functon for the system (1.1). Wth the help of ths theorem one can check, whether a system has the ISS property: the exstence of an ISS-Lyapunov functon for the system s suffcent and necessary for ISS. Note that the ISS-Lyapunov gan γ ISS and the gan γ ISS n the defnton of ISS are dfferent n general. An equvalent defnton of an ISS-Lyapunov functon can be obtaned, replacng

19 Chapter 1. Prelmnares 19 (1.5) by V (x) f(x, u) ˆγ ISS ( u ) α(v (x)), where ˆγ ISS K, α P, whch s called the dsspatve Lyapunov form, see [119], for example. 1.2 Interconnected systems Many systems n applcatons are networks of subsystems, whch are nterconnected. Ths means that the evoluton of a subsystem could depend on the states of other subsystems and external nputs. Analyzng such a network n vew of stablty the noton of ISS s useful, because t takes (nternal and external) nputs of a system nto account. The queston s, under whch condton the network possesses the ISS property and how t can be checked? For the purpose of ths work, we consder n 2 nterconnected subsystems of the form ẋ (t) =f (x 1 (t),...,x n (t),u(t)), =1,...,n, (1.6) where n N, x R N, N N, u L (R +, R m ), f : R n j=1 Nj+m R N. We assume that the functon f satsfes the condtons n Theorem to have exstence and unqueness of a soluton of a subsystem for all =1,...,n. Wthout loss of generalty we consder the same nput functon u for all subsystems n (1.6). One can use a projecton P such that u s the nput of the th subsystem and f (...,u)= f (...,P (u)) = f (...,u ) wth u =(u 1,...,u n ) T, see also [25]. The ISS property for subsystems s the followng: The -th subsystem of (1.6) s called LISS, f there exst constants ρ,ρ j,ρ u > 0 and functons γj ISS, γiss K and β KLsuch that for all ntal values x 0 ρ, all nputs x j [0, ) ρ j, u ρ u and all t R + t holds { ( x (t) max β x 0,t ) ) }, max j γiss j ( x j [0,t],γ ISS ( u ). (1.7) γj ISS are called gans. If ρ = ρ j = ρ u =, then the -th subsystem of (1.6) s called ISS. By replacng (1.7) by ( ) ) x (t) β x 0,t + γj ( x ISS j [0,t] + γ ISS ( u ) (1.8) j we get an equvalent formulaton of ISS for subsystems. We refer to ths as the summaton formulaton and (1.7) as the maxmum formulaton of ISS. Note that β and the gans n (1.7) and (1.8) are dfferent n general, but we use the same notaton for smplcty. Note that n the ISS estmaton (1.7) or (1.8) the nternal and external nputs of a subsystem are taken nto account. In contrast to the ISS ) estmaton (1.2) or (1.3) for a sngle system, ths results n addng the gans γj ( x ISS j [0,t] to the ISS estmaton of the th subsystem, where the ndex j denotes the jth subsystem that s connected to the th subsystem. Also, an ISS-Lyapunov functon for the th subsystem can be gven, where the subsystems have to be taken nto account, whch are connected to the th subsystem. It reads as follows:

20 Interconnected systems We assume that for each subsystem of the nterconnected system (1.6) there exsts a functon V : D R + wth D R N open, whch s locally Lpschtz contnuous and postve defnte. Then, the functon V s called a LISS-Lyapunov functon of the -th subsystem of (1.6), f V satsfes the followng two condtons: There exst functons ψ 1,ψ 2 K such that ψ 1 ( x ) V (x ) ψ 2 ( x ), x D (1.9) and there exst γ j ISS, γiss K, α Pand constants ρ,ρ j,ρ u > 0 such that B(0,ρ ) D and wth x =(x T 1,...,xT n ) T t holds V (x ) max { max j γiss j (V j (x j )), γ ISS ( u ) } V (x ) f (x, u) α (V (x )) (1.10) for almost all x B(0,ρ ), x j ρ j, u ρ u. If ρ = ρ j = ρ u =, then V s called an ISS-Lyapunov functon of the -th subsystem of (1.6). Functons γ j ISS are called (L)ISS- Lyapunov gans. Note that an equvalent formulaton of an ISS-Lyapunov functon can be obtaned, f we replace (1.10) by V (x ) j γ j ISS (V j (x j )) + γ ISS ( u ) V (x ) f (x, u) ᾱ (V (x )), (1.11) where γ j ISS, γiss Kand ᾱ P. We consder an nterconnected system of the form (1.6) as one sngle system (1.1) wth x = ( x T ) T ( 1,...,xT n,f(x, u) = f1 (x, u) T,...,f n (x, u) T ) T and call t overall or whole system. It s not guaranteed that the overall system possesses the ISS property even f all subsystems are ISS. A well-developed condton to verfy ISS and to construct a Lyapunov functon for the whole system s a small-gan condton, see [25, 98, 28], for example. To ths end, we collect all the gans γ j ISS whch defnes a map Γ: R n + n a matrx, called gan-matrx Γ:=( γ j ISS) n n,,j =1,...,n, γ ISS R n + by ( Γ(s) := max γ ISS j 1j (s j ),...,max j 0, ) T γ nj ISS (s j ),s R n +. (1.12) Note that the matrx Γ descrbes n partcular the nterconnecton structure of the network. Moreover, t contans nformaton about the mutual nfluence between the subsystems, whch can be used to verfy the (L)ISS property of networks. If we use (1.11) nstead of (1.10), we collect the gans n the matrx Γ:=( γ j ISS) n n,,j= 1,...,n, γ ISS 0, whch defnes a map Γ: R n + R n + by Γ(s) := γ 1j ISS (s j ),..., γ nj ISS (s j ),s R n +. (1.13) j j For the stablty analyss of the whole system n vew of LISS, we wll use the followng condton (see [27]): we say that a gan-matrx Γ satsfes the local small-gan condton (LSGC) on [0,w ], w R n +,w > 0, provded that Γ(w ) <w and Γ(s) s, s [0,w ],s 0. (1.14) T

21 Chapter 1. Prelmnares 21 Notaton denotes that there s at least one component {1,...,n} such that Γ(s) <s. In vew of ISS, we say that Γ satsfes the small-gan condton (SGC) (see [98]) f Γ(s) s, s R n +\{0}. (1.15) If we consder the summaton formulaton of ISS or ISS-Lyapunov functons, respectvely, the SGC s of the form (see also [25]) ( ) Γ D (s) s, s R n + \{0}, (1.16) where D : R n + R n + s a dagonal operator defned by (Id +ϖ)(s 1 ) D (s) :=.,s Rn +,ϖ K. (Id +ϖ)(s n ) For smplcty, we wll use Γ for a matrx defned by (1.12), usng the maxmum formulaton or defned by (1.13), usng the summaton formulaton. Note that by γj ISS K {0} and for v, w R n + we get v w Γ(v) Γ(w). Remark The SGC (1.15) s equvalent to the cycle condton (see [98], Lemma for detals). A k-cycle n a matrx Γ = (γ j ) n n s a sequence of K functons (γ 0 1,γ 1 2,...,γ k 1 k ) of length k wth 0 = k. The cycle condton for a matrx Γ s that all k-cycles of Γ are contractons,.e., γ 0 1 γ 1, 2... γ k 1, k < Id, for all 0,..., k {1,...,n} wth 0 = k and k n. See [98] and [57] for further detals. To recall the Lyapunov versons of the small-gan theorem for the LISS and ISS property from [28] and [27], we need the followng: Defnton A contnuous path σ K n s called an Ω-path wth respect to Γ, f () for each, the functon σ 1 s locally Lpschtz contnuous on (0, ); () for every compact set P (0, ) there are constants 0 <K 1 <K 2 such that for all ponts of dfferentablty of σ 1 and =1,...,n we have 0 <K 1 (σ 1 ) (r) K 2, r P ; (1.17) () t holds Γ(σ(r)) <σ(r), r >0. (1.18)

22 Interconnected systems More detals about an Ω-path can be found n [98, 99, 28]. The followng proposton s useful for the constructon of an ISS-Lyapunov functon for the whole system. Proposton Let Γ (K {0}) n n be a gan-matrx. If Γ satsfes the smallgan condton (1.15), then there exsts an Ω-path σ wth respect to Γ. IfΓ satsfes the SGC n the form (1.16), then there exsts an Ω-path σ, where Γ(σ(r)) <σ(r) s replaced by (Γ D)(σ(r)) <σ(r), r >0. The proof can be found n [28], Theorem 5.2, see also [98, 99], however only the exstence s proved n these works. In [20], Proposton 3.4., t was shown how to construct a fnte but arbtrary long path. For the case that Γ satsfes the LSGC (1.14) a strctly ncreasng path σ :[0, 1] [0,w ] exsts, whch satsfes Γ(σ(r)) < σ(r), r (0, 1]. σs pecewse lnear and satsfes σ(0) = 0, σ(1) = w, see Proposton 4.3 n [28], Proposton 5.2 n [99]. Now, we recall the man results of [27] and [28]. They show under whch condtons the overall system possesses the (L)ISS property. Moreover, an explct constructon of the (L)ISS-Lyapunov functon for the whole system s gven. Theorem Let V be an ISS-Lyapunov functon for the -th subsystem n (1.6), for all = 1,...,n. LetΓ be a gan-matrx and satsfes the SGC (1.15). Then, the whole system of the form (1.1) s ISS and the ISS-Lyapunov functon of the overall system s gven by V (x) = max σ 1 (V (x )). The proof can be found n [23], Theorem 6 or n a generalzed form n [28], Corollary 5.5. A verson usng LISS s gven by the followng: Theorem Assume that each subsystem of (1.6) admts an LISS-Lyapunov functon and that the correspondng gan-matrx Γ satsfes the LSGC (1.14). Then, the whole system of the form (1.1) s LISS and the LISS-Lyapunov functon of the overall system s gven by V (x) = max σ 1 (V (x )). The proof can be found n [27], Theorem 5.5. An approach for a numercal constructon of LISS-Lyapunov functons can be found n [33]. The mentoned theorems provde tools how to check, f a network possesses the ISS property: we have to fnd ISS-Lyapunov functons and the correspondng gans for the subsystems. If the gans satsfy the small-gan condton, then the whole system s ISS. In the followng chapters, we use the mentoned tools for the stablty analyss, observer desgn and control of nterconnected systems. Moreover, tools for the stablty analyss of networks of tme-delay systems and of networks of mpulsve systems wth tme-delays are derved. Wth all these notatons and consderatons of ths chapter, we are able to formulate and prove the man results of ths work n the next chapters.

23 Chapter 2 Input-to-state dynamcal stablty (ISDS) In ths chapter, the noton of nput-to-state dynamcal stablty (ISDS) s descrbed and as the man result of ths chapter, we prove an ISDS-Lyapunov small-gan theorem. The stablty noton ISDS was ntroduced n [35], further nvestgated n [36] and some local propertes studed n [40]. ISDS s equvalent to ISS, however, one advantage of ISDS over ISS s that the bound for the trajectores takes essentally only the recent values of the nput u nto account and n many cases t gves a better bound for trajectores due to the memory fadng effect of the nput u. Smlar to ISS systems, the ISDS property of system (1.1) s equvalent to the exstence of an ISDS-Lyapunov functon for system (1.1), see [36]. Also a 0-GAS small-gan theorem for two nterconnected systems wth the nput u =0can be found n [36]. Another advantage of ISDS over ISS s that the gans n the trajectory based defnton of ISDS are the same as n the defnton of the ISDS-Lyapunov functon, whch s n general not true for ISS systems. In ths chapter, we extend the result for nterconnected ISS systems to the case of ISDS systems. In partcular, we provde a tool for the stablty analyss of networks n vew of ISDS. Ths s a small-gan theorem for n N nterconnected ISDS systems of the form (1.6) wth a constructon of an ISDS-Lyapunov functon as well as the rates and gans of the ISDS estmaton for the entre system. Moreover, we derve decay rates of the trajectores of n N nterconnected ISDS systems and the trajectory of the entre system wth the external nput u =0. These results are compared to an example n [36] for n =2nterconnected systems wth u =0. The next secton ntroduces the noton of ISDS for sngle systems of the form (1.1). Secton 2.2 contans the man result of ths chapter. Examples are gven n Secton

24 ISDS for sngle systems 2.1 ISDS for sngle systems We consder systems of the form (1.1). For the ISDS property we defne the class of functons KLD by KLD := {μ KL μ(r, t + s) =μ(μ(r, t),s), r, t, s 0}. Remark The condton μ(r, t + s) =μ(μ(r, t),s) mples μ(r, 0) = r, r 0. To show ths, suppose that there exsts r 0 such that μ(r, 0) r. Then μ(r, 0) = μ(r, 0+0)=μ(μ(r, 0), 0) μ(r, 0), whch s a contradcton. The last nequalty follows from the strct monotoncty of μ wth respect to the frst argument. Ths shows the asserton. The noton of ISDS was ntroduced n [35] and t s as follows: Defnton (Input-to-state dynamcal stablty (ISDS)). The system (1.1) s called nput-to-state dynamcally stable (ISDS), f there exst μ KLD, η, γ ISDS K such that for all ntal values x 0 and all nputs u t holds x(t; x 0,u) max{μ(η( x 0 ),t), ess sup τ [0,t] μ(γ ISDS ( u(τ) ),t τ)} (2.1) for all t R +. μ s called decay rate, η s called overshoot gan and γ ISDS s called robustness gan. Remark One obtans an equvalent defnton of ISDS f one replaces the Eucldean norm n (2.1) by any other norm. Moreover, t can be checked that all results n [36] and [35] hold true, f one uses a dfferent norm nstead of the Eucldean one. It was shown n [35], Proposton () that ISDS s equvalent to ISS n the maxmum formulaton (1.2). Note that n contrast to ISS, the ISDS property takes essentally only the recent values of the nput u nto account and past values of the nput wll be forgotten by tme, whch s also known as the memory fadng effect. In partcular, t follows mmedately from (2.1): Lemma If the system (1.1) s ISDS and lm sup t lm x(t; x 0,u) =0. t Proof. Snce (1.1) s ISDS we have x(t; x 0,u) max{μ(η( x 0 ),t), ess sup τ [0,t] = max{μ(η( x 0 ),t), ess sup τ [0, t 2] u(t) =0, then t holds μ(γ ISDS ( u(τ) ),t τ)} μ(γ ISDS ( u(τ) ),t τ), ess sup τ [ t 2,t] μ(γ ISDS ( u(τ) ),t τ)} max{μ(η( x 0 ),t),μ(γ ISDS ( u t [0, 2] ), t ), ess sup μ(γ ISDS ( u(τ) ), 0)}. 2 τ [ t,t] 2

25 Chapter 2. Input-to-state dynamcal stablty (ISDS) 25 It holds lm sup u(t) =0and u s essentally bounded,.e., there exsts a K R + such t that u [0,t] K, for all t>0. Furthermore, for all ε>0 there exsts a T > 0 such that for all τ [ T 2,T] t holds ess sup τ [ T,T] γisds ( u(τ) ) <ε. Wth these consderatons, the 2 KLD-property of μ and Remark we get lm t x(t; x 0,u) lm max{μ(η( x 0 ),t),μ(γ ISDS ( u t t [0, 2] ), t ), ess sup γ ISDS ( u(τ) )} 2 τ [ t,t] 2 max{ lm μ(γ ISDS (K), t ), lm ess sup γ ISDS ( u(τ) )} =0. t 2 t τ [ t,t] 2 In the rest of the thess, we assume the functons μ, η and γ ISDS to be C n R + R or R +, respectvely. Ths regularty assumpton s not restrctve, because for non-smooth rates and gans one can fnd smooth functons arbtrarly close to the orgnal ones, whch was shown n [35], Appendx B. As we know that Lyapunov functons are an mportant to tool to verfy the ISS property of systems of the form (1.1), ths s also the case for the ISDS property. Defnton (ISDS-Lyapunov functon). Gven ε>0, a functon V : R N R +, whch s locally Lpschtz contnuous on R N \{0}, s called an ISDS-Lyapunov functon of the system (1.1), f there exst η, γ ISDS K, μ KLDsuch that t holds x 1+ε V (x) η ( x ), x RN, (2.2) V (x) >γ ISDS ( u ) V (x) f(x, u) (1 ε) g (V (x)) (2.3) for almost all x R N \{0} and all u, where μ solves the equaton for a locally Lpschtz contnuous functon g : R + R +. d μ(r, t) = g (μ (r, t)), r,t > 0 (2.4) dt The equvalence of ISDS and the exstence of a smooth ISDS-Lyapunov functon was proved n [36]. Here, we use locally Lpschtz contnuous Lyapunov functons, whch are dfferentable almost everywhere by Theorem of Rademacher (Theorem 1.0.3). Theorem The system (1.1) s ISDS wth μ KLDand η, γ ISDS K, f and only f for each ε>0 there exsts an ISDS-Lyapunov functon V. Proof. Ths follows by Theorem 4, Lemma 16 n [36] and Proposton n [35]. Remark Note that for a system, whch possesses the ISDS property, t holds that the decay rate μ and gans η, γ ISDS n Defnton are exactly the same as n Defnton Recall that the gans of the defnton of ISS (Defnton 1.1.1) are dfferent n general from the ISS-Lyapunov gans n Defnton

26 ISDS for sngle systems In order to have ISDS-Lyapunov functons wth more regularty, one can use Lemma 17 n [36], whch shows that for a locally Lpschtz functon V there exsts a smooth functon Ṽ arbtrary close to V. To demonstrate the advantages of ISDS over ISS, we consder the followng example: Example Consder the system ẋ(t) = x(t)+u(t), (2.5) x R, t R + wth a gven ntal value x 0. The nput s chosen as { 4, 0 t 10, u(t) = 0, otherwse. From the general equaton for the soluton of lnear systems, namely x(t; x 0,u)=e A(t t 0) x 0 + t t 0 e A(t s) Bu(s)ds, we get wth t 0 =0 x(t; x 0,u) x 0 e t + u, whch mples that the system (2.5) has the ISS property wth β( x 0,t) = x 0 e t and γ ISS ( u )= u. The estmaton s dsplayed n the Fgure 2.1 wth x 0 =0.1. To verfy the ISDS property, we use ISDS-Lyapunov functons. We choose V (x) = x as a canddate for the ISDS-Lyapunov functon. For any ε>0 and by the choce γ ISDS ( u ) := 1 δ u, wth gven 0 <δ<1 we obtan (1 + ε)γ ISDS ( u ) V (x) V (x) f(x, u) 1+ε2 +δ δε 1 ε 2 x (1 ε)(1 δ) x. By g (r) :=(1 δ) r we get μ(r, t) =e (1 δ)t r (as soluton of μ = g(μ)) and hence, the system (2.5) has the ISDS property. Note that the choce δ close to 1 results n a sharp gan γ ISDS but slow decay rate μ (Fgure 2.2 wth δ = and x 0 =0.1). In contrast, by a smaller choce δ ths results n more conservatve gan γ ISDS but faster decay rate μ (Fgure 2.3 wth δ = 3 4 and x 0 =0.1). Fgure 2.1: ISS estmaton wth x 0 =0.1.

27 Chapter 2. Input-to-state dynamcal stablty (ISDS) 27 Fgure 2.2: ISDS estmaton wth δ = , x 0 =0.1. Fgure 2.3: ISDS estmaton wth δ = 3 4, x 0 =0.1. From Fgures , we perceve that the ISDS estmaton tends to zero, f the nput tends to zero n contrast to the ISS estmaton. Ths property of the ISDS estmaton s known as the memory fadng effect. In the next secton, we provde an ISDS small-gan theorem for nterconnected systems wth a constructon of an ISDS-Lyapunov functon for the whole system. 2.2 ISDS for nterconnected systems We consder nterconnected systems of the form (1.6). The ISDS property for subsystems reads as follows: The -th subsystem of (1.6) s called ISDS, f there exsts a KLD-functon μ and functons η, γ ISDS and γj ISDS K {0},,j = 1,...,n wth γ ISDS = 0 such that the soluton x (t; x 0,u)=x (t) for all ntal values x 0 and all nputs x j,j, u satsfes { } x (t) max μ (η ( x 0 ),t), max ν j(x j,t),ν (u, t) (2.6) j for all t R +, where ν (u, t) := ess sup τ [0,t] ν j (x j,t):= μ (γ ISDS ( u(τ) ),t τ), sup μ (γj ISDS ( x j (τ) ),t τ) τ [0,t], j =1,...,n. γj ISDS,γ ISDS are called (nonlnear) robustness gans. To show the ISDS property for networks, we need the gan-matrx Γ ISDS, whch s defned ( by Γ ISDS := γ ISDS j ) n n wth γisds 0,, j =1,...,n and defned by (1.12). Defnton For vector valued functons x =(x T 1,...,xT n ) T : R + R n =1 N wth x : R + R N and tmes 0 t 1 t 2, t R + we defne x(t) := ( x 1 (t),..., x n (t) ) T R n + and x [0,t] accordngly.

28 ISDS for nterconnected systems For u R m, t R + and s R n + we defne γ ISDS ( u(t) ) := (γ1 ISDS ( u(t) ),...,γn ISDS ( u(t) )) T R n +, μ(s, t) := (μ 1 (s 1,t),...,μ n (s n,t)) T R n +, η(s) := (η 1 (s 1 ),...,η n (s n )) T R n +. Now, we can rewrte condton (2.6) for all subsystems n a compact form x(t) max { μ ( η ( x 0 ),t ), sup τ [0,t] } μ (Γ ISDS ( x(τ) ),t τ), ess sup μ( γ ISDS ( u(τ) ),t τ) τ [0,t] (2.7) for all t R +. Note that the maxmum, the supremum and the essental supremum used n (2.7) for vectors are taken component-by-component. For the ISDS property, from (2.7), usng the KLD-property of μ and wth Γ ISS := Γ ISDS, γ ISS := γ ISDS, β(r, t) := μ( η(r),t) we get x(t) max { β ( x 0,t ), Γ ISS ( x [0,t] ), γ ISS ( u ) }. Ths mples that each subsystem of (1.6) s ISS and provded that Γ ISDS satsfes the SGC (1.15), also Γ ISS satsfes the SGC (1.15),.e., the nterconnecton s ISS and hence ISDS. However, we loose the quanttatve nformaton about the rate and gans of the ISDS estmaton for the whole system n such a way. In order to conserve the quanttatve nformaton of the ISDS rate and gans of the overall system, we utlze ISDS-Lyapunov functons. For subsystems of the form (1.6) they read as follows: We assume that for each subsystem of (1.6) there exsts a functon V : R N R +, whch s locally Lpschtz contnuous and postve defnte. Gven ε > 0, a functon V : R N R +, whch s locally Lpschtz contnuous on R N \{0} s an ISDS-Lyapunov functon of the -th subsystem n (1.6), f t satsfes: () there exsts a functon η K such that for all x R N t holds x V (x ) η ( x ); (2.8) 1+ε () there exst functons μ KLD,γ ISDS K {0}, γj ISDS K {0},j=1,...,n, j such that for almost all x R N \{0}, all nputs x j,j and u t holds V (x ) > max{γ ISDS ( u ), max j γisds j (V j (x j ))} V (x )f (x, u) (1 ε ) g (V (x )), (2.9) where μ KLDsolves the equaton d dt μ (r, t) = g (μ (r, t)), r,t > 0 for some locally Lpschtz contnuous functon g : R + R +. Now, we state the man result of ths chapter, whch provdes a tool to check whether a network possesses the ISDS property. Moreover, the decay rate and the gans of the ISDS estmaton for the network can be constructed explctly.

29 Chapter 2. Input-to-state dynamcal stablty (ISDS) 29 Theorem Assume that each subsystem of (1.6) has the ISDS property. Ths means that for each subsystem and for each ε > 0 there exsts an ISDS-Lyapunov functon V, whch satsfes (2.8) and (2.9). Let Γ ISDS be gven by (1.12), satsfyng the small-gan condton (1.15) and let σ K n be an Ω-path from Proposton wth Γ=Γ ISDS. Then, the whole system (1.1) has the ISDS property and ts ISDS-Lyapunov functon s gven by V (x) =ψ (max 1 { σ 1 (V (x )) }) (2.10) wth rates and gans g(r) =(ψ 1 ) (ψ(r)) mn η(r) =ψ 1 { (max σ 1 (η (r)) } ), γ ISDS (r) =ψ 1 (max { (σ 1 ) (σ (ψ(r)))g (σ (ψ(r))) }, { σ 1 (γ ISDS (r)) } ), (2.11) ( where r>0 and ψ ( x ) = mn σ 1 x n ). The proof of Theorem follows the dea of the proof of Theorem 5.3 n [28] and correspondng results n [24] wth changes due to the constructon of the gans and of the rate of the whole system. Proof. Let 0 x = ( x T 1,...,xT n ) T. We defne { V (x) :=max σ 1 (V (x )) } {, η( x ) :=max σ 1 where V satsfes (2.8) for =1,...,n. Note that σ 1 for some j {1,...,n}, then t holds max σ 1 ( x 1+ε ) max where ε := max ε and we have σ 1 ( x 1+ε ) σ 1 j (η ( x )) }, ψ( x ) :=mn ( ) σ 1 x n, K. Let j be such that x = x j ( ) xj 1+ε mn σ 1 ( ) x n(1+ε) (2.12) ( ) x ψ V (x) η( x ). (2.13) 1+ε Note that V s locally Lpschtz contnuous and hence t s dfferentable almost everywhere. We defne I := { {1,...,n} V (x) = { σ 1 (V (x )) } max j,j {σj 1 (V j (x j ))}}. Fx an { } I. Let γ ISDS ( u ) := max j σj 1 (γj ISDS ( u )), j =1,...,n. Assume V (x) > γ ISDS ( u ). Then, V (x )=σ (V (x)) >σ (σ 1 From () n Defnton we have V (x )=σ (V (x)) > max j γisds j Thus, for almost all x R N (2.9) mples (γ ISDS ( u ))) = γ ISDS ( u ). (σ j (V (x))) max j γisds j (V j (x j )). V (x)f(x, u) (1 ε ) ( σ 1 ) (V (x ))g (V (x )) = (1 ε ) g (V (x)),

30 ISDS for nterconnected systems where g (r) := ( σ 1 ) (σ (r))g (σ (r)) s postve defnte and { locally Lpschtz. } As ndex was arbtrary n these consderatons, wth γ ISDS ( u ) = max j σj 1 (γj ISDS ( u )) and ḡ(r) := mn g (r), ε= max ε the condton (2.3) for the functon V s satsfed. From (2.13) we get x 1+ε ψ 1 ( V (x) ) ψ 1 ( η ( x )) and we defne V (x) :=ψ 1 ( V (x) ) as the ISDS-Lyapunov functon canddate of the whole system wth η ( x ) :=ψ 1 ( η ( x )). Note that ψ 1 K and V (x) s locally Lpschtz contnuous. By the prevous calculatons for V (x) t holds V (x) >ψ 1 ( γ ISDS ( u )) =: γ ISDS ( u ) V (x) (1 ε)g (V (x)), a.e., where g(r) :=(ψ 1 ) (ψ(r)) ḡ (ψ(r)) s locally Lpschtz contnuous. Altogether, V (x) satsfes (2.2) and (2.3). Hence, V (x) s the ISDS-Lyapunov functon of the whole system and by applcaton of Proposton the whole system has the ISDS property. Ths theorem provdes a tool how to check, whether a network possesses the ISDS property: one has to fnd the ISDS-Lyapunov functons and gans of the subsystems and has to check, f the small-gan condton s satsfed. Moreover, the theorem gves an explct constructon of the ISDS-Lyapunov functon and the correspondng rate and gans. In the followng, we present a corollary, whch s smlar to Theorem 10 n [36] for two coupled systems and covers n N coupled systems, where the rates and gans defned n Theorem are used. Here, we get decay rates for the norm of the soluton of the whole system and for each subsystem of n coupled systems wth external nput u =0. Corollary Consder the system (1.6) and assume that all subsystems have the ISDS property wth decay rates μ and gans η, γ ISDS and γj ISDS,,j =1,...,n, j. If the small-gan condton (1.15) s satsfed, then the coupled system ẋ = ẋ 1. ẋ n = f 1 (x 1,...,x n ). f n (x 1,...,x n ) = f(x) (2.14) s 0-GAS wth ( x j (t) x(t) μ (ψ 1 max { ( ( σ 1 η x 0 ))}) ),t (2.15) for, j =1,...,n,allt R +, wth functons μ, σ, ψ and η from Theorem Remark Note that for large n functon ψ n (2.11) becomes small and hence the rates and gans defned by ψ 1 become large whch s not desred n applcatons. To avod ths knd of conservatveness one can use the maxmum norm x for the states n the above defntons and n Theorem and Corollary Ths s possble as we have noted n Remark In ths case, the dvson by n n (2.12) can be avoded and we get (2.11) wth ψ ( x )=mn σ 1 ( x ). Ths s used n our examples below.

31 Chapter 2. Input-to-state dynamcal stablty (ISDS) 31 Unfortunately, we cannot compare drectly the estmaton of Theorem 10 n [36] wth our estmaton (2.15), snce another approach for estmatons of the trajectores for two coupled systems was used n [36]. The extenson of ths approach to n>2 seems to be hardly possble. Our approach allows to consder n nterconnected systems. In the frst example of the next secton, we compare our result for two coupled systems to the result n [36]. 2.3 Examples To compare Theorem 10 n [36] wth Corollary for the case of two subsystems, we consder Example 12 gven n [36]. Example Consder two nterconnected systems ẋ 1 (t) = x 1 (t)+ x3 2 (t) 2, ẋ 2 (t) = x 3 2(t)+x 1 (t). As n [36] we choose V = x and γ 1 (r) = 2 3 r3, γ 2 (r) = r, η 1,η 2 = Id, g 1 (r) = 1 4 r, g 2(r) = 1 4 r3. It s easy to check that the small-gan condton s satsfed and an Ω-path can be chosen by σ 1 (r) =Id, σ 2 (r) = r.forx0 1 = x0 2 =2the soluton x was calculated numercally. The plot of x as well as ts estmatons by (2.15) and from [36] are shown on Fgure 2.4. To compare our estmaton wth [36], we plot the ISDS estmaton n Example 12 n [36] wth respect to the maxmum norm for states usng Remark 11 n [36]. The sold (dashed) curve s the estmaton of x by Corollary ([36]). Both estmatons tend Fgure 2.4: x and estmatons wth help of Corollary (sold curve) and Example 12 n [36] (dashed curve) to zero as well as the trajectory and provde nearly the same estmate for the norm of the trajectory as t should be expected.

32 Examples The advantage of our approach s that t can be appled for larger nterconnectons. The followng example llustrates the applcaton of Theorem for a constructon of an ISDS- Lyapunov functon for the case n 2. Example Consder n N nterconnected systems of the form ẋ 1 (t) = a 1 x 1 (t)+ n j>1 1 n b 1jx 2 j(t)+ 1 n u(t), ẋ (t) = a x (t)+ 1 n b 1 x1 (t)+ n j>1,j (2.16) 1 n b jx j (t)+ 1 n u(t), =2,...,n, for b j [0, 1), a =(1+ε ), ε (1, ) and any nput u R m. We choose V (x )= x as an ISDS-Lyapunov functon canddate for the -th subsystem, =1,...,n and defne γ1j ISDS (r) :=b 1j r 2, j =2,...,n, γj1 ISDS (r) :=b j1 r, j =2,...,n, γj ISDS (r) :=b j r,, j =2,...,n, j, γ ISDS (r) :=r, =1,...,n, ( ) Γ ISDS := γj ISDS,,j =1,...,n, γisds 0, η (r) :=r and μ (r, t) =e εt r as soluton n n of d dt μ (r, t) = g (μ (r, t)) wth g (r) :=ε r. We obtan that V s an ISDS-Lyapunov functon of the -th subsystem. To check whether the small-gan condton s satsfed, we use the cycle condton, whch s satsfed (ths can be easly verfed). We choose σ(s) =(σ 1 (s),...,σ n (s)) T wth σ 1 (s) :=s 2 and σ j (s) :=s, j =2,...,n for s R +, whch s one of the possbltes of choosng σ. Then, σ s an Ω-path, whch can be easly checked. In partcular, σ satsfes Γ ISDS (σ(s)) <σ(s), s >0. Now, by applcaton of Theorem the whole system s ISDS and the ISDS-Lyapunov functon s gven by ) V (x) =ψ (max 1 σ 1 ( x ) { r, r 1, wth ψ(r) = mn σ 1 (r) =. The gans and rates of the ISDS estmaton and r, r < 1 ISDS-Lyapunov functon, respectvely, are gven by (2.11). Furthermore, f u(t) 0 then by Corollary the whole system s 0-GAS and the decay rate s gven by (2.15). In the followng, we llustrate the trajectory and the ISDS estmaton for a system consstng of subsystems of the form (2.16) for n =3. We choose a = 11 10,b j = 1 2,,j=1, 2, 3, j, u(t) =e t as the nput and the ntal values x 0 1 =0.5, x0 2 =0.8 and x0 3 =1.2. Then, we calculate the ISDS estmaton of the whole system as descrbed above and get x(t) max{μ((x 0 3) 2,t), ess sup τ [0,t] μ( u(τ),t τ)}. Ths estmaton s dsplayed n the Fgure 2.5 (dashed lne). To verfy whether the norm of the

33 Chapter 2. Input-to-state dynamcal stablty (ISDS) 33 Fgure 2.5: x and ISDS estmaton of the whole system consstng of n =3subsystems of the form (2.16). trajectory of the whole system s below the ISDS estmaton we solve the system of the form (2.16) for n =3numercally. The norm of the resultng trajectory of the whole system s also dsplayed n the Fgure 2.5. We see, f the nput u(t) tends to zero, the ISDS estmaton tends to zero as well, whereas n the case of ISS ths s not true. Also, the norm of the soluton tends to zero and t s below the ISDS estmaton. In the next chapter, the dea of ISDS and ts advantages are transferred to observer desgn for sngle systems, for subsystems of nterconnectons and for whole networks. Furthermore, we combne the ISDS property wth model predctve control n Chapter 6.

34 Examples

35 Chapter 3 Observer and quantzed output feedback stablzaton In ths chapter, we ntroduce the noton of quas-isds reduced-order observers and use error Lyapunov functons to desgn such observers for sngle systems. Consderng nterconnected systems we desgn quas-iss/isds observers for each subsystem and the whole system under a small-gan condton. Ths s appled to stablzaton of systems subject to quantzaton. We consder systems of the form (1.1) wth outputs ẋ = f(x, u), y = h(x), (3.1) where y R P s the output and functon h : R N R P s contnuously dfferentable wth locally Lpschtz dervatve (called a CL 1 functon). In addton, t s assumed that h(0) = 0 holds. In practce, observers are used for systems, where the state or parts of the state can not be measured due to uneconomc measurement costs or physcal crcumstances lke hgh temperatures, where no measurement equpment s avalable, for example. They are also used n cases, where the output of a system s dsturbed and for stablzaton of a system, for example. There, a control law subject to stablze a system s desgned usng the estmated state of the system generated by the observer based on the dsturbed output. Ths can lead to an unbounded growth of the state estmaton error and therefore to a desgn of a control law, whch does not stablzes the system. A state observer for the system (3.1) s of the form ˆξ = F (ȳ, ˆξ,u), ˆx = H(ȳ, ˆξ,u), (3.2) where ˆξ R L s the observer state, ˆx R N s the estmate of the system state x and ȳ R P s the measurement of y that may be dsturbed by d: ȳ = y + d, where d L (R +, R P ). The functon F : R P R L R m R L s locally Lpschtz n ȳ and ˆξ unformly n u and functon 35

36 36 H : R P R L R m R N s a CL 1 functon. In addton, t s assumed that F (0, 0, 0)=0and H(0, 0, 0)=0holds. We denote the state estmaton error by x =ˆx x. We are nterested under whch condtons the desgned observer guarantees that the state estmaton error s ISS or ISDS. The used stablty propertes for observers are based on ISS and ISDS, and are called quas-iss and quas-isds, respectvely. Inspred by the work [110], where the noton of quas-iss reduced-order observers was ntroduced and the advantages of ISDS over ISS, nvestgated n Chapter 2, ths motvates the ntroducton of the quas-isds property for observers, where the approaches of reducedorder observers and the ISDS property are combned. The property has the advantage that the recent dsturbance of the output of the system s taken nto account. We nvestgate under whch condtons a quas-isds reduced-order observer can be desgned for sngle nonlnear systems, where error Lyapunov functons (see [88, 60]) are used. The desgn of observers n the context of ths thess was nvestgated n [112, 60, 72, 61, 110], for example, and remarks on the equvalence of full order and reduced-order observers can be found n [111]. Consderng nterconnected systems t s desrable to have observers for each of the subsystems and the whole network. Here, we desgn quas-iss/isds reduced-order observers for each subsystem of an nterconnected system, from whch an observer for the whole system can be desgned under a small-gan condton. Furthermore, the problem of stablzaton of systems s nvestgated and we apply the presented approach to quantzed output feedback stablzaton for sngle and nterconnected systems. The goal of stablzng a system s an mportant problem n applcatons. Many approaches were performed durng the last years and the desgn of stablzng feedback laws s a popular research area, whch s lnked up wth many applcatons. The stablzaton usng output feedback quantzaton was nvestgated n [7, 70, 62, 63, 60, 71, 72, 110], for example. A quantzer s a devce, whch converts a real-valued sgnal nto a pecewse constant sgnal,.e., t maps R P nto a fnte and dscrete subset of R P. It may affect the process output or may also affect the control nput. Adaptng the quantzer wth a so-called zoom varable ths leads to dynamc quantzers, whch have the advantage that asymptotc stablty for sngle and nterconnected systems can be acheved under certan condtons. Ths chapter s organzed as follows: The noton of quas-isds observers s ntroduced n Secton 3.1, where the desgn of such an observer for sngle systems under the exstence of an error ISDS-Lyapunov functon s performed. Secton 3.2 contans all the results for the quas-iss/isds observer desgn accordng to nterconnected systems. The applcaton of the results to quantzed output feedback stablzaton for sngle and nterconnected systems can be found n Secton 3.3.

37 Chapter 3. Observer and quantzed output feedback stablzaton Quas-ISDS observer for sngle systems In ths secton, we ntroduce quas-isds observers and gve a motvatng example for the ntroducton. Then, we show that the reduced-order observer desgned n [110], Theorem 1, s a quas-isds observer provded that an error ISDS-Lyapunov functon exsts. We recall the defnton of quas-iss observers from [110], whch guarantee that the norm of the state estmaton error s bounded for all tmes. Defnton (Quas-ISS observer). The system (3.2) s called a quas-iss observer for the system (3.1), f there exsts a functon β KLand for each K>0, there exsts a functon γ K ISS K such that whenever u [0,t] K and x [0,t] K. x(t) max{ β( x 0,t), γ ISS K ( d [0,t] )}, t R + Modfyng ths defnton by usng the dea of the ISDS property to transfer the advantages of ISDS over ISS to observers we defne quas-isds observers: Defnton (Quas-ISDS observers). The system (3.2) s called a quas-isds observer for the system (3.1), f there exst functons μ KLD, η K and for each K>0 a functon γ K ISDS K such that x(t) max{ μ( η( x 0 ),t), ess sup τ [0,t] whenever u [0,t] K and x [0,t] K. μ( γ ISDS K ( d(τ) ),t τ)}, t R + Recallng that ISDS possesses the memory fadng effect, the motvaton for the ntroducton of quas-isds observers s the followng: quas-isds observers take the recent dsturbance of the measurement nto account, whereas a quas-iss observer takes nto account the supremum norm of the dsturbance. The advantage wll be llustrated by the followng example. Example Consder the system as n Example 1 n [110] ẋ = x + x 2 u, y = x, (3.3) where ˆx = ˆx + y 2 u s an observer. We consder the perturbed measurement ȳ = y + d, wth d = e t Then, the error dynamcs becomes x = x +2xud + ud 2. Ths system s ISS and ISDS from d to x when u and x are bounded. Let u 1 be constant, then the estmatons of the error dynamcs are dsplayed n the Fgure 3.1 for x 0 = x 0 =0.3. The ISS estmate s chosen equal to 1, snce β( x 0,t) γ K ISS( d[0,t] ) for a suffcent functon β, d =1, x 0 small enough and wth γ K ISS =Id. The ISDS estmaton follows by choosng

38 Quas-ISDS observer for sngle systems Fgure 3.1: Dsplayng of the trajectory, error, quas-iss and quas-isds estmate of the system (3.3). γ K ISDS ( d(τ) ) = (d(τ))2 1 ɛ and μ(r, t) =e ɛ(t) r, r 0 and ɛ =0.1. Here, the quas-iss estmaton takes the maxmal value of d nto account, whereas the quas-isds estmaton possesses the so-called memory-fadng effect. Usng a quas-isds observer, t provdes a better estmate of the norm of the state estmaton error n contrast to the usage of a quas-iss observer. In the followng, we focus on the desgn of reduced-order observers. We assume that systems of the form (3.1) can be dvded nto one part, where the state can be measured and a second part, where the state can not be measured. The practcal meanng s the followng: for systems t can be uneconomc to measure all of the systems state, because the measurement equpment or the runnng costs for the measurement are very expensve, for example. Therefore, a part of the state s measured and the other part has to be estmated. Here, we use quas-iss/isds reduced-order observers for the state estmaton, where only the part of the state s estmated that s not measured. We assume that there exsts a global coordnate change z = φ(x) such that the system (3.1) s globally dffeomorphc to a system wth lnear output of the form ż = [ ż1 ż 2 ] [ ] f1 (z 1,z 2,u) = = f(z,u), f 2 (z 1,z 2,u) (3.4) y = z 1, where z 1 R P and z 2 R N P. For the constructon of observers we need the followng assumpton, where we use reducedorder error Lyapunov functons. Error Lyapunov functons were frst ntroduced n [88] and n [60] the equvalence of the exstence of an error Lyapunov functon and the exstence of an observer was shown. Assumpton (Error ISS-Lyapunov functon). There exst a CL 1 functon l : RP R N P,aC 1 functon V : R N P R +, called an error ISS-Lyapunov functon, and functons

39 Chapter 3. Observer and quantzed output feedback stablzaton 39 α K, =1,...,4 such that for all e R N P, z R N and u R m α 1 ( e ) V (e) α 2 ( e ), (3.5) V (e) e α 4 (V (e)), (3.6) ( V (e) [ e f 2 (z 1, ẽ, u)+ l(z 1) f1 (z 1, ẽ, u)] [ z f 2 (z 1,z 2,u)+ l(z ) 1) f1 (z 1,z 2,u)] 1 z 1 (3.7) α 3 (V (e)), ẽ := e + z 2 and there exsts a functon α K such that α(s)α 4 (s) α 3 (s), s R +. Remark Note that n [110] the propertes of an error Lyapunov functon are slghtly dfferent, namely for α 3, α 4 K V (e) e α 3 ( e ), α 1 ( e ) V (e) α 2 ( e ), V (e) e α 4 ( e ), ( [ f 2 (z 1, ẽ, u)+ l(z 1) z f1 1 (z 1, ẽ, u)] [ f ) 2 (z 1,z 2,u)+ l(z 1) z f1 1 (z 1,z 2,u)] whch are equvalent to (3.5), (3.6) and (3.7). Now, the followng lemma can be stated, whch was proved n [110]. It shows, how a quas-iss observer for the system (3.4) can be desgned, provded that an error ISS-Lyapunov functon exsts. Lemma Under Assumpton 3.1.4, the system ˆξ = f 2 (ȳ, ˆξ l(ȳ),u)+ l(ȳ) f1 (ȳ, z ˆξ l(ȳ),u), 1 ẑ 1 =ȳ, ẑ 2 = ˆξ l(ȳ) (3.8) becomes a quas-iss reduced-order observer for the system (3.4), where ˆξ R N P s the observer state and ẑ 1, ẑ 2 are the estmates of z 1 and z 2, respectvely, and ȳ = y + d = z 1 + d, whch s the measurement of z 1 dsturbed by d. In order to use quas-isds observers we adapt Assumpton accordng to the ISDS property: Assumpton (Error ISDS-Lyapunov functon). Let ε>0 be gven. There exst a CL 1 functon l : R P R N P,aC 1 functon V : R N P R +, called an error ISDS-Lyapunov functon, functons α, η K and μ KLDsuch that for all e R N P, z R N and u R m e 1+ε V (e) η( e ), (3.9) V (e) e α(v (e)), (3.10) ( V e (e) [ f 2 (z 1, ẽ, u)+ l(z 1) f1 (z 1, ẽ, u)] [ z f 2 (z 1,z 2,u)+ l(z ) 1) f1 (z 1,z 2,u)] 1 z 1 (3.11) (1 ε) g(v (e)),

40 Quas-ISDS observer for sngle systems ẽ := e + z 2, where μ solves the equaton d dt μ(r, t) = g ( μ (r, t)), r,t 0 for a locally Lpschtz contnuous functon g : R + R + and there exsts a functon ᾱ K such that ᾱ(s)α(s) (1 ε)g(s), s R +. (3.12) The next theorem s a counterpart of Lemma It provdes a desgn of a quas-isds reduced-order observer for the system (3.4) provded that an error ISDS-Lyapunov functon exsts. Theorem Under Assumpton the system (3.8) becomes a quas-isds reducedorder observer for the system (3.4). The proof goes along the lnes of the proof of Lemma n [110] wth correspondng changes accordng to Defnton and Assumpton 3.1.7: Proof. We defne ξ := z 2 + l(z 1 ) and convert the system (3.4) nto ż 1 = f 1 (z 1,ξ l(z 1 ),u), y = z 1, ξ = f 2 (z 1,ξ l(z 1 ),u)+ l(z 1) f 1 (z 1,ξ l(z 1 ),u)=:f (z 1,ξ,u). z 1 Wth ths F, the dynamc of the observer state n (3.8) can be wrtten as ˆξ = F (ȳ, ˆξ,u), where ȳ = y + d and d s the measurement dsturbance. Let e := ˆξ ξ and from Assumpton we obtan V (e) = = V (e) e V (e) e ( f 2 (z 1 + d, ˆξ l(z 1 + d),u) f 2 (z 1,ξ l(z 1 ),u) + l(z 1+d) z 1 f 1 (z 1 + d, ˆξ ) l(z 1 + d),u) l(z 1) z 1 f 1 (z 1,ξ l(z 1 ),u) ( f 2 (z 1 + d, ˆξ l(z 1 + d),u) f 2 (z 1 + d, ξ l(z 1 + d),u) + l(z 1+d) z 1 f 1 (z 1 + d, ˆξ l(z 1 + d),u) l(z 1+d) z 1 f 1 (z 1 + d, ξ l(z 1 + d),u) V (e) + (F (ȳ, ξ, u) F (y, ξ, u)) e (1 ε) g( e )+α( e )(F (ȳ, ξ, u) F (y, ξ, u)). In [32], Lemma A.14, t was shown that there exst a contnuous postve functon ρ and γ K such that F (ȳ, ξ, u) F (y, ξ, u) ρ(y, ξ, u)γ( d ) and t follows for an arbtrary δ (0, 1) wth (3.12) ( ) ρ(y, ξ, u)γ( d ) V (e) ᾱ 1 1 δ V (e) (1 ε)ḡ(v (e)), )

41 Chapter 3. Observer and quantzed output feedback stablzaton 41 where ḡ(r) :=δg(r), r >0. By Theorem n [35] and ts proof ths s equvalent to { } e(t) max μ( η( e(0) ), t), ess sup μ( γ K ISDS ( d(τ) ),t τ) (3.13) τ [0,t] under z [0,t] K, u [0,t] K, where γ K ISDS K s parametrzed by K. Now, we have ( ) ( ) ( ) z1 ẑ1 z 1 d z = := =. z 2 ẑ 2 z 2 e (l(ȳ) l(z 1 )) By θ K K, parametrzed by K such that l(z 1 + d) l(z 1 ) θ K ( d ), z 1 K t follows z e + d + θ K ( d ) and e z 2 + θ K ( d ). (3.14) Overall, combnng (3.13) and (3.14) we have { z max max { μ( η( e(0) ), t), ess sup τ [0,t] μ( η( e(0) ), t), ess sup τ [0,t] μ( γ ISDS K ( d(τ) ),t τ) } + d(t) + θ K ( d(t) ) } μ( γ K ISDS ( d(τ) ),t τ) + χ K ( d(t) ) where χ K (s) :=s + θ K (s), s 0. Snce μ s a KLD-functon t follows { } z max 2 μ( η( e(0) ), t), ess sup 2 μ( γ K ISDS ( d(τ) ),t τ), ess sup 2 μ(χ K ( d(τ) ),t τ) τ [0,t] τ [0,t] { } max 2 μ( η( z 2 (0) + θ K ( d(0) )),t), ess sup 2 μ( γ K ISDS ( d(τ) ),t τ), τ [0,t] where γ K ISDS (s) :=max{ γisds K (s),χ K(s)}, and we used (3.14) and the nequalty α(a + b) max{α(2a),α(2b)} for α K,a,b 0. Furthermore, we have { } z max 2 μ( η(2 z 2 (0) ),t), 2 μ( η(2θ K ( d(0) )),t), ess sup 2 μ( γ K ISDS ( d(τ) ),t τ) τ [0,t] { max 2 μ( η(2 z 2 (0) ),t), ess sup 2 μ( η(2θ K ( d(τ) )),t τ), τ [0,t] } ess sup τ [0,t] 2 μ( γ ISDS K ( d(τ) ),t τ) { max 2 μ( η(2 z 2 (0) ),t), ess sup τ [0,t] where γ K ISDS(s) := max{ η(2θ K(s)), γ K ISDS η(s) := η(2s) t follows { z max μ( η( z(0) ), t), ess sup τ [0,t] 2 μ( γ ISDS K ( d(τ) ),t τ) for z [0,t] K, u [0,t] K, whch proves the asserton. }, (s)}. Fnally, by defnton of μ(r, t) :=2 μ(r, t) and μ( γ ISDS K ( d(τ) ),t τ) },

42 Quas-ISS and quas-isds observer for nterconnected systems Remark From Chapter 2 the decay rate and the gan of the defnton of ISDS are the same as the ones usng ISDS-Lyapunov functons. Note that ths s not the case for the defnton of a quas-isds observer and usng error ISDS-Lyapunov functons. It remans as an open topc to nvestgate f t s possble to use a dfferent error ISDS-Lyapunov functon, from whch the nformaton about the decay rate and the gans of the error ISDS-Lyapunov functon can be preserved for the quas-isds estmaton. In the next secton, we are gong to extend the noton of quas-iss/isds observers for nterconnected systems and provde tools to desgn such knd of observers. 3.2 Quas-ISS and quas-isds observer for nterconnected systems We consder n N nterconnected systems of the form (1.6) wth outputs ẋ = f (x 1,...,x n,u ), y = h (x ), (3.15) =1,...,n, where u R M are control nputs, y R P are the outputs and functons h : R N R P are CL 1 functons. In addton, t s assumed that h (0) = 0. The state observer of the th subsystem s of the form ˆξ = F (ȳ 1,...,ȳ n, ˆξ 1,...,ˆξ n,u ), ˆx = H (ȳ 1,...,ȳ n, ˆξ 1,...,ˆξ n,u ), (3.16) =1,...,n, where ˆξ R L s the observer state of the th subsystem, ˆx R N s the estmate of the system state x and ȳ R P s the measurement of y that may be dsturbed by d : ȳ = y + d, d L (R +, R P ). The functon F : R P R L R M R L, P = P,L= L s locally Lpschtz n ȳ =(ȳ 1,...,ȳ n ) T and ˆξ =(ˆξ 1,...,ˆξ n ) T unformly n u and functon H : R P R L R M R N s a CL 1 functon. In addton, t s assumed that F (0, 0, 0)=0and H (0, 0, 0)=0holds. We denote the state estmaton error of the th subsystem by x := ˆx x. It s nfluenced by the state estmaton error as well as by the measurement error of the jth subsystem whch s connected to the th subsystem. Therefore, they have to be taken nto account for the estmaton of the norm of the state estmaton error of the th subsystem. The quas- ISS/ISDS property for nterconnected systems reads as follows: 1. The th subsystem of (3.16) s a quas-iss observer for the th subsystem of (3.15), f there exsts a functon β KLand for each K > 0 there exst functons ( γ K ) ISS, ( γ K j )ISS K,j=1,...,n, j such that x (t) { max } β ( x 0 ),t), max j ( γk j )ISS ( x j [0,t] ), max j ( γk j )ISS ( (d j ) [0,t] ), ( γ K ) ISS ( (d ) [0,t] ), whenever (u ) [0,t] K and x j [0,t] K,j=1,...,n.

43 Chapter 3. Observer and quantzed output feedback stablzaton The th subsystem of (3.16) s a quas-isds observer for the th subsystem of (3.15), f there exst functons μ KLD, η K and for each K > 0 there exst functons ( γ K ) ISDS, ( γ K j )ISDS K,j=1,...,n, j such that { } x (t) max μ ( η ( x 0 ),t), max ν j ( x j,t), max ν j (d j,t), ν (d,t) j j whenever (u ) [0,t] K and x j [0,t] K,j=1,...,n, where ν j ( x j,t):=sup μ (( γ K j )ISDS ( x j (τ) ),t τ), τ [0,t] ν j (d j,t):=ess sup τ [0,t] ν (d,t):=ess sup τ [0,t] μ (( γ K j )ISDS ( d j (τ) ),t τ), μ (( γ K ) ISDS ( d (τ) ),t τ). We assume that there exsts a global coordnate change z = φ (x ) wth x =(x T 1,...,xT n ) T such that the th subsystem of (3.15) s globally dffeomorphc to a system wth lnear output of the form [ ] [ ] ż1 f1 (z 11,...,z 1n,z 21,...,z 2n,u ) ż = = = f (z 1,...,z n,u ), ż 2 f 2 (z 11,...,z 1n,z 21,...,z 2n,u ) (3.17) y = z 1 for =1,...,n, where z 1 R P and z 2 R N P. For the desgn of a quas-iss/isds observer for each subsystem of (3.15) we assume the followng: Assumpton For each =1,...,n there exst a CL 1 functon l : R P R N P, a C 1 functon V : R N P R +, functons α 1,α 2,α 3,α 4,γ j K, j =1,...,n, j such that for all e R N P, z 1 R P, z 2 R N P and u R M t holds α 1 ( e ) V (e ) α 2 ( e ), V e (e ) α 4 (V (e )), and whenever V (e ) max j γ j (V j (e j )) holds, t follows ([ V (e ) e α 3 (V (e )) ] (z 1 )f 1 (z 1,e+ z 2,u ) z 1 ]) f 2 (z 1,e+ z 2,u )+ l [ f 2 (z 1,z 2,u )+ l (z 1 )f 1 (z 1,z 2,u ) z 1 z 1 =(z11 T,...,zT 1n )T, z 2 =(z21 T,...,zT 2n )T, e + z 2 =(e 1 + z 21,...,e n + z 2n ) T and there exsts α K such that α (s)α 4 (s) α 3 (s), s R +.

44 Quas-ISS and quas-isds observer for nterconnected systems 2. Let ε be gven. For each =1,...,n there exst a CL 1 functon l : R P R N P,a C 1 functon V : R N P R +, functons α, η, γ j K, j =1,...,n, j and μ KLDsuch that for all e R N P, z 1 R P, z 2 R N P and u R M t holds e 1+ε V (e ) η ( e ), V e (e ) α (V (e )), and whenever V (e ) > max j γ j (V j (e j )) holds, t follows ([ V (e ) f 2 (z 1,e+ z 2,u )+ l ] (z 1 )f 1 (z 1,e+ z 2,u ) e z 1 [ f 2 (z 1,z 2,u )+ l ]) (z 1,z 2,u ) z 1 (1 ε )g (V (e )), where μ solves the equaton d dt μ (r, t) = g ( μ (r, t)), r,t > 0 for a locally Lpschtz contnuous functon g : R + R + and there exsts ᾱ K such that ᾱ (s)α (s) (1 ε )g (s), s R +. The next theorem s a counterpart of Lemma and Theorem for the desgn of a quas-iss/isds reduced-order observer for a subsystem of an nterconnected system. Theorem Under Assumpton 3.2.1, pont 1., the system ˆξ = f 2 (ȳ 1,...,ȳ n, ˆξ 1 l 1 (ȳ 1 ),...,ˆξ n l n (ȳ n ),u ) ẑ 1 =ȳ, + l z 1 (ȳ )f 1 (ȳ 1,...,ȳ n, ˆξ 1 l 1 (ȳ 1 ),...,ˆξ n l n (ȳ n ),u ) ẑ 2 = ˆξ l (ȳ ) (3.18) becomes a quas-iss reduced-order observer for the th subsystem of (3.17). 2. Under Assumpton 3.2.1, pont 2., the system (3.18) becomes a quas-isds reduced-order observer for the th subsystem of (3.17). Proof. The proof goes along the proof of Theorem wth changes accordng to the quas- ISS/ISDS property for nterconnected systems and Assumpton We defne ξ := z 2 + l (z 1 ). Then, ż 1 = f 1 (z 11,...,z 1n,ξ 1 l 1 (z 11 ),...,ξ n l n (z 1n ),u ), y = z 1 ξ = f 2 (z 11,...,z 1n,ξ 1 l 1 (z 11 ),...,ξ n l n (z 1n ),u ) + l (z 1 )f 1 (z 1,...,z 1n,ξ 1 l 1 (z 11 ),...,ξ n l n (z 1n ),u ) z 1 =: F (z 11,...,z 1n,ξ 1,...,ξ n,u ).

45 Chapter 3. Observer and quantzed output feedback stablzaton 45 The reduced-order observer (3.18) s wrtten as ˆξ = F (ȳ 1,...,ȳ n, ˆξ 1,...,ˆξ n,u ). Let e := ˆξ ξ. We use the shorthand for j =1, 2 ˆf d j = f j (y 1 + d 1,...,y n + d n, ˆξ 1 l 1 (y 1 + d 1 ),...,ˆξ n l n (y n + d n ),u ), f j = f j (y 1,...,y n,ξ 1 l 1 (y 1 ),...,ξ n l n (y n ),u ), then we have whenever V (e ) max j γ j (V j (e j )) holds, t follows V (e )= V ([ (e ) ˆf 2 d + l ] [ (z 1 + d ) e z ˆf 1 d f 2 + l ]) (z 1 )f 1 1 z 1 α 3 (V (e )) + V (e )(F (ȳ 1,...,ȳ n,ξ 1,...,ξ n,u ) F (y 1,...,y n,ξ 1,...,ξ n,u )) e α 3 (V (e )) + α 4 (V (e ))ρ (y 1,...,y n,ξ 1,...,ξ n,u ) max γ ( d j ), j where γ Kand ρ s a contnuous postve functon such that F (ȳ 1,...,ȳ n,ξ 1,...,ξ n,u ) F (y 1,...,y n,ξ 1,...,ξ n,u ) ρ (y 1,...,y n,ξ 1,...,ξ n,u ) max γ ( d j ) j holds, whose exstence can be shown usng the results n the appendx n [32]: F (ȳ 1,...,ȳ n,ξ 1,...,ξ n,u ) F (y 1,...,y n,ξ 1,...,ξ n,u ) ρ (y 1,...,y n,ξ 1,...,ξ n,u ) γ ( (d 1,...,d n ) T ) ρ (y 1,...,y n,ξ 1,...,ξ n,u ) γ (max n dj ) j =: ρ (y 1,...,y n,ξ 1,...,ξ n,u )γ (max d j ). j It follows that for an arbtrary δ (0, 1), wehave ( ) V (e ) α 1 (1 δ ) 1 ρ (y 1,...,y n,ξ 1,...,ξ n,u ) max γ ( d j ) j V δ α 3 (V (e )). Under the condtons that z j [0,t] K and (u ) [0,t] K, j =1,...,n t can be shown by standard arguments that there exst a functon β KL, functons γ K, γ K j K, j = 1,...,n, j such that { } e (t) max β ( e 0 ),t), max j γk j ( (e j) [0,t] ), max γ K j ( (d j ) [0,t] ), t R +. (3.19) Recallng (3.18), we have that z d + e + θ K ( d ) and e z 2 + θ K ( d ), where θ K ( d ) s a class-k functon, parametrzed by K such that l(z 1 + d ) l(z 1 ) θ K ( d ) when z 1 K. Together wth (3.19) we obtan { z (t) max β ( e 0 ),t), max j γk j ( (e j) [0,t] ), max j { max 3 β (( z θ K ( d 0 )),t), 3 max j γk 3 max j γk ( (d j ) [0,t] ), 3χ K ( (d ) [0,t] ) } γ K ( (d j ) [0,t] ) + θ K ( d (t) )+ d (t) j ( z 2j [0,t] + θ Kj ( (d j ) [0,t] )), },

46 Quas-ISS and quas-isds observer for nterconnected systems where χ K (r) :=max{ γ K (r),θ K (r),r}. Byα(a+b) max{α(2a),α(2b)} for α Kwe have that z (t) { max 3 β (2 z 2 ),t), 0 3 β (2θ K ( d 0 )),t), 3 max j γk j (2 z 2j [0,t] ), } 3 max j γk j (2θ K j ( (d j ) [0,t] )), 3 max j γk ( (d j ) [0,t] ), 3χ K ( (d ) [0,t] ) { } max β ( z 0 ),t), max j ( γk j )ISS ( z j [0,t] ), max j ( γk j )ISS ( (d j ) [0,t] ), ( γ K ) ISS ( (d ) [0,t] ), z j [0,t] K and (u ) [0,t] K,j=1,...,n, where β (r, t) :=3 β (2r, t), { } ( γ K j )ISS (r) := max 3 γ K j (2r), 3 γk j (2θ K j (r), 3 γ K (r), { ( γ K ) ISS (r) := max 3 β } (2θ K (r)), 0), 3χ K (r). Ths proves that the system (3.18) s a quas-iss reduced-order observer for the th subsystem. The proof for the quas-isds reduced-order observer for the th subsystem follows the same steps as for a quas-iss reduced-order observer. Now, f we defne P := P, N := N, m := M, z := (z1 T,...,zT n ) T R N, z 1 := (z11 T,...,zT 1n )T R P, z 2 := (z21 T,...,zT 2n )T R P N, u := (u T 1,...,uT n ) T R m, d = (d T 1,...,dT n ) T and f := (f1 T,...,fT n ) T, f1 := (f11 T,...,fT 1n )T, f2 := (f21 T,...,fT 2n )T, then the system (3.17) can be wrtten as a system of the form (3.4). Now, we nvestgate under whch condtons a quas-iss/isds observer for the whole system can be desgned. We collect all gans ( γ K j )ISS n a gan-matrx Γ ISS, whch defnes a map as n (1.12).We defne Γ ISDS accordngly. Wth these consderatons a quas-iss/isds reduced-order observer for the overall system can be desgned. Theorem Consder a system of the form (3.17). 1. Assume that Assumpton 3.2.1, pont 1. and Theorem 3.2.2, pont 1. hold true for ISS each =1,...,n. If Γ satsfes the SGC (1.15), then the reduced-order error Lyapunov functon V as n Assumpton s gven by V = max {σ 1 (V )}, where σ =(σ 1,...,σ n ) T s an Ω-path from Proposton and the quas-iss reduced-order observer for the overall system s gven by ˆξ =(ˆξ T 1,...,ˆξ T n ) T, (3.20) and ˆ ξ = f 2 (ȳ, ˆξ l(ȳ),u)+ l z 1 (ȳ)f 1 (ȳ, ˆξ,u), ẑ 1 =ȳ, ẑ 2 = ˆξ l(ȳ). (3.21)

47 Chapter 3. Observer and quantzed output feedback stablzaton Assume that Assumpton 3.2.1, pont 2. and Theorem 3.2.2, pont 2. hold true for each = 1,...,n. If Γ ISDS satsfes the SGC (1.15), then the reduced-order error Lyapunov functon V as n Assumpton s gven by V = max {σ 1 (V )}, where σ =(σ 1,...,σ n ) T s an Ω-path from Proposton and the quas-isds reduced-order observer for the overall system s gven by (3.20) and (3.21). Proof. Ad pont 1.: We defne the error Lyapunov functon canddate of the whole system by V (e) = max {σ 1 (V (e ))}, e =(e 1,...,e n ) T. We have to verfy the condtons from Assumpton 3.1.4, from whch wth the help of Lemma the observer defned by (3.20) becomes a quas-iss reduced-order observer for the whole system of the form (3.4). Let I := { {1,...,n} V (e) = 1 s V (e ) max j,j { 1 s j (V j (e j ))}}. Fx an I. Let e := ˆξ ξ. From (1.18) t follows V (e )=σ (V (e)) > max j ( γk j )ISS (σ (V (e))) = max j ( γk j )ISS (V (e )) We observe that there exst c 1 > 0,c 2 > 0 such that V (e) mn σ 1 (α 1 (c 1 e )) =: α 1 ( e ) and V (e) max σ 1 (α 2 (c 2 e )) =: α 2 ( e ). Now, wth (1.17) t holds for each V (e) e = σ 1 (V (e )) e = (σ 1 ) (V (e )) V (e ) e K 2α 4 (V (e )) α 4 (V (e)), where α 4 (r) :=max K 2 α 4 (σ (r)), K 2 s from (1.17) and (3.6) s satsfed. Furthermore, we have for each that t holds V (e) =(σ 1 ) (V (e )) V (e ) ė e =(σ 1 ) (V (e )) V (e ) ([f 2 (z 1,e 1 + z 21,...,e n + z 2n,u ) e + l ] (z 1 )f 1 (z 1,e 1 + z 21,...,e n + z 2n,u ) z 1 [ f 2 (z 1,z 2,u )+ l ]) (z 1 )f 1 (z 1,z 2,u ) z 1 K 1 α 31 (V (e )) α 3 (V (e)), where α 3 (r) := mn K 1 α 3 (σ (r)), K 1 s from (1.17) and (3.7) s satsfed. Fnally, we choose a functon α K such that α(s)α 4 (s) α 3 (s) and all the condtons from Assumpton are satsfed and by applcaton of Lemma a quas-iss reduced-order observer of the whole system can be desgned. Ad pont 2.: We defne { V (e) :=max σ 1 (V (e )) }, { η( e ) :=max σ 1 (η ( e )) }, ( ) e ψ ( e ) :=mnσ 1, n (3.22)

48 Applcatons where V satsfes the condtons n Assumpton 3.2.1, pont 2. for =1,...,n. Let j be such that e = e j for some j {1,...,n}, then we have ( ) ( ) ( ) ( ) max σ 1 e max σ 1 e 1+ε σj 1 ej mn σ 1 e 1+ε 1+ε (3.23) n(1 + ε) where ε := max ε and we obtan ( ) e ψ V (e) η( e ). (3.24) 1+ε V (e) Then, e α(v (e)) holds wth α(r) := max K 2 α (σ r), r 0, where K 2 s from (1.17). Furthermore, we have V (e) (1 ε)ḡ(v (e)) wth ε = max ε and ḡ(r) :=mn K 1 g (σ r), r 0 s postve defnte and locally Lpschtz, where K 1 s from (1.17). From (3.24) we get e 1+ε ψ 1 ( V (e) ) ψ 1 ( η ( e )) and we defne V (e) :=ψ 1 ( V (e) ) as the reduced-order error Lyapunov functon canddate wth η ( e ) :=ψ 1 ( η ( e )). By the prevous calculatons for V (e) t holds V (e) (1 ε)g (V (e)), where g(r) :=(ψ 1 ) (ψ(r)) ḡ (ψ(r)) s locally Lpschtz contnuous. Altogether, V (e) satsfes all condtons n Assumpton Hence, V (e) s the reduced-order error Lyapunov functon of the whole system and by applcaton of Theorem a quas-isds reduced-order observer of the whole system can be desgned. Remark Note that for large n the functon ψ n (3.22) becomes small and hence the rates and gans of the quas-isds property defned by ψ 1 become large, whch s not desred n applcatons. To avod ths knd of conservatveness one can use the maxmum norm nstead of the Eucldean one n the defntons above and n Theorem In ths case, the dvson by n n (3.23) can be avoded and we get (3.22) wth ψ ( e )=mn σ 1 ( e ). 3.3 Applcatons In ths secton, we nvestgate the stablzaton of sngle and nterconnected systems subject to quantzaton. At frst, we consder sngle systems and combne the quantzed output feedback stablzaton wth the ISDS property as n Chapter V n [110] for the ISS property. Then, we consder nterconnected systems and gve a counterpart to Proposton 1 n [110] for such knd of systems. Furthermore, we nvestgate dynamc quantzers, where the quantzers can be adapted by a zoomng varable. Ths leads to asymptotc stablty of the overall closed-loop system.

49 Chapter 3. Observer and quantzed output feedback stablzaton 49 We consder a sngle system of the form (3.4). By an output quantzer we mean a pecewse constant functon q : R P Q, where Q s a fnte and dscrete subset of R P. The quantzaton error s denoted by d := q(y) y. (3.25) We assume that there exsts M>0, the quantzer s range, and Δ > 0, the error bound, such that y M mples d Δ. Ths property s referred to saturaton n the lterature, see [71], for example. Now, suppose that Assumpton holds and a quas-isds observer has been desgned as n Lemma Wth d as n (3.25) the observer acts on the quantzed output measurements ȳ = q(y). Furthermore, suppose that a controller s gven n the form u = k(z). We can now defne a quantzed output feedback law by u = k(ẑ) =k(z + z), where ẑ s the state estmate generated by the observer and z =ẑ z s the state estmaton error. We mpose on the feedback law: Assumpton The system ż = f(z,k(ẑ)) = f(z,k(z + z)) s ISDS,.e., { } z(t) max ˆμ(ˆη( z 0 ),t), ess sup ˆμ(ˆγ ISDS ( z(τ) ),t τ) τ [0,t] (3.26) for some ˆμ KLD, ˆη and ˆγ ISDS K. For a detaled dscusson for the case of an ISS controller we refer to [72]. The overall closed-loop system obtaned by combnng the plant, represented n the form (3.4) after a sutable coordnate change, the observer and the control law can be wrtten as [ ] [ ] ż1 f1 (z 1,z 2,k(ẑ)) ż = =, ż 2 f 2 (z 1,z 2,k(ẑ)) [ ] [ ] ẑ1 q(z 1 ) ẑ = =, ẑ 2 ˆξ l(q(z 1 )) ˆξ = f 2 (q(z 1 ), ˆξ l(q(z 1 )),k(ẑ)) + l (q(z 1 ))f 1 (q(z 1 ), z ˆξ l(q(z 1 )),k(ẑ)). 1 We know from the equaton (3.13) n the proof of Theorem that for e = ˆξ ξ, where ξ = z 2 + l(z 1 ), the bound { } e(t) max μ( η( e 0 ),t), ess sup μ( γ ISDS ( d(τ) ),t τ) τ [0,t] holds. Combnng ths wth (3.26) and z d + e + θ K ( d ), θ K K, we obtan the estmaton ( ) { ( ( ( ) ) ) } z(t) ˆξ(t) max z0 μ η,t,ν(d, t) ˆξ 0

50 Applcatons for z [0,t] K, u[0,t] = k(ẑ)[0,t] K and μ KLD, η K, where ν(d, t) :=ess sup τ [0,t] μ(γ ISDS ( d(τ) ),t τ), γ ISDS K. Ths follows accordng to the lnes n the proof of Theorem Now, we consder an nterconnected system of the form (3.15) or (3.17), respectvely. The output quantzer of the th subsystem s gven by q : R P Q, where Q s a fnte subset of R P, the quantzaton error by d := q (y ) y, the quantzer s range M > 0 by y M, whch mples d Δ, where Δ > 0 s the error bound. We suppose that Assumpton holds and an observer for the th subsystem has been desgned by Theorem 3.2.2, whch acts on the quantzed output measurements y = q (y ). Suppose that a controller of the th subsystem s gven by u = k (z ) and the quantzed output feedback law s defned by u := k (ẑ )=k (z + z ), where ẑ s the state estmate generated by the observer and z =ẑ z s the state estmaton error. In the rest of the secton we suppose the followng: Assumpton The th subsystem ż = f (y 1,...,z,...,y n,k (ẑ )) s ISS,.e., { } z (t) max ˆβ ( z 0,t), max (ˆγ j) ISS ( ( z j ) [0,t] ), (ˆγ ) ISS ( ( z ) [0,t] ) j for some ˆβ KLand (ˆγ j ) ISS, (ˆγ ) ISS K. 2. The th subsystem ż = f (y 1,...,z,...,y n,k (ẑ )) s ISDS,.e., { } z (t) max ˆμ (ˆη ( z 0 ),t), max ˆν j ( z j,t), ˆν ( z,t) j for some ˆμ KLD, ˆη K and ˆν j ( z j,t):=ess sup τ [0,t] ˆν ( z,t):=ess sup τ [0,t] where (ˆγ j ) ISDS, (ˆγ ) ISDS K. ˆμ ((ˆγ j ) ISDS ( z j (τ) ),t τ), ˆμ ((ˆγ ) ISDS ( z (τ) ),t τ), It can be verfed, f Assumpton 3.3.2, pont 1. holds for the th subsystem of the overall closed-loop system obtaned by combnng the plant, the observer and the control law that t holds ( ) { ( ) } z (t) ˆξ (t) max z 0 β ˆξ 0,t, max j (γk j )ISS ( (d j ) [0,t] ), (γ K ) ISS ( (d ) [0,t] ), (3.27) for β KL, (γ K j )ISS (k (ẑ )) [0,t] K. and (γ K ) ISS K, z j [0,t] K for j =1,...,n, and (u ) [0,t] =

51 Chapter 3. Observer and quantzed output feedback stablzaton 51 If Assumpton 3.3.2, pont 2. holds for the th subsystem of the overall closed-loop system obtaned by combnng the plant, the observer and the control law, then t holds ( ) ( ( ) ) } z (t) {μ ˆξ (t) max z 0 η ˆξ 0,t, max ν j(d j,t),ν (d,t) (3.28) j for μ KLD, η, (γ K j )ISDS, (γ K ) ISDS K, z j [0,t] K, j =1,...,n and (u ) [0,t] = (k (ẑ )) [0,t] K, where ν j (d j,t):=ess sup τ [0,t] ν (d,t):=ess sup τ [0,t] μ ((γ K j )ISDS ( d j (τ) ),t τ), μ ((γ K ) ISDS ( d (τ) ),t τ). Furthermore, we can show that f the small-gan condton (1.15) s satsfed for Γ ISS = ((γ K j )ISS ) n n or Γ ISDS =((γ K j )ISDS ) n n, respectvely, whch defnes a map as n (1.12), then for the overall system t holds 1. ( z(t) ˆξ(t) ) max { ( β z 0 ˆξ 0 ) },t, (γ K ) ISS ( d [0,t] ), (3.29) 2. ( z(t) ˆξ(t) ) { ( ( max μ η z 0 ˆξ 0 ) ),t, ess sup τ [0,t] } μ ((γ K ) ISDS ( d(τ) ),t τ). (3.30) Let κ l K wth the property { l (z 1 ) κ l ( z 1 ), z 1, κ u K such that k (z ) κ u ( z ), z and defne K := max M,κ u (2 M +Δ + κ l ( M } +Δ )). Wth z 1 =(z11 T,...,zT 1n )T we gve a counterpart of Proposton 1 n [110] for nterconnected systems, whch provdes estmatons of the norm of the systems state and the observer state usng quantzed output feedbacks. Proposton Let γ K j (γ K j )ISDS (γ K j )ISS and γ K (γ K ) ISDS (γ K ) ISDS. 1. Assume, max{max j γ K j (Δ j),γ K (Δ )} M and ( ) z 0 <E0 (3.31) ˆξ 0 where E 0 > 0 s such that β (E 0, 0) = μ ( M, 0) = M. Then, the correspondng soluton of the th subsystem of the overall closed-loop system satsfes lm sup t ( ) z (t) ˆξ (t) max { } max j γk j (Δ j),γ K (Δ ). (3.32)

52 Applcatons 2. Assume that 1. holds for all =1,...,n. Defne M := max M, Δ := max Δ, K := max K and suppose that Γ=(γ K j ) n n satsfes the small-gan condton (1.15). Then, the correspondng soluton of the overall closed-loop system satsfes ( ) z(t) lm sup t ˆξ(t) γk (Δ). (3.33) Proof. Ad pont 1.: Note ( that t ) holds y (t) = z 1 (t) M, q (z 1 (t)) z 1 (t) = d (t) z (t) Δ. As long as t holds ˆξ (t) M, we have z (t) M K and u (t) = k (ẑ (t)) κ u ( ẑ (t) ) κ u ( q (z 1 (t)) + ˆξ (t) + l (q (z 1 (t))) ) κ u ( M +Δ + M + κ l ( M +Δ )) K. { ( ) } z (t) Defne T := sup t 0: ˆξ (t) < M. Ths s well-defned, snce (3.31) and β (E 0, 0) E0 = μ (E 0, 0) holds. It follows that (3.27) or (3.28), respectvely, s true for t [0,T] and we obtan ( ) z (t) ˆξ (t) < M, t [0,T] (3.34) usng the requrements of ths ( proposton. ) Now, assume that T s fnte. Therefore, there must exsts a T z (T ) such that ˆξ (T ) = M. But from (3.27) or (3.28), respectvely, and ( ) z (T ) from the above calculatons t holds ˆξ (T ) < M, whch contradcts the assumpton that T s fnte. It follows that T s nfnte and from the fact that z and ˆξ are contnuous the estmaton (3.34) holds for all t 0. Now, snce β KLfor every ɛ>0 there exsts T (ɛ) such that ( ( ) ) z 0 β ˆξ 0,t ɛ, t T (ɛ) and therefore ( z (t) ˆξ (t) ) max { } max j γk j (Δ j),γ K (Δ ), t T (ɛ), whch proves pont 1. Ad pont 2.: Ths follows by the same steps as for the proof of pont 1. usng (3.29) or (3.30), respectvely, under the small-gan condton Dynamc quantzers Now, we are gong to mprove the mentoned results n order to get smaller bounds (3.32) and (3.33) usng a dynamc quantzer, see [70, 72]. Here, we obtan asymptotc convergence

53 Chapter 3. Observer and quantzed output feedback stablzaton 53 n (3.32) and (3.33) and we use the zoomng-n strategy. We consder sngle systems of the form (3.4) and the dynamc quantzer q λ (y) :=λq( y λ ), where λ>0 s the zoom varable. The range of ths quantzer s Mλ and Δλ s the quantzaton error. If we ncrease λ ths s referred to the zoomng-out strategy and corresponds to a larger range and quantzaton error and by a decreasng λ we obtan a smaller range and smaller quantzaton error, referred to the zoomng-n strategy. The parameter λ can be updated contnuously, but the update of λ for dscrete nstants of tme has some advantages, see [70]. Usng a dscrete tme the dynamcs of the system change suddenly and s referred to hybrd feedback stablzaton, whch was nvestgated n [70]. Consderng nterconnected systems of the form (3.15) or (3.17), respectvely, the dynamc quantzer of the th subsystem s defned by q λ (y ):=λ q ( y λ ),λ > 0 wth range M λ and quantzaton error Δ λ. Note that to get contracton of the bound (3.32) t must hold that max{max j γ K j (Δ j),γ K (Δ )} E 0 and β (max{max j γ K j (Δ j),γ K (Δ )}, 0) = μ (max{max j γ K j (Δ j),γ K (Δ )}, 0) < M. Usng ths, we can fnd a λ < 1 such that β (max{max j γk j (Δ j),γ K (Δ )}, 0) = μ (max{max j γk j (Δ j),γ K (Δ )}, 0) < M λ. Now, for E λ > 0 such that β (E λ ( ), 0) = μ ( M λ, 0) = M λ there s a tme t for whch z ( t) { ˆξ ( t) <Eλ. Defne K λ := max M λ,κ u (2 M λ +Δ λ + κ l ( M } λ +Δ λ )) and applyng the same analyss as n Proposton 3.3.3, we obtan a smaller bound n (3.32): ( ) z (t) λ lm sup t ˆξ (t) max{max j γk j (Δ j λ ),γ Kλ (Δ λ )}. (3.35) Then, we can choose a smaller value of λ and repeat the procedure. Theoretcally, we can decrease λ to 0 and we obtan asymptotc convergence of the th subsystem. Practcally, the choce of the sze of λ depends on lmtatons, whch determnes the sze of the bound (3.35), see [72]. The descrbed procedure can be appled to the overall closed-loop system and we also obtan a smaller bound n (3.33) provded that Γ=(γ Kλ j ) n n satsfes the small-gan condton. If we can decrease λ to 0, for all =1,...,n, we obtan asymptotc convergence of the overall closed-loop system. We summarze the observatons n the followng corollary: Corollary Under a dynamc quantzer of the form q λ (y ):=λ q ( y λ ),λ > 0, t holds for the correspondng soluton of the th subsystem of the overall closed-loop system: ( ) z (t) λ 0, lm sup t ˆξ (t) 0.

54 Applcatons 2. Assume that 1. holds for all =1,...,n. Defne λ := max λ.ifγ=(γ Kλ j ) n n satsfes the small-gan condton, then for the correspondng soluton of the overall closed-loop system t holds: ( ) z(t) λ 0, lm sup t ˆξ(t) 0. Proof. Ad pont 1.: Applyng Proposton 3.3.3, pont 1. by usng the dynamc quantzer q λ (y )=λ q ( y λ ),λ > 0, we obtan (3.35). If λ 0, the asserton follows. Ad pont 2.: Ths follows, usng Proposton 3.3.3, pont 2. Consderng another type of systems, namely systems wth tme-delays, we provde tools to analyze such nterconnectons n vew of ISS n the next chapter.

55 Chapter 4 ISS for tme-delay systems In ths chapter, we state two man results, an ISS-Lyapunov-Razumkhn and an ISS-Lyapunov-Krasovsk small-gan theorem for general networks wth tme-delays. They provde tools how to check, whether a network possesses the ISS property. As an applcaton, a scenaro of a producton network wth transportatons s analyzed n vew of ISS. Consderng systems of the form (1.1) descrbed by ODEs, the future state of the system s ndependent of the past states. However, n applcatons the future state of a system can depend on past values of the state, see [44]. The systems that nclude such knd of delays are called tme-delay systems (TDS) and they have applcatons n many areas such as bology, economcs, mechancs, physcs and socal scences [5, 65], for example. In producton networks, the tme needed for transportaton of parts from one locaton to another can be consdered as tme-delay, for example. Accordng to [44, 48], the dynamcs of TDS can be modeled usng retarded functonals dfferental equatons of the form ẋ(t) =f(x t,u(t)), t R +, x 0 (τ) =ξ(τ), τ [ θ, 0], (4.1) where x R N,u L (R +, R m ) and represents the rght-hand sde dervatve. θ s the maxmum nvolved delay and f : C ( [ θ, 0] ; R N) R m R N s locally Lpschtz contnuous on any bounded set. Ths guarantees that the system (4.1) admts a unque soluton on a maxmal nterval [ θ, b), 0 <b +, whch s locally absolutely contnuous, see [44], Secton 2.6. We denote the soluton by x(t; ξ,u) or x(t) for short, satsfyng the ntal condton x 0 ξ for any ξ C([ θ, 0], R N ). Durng the last ffty years, many nterestng mathematcal problems accordng to TDS were nvestgated and n partcular a lot of results accordng to stablty of TDS were obtaned, see for example [44, 65, 42, 82]. A common tool for the stablty analyss of TDS are Lyapunov functonals, ntroduced by Krasovsk n [67], whch are a natural generalzaton of a Lyapunov functon for systems wthout tme-delays. The constructon of such a functonal s more challengng n contrast to the usage of a Lyapunov functon. Therefore, another approach to nvestgate TDS n vew of stablty s a Lyapunov-Razumkhn functon, see [44], Chapter

56 56 Overvews of exstng stablty results for TDS can be found n [44, 42]. Investgatng stablty and the problem of stablzaton of lnear tme-delay systems, one can fnd the exstng results n [82]. Consderng nonlnear contnuous-tme TDS wth external nputs and the noton of ISS, t was shown n [121] that the exstence of an ISS-Lyapunov-Razumkhn functon s suffcent for the ISS property under the assumpton that the nternal Lyapunov gan s less than the dentty functon. For dscrete-tme TDS such type of a theorem was proved n [75]. Usng Lyapunov-Krasovsk functonals, a theorem, whch mples ISS for contnuous-tme TDS under the exstence of an ISS-Lyapunov-Krasovsk functonal, was proved n [87]. However, the necessty of the exstence of such a Lyapunov functon or functonal for the ISS property s not proved yet. Note that a TDS can be unstable even f the accordng delay-free system s 0-GAS or ISS, for example. To demonstrate ths, we consder the followng example: Example Consder the system ẋ(t) = x(t τ), x 0 (s) =ξ(s), s [ τ,0], where x R, τ>0 s the delay and ξ :[ τ,0] R s the ntal (contnuous) functon of the system. Obvously, the delay-free system ẋ = x s 0-GAS for any ntal value x 0 R. Also, for tme-delays τ 1.5 the system s 0-GAS as shown n Fgure 4.1 wth ξ 10. If τ =2 and ξ 10 are chosen, then the system becomes unstable as t s dsplayed n Fgure 4.2. Fgure 4.1: Systems behavor wth τ =1.5. Fgure 4.2: Systems behavor wth τ =2. We are nterested n the ISS property for nterconnectons of systems wth tme-delays. In ths chapter, we utlze on the one hand ISS-Lyapunov-Razumkhn functons and on the other hand ISS-Lyapunov-Krasovsk functonals to prove that a network of ISS systems wth tme-delays has the ISS property under a small-gan condton, provded that each subsystem has an ISS-Lyapunov-Razumkhn functon or an ISS-Lyapunov-Krasovsk functonal,

57 Chapter 4. ISS for tme-delay systems 57 respectvely. To prove ths, we construct the ISS-Lyapunov-Razumkhn functon and ISS- Lyapunov-Razumkhn functonal, respectvely, and the correspondng gans of the whole system. Ths chapter s organzed as follows: In Secton 4.1, we nvestgate the (L)ISS property of TDS and prove an ISS-Lyapunov-Razumkhn theorem. The ISS small-gan theorems for nterconnected tme-delay systems can be found n Secton 4.2, where Subsecton contans the ISS-Lyapunov-Razumkhn type theorem and Subsecton the ISS-Lyapunov- Krasovsk type theorem. In Secton 4.3, we apply the mentoned results to logstc systems, namely producton networks. Fnally, Secton 7.3 concludes the chapter wth a summary and an outlook. 4.1 ISS for sngle tme-delay systems The noton of ISS for TDS reads as follows: Defnton ((L)ISS for TDS). The system (4.1) s called LISS n maxmum formulaton, f there exst ρ>0,ρ u > 0, β KLand γ Ksuch that for all ntal functons ξ satsfyng ξ [ θ,0] ρ and all nputs u satsfyng u ρ u t holds { } x(t) max β( ξ [ θ,0],t),γ( u ) (4.2) for all t R +. If we replace (4.2) by x(t) β( ξ [ θ,0],t)+γ( u ), then the system (4.1) s called ISS n summaton formulaton. If ρ = ρ u =, then the system (4.1) s called ISS (n maxmum or summaton formulaton). To check, whether a TDS possesses the ISS property, we defne (L)ISS-Lyapunov-Razumkhn functons, ntroduced n [121]. Defnton ((L)ISS-Lyapunov-Razumkhn functon). A locally Lpschtz contnuous functon V : D R +, wth D R N open, s called a LISS-Lyapunov-Razumkhn functon of the system (4.1), f there exst ρ>0,ρ u > 0, ψ 1,ψ 2 K, χ d,χ u,α Ksuch that B(0,ρ) D and the followng condtons hold: ψ 1 ( φ(0) ) V (φ(0)) ψ 2 ( φ(0) ), φ(0) D, (4.3) V (φ(0)) χ d ( V d (φ) [ θ,0] ) + χ u ( u ) D + V (φ(0)) α(v (φ(0))) (4.4) for all φ [ θ,0] ρ, φ C ( [ θ, 0] ; R N) and all u ρ u, where V d : C ( [ θ, 0] ; R N) C ([ θ, 0] ; R + ) wth V d (x t )(τ) :=V (x(t + τ)), τ [ θ, 0]. D + V (x(t)) denotes the upper rght-hand sde dervatve of V along the soluton x(t), whch s defned as D + V (x(t)) := lm sup h 0 + V (x(t+h)) V (x(t)) h. If ρ = ρ u =, then the functon V s called an ISS-Lyapunov-Razumkhn functon.

58 ISS for sngle tme-delay systems Note that we get an equvalent defnton of a (L)ISS-Lyapunov-Razumkhn functon, f we replace (4.4) by { ( V V (φ(0)) max χ d d (φ) [ θ,0] ) }, χ u ( u ) D + V (φ(0)) α(v (φ(0))), (4.5) where χ d, χ u, α Kand χ d,χ u,α are dfferent, n general. Wth ths defnton we state that the exstence of a LISS-Lyapunov-Razumkhn functon mples LISS: Theorem If there exsts a LISS-Lyapunov-Razumkhn functon V for system (4.1) and χ d (s) <s, s R +, s > 0, then the system (4.1) s LISS n summaton formulaton. If an LISS-Lyapunov-Razumkhn functon wth (4.5) and χ d (s) <s, s R +, s > 0 s used, then the system (4.1) s LISS n maxmum formulaton. Proof. We use (4.5) and prove the LISS property n maxmum formulaton. The proof usng the summaton formulaton follows smlarly. We use the dea of the proof of Theorem 1 n [121] wth accordng changes to the LISS property. Usng a standard comparson prncple (see [74], Lemma 4.4 or [121], Lemma 1), we obtan from (4.5) that t holds β KL. It holds { ( V (x(t)) max β(v (x 0 ),t), χ d sup s t sup s t ) V d (x s ) [ θ,0] { ( V ) V d (x s ) max β d (x 0 ), 0 [ θ,0] [ θ,0] }, χ u ( u ), (4.6) }, sup V (x(s)), (4.7) s t takng the supremum over [0,t] n (4.6) and nsertng t nto (4.7), we obtan, usng χ d (s) < s, s R + and the fact that for all b 1,b 2 > 0 from b 1 max{b 2, χ d (b 1 )} t follows b 1 b 2 : sup s t { ( V ) V d (x s ) max β d (x 0 ), 0 [ θ,0] [ θ,0], χ d (sup s t ) V d (x s ) [ θ,0] }, χ u ( u ) and therefore { x(t) max ψ1 1 ( β(ψ } 2 ( ξ [ θ,0] ), 0)),ψ1 1 ( χu ( u )), t 0, for all ξ [ θ,0] ρ, ξ C ( [ θ, 0] ; R N) and all u ρ u. Gven ε > 0 and defne κ := max{ β(ψ 2 (ρ), 0), χ u (ρ u )}. Note that t holds sup t s V d (x s ) [ θ,0] κ, whch mples V d (x 0 ) [ θ,0] κ. Let δ 2 > 0 be such that β(κ, δ 2 ) ψ 1 (ε) and let δ 1 >θ. Then, by the estmate (4.6) we have sup t s δ 1 +δ 2 V d (x s ) sup V (x(s)) [ θ,0] t s+δ 2 ( max {ψ 1 (ε), χ d sup s [0,t] V d (x s ) ) }, χ u ( u ). [ θ,0]

59 Chapter 4. ISS for tme-delay systems 59 It follows sup t s 2(δ 1 +δ 2 ) ( V d (x s ) max {ψ 1 (ε), χ d [ θ,0] sup s [δ 1 +δ 2,t] ( max {ψ 1 (ε), ( χ d ) 2 sup s [0,t] ) } V d (x s ), χ u ( u ) [ θ,0] V d (x s ) ) }, χ u ( u ). [ θ,0] Snce χ d <d, there exsts a number ñ N, whch depends on κ and ε such that By nducton we conclude that and fnally we obtan sup t s ñ(δ 1 +δ 2 ) ( χ d )ñ(κ) max {ψ 1 (ε), χ u ( u )}. V d (x s ) max {ψ 1 (ε), χ u γ u ( u )} [ θ,0] x(t) max { ε, ψ 1 1 ( χu ( u )) }, t ñ(δ 1 + δ 2 ) (4.8) for all ξ [ θ,0] ρ, ξ C ( [ θ, 0] ; R N) and all u ρ u. Usng the same technque for the constructon of β as n the proof of Lemma 3.10 n [17], we obtan x(t) max { β( ξ [ θ,0],t),γ( u ) }, for all ξ [ θ,0] ρ, and all u ρ u, where γ(r) :=ψ 1 1 ( χu (r)), r 0. Note that n [121] a smlar theorem as Theorem for the ISS property was proved, but the ISS property defned n [121] dffers from that n Defnton For systems wthout tme-delays these defntons are equvalent, see [120]. Untl now, t s an open problem whether these defntons are equvalent for tme-delay systems. A crucal step needed for the proof s that the combnaton of global stablty and the asymptotc gan property s equvalent to ISS, whch s not proved yet, see [123, 122]. Nevertheless, the mentoned result n [121] s also vald usng the ISS property n Defnton Theorem If there exsts an ISS-Lyapunov-Razumkhn functon V for system (4.1) and χ d (s) <s, s R +, s > 0, then the system (4.1) s ISS n summaton formulaton. If an ISS-Lyapunov-Razumkhn functon wth (4.5) s used, then the system (4.1) s ISS n maxmum formulaton, f χ d (s) <s, s R +, s > 0. The proof can be found n [121] wth changes accordng to the constructon of the KLfuncton β as n the proof of Theorem and the proof of Lemma 3.10 n [17]. Another approach to check, whether a system of the form (4.1) has the ISS property was ntroduced n [87]. There, ISS-Lyapunov-Krasovsk functonals are used. Gven a locally Lpschtz contnuous functonal V : C ( [ θ, 0] ; R N) R +, the upper rght-hand sde dervatve

60 ISS for sngle tme-delay systems D + V of the functonal V along the soluton x(t; ξ,u) s defned accordng to [44], Chapter 5.2: ( ( D + 1 V (φ, u) := lm sup h V x t+h) ) V (φ), (4.9) h 0 + where x t+h C ( [ θ, 0] ; R N) s generated by the soluton x(t; φ, u) of ẋ(t) =f(x t,u(t)), t (t 0,t 0 + h) wth x t 0 := φ C ( [ θ, 0] ; R N). Remark Note that n contrast to (4.9), the defnton of D + V n [87] s slghtly dfferent, snce the functonal s assumed to be only contnuous and n ths case, D + V can take nfnte values. Nevertheless, the results n [87] also hold true, f the functonal s chosen to be locally Lpschtz contnuous, accordng to the results n [85], [86] and usng (4.9). By a, we ndcate any norm n C ( [ θ, 0] ; R N) such that for some postve reals c 1,c 2 the followng nequaltes hold c 1 φ(0) φ a c 2 φ [ θ,0], φ C ( [ θ, 0] ; R N). Defnton (ISS-Lyapunov-Krasovsk functonal). A locally Lpschtz contnuous functonal V : C ( [ θ, 0] ; R N) R + s called an ISS-Lyapunov-Krasovsk functonal for the system (4.1), f there exst functons ψ 1,ψ 2 K and functons χ, α Ksuch that ψ 1 ( φ(0) ) V (φ) ψ 2 ( φ a ), (4.10) V (φ) χ ( u ) D + V (φ, u) α (V (φ)), (4.11) for all φ C ( [ θ, 0] ; R N), u L (R +, R m ). The next theorem s a counterpart to Theorem wth accordng changes to Lyapunov- Krasovsk functonals. Theorem If there exsts an ISS-Lyapunov-Krasovsk functonal V for the system (4.1), then the system (4.1) has the ISS property. Proof. Ths follows by Theorem 3.1 n [87] wth functons ρ and α 3 used there defnng ρ := ψ2 1 χ and D + V (φ, u) α 3 ( φ a ) α(v (φ)), where α := α 3 ψ 1 2 and the functonal s chosen locally Lpschtz contnuous accordng to the results n [85], [86]. Remark We conjecture that an LISS verson of Theorem can be stated. The proof wll follow the lnes of the one of Theorem 3.1 n [87] wth accordng changes to the LISS property. We skp the formulaton of the theorem and ts proof.

61 Chapter 4. ISS for tme-delay systems 61 Remark By an ISS-Lyapunov-Razumkhn functon and an ISS Lyapunov-Krasovsk functonal, there exsts two tools to check, whether a TDS has the ISS property. For some systems, the usage of a Lyapunov-Krasovsk functonal s more challengng n contrast to the usage of a Lyapunov-Razumkhn functon, because the constructon of such a functonal s not always an easy task. An approach to construct Lyapunov-Krasovsk functonals based on decomposton of a TDS can be found n [51]. More examples can be found n [87, 44]. In the next secton, we consder nterconnected TDS and develop tools to check, whether a network of TDS has the ISS property. 4.2 ISS for nterconnected tme-delay systems In ths secton, we provde ISS small-gan theorems for nterconnected TDS usng ISS- Lyapunov-Razumkhn functons and ISS-Lyapunov-Krasovsk functonals. We consder n N nterconnected TDS of the form ẋ (t) =f ( x t 1,...,x t n,u(t) ),=1,...,n, (4.12) where x t C ( [ θ, 0] ; R N ),x t (τ) :=x (t + τ), τ [ θ, 0],x R N and u L (R +, R m ). θ denotes the maxmal nvolved delay and x t j,j can be nterpreted as nternal nputs of the th subsystem. The functonals f : C ( [ θ, 0] ; R N ( ) 1)... C [ θ, 0] ; R N n R m R N are locally Lpschtz contnuous on any bounded set. We denote the soluton of a subsystem by x (t; ξ,u) or x (t) for short, satsfyng the ntal condton x 0 ξ for any ξ C([ θ, 0], R N ). The (L)ISS property for a subsystem of (4.12) reads as follows: The -th subsystem of (4.12) s LISS, f there exst β KL, γj d,γu K {0}, j =1,...,n, j and ρ j > 0,ρu > 0, j =1,...,n such that for all ξ [ θ,0] ρ, x j [ θ, ) ρ j,j, u ρu and for all t R + t holds { } x (t) max β ( ξ [ θ,0],t), max j,j γd j( x j [ θ,t] ),γ u ( u ). (4.13) If ρ j = ρu =, then the -th subsystem of (4.12) s called ISS. Ths s referred to (L)ISS n maxmum formulaton. We get an equvalent formulaton, f we replace (4.13) by x (t) β ( ξ [ θ,0],t)+ j,j γ d j( x j [ θ,t] )+γ u ( u ), where we use the same functon β and same gans for smplcty. If we defne N := N, x := (x T 1,...,xT n ) T and f := (f1 T,..., ft n ) T, then (4.12) can be wrtten as a system of the form (4.1), whch we call the whole system. We nvestgate under whch condtons the whole system has the ISS property and utlze Lyapunov-Razumkhn functons as well as Lyapunov-Krasovsk functonals.

62 ISS for nterconnected tme-delay systems Lyapunov-Razumkhn approach A locally Lpschtz contnuous functon V : D R +, wth D R N open, s an LISS- Lyapunov-Razumkhn functon for the -th subsystem of (4.12), f there exst functons V j, j =1,...,n, whch are contnuous, postve defnte and locally Lpschtz contnuous on R N j D j \{0}, ρ j > 0,ρ u > 0, functons ψ 1,ψ 2 K, χ u K {0}, χ d j K {0}, α K, j =1,...,n, such that B(0,ρ ) D and the followng condtons hold: ψ 1 ( φ (0) ) V (φ (0)) ψ 2 ( φ (0) ), φ (0) D, (4.14) { ( V ) } V (φ (0)) max max χ d d j j (φ j ), χ u ( u ) j [ θ,0] (4.15) D + V (φ (0)) α (V (φ (0))), for all φ j [ θ,0] ρ j, φ j C ( [ θ, 0] ; R N j), j =1,...,n and all u ρ u. (4.15) can be replaced by V (φ (0)) ( V ) χ d d j j (φ j ) + χ u ( u ) D + V (φ (0)) α (V (φ (0))), (4.16) [ θ,0] j where χ d j,χu,α Kand χ d j, χu, α are dfferent, n general. Wthout loss of generalty, we use χ d j and χu for the summaton and maxmum formulaton. If ρ j = ρu =, then V s called an ISS-Lyapunov-Razumkhn functon for the -th subsystem of (4.12). We collect all the gans n a gan-matrx Γ:=(χ d j ) n n,,j =1,...,n, whch defnes a map Γ:R n + R n + by ( Γ(s) = max j ) T χ d 1j(s j ),...,max χ d nj(s j ),s R n + j usng (4.15) and consderng (4.16) by Γ(s) = χ d 1j(s j ),..., j j T χ d nj(s j ),s R n +. (4.17) Recall that we get for v, w R n +: v w Γ(v) Γ(w). In contrast to nterconnected delay-free systems the gan-matrx Γ usng the Razumkhn approach has not necessarly zero entres on the man dagonal. To check, whether a network of TDS possesses the ISS property, we state the followng theorem. It provdes a tool utlzng ISS-Lyapunov-Razumkhn functons to verfy ISS of a network under a small-gan condton. Theorem (ISS-Lyapunov-Razumkhn theorem for general networks wth tme-delays) Consder the nterconnected system (4.12). Each subsystem has an ISS-Lyapunov-Razumkhn functon V wth summaton formulaton (4.16). If the correspondng gan-matrx Γ, gven by (4.17), satsfes the small-gan condton (1.16), then the functon V (x) =max{σ 1 (V (x ))}

63 Chapter 4. ISS for tme-delay systems 63 s the ISS-Lyapunov-Razumkhn functon for the whole system of the form (4.1), whch s ISS n summaton formulaton, where σ =(σ 1,...,σ n ) T s an Ω-path as n Defnton The gans are gven by χ d (r) :=max j χ u (r) :=maxλ 1 (χ u (r)) σj 1 ((χ d j) 1 ((Id + ϖ 2 ) 1 )(χ d j(σ j (r)))), for r 0, where λ(r) :=mn k λ k (r), λ k (r) := ϖ 2 ( χ d kj (σ k(r))). Remark If we consder n Theorem ISS-Lyapunov-Razumkhn functons V wth maxmum formulaton (4.15) and Γ satsfes the SGC (1.15), then the whole system s ISS n maxmum formulaton and the Lyapunov gans are gven by χ d (r) :=max,j χ u (r) :=max σ 1 (χ d j(σ j (r))), σ 1 (χ u (r)). Proof. All subsystems of (4.12) have an ISS-Lyapunov-Razumkhn functon V,=1,...,n,.e., V satsfes (4.14) and (4.16). From the small-gan condton (1.16) for Γ, gven by (4.17), t follows that there exsts an Ω-path σ =(σ 1,...,σ n ) T as n Defnton and Proposton Note that σ 1 K,=1,...,n. Let 0 x =(x T 1,...,xT n ) T. We defne V (x) :=max{σ 1 (V (x ))} as the ISS-Lyapunov-Razumkhn functon canddate for the overall system. Note that V s locally Lpschtz contnuous. V satsfes (4.3), whch can be easly checked defnng ψ 1 (r) := mn σ 1 (ψ 1 ( r n )), ψ 2 (r) :=max σ 1 (ψ 2 (r)), r>0 and usng the condton (4.14). Let I := { {1,...,n} V (x) =σ 1 (V (x )) max j,j {σj 1 (V j (x j ))}}. Fx an I. We defne χ d (r) :=max j σj 1 ((χ d j ) 1 ((Id + ϖ 2 ) 1 )(χ d j (σ j(r)))), χ u (r) :=max λ 1 (χ u (r)), r>0, where λ(r) := mn k λ k (r), λ k (r) := ϖ 2 ( χ d kj (σ k(r))) and assume ( V ) V (x(t)) χ d d (x t ) + χ u ( u(t) ). [ θ,0] Note that χ d (r) <r. It follows from ( Γ D ) (σ(r)) <σ(r), r >0 that t holds From (4.16) we obtan V (x (t)) = σ (V (x(t))) > (Id +ϖ) n j=1 D + V (x(t)) = D + σ 1 χ d j n χ d j(σ j (V (x(t)))) j=1 ( V ) d j (x t j) + χ u ( u(t) ). [ θ,0] (V (x (t)))=(σ 1 ) (V (x (t)))d + V (x (t)) (σ 1 ) (V (x (t)))α (V (x (t))) = ᾱ (V (x(t))),

64 ISS for nterconnected tme-delay systems where ᾱ (r) :=(σ 1 ) (σ (r))α (σ (r)), r > 0. By defnton of α := mn ᾱ, the functon V satsfes (4.4). All condtons of Defnton are satsfed and V s the ISS-Lyapunov- Razumkhn functon of the whole system of the form (4.1). By Theorem 4.1.4, the whole system s ISS n summaton formulaton. Usng the maxmum formulaton n (4.15), (4.5) and of the ISS property, the proof follows the same steps as for the summaton formulaton. For many applcatons, an LISS small-gan theorem for TDS usng Lyapunov-Razumkhn functons wll be benefcal due to restrctons on the ntal value or the nput functon n practce. Ths s provded n the next corollary. Corollary Consder the nterconnected system (4.12) and assume that each subsystem has a LISS-Lyapunov-Razumkhn functon V (x ) n summaton or maxmum formulaton, for all φ j [ θ,0] ρ j, φ j C ( [ θ, 0] ; R N j) wth φj (0) = x j, j =1,...,n and all u ρ u. If the correspondng gan-matrx Γ satsfes the LSGC (1.14), then the functon V as defned n Theorem s the LISS-Lyapunov-Razumkhn functon for the whole system wth φ = (φ 1,...,φ n ) T, φ [ θ,0] ρ, φ C ( [ θ, 0] ; R N ), where ρ := mn{w,ψ 1 2 (w ), mn j ρ j } such that ρ := mn ρ and ρ u := mn ρ u. Furthermore, the whole system s LISS n summaton or maxmum formulaton, respectvely. Proof. Snce Γ satsfes the LSGC (1.14), we know that there exsts a strctly ncreasng path σ :[0, 1] [0,w ], whch satsfes Γ(σ(r)) <σ(r), r (0, 1] and σ(0)=0, σ(1) = w. The LISS-Lyapunov-Razumkhn functon canddate V (x) := max {σ 1 (V (x ))} s well defned, because φ [ θ,0] ψ2 1 (w ),φ C ( [ θ, 0] ; R N ) wth φ (0) = x mples V (x ) w. Followng the same steps as n the proof of Theorem 4.2.1, we conclude that V (x) s the LISS-Lyapunov-Razumkhn functon for the whole system. In the next subsecton, we prove a small-gan theorem usng Lyapunov-Krasovsk functonals for subsystems to check, whether a network of TDS possesses the ISS property Lyapunov-Krasovsk approach The Krasovsk functonals for subsystems are as follows: A locally Lpschtz contnuous functonal V : C([ θ, 0]; R N ) R + s an ISS-Lyapunov- Krasovsk functonal of the -th subsystem of (4.12), f there exst functonals V j,j=1,...,n, whch are postve defnte and locally Lpschtz contnuous on C([ θ, 0]; R N j ), functons ψ 1,ψ 2 K, χ j, χ K {0}, α K, j =1,...,n, j such that ψ 1 ( φ (0) ) V (φ ) ψ 2 ( φ a ), φ C ( [ θ, 0], R N ) (4.18) V (φ ) max {max j,j χ j (V j (φ j )), χ ( u )} D + V (φ,u) α (V (φ )), (4.19) for all φ C ( [ θ, 0], R N ), u L (R +, R m ), =1,...,n. We get an equvalent formulaton, f we replace (4.19) by V (φ ) j,j χ j (V j (φ j )) + χ ( u ) D + V (φ,u) α (V (φ )), (4.20)

65 Chapter 4. ISS for tme-delay systems 65 where the gans are dfferent n general. Wthout loss of generalty, we use χ j and χ for the summaton and maxmum formulaton. The gan-matrx s defned by Γ:=(χ j ) n n, χ 0, =1,...,n and the map Γ:R n + R n + s defned by (1.12) usng (4.19) and consderng (4.20) t s defned by (1.13). The next theorem s the second man result of ths chapter. It s a counterpart of Theorem and t provdes a tool how to verfy the ISS property for networks of TDS usng ISS-Lyapunov-Krasovsk functonals. Theorem (ISS-Lyapunov-Krasovsk theorem for general networks wth tme-delays) Consder an nterconnected system of the form (4.12). Assume that each subsystem has an ISS-Lyapunov-Krasovsk functonal V, whch satsfes the condtons (4.18) and (4.20), = 1,...,n. If the correspondng gan-matrx Γ satsfes the small-gan condton (1.16), then V (φ) :=max{σ 1 (V (φ ))} s the ISS-Lyapunov-Krasovsk functonal for the whole system of the form (4.1), whch s ISS n summaton formulaton, where σ =(σ 1,...,σ n ) T s an Ω-path as n Defnton and φ =(φ,...,φ n ) T C([ θ, 0]; R N ). The Lyapunov gan s gven by χ(r) := max λ 1 (χ (r)) wth λ := mn k=1,...,n λ k, λ k (r) :=ϖ( n j=1,k j χ kj(σ j (r))). Remark If we consder n Theorem ISS-Lyapunov-Krasovsk functonals V wth maxmum formulaton (4.19) and Γ satsfes the SGC (1.15), then the whole system s ISS n maxmum formulaton and the Lyapunov gan s gven by χ(r) :=max σ 1 (χ (r)). Proof. All subsystems of (4.12) have an ISS-Lyapunov-Krasovsk functonal V,=1,...,n,.e., V satsfes (4.18) and (4.20). From the small-gan condton (1.16) for Γ, there exsts an Ω-path σ =(σ 1,...,σ n ) T. Let 0 x t =((x t 1 )T,...,(x t n) T ) T C([ θ, 0]; R N ). We defne V (x t ):=max (V (x t ))} as the ISS-Lyapunov-Krasovsk functonal canddate of the whole system. Note that V s {σ 1 locally Lpschtz. V satsfes (4.10), by defnton of ψ 1 (r) := mn σ 1 (ψ 1 ( r n )), ψ 2 (r) := max σ 1 (ψ 2 (r)), r>0 and usng (4.18). Let I := { {1,...,n} V (x t )= { σ 1 I. From (Γ D)(σ(r)) <σ(r), for all r>0 we get σ (r) > (Id +ϖ) χ j (σ j (r)) j,j σ (r) χ j (σ j (r)) >ϖ j,j ( V (x t ))} max j,j {σj 1 (V j (x t j ))}}. Fx an n j=1, j χ j (σ j (r)) =: λ (r). Defne λ := mn λ and assume V (x t ) λ 1 (χ ( u )). Then, t follows λ(v (x t )) χ ( u ) σ (V (x t )) j,j χ j (σ j (V (x t ))) >χ ( u )

66 Applcatons n logstcs and we get V (x t )=σ (V (x t )) >χ ( u )+ j,j χ j (σ j (V (x t ))) = χ ( u )+ j,j χ j (V j (x t j)). From (4.20) we obtan D + V (x t,u)=(σ 1 ) (V (x t ))D + V (x t,u) (σ 1 ) (V (x t ))α (V (x t )) = ᾱ (V (x t )), where ᾱ (r) :=(σ 1 ) (σ (r))α (σ (r)), r 0. By defnton of χ := max λ 1 χ and α := mn α, the functon V satsfes (4.11). All condtons of Defnton are satsfed and V s the ISS-Lyapunov-Krasovsk functonal of the whole system of the form (4.1). By Theorem the whole system s ISS n summaton formulaton. The case n the maxmum formulatons follows the same steps as for the summaton formulaton. 4.3 Applcatons n logstcs In ths secton, we present an applcaton of the Lyapunov-Razumkhn approach, namely the applcaton of Theorem 4.2.1, for the nvestgaton of LISS of logstcs networks. A typcal example of a logstc network s a producton network. Such an nterconnected system descrbes a company or cross-company owned network wth geographcally dspersed plants [126], whch are connected by transport routes. By an ncreasng number of plants and logstc objects, a central control of the network becomes challengng. Autonomous control can help to handle such complex networks, see [104, 107, 106, 105], for example. Autonomous control polces allow objects of a network, such as vehcles, contaners or machnes, for example, to decde and to execute ther own decsons based on some gven rules and avalable local nformaton to route themselves through a network. It can be seen as a paradgm shft from centralzed to decentralzed control n logstcs. Due to economc crcumstances, such as hgh nventory costs, for example, an unbounded growth of the number of logstc objects such as parts or orders, s undesred n logstc networks. The stablty analyss of logstc networks allows to get knowledge of the dynamcal propertes of a network and to desgn stable networks to avod negatve economc outcomes. For producton networks wthout takng transportatons nto account, there exst modelng approaches and the correspondng stablty analyss for certan scenaros, see [15, 103, 12, 13], for example. In [103], a dual approach for the dentfcaton of stablty regons of producton networks was presented: based on the mathematcal stablty analyss, the dentfed stablty regons were refned by an engneerng smulaton approach. The beneft of the dual approach s that t saves much tme n contrast to a pure smulaton approach. Consderng producton networks wth transportatons, the modelng was performed n [16, 12, 102, 14], for example, whch we use n the next subsecton. Transportatons can be modeled usng TDS. In the next subsecton, we provde a modelng approach to descrbe a certan scenaro of a producton network. We show, how a stablty analyss can be performed usng Theorem The autonomous control method, whch we use n the next subsecton,

67 Chapter 4. ISS for tme-delay systems 67 was modeled n [12, 13, 102]. The mpact of the presence of delays and autonomous control methods was nvestgated n [13], where also breakdowns of machnes or plants were consdered and analyzed n vew of stablty A certan scenaro Consder a producton network that processes ron ore and that conssts of four plants: one ron ore mne, one ron steel manufacturer, one car manufacturer and one tool manufacturer. x R + s measured as the amount of ron ore wthn the th locaton,.e., for one car a certan amount of ron ore s needed. The mne, locaton 1, produces ron ore and sends t to the steel manufacturer, locaton 2. The produced steel s send to the car or tool manufacturer, locatons 3 and 4, accordng to an autonomous control method. From there, one thrd each of the producton of cars and tools are send to the mne, whch need them for producton. The reflux of cars and tools can be nterpreted as a model for a recyclng process. All plants can send and get some materal to/from plants or customers outsde the network. The nput of each locaton s denoted by u. The scenaro s dsplayed n Fgure 4.3. Fgure 4.3: Scenaro of a producton network. The producton rates p (x ) Kof the locatons are chosen as p (x )=α x, α R +, whch means n practce that the plants can ncrease ther producton arbtrarly. The processed materal s send to other subsystems of the network wth the rate c j (x(t))p (x (t)), where c j (x) R +, j are some postve dstrbuton coeffcents, or to customers outsde the network. We nterpret the constant dstrbuton coeffcents as central plannng and varable dstrbuton coeffcents can be used for some autonomous control method, e.g., the queue length estmator (QLE), see [106], for example. The QLE polcy enables parts n a producton system to estmate the watng and processng tmes of dfferent alternatve processng resources. It uses exclusvely local nformaton to evaluate the states of the alternatves. The dstrbuton rates representng the QLE are gven by c j (x(t τ j )) := 1 x (t τ j )+ε 1 k x k (t τ j )+ε,

68 Applcatons n logstcs where k s the ndex of the subsystems whch get materal from subsystem j. It holds 0 c j 1. ε>0 s nserted to let the fractons be well-defned. The nterpretaton of c j s the followng: f the queue length of the th subsystem s small, then more materal wll be send to subsystem n contrast to the case where x s large and c j s small. The tme needed for the transportaton of materal from the jth to the th locaton s denoted by τ j R +. Then, the dynamcs of the th subsystem can be descrbed by retarded dfferental equatons (4.12) as ẋ 1 (t) = 1 3 α 3 x3 (t τ 31 )+ 1 3 α 4 x4 (t τ 41 )+u 1 (t) α 1 x1 (t), ẋ 2 (t) = 6 10 α 1 x1 (t τ 12 )+u 2 (t) α 2 x2 (t), ẋ 3 (t) = ẋ 4 (t) = 1 x 3 (t τ 23 )+ε 1 x 3 (t τ 23 )+x 4 (t τ 23 )+ε 1 x 4 (t τ 24 )+ε 1 x 3 (t τ 24 )+x 4 (t τ 24 )+ε 6 10 α 2 x2 (t τ 23 )+u 3 (t) α 3 x3 (t), 6 10 α 2 x2 (t τ 24 )+u 4 (t) α 4 x4 (t), where x (τ) R +,τ [ θ, 0], θ:= max{τ 31,τ 12,τ 23,τ 24 } are gven. To apply Theorem 4.2.1, we have to fnd LISS-Lyapunov-Razumkhn functons and to check, whether the small-gan condton s satsfed. We choose V (x )= x = x as a LISS-Lyapunov-Razumkhn functon canddate for the th subsystem. Obvously, V (x ) satsfes the condton (4.14). To prove that the condton (4.15) holds, we choose the functons χ d j and χu as ( χ d α 2 1j(s) := j α 1 (1 ε 1 )) s, j =3, 4, ( 2 χ d kl (s) := 9α l 10α k (1 ε k )) s, kl {21, 32, 42}, ( 2 χ u 1 (s) := 3 s),=1, 2, 3, 4, α (1 ε ) wth 0 <ε < 1. We nvestgate the frst subsystem and t holds ( ) x 1 χ d 1j Vj d (x t j) [ θ,0] 1 3 α j Vj d(xt j ) [ θ,0] 1 3 α 1(1 ε 1 ) x 1, x 1 χ u 1 ( u 1 ) u α 1(1 ε 1 ) x 1. Usng 0 c j 1 t follows V 1 (x 1 (t)) max{max j χ d 1j ( V j d(xt j ) [ θ,0]), χ u 1 ( u 1(t) )} D + V 1 (x 1 (t)) = 1 3 α 3 x3 (t τ 31 )+ 1 3 α 4 x4 (t τ 41 )+u 1 (t) α 1 x1 (t) ε 1 α 1 x1 (t) =: α 1 (V 1 (x 1 (t))). We conclude that V 1 s the LISS-Lyapunov-Razumkhn functon for the frst subsystem. By smlar calculatons, we conclude that V are the LISS-Lyapunov-Razumkhn functons for the subsystems. To verfy whether the small-gan condton s satsfed, we use the cycle condton (see Remark 1.2.1) and t holds wth the choce ε = χ d 21( χ d 32( χ d 13(s))) = ( 9α 1 10α 2 (1 ε 2 ) )2 9α ( 2 10α 3 (1 ε 3 ) )2 ( α 3 α 1 (1 ε 1 ) )2 s<s,

69 Chapter 4. ISS for tme-delay systems 69 and χ d 21 ( χd 42 ( χd 14 (s))) <sby smlar calculatons. Thus, the cycle condton s satsfed and the small gan condton also. By Theorem 4.2.1, the whole network s LISS for all x (τ) R +,τ [ θ, 0] and all nputs u R +. In the next chapter, we transfer the tools of ths chapter to mpulsve systems wth tmedelays and ther nterconnecton.

70 Applcatons n logstcs

71 Chapter 5 ISS for mpulsve systems wth tme-delays Another type of systems are mpulsve systems. They combne contnuous and dscontnuous dynamcs n one system, see [3, 43], for example. The contnuous dynamcs s typcally descrbed by dfferental equatons and the dscontnuous behavor conssts of nstantaneous state jumps that occur at gven tme nstants, also referred to as mpulses. Impulsve systems are closely related to hybrd systems [43] and swtched systems [113] and have a wde range of applcatons n network control, engneerng, bologcal or economcal systems, see [3, 124, 43, 113], for example. An mpulsve system s of the form ẋ(t) =f(x(t),u(t)), t t k,k N, x(t) =g(x (t),u (t)), t= t k,k N, (5.1) where t 0 t R +, x R N, u L (R +, R m ) and {t 1,t 2,t 3,...} s a strctly ncreasng sequence of mpulse tmes n (t 0, ) for some ntal tme t 0 <t 1. The mpulse tmes are ndependent from the state of the system. The generalzaton that ncludes state-dependent resettng of the systems state s known as hybrd systems, see for example [43, 100, 34, 66]. Impulsve systems can be vewed as a subclass of hybrd systems as proposed n [100]. The set of mpulse tmes s assumed to be ether fnte or nfnte and unbounded, and mpulse tmes t k have no fnte accumulaton pont. Gven a sequence {t k } and a par of tmes s, t satsfyng t 0 s<t, N(t, s) denotes the number of mpulse tmes t k n the sem-open nterval (s, t]. Furthermore, f : R N R m R N, g : R N R m R N, where we assume that f s locally Lpschtz. All sgnals (x and nputs u) are assumed to be rght-contnuous and to have left lmts at all tmes and we denote x := lm s t x(s). These requrements assure that a unque soluton of the mpulsve system (5.1) exsts, whch s absolutely contnuous between mpulses, see [43], Chapter 2.2. We denote the soluton by x(t; x 0,u) or x(t) for short for any ntal value x(t 0 )=x 0. Investgatng the stablty of mpulsve systems, t turns out that a condton on the 71

72 72 frequency of the mpulse tmes s helpful, f one of the contnuous or dscrete dynamcs destablzes the system. To clarfy ths, we provde the followng example. Example We consder the system ẋ(t) = 2 5 x(t), t t k, k N, x(t) =2x (t), t = t k, k N, x R and choose x(0)=1. Note that the contnuous dynamcs s 0-GAS, but the dscrete dynamcs destablzes the system. Let the mpulse tmes be gven by t k =2k, k =1, 2,... Then, we observe the stable behavor of the system dsplayed n Fgure 5.1. If the mpulse tmes occur more frequently, for example t k = k, k =1, 2,..., then the trajectory tends to nfnty, as shown n Fgure 5.2. Fgure 5.1: Systems behavor wth t k =2k. Fgure 5.2: Systems behavor wth t k = k. An example wth unstable contnuous dynamcs and stable dscrete dynamcs can be formulated smlarly. The example llustrates the mportance of the frequency of the mpulse tmes and motvates the ntroducton of a dwell-tme condton to check, whether a system s stable, as n [46, 45, 10], for example. In ths chapter, we study the ISS property of mpulsve systems wth tme-delays and ther nterconnectons usng exponental ISS-Lyapunov-Razumkhn functons and exponental ISS- Lyapunov-Krasovsk functonals. The ISS property and the ISS property for sngle systems wthout tme-delays were studed n [45] and n [10, 76] for sngle non-autonomous tme-delay systems. There, suffcent condtons, whch assure ISS and ISS of an mpulsve system, were derved usng exponental ISS-Lyapunov(-Razumkhn) functons, where n [76] multple Krasovsk functonals are used. A recent paper nvestgates the ISS property of dscrete-tme mpulsve systems wth tme-delays usng a Razumkhn approach, see [128]. In [45], the average dwell-tme condton was used, whereas n [10] a fxed dwell-tme condton was utlzed. The average dwell-tme condton was ntroduced n [46] for swtched systems. Ths condton consders the average of mpulses over an nterval, whereas the fxed dwell-tme condton consders the (mnmal or maxmal) nterval between two mpulses.

73 Chapter 5. ISS for mpulsve systems wth tme-delays 73 We provde a Lyapunov-Krasovsk type theorem and a Lyapunov-Razumkhn type ISS theorem usng the average dwell-tme condton for sngle mpulsve systems wth tme-delays. The proofs use the dea of the proof of [45], [121] and [120]. For the Razumkhn type ISS theorem we requre as an addtonal condton that the Lyapunov gan fulflls a small-gan condton. In contrast to the Razumkhn type theorem from [10], we consder autonomous tme-delay systems and the average dwell-tme condton. Our theorem allows to verfy the ISS property for larger classes of mpulse tme sequences, however, we have used an addtonal techncal condton on the Lyapunov gan n our proofs. Consderng nterconnected mpulsve systems, our man goal s to fnd suffcent condtons whch assure ISS of such nterconnectons wthout tme-delays. To ths end, we use the approach used for contnuous networks. We prove that under a small-gan condton wth lnear gans and a dwell-tme condton a network of mpulsve subsystems possesses the ISS property. We construct the exponental ISS-Lyapunov-Razumkhn and ISS-Lyapunov- Krasovsk functon(al)s and the correspondng gans of the whole system. In Secton 5.1, sngle mpulsve systems wth tme-delays are consdered. Subsecton presents the Lyapunov-Krasovsk approach and Subsecton presents the Lyapunov- Razumkhn approach. The ISS property for nterconnectons of mpulsve systems wth tme-delays s nvestgated n Secton 5.2. The tools to check whether a network possesses the ISS property can be found n Subsecton for the Lyapunov-Razumkhn approach and n Subsecton for the Lyapunov-Krasovsk approach. An example s gven n Secton Sngle mpulsve systems wth tme-delays We consder sngle mpulsve system wth tme-delays of the form ẋ(t) =f(x t,u(t)), t t k,k N, x(t) =g((x t ),u (t)), t= t k,k N, (5.2) where we make the same assumptons as n the delay-free case, where f : PC ( [ θ, 0] ; R N) R m R N s locally Lpschtz, g : PC ( [ θ, 0] ; R N) R m R N and we denote (x t ) := lm s t x s. We assume that the regularty condtons (see e.g., [4]) for the exstence and unqueness of a soluton of system (5.2) are satsfed. We denote the soluton by x(t; ξ,u) or x(t) for short for any ξ PC([ θ, 0], R N ) that exsts n a maxmal nterval [ θ, b), 0 <b +, satsfyng the ntal condton x t 0 = ξ. The soluton s pecewse rght-contnuous for all t t 0 and t s contnuous at each t t k,t t 0. The ISS property s redefned wth respect to mpulsve systems wth tme-delays, see [10]: Defnton (ISS for mpulsve systems wth tme-delays). Suppose that a sequence {t k } s gven. We call the system (5.2) or (5.19) ISS, f there exst functons β KL, γ u K, such that for every ntal condton ξ PC([ θ, 0], R N ) and every nput u t holds x(t) max{β( ξ [ θ,0],t t 0 ),γ u ( u [t0,t])}, t t 0. (5.3)

74 Sngle mpulsve systems wth tme-delays The mpulsve system (5.2) or (5.19) s unformly ISS over a gven class S of admssble sequences of mpulse tmes, f (5.3) holds for every sequence n S, wth functons β and γ u, whch are ndependent on the choce of the sequence. Remark Note that we get an equvalent defnton of ISS, f we use the summaton formulaton nstead of the maxmum formulaton n Defnton In the followng, we only use the maxmum formulaton. The results usng the summaton formulaton follow equvalently. We use the tools, ntroduced n Chapter 4, namely Lyapunov-Krasovsk functonals and Lyapunov-Razumkhn functons, to check whether an mpulsve tme-delay system has the ISS property The Lyapunov-Razumkhn methodology At frst, we gve the defnton of an exponental ISS-Lyapunov-Razumkhn functon: Defnton (Exponental ISS-Lyapunov-Razumkhn functons). A functon V : R N R + s called an exponental ISS-Lyapunov-Razumkhn functon for the system (5.2) wth rate coeffcents c, d R, fv s locally Lpschtz contnuous and there exst functons ψ 1,ψ 2, γ d, γ u K such that ψ 1 ( φ(0) ) V (φ(0)) ψ 2 ( φ(0) ), φ(0) R N and whenever V (φ(0)) max{γ d ( V d (φ) [ θ,0] ),γ u ( u )} holds, t follows D + V (φ(0)) cv (φ(0)) and (5.4) V (g(φ, u)) e d V (φ(0)), (5.5) for all φ PC ( [ θ, 0] ; R N) and u R m, where V d : PC ( [ θ, 0] ; R N) PC([ θ, 0] ; R + ) s defned by V d (x t )(τ) :=V (x(t + τ)), τ [ θ, 0]. Roughly speakng, the condton (5.4) states that f c s postve, then the functon V decreases along the soluton x(t) at t. On the other hand, f c<0 then the functon V can ncrease along the soluton x(t) at t. Condton (5.5) states that f d s postve, then the jump (mpulse) decreases the magntude of V. On the other hand, f d<0 then the jump (mpulse) ncrease the magntude of V. Remark Note that n [10] the condtons (5.4) and (5.5) are n the dsspatve form. By Proposton 2.6 n [8], the condtons n dsspatve form are equvalent to the condtons n mplcaton form, used n Defnton 5.1.3, where the coeffcents c, d are dfferent n general. For the man result of ths subsecton, we need the followng: Defnton Assume that a sequence {t k } s gven. We call the system (5.2) or (5.19) globally stable (GS), f there exst functons ϕ, γ K such that for all ξ PC([ θ, 0], R N ) and all u t holds x(t) max{ϕ( ξ [ θ,0] ),γ( u [t0,t])}, t t 0. (5.6)

75 Chapter 5. ISS for mpulsve systems wth tme-delays 75 The mpulsve system (5.2) or (5.19) s unformly GS over a gven class S of admssble sequences of mpulse tmes, f (5.6) holds for every sequence n S, wth functons ϕ and γ u, whch are ndependent on the choce of the sequence. To prove a Razumkhn type theorem for the verfcaton of ISS for mpulsve tme-delay systems, we need the followng characterzaton of the unform ISS property: Lemma The system (5.2) or (5.19) s unformly ISS over S, f and only f t s unformly GS over S and there exsts γ Ksuch that for each ɛ>0, η x R +, η u R + there exsts T 0 (whch does not depend on the choce of the mpulse tme sequence from S) such that ξ [ θ,0] η x and u [t0, ) η u mply x(t) max{ɛ, γ( u [t0,t])}, for all t T + t 0. The proof can be found n [17]. Now, we prove as a man result of ths secton the followng theorem. It provdes a tool to check whether a sngle mpulsve system wth tmedelays possesses the ISS property usng exponental Lyapunov-Razumkhn functons. Theorem (Lyapunov-Razumkhn type theorem). Let V be an exponental ISS- Lyapunov-Razumkhn functon for the system (5.2) wth c, d R, d 0. For arbtrary constants μ, λ > 0, let S[μ, λ] denote the class of mpulse tme sequences {t k } satsfyng the average dwell-tme condton dn(t, s) (c λ)(t s) μ, t s t 0. (5.7) If γ d satsfes γ d (r) <e μ d r, r > 0, then the system (5.2) s unformly ISS over S[μ, λ]. For d =0the jumps do not nfluence the stablty of the system and the system wll be ISS, f the correspondng contnuous dynamcs has the ISS property. Ths case was nvestgated more detaled n [45], Secton 6. Note that the condton (5.7) guarantees stablty of an mpulsve system even f the contnuous or dscontnuous behavor s unstable. For example, f the contnuous behavor s unstable, whch means c<0, then ths condton assumes that the dscontnuous behavor has to stablze the system (d >0) and the jumps have to occur often enough. Conversely, f the dscontnuous behavor s unstable (d <0) and the contnuous behavor s stable (c >0) then the jumps have to occur rarely, whch stablzes the system. Proof. From (5.4) we have for any two consecutve mpulses t k 1,t k, for all t (t k 1,t k ) V (x(t)) max{γ d ( V d (x t ) [ θ,0] ),γ u ( u(t) )} D + V (x(t)) cv (x(t)) (5.8) and smlarly wth (5.5) for every mpulse tme t k V (x(t k )) max{γ d ( V d ((x t k ) ) [ θ,0] ),γ u ( u (t k ) )} V (g((x t k ),u (t k ))) e d V (x(t k )). (5.9)

76 Sngle mpulsve systems wth tme-delays Because of the rght-contnuty of x and u, there exsts a sequence of tmes t 0 := t 0 < t 1 < t 1 < t 2 < t 2 <...such that for =0, 1,... we have ( ) } V (x(t)) max {γ d sup V d (x r ),γ u ( u [t0 [ θ,0],t]), t [ t, t +1 ) (5.10) r [t 0,t] and for all =1, 2,... t holds ( V (x(t)) max {γ d sup r [t 0,t] ) V d (x r ) [ θ,0] },γ u ( u [t0,t]), t [ t, t ), (5.11) where ths sequence breaks the nterval [t 0, ) nto a dsjont unon of subntervals. Suppose t 0 < t 1, so that [t 0, t 1 ) s nonempty. Otherwse we can contnue the proof n the lne below (5.13). Between any two consecutve mpulses t k 1,t k [t 0, t 1 ] from (5.10) and (5.8) we have D + V (x(t))) cv (x(t)), for all t (t k 1,t k ) and therefore V (x (t k )) e c(t k t k 1 ) V (x(t k 1 )). From (5.9) and (5.10), we have V (x(t k )) e d V (x (t k )). Combnng ths, t follows V (x(t k )) e d c(t k t k 1 ) V (x(t k 1 )) and by the teraton over the N(t, t 0 ) mpulses on [t 0,t], we obtan the bound V (x(t)) e dn(t,t 0) c(t t 0 ) V (x(t 0 )), t [t 0, t 1 ]. (5.12) Usng the dwell-tme condton (5.7), we get V (x(t)) e μ λ(t t 0) V (x(t 0 )), t [t 0, t 1 ]. (5.13) For any subnterval of the form [ t, t ), =1, 2,..., we have (5.11) as a bound for V (x(t)). Now, consder two cases. Let t be not an mpulse tme, then (5.11) s a bound for t = t. Consder the subnterval [ t, t +1 ). Repeatng the argument used to establsh (5.13), wth t n place of t 0 and usng (5.11) wth t = t we get for all t ( t, t +1 ] V (x(t)) e μ λ(t t ) V (x( t )) e μ max γ d sup V d (x r ),γ u ( u [t0, t [ θ,0] ] ). r [t 0, t ] Now, let t be an mpulse tme. Then, we have V (x( t )) e d max γ d sup V d (x r ) r [t 0, t ] [ θ,0],γ u ( u [t0, t ] ) (5.14) and n ether case ( V (x(t)) e d max {γ d sup r [t 0,t] ) V d (x r ) [ θ,0] },γ u ( u [t0,t]), t [ t, t ]. (5.15)

77 Chapter 5. ISS for mpulsve systems wth tme-delays 77 Repeatng the argument used to establsh (5.13), wth t n place of t 0 and usng (5.15) wth t = t we get V (x(t)) e μ λ(t t ) V (x( t )) e μ+ d max γ d sup V d (x r ),γ u ( u [t0, t [ θ,0] ] ), r [t 0, t ] for all t ( t, t +1 ], 1. Overall, we obtan for all t t 0 ( ) } V (x(t)) max {e μ λ(t t0) V (x(t 0 )),e μ+ d γ d sup V d (x r ),e μ+ d γ u ( u [t0 [ θ,0],t]). r [t 0,t] (5.16) Now, t holds { V } V sup d (x s ) max d (x t 0 ), sup V (x(s)). (5.17) t s t 0 [ θ,0] [ θ,0] t s t 0 We take the supremum over [t 0,t] n (5.16) and nsert t nto (5.17). Then, usng γ d (r) < e μ d r and the fact that for all b 1,b 2 > 0 from b 1 max{b 2,e μ+ d γ d (b 1 )} t follows b 1 b 2, we obtan { } sup V d (x s ) [ θ,0] max e μ+ d ψ 2 ( ξ [ θ,0] ),e μ+ d γ u ( u [t0,t]) t s t 0 and therefore { } x(t) max ψ1 1 (eμ+ d ψ 2 ( ξ [ θ,0] )),ψ1 1 (eμ+ d γ u ( u [t0,t])), t t 0, whch means that the system (5.2) s unformly GS over S[μ, λ]. Note that ϕ( ) :=ψ1 1 (eμ+ d ψ 2 ( )) s a K -functon. Now, for gven ɛ, η x,η u > 0 such that ξ [ θ,0] η x, u [t0, ) η u let κ := max { e μ+ d ψ 2 (η x ),e μ+ d γ u (η u ) }. It holds sup t s t0 V d (x s ) [ θ,0] κ. Let ρ 2 > 0 be such that e λρ 2 κ ψ 1 (ɛ) and let ρ 1 > θ. Then, by the estmate (5.16) we have sup V d (x s ) [ θ,0] sup V (x(s)) t s t 0 +ρ 1 +ρ 2 t s t 0 +ρ 2 ( ) } max {ψ 1 (ɛ),e μ+ d γ d sup V d (x r ),e μ+ d γ u ( u [t0 [ θ,0],t]). r [t 0,t] Replacng t 0 by t 0 + ρ 1 + ρ 2 n the prevous nequalty we obtan sup V d (x s ) [ θ,0] t s t 0 +2(ρ 1 +ρ 2 ) ( ) } max {ψ 1 (ɛ),e μ+ d γ d sup V d (x r ),e μ+ d γ u ( u [t0 +ρ 1 +ρ 2 r [t 0 +ρ 1 +ρ 2,t] [ θ,0],t]) ( ) } max {ψ 1 (ɛ), (e μ+ d γ d ) 2 sup V d (x r ),e μ+ d γ u ( u [t0 [ θ,0],t]). r [t 0,t]

78 Sngle mpulsve systems wth tme-delays Snce e μ+ d γ d <Id, there exsts a number ñ N, whch depends on κ and ɛ such that { } (e μ+ d γ d )ñ(κ) :=(e μ+ d γ d )... (e μ+ d γ d )(κ) max } {{ } ñ tmes ψ 1 (ɛ),e μ+ d γ u ( u [t0,t]). By nducton, we conclude that { } sup t s t 0 +ñ(ρ 1 +ρ 2 ) V d (x s ) [ θ,0] max ψ 1 (ɛ),e μ+ d γ u ( u [t0,t]), and fnally, we obtan { } x(t) max ɛ, ψ1 1 (eμ+ d γ u ( u [t0,t])), t t 0 +ñ(ρ 1 + ρ 2 ). (5.18) Thus, the system (5.2) satsfes the second property from Lemma 5.1.6, whch mples that a system of the form (5.2) s unformly ISS over S[μ, λ]. Note that the condton γ d (r) <e μ d r, r > 0 means that the gan s connected wth the average dwell-tme condton (5.7). As we wll see, ths s also the case for nterconnected mpulsve tme-delay systems. Also note that the condton γ d (r) <e μ d r, r > 0 leads to some knd of conservatveness n contrast to the condton γ d (r) <r, r>0 for tme-delay systems wthout mpulses. To relax ths condton, one can get rd of the term e d n the prevous condton, whch we quote n the followng theorem. Theorem Consder all assumptons from Theorem 5.1.7, wth γ d satsfyng γ d (r) < e μ r, r > 0. Then, the system (5.2) s unformly ISS over S[μ, λ]. The proof wth the technque to get rd of the term e d can be found n [17]. The dea s that n the proof of Theorem 5.1.7, namely n equaton (5.14), the set of tme sequences S [μ, λ] s consdered for whch the average dwell-tme condton holds wth the number of jumps n the compact nterval [s, t], defned by N (t, s), nstead of the number of jumps N(t, s) n the nterval (s,t]. It can be shown that S [μ, λ] =S[μ, λ] and that the estmatons around (5.14) hold true wth an addtonal multpler e d n (5.14), such that e d e d =1and the term e d s not necessary n the estmatons of the proof of Theorem Remark Another Razumkhn type theorem (for non-autonomous systems) has been proposed n [10]. There, a so-called fxed dwell-tme condton was used to characterze the class of mpulse tme sequences, over whch the system s unformly ISS. In contrast to Theorems 1 and 2 from [10], we prove the Razumkhn type theorem (Theorem 5.1.7) over the class of sequences, whch satsfy the average dwell-tme condton. Ths class s larger than the class of sequences, whch satsfy the fxed dwell-tme condton. However, the small-gan condton, that we have used n ths thess, γ d (r) <e μ r, r > 0 or γ d (r) <e μ d r, r > 0, respectvely, s stronger than that from [10]. In the next subsecton, we gve a counterpart to Theorem or Theorem 5.1.8, respectvely, usng exponental Lyapunov-Krasovsk functonals.

79 Chapter 5. ISS for mpulsve systems wth tme-delays The Lyapunov-Krasovsk methodology We consder another type of mpulsve systems wth tme-delays of the form ẋ(t) =f(x t,u(t)), t t k,k N, x t = g((x t ),u(t)), t= t k,k N, (5.19) where we make the same assumptons as before and the functonal g s now a map from PC ( [ θ, 0] ; R N) R m nto PC ( [ θ, 0] ; R N). Accordng to [48], Secton 2, the ntal state and the nput together determne the evoluton of a system accordng to the rght-hand sde of the dfferental equaton. Therefore, for tme-delay systems, we denote the state by the functon x t PC([ θ, 0], R N ) and we change the dscontnuous behavor n (5.19): n contrast to the system (5.2), at an mpulse tme t k not only the pont x(t k ) jumps, but all the states x t n the nterval (t k θ, t k ]. Due to ths change, the Lyapunov-Razumkhn approach cannot be appled. In ths case, we propose to use Lyapunov-Krasovsk functonals for the stablty analyss of systems of the form (5.19). Another approach usng Lyapunov functonals can be found n [109]. There, Lyapunov functonals for systems of the form (5.2) wth zero nput are used for stablzaton results of mpulsve systems, where the defnton of such a functonal s dfferent to the approach presented here accordng to mpulse tmes. Defnton (Exponental ISS-Lyapunov-Krasovsk functonals). A functonal V : PC ( [ θ, 0] ; R N) R + s called an exponental ISS-Lyapunov-Krasovsk functonal wth rate coeffcents c, d R for the system (5.19), fv s locally Lpschtz contnuous, there exst ψ 1,ψ 2 K such that ψ 1 ( φ(0) ) V (φ) ψ 2 ( φ a ), φ PC ( [ θ, 0] ; R N) (5.20) and there exsts a functon γ Ksuch that whenever V (φ) γ( u ) holds, t follows for all φ PC ( [ θ, 0] ; R N) and u R m. D + V (φ, u) cv (φ) and (5.21) V (g(φ, u)) e d V (φ), (5.22) Note that the rate coeffcents c, d are not requred to be non-negatve. The followng result s a counterpart of Theorem 1 n [45] and Theorems 1 and 2 n [10] for mpulsve systems wth tme-delays usng the Lyapunov-Krasovsk approach. Theorem (Lyapunov-Krasovsk type theorem). Let V be an exponental ISS- Lyapunov-Krasovsk functonal for the system (5.19) wth c, d R, d 0. For arbtrary constants μ, λ R +,lets[μ, λ] denote the class of mpulse tme sequences {t k } satsfyng the dwell-tme condton (5.7). Then, the system (5.19) s unformly ISS over S[μ, λ]. Proof. From (5.21) we have for any two consecutve mpulses t k 1,t k, for all t (t k 1,t k ) V (x t ) γ( u(t) ) D + V (x t,u(t)) cv (x t ) (5.23)

80 Sngle mpulsve systems wth tme-delays and smlarly wth (5.22) for every mpulse tme t k V (x t k ) γ( u (t k ) ) V (g((x t k ),u (t k ))) e d V (x t k ). (5.24) Because of the rght-contnuty of x and u there exsts a sequence of tmes t 0 := t 0 < t 1 < t 1 < t 2 < t 2 <...such that we have V (x t ) γ( u [t0,t]), t [ t, t +1 ),=0, 1,..., (5.25) V (x t ) γ( u [t0,t]), t [ t, t ),=1, 2,..., (5.26) where ths sequence breaks the nterval [t 0, ) nto a dsjont unon of subntervals. Suppose t 0 < t 1, so that [t 0, t 1 ) s nonempty. Otherwse we can contnue the proof n the lne below (5.27). Between any two consecutve mpulses t k 1,t k (t 0, t 1 ] wth (5.25) and (5.23) we have D + V (x t,u(t)) cv (x t ), for all t (t k 1,t k ) and therefore V ((x t k ) ) e c(t k t k 1 ) V (x t k 1 ). From (5.24) and (5.25) we have V (x t k) e d V ((x t k) ). Combnng ths, t follows V (x t k ) e d e c(t k t k 1 ) V (x t k 1 ) and by teraton over the N(t, t 0 ) mpulses on (t 0,t] we obtan the bound Usng the dwell-tme condton (5.7), we get V (x t ) e dn(t,t 0) c(t t 0 ) V (ξ), t (t 0, t 1 ]. V (x t ) e μ λ(t t 0) V (ξ), t (t 0, t 1 ]. (5.27) Now, on any subnterval of the form [ t, t ), we already have (5.26) as a bound. If t s not an mpulse tme, then (5.26) s a bound for t = t.if t s an mpulse tme, then we have and n ether case V (x t ) e d γ( u [t0, t ] ) V (x t ) e d γ( u [t0,t]), t [ t, t ], 1, (5.28) where ths bound holds for all t t,f t =. Now, consder any subnterval of the form [ t, t +1 ), 1. Repeatng the argument used to establsh (5.27) wth t n place of t 0 and usng (5.28) wth t = t, we get V (x t ) e μ λ(t t ) V (x t ) e μ+ d γ( u [t0, t ] ) for all t ( t, t +1 ], 1. Combnng ths wth (5.27) and (5.28), we obtan V (x t ) max{e μ λ(t t 0) V (ξ),e μ+ d γ( u [t0,t])}, t t 0. By defnton of β(r, t t 0 ):=ψ1 1 (eμ λ(t t0) ψ 2 ( cr)) and γ u (r) :=ψ1 1 (eμ+ d γ(r)) the unform ISS property follows from (5.20). Note that β and γ u do not depend on the partcular choce of the tme sequence and therefore unformty s clear. In the next secton, we nvestgate nterconnected mpulsve tme-delay systems n vew of stablty.

81 Chapter 5. ISS for mpulsve systems wth tme-delays Networks of mpulsve systems wth tme-delays We consder n nterconnected mpulsve systems wth tme-delays of the form ẋ (t) =f (x t 1,...,x t n,u (t)), t t k, x (t) =g ((x t 1),...,(x t n),u (t)), t= t k, (5.29) where the same assumptons on the system as n the delay-free case are consdered wth the followng dfferences: We denote x t (τ) :=x (t + τ), τ [ θ, 0] and (x t ) (τ) :=x (t + τ) := lm s t x (s + τ), τ [ θ, 0]. Furthermore, f : PC([ θ, 0], R N 1 )... PC([ θ, 0], R Nn ) R M R N,andg : PC([ θ, 0], R N 1 )... PC([ θ, 0], R Nn ) R M R N, where we assume that f,=1,...,n are locally Lpschtz. If we defne N := N, m := M, x =(x T 1,...,xT n ) T, u =(u T 1,...,uT n ) T, f = (f1 T,...,fT n ) T and g =(g1 T,...,gT n ) T, then (5.29) becomes a system of the form (5.2). The ISS property for systems wth several nputs and tme-delays s as follows: Suppose that a sequence {t k } s gven. The th subsystem of (5.29) s ISS, f there exst β KL, γ j,γ u K {0} such that for every ntal condton ξ and every nput u t holds x (t) max{β ( ξ [ θ,0],t t 0 ), max γ j( x j [t0 θ,t]),γ u ( u [t0,t])} (5.30) j,j for all t t 0. The th subsystem of (5.29) s unformly ISS over a gven class S of admssble sequences of mpulse tmes, f (5.30) holds for every sequence n S, wth functons β,γ j and γ u that are ndependent of the choce of the sequence. In the followng, we present tools to analyze systems of the form (5.29) n vew of ISS: exponental ISS-Lyapunov-Razumkhn functons and exponental ISS-Lyapunov-Krasovsk functonals for the subsystems The Lyapunov-Razumkhn approach Assume that for each subsystem of the nterconnected system (5.29) there s a gven functon V : R N R +, whch s contnuous, postve defnte and locally Lpschtz contnuous on R N \{0}. For =1,...,n the functon V s an exponental ISS-Lyapunov-Razumkhn functon of the th subsystem of (5.29), f there exst ψ 1,ψ 2 K, γ u K {0}, γ j K {0}, j=1,...,n and scalars c,d R, such that ψ 1 ( φ (0) ) V (φ (0)) ψ 2 ( φ (0) ), φ (0) R N (5.31) and whenever V (φ (0)) max{max j γ j ( V d j (φ j) [ θ,0] ),γ u ( u )} holds, t follows D + V (φ (0)) c V (φ (0)) (5.32) for all φ =(φ T 1,...,φT n ) T PC([ θ, 0], R N ) and u R M, where V d j : PC ( [ θ, 0] ; R N j) PC([ θ, 0] ; R + ) wth V d j (xt j )(τ) :=V j(x j (t + τ)), τ [ θ, 0] and t holds V (g (φ 1,...,φ n,u )) max{e d V (φ (0)), max γ j ( Vj d (φ j ) [ θ,0] ),γ u ( u )}, (5.33) j

82 Networks of mpulsve systems wth tme-delays for all φ PC([ θ, 0], R N ) and u R M. Another formulaton can be obtaned by replacng (5.33) by V (φ (0)) max{max γ j ( Vj d (φ j ) [ θ,0] ), γ u ( u )} V (g (φ, u )) e d V (φ (0)), j where γ j, γ u K. In the followng, we assume that the gans γ j are lnear and we denote throughout the chapter γ j (r) =γ j r, γ j,r 0. All the gans are collected n a matrx Γ:=(γ j ) n n,, j =1,...,n, whch defnes a map as n (1.12). Snce the gans are lnear, the small-gan condton (1.15) s equvalent to Δ(Γ) < 1, (5.34) where Δ s the spectral radus of Γ, see [24, 98]. Note that Δ(Γ) < 1 mples that there exsts a vector s R n, s>0 such that Γ(s) <s. (5.35) Now, we state one of the man results: the ISS small-gan theorem for nterconnected mpulsve systems wth tme-delays and lnear gans. We construct the Lyapunov-Razumkhn functon and the gan of the overall system under a small-gan condton, whch s here of the form Γ(s) mn{e μ,e d μ }s, s R n +\{0} s R n +\{0} : Γ(s) < mn{e μ,e d μ }s, (5.36) where μ s from the dwell-tme condton (5.7) and d := mn d. Theorem Assume that each subsystem of (5.29) has an exponental ISS-Lyapunov- Razumkhn functon wth c,d R, d 0and gans γ u, γ j, where γ j are lnear. Defne c := mn c and d := mn d. If for some μ>0 the operator Γ satsfes the small-gan condton (5.36), then for all λ>0 the whole system (5.2) s unformly ISS over S[μ, λ] and the exponental ISS-Lyapunov-Razumkhn functon s gven by where s =(s 1,...,s n ) T s from (5.36). The gans are gven by V (x) :=max{ 1 s V (x )}, (5.37) γ d (r) :=max{e d, 1} max k,j γ u (r) :=max{e d, 1} max 1 s k γ kj s j r, 1 s γ u (r). Proof. Let 0 x =(x T 1,...,xT n ) T. We defne V (x) as n (5.37) and show that V s the exponental ISS-Lyapunov-Razumkhn functon for the overall system. Note that V s locally Lpschtz contnuous and satsfes (5.31), whch can be easly checked. Let I := { {1,...,n} V (x) = 1 s V (x ) max j,j { 1 s j (V j (x j ))}}. Fx an I.

83 Chapter 5. ISS for mpulsve systems wth tme-delays We defne the gans γ d (r) := max k,j s k γ kj s j r, γ u (r) := max s γ u (r), r>0 and assume V (x(t)) max{ γ d ( V d (x t ) [ θ,0] ), γ u ( u(t) )}. It follows V (x (t)) s max{max k,j Then, from (5.32) we obtan 1 γ kj s j V d (x t 1 ) s [ θ,0], max k s γ u ( u(t) )} max{max γ j Vj d (x t j) [ θ,0],γ u ( u (t) )}. j D + V (x(t)) = D + 1 s V (x (t)) 1 s c V (x (t)) = c V (x(t)). We have shown that for c = mn c the functon V satsfes (4.5) wth γ d, γ u. Defnng d := mn d and usng (5.33) t holds V (g(x t,u(t))) = max{ 1 s V (g (x t 1,...,x t n,u (t)))} max { 1 s max{e d V (x (t)), max j γ j V d j (x t j) [ θ,0],γ u ( u (t) )}} max { 1,j j s e d s V (x(t)), 1 s γ j s j V d (x t ) [ θ,0], 1 s γ u ( u (t) )} max{e d V (x(t)), γ d ( V d (x t ) [ θ,0] ), γ u ( u(t) )}. Defnng γ d (r) := e d γ d (r), γ u (r) := e d γ u (r) and assumng that t holds V (x(t)) max{ γ d ( V d (x t ) [ θ,0] ), γ u ( u(t) )}, t follows V (g(x t,u(t))) max{e d V (x(t)), γ d ( V d (x t ) [ θ,0] ), γ u ( u(t) )} = max{e d V (x(t)),e d γ d ( V d (x t ) [ θ,0] ),e d γ u ( u(t) )} e d V (x(t)),.e., V satsfes the condton (5.5) wth γ d, γ u. Now, defne γ d (r) :=max{ γ d (r), γ d (r)} and γ u (r) := max{ γ u (r), γ u (r)}. Then, V satsfes (4.5) and (5.5) wth γ d,γ u. By (5.36) t holds γ d (r) = max{ γ d (r),e d γ d (r)} < max{mn{e μ,e d μ }, mn{e d μ,e μ }}r = e μ r. All condtons of Defnton are satsfed and V s the exponental ISS-Lyapunov- Razumkhn functon of the whole system of the form (5.2). We can apply Theorem and the whole system s unformly ISS over S[μ, λ] The Lyapunov-Krasovsk approach Let us consder n nterconnected mpulsve subsystems of the form ẋ (t) =f (x t 1,...,x t n,u (t)), t t k, x t = g ((x t 1),...,(x t n),u (t)), t= t k, (5.38) k N, =1,...,n, where we make the same assumptons as n the prevous subsectons and the functonals g are now maps from PC ( [ θ, 0] ; R N ( ) 1)... PC [ θ, 0] ; R N n R M nto PC ( [ θ, 0] ; R N ).

84 Networks of mpulsve systems wth tme-delays Note that the ISS property for systems of the form (5.38) s the same as n (5.30). If we defne N, m, x, u, f and g as before, then (5.38) becomes a system of the form (5.19). Exponental ISS-Lyapunov-Krasovsk functonals of systems wth several nputs are as follows: Assume that for each subsystem of the nterconnected system (5.38) there s a gven functonal V : PC ( [ θ, 0] ; R N ) R+, whch s locally Lpschtz contnuous and postve defnte. For =1,...,n the functonal V s an exponental ISS-Lyapunov-Krasovsk functonal of the th subsystem of (5.38), f there exst ψ 1,ψ 2 K, γ K {0}, γ j K {0}, γ 0,,j=1,...,n and scalars c,d R such that ψ 1 ( φ (0) ) V (φ ) ψ 2 ( φ a ), φ PC ( [ θ, 0] ; R N ) and whenever V (φ ) max{max j,j γ j (V j (φ j )),γ ( u )} holds, t follows and D + V (φ,u ) c V (φ ) (5.39) V (g (φ, u )) max{e d V (φ ), max j,j γ j(v j (φ j )),γ ( u )}, (5.40) for all φ PC ( [ θ, 0] ; R N ),φ=(φ T 1,...,φ T n ) T and u R M. A dfferent formulaton can be obtaned by replacng (5.40) by V (φ ) max{max j,j γ j(v j (φ j )), γ ( u )} V (g (φ, u )) e d V (φ ), where γ j, γ K. Furthermore, we defne the gan-matrx Γ=(γ j ) n n wth γ 0. The next result s an ISS small-gan theorem for mpulsve systems wth tme-delays usng the Lyapunov-Krasovsk methodology. Ths theorem allows to construct an exponental ISS- Lyapunov-Krasovsk functonal and the correspondng gan for the whole nterconnecton under a dwell-tme and a small-gan condton. Theorem Assume that each subsystem of (5.38) has an exponental ISS-Lyapunov- Krasovsk functonal V wth correspondng gans γ,γ j, where γ j are lnear, and rate coeffcents c, d, d 0. Defne c := mn c and d := mn {d, ln( s j,j, j s γ j )}. If Γ satsfes the small-gan condton (1.15), then the mpulsve system (5.19) s unformly ISS over S[μ, λ], μ, λ > 0 and the exponental ISS-Lyapunov-Krasovsk functonal s gven by V (φ) :=max{ 1 s V (φ )}, (5.41) where s =(s 1,...,s n ) T s from (5.35), φ PC ( [ θ, 0] ; R N). The gan s gven by γ(r) := max{e d 1, 1} max s γ (r). Proof. Let 0 x t =((x t 1 )T,...,(x t n) T ) T and let V be defned by V (x t ):=max { 1 s (V (x t ))}. 1 Defne the ndex set I := { {1,...,n} s V (φ ) max j,j { 1 s j (V j (φ j ))}}, where φ PC ( [ θ, 0] ; R N ),φ=(φ T 1,...,φ T n ) T. Fx an I.

85 Chapter 5. ISS for mpulsve systems wth tme-delays 85 It can be easly checked that the condton (5.20) s satsfed. 1 We defne γ(r) :=max s γ (r), r>0 and assume V (x t ) γ( u(t) ). It follows from (5.35) that t holds V (x t )=s V (x t ) max{max γ j s j V (x t ),s j γ( u(t) )} max{max γ j V j (x t j),γ ( u (t) )}. j Then, from (5.39) we obtan D + V (x t,u(t)) = 1 s D + V (x t,u (t)) 1 s c V (x t )= c V (x t ). By defnton of c = mn c, the functon V satsfes (5.21) wth γ. Wth d := mn,j, j {d, ln( s j s γ j )} and usng (5.40) t holds V (g(x t,u(t))) = max{ 1 s V (g (x t 1,...,x t n,u (t)))} max{ 1 s max{e d V (x t ), max γ jv j (x t j),γ ( u (t) )}} j,j max { 1,j j s e d s V (x t ), 1 s γ j s j V (x t ), 1 s γ ( u (t) )} max{e d V (x t ), γ( u(t) )}. Defne γ(r) :=e d γ(r). If t holds V (x t ) γ( u(t) ), t follows V (g(x t,u(t))) max{e d V (x t ), γ( u(t) )} = max{e d V (x t ),e d γ( u(t) )} e d V (x t ) for all x t and u and V satsfes (5.22) wth γ. Defne γ(r) :=max{e d 1, 1} max s γ (r), then we conclude that V s an exponental ISS-Lyapunov-Krasovsk functonal for the overall system of the form (5.19) wth rate coeffcents c, d and gan γ. We can apply Theorem and the overall system s unformly ISS over S[μ, λ], μ, λ > 0. In the next secton, we gve an example of a networked control system and we apply Theorem to check whether the system has the ISS property. 5.3 Example Networked control systems (NCS) are spatally dstrbuted systems, where sensors, actuators and controllers communcate through a shared band-lmted dgtal communcaton network. They are appled n moble sensor networks, remote surgery, haptcs collaboraton over the Internet or automated hghway systems and unmanned vehcles, see [47] and the references theren. Stablty analyss for such knds of networks were performed n [125], [83] and [45], for example. Here, we consder a class of NCS gven by an nterconnecton of lnear systems wth tme-delays. The th subsystem s descrbed as follows: ẋ (t) = a x (t)+ a j x j (t τ j )+b ν (t), j,j (5.42) y (t) = x (t)+μ (t),

86 Example where x R, =1,...,n, τ j [0,θ] s a tme-delay of the nput from the other subsystems wth maxmum nvolved delay θ>0 and ν L (R +, R) s an nput dsturbance. y R s the measurement of the system, dsturbed by μ and a j,a,b > 0 are some parameters to descrbe the nterconnecton. {t 1,t 2,...} s a sequence of tme nstances at whch measurements of x are sent. It s allowed to send only one measurement per each tme nstant. Between the sendng of new measurements the estmate ˆx of x s gven by ˆx (t) = a ˆx (t)+ j,j a j ˆx j (t τ j ),t {t 1,t 2,...}. (5.43) At tme t k the node k gets access to the measurement y k of x k and all other nodes stay unchanged: { y ˆx (t k )= k (t k ), = k, ˆx (t k), k. The estmaton error of the th subsystem s defned as n Secton 3.2 by e := ˆx x. The dynamcs of e can be then gven by the followng mpulsve system: ė (t) = a e (t)+ j,j a j e j (t τ j ) b ν (t), t t k,k N, (5.44) { μ e (t k )= k (t k ), = k, e (t k), k. (5.45) The decson whch measurement of a subsystem wll be sent, s performed usng some protocol, see [83], for example. To verfy that the error dynamcs of the whole nterconnected system (5.44), (5.45) has the unformly ISS property, we show that there exsts an exponental ISS-Lyapunov-Razumkhn functon for each subsystem and that the small-gan condton (5.36) and the dwell-tme condton (5.7) are satsfed. At frst, we wll defne an ISS-Lyapunov-Razumkhn functon canddate for each subsystem. Consder the functon V (e ):= e. If t = t k, then V (g (e )) max{ e, μ } = max{e d V (e ), μ } wth d =0. Consder now the case t t k.if then we have { e max max j,j n a j a ɛ max V j(e j (s)),n b } ν,ɛ [0,a ), t θ s t a ɛ D + V (e )=( a e + j,j a j e j (t τ j ) b ν ) sgn e a e + j,j a j e j (t τ j ) + b ν a e +(a ɛ ) e = ɛ e = ɛ V (e )=: c V (e ) Thus, the functon V (e )= e s an exponental ISS-Lyapunov-Razumkhn functon for the th subsystem wth c = ɛ, d =0, γ j (r) =n a j a ɛ r and γ (r) = max{1,n b a ɛ }r.

87 Chapter 5. ISS for mpulsve systems wth tme-delays 87 To prove ISS of the whole error system, we need to check the dwell-tme condton (5.7) and the small-gan condton (5.36), see Theorem Let us check condton (5.7). We have d = mn d =0, c = mn c = mn ɛ > 0. Takng 0 <λ c and any μ>0, the dwell-tme condton s satsfed for any t s 0 and tme sequence {t k }: dn(t, s) (c λ)(t s) = (c λ)(t s) 0 <μ. The fulfllment of the small gan condton (5.36) can be checked by slghtly modfyng the cycle condton: for all (k 1,..., k p ) {1,..., n} p, where k 1 = k p, t holds γ k1 k 2 γ k2 k 3... γ kp 1 k p <e μ Id, where n ths example we can choose μ arbtrarly small. Let us check ths condton for the followng parameters: n =3, b =1, ν 2, ɛ =0.1, =1, 2, 3; τ j =0.03,, j =1, 2, 3, j; e(s) =(0.9; 0.3; 0.6) T, s [ θ, 0]; a 1 =1, a 2 =2, a 3 =0.5, A := (a j ) 3 3 = The system uses a TOD-lke protocol [83] and t sends measurements at tme nstants t k = 0.1k, k N. The gan matrx Γ s then gven by Γ:=(γ j ) 3 3 = It s easy to check that all the cycles are less than the dentty functon multpled by e μ, because μ can be chosen arbtrarly small. Thus, the cycle condton s satsfed and by applcaton of Theorem the error system (5.44), (5.45) s unformly ISS. The trajectory of the Eucldean norm of the error s gven n Fgure 5.3. It remans as an open research topc, not analyzed n ths thess, to nvestgate the nfluence of the nput dsturbances to the error systems n more detal as well as to nvestgate the nfluence to the error system of the usage of dfferent protocols (see [83]). In the next chapter, we nvestgate on the one hand the ISDS property of MPC for sngle and nterconnected systems, and on the other hand we analyze the ISS property of MPC for sngle and nterconnected TDS. The tools, presented n Chapter 2 and Chapter 4 are used for MPC.

88 Example Fgure 5.3: The trajectory of the Eucldean norm of the error of the networked control system.

89 Chapter 6 Model predctve control In ths chapter, we ntroduce the ISDS property for MPC of sngle and nterconnected systems and we ntroduce the ISS property for MPC of sngle and nterconnected TDS. We derve condtons whch assure ISDS and ISS, respectvely, for such systems usng an MPC scheme and usng the results of Chapter 2 and Chapter 4, respectvely. The approach of MPC started n the late 1970s and spread out n the 1990s by an ncreasng usage of automaton processes n the ndustry. It has a wde range of applcatons, see the survey papers [89, 90]. The am of MPC s to control a system to follow a certan trajectory or to steer the soluton of a system nto an equlbrum pont under constrants and unknown dsturbances. Addtonally, the control should be optmal n vew of defned goals, e.g., optmal regardng tme or effort. We consder systems of the form (1.1) wth dsturbances, ẋ(t) =f(x(t),w(t),u(t)), (6.1) where w W L (R +, R P ) s the unknown dsturbance and W s a compact and convex set contanng the orgn. The nput u s a measurable and essentally bounded control subject to nput constrants u U, where U R m s a compact and convex set contanng the orgn n ts nteror. The functon f has to satsfy the same condtons as n Chapter 1 to assure that a unque soluton exsts, whch s denoted by x(t; x 0,w,u) or x(t) n short. Note that the majorty of the works regardng MPC consder dscrete-tme systems. One can derve a dscrete-tme system from a contnuous-tme system usng sampled-data systems, see [84], for example. The prncple of MPC s the followng: at samplng tme t = kδ, k N, Δ > 0, the current state of a system of the form (6.1) s measured, whch s x(t). Ths s used as an ntal value to predct the trajectory of the system untl the tme t + T, where T>0 s the fnte predcton horzon, wth an arbtrary control u. Let us frst assume w 0. Then, the cost functon defned by J(x(t),u; t, T ):= t+t t l(x(t ),u(t ))dt (6.2) 89

90 90 wll be mnmzed wth respect to the control u subject to constrants x X, u U, where X R N s a compact and convex set contanng the orgn n ts nteror. The functon l : R N R m R + s the stage cost and t s a penalty of the dstance of the state from the equlbrum pont f(0, 0, 0)=0and a penalty of the control effort. A popular choce of the stage cost s l(x, u) = x 2 + c q u 2, for example, where c q > 0 s a weghtng coeffcent of the control. Ths control strategy s referred to as open-loop MPC [79, 92, 84]. A sutable approach to stablze a system wth mplemented control obtaned by an MPC scheme s that the termnal predcted state of the system should satsfy a termnal constrant, namely x(t + T ) Ω, where Ω R N s the termnal regon. Then, we add the addtonal term V f (x(t + T )) to the cost functon of the MPC scheme, where V f :Ω R +. The soluton of ths optmzaton problem s a control n feedback form u (t) :=π(t, x(t)) and u (t ), t [t, t +Δ] s appled to the system. We assume that the feedback π Π s essentally bounded, locally Lpschtz n x and measurable n t. The set Π R m s assumed to be compact and convex contanng the orgn n ts nteror. At the samplng tme t +Δ, the state of the system wll be measured and the procedure starts agan movng the predcton horzon T. Overall, an optmal control u (t) for t R + s obtaned. A control law n feedback form or closed-loop control s a functon π : R + R N Π and t s appled to the system by u( ) =π(,x( )). Illustratons and more detals of ths procedure as well as an overvew about MPC can be found n the books [78, 9, 38] and the PhD theses [92, 84, 68], for example. We are nterested n stablty of MPC. It was shown n [91] that the applcaton of the control obtaned by an MPC scheme to a system does not guarantee that a system wthout dsturbances s asymptotcally stable. For stablty of a system n applcatons t s desred to analyze under whch condtons stablty of a system can be acheved usng an MPC scheme. An overvew about exstng results regardng stablty and MPC for systems wthout dsturbances can be found n [81] and recent results are ncluded n [92, 84, 68, 38]. To desgn stablzng MPC controllers for nonlnear systems, a general framework can be found n [31]. On the one hand, to assure stablty, there exst MPC schemes wth stablzng constrants, see [78, 31, 9, 92, 68], for example. On the other hand condtons were derved n [37, 39, 41] to assure asymptotc stablty of unconstraned nonlnear MPC schemes. Takng the unknown dsturbance w Wnto account, MPC schemes whch guarantee ISS were developed. Frst results can be found n [80] regardng ISS for MPC of nonlnear dscrete-tme systems. Furthermore, results usng the ISS property wth ntal states from a compact set, namely regonal-iss, are gven n [79, 92]. In [73, 69], an MPC scheme whch guarantees ISS usng the so-called mn-max approach was gven. The approach uses a closed-loop formulaton of the optmzaton problem to compensate the effect of the unknown dsturbance. It takes the worst case nto account,.e., t mnmzes the cost functon wth respect to the control law π and maxmzes the cost functon wth respect to the dsturbance w: mn π max w J(x(t),π,w; t, T ):=mn π max w t+t t (l(x(t ),π(t,x(t )),w(t )))dt.

91 Chapter 6. Model predctve control 91 The usage of ths approach s motvated by the crcumstance that the soluton of the openloop MPC strategy can be extremely conservatve and t can provde a small robust output admssble set (see [93], Defnton 11, for example). Stable MPC schemes for nterconnected systems were nvestgated n [93, 92, 41], where n [93, 92] condtons to assure ISS of the whole system were derved and n [41] asymptotcally stable MPC schemes wthout termnal constrants were provded. Note that n [41], the subsystems are not drectly connected, but they exchange nformaton over the network to control themselves accordng to state constrants. However, for networks there exst several approaches for MPC schemes due to the nterconnected structure of the system. Ths means that large-scale systems can be dffcult to control wth a centralzed control structure, due to computatonal complexty or due to communcaton bandwdth lmtatons, for example, see [101]. A revew of archtectures for dstrbuted and herarchcal MPC can be found n [101]. Consderng TDS and MPC, recent results for asymptotcally stable MPC schemes of sngle systems can be found n [29, 96, 95]. In these works, contnuous-tme TDS were nvestgated and condtons were derved, whch guarantee asymptotc stablty of a TDS usng a Lyapunov-Krasovsk approach. Moreover, wth the help of Lyapunov-Razumkhn arguments t was shown, how to determne the termnal cost and termnal regon, and to compute a locally stablzng controller. For the mplementaton of MPC schemes n applcatons, numercal algorthms were developed, see [84, 68, 38] for example. Ths chapter provdes two new drectons n MPC: at frst, we combne the ISDS property wth MPC for sngle and nterconnected systems. Then, ISS of MPC for TDS s nvestgated. Condtons are derved such that sngle systems and whole networks wth an optmal control obtaned by an MPC scheme have the ISDS or ISS property, respectvely. The result of Chapter 2, the ISDS small-gan theorem for networks, Theorem 2.2.2, s appled as well as the result of Chapter 4, namely Theorem The ISS property for MPC schemes of TDS has not been nvestgated so far as well as MPC for nterconnected TDS. The approach of ISS for TDS provdes the calculaton of an optmal control for TDS wth unknown dsturbances, whch was not done before. We use the mn-max closed-loop MPC strategy and Lyapunov-Krasovsk arguments. The advantage of the usage of ISDS over ISS for MPC s that the ISDS estmaton takes only recent values of the dsturbance nto account due to the memory fadng effect, see Chapter 2. In partcular, f the dsturbance tends to zero, then the ISDS estmaton tends to zero. Moreover, the decay rate can be derved usng ISDS-Lyapunov functons. Ths nformaton can be useful for applcatons of MPC. For example, consder two arplanes whch are flyng to each other. To avod a collson at a certan tme T we use MPC to calculate optmal controls for both planes under constrants and unknown dsturbances. The dsturbances are turbulences caused by wnds, whch nfluence the alttude of both planes. The constrant s that the alttude of the planes at tme T are not equal takng the dsturbances nto account. Therefore, the ISDS or ISS estmaton are used for checkng the

92 ISDS and MPC constrant. Assume that at the tme t, where t s a tme before the tme T, the dsturbances,.e., the wnds, are large. Closer to T we assume that the dsturbances tend to zero. In practce, the nformaton about ths crcumstance can be obtaned from the weather forecast. Then, the followng observatons can be made: The ISS estmaton takes the supremum of the dsturbances nto account and for large dsturbances the estmaton s also large. Hence, the control takng nto account the collson constrant usng the ISS estmaton, s conservatve and more effort for the control s used than t s needed. In contrast to ths, the control calculated by an MPC scheme under the collson constrant usng the ISDS estmaton does not need so much effort as the control usng the ISS estmaton, because we assume that the dsturbances tend to zero. By the control wth less effort, jet fuel could be saved, for example. Ths chapter s organzed as follows: Secton 6.1 ntroduces the ISDS property for MPC of nonlnear systems. A result wth respect to ISDS for MPC of sngle systems s provded n Subsecton and a result for nterconnected systems s ncluded n Subsecton In Secton 6.2, tme-delay systems and ISS for MPC are consdered, where Subsecton nvestgates the ISS property for MPC of sngle systems and Subsecton analyzes ISS for MPC of networks. 6.1 ISDS and MPC In ths secton, we combne ISDS and MPC for nonlnear sngle and nterconnected systems. Condtons are derved, whch assure ISDS of a system obtaned by applcaton of the control to the system (6.1), calculated by an MPC scheme Sngle systems We consder systems of the form (6.1) and we use the mn-max approach to calculate an optmal control: to compensate the effect of the dsturbance w, we apply a feedback control law π(t, x(t)) to the system. An optmal control law s obtaned by solvng the fnte horzon optmal control problem (FHOCP), whch conssts of mnmzaton of the cost functon J wth respect to π(t, x(t)) and maxmzaton of the cost functon J wth respect to the dsturbance w: Defnton (Fnte horzon optmal control problem (FHOCP)). Let 1 >ε>0 be gven. Let T > 0 be the predcton horzon and u(t) = π(t, x(t)) be a feedback control law. The fnte horzon optmal control problem for a system of the form (6.1) s formulated as mn max J( x π 0,π,w; t, T ) w t+t := mn max (1 ε) (l(x(t ),π(t,x(t ))) l w (w(t )))dt + V f (x(t + T )) π w t

93 Chapter 6. Model predctve control 93 subject to ẋ(t )=f(x(t ),w(t ),u(t )), x(t) = x 0, t [t, t + T ], x X, w W, π Π, x(t + T ) Ω R N, where x 0 R N s the ntal value of the system at tme t, the termnal regon Ω s a compact and convex set wth the orgn n ts nteror and π(t, x(t)) s essentally bounded, locally Lpschtz n x and measurable n t. l l w s the stage cost, where l : R N R m R + penalzes the dstance of the state from the equlbrum pont 0 of the system and t penalzes the control effort. l w : R P R + penalzes the dsturbance, whch nfluences the systems behavor. l and l w are locally Lpschtz contnuous wth l(0, 0) = 0, l w (0) = 0, and V f :Ω R + s the termnal penalty. The FHOCP wll be solved at the samplng nstants t = kδ, k N, Δ R +. The optmal soluton s denoted by π (t,x(t ); t, T ) and w (t ), t [t, t + T ]. The optmal cost functon s denoted by J ( x 0,π,w ; t, T ). The control nput to the system (6.1) s defned n the usual recedng horzon fashon as u(t )=π (t,x(t ); t, T ), t [t, t +Δ]. In the followng, we need some defntons: Defnton A feedback control π s called a feasble soluton of the FHOCP at tme t, f for a gven ntal value x 0 at tme t the feedback π(t,x(t )), t [t, t + T ] controls the state of the system (6.1) nto Ω at tme t + T,.e., x(t + T ) Ω, for all w W. A set Ω R N s called postvely nvarant, f for all x 0 Ω a feedback control π keeps the trajectory of the system (6.1) n Ω,.e., x(t; x 0,w,π) Ω, t (0, ), for all w W. To prove that the system (6.1) wth the control obtaned by solvng the FHOCP has the ISDS property, we need the followng: Assumpton There exst functons α l,α w K, where α l s locally Lpschtz contnuous such that l(x, π) α l ( x ), x X, π Π, l w (w) α w ( w ), w W.

94 ISDS and MPC 2. The FHOCP n Defnton admts a feasble soluton at the ntal tme t =0. 3. There exsts a controller u(t) = π(t, x(t)) such that the system (6.1) has the ISDS property. 4. For each 1 >ε>0 there exsts a locally Lpschtz contnuous functon V f (x) such that the termnal regon Ω s a postvely nvarant set and we have V f (x) η( x ), x Ω, (6.3) V f (x) (1 ε)l(x, π)+(1 ε)l w (w), f.a.a. x Ω, (6.4) where η K, w W and V f denotes the dervatve of V f along the soluton of system (6.1) wth the control u π from pont 3. of ths assumpton. 5. For each suffcently small ε>0 t holds t+t (1 ε) l(x(t ),π(t,x(t )))dt x(t) 1+ε. (6.5) t 6. The optmal cost functon J ( x 0,π,w ; t, T ) s locally Lpschtz contnuous. Remark In [92], t s dscussed that a dfferent stage cost, for example by the defnton of l s := l l w, can be used for the FHOCP. In vew of stablty, the stage cost l s has to fulfll some addtonal assumptons, see [92], Chapter 3.4. Remark The assumpton (6.5) s needed to assure that the cost functon satsfes the lower estmaton n (2.2). However, we dd not nvestgated whether ths condton s restrctve or not. In case of dscrete-tme systems and the accordng cost functon, the assumpton (6.5) s not necessary, see the proofs n [80, 79, 92, 73, 69]. The followng theorem establshes ISDS of the system (6.1), usng the optmal control nput u π obtaned from solvng the FHOCP. Theorem Consder a system of the form (6.1). Under Assumpton 6.1.3, the system resultng from the applcaton of the predctve control strategy to the system, namely ẋ(t) = f(x(t),w(t),π (t, x(t))), t R +, x(0) = x 0, possesses the ISDS property. Remark Note that the gans and the decay rate of the defnton of the ISDS property, Defnton 2.1.2, can be calculated usng Assumpton 6.1.3, as t s partally dsplayed n the followng proof. Proof. We show that the optmal cost functon J ( x 0,π,w ; t, T ) =: V ( x 0 ) s an ISDS- Lyapunov functon, followng the steps: the control problem admts a feasble soluton π for all tmes t>0, J ( x 0,π,w ; t, T ) satsfes the condtons (2.2) and (2.3).

95 Chapter 6. Model predctve control 95 Then, by applcaton of Theorem 2.1.6, the ISDS property follows. Let us proof feasblty: we suppose that a feasble soluton π(t,x(t )), t [t, t + T ] at tme t exsts. For Δ > 0, we construct a control by { π(t ˆπ(t,x(t,x(t )), t [t +Δ,t+ T ], )) = π(t,x(t )), t (t + T,t + T +Δ], (6.6) where π s the controller from Assumpton 6.1.3, pont 3. Snce π controls x(t +Δ) nto x(t + T ) Ω and Ω s a postvely nvarant set, π(t,x(t )) keeps the systems trajectory n Ω for t + T < t t + T +Δ under the constrants of the FHOCP. Ths means that from the exstence of a feasble soluton for the tme t, we have a feasble soluton for the tme t +Δ. Snce, we assume that a feasble soluton for the FHOCP at the tme t =0exsts (Assumpton 6.1.3, pont 2.), t follows that a feasble soluton exsts for every t>0. We replace π n (6.6) by π. Then, t follows from (6.4) that t holds J ( x 0,π,w ; t, T +Δ) J( x 0, ˆπ, w ; t, T +Δ) =(1 ε) +(1 ε) t+t (l(x(t ),π (t,x(t ); t, T )) l w (w (t )))dt t t+t +Δ t+t + V f (x(t + T + Δ)) (l(x(t ),π(t,x(t ))) l w (w (t )))dt = J ( x 0,π,w ; t, T ) V f (x(t + T )) + V f (x(t + T + Δ)) +(1 ε) t+t +Δ t+t J ( x 0,π,w ; t, T ). From ths and wth (6.3) t holds (l(x(t ),π(t,x(t ))) l w (w (t )))dt J ( x 0,π,w ; t, T ) J ( x 0,π,w ; t, 0) = V f ( x 0 ) η( x 0 ). Now, wth Assumpton 6.1.3, pont 5., we have V ( x 0 ) J( x 0,π, 0; t, T ) (1 ε) t+t t l(x(t ),π (t,x(t )))dt x 0 1+ε. Ths shows that J satsfes (2.2). Now, denote x 0 := x(t + h). From J ( x 0,π,w ; t, T + Δ) J ( x 0,π,w ; t, T ) we get (1 ε) (1 ε) and therefore t+h t t+h t (l(x(t ),π (t,x(t ); t, T )) l w (w (t )))dt + J ( x 0,π,w ; t + h, T +Δ h) (l(x(t ),π (t,x(t ); t, T )) l w (w (t )))dt + J ( x 0,π,w ; t + h, T h), J ( x 0,π,w ; t + h, T +Δ h) J ( x 0,π,w ; t + h, T h). (6.7)

96 ISDS and MPC Now, we show that J satsfes the condton (2.3). Note that by Assumpton 6.1.3, pont 6., J s locally Lpschtz contnuous. Wth (6.7) t holds J ( x 0,π,w ; t, T ) =(1 ε) (1 ε) Ths leads to t+h t t+h t (l(x(t ),π (t,x(t ); t, T )) l w (w (t )))dt + J ( x 0,π,w ; t + h, T h) (l(x(t ),π (t,x(t ); t, T )) l w (w (t )))dt + J ( x 0,π,w ; t + h, T ). J ( x 0,π,w ; t + h, T ) J ( x 0,π,w ; t, T ) h 1 t+h h (1 ε) (l(x(t ),π (t,x(t ); t, T )) l w (w (t )))dt. t For h 0 and usng the frst pont of Assumpton we obtan V ( x 0 ) (1 ε)α l ( x 0 )+(1 ε)α w ( w ), f.a.a. x 0 X, w W. By defnton of γ(r) :=η(α 1 l (2α w (r))) and g(r) := 1 2 α l(η 1 (r)), r 0 ths mples V ( x 0 ) >γ( w ) V ( x 0 ) (1 ε)g(v ( x 0 )), where the functon g s locally Lpschtz contnuous. Lyapunov functon for the system We conclude that J s an ISDS- ẋ(t) =f(x(t),w(t),π (t, x(t))) and by applcaton of Theorem the system has the ISDS property. In the next subsecton, we transform the analyss of ISDS for MPC of sngle systems to nterconnected systems Interconnected systems We consder nterconnected systems wth dsturbances of the form ẋ (t) =f (x 1 (t),...,x n (t),w (t),u (t)), =1,...,n, (6.8) where u R M, measurable and essentally bounded, are the control nputs and w R P are the unknown dsturbances. We assume that the states, dsturbances and nputs fulfll the constrants x X,w W,u U,=1,...,n, where X R N, W L (R +, R P ) and U R M are compact and convex sets contanng the orgn n ther nteror. Now, we are gong to determne an MPC scheme for nterconnected systems. An overvew of exstng dstrbuted and herarchcal MPC schemes can be found n [101]. The used scheme

97 Chapter 6. Model predctve control 97 n ths thess s nspred by the mn-max approach for sngle systems as n Defnton 6.1.1, see [73, 69]. At frst, we determne the cost functon of the th subsystem by J ( x 0, (x j ) j,π,w ; t, T ) := (1 ε ) t+t t (l (x (t ),π (t,x(t ))) (l w ) (w (t )) l j (x j (t )))dt +(V f ) (x (t + T )), j where 1 >ε > 0, x 0 X s the ntal value of the th subsystem at tme t and π Π s a feedback, essentally bounded, locally Lpschtz n x and measurable n t, where Π R M s a compact and convex set contanng the orgn n ts nteror. l (l w ) l j s the stage cost, where l : R N R M R +. (l w ) : R P R + penalzes the dsturbance and l j : R N j R + penalzes the nternal nput for all j =1,...,n, j. l, (l w ) and l j are locally Lpschtz contnuous functons wth l (0, 0) = 0, (l w ) (0) = 0, l j (0) = 0, and (V f ) :Ω R + s the termnal penalty of the th subsystem, Ω R N. In contrast to sngle systems we add the terms l j (x j ), j to the cost functon due to the nterconnected structure of the subsystems. Here, two problems arse: the formulaton of an optmal control problem for each subsystem and the calculaton/determnaton of the nternal nputs x j,j. We conserve the mnmzaton of J wth respect to π and the maxmzaton of J wth respect to w as n Defnton for sngle systems. In the sprt of ISS/ISDS, whch treat the nternal nputs as dsturbances, we maxmze the cost functon wth respect to x j, j (worst-case approach). Snce we assume that x j X j, we get an optmal soluton π,w,x j,j of the control problem. The drawbacks of ths approach are that, on the one hand, we do not use the systems equatons (6.8) to predct x j, j and, on the other hand, the computaton of the optmal soluton could be numercally neffcent, especally f the number of subsystems n s huge or/and the sets X are large. Moreover, takng nto account the worst-case approach, the maxmzaton over x j, the obtaned optmal control π for each subsystem could be extremely conservatve, whch leads to extremely conservatve ISS or ISDS estmatons. To avod these drawbacks of the maxmzaton of J wth respect to x j, j, one could use the systems equatons (6.8) to predct x j,j nstead. A numercally effcent way to calculate the optmal solutons π,w of the subsystems s a parallel calculaton. Due to nterconnected structure of the system the nformaton about systems states of the subsystems should be exchanged. But ths exchange of nformaton causes that an optmal soluton π,w could not be calculated. To the best of our knowledge, no theorem s proved that provdes the exstence of an optmal soluton of the optmal control problem usng such a parallel strategy. We conclude that a parallel calculaton can not help n our case. Another approach of an MPC scheme for networks s nspred by the herarchcal MPC scheme n [97]. One could use the predctons of the nternal nputs x j, j as follows: at samplng tme t = kδ, k N, Δ > 0 all subsystems calculate the optmal soluton teratvely.

98 ISDS and MPC Ths means that for the calculaton the optmal soluton for the th subsystem, the currently optmzed trajectores of the subsystems 1,..., 1 wll be used, denoted by x opt,kδ p, p = 1,..., 1, and the optmal trajectores of the subsystems +1,...,n of the optmzaton at samplng tme t =(k 1)Δ wll be used, denoted by xp opt,(k 1)Δ,p= +1,...,n. The advantage of ths approach would be that the optmal soluton s not that much conservatve as the mn-max approach and the calculaton of the optmal soluton could be performed n a numercally effcent way, due to the usage of the model to predct the optmal trajectores and that the maxmzaton over x j,j wll be avoded. The drawback s that the optmal cost functon of each subsystem depends on the trajectores x opt, j, j usng ths herarchcal approach. Then, to the best of our knowledge, t s not possble to show that the optmal cost functons are ISDS-Lyapunov functons of the subsystems, whch s a crucal step for provng ISDS of a subsystem or the whole network, because no helpful estmatons for the Lyapunov functon propertes can be performed due to the dependence of the optmal cost functons of the trajectores x opt, j,j. The FHOCP for the th subsystem reads as follows: subject to mn π max w max J ( x 0, (x j ) j,π,w ; t, T ) (x j ) j ẋ (t )=f (x 1 (t ),...,x n (t ),w (t ),u (t )), t [t, t + T ], x (t) = x 0, x j X j,j=1,...,n, w W, π Π, x (t + T ) Ω R N, where the termnal regon Ω s a compact and convex set wth the orgn n ts nteror. The resultng optmal control of each subsystem s a feedback control law,.e., u (t) = π (t, x(t)), where x = (xt 1,...,xT n ) T R N, N = N and π (t, x (t)) s essentally bounded, locally Lpschtz n x and measurable n t, for all =1,...,n. To show that each subsystem and the whole system have the ISDS property usng the mentoned dstrbuted MPC scheme, we suppose for the th subsystem of (6.8): Assumpton that 1. There exst functons α l,αw,α j K, j =1,...,n, j such l (x,π ) α l ( x ), x X, π Π, (l w ) (w ) α w ( w ), w W, l j (x j ) α j (V j (x j )), x j X j, j =1,...,n, j. 2. The FHOCP admts a feasble soluton at the ntal tme t =0.

99 Chapter 6. Model predctve control There exsts a controller u (t) =π (t, x(t)) such that the th subsystem of (6.8) has the ISDS property. 4. For each 1 >ε > 0 there exsts a locally Lpschtz contnuous functon (V f ) (x ) such that the termnal regon Ω s a postvely nvarant set and we have (V f ) (x ) η ( x ), x Ω, ( V f ) (x ) (1 ε )l (x,π )+(1 ε )(l w ) (w )+(1 ε ) j l j(x j ), for almost all x Ω, where η K, w W and ( V f ) denotes the dervatve of (V f ) along the soluton of the th subsystem of (6.8) wth the control u π from pont 3. of ths assumpton. 5. For each suffcently small ε > 0 t holds (1 ε ) t+t t l (x (t ),π (t,x(t ))) j l j (x j (t ))dt x(t) 1+ε (6.9) 6. The optmal cost functon J ( x0, (x j) j,π,w ; t, T ) s locally Lpschtz contnuous. Now, we can state that each subsystem possesses the ISDS property usng the mentoned MPC scheme. Theorem Consder an nterconnected system of the form (6.8). Let Assumpton be satsfed for each subsystem. Then, each subsystem resultng from the applcaton of the control obtaned by the FHOCP for each subsystem to the system, namely ẋ (t) =f (x 1 (t),...,x n (t),w (t),π (t, x(t))), t R +, x 0 = x (0), possesses the ISDS property. Proof. Consder the th subsystem. We show that the optmal cost functon V ( x 0 ) := We ab- J ( x0, (x j) j,π,w ; t, T ) s an ISDS-Lyapunov functon for the th subsystem. brevate x j =(x j ) j. By followng the steps of the proof of Theorem 6.1.6, we conclude that there exsts a feasble soluton for all tmes t > 0 and that by (6.9) the functonal V ( x 0 ) satsfes the condton x 0 (1+ε ) V ( x 0 ) η ( x 0 ), usng x 0 x 0. Note that by Assumpton 6.1.8, pont 6., J We have that t holds s locally Lpschtz contnuous. V ( x 0 ) (1 ε )α l (η 1 (V ( x 0 )))+(1 ε )α w ( w )+(1 ε ) j α j (V j (( x 0 j))) and equvalently V ( x 0 ) (1 ε )α l (η 1 (V ( x 0 )))+(1 ε ) max{nα w ( w ), max j nα j(v j (( x 0 j)))},

100 ISS and MPC of tme-delay systems whch mples V ( x 0 ) > max{γ ( w ), max j γ j(v j (( x 0 )))} V ( x 0 ) (1 ε )g (V ( x 0 )), for almost all x 0 X and all w W, where γ (r) := η ((α l) 1 (2nα w(r))), γ j(r) := η ((α l) 1 (2nα j (r))) and g (r) := 1 2 αl (η 1 (r)), where g s locally Lpschtz contnuous. Snce, can be chosen arbtrarly, we conclude that each subsystem has an ISDS-Lyapunov functon. It follows that each subsystem has the ISDS property. To nvestgate whether the whole system has the ISDS property, we collect all functons γ j n a matrx Γ:=(γ j ) n n, γ 0, whch defnes a map as n (1.12). Usng the small-gan condton for Γ, the ISDS property for the whole system can be guaranteed: Corollary Consder an nterconnected system of the form (6.8). Let Assumpton be satsfed for each subsystem. If Γ satsfes the small-gan condton (1.15), then the whole system possesses the ISDS property. Proof. Each subsystem has an ISDS-Lyapunov functon wth gans γ j. Ths follows from Theorem The matrx Γ satsfes the SGC and all assumptons of Theorem are satsfed. It follows that wth x = (x T 1,...,xT n ) T, w = (w1 T,...,wT n ) T and π (,x( )) = ((π1 (,x( )))T,...,(πn(,x( ))) T ) T, the whole system of the form has the ISDS property. ẋ(t) =f(x(t),w(t),π (t, x(t))) In the next secton, we nvestgate the ISS property for MPC of TDS. 6.2 ISS and MPC of tme-delay systems Now, we ntroduce the ISS property for MPC of TDS. We derve condtons to assure that a sngle system, a subsystem of a network and the whole system possess the ISS property applyng the control obtaned by an MPC scheme for TDS Sngle systems We consder systems of the form (4.1) wth dsturbances, ẋ(t) =f(x t,w(t),u(t)), t R +, x 0 (τ) =ξ(τ), τ [ θ, 0], (6.10) where w W L (R +, R P ) s the unknown dsturbance and W s a compact and convex set contanng the orgn. The nput u s an essentally bounded and measurable control subject to nput constrants u U, where U R s a compact and convex set contanng the orgn

101 Chapter 6. Model predctve control 101 n ts nteror. The functon f has to satsfy the same condtons as n Chapter 4 to assure that a unque soluton exsts, whch s denoted by x(t; ξ,w,u) or x(t) n short. The am s to fnd an (optmal) control u such that the system (6.10) has the ISS property. Due to the presence of dsturbances, we apply a feedback control structure, whch compensates the effect of the dsturbance. Ths means that we apply a feedback control law π(t, x t ) to the system. In the rest of ths secton, we assume that π(t, x t ) Π s essentally bounded, locally Lpschtz n x t and measurable n t. The set Π R m s assumed to be compact and convex contanng the orgn n ts nteror. We obtan an MPC control law by solvng the control problem: Defnton (Fnte horzon optmal control problem wth tme-delays (FHOCPTD)). Let T be the predcton horzon and π(t, x t ) be a feedback control law. The fnte horzon optmal control problem wth tme-delays for a system of the form (6.10) s formulated as mn π subject to max w J( ξ,π,w; t, T ):=mn π max w t+t t (l(x(t ),π(t,x t )) l w (w(t )))dt + V f (x t+t ) ẋ(t )=f(x t,w(t ),u(t )), t [t, t + T ], x(t + τ) = ξ(τ), τ [ θ, 0], x t X, w W, π Π, x t+t Ω C([ θ, 0], R N ), where ξ C([ θ, 0], R N ) s the ntal functon of the system at tme t, the termnal regon Ω and the state constrant set X C([ θ, 0], R N ) are compact and convex sets wth the orgn n ther nteror. l l w s the stage cost, where l : R N R m R + and l w : R P R + are locally Lpschtz contnuous wth l(0, 0) = 0, l w (0) = 0, and V f :Ω R + s the termnal penalty. The control problem wll be solved at the samplng nstants t = kδ, k N, Δ R +. The optmal soluton s denoted by π (t,x t ; t, T ) and w (t ), t [t, t + T ] and the optmal cost functonal s denoted by J ( ξ,π,w ; t, T ). The control nput to the system (6.10) s defned n the usual recedng horzon fashon as u(t )=π (t,x t ; t, T ), t [t, t +Δ]. Defnton A feedback control π s called a feasble soluton of the FHOCPTD at tme t, f for a gven ntal functon ξ at tme t the feedback π(t,x t ), t [t, t + T ] controls the state of the system (6.10) nto Ω at tme t + T,.e., x t+t Ω, for all w W.

102 ISS and MPC of tme-delay systems A set Ω C([ θ, 0], R N ) s called postvely nvarant, f for all ntal functons ξ Ω a feedback control π keeps the trajectory of the system (6.10) n Ω,.e., x t Ω, t (0, ), for all w W. For the goal of ths secton, establshng ISS of TDS wth the help of MPC, we need the followng: Assumpton There exst functons α l,α w K such that l(φ(0),π) α l ( φ a ), φ X, π Π, l w (w) α w ( w ), w W. 2. The FHOCPTD n Defnton admts a feasble soluton at the ntal tme t =0. 3. There exsts a controller u(t) =π(t, x t ) such that the system (6.10) has the ISS property. 4. There exsts a locally Lpschtz contnuous functonal V f (φ) such that the termnal regon Ω s a postvely nvarant set and for all φ Ω we have V f (φ) ψ 2 ( φ a ), (6.11) D + V f (φ, w) l(φ(0),π)+l w (w), (6.12) where ψ 2 K, w Wand D + V f denotes the upper rght-hand sde dervate of the functonal V along the soluton of (6.10) wth the control u π from pont 3. of ths assumpton. 5. There exsts a K functon ψ 1 such that for all t>0 t holds t+t t l(x(t ),π(t,x t ))dt ψ 1 ( ξ(0) ), ξ(0) = x(t). (6.13) 6. The optmal cost functonal J ( ξ,π,w ; t, T ) s locally Lpschtz contnuous. Now, we can state a theorem that assures ISS of MPC for a sngle tme-delay system wth dsturbances. Theorem Let Assumpton be satsfed. Then, the system resultng from the applcaton of the predctve control strategy to the system, namely ẋ(t) =f(x t,w(t),π (t, x t )), t R +, x 0 (τ) =ξ(τ), τ [ θ, 0], possesses the ISS property. Proof. The proof goes along the lnes of the proof of Theorem wth accordng changes to tme-delays and functonals,.e., we show that the optmal cost functonal V ( ξ) := J ( ξ,π,w ; t, T ) s an ISS-Lyapunov-Krasovsk functonal.

103 Chapter 6. Model predctve control 103 For a feasble soluton for all tmes t>0, we suppose that a feasble soluton π(t,x t ),t [t, t + T ] at tme t exsts. We construct a control by { π(t ˆπ(t,x t ), t [t +Δ,t+ T ],,x t )= π(t,x t ), t (t + T,t + T +Δ], (6.14) where π s the controller from Assumpton 6.2.3, pont 3, and Δ > 0. π steers x t+δ nto x t+t Ω and Ω s a postvely nvarant set. Ths means that π(t,x t ) keeps the system trajectory n Ω for t + T < t t + T +Δ under the constrants of the FHOCPTD. Ths mples that from the exstence of a feasble soluton for the tme t, we have a feasble soluton for the tme t +Δ. From Assumpton 6.2.3, pont 2., there exsts a feasble soluton for the FHOCPTD at the tme t =0and t follows that a feasble soluton exsts for every t>0. Replacng π n (6.14) by π, t follows from (6.12) that t holds J ( ξ,π,w ; t, T +Δ) J( ξ, ˆπ, w ; t, T +Δ) = t+t t (l(x(t ),π (t,x t ; t, T )) l w (w (t )))dt + + V f (x t+t +Δ ) = J ( ξ,π,w ; t, T ) V f (x t+t )+V f (x t+t +Δ )+ J ( ξ,π,w ; t, T ) and wth (6.11) ths mples t+t +Δ t+t t+t +Δ t+t (l(x(t ),π(t,x t )) l w (w (t )))dt (l(x(t ),π(t,x t )) l w (w (t )))dt J ( ξ,π,w ; t, T ) J ( ξ,π,w ; t, 0) = V f ( ξ) ψ 2 ( ξ a ). For the lower bound, t holds V ( ξ) J( ξ,π, 0; t, T ) t+t t l(x(t ),π (t,x t ))dt and by (6.13) we have V ( ξ) ψ 1 ( ξ(0) ). Ths shows that J satsfes (4.10). Now, we use the notaton x t (τ) := ξ(τ), τ [ θ + t, t]. Wth J (x t,π,w ; t, T +Δ) J (x t,π,w ; t, T ) we have t+h t t+h t (l(x(t ),π (t,x t ; t, T )) l w (w (t )))dt + J (x t+h,π,w ; t + h, T +Δ h) (l(x(t ),π (t,x t ; t, T )) l w (w (t )))dt + J (x t+h,π,w ; t + h, T h). Ths mples J (x t+h,π,w ; t + h, T +Δ h) J (x t+h,π,w ; t + h, T h). (6.15)

104 ISS and MPC of tme-delay systems Note that by Assumpton 6.2.3, pont 6., J s locally Lpschtz contnuous. Wth (6.15) t holds = J (x t,π,w ; t, T ) t+h t t+h whch leads to t (l(x(t ),π (t,x t ; t, T )) l w (w (t )))dt + J (x t+h,π,w ; t + h, T h) (l(x(t ),π (t,x t ; t, T )) l w (w (t )))dt + J (x t+h,π,w ; t + h, T ), J (x t+h,π,w ; t + h, T ) J (x t,π,w ; t, T ) h 1 h t+h t (l(x(t ),π (t,x t ; t, T )) l w (w (t )))dt. Let h 0 + and usng the frst pont of Assumpton we get By defnton of χ(r) :=ψ 2 (α 1 l D + V (x t,w ) α l ( x t a )+α w ( w ). (2α w (r))) and α(r) := 1 2 α l(ψ2 1 (r)), r 0 ths mples V (x t ) χ( w ) D + V (x t,w ) α(v (x t )),.e., J satsfes the condton (4.11). We conclude that J s an ISS-Lyapunov-Krasovsk functonal for the system ẋ(t) =f(x t,w(t),π (t, x t )) and by applcaton of Theorem the system has the ISS property. Now, we consder nterconnectons of TDS and provde condtons such that the whole network wth an optmal control obtaned from an MPC scheme has the ISS property Interconnected systems We consder nterconnected systems wth tme-delays and dsturbances of the form ẋ (t) = f ( x t 1,...,x t n,w (t),u (t) ),=1,...,n, (6.16) where u R M are the essentally bounded and measurable control nputs and w R P are the unknown dsturbances. We assume that the states, dsturbances and nputs fulfll the constrants x X,w W,u U,=1,...,n, where X C([ θ, 0], R N ), W L (R +, R P ) and U R M contanng the orgn n ther nteror. are compact and convex sets

105 Chapter 6. Model predctve control 105 We assume the same MPC strategy for nterconnected TDS as n Subsecton The FHOCPTD for the th subsystem of (6.16) reads as mn π := mn π max w max w subject to max J ( ξ, (x j ) j,π,w ; t, T ) (x j ) j max (x j ) j t+t t (l (x (t ),π (t,x t )) (l w ) (w (t )) l j (x j (t )))dt +(V f ) (x t+t ) j ẋ (t )=f (x t 1,...,x t n,w (t ),u (t )), t [t, t + T ], x (t + τ) = ξ (τ), τ [ θ, 0], x j X j,j=1,...,n, w W, π Π, x t+t Ω C([ θ, 0], R N ), where ξ X s the ntal functon of the th subsystem at tme t, the termnal regon Ω s a compact and convex set wth the orgn n ts nteror. π (t, x t ) s essentally bounded, locally Lpschtz n x and measurable n t and Π R M s a compact and convex sets contanng the orgn n ts nteror. l (l w ) l j s the stage cost, where l : R N R M R +. (l w ) : R P R + penalzes the dsturbance and l j : R N j R + penalzes the nternal nput for all j =1,...,n, j. l, (l w ) and l j are locally Lpschtz contnuous functons wth l (0, 0) = 0, (l w ) (0) = 0, l j (0) = 0, and (V f ) :Ω R + s the termnal penalty of the th subsystem. We obtan an optmal soluton π, (x j) j, w, where the control of each subsystem s a feedback control law, whch depends on the current states of the whole system,.e., u (t) =π (t, xt ), where x t =((x t 1 )T,...,(x t n) T ) T C([ θ, 0], R N ), N = N. For the th subsystem of (6.16) we suppose: Assumpton that 1. There exst functons α l,αw,α j K, j =1,...,n, j such l (φ (0),π ) α l ( φ a ), φ C([ θ, 0], R N ), π Π, (l w ) (w ) α w ( w ), w W, l j (φ j (0)) α j (V j (φ j )), φ j C([ θ, 0], R N j ), j =1,...,n, j. 2. The FHOCPTD admts a feasble soluton at the ntal tme t =0. 3. There exsts a controller u (t) =π (t, x t ) such that the th subsystem of (6.16) has the ISS property.

106 ISS and MPC of tme-delay systems 4. There exsts a locally Lpschtz contnuous functonal (V f ) (φ ) such that the termnal regon Ω s a postvely nvarant set and for all φ Ω we have (V f ) (φ ) ψ 2 ( φ a ), D + (V f ) (φ,w ) l (φ (0),π )+(l w ) (w )+ j l j(φ j (0)), where ψ 2 K, φ j C([ θ, 0], R N j ), j =1,...,n and w W. D + (V f ) denotes the upper rght-hand sde dervate of the functonal (V f ) along the soluton of the th subsystem of (6.16) wth the control u π from pont 3. of ths assumpton. 5. For each, there exsts a K functon ψ 1 such that for all t>0 t holds t+t t l (x (t ),π (t,x t ))dt ψ 1 ( ξ(0) ), ξ(0) = x(t). (6.17) 6. The optmal cost functonal J ( ξ, (x j ) j,π,w ; t, T ) s locally Lpschtz contnuous. Now, we state that each subsystem of (6.16) has the ISS property by applcaton of the optmal control obtaned by the FHOCPTD. Theorem Consder an nterconnected system of the form (6.16). Let Assumpton be satsfed for each subsystem. Then, each subsystem resultng from the applcaton of the predctve control strategy to the system, namely ẋ (t) =f (x t 1,...,xt n,w (t),π (t, xt )), t R +, x 0 (τ) =ξ (τ), τ [ θ, 0], possesses the ISS property. Proof. Consder the th subsystem. We show, that the optmal cost functonal V ( ξ ):= J ( ξ, (x j ) j,π,w ; t, T ) s an ISS-Lyapunov-Krasovsk functonal for the th subsystem. We abbrevate x t j =((x j) t j ). Followng the lnes of the proof of Theorem 6.2.4, we have that there exsts a feasble soluton of the th subsystem for all tmes t>0 and that the functonal V ( ξ ) satsfes the condton ψ 1 ( ξ (0) ) V ( ξ ) ψ 2 ( ξ a ), usng (6.17) and ξ(0) ξ (0). Note that by Assumpton 6.2.5, pont 6., J s locally Lpschtz contnuous. We arrve that t holds D + V (x t,w ) α l (ψ 1 2 (V (x t ))) + α w ( w )+ j α j (V j (x t j)). Ths s equvalent to whch mples D + V (x t,w ) α l (ψ 1 2 (V (x t ))) + max{nα w ( w ), max j nα j(v j (x t j))}, V (x t ) max{ χ ( w ), max j χ j (V j (x t j))} D + V (x t,w ) ᾱ l (V (x t )),

107 Chapter 6. Model predctve control 107 where χ (r) :=ψ 2 ((α l ) 1 (2nα w (r))), χ j (r) :=ψ 2 ((α l ) 1 (2nα j (r))), ᾱ l (r) := 1 2 αl (ψ 1 2 (r)). Ths can be shown for each subsystem and we conclude that each subsystem has an ISS-Lyapunov-Krasovsk functonal. It follows that the th subsystem s ISS n maxmum formulaton. We collect all functons χ j n a matrx Γ:=( χ j ) n n, χ 0, whch defnes a map as n (1.12). Usng the small-gan condton for Γ, t follows from Theorem 6.2.6: Corollary Consder an nterconnected system of the form (6.16). Let Assumpton be satsfed for each subsystem. If Γ satsfes the small-gan condton (1.15), then the whole system possesses the ISS property. Proof. We know from Theorem that each subsystem of (6.16) has an ISS-Lyapunov- Krasovsk functonal wth gans χ j. Snce, the matrx Γ satsfes the SGC, all assumptons of Theorem are satsfed and by Remark the whole system of the form ẋ(t) =f(x t,w(t),π (t, x t )) s ISS n maxmum formulaton, where x t =((x t 1 )T,...,(x t n) T ) T, w =(w1 T,...,wT n ) T π (t, x t )=((π1 (t, xt )) T,...,(πn(t, x t )) T ) T. and The next chapter summarzes the thess and open questons for future research actvtes are lsted.

108 ISS and MPC of tme-delay systems

109 Chapter 7 Summary and Outlook We have provded several tools for dfferent knds of systems to analyze them n vew of stablty, to desgn observers and to control them. Ths thess has extended the analyss toolbox of nonlnear sngle and nterconnected systems usng Lyapunov methods. The results can be used for a wde range of applcatons from dfferent areas. Moreover, the theory presented here can be seen as a startng pont for further nvestgatons. In ths chapter, we summarze all man results of ths thess and propose several topcs for future research actvtes as well as open questons. 7.1 ISDS Consderng networks of nterconnected ISDS subsystems, we have shown that they possess the ISDS property, f the small-gan condton (1.15) s satsfed. In ths case, we have provded explct expressons for an ISDS-Lyapunov functon and the correspondng rates and gans of the entre nterconnecton, whch s Theorem As an applcaton of ths result, we have nvestgated a system of nterconnectons wth zero external nput and we have derved decay rates of the subsystems and the entre system, see Corollary An example wth two systems taken from [36] compares the resultng estmates of the norm of a trajectory obtaned by [36] and by (2.15), see Example Another example (Example 2.3.2) wth n nterconnected ISDS systems has llustrated the applcaton of our man result. The ISDS property wth ts advantages over ISS s not deeply nvestgated and used n applcatons and only few works exst n the research lterature accordng to ths topc untl now. The advantages of ISDS could be used n practce for a wde range of applcatons to obtan (economc) benefts by the analyss of dynamcal systems. For example, n producton networks, ISDS can help to desgn systems avodng overdmensoned producton lnes or warehouses. Possble research actvtes regardng ISDS can be, for example: the nvestgaton of a local varant of ISDS (see [40]) for networks and applcatons to producton networks, for example. A small-gan theorem usng LISDS-Lyapunov functons of the subsystems smlar to Theorem can be proved and the queston arses, f ρ, ρ u n the defnton of local ISDS 109

110 Observer and quantzed output feedback stablzaton (LISDS) accordng to LISS are equal to ρ, ρ u usng LISDS-Lyapunov functons. Smlar to ISS (see [116, 1]), an ntegral varant of ISDS, namely ISDS, could be nvestgated. Snce, the set of ISS systems s larger than the set of ISS systems, ths s also expected for ISDS systems n comparson to ISDS systems. The benefts of ISDS over ISDS or ISS for applcatons and the characterzaton of ISDS by Lyapunov functons should be analyzed. As ISS was nvestgated for networks of dscrete-tme systems n [53, 54], ths could also be done for the ISDS property and smlar theorems presented here, can be adapted for such class of systems. Ths can be used for MPC, for example. Furthermore, the ISDS property can be defned for tme-delay systems and the nfluence of the presence of delays to the decay rate could be analyzed. Also, all the theory accordng to ISDS could be developed for TDS. Ths would ncrease the range of possble applcatons of ISDS. We have transfered one of the advantage of ISDS over ISS, namely the memory fadng effect, to observer desgn: 7.2 Observer and quantzed output feedback stablzaton We have ntroduced the quas-isds property for observers (Defnton 3.1.2), whch man advantage over the quas-iss property s the memory fadng effect due to measurement dsturbances. Ths was demonstrated n Example A quas-isds observer was desgned (Theorem 3.1.8) usng error ISDS-Lyapunov functons. Consderng networks, we have shown how to desgn quas-iss/isds reduced-order observers for subsystems of nterconnected systems n Theorem They were used to desgn a quas-iss/isds reduced-order observer for the overall system under a small-gan condton (Theorem 3.2.3). As an applcaton, we have shown that quantzed output feedback stablzaton for a subsystem s achevable, under the assumptons that the subsystem possesses a quas-iss/isds reduced-order observer and a state feedback controller provdng ISS/ISDS wth respect to measurement errors (Proposton 3.3.3, pont 1.). If ths holds for all subsystems of the large-scale system and the small-gan condton s satsfed, then quantzed output feedback stablzaton s also achevable for the overall system (Proposton 3.3.3, pont 2.). The obtaned bounds can be mproved by usng dynamc quantzers. We have shown that asymptotc convergence can be acheved for each subsystem and for the overall system provded that a small-gan condton s satsfed. In future, t would be nterestng to study the desgn of nonlnear output feedback control or nonlnear observers to satsfy the small-gan condton. The applcaton of the results n ths thess to the desgn of dynamc quantzed nterconnected control systems could be nvestgated. Also, the desgn of the observers ntroduced n ths thess for systems wth tme-delays s of nterest for future research actvtes as far as tme-delays occur n many real-world applcatons. Moreover, t could be nvestgated, f the decay rate and the gans of the quas-isds property can be drectly obtaned by the usage of a dfferent error ISDS-Lyapunov functon n opposte to the one used n Assumpton and adaptng/mprovng the proof of

111 Chapter 7. Summary and Outlook 111 Theorem We have nvestgated another type of systems: 7.3 ISS for TDS For networks of tme-delay systems, we have proved two theorems: an (L)ISS-Lyapunov- Razumkhn and an ISS-Lyapunov-Krasovsk small-gan theorem. They provde a tool how to check whether a network of TDS has the (L)ISS property, usng a small-gan condton and (L)ISS-Lyapunov-Razumkhn functons or ISS-Lyapunov-Krasovsk functonals, respectvely. Furthermore, we have shown how to construct the (L)ISS-Lyapunov-Razumkhn functon, the ISS-Lyapunov-Krasovsk functonal and the correspondng gans of the whole system, see Theorem and Theorem As an applcaton of the results, we have consdered a detaled scenaro of a producton network and we have analyzed t, usng the presented tools n ths chapter, see Secton 4.3. Further research actvtes could be the defnton of ISDS for TDS and the nvestgaton of ts characterzaton by ISDS-Lyapunov-Razumkhn functons and ISDS-Lyapunov-Krasovsk functonals. The nfluence of the decay rate of the ISDS estmaton by tme-delays could be analyzed. The presented results of ths work can be further extended and an ISDS small-gan theorem for networks of TDS could be proved. For systems wthout tme-delays, a converse ISS-Lyapunov theorem was proved n [119, 74]. Ths s not done yet for TDS usng ISS-Lyapunov-Razumkhn functons or ISS- Lyapunov-Krasovsk functonals, respectvely, and remans as an open queston. Another topc to be analyzed s the equvalence of ISS and 0-GAS+GS, see [120] for systems wthout tme-delays. Consderng tme-delays, ths s not proved yet and new technques should be developed for the proof. For further detals, see [123, 122], for example. In practce, consderng producton networks t can happen that machnes wthn a plant wll break down, for example. Usng autonomous control methods, the network s stable despte the breakdowns. Due to the dea of autonomous control, t s not necessary to montor the producton process such that breakdowns wll not dscovered or not dscovered very fast. By an ncreasng number of breakdowns, the performance of the network wll decrease, whch causes economc drawbacks. To overcome these negatve outcomes, one can use a fault detecton approach (see [129, 52, 2, 6], for example) to detect breakdowns n a network usng autonomous control methods. An observer-based approach for the fault detecton can be used (see [130], for example) as well as autonomous control methods. Once a fault n the network s observed/detected, a message to the mechancs of the plant wll be sent mmedately such that the breakdown could be repared. In practce, ths would help to treat autonomously controlled networks wth tme-delays and breakdowns or other dsturbances and would help to avod negatve economc effects. We have used the presented tools n ths chapter for the stablty analyss of mpulsve systems wth tme-delays:

112 ISS for mpulsve systems wth tme-delays 7.4 ISS for mpulsve systems wth tme-delays Consderng mpulsve systems wth tme-delays, we have ntroduced the Lyapunov-Krasovsk and the Lyapunov-Razumkhn methodology for establshng ISS of sngle mpulsve systems, see Theorem and Theorem Then, we have nvestgated networks of mpulsve subsystems wth tme-delays. We have proved ISS small-gan theorems, whch guarantees that the whole network has the ISS property under a small-gan condton wth lnear gans and a dwell-tme condton usng the Lyapunov-Razumkhn (Theorem 5.2.1) and Lyapunov- Krasovsk (Theorem 5.2.2) approach. To prove ths, we have constructed the Lyapunov functon(al)s and the correspondng gans of the whole system. These theorems provde tools to check, whether a network of mpulsve tme-delay systems possesses the ISS property. It seems that the usage of general Lyapunov functons nstead of exponental Lyapunov functons for sngle systems could be possble, f the dwell-tme condton s formulated n a dfferent way. Then, for an nterconnecton one can also use general Lyapunov functons and general gans nstead of only lnear gans. Ths should be nvestgated more detaled. Consderng nterconnected mpulsve systems, there s a relatonshp between the choce of the gans satsfyng the small-gan condton and the dwell-tme condton. Ths s an nterestng topc for a more detaled nvestgaton. It could be nvestgated, how to develop tools for the stablty analyss of nterconnectons, f the mpulse sequences of subsystems are dfferent. Also, a proof wth nonlnear gans nstead of lnear ones could be performed. Impulsve systems usng a dynamcal dwell-tme condton, presented n [127] for swtched systems, could be nvestgated. There, the dwell-tme condton s formulated usng Lyapunov functons provng asymptotc stablty. Ths approach can be adapted to mpulsve systems and the ISS property, where the benefts of that approach could be nvestgated. Snce mpulsve systems are closely connected to hybrd systems, tme-delays could be ntroduced to hybrd systems and such nterconnectons. Also, the ISDS property for mpulsve systems could be nvestgated. Accordng to [45, 10], one can analyze the ISS property for nterconnected mpulsve systems wth and wthout tme-delays. The analyss of the ISDS property have motvated the nvestgaton of ISDS for MPC of sngle systems and networks. Moreover, the tool of a Lyapunov-Krasovsk functonal have been used for the analyss of ISS for MPC of sngle systems and networks wth tme-delays: 7.5 MPC We have combned the ISDS property wth MPC for nonlnear contnuous-tme systems wth dsturbances. For sngle systems, we have derved condtons such that by applcaton of the control obtaned by an MPC scheme to the system, t has the ISDS property, see Theorem Consderng nterconnected systems, we have proved that each subsystem possesses the ISDS property usng the control of the proposed MPC scheme, whch s Theorem Usng a small-gan condton, we have shown n Corollary that the whole network has

113 Chapter 7. Summary and Outlook 113 the ISDS property. For the proof, we have used one of the results of ths thess, namely Theorem Consderng sngle systems wth tme-delays, we have proved n Theorem that a TDS has the ISS property usng the control obtaned by an MPC scheme, where we have used ISS-Lyapunov-Krasovsk functonals. For nterconnected TDS, we have establshed a theorem, whch guarantees that each closed subsystem obtaned by applcaton of the control obtaned by a mn-max MPC scheme has the ISS property, see Theorem From ths result and usng Theorem 4.2.4, we have shown that the whole network wth tme-delays has the ISS property under a small-gan condton, see Corollary Note that the results presented here are frst steps of the approaches of ISDS for MPC and ISS for MPC of TDS. More detaled studes should be done n these drectons, especally n applcatons of these approaches. An nterestng topc for nvestgatons s the usage of an MPC scheme for nterconnected systems, see the dscusson n Subsecton It should be nvestgated f and how a dfferent MPC scheme whch guarantees ISS or ISDS, respectvely, can be formulated such that the drawbacks of the proposed mn-max approach wll be elmnated. Besdes the proposed closed-loop MPC scheme, condtons for open-loop MPC schemes to ISDS or ISS of TDS, respectvely, could be derved. The dfferences of both schemes should be analyzed and appled n practce. Snce we have focused on theoretcal results regardng MPC, t remans to develop numercal algorthms for the mplementaton of the proposed schemes, as n [84, 38], for example. It could be analyzed, f and how other exstng algorthms could be used or how they should be adapted for mplementaton for the results presented n ths thess. Fnally, one can nvestgate networks, where the subsystems are not drectly nterconnected but they are able to exchange nformaton to control themselves n dependence of the other subsystems to fulfll possble constrants. For ths case, a dstrbuted MPC scheme as n [97, 41] should be used to calculate an optmal control for the subsystems.

114 MPC

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124 124 Bblography

125 Index B(x, r), 15 C ( [t 1,t 2 ];R N),15 PC ( [t 1,t 2 ];R N),15 Ω-path, 21 R n +,15 R +,15 K, 16 KL, 16 L, 16 P, 16 σ, 21 x t,15 K,16 #I, 15 Banach space, 15 Carathéodory condtons, 17 cost functon, 92 cycle condton, 21 dsspatve Lyapunov form, 19 dynamc quantzer, 52 essental supremum norm, 15 Eucldean norm, 15 feasble soluton, 93, 101 feedback control law, 92, 101 FHOCP, 92 FHOCPTD, 101 Gan-matrx, 20 gradent, 15 Impulsve systems, 71 Impulsve systems wth tme-delays, 73, 79 ntal value, 16 nput, 16 nterconnected system, 19 ISDS small-gan theorem, 29 ISS-Lyapunov-Krasovsk small-gan theorem, 65 ISS-Lyapunov-Razumkhn small-gan theorem, 62 local small-gan condton, 20 Locally Lpschtz contnuous, 16 locally Lpschtz contnuous, 16 Lyapunov tools error ISDS-Lyapunov functon, 39 error ISS-Lyapunov functon, 38 exponental ISS-Lyapunov-Krasovsk functonal, 79 exponental ISS-Lyapunov-Razumkhn functon, 74 ISDS-Lyapunov functon, 25 ISS-Lyapunov functon, 18 ISS-Lyapunov-Krasovsk functonal, 60 LISS-Lyapunov-Razumkhn functon, 57 networked control systems, 85 Networks of mpulsve systems wth tme-delays, 81, 83 Networks of tme-delay systems, 61 optmal cost functon, 93 optmal cost functonal, 101 Ordnary dfferental equatons, 16 partal order, 15 postve defnte, 16 postvely nvarant, 93,

126 126 Index predcton horzon, 92, 101 Producton network, 66 Producton network scenaro, 67 projecton, 15 QLE, 67 quantzer, 49 quas-isds observer, 37 quas-iss observer, 37 small-gan condton, 21 soluton of a system, 17 Stablty 0-GAS, 18 ISDS, 24 ISS, 17 LISS, 17 stage cost, 93, 101 state of a system, 16 TDS, 55 termnal penalty, 93, 101 Theorem of Rademacher, 16 Tme-delay systems, 55 upper rght-hand sde dervatve, 57, 59