Robust output feedbackstabilization via risksensitive control


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1 Automatica 38 22) Robust output feedbackstabilization via risksensitive control Valery A. Ugrinovskii, Ian R. Petersen School of Electrical Engineering, Australian Defence Force Academy, Canberra AC 26, Australia Received 2 February 2; received in revised form 24 October 2; accepted 2 November 2 Abstract We consider a problem of robust linear quadratic Gaussian LQG) control for discretetime stochastic uncertain systems with partial state measurements. For a nitehorizon case, the problem was recently introduced by Petersen et al. IEEE rans. Automat. Control 45 2) 398). In this paper, an innite horizon extension of the results of Petersen et al. IEEE rans. Automat. Control 45 2) 398) is discussed. We show that for a broad class of uncertain systems under consideration, a controller constructed in terms of the solution to a specially parameterized risksensitive stochastic control problem absolutely stabilizes the stochastic uncertain system.? 22 Elsevier Science Ltd. All rights reserved. Keywords: Robust control; LQG control; Stochastic control; Stochastic risksensitive control. Introduction For systems with partial state measurements, the linear quadratic Gaussian LQG) methodology has proved to be a useful technique for designing output feedbackcontrollers. However, the development of a robust version of the LQG technique has been a challenging problem. It is known that LQG controllers may lead to very poor robustness in terms of gain and phase robustness margins Doyle, 978). herefore, considerable eorts have recently been undertaken to develop a robust version of the LQG control synthesis methodology. For dierent uncertainty models, this problem has been attacked using the H and mixed H 2 =H control approach e.g., see Mustafa & Bernstein, 99; Haddad et al., 99; Basar & Bernhard, 995), the guaranteed cost and quadratic stabilization approach Fu, de Souza, & Xie, 99), the minimax optimization approach based on sum quadratic constraints Moheimani, Savkin, & Petersen, 995). It has recently been demonstrated in Petersen, James, and Dupuis 2a) that one possibility to enhance the his workwas ported by he Australian Research Council and he Defence Science and echnology Organization. his paper was not presented at any IFAC meeting. Corresponding author. el.: ; fax: addresses: V.A. Ugrinovskii), I.R. Petersen). robustness of an LQG controller is to use a risksensitive control approach to the controller synthesis. he risksensitive control approach has a number of attractive features. Apart from the enhanced robustness of a controller, this approach leads to a tractable minimax design procedure which can be viewed as a direct extension of the existing LQG control technique, see Petersen, Ugrinovskii, and Savkin 2b). Note that the results in Petersen et al. 2a) which concern the output feedbackrobust control of linear systems, are incomplete in that they address the nitehorizon control problem and lead to timevarying controllers, whereas in practical robust controller design problems, timeinvariant controllers are most useful. he derivation of timeinvariant output feedbackrobust controllers requires that an innitehorizon version of the partial information minimax optimal control problem of Petersen et al. 2a) be considered. Although a steadystate version of the minimax optimal controller was discussed by Petersen et al. 2a), the stabilizing properties of this controller were not addressed. In this paper, we rigorously address an innitehorizon robust control problem which can be regarded as an extension of the problems considered by Petersen et al. 2a) to the innite time horizon case. he contribution of this paper is to show that the resulting optimal control schemes guarantee the absolute stability of the closed loop system against the class of admissible uncertainty perturbations under consideration; see also Ugrinovskii and Petersen 2a), Petersen et al. 2b) and Dupuis et al. 998) in the continuoustime case. 598/2/$  see front matter? 22 Elsevier Science Ltd. All rights reserved. PII: S 598)2886
2 946 V.A. Ugrinovskii, I.R. Petersen / Automatica 38 22) he main result of the paper is a robust LQG control synthesis procedure based on a pair of discretetime algebraic Riccati equations arising in a risksensitive optimal control; see Glover and Doyle 988). We show that solutions to a certain specially parameterized risksensitive control problem provide us with a controller which guarantees an optimal upper bound on the timeaveraged performance of the closedloop system in the presence of admissible uncertainties. his result is analogous to the corresponding continuoustime result of Ugrinovskii and Petersen 2a). 2. Denitions We adopt the stochastic uncertain system model introduced in Petersen et al. 2a). his model includes a nominal system model and also a description of a class of admissible uncertain perturbations. In this section, we present a modication of the model of Petersen et al. 2a) which is required in order to obtain an innitehorizon minimax LQG control result. he nominal system and measurement dynamics are described by the following equation: x t+ = Ax t + B u t + B 2 w t+ ; z t = C x t + D u t ; y t+ = C 2 x t + v t+ : ) In the above equations x t R n is the state, z t R p is the uncertainty output, and y t R q is the measured output. Also, w := {w t } t= ;v := {v t } t= are two sequences of i.i.d. Gaussian variables with zero mean and covariance matrices W ;, respectively. It is assumed that w t R r ;v t R q ;t=; 2;:::.WeletP w ;P v denote marginal probability measures of w t ;v t. he sequence w corresponds to the system noise, while the sequence v is referred to as the measurement noise. For simplicity, we pose that the reference system noise and the reference measurement noise are independent. All coecients in Eqs. ) are assumed to be constant matrices of corresponding dimensions. Also for the sake of simplicity, we assume that C D =. he initial state of system ) is a Gaussian random vector x R n with mean x and nonsingular covariance matrix Y. he corresponding Gaussian probability measure associated with x is denoted P x. he random variable x is assumed to be independent of the reference noise sequences w ;v. he joint probability measure on the initial condition vector x and the reference noise sequences w and v is hroughout the paper, t ; t will denote sequences of random variables as follows t := { t; t+ ;:::; }; t := { t; t+ ;:::}. denoted by P : P dx dw dv ) = P x dx ) P w dw t ) P v dv t ): t= his probability measure is dened on measurable sets of the ltration {F ;=; ;:::}. For each =; ;:::;the algebra F is generated by the initial condition x of system ) and the noise sequences w, v and is completed by including all sets of P probability zero. he sets of the algebra F are subsets of the noise space W, W = {x ;w ;v )}: From Kolmogorov s theorem, the family of probability measures P gives rise to a probability measure P dened on measurable space W; F); F = {F ;=; ;:::}. 2.. Stochastic uncertain system We now introduce an uncertainty model. he uncertainty will be modeled in terms of perturbations of corresponding joint probability distributions of the noise inputs w, v and the initial condition of the system. It is demonstrated in Section 2.3 that such a model provides a meaningful description of stochastic uncertain systems modeled using a linear fractional transformation LF); also, see Petersen et al. 2a,b) and Ugrinovskii and Petersen 999). he rigorous denition of our uncertainty model is the following. Let Q := {Q ;Q 2 ;:::} be a collection of conditional probability measures referred to as an uncertainty; i.e., for each t =; 2;:::; Q t dw t dv t x ;w t ;v t ) is a probability measure dened on measurable subsets of R r R q.itisassumed that Q t dw t dv t x ;w t ;v t ) Pdw t dv t x ;w t ;v t ); t =; 2;::: 2) Here, the notation means that the probability measure is absolutely continuous with respect to the reference probability measure. Using the collection Q, the joint probability measure of the initial condition vector x and the perturbed noise sequences w and v is dened by Q dx dw dv ) = P x dx ) t= Q t dw t dv t x ;w t ;v t ): t= It follows from Eq. 2) that Q P. Also, for any F measurable set W; Q )=Q ). As in Petersen et al. 2a), we will require that each uncertainty Q = {Q ;Q 2 ;:::} has the property hq P ) for all :
3 Here, hq P ) is the relative entropy functional; e.g., see Dupuis and Ellis 997). his requirement will allow us to rule out the possibility of singular uncertain probability measure perturbations. he class of uncertain systems considered in this paper comprises the systems of the form ) in which the marginal conditional probability measures of the noise inputs w, v are required to satisfy the above requirements. he set of such uncertainties Q will be denoted as Q. From the denition and properties of the relative entropy functional, it follows that the set Q is convex. V.A. Ugrinovskii, I.R. Petersen / Automatica 38 22) Relative entropy constraint on stochastic uncertainty LF uncertainty models require that a magnitude constraint be imposed on the uncertainty. For example, it is often required that the unmodeled dynamics transfer function has a bounded H norm. As shown in Ugrinovskii and Petersen 999, 2a) and Petersen et al. 2b), for stochastic LF systems with additive noise, the constraint on the H norm of the uncertainty transfer function can be replaced by a constraint on the relative entropy of the associated perturbation probability measure. For discrete time systems, this leads to an extension of the sum quadratic constraint uncertainty description Moheimani et al., 995); see Section 2.3. Let E Q denote the expectation with respect to the probability measure Q. Denition. Let d be a given positive constant. A collection of conditional probability measures Q Q is said to dene an admissible uncertainty if the following stochastic uncertainty constraint is satised: hq P ) 6 d + 2 EQ z s 2 + ) 3) for all =; 2;:::.In3);z t is the uncertainty output dened by Eq. ). Also; ) as. We denote the set of uncertainties Q Q satisfying condition 3) by. he corresponding probability measures Q are also called admissible probability measures. Observe that the set is not empty. Indeed, consider the reference probability measure P. Since the Gaussian random variables w t, v t are independent, the corresponding reference conditional probability measures are simply the marginal probability measures: P t dw t dv t x ;w t ;v t )=P w dw t ) P v dv t ): Also, the fact that hp P ) = and the condition d imply that the corresponding collection of conditional probabilities P := {P ;P 2 ;:::} is admissible, P. Note that in this case, constraint 3) is satised strictly. Hence, P is an interior point of the set. Fig.. An uncertain control system A connection between uncertainty input signals and uncertain probability measures In order to give a further insight into Denition, we show that the uncertainty class introduced in Section 2.2 includes uncertainty models which often arise in control systems such as H normbounded uncertainty and bounded exogenous uncertainty. Consider an uncertain system shown in Fig.. he system is described by the equations x t+ = Ax t + B u t + B 2 t z t = C x t + D u t ; y t+ = C 2 x t + t + ex t + w t+ ); + ex t +ṽ t+ ); 4) and is driven by the system and measurement noises w t, ṽ t, exogenous disturbance processes ex t, ex t and uncertainty inputs t, t which are generated by a stable linear timeinvariant uncertainty. he latter is due to the presence of unmodeled dynamics which are described by a stable transfer function z). It is assumed that! ; W =2 e j! ) =2 6 2 : 5) Condition 5) constitutes a H type norm bound on the size of unmodeled dynamics in system 4). System 4) is considered in a complete probability space W; F; Q); w,ṽ are Gaussian white noise processes in this probability space, E Q w t w t =W, E Qṽ t ṽ t =. It follows from 5) that the processes, are adapted to the ltration {F ;=; ;:::;} generated by the noise inputs
4 948 V.A. Ugrinovskii, I.R. Petersen / Automatica 38 22) and W =2 t 2 + =2 t 2 ) 6 2 E Q z t 2 : E Q Also, it is assumed that the exogenous disturbances ex, ex are adapted to the ltration {F ;=; ;:::;} and E Q W =2 ex t 2 + =2 ex t 2 ) 6 d + ); 6) where d is a given constant, and ) as. Condition 6) imposes a bound on the size of uncertain exogenous perturbations in system 4). In particular, one can see the role of the constant d from Denition in this example. he introduction of this constant reects the fact that the power seminorm of the input disturbances does not exceed d. he above constraints on the exogenous disturbance and unmodeled dynamics lead to an uncertainty constraint which can be represented in the form of the relative entropy uncertainty constraint 3). Indeed, the above assumptions imply that the processes ex, ex t := t + ex t ; t := t + ex t satisfy the condition 2 E Q W =2 t 2 + =2 t 2 ) 6 d + 2 E Q z t 2 + ): 7) In order to express constraint 7) in the form of condition 3), we introduce the probability measure transformation dp = exp { d Q tw w t+ 2 } F tw t exp { t ṽ t+ 2 } t t : 8) Using the discretetime version of the Girsanov s theorem Elliott, Aggoun, & Moore, 994) we conclude that under the probability measure P the sequences w ;v dened by the equations w t+ = w t+ + t ; v t+ =ṽ t+ + t ; t =; ; 2;::: : 9) are the sequences of i.i.d. Gaussian random variables. Furthermore, when considered with respect to the probability measure P, system 4) becomes a system of the form ) driven by the i.i.d. Gaussian white noises w t ;v t. Finally, using the chain rule Dupuis & Ellis, 997), we obtain h Q P )= 2 E Q W =2 t 2 + =2 t 2) : Here, Q and P denote the restrictions of the probability measures Q and P to W; F ). hus, from 7), condition 3) of Denition 2 follows. herefore, the uncertain system considered in this section belongs to the class of uncertain systems dened in Denition. It can be shown in a similar fashion that Denition encompasses some other important classes of uncertainty arising in control systems such as, for example, the conebounded uncertainty Absolutely stabilizing control In this paper, our attention will be restricted to linear outputfeedbackcontrollers of the form ˆx t+ = A c ˆx t + B c y t+ ; u t = C c ˆx t ; ) where ˆx R ˆn is the state of the controller, A c R ˆn ˆn, B c R ˆn q, and C c R m ˆn. Let U denote this class of linear controllers. Note that controller ) is adapted to the ltration {Y t ;t } generated by the observation process y; Y t = {y s ;s=;:::;t}. he closedloop system corresponding to system ) and controller ) is described by a linear dierence equation of the form x t+ = A x t + B w t+ ; z t = C x t ; u t = C c x t : ) In Eq. ), x =x ˆx R n+ˆn and w t =w t v t are the state and the noise input of the closed loop system. Also, the following notation is used: A B C c B2 A = ; B = ; B c C 2 A c B c C =C D C c : 2) In the sequel, we will consider a subclass of controllers ) which satisfy some additional stabilizability and observability requirements. We say that a controller K belongs to the class U U if for this controller, the corresponding matrix pair A; B) is stabilizable 2 and the pair A c ;C c )is observable. It was shown in Petersen et al. 2a) that in the steadystate case, the minimax optimal LQG controller has the structure given in equation ). As mentioned above, we wish to investigate the stabilizing properties of this optimal controller as. First, we introduce a denition 2 For example, this condition holds if the matrix pairs A; B 2 ) and A c;b c) are stabilizable.
5 V.A. Ugrinovskii, I.R. Petersen / Automatica 38 22) of stabilizability which allows for uncertain control systems driven by additive noise whose solutions are not square summable. Denition 2. A controller K of the form ) is said to be an absolutely stabilizing controller for the uncertain system ) with uncertainty satisfying the relative entropy constraint 3); if the system state process x ; the controller state process ˆx and the control process u dened by the closedloop system ) corresponding to this controller satisfy the following condition. here exist constants c ; c 2 such that for any admissible uncertainty Q ; x t 2 + ˆx t 2 + u t 2 ) E Q +hq P ) 6 c + c 2 d: 3) In particular, for the uncertain system shown in Fig. above property means that in a stable closedloop system, timeaverage norms of the system state, controller state as well as control input and uncertainty inputs are bounded regardless the value of the transfer function z) and disturbances ex, ex. In the special case of the uncontrolled uncertain system ), 3), the above denition reduces to the property of absolute stability the system. hat is, the uncertain system ), 3) with u t = is absolutely stable if for any admissible uncertainty Q, x t 2 + hq P ) 6 c + c 2 d: E Q 4) Lemma. Suppose the stochastic nominal closedloop system ) is mean square stable; i.e. E x t 2 : 5) Also; pose the pair A; B) is stabilizable. hen; the matrix A must be stable. he proof of this lemma is identical to the proof of the corresponding continuoustime result in Ugrinovskii and Petersen 2a,b) and Petersen et al. 2b). 3. he main results he steadystate minimax LQG control problem considered in Petersen et al. 2a) was to nd a controller which attained an upper value inf J K; Q) 6) K Q of the cost functional J K; Q) = 2 EQ Fx t ;u t ); Fx; u):=x Rx + u Gu; 7) R and G are symmetric positive denite matrices, R R n n, G R m m. Here, x t denotes the solution to system ) corresponding to a given controller K and an admissible uncertainty Q. he minimax optimal controller for problem ), 3), 7) proposed in Petersen et al. 2a) was constructed using a pair of parameter dependent algebraic Riccati equations X = A X + B G + D D ) B B 2WB 2) A + R + C C ; 8) Y = A Y + C 2 C 2 ) R C C A + B 2 WB 2: 9) Here, is a positive constant which was chosen as follows. For each such that the Riccati equations 8), 9) admit positive denite stabilizing solutions satisfying the conditions Y + C 2 C 2 R C C ; X B 2WB 2 ; X Y 2) dene the quantity V = det 2 log I ) Y R + C C ) det 2 log I X I ) ) Y X ; 2) where := K + C 2 I ) ) Y R + C C ) Y C K ; K := A Y + C 2 C 2 ) R C C C 2 : 22) he set of the parameters satisfying the above requirement is denoted. In order to obtain the minimax optimal LQG controller, the parameter is chosen to achieve the inmum in inf V + d): 23) Using this optimal value of, the corresponding minimax optimal steadystate controller K is dened by the
6 95 V.A. Ugrinovskii, I.R. Petersen / Automatica 38 22) equations u t = L ˆx t ; 24) L := G + D D ) B X + B G + D D ) B B 2WB 2 A I ) Y X ; where ˆx t was generated by the lter ˆx t+ = A ˆx t + B u t + Ky t+ C 2 ˆx t ) + A Y + C 2 C 2 ) R C C R + C C )ˆx t ; ) ˆx =x : 25) he above solution to the minimax optimal LQG control problem was obtained in Petersen et al. 2a) by letting in the corresponding nite horizon problem. However, as we let approach, two important issues arise which were not addressed in Petersen et al. 2a). ) he ability of an optimal controller to stabilize the system is an important issue in optimal control on an in nite time interval. In our case, it is desired that the resulting minimax optimal controller stabilizes the system ) for all admissible uncertainties Q. 2) It was shown in Petersen et al. 2a) that on a nite interval, the minimax optimal LQG controller exists if and only if associated parameter dependent dierence Riccati equations have positive denite solutions. In the innite horizon case, the design methodology outlined in Petersen et al. 2a) provides only a sucient condition for the minimax optimal controller to exist. However, it is often useful to know that the only if claim holds as well. Necessary conditions usually give a good indication as to how conservative the proposed controller is. hese issues are addressed in the statements given below. heorem shows that the existence of stabilizing solutions to the algebraic Riccati equations 8) and 9) is a sucient as well as a necessary condition for the innitehorizon minimax optimal LQG control problem 26) to have a solution. Furthermore, heorem 2 shows that the minimax LQG controller proposed in Petersen et al. 2a) is an absolutely stabilizing controller. heorem. i) If the set is nonempty, then the minimax optimal control problem inf J K; Q) 26) K U Q has a nite value. ii) Conversely, if there exists an absolutely stabilizing controller K U which attains the inmum in 26); then the set is not empty. he innitehorizon version of the minimax optimal LQG control result of Petersen et al. 2a) now follows from heorem. heorem 2. Suppose that and attains the inmum in 23). hen; the corresponding controller K = K dened by 24); 25); guarantees that the worst case of the cost functional 7) does not exceed the value 23). Furthermore; this controller is an absolutely stabilizing controller for stochastic uncertain system ) satisfying relative entropy constraint 3). Remark. Note that the controller resulting from heorem 2 is the steadystate limit of the nite horizon minimax optimal controller proposed in the reference Petersen et al. 2a). Also; the worstcase performance 23) is the steadystate limit of the performance guaranteed by the nitehorizon minimax optimal controller. he proofs of the above theorems will be given in Sections 3.3 and Preliminary remarks he proof of heorem relies on a duality relationship between free energy and relative entropy established in Dupuis and Ellis 997) and Dai Pra, Meneghini and Runggaldier 996). Associated with system ), consider the parameter dependent risksensitive cost functional I ; K)= {exp log E 2 } F x t ;u t ) ; 27) where is a given constant, K is a controller dened by Eq. ). Also, F x; u):=x R + C C )x + u G + D D )u: 28) When applied to system ) and the risksensitive cost functional 27), the duality result states that for each admissible controller K, I ; K) = Q : hq P ) 2 EQ F x t ;u t ) : hq P ) 29) he use of the above duality result is a key step in this paper in that it enables us to replace a guaranteed cost control problem by the following risksensitive optimal control problem:
7 V.A. Ugrinovskii, I.R. Petersen / Automatica 38 22) Find an output feedbackcontroller of form ) which attains the inmum in inf lim I ;K): 3) K In this paper, we use the solution to this risksensitive control problem presented in Whittle 99). Note that similar results concerning problem 3) were obtained in Iglesias, Mustafa, and Glover 99) and Glover and Doyle 988) 3 where the problem was shown to be equivalent to a problem of constructing an output feedbackcontroller which maximizes the entropy functional I d K;)= log deti H e j! )H e j! )) d! 2 3) over the set of stabilizing output feedbackcontrollers satisfying the following bound: H := H e j! )H e j! ) : 32)! ; Here, H s) is the transfer function of the closed loop system dened by the equations x t+ = Ax t + B u t +B 2 W =2 t+ ; 33) R =2 z t = x t + G =2 u t; C D y t+ = C 2 x t + =2 t+ ; and a stabilizing linear controller u t =Ky t ). In particular, for a linear controller of the form ), the transfer function H s) is given by R =2 H s) := G =2 C c si A) C D C c B2 W =2 : 34) B c =2 It follows from the results of Iglesias et al. 99), Glover and Doyle 988) and Whittle 99) that the solution to the above maximum entropy problem and equivalently, the solution to the risksensitive control problem 3) is given by the central solution to the corresponding H control problem; see Basar and Bernhard 995). 4 his controller is dened by Eqs. 24) and 25) and involves positive denite solutions 3 Iglesias et al. 99), Glover and Doyle 988) consider a slightly dierent information pattern which allows the control u t to instantaneously access y t+. his leads to an optimal controller which is not strictly causal. 4 Also, the links between the problem considered in this paper and H control were discussed in Petersen et al. 2a). to the Riccati equations 8) and 9). Furthermore, the minimum risksensitive cost corresponding to this controller is given by Eq. 2) Absolutely stabilizing properties of risksensitive control In this section, we establish the absolutely stabilizing properties of the risksensitive controller. Lemma 2. Let K be a controller which guarantees a nite risksensitive cost: V K := lim I ;K) : hen; the controller K is an absolutely stabilizing controller for the stochastic uncertain system ) satisfying the relative entropy constraint 3). Furthermore; J K; Q) 6 V K + d: 35) Q Proof. We wish to prove that the controller K satises condition 3) of Denition 2. Consider an uncertain system with an admissible uncertainty Q governed by the controller K. Recall that the system is described by Eq. ) where the control process u is generated by the given controller K. Also, the joint probability distribution of the noise inputs w, v and the initial condition x is dened by the given collection of conditional probability measures Q. Note that since the uncertainty Q is admissible, then the corresponding probability measures Q have the property hq P ) for every =; 2;:::. his allows us to apply duality condition 29). From this equation, we obtain + { 2 EQ 2 EQ Fx s ;u s ) } z s 2 hq P ) 6 V K : 36) Furthermore since Q, satisfaction of Eq. 35) follows from 36) and 3). We now show that the controller K is absolutely stabilizing. Since the matrices R and G are positive denite, then inequality 35) implies EQ x s 2 + u s 2 ) 6 V K + d); 37) where is a constant which depends only on R and G. Next, we show that there exist constants c ;c 2 such that hq P ) c + c 2 d: 38)
8 952 V.A. Ugrinovskii, I.R. Petersen / Automatica 38 22) o this end, we note that for any Q, t t hqt P t ) 6 d + ˆ + t 2t EQ z s 2 : 39) t t Here, ˆ := t t) as. Inequality 39) takes into account the fact that the probability measure Q t satises the relative entropy constraint 3). herefore, inequalities 37) and 39) imply hq P ) 6 d + 6 c + c 2 d; 2 EQ z s 2 where the constants c, c 2 are dened by V K,,, C, D, and hence independent of Q. o complete the proof, it remains to prove that the process ˆx satises condition 3) of Denition 2. Indeed, consider the uncertain closed loop system ) corresponding to the controller K and an admissible uncertainty Q. Using the duality relation between free energy and relative entropy established in Dupuis and Ellis 997) and Dai Pra et al. 996), we obtain Q 2 EQ x t 2 hq P ) 6 Q :hq P ) 2 EQ x t 2 hq P ) { } =log E exp x t 2 : 2 From this equation, it follows that for any Q, 2 EQ x t 2 6 hq P ) { } + lim log E exp x t 2 : 4) 2 Note that the process x on the lefthand side of Eq. 4) corresponds to the uncertain system ) while the process x that appears in the second term on the righthand side of Eq. 4) is generated by the nominal closed loop system ). Since this nominal system is driven by the Gaussian process w, one can choose suciently small such that the second term on the righthand side of Eq. 4) is nite; e.g., see Glover and Doyle 988) and Whittle 99). his term is independent of Q. Since we have already proved 38), then condition 3) follows. Remark 2. It follows from Lemma 2 that the controller solving the risksensitive control problem 3) is an absolutely stabilizing controller for the uncertain system under consideration. Lemma 3. Suppose 5 that for any K U J K; Q)= : 4) Q Q If there exists an absolutely stabilizing controller K U such that J K; Q) c ; 42) Q then there exists a such that V. Proof. Since the given controller K U absolutely stabilizes the uncertain system ); 3); then condition 3) of Denition 2 is satised. Note that it follows from 3) and Lemma that the matrix A corresponding to the controller K is stable. Also; condition 3) implies that there exists a positive constants c ; such that for all Q ; lim inf 2 EQ + lim inf Fx s ;u s ) 2 EQ x s 2 + ˆx s 2 ) 6 c: 43) Consider the functionals G Q) := c lim inf lim inf G Q):= d lim inf 2 EQ 2 EQ Fx s ;u s ) x s 2 + ˆx s 2 ); 2 EQ z s 2 hq P ) : 44) It is readily proved that satisfaction of condition 43) implies that the following condition is satised: If G Q) 6 then G Q) : 45) he proof of this fact follows along the same lines as the proof of the corresponding fact in Ugrinovskii and Petersen 2a); see also Petersen et al. 2b). Furthermore; the set of uncertainties satisfying the condition G Q) 6 has an interior point; see the remarkfollowing Denition. Also; it follows from the properties of the relative entropy functional that the functionals G ) and G ) are convex. 5 Condition 4) is similar to a corresponding condition in Petersen et al. 2a). his condition was introduced in Petersen et al. 2a) to rule out the case = in the proof of heorem 3:. In the continuoustime case, this condition holds if the pair A; B) corresponding to the given controller K U is controllable; e.g., see Ugrinovskii and Petersen 2a) and Petersen et al. 2b). Obviously, in the discrete time case, condition 4) can be replaced by a similar controllability condition.
9 V.A. Ugrinovskii, I.R. Petersen / Automatica 38 22) We have now veried all of the conditions needed to apply the Lagrange multiplier result e.g.; see Luenberger; 969). Indeed; heorem on p. 27 of Luenberger 969) implies that there exists a constant such that lim inf + EQ lim inf + lim inf 2 Fx t;u t ) 2 EQ 2 EQ x t 2 + ˆx t 2 ) z t 2 hq P ) 6 c d: 46) for all Q Q. Also; condition 4) rules out the possibility that =. hus;. Condition 46) implies the satisfaction of condition 32) for the transfer function H s) corresponding to the system 33) and the given controller K. his claim can be established using the same arguments as those used in proving the corresponding fact in Ugrinovskii and Petersen 2b) and Petersen et al. 2b). For the sake of brevity, we only outline the proof. Consider an augmented version of the closed loop system corresponding to system 33) and the given controller K: B2 x t+ = W =2 A x t + t ; 47) B c =2 R =2 G =2 C c z ;t = C D C c x t : I I his system is posed to be driven by a deterministic input disturbance which has a nite autocorrelation matrix and also has a power spectral density function Makila, Partington, & Norlander 998; Ljung, 987). We denote the set of such signals by P +. he transfer function of system 47) is denoted H ; z). We will establish condition 32) by showing that H ; 6 : 48) he proof is by establishing a contradiction. First, we observe that the failure of condition 48) to hold must lead to the existence of a sequence of deterministic inputs { N ;N =; 2;:::} P+ such that z ;t 2 t N 2 ) N 49) 2 for all ; N), where is a suciently small constant. hen, we show that the satisfaction of condition 49) must lead to a contradiction with 46). Indeed, for each input N, a collection of conditional probability measures Q N = {Q N ;QN 2 ;:::} Q can be dened using a corresponding probability measure transformation; see Section 2.3. o this end, the input N has to be partitioned as follows: N t W =2 = N =2 t : N t Also as in Section 2.3, the probability measures Q N; are dened, hq N; P )= 2 W =2 N t 2 + =2 N t 2 : 5) Next, it is shown using 49) that the closed loop system ) driven by the deterministic inputs N, N and considered on the probability space W; F ;Q N; ), satises the following condition: N; M N := lim 2 EQ Fx t;u t )+ x t 2 Q Q + z t 2 W =2 N t 2 =2 N t 2 N: 5) Letting N in Eq. 5) and using Eq. 5), we obtain the following contradiction with 46): lim inf EQ 2 Fx t;u t ) + lim inf + lim inf 2 EQ 2 EQ x t 2 z t 2 hq P ) M N = : N he above contradiction shows that condition 48) must hold. Since condition 48) implies 32), the lemma now follows from condition 32) and the results of Glover and Doyle 988) and Whittle 99) Proof of heorem Part i) of the theorem follows from Lemma 2. Indeed, if, then the corresponding Riccati equations 8) and
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