Feedback-Feedforward Scheduling of Control Tasks

Transcription

1 Real-Time Sysems (Special Issue on Conrol-Theoreical Approaches o Real-Time Compuing). To appear in Feedback-Feedforward Scheduling of Conrol Tasks Anon Cervin, Johan Eker, Bo Bernhardsson, Karl-Erik Årzén Deparmen of Auomaic Conrol Lund Insiue of Technology Box 8, SE Lund, Sweden anon@conrol.lh.se Absrac A scheduling archiecure for real-ime conrol asks is proposed. The scheduler uses feedback from execuion-ime measuremens and feedforward from workload changes o adjus he sampling periods of he conrol asks so ha he combined performance of he conrollers is opimized. The performance of each conroller is described by a cos funcion. Based on he soluion o he opimal resource allocaion problem, explici soluions are derived for linear and quadraic approximaions of he cos funcions. I is shown ha a linear rescaling of he nominal sampling frequencies is opimal for boh of hese approximaions. An exensive invered pendulum example is presened, where he performance obained wih open-loop, feedback, combined feedback and feedforward scheduling, and earlies-deadline firs scheduling are compared. The performance under earliesdeadline firs scheduling is explained by sudying he behavior of periodic asks under overload condiions. I is shown ha he average values of he sampling periods equal he nominal periods, rescaled by he processor uilizaion.. Inroducion There is currenly a rend owards more flexible realime conrol sysems. Combining scheduling heory and conrol heory opens up he possibiliy o achieve higher CPU-resource uilizaion and beer conrol performance. To obain he bes resuls, co-design of he scheduler and he conrollers is necessary. Conrol asks are generally viewed by he scheduling communiy as hard real-ime asks wih fixed sampling periods and known wors-case execuion imes (WCETs). Upon closer inspecion, neiher of hese assumpions need necessarily be rue. For insance, many conrol algorihms are quie robus agains variaions in sampling period and inpuoupu laency. In some siuaions i is also possible o le he conroller acively compensae for he variaions by, e.g., recalculaing he conroller parameers. Hence, he iming nondeerminism can be viewed as a disurbance or uncerainy affecing he conrol loop. Conrollers can also be designed o swich beween differen modes wih differen execuion imes and/or differen sampling periods. An on-line scheduler ha uses feedback o dynamically adjus he conrol ask aribues in order o opimize he global conrol performance can be inerpreed as a conroller in iself, i.e., as a feedback scheduler. The conrol performance, or Qualiy-of-Conrol (QoC), can, hence, be inerpreed as a qualiy-ofservice measure. In his work, we presen a feedback scheduler for conrol asks ha aemps o keep he CPU uilizaion a a high level, avoid overload, and disribue he compuing resources among he conrol asks so ha he oal conrol performance is opimized. While we wan o keep he number of missed deadlines as low as possible, conrol performance is our primary objecive. Thus, conrol asks, in our view, fall in a caegory somewhere beween hard and sof real-ime asks. The known-wcet assumpion is relaxed by he use of feedback from execuionime measuremens. We also inroduce feedforward o furher improve he regulaion of he uilizaion. The feedforward acion can be inerpreed as an admission conroller. The aricle is based on he feedback scheduling approach in [Cervin and Eker, 2000]. There he problem of scheduling a se of hybrid conrol asks was sudied. The hybrid conrollers swiched beween differen modes wih differen execuion-ime characerisics. The feedback scheduler adjused he conroller ask periods hrough a simple linear rescaling wihou any direc connecion o conrol performance opimizaion. In his paper we insead sudy he problem of scheduling a se of linear conroller asks. The performance of each conroller is described by a cos funcion. By saring and sopping he differen conrol asks he CPU load is varied. The sampling periods are adjused so ha he oal cos of he conrollers is minimized. In [Eker e al., 2000] a feedback scheduler for he special case of linear-quadraic (LQ) conrol asks

2 was presened. An LQ-conroller is a linear saefeedback conroller ha is designed o minimize a quadraic cos crierion. Formulas for he LQ cos funcion and is derivaive wih respec o he sampling period were given. The scheduling problem was formulaed as a recursive opimizaion problem based on he exac formulas for he derivaives of he cos funcions. Due o he high compuaional coss involved, an approximae version of he scheduler was also developed. The cos funcions were approximaed by quadraic funcions of he sampling periods, and explici expressions for he opimal sampling periods were derived. Tha resul is uilized in his paper. Furhermore, we allow for a se of general linear conrollers and also sudy he resuls obained when he cos funcions are approximaed by linear funcions of he sampling periods. On he implemenaion side, boh prioriy-based and earliesdeadline-firs (EDF) based approaches are sudied. A new resul concerning he properies of EDF scheduling for periodic conrol asks during overload condiions is presened.. Ouline of he paper Addiional moivaion and a survey of relaed work is conained in Secion 2. The CPU-resource allocaion problem for a se of conrol asks is reaed in Secion 3. The srucure of he proposed scheduling archiecure is described in Secion 4. The feedback scheduling example is presened in Secion 5. A comparison is made beween open-loop scheduling, feedback-based scheduling and combined feedback and feedforward scheduling. Also, ordinary earliesdeadline-firs scheduling is sudied. A new resul concerning he properies of EDF schedules for periodic asks during overload condiions is presened in Secion 6. The conclusions are given in Secion Background Mos conrol sysems are embedded sysems where he compuer is a componen in a larger engineering sysem. The conrol sysem is ofen implemened on a microprocessor using a real-ime kernel or a real-ime operaing sysem. The real-ime kernel or he operaing sysem uses muliprogramming o muliplex he execuion of he asks on he CPU. The CPU ime, hence, consiues a shared resource which he asks compee for. To guaranee ha he ime requiremens and ime consrains of he individual asks are all me, i is necessary o schedule he usage of he shared resource. In radiional hard real-ime compuing models, i is assumed ha conrol asks fi he simple ask model, i.e., he asks are periodic, or can be ransformed o periodic asks, wih a fixed period, a known worscase bound on he execuion ime (WCET), anda hard deadline. The fixed-period assumpion of he simple ask model has also been widely adoped by he conrol communiy and has, e.g., resuled in he developmen of he sampled compuer-conrol heory wih is assumpion on deerminisic, equi-disan sampling. Upon closer inspecion i is quie clear ha many of he assumpions of he simple model are oo resricive. Firs, he assumpions do no allow us o use low-cos general purpose hardware and off-heshelf operaing sysems, which in general are no able o give any guaranees on deerminism. These sysems are, ypically, designed o achieve good average performance raher han guaraneed wors-case performance. They ofen inroduce significan nondeerminism in ask scheduling. For compuaioninensive high-end applicaions, he large variabiliy in execuion ime caused by modern hardware archiecure also becomes visible. The effec of his on he conrol loop is jier in sampling period and conrol delay (inpu-oupu laency). In order o mainain good conrol performance i is imporan o compensae on-line for he variaions. A requiremen for his is ha he necessary iming informaion is provided by he real-ime kernel. The assumpions of he simple model are also overly resricive wih respec o he characerisics of many conrol loops. Many conrol loops are no periodic, or hey may swich beween a number of differen fixed sampling periods. Conrol loop deadlines are no always hard. On he conrary, many conrollers are quie robus owards variaions in sampling period and response ime. Hence, i is quesionable wheher i is necessary o model hem as hard deadline asks. I is also in many cases possible o compensae on-line for he variaions by, e.g., recompuing he conroller parameers. Obaining an accurae value for he WCET is generally a difficul problem. Measuring WCET always implies he risk of underesimaion, whereas analyical execuion ime analysis ools are sill rare. An alernaive way may be o insead measure he acual execuion ime every ask invocaion and o adjus he ask parameers accordingly. Finally, i is also possible o consider conrol sysems ha are able o do a radeoff beween he available compuaion ime, i.e., how long he conroller may spend calculaing he new conrol signal, and he conrol loop performance. On-line ask aribue adjusmen mechanisms can in cerain siuaions be viewed as conrollers in hemselves. Using feedback from on-line measuremens of, e.g., acual execuion imes, he resource uilizaion of he differen asks is changed. If he asks are conrol asks, feedback from he conrol performance can also be used o, e.g., opimize conrol performance under given resource uilizaion

3 consrains. The feedback scheduler hen solves an opimizaion problem o perform he ask aribue adjusmen. The use of on-line opimizaion as a way of realizing feedback conrol is quie common in advanced conrol sysems. One example is model predicive conrol (MPC) where a convex opimizaion problem is solved in each sample. 2. Relaed Work The relaed work falls ino hree caegories: inegraed conrol and real-ime sysem design, qualiyof-service approaches in real-ime sysems, and flexible and adapive real-ime sysem algorihms and archiecures. In [Seo e al., 996], sampling period selecion for a se of conrol asks is considered. The performance of a ask is given as funcion of is sampling frequency, and an opimizaion problem is solved o find a se of opimal ask periods. Co-design of real-ime conrol sysems is also considered in [Ryu e al., 997], where he performance parameers are expressed as funcions of he sampling periods and he inpuoupu laencies. [Shin and Meissner, 999] deals wih on-line rescaling and relocaion of conrol asks in a muli-processor sysem. The second area of relaed work is on qualiy-ofservice (QoS) aware real-ime sofware, where a sysem s resource allocaion is adjused on-line in order o maximize he performance in some respec. In [Li and Nahrsed, 998] a general framework is proposed for conrolling he applicaion requess for sysem resources using he amoun of allocaed resources for feedback. I is shown ha a PID (proporional-inegral-derivaive) conroller can be used o bound he resource usage in a sable and fair way. In [Abeni and Buazzo, 999] ask models suiable for mulimedia applicaions are defined. Two of hese use PI conrol feedback o adjus he reserved fracion of CPU bandwidh. The resource allocaion scheme Q-RAM is presened in [Rajkumar e al., 997]. Several asks are compeing for finie resources, and each ask is associaed wih a uiliy value, which is a funcion of he assigned resources. The sysem disribues he resources beween he asks o maximize he oal uiliy of he sysem. In [Abdelzaher e al., 997] a QoS renegoiaion scheme is proposed as a way o allow graceful degradaion in cases of overload, failures or violaion of pre-runime assumpions. The mechanism permis cliens o express, in heir service requess, a range of QoS levels hey can accep from he provider, and he perceived uiliy of receiving service a each of hese levels. The approach is demonsraed on an auomaed fligh-conrol sysem. Conrol-heoreical approaches for QoS adapaion are also presened in [Abdelzaher and Shin, 999]. The hird area relaes o he wealh of flexible scheduling algorihms available. An ineresing alernaive o linear ask rescaling is given in [Buazzo e al., 998], where an elasic ask model for periodic asks is presened. The relaive sensiiviy of asks o rescaling are expressed in erms of elasiciy coefficiens. Schedulabiliy analysis of he sysem under EDF scheduling is given. Task aribue adjusmen sraegies are also presened in [Nakajima, 998; Kuo and Mok, 99; Kosugi e al., 994; Nakajima and Tezuka, 994; Lee e al., 996]. The idea of using feedback in scheduling has o some exen been used previously in general purpose operaing sysems in he form of muli-level feedback queue scheduling [Kleinrock, 970; Blevins and Ramamoorhy, 976; Poier e al., 976]. However, his has mosly been done in an ad-hoc way. Relaed o our work, [Sankovic e al., 999; Lu e al., 999] presen a scheduling algorihm, he FC-EDF, ha explicily uses feedback in combinaion wih EDF scheduling. A PID conroller regulaes he deadline miss-raio for a se of sof real-ime asks wih varying execuion imes, by adjusing heir requesed CPU uilizaion. I is assumed ha asks can change heir CPU consumpion by execuing differen versions of he same algorihm. An admission conroller is used o accommodae larger changes in he workload. In [Lu e al., 2000] he same approach is exended. An addiional PID conroller is added ha insead conrols he CPU uilizaion. The wo conrollers are combined using a min-approach. The resuling hybrid conroller scheme, named FC-EDF 2, gives good performance boh during seady-sae and under ransien condiions. Alhough relaed o he work presened here, here are imporan differences. In our approach he asks ha are scheduled are conrollers, conrolling some physical plans. The performance, or Qualiy-of-Conrol (QoC) is explicily used by he feedback scheduler o opimize he oal performance of all he conroller asks. In order o evaluae how he scheduling of conroller asks influence he conrol performance i is necessary o have simulaors ha allow join simulaion of coninuous-ime plan dynamics, discree-ime conrollers, and he real-ime scheduling of he corresponding conroller asks. The simulaions in he curren paper are based on he Malab/Simulink oolbox presened in [Eker and Cervin, 999]. A similar ool is presened in [Palopoli e al., 2000]. An exensive survey wih addiional references o oher relaed work in he area of conrol and CPU scheduling can be found in [Årzén e al., 999].

4 3. Conrol Loop Resource Allocaion In his secion, we ake a higher-level, idealized view of a feedback scheduling sysem and sudy he problem of disribuing limied compuing resources o a number of conrol loops. For now, i is assumed ha lower-level mechanisms in he real-ime sysem can provide he feedback informaion needed, and ha he available compuing resources can be divided in an exac way. We sudy he problem of scheduling several independen conrol asks on a shared processing uni. The plans are described by linear coninuous-ime sysems, he conrollers are described by linear discreeime sysems, and he performance of he conrol loops are measured using quadraic performance crieria. The goal is o assign sampling periods o he conrol loops such ha he overall conrol performance is opimized subjec o he schedulabiliy consrain. The effecs of conrol delay and jier on he conrol performance are negleced a his sage. The nex secion inroduces a sandard mahemaical model of a compuer-conrolled sysem. More deails can be found in exbooks on sampled-daa conrol heory, e.g. [Åsröm and Wienmark, 997]. 3. Conrol Sysem Descripion Each physical process (or plan) o be conrolled is assumed o be described by a se of linear differenial equaions, which give he relaionship beween he inpu (he conrol signal) and he oupu (he measuremen signal). A sandard mahemaical represenaion of such a sysem is dx() = Ax()+Bu()+Gw() d y()=cx()+v() () Here, x is a vecor of sae variables (he sae of he plan), u is he conrol signal, y is he measuremen signal, and w and v are uncorrelaed zero-mean Gaussian whie noises which disurb he saes and he measuremens. A, B, C, and G are consan marices ha describe he dynamics of he plan. The goal of he conroller is o keep he plan sae as close as possible o zero (which is assumed o be he desired sae) in he presence of noise and oher disurbances. As he plan exiss in he real world, i is described in coninuous ime. On he oher side, he conroller is implemened in a compuer and is herefore inherenly a discree-ime sysem. A general linear conroller can be described by a se of linear difference equaions, x c k+ = Φc x c k + Γc y k u k = C c x c k + Dc y k (2) u k u( ) u( ) Hold u k D-A Process Compuer y( ) A-D y( ) Sampler y k Figure Relaionships among he coninuous and he discree signals in he conrol loop. Here, x c is he sae vecor of he conroller, and Φ c, Γ c, C c, and D c are marices which describe he dynamics of he conroller. Depending on how he conroller is designed, hese marices may or may no be funcions of he sampling period. In he inerconnecion beween he coninuous-ime and he discree-ime sysem, y is sampled and u is held a a consan value unil he nex updae, see Figure. The conroller could be designed using a large number of echniques, for insance loop shaping, pole placemen, or using opimal conrol heory. There is also he choice beween direc discree design or ranslaion from coninuous design o discree implemenaion. The linear conroller descripion (2) covers many common conrol principles, including PID conrol, sae feedback conrol from an observer, lead-lag filering, ec. 3.2 Evaluaing Conrol Performance A good conrol design should fulfill a large number of crieria, including fas disurbance rejecion, good robusness owards plan variaions, low noise sensiiviy, ec. In his work, however, we focus on sampling period selecion, and we choose a simple performance crierion ha can be used o direcly compare he performance of he conrol loops under differen sampling periods. The performance of a conroller is measured using a quadraic cos crierion, { T J = lim T T E ( x T ()Q x()+u T ()Q 2 u() ) } d v,w 0 (3) The cos J can be inerpreed as a weighed sum of he saionary variance of he plan sae and he conrol signal. The consan marices Q and Q 2 are weighs ha are chosen for each plan. A large value of J indicaes large deviaions from he desired sae or large conrol signals, and is hus worse han a lower value of J. An infinie cos implies ha he conrol sysem is unsable. y k

5 J (a) h J h Figure 2 Cos funcions for wo differen plans, (a) Invered pendulum, G(s) =,(b)dc servo, G(s) = s 2 s(s+). Evaluaion of he cos (3) for a given plan, a given conroller, and a given sampling period h is a sraigh-forward ask ha involves hree compuaional seps. Firs, he plan descripion and he cos funcion are discreized. Second, he closed-loop sysem is formulaed and he saionary variance of he saes in he sysem is compued. Third, he cos J is obained as a linear weighing of he sae variance. More deails abou he sampling of a plan and a cos funcion can be found in [Åsröm and Wienmark, 997]. 3.3 LQG-Conrol In he examples in his paper we use LQG (linearquadraic-gaussian) conrollers. An LQG-conroller is an opimal conroller, explicily designed o minimize he crierion (3). The opimal conrol parameers depend on he sampling period, and hey are obained using direc discree design. Sandard conrol design sofware (e.g. MATLAB) is used o compue he LQG-conroller parameers. 3.4 Cos Funcions A conroller can normally give saisfacory performance wihin a range of sampling periods. Compuing he cos J for such a range we obain a cos funcion. The cos funcion J(h) will have differen shapes for differen plans and differen conrollers, bu in general i is an increasing funcion. In Figure 2, LQG-cos funcions have been calculaed for wo differen plans, an invered pendulum and a DC servo. For small sampling periods, he cos increases linearly in boh cases. I should be poined ou ha cos funcions are no necessarily increasing funcions. If he conroller has fixed parameers and is designed for a nominal sampling period, he conroller migh perform worse wih boh shorer and longer sampling periods. Cos funcions are no necessarily convex funcions eiher. An example of a very irregular cos funcion is given in [Eker e al., 2000]. (b) 3.5 Performance Opimizaion The feedback scheduler should conrol he workload of he processor by adjusing he sampling periods of he conrollers. A he same ime, i should opimize he overall conrol performance. This is formulaed as an opimizaion problem. Given n conrol asks wih average execuion imes C = [C... C n ] T and sampling periods h =[h... h n ] T, he feedback scheduler should solve he problem subjec o min h J = n J i (h i ) i= (4) n C i /h i U sp i= where U sp is he desired processor uilizaion level. This problem has nonlinear consrains. To ge linear consrains, he coss are recas as funcions of he sampling frequencies, f =[f... f n ] T, The problem is now wrien V i (f)=j i (/h) (5) min f V = n V i ( f i ) i= (6) subjec o C T f U sp If he funcions V(f) =[V (f )... V n (f n )] T are decreasing and convex, he opimal soluion f = [ f... fn ] T fulfills he Kuhn-Tucker condiions V f ( f)+λc=0 λ(u sp C T f)=0 λ 0 (7) where V f is he gradien and λ is he Lagrange muliplier. Solving he opimizaion problem exacly can be very ime-consuming, especially if he cos funcions V i ( f ) are non-convex. Jus evaluaing a cos funcion in a single poin involves a large amoun of compuaions. If he resource allocaion problem is o be solved by an on-line opimizer, he cos funcions for he plans mus be compued off-line and hen approximaed by simpler funcions. A quadraic approximaion was suggesed in [Eker e al., 2000]. Here, we also presen a linear approximaion. The soluion o he simplified opimizaion problem can in boh cases be inerpreed as a simple linear rescaling of a se of nominal sampling periods.

6 Quadraic Approximaion Assume ha he cos funcions can be approximaed by or, equivalenly, J i (h) =α i +β i h 2 (8) V i ( f )=α i +β i (/f) 2 (9) Applying he Kuhn-Tucker condiions (7) yields he explici soluion ( β ) i /3 U sp f i = C n (0) i j= C2/3 j β /3 j Noice ha he consans α i can be disregarded, i.e., i is sufficien o esimae he curvaure of he cos funcions. Linear Approximaion Assume ha he cos funcions can be approximaed by or, equivalenly, J i (h)=α i +γ i h () V i ( f )=α i +γ i /f (2) This ofen seems o be a beer approximaion han (9), especially for open-loop sable plans, or any plans sampled reasonably fas. Applying he Kuhn- Tucker condiions (7) yields he explici soluion ( γ ) /2 i U f i = sp C n i j= (C (3) jγ j ) /2 Noice ha he consans α i can be disregarded, i.e., i is sufficien o esimae he slope of he cos funcions. Inerpreaion as Simple Rescaling Boh he quadraic and he linear cos funcion approximaions yield quie simple, explici formulas for opimal ask frequency assignmen, ha could be used on-line in a feedback scheduler. If a new ask arrives, or if he execuion ime of a conroller suddenly changes, new sampling periods could be calculaed using (0) or (3). However, no even ha amoun of calculaions is really needed. In boh cases, i can be noed ha each ask receives a share of he CPU ha is proporional o a ask consan. In he quadraic case, he proporionaliy consan is (β i /C i ) /3,and in he linear case i is (γ i /C i ) /2. The raios beween he opimal sampling periods are hus consan and do no depend on he available resources or he number of curren asks in he sysems. This implies ha, if he nomial sampling periods have been chosen wisely, opimal feedback scheduling can be performed by simple rescaling of he ask periods. This is formulaed in he following heorem: THEOREM If he cos funcions of he curren asks in he sysem can be described by eiher a) quadraic funcions of he sampling period, Eq. (8), orbyb)linear funcions of he sampling period, Eq. (), and if nominal sampling frequencies f 0 = [ f 0... f 0n ] are chosen in proporion o a) (β i /C i ) /3 or b) (γ i /C i ) /2, hen simple rescaling of f 0 o mee he uilizaion consrain is opimal wih respec o he oal conrol performance J. Proof: Follows from he proporionaliy argumen above. Addiional Consrains I is possible o add more consrains o he approximae opimizaion problem and sill reain a simple soluion. Firs, one can le he nominal sampling periods f 0 be minimal sampling periods. If C T f 0 U sp,hen he nominal periods are used, oherwise hey are rescaled. This consrain prevens he CPU from being fully loaded when i is no necessary from a conrol performance poin of view. Second, one can impose maximum sampling periods o some asks. This leads o an ieraive soluion (linear programming), where he remaining asks are rescaled unil all consrains are me. 4. A Feedback-Feedforward Scheduling Archiecure To implemen he feedback scheduling principle given in he previous secion, a feedback-feedforward scheduling archiecure is developed. I is assumed ha he conrol asks can swich beween differen modes, wih possibly very differen execuion-ime demands. Hybrid conrollers ha swich beween differen conrol algorihms depending on he sae of he process, exernal signals, ec., are a large class of conrollers ha exhibi such behavior. The execuion imes of he algorihms in he differen modes are no known exacly, bu mus be esimaed. The cos funcions may also change beween he differen modes. The srucure of he feedback scheduler is shown in Figure 3. A se of conrol asks generae jobs ha are fed o a run-ime dispacher. The scheduler receives feedback informaion abou he acual execuion ime, c i, of he jobs. I also receives feedforward informaion from conrol asks when hey swich mode. In his way, he scheduler can pro-ac raher han reac o sudden changes in he workload. Also, he scheduler may change he parameers in he opimizaion rouine according o he curren modes of he conrollers. The scheduler aemps o keep he CPU uilizaion, U, as close as possible o a uiliza-

7 U sp Feedback Scheduler mode changes h Tasks jobs Dispacher c i, U of is curren period and adjus is parameers accordingly. On-line recalculaions are ofen oo cosly, so parameers for a range of sampling periods mus be calculaed off-line and sored in a able. Figure 3 Block diagram of he feedback-feedforward scheduling srucure. ion se-poin, U sp. This is done by manipulaing he sampling periods, h. 4. Design Consideraions The feedback scheduler can be implemened in one of wo main ways. The firs opion is o design i as an inheren mechanism, alongside he ask dispacher, in he real-ime operaing sysem. The oher way, which we discuss here, is o implemen i as an applicaion ask, execuing in parallel wih he conrol asks. The advanage of his approach is ha i could be implemened on op of exising real-ime operaing sysems. A number of oher addiional design consideraions exis: The uilizaion se-poin, U sp, mus be chosen. The choice will depend on he scheduling policy of he dispacher and on he sensiiviy of he conrollers o missed deadlines. A oo low se-poin will give low resource uilizaion and poor conrol performance. A oo high se-poin, on he oher hand, may cause asks o suffer from emporary overruns. Noice ha he well-known, guaraneed uilizaion bounds of 00 % for deadline-driven scheduling and 69 % for prioriy-driven scheduling [Liu and Layland, 973] are no valid in his conex, since he assumpions abou known, fixed WCETs and fixed periods are violaed. Also, recen research, e.g. [Palopoli e al., 2000], showsha operaion close o or even above he level U = may give very good conrol performance in some cases, despie a large number of missed deadlines. The feedback scheduler will execue as a periodic ask in he sysem, and is period, h FBS, mus be chosen. A shor period will give good conrol of he uilizaion bu also consume much of he available resources. A longer period will consume less resources, bu make he scheduler respond slower o load disurbances. If he conroller parameers depend on he sampling period, he conroller mus be aware 4.2 Conrolling he Uilizaion The feedback scheduler conrols he processor uilizaion by assigning ask periods ha opimize he overall conrol performance. However, his requires ha esimaes of he execuion imes of he asks, Ĉ i, are available. I is assumed ha he real-ime operaing sysem can monior he execuion-ime of individual jobs. An esimae Ĉi is obained from filered job execuion-ime measuremens, Ĉ i (k) =λĉi(k )+( λ)c i (4) where λ is a forgeing facor. Seing λ closeo resuls in a smooh, bu slow esimae. A λ close o 0 gives a faser esimae, bu i will also capure more of he high-frequency execuion-ime noise. This is a ypical rade-off in conrol design. As deailed in Secion 3.5, near-opimal ask periods can be calculaed by rescaling of a se of nominal sampling periods. Firs, he esimaed nominal requesed uilizaion is compued by Û 0 = n i= Ĉ i h 0i (5) Here, h 0i can be a funcion of he mode of he conroller, if he differen modes should have differen cos funcions. Second, new ask periods are calculaed by he rescaling h i = h 0i U sp Û 0 (6) Addiional consrains (minimum and maximum sampling periods) can also be deal wih, see Secion Feedforward Informaion The role of he feedforward informaion from he conrollers o he feedback scheduler is hree-fold. Firs, he scheduler can reac o sudden changes in he workload by execuing an exra ime in connecion wih a mode change. The conroller, which ypically execues more frequenly han he feedback scheduler, is responsible for signaling he scheduler as soon as a mode swich condiion has been deeced. If he mode change is likely o increase he workload, and if he swiching ime iself is no criical, he conroller could delay he swich one or several sampling periods while he scheduler recalculaes he periods.

8 Such a non-criical mode swich could for insance be he resul of an operaor enering a new sepoin for he conroller. Second, he scheduler needs o keep rack of he modes of he conrollers in order o compue opimal ask periods according o he cos funcions. The conroller can communicae he new nominal sampling period a he mode swich, or he scheduler can keep a able of he nominal sampling periods for all he conroller modes. Third, he feedforward informaion allows he scheduler o run separae execuion-ime esimaors in he differen modes. The forgeing facor λ can hen be chosen according o he execuion-ime variabiliy wihin each mode. A a mode change, he scheduler can immediaely swich o he curren esimae in he new mode. This furher improves he regulaion of he uilizaion a he mode changes. 5. A Feedback Scheduling Example As an example of feedback scheduling, we sudy he problem of simulaneously sabilizing four invered pendulums. The pendulum is a common benchmark process because of is open-loop insabiliy, which makes i very sensiive o disurbances, including iming fauls. Differen scheduling approaches are evaluaed by co-simulaion of he scheduler, he conrol asks, and he pendulums. By simulaing he execuion of he asks, he effecs of conrol delay and jier (due o varying execuion imes and scheduling) on he conrol performance are also capured in he resuls. 5. Plans and Conrollers Each pendulum is described by he sysem (see Secion 3.) [ ] [ ] dx() 0 0 = x()+ u()+v() d ω ω 2 0 (7) y()=[ 0]x()+e() where ω 0 is he naural frequency of he pendulum, and he noise variances are given as Ev 2 = /ω 0 and Ee 2 = 0 4. The four pendulums have differen lenghs, which correspond o differen frequencies: ω 0 =[ ]. LQG-conrollers for he plans are calculaed according o he design weighs [ 0 Q = 0 0 ], Q 2 = (8) This corresponds o minimizaion of he cos crierion { } T J = lim T T E ( y 2 ()+u 2 ())d (9) 0 J h Figure 4 The cos funcions for he four invered pendulums, () (4). The circles indicae he nominal sampling periods. i.e., he conroller should aemp o minimize he sum of he variance of he oupu signal and he conrol signal. The coss funcions for he four pendulums shown in Figure 4. I is seen ha he cos funcions can be reasonably well approximaed by linear funcions (see Secion 3.5), The esimaed slopes are (4) (3) (2) () J i (h)=α i +γ i h. (20) γ =[ ]. (2) I can be noed ha, he higher he naural frequency of he pendulum is, he more sensiive he conroller is owards an increase in he sampling period. This is quie inuiive. Nominal sampling periods are chosen according o he soluion of he performance opimizaion problem and according o he rule of humb [Åsröm and Wienmark, 997] ha saes ha he sampling period should be chosen such ha 0.2 < ω 0 h < 0.6. The resuling nominal periods are h 0 =[ ] ms (22) These periods have been indicaed in Figure 4. The opimal coss associaed wih hese sampling periods are J 0 =[ ] (23) These are he expeced coss if he conrollers could really execue a heir nominal sampling periods, wih zero delay and zero jier. Implemened in a compuer, he conrollers will suffer from various amouns of sampling jier, inpu-oupu delay, and oupu jier, and he resuling cos will be higher.

9 p(c i ) c i [ms] Figure 5 Probabiliy disribuion of he conrol ask execuion ime. To allow for fas changes beween differen sampling periods during run-ime, he conroller parameers Φ c, Γ c, C c,andd c are calculaed off-line for a range of differen sampling periods for each pendulum conroller and sored in a able. 5.2 The Experimens Each of he conrollers has wo differen modes: on and off. When on, he average ask execuion ime is C = 5.5 ms. The conroller code consiss of hree pars. Firs, A-D conversion is requesed, i.e. he oupu of he plan is sampled. Second, a new conrol signal is compued according o he conrol algorihm. Third, D-A conversion is requesed, i.e. he new conrol signal is sen ou o he plan. The oal execuion ime of he ask is assumed o vary according o he probabiliy disribuion shown in Figure 5. When he conroller is urned off, he execuion ime is zero. A he sar of he experimen, = 0, Conrollers and 2 are on, while Conrollers 3 and 4 are off. A = 2, Conroller 3 swiches on, and a = 4, Conroller 4 also swiches on. The four conrollers run in parallel unil = 6. I is iniially assumed ha he feedback scheduler and he asks are implemened in a prioriypreempive real-ime sysem. The feedback scheduler is given he highes prioriy while he conrol asks are assigned rae-monoonic prioriies. The reason for sudying he prioriy-preempive seing is clariy under rae-monoonic scheduling i is easy o predic he effecs of overloads. In his case, Conroller will be given he lowes prioriy and will hus suffer he mos during an overload. I is assumed ha he execuion ime of he feedback scheduling ask is C FBS = 2 ms. Is period is chosen as h FBS = 200 ms and he uilizaion sepoin is chosen as U sp = 0.85 o yield good conrol performance and no oo many missed deadlines. The execuion-ime esimaion forgeing facor is chosen as λ = This gives smooh esimaes bu will cause he scheduler o reac slowly o mode swiches if he feedforward acion is no used. The experimen is repeaed for differen scheduling approaches. Firs, open-loop scheduling is aemped. Then, feedback scheduling wihou he feedforward mechanism is ried. Then, feedbackfeedforward scheduling is sudied. In he end, openloop EDF scheduling is also sudied. I is imporan o specify he behavior of he periodic asks in he case of missed deadlines. No maer when a ask finishes, he nex release ime is se o be he curren release ime plus he assigned period of he ask. Thus, a ask which has missed many deadlines may have a release ime which is far back in ime compared o he acual saring ime of he ask. This kind of implemenaion of periodic asks penalizes especially he low-prioriy asks in he case of overload under fixed-prioriy dispaching. To measure he performance of a conroller, he accumulaed cos is recorded, J i () = 0 (y 2 (τ)+u 2 (τ))dτ. (24) Compared o he crierion in Eq. (9), his quaniy is no rescaled by he ime horizon. If he pendulum falls down, he accumulaed cos is se o infiniy. The pendulums are subjeced o idenical sequences of process noise and measuremen noise in all simulaions. The execuion imes also consis of idenical random sequences in all cases. During each experimen, he schedule, i.e. he execuion race, is also recorded, ogeher wih an esimae of he curren requesed uilizaion. This quaniy is compued as n c i U req =, (25) h i where c i is he execuion ime of he laes invocaion of ask i and h i is he sampling period currenly assignedoaski. Proper regulaion of he uilizaion should keep his quaniy approximaely equal o or less han U sp. 5.3 The Simulaion Environmen The complee se-up, including he scheduler, he asks, and he pendulums are simulaed using he Malab/Simulink-based ool described in [Eker and Cervin, 999]. The op level view of he simulaion model is shown in Figure 6. Here, he performance of he pendulums can be sudied. Opening he Compuer block, deails abou he execuion may be sudied, see Figure 7. The Real-Time Kernel block simulaes a ick-based, preempive kernel wih an arbirary, user-defined i=

10 y u y2 u2 y3 u3 y4 u4 Compuer y u J Pendulum y u J Pendulum 2 y u J Pendulum 3 y u J Pendulum 4 J J2 J3 J4 A = 2, Task 3 sars o execue. Togeher wih Task 2, he wo asks consume C/h 0 + C/h 02 = 0.85 CPU, which leaves only 0.5 CPU o Task, which is he lowes-prioriy ask. The resuling average period is h = C/0.5 = 37 ms, which is acually sufficien o sabilize he pendulum, alhough he cos increases more rapidly. A = 4, Task 4 is urned on. Togeher wih Task 3, he wo asks reques C/h 03 + C/h 04 =.0 CPU, and his blocks Tasks and 2 compleely. The resul is ha boh pendulums fall down. Figure 6 Top-level of he simulaion model of he feedback scheduling sysem. y 2 y2 3 y3 4 y4 Real Time Kernel m u 2 u2 3 u3 4 u4 Req Uilizaion Schedule Figure 7 Opening up he Compuer block in Figure 6, deails abou he execuion may be sudied. dispach policy. In he experimens, he ick-size is se o 0.5 ms. The conrollers and he feedback scheduler are implemened as funcions ha he kernel execues repeaedly during a simulaion. The scheduler and he conrollers communicae informaion using message-passing primiives in he simulaed kernel. 5.4 Simulaion Resuls The simulaion resuls in he four differen scheduling cases are presened and discussed below. Open-Loop Scheduling Under open-loop scheduling, he conrollers aemp o execue a heir nominal sampling periods. The accumulaed coss for he pendulums are shown in Figure 8, he requesed processor uilizaion is shown in Figure 9, and a close-up of schedule a = 4 is shown in Figure 0. From ime = 0, Tasks and 2 consume (in average) C/h 0 + C/h 02 = 0.72 CPU and conrol performance is good. The accumulaed coss are expeced o increase by abou 3 per second according o he opimal coss in (23), and his holds in pracice as well. Feedback Scheduling Under feedback scheduling, he asks sars o execue a heir nominal sampling periods when urned on. The feedback scheduler hen adjuss he periods every 200 ms. The accumulaed coss for he pendulums are shown in Figure, he requesed processor uilizaion is shown in Figure 2, and a close-up of schedule a = 4 is shown in Figure 3. A = 2, Task 3 is urned on and he requesed uilizaion raises above he uilizaion sepoin of A = 2., he feedback scheduler adjuss he periods, and he overload condiion is evenually removed. The same hing is repeaed a = 4when Task 4 is urned on. The ransien overload causes he performance of Conroller o degrade, bu he siuaion is soon sabilized. The slow response is due o he forgeing facor λ = A lower value would correc he overload siuaion faser, bu i would also increase he variance of he uilizaion in saionariy. Feedback-Feedforward Scheduling Under feedback-feedforward scheduling, asks ha swich modes also immediaely acivae he feedback scheduler. The accumulaed coss for he pendulums are shown in Figure 4, he requesed processor uilizaion is shown in Figure 5, and a close-up of schedule a = 4 is shown in Figure 6. The resuls are similar o hose under feedback scheduling, excep a he mode changes. Here, he overloads are avoided hanks o immediae rescaling of he ask periods a = 2 and = 4. The ransiens are avoided, and he performance of all he conrollers is good hroughou. Open-Loop EDF Scheduling Under openloop EDF scheduling, all asks aemp o execue a heir nominal periods. The accumulaed coss for he pendulums are shown in Figure 7 and a closeup of schedule a = 4 is shown in Figure 8. The requesed processor uilizaion is idenical o he one under open-loop rae-monoonic scheduling, shown in Figure 9.

11 Sraegy J J 2 J 3 J 4 Ji Ideal (compued expeced value) Open-loop scheduling 3 5 Feedback scheduling Feedback-feedforward scheduling Open-loop EDF scheduling Table Final accumulaed coss for he four pendulums under differen scheduling sraegies. The symbol indicaes ha he pendulum has fallen down. Alhough he sysem is overloaded from = 2, he performance of Conrollers 3 is good hroughou. ThereasonishaordinaryEDFschedulingacsas a naural period rescaling mechanism in overload siuaions. This propery is discussed furher in Secion 6. Task 4 experiences some problems, however. When i is released a = 4, he sysem has been overloaded for wo seconds. This means ha he absolue deadlines of Tasks 3 lie somewhere backwards in ime, and hey will have prioriy over Task 4, which iniially has he absolue deadline 4 + h 04 = The resul is ha Task 4 is blocked unil around = 4.35, before which he pendulum has fallen down. Summary of Resuls and Discussion The final accumulaed coss for he four pendulums in he differen cases are summarized in Table. Feedback-feedforward scheduling gives he bes overall conrol performance, alhough open-loop EDF scheduling does a quie good job a scheduling Tasks 3. The reason ha he conrol performance is slighly beer under feedback-feedforward scheduling han under open-loop EDF scheduling is ha, in he EDF case, he conrollers do no adjus heir parameers according o he acual sampling periods. Also indicaed in he able are he ideal expeced accumulaed coss of he conrollers. They are rivially compued from knowing he opimal coss (23) and he duraion ha each conroller is running. The rue expeced cos, in saionariy, of he conroller in he real-ime sysem can also, wih some resricions, be numerically calculaed using analysis echniques from he heory of conrol sysems wih random delays [Nilsson, 998]. The ransien behavior of he conrollers a he mode changes is much harder o analyze, and here we mus rely on simulaions. 6. EDF as a Feedback Scheduling Mechanism? As seen in he simulaions in he previous secion, he performance of he conrollers under open-loop EDF scheduling was quie good, despie he sysem being permanenly overloaded and all deadlines being missed. This can be explained by he following heorem, which o he bes of he auhors knowledge has no been formulaed before: THEOREM 2 Assume a se of n periodic asks, where each ask i is described by a fixed period, T i, a fixed execuion ime, C i, a relaive deadline, D i, and a release offse, O i.thejobsofheasksarescheduledaccording o heir absolue deadlines (i.e., EDF scheduling). If U = n j= C j /T j >, hen he average acual resuling ask period of ask i in saionariy, T i, will be given by T i = T i U. Proof: See appendix A. Theorem and Theorem 2 aken ogeher gives he following: COROLLARY An ordinary EDF scheduler can be inerpreed, in saionariy, as an opimal feedback scheduler for conrol asks ha have cos funcions ha can be described by quadraic or linear funcions of he sampling period. However, his resul ress upon several assumpions, and here are pifalls. Firs, i is assumed ha he conroller samples he plan when i sars o execue, and no when i is released. This way, he conrol delay will be bounded, even hough he ask laency (he finish ime minus he release ime) migh approach infiniy. Second, i is assumed ha jier has only negligible impac on he conrol performance. This may no be rue during a permanen overload siuaion where he asks sar o execue non-preempively. While he average period of a ask is given by Theorem 2, he jier may be unbounded because of he non-preempive execuion paern. In he feedback scheduling example in he previous secion his was no a problem, since all he asks had execuion imes and periods of he same magniude. As seen in he example, problems may occur when asks swich mode (and his is when feedback scheduling is really needed). Since asks are scheduled using old deadlines, i will ake ime for a resource redisribuion o have effec. One soluion would be o rese he release ime of all asks o he curren ime immediaely following a mode change.

12 Anoher problem wih he open-loop EDF approach is ha he period informaion is no communicaed o he conrollers. Thus, hey canno use he correc conrol parameers, and his degrades he performance o some degree. This could be correced by leing he conrol asks measure heir own acual periods. 7. Conclusions A scheduler archiecure has been proposed ha combines feedback and feedforward acion in order o opimize conrol performance while mainaining high resource uilizaion. The feedback par relaxes he requiremen on known execuion-ime bounds for muliasking conrol sysems. The feedforward par allows for rapid adapaion o changing load condiions. The conrol performance, or Qualiy-of-Conrol (QoC), is considered as a qualiy-of-service measure ha should be maximized. Opimal adjusmen sraegies for he conroller ask periods have been derived for he cases when he cos funcion is a quadraic funcion of he sampling period and when i is a linear funcion of he sampling period. The adjusmen sraegy uses linear rescaling, making i compuaionally efficien, and hence, possible o use on-line. The differen sraegies have been evaluaed on a realisic simulaion example. The proposed approach gives subsanially beer resuls han wha is achieved using classical open-loop scheduling mehods. A new resul for periodic asks wih EDF scheduling under overload condiions makes i possible o, in cerain siuaions, o inerpre a plain EDF dispacher as a feedback scheduler for conrol asks. Acknowledgmens This work has been funded by ARTES (he Swedish real-ime sysems research iniiaive), by LUCAS (he Lund Universiy cener for applied sofware research), and by TFR. References Abdelzaher, T., E. Akins, and K. Shin QoS negoiaion in real-ime sysems, and is applicaion o fligh conrol. In Proceedings of he 3rd IEEE Real-Time Technology and Applicaions Symposium, pp Abdelzaher, T. and K. Shin QoS provisioning wih qconracs in web and mulimedia servers. In Proceedings of he 20h IEEE Real-Time Sysems Symposium, pp Abeni, L. and G. Buazzo Adapive bandwidh reservaion for mulimedia compuing. In Proceedings of he 6h Inernaional Conference on Real-Time Compuing Sysems and Applicaions, pp Årzén, K.-E., B. Bernhardsson, J. Eker, A. Cervin, K. Nilsson, P. Persson, and L. Sha Inegraed conrol and scheduling. Technical Repor TFRT- 7586, Deparmen of Auomaic Conrol, Lund Insiue of Technology, Lund, Sweden. Åsröm, K. J. and B. Wienmark Compuer- Conrolled Sysems, 3rdedn.PreniceHall. Blevins, P. and C. Ramamoorhy Aspecs of a dynamically adapive operaing sysem. IEEE Transacions on Compuers 25(7): Buazzo, G., G. Lipari, and L. Abeni Elasic ask model for adapive rae conrol. In Proceedings of he 9h IEEE Real-Time Sysems Symposium, pp Cervin, A. and J. Eker Feedback scheduling of conol asks. In Proceedings of he 39h IEEE Conference on Decision and Conrol, pp Eker, J. and A. Cervin A Malab oolbox for real-ime and conrol sysems co-design. In Proceedings of he 6h Inernaional Conference on Real-Time Compuing Sysems and Applicaions, pp Eker, J., P. Hagander, and K. Erik Årzén A feedback scheduler for real-ime conrol asks. Conrol Engineering Pracice 8(2): Kleinrock, L A coninuum of ime-sharing scheduling algorihms. In AFIPS Conference Proceedings, Spring Join Compuer Conference, pp Kosugi, N., K. Takashio, and M. Tokoro Modificaion and adjusmen of real-ime asks wih rae monoonic scheduling algorihm. In Proceedings of he 2nd Workshop on Parallel and Disribued Sysems, pp Kuo, T.-W. and A. Mok. 99. Load adjusmen in adapive real-ime sysems. In Proceedings of he 2h IEEE Real-Time Sysems Symposium, pp Lee, C., R. Rajkumar, and C. Mercer Experiences wih processor reservaion and dynamic QoS in Real-Time Mach. In Proceedings of Mulimedia Japan 96. Li, B. and K. Nahrsed A conrol heoreical model for qualiy of service adapaions. In Proceedings of 6h Inernaional Workshop on Qualiy of Service, pp

13 Liu, C. L. and J. W. Layland Scheduling algorihms for muliprogramming in a hard-realime environmen. Journal of he ACM 20(): Lu, C., J. Sankovic, T. Abdelzaher, G. Tao, S. Son, and M. Marley Performance specificaions and merics for adapive real-ime sysems. In Proceedings of he 2s IEEE Real-Time Sysems Symposium, pp Lu, C., J. Sankovic, G. Tao, and S. H. Son Design and evaluaion of a feedback conrol EDF scheduling algorihm. In Proceedings of he 20h IEEE Real-Time Sysems Symposium, pp Nakajima, T Resource reservaion for adapive QoS mapping in Real-Time Mach. In Proceedings of he 6h Inernaional Workshop on Parallel and Disribued Real-Time Sysems, pp Nakajima, T. and H. Tezuka A coninuous media applicaion supporing dynamic QoS conrol on Real-Time Mach. In Proceedings of ACM Mulimedia 94, pp Nilsson, J Real-ime conrol sysems wih delays. Ph.D. Thesis TFRT-049, Deparmen of Auomaic Conrol, Lund Insiue of Technology, Lund, Sweden. Palopoli, L., L. Abeni, and G. Buazzo Realime conrol sysem analysis: An inegraed approach. In Proceedings of he 2s IEEE Real- Time Sysems Symposium, pp Poier, D., E. Gelenbe, and J. Lenfan Adapive allocaion of cenral processing uni quana. Journal of he ACM 23(): Rajkumar, R., C. Lee, J. Lehoczky, and D. Siewiorek A resources allocaion model for QoS managemen. In Proceedings of he 8h IEEE Real- Time Sysems Symposium, pp Ryu, M., S. Hong, and M. Saksena Sreamlining real-ime conroller design: From performance specificaions o end-o-end iming consrains. In Proceedings of he 3rd IEEE Real- Time Technology and Applicaions Symposium, pp Seo, D., J. P. Lehoczky, L. Sha, and K. G. Shin On ask schedulabiliy in real-ime conrol sysems. In Proceedings of he 7h IEEE Real- Time Sysems Symposium, Washingon, DC, pp Shin, K. and C. Meissner Adapaion of conrol sysem performance by ask reallocaion and period modificaion. In Proceedings of he h Euromicro Conference on Real-Time Sysems, pp Sankovic, J. A., C. Lu, S. H. Son, and G. Tao The case for feedback conrol real-ime scheduling. In Proceedings of he h Euromicro Conference on Real-Time Sysems, pp. 20. A. Proof of Theorem 2 The release ime of job number k of ask i is kt i + O i. The deadline of job number k of ask i is kt i +O i +D i. Le k i () be he number of finished jobs of ask i a ime. Afer a while, CPU is never idle, herefore, I idle ()+ i (k i ()C i +e i ()) = (26) where I idle () is he accumulaed idle ime, e i is how long he curren invocaion of ask i has execued. Boh I idle () and e i are bounded and 0 e i C i. Furhermore, due o he overload siuaion, asks are finished in he order of heir deadlines. Therefore, k l ()T l + D l + O l (k i ()+)T i +D i +O i (27) (Oherwise ask l would no have finished job number k l () before ask i finished job number k i ()+). Symmerically, k i ()T i + D i + O i (k l ()+)T l +D l +O l (28) The wo equaions above give Hence, k l ()T l + D l + O l T i k i ()T i + D i + O i (k l ()+)T l +D l +O l (29) k l ()T l + D l T i + O l k l ()T l k i()t i + D i + O i k l ()T l (k l()+)t l +D l +O l k l ()T l (30) Here, he limi of he lef-hand side and he righhand side are boh equal o one, so, k j ()T j lim = (3) k l ()T l Rearranging he erms in Eq. (26) and leing,

14 we have = lim = lim Hence, = lim k i ()C i i i j k j()t j = lim k i ()T i k i ()T i C i T i i k j()t j j k i()t i i k i ()T i n C i = lim n T i i = lim k i ()T i i C i T i k i ()T i C i j k j()t j T i k j()t j j k i()t i C i T i (32) Accumulaed Cos Accumulaed coss under open-loop schedul- Figure 8 ing. J J2 J3 J4 T i = lim k i () = T C j i = T i U (33) T j j.8.6 Ureq (full) and Usp (dashed) Figure 9 Requesed uilizaion under open-loop scheduling. Schedule (high=running, medium=preemped, low=sleeping) FBS Task 4 Task 3 Task 2 Task Figure 0 Close-up of schedule a = 4 under openloop scheduling.

15 Accumulaed Cos J J2 J3 Accumulaed Cos J J2 J3 0 0 J4 J Accumulaed coss under feedback schedul- Figure ing. Figure 4 Accumulaed coss under feedbackfeedforward scheduling Ureq (full) and Usp (dashed) Ureq (full) and Usp (dashed) Figure 2 Requesed uilizaion under feedback scheduling Figure 5 Requesed uilizaion under feedbackfeedforward scheduling. Schedule (high=running, medium=preemped, low=sleeping) Schedule (high=running, medium=preemped, low=sleeping) FBS FBS Task 4 Task 4 Task 3 Task 3 Task 2 Task 2 Task Task Figure 3 scheduling. Close-up of schedule a = 4 under feedback Figure 6 Close-up of schedule a = 4 under feedbackfeedforward scheduling.

16 50 40 Accumulaed Cos J4 J2 J J Figure 7 Accumulaed coss under open-loop EDF scheduling. Schedule (high=running, medium=preemped, low=sleeping) FBS Task 4 Task 3 Task 2 Task Figure 8 Close-up of schedule a = 4 under openloop EDF scheduling.