On SelfT riggered ullInformation Hinnit Con trollers Mic hael Lemmon Thidapat Chan tem Xiaob Sharon Hu and Matthew Zysk wski Univ ersit of Notre Dame Departmen of Electrical Engineering Notre Dame IN

Presentation on theme: "On SelfT riggered ullInformation Hinnit Con trollers Mic hael Lemmon Thidapat Chan tem Xiaob Sharon Hu and Matthew Zysk wski Univ ersit of Notre Dame Departmen of Electrical Engineering Notre Dame IN"— Presentation transcript:

Page 1
On Self-T riggered ull-Information H-innit Con trollers Mic hael Lemmon Thidapat Chan tem Xiaob Sharon Hu and Matthew Zysk wski Univ ersit of Notre Dame, Departmen of Electrical Engineering, Notre Dame, IN 46556, USA. lemmon,mzyskows@nd.edu Univ ersit of Notre Dame, Departmen of Computer Science and Engineering, Notre Dame, IN 46556, USA. Sharon.Hu.8,tchantem@nd.edu Abstract. self-triggered con trol task is one in whic the task deter- mines its next release time. It has een conjectured that self-triggering can relax the requiremen ts on real-time sc heduler while main

taining application (i.e. con trol system) erformance. This pap er presen ts pre- liminary results supp orting that conjecture for self-triggered real-time system implemen ting full-information con trollers. Release times are selected to enforce upp er ounds on the induced gain of linear feed- bac con trol system. These release times are treated as requests the system sc heduler, whic then assigns actual release times using But- tazzo’s elastic sc heduling algorithm. Preliminary exp erimen tal results from Matlab stateo sim ulink mo del demonstrated remark able ro- bustness to sc

heduling dela ys induced real-time sc hedulers. These results sho that self-triggered con trollers are indeed able to main tain acceptable lev els of application erformance during prolonged erio ds of pro cessor erloading. In tro duction Computer con trolled systems are often implemen ted using erio dic real-time tasks. This approac can lead to signican er-pro visioning of the real-time system since task erio ds are determined the orst case time in terv al assur- ing closed lo op system stabilit In recen ears, um er of researc hers ha prop osed ap erio dic task mo dels in whic tasks

are either event-trigger [1] or self-trigger [2] con trollers. Ev en t-triggered con trol systems are systems whose con trol tasks are triggered some async hronous \ev en t" within the con trol lo op. These ev en ts are usually generated when an error signal crosses sp eci- ed threshold. The notion of ev en t-triggered feedbac [1] has app eared under ariet of names, suc as in terrupt-based feedbac [3], Leb esgue sampling [4], async hronous sampling [5], or state-triggered feedbac [6]. Except for rela or pulse-width mo dulated feedbac k, ev en t-triggered feedbac can impractical since

it requires in tegrating an analog ev en detector in to the The authors gratefully ac kno wledge the partial nanical supp ort of the National Science oundation (NSF-CNS-0410771)
Page 2
ph ysical plan t. more pragmatic approac for implemen ting ap erio dic feedbac is found in the self-triggered task mo del of elasco et al. [2]. In self-triggered systems, the con trol task determines its next release time based on samples of the state gathered at the curren release time. Self-triggered task mo dels, therefore, can implemen ted in existing computer con trolled system without the

need for an sp ecial analog ev en t-detectors. This pap er presen ts exp erimen tal results examining the erformance of self-triggered con trol system. Our system’s con trol tasks select sampling eri- ds in that guaran tees the closed-lo op system’s induced gain satises sp ecied ound. then consider real-time system that sc hedules ul- tiple self-triggered con trol tasks using traditional earliest-deadline-rst (EDF) sc heduling and Buttazzo’s elastic sc heduling algorithm [7]. Our implemen ta- tion of Buttazzo’s sc heduler relies on utilization constrain similar to that

originally suggested Chan tem et al. [8]. Preliminary sim ulation results for Sim ulink/Stateo mo del of real-time system con trolling three in erted en- dulums sho ed that the con trol system’s erformance under self-triggering as remark ably insensitiv to the yp of sc heduler used the real-time system. While preliminary these results strongly suggest that self-triggering can pro- vide aluable of ensuring con trol system erformance in cases where the sc heduler is unable to pro vide hard real-time guaran tees on job completion. Prior ork on Sample erio Selection This section

briey reviews some of the prior ork on sample erio selection. Sample erio selection for ap erio dic real-time systems requires detailed anal- ysis of the system’s in tersample eha vior. This usually in olv es studying can- didate Ly apuno function as as done in Zheng et al. [9] for class of nonlin- ear sampled-data systems. Nesic et al. [10] used input-to-state stabilit (ISS) tec hniques to ound the in tersample eha vior of nonlinear systems [10]. This approac as used abuada et al. [6] to estimate sampling erio ds for class of nonlinear ev en t-triggered con trol systems. All of the

aforecited orks selected sampling erio ds to preserv some mea- sure of the con trol system’s stabilit whether this is asymptotic stabilit or input-to-state stabilit Applications, ho ev er, also need to ensure some mini- um lev el of con trol system erformance. Early ork concerning the co-design of con trol systems and real-time systems view ed this as sc hedulabilit prob- lem in whic sampling erio ds ere selected to solv the follo wing optimization problem, minimize: enalt on Con trol erformance with resp ect to: Sampling erio ds sub ject to: closed lo op stabilit task set sc hedulabilit (1)

Early statemen ts of this problem ma found in Seto et al. [11] with more recen studies in [12] and [13]. The enalt function used in the ab problem
Page 3
is often erformance index for an innite-horizon optimal con trol. The problem face here, ho ev er, is that suc erformance indices [5] are rarely monotone functions of the sampling erio d. So it can ery dicult to iden tify \optimal" sampling erio ds for the ab problem. This pap er uses Ly apuno tec hniques to select sampling erio ds that ound the in tersample eha vior of the system. This is similar to the approac hes

in [10], [9], and [6]. But rather than simply assuring closed lo op stabilit select sam- pling erio ds to adjust the induced gain of the system. This approac allo ws us to mak the system resp onsiv to ariations in the in tensit of the input dis- turbances driving the system. In the presence of high-in tensit disturbances, for example, the system can reduce its gain to eep its output signal elo some sp ecied threshold. In the presence of lo w-in tensit disturbances, it can then relax that gain and still ensure that the output remains elo the same sp ec- ied threshold. The

ariations in disturbance in tensit are therefore mirrored in ariations of the system gain whic in turn result in large ariations in the sampling erio d. This means that during erio ds of lo disturbance in tensit the erage sampling erio can uc longer than during erio ds of high disturbance in tensit The ounds on in tersample eha vior ensuring sp ecied gain are discussed in section 4. Because our con trollers enforce sp ecied in- duced gain, conne our atten tion to linear full-information con trollers for whic can generate tigh ounds on the system’s in tersample eha vior.

System Mo del The real-time system considered in this pap er consists of dynamical systems (called plan ts) that are con trolled tasks running on single pro cessor. Eac task samples (S) the system state, computes state feedbac con trol, and outputs that con trol to the plan through zero-order hold (H). The state of the th plan satises the initial alue problem, (2) (0) for and The function is an uncon- trolled and ounded disturbance. The function is the con trol input generated the th real-time con trol task. and are appropriately dimensioned matrices. The th plan t’s con trol, is

generated task ask is asso ciated with sequence of release times =1 The time is that time when the th job of task is ailable for execution. The task set is said to sync hronous if [0] is the same for all The erio for the th job of task is denoted as 1] (3) If is constan for all then the task is said to erio dic. task that is not erio dic is said to sp oradic
Page 4
The task set is asso ciated with sc heduling function This function tak es the alue at time when task is executing at that time. The nishing time for job of task is denoted as and is formally dened as max

1] (4) where lim and lim are the left and righ hand limits of at resp ectiv ely The th task’s orst-case execution time (W CET) is denoted as The task’s relativ deadline is denoted as task set is said to sc hedulable if there exist sequences of release times =1 and sc heduling function suc that (5) for all and The th task computes the con trol for the th plan t. This con trol is assumed to state feedbac con trol la of the form ]) (6) for 1]) where Note that the con trol output is constan et een nishing times and the alue of that constan is determined the system state at the job’s

release time, ]. Sample erio Selection for Induced Gain This section states the pap er’s main result concerning sample erio selection enforcing sp ecied ound on the closed-lo op system’s induced gain. conne our atten tion to the eha vior of single plan et een consecutiv release times. therefore drop the task index, without loss of generalit assume, for the purp ose of analytic simplicit that the con trol satises equation for all et een consecutiv ele ase times rather than con- secutiv nishing times This simplication is done for analytical con enience at

the exp ense of some loss in generalit The follo wing theorem pro vides conditions whic guaran tee that the induced gain from the plan t’s disturbance to its state less than sp ecied ositiv um er, Theorem 1. et denote the sample d-data ontr ol system given by quations and 6. Assume the ontr ol gain is wher is ositive symmetric matrix that satises the algebr aic ic ati quation, (7)
Page 5
for some et denote the system’s state at ele ase time If the system state satises k (8) for al 1]) and then the induc gain of is less than Pr of. The directional deriv

ativ of is Ax Using the standard completing the square argumen and the Riccati equation 7, rewrite the ab equation as where (9) Note that if k then the ab inequalities imply that k whic is sucien to ensure that the induced gain of is less than Remark: The matrix in equation can view ed as collection of rank- one erturbations of the blo diagonal matrix diag 0). can use this observ ation to sho that )) where is the largest eigen alue of So if can ensure that )) then can again guaran tee that the conditions in theorem are satised. This is more conserv ativ

condition than the one in theorem and it is similar to the switc hing condition used in [6]. Therefore using an analysis similar to that in [6] can sho that the \sampling erio d" is ounded elo ositiv constan t. In general, ho ev er, this lo er ound can an extremely conserv ativ estimate of the sampling erio d.
Page 6
After the task’s release, e’re in terested in appro ximating the in terv al er whic can guaran tee the condition in theorem 1. This time in terv al can tak en as an estimate of the task’s next release time, whic treat as the task erio d. reasonable appro ximation for

this erio is obtained in tegrating the dieren tial equation Ax (10) Let At the transition matrix for so can easily see that At As ds (11) Because and are kno wn, can ev aluate the matrix function ). Sc hedulabilit with Deadlines less than erio ds The preceding section suggests that if our task retriggers itself so equation is alw ys satised, then our sampled-data system can guaran tee the system’s closed lo op gain is less than This ma only happ en, ho ev er, if the released tasks can meet their real-time deadlines. Earliest deadline rst (EDF) sc hedulers are frequen

tly used in erio dically triggered real-time con trol systems. In the self- triggered system, ho ev er, release times will ary dep ending up on the system’s curren state. As result need to consider sc heduler that can adjust its task erio ds while assuring EDF sc hedulabilit and the \minim um" task erio required the application. The elastic task mo del of Buttazzo et al., [7], is opular metho of adjusting task erio ds. The elastic task mo del uses mec hanical analogy to dev elop an algorithm for adjusting task erio ds. This analogy views tasks as eing in terconnected \springs". The length of

the spring represen ts the task’s utilization and the spring constan represen ts that task’s resistance to hanging its utilization. Buttazzo’s elastic task mo del as extended Caccamo et al. [14] to handle uncertain ties in computation time. later pap er [15] sho ed ho to mo dify Buttazzo’s algorithm to handle additional resource constrain ts. Hu et al [16] sho ed that Buttazzo’s elastic sc heduling algorithm can view ed as minimizing task set’s summed squared utilization sub ject to the Liu-La yland EDF sc hedulabilit condition [17]. In our system, task deadlines will alw ys signican

tly less than task e- rio ds. This is needed to ensure short time dela et een the state sampling and the release of the con trol signal. Suc task sets are sc hedulable under EDF if and only if =1 (12)
Page 7
for all where i;k i;k This condition is pro en Baruah et al. [18] using pro cessor demand analysis. Pro cessor demand analysis requires that the total pro cessor demand of all released tasks in an in terv al is less than or equal to the total pro cessing er ailable in that in terv al. or task sets in whic the deadline is less than the erio d, the th task’s pro cessor demand er

time in terv al [0 is (0 The condition in equation 12 is simply the sum of all demands, (0 ), er whic ust hec ed for all ossible future releases of the tasks. In our application, release times are recomputed eac time the task is called and so the task set is really not erio dic. As result only need to hec equation 12 er subset of namely those times that are less than or equal to the next release time. This idea as used in Chan tem et al. [8] to prop ose heuristic generalization of Buttazzo’s elastic sc heduling algorithm. The follo wing theorem in tro duces sc hedulabilit condition similar to

that used in [8] whic can directly used with Buttazzo’s algorithm. Theorem 2. Consider task set in which al tasks ar ele ase at time As- sume that the task set is sorte in or der of non-de cr asing elative de ad lines +1 and let =1 set of ounds on the task erio that ar gener ate cursively fr om =1 (13) +1 +1 +1 =1 =1 +1 (14) for et arg min then the task set wil miss no de ad lines over the interval fr om [0 if for al and =1 =1 (15) Pr of. pro this theorem need to demonstrate that the pro cessor demand satises =1
Page 8
er in terv als min gg Essen tially this means that the

pro cessor demand is satised et een the con- secutiv release times. When the pro cessor demand can written as triangular system of algebraic equations (16) =1 (17) =1 (18) (19) in whic the th term has the form, =1 =1 This is triangular system of equations that can solv recursiv ely for In particular the second equation (eqn. 17) yields equation 13. Applying this in recursiv manner yields equation 14. So if these equations are satisied then can guaran tee the pro cessor demand is sucien for time in terv als equal to the task deadlines. No consider for an arbitrary and

assume that for all Then clearly =1 =1 where used the fact that 1. can no rearrange our last inequalit to obtain =1 =1 =1 The lefthand side of the ab inequalit can ounded as =1 =1 =1 =1
Page 9
By the assumption in equation 15 can see that =1 =1 =1 =1 So if the ab condition holds can ensure that =1 whic is the inequalit required to ensure that the pro cessor demand is satised prior to the giv en release time. ho ose to complete the pro of. Self-triggered Real-time Con trollers This section discusses ho theorems and are used to elastically sc hedule self-triggered con trol

tasks. Up on release, the th task umerically in tegrates equation 10 forw ard to determine serv es as the task’s desired next release time. The sc heduler handles as request for the sp ecied task erio d. If the sc heduler simply gran ts the task’s requested erio d, sa it is rigid task sc heduler. If the sc heduler adjusts the requested erio as is done in Buttazzo’s algorithm, then sa the task sc heduler is elastic Let =T denote the utilization requested the th task. or the sim ulations in section 7, our elastic task sc heduler assigns task utilization, =T in manner that solv es the

follo wing optimization problem. minimize: =1 sub ject to: min =T =1 (20) where is the minim um individual task utilization for closed lo op stabilit is the minim um task erio required in theorem 2, =T is the task’s requested utilization and min =1 is the upp er ound on the total utilization giv en in equation 15 of theorem 2. This optimization problem seeks to minimize the squared dierence et een task’s desired utilization, and its actual utilization, The allo cation is done sub ject to constrain ts in theorem 2. These constrain ts require that the allo cated utilizations assure

closed lo op stabilit while remaining sc hedulable under EDF. It is imp ortan to note that the optimization problem in equation 20 is pre- cisely the problem considered in Hu et al. [16] and Chan tem et al. [8]. In those pap ers it as sho wn that Buttazzo’s elastic sc heduling algorithm [15] actually assigns task erio ds in that alw ys satises the optimization problem in equation 20. So in our prop osed self-triggered system, can simply use But- tazzo’s algorithm directly to allo cate task utilization.
Page 10
Sim ulation Results This section presen ts preliminary sim ulation

results for the real-time con trol of three iden tical in erted endulums that are con trolled three con trol tasks running on the same pro cessor. The con trollers are full-information con- trollers. The plan ts are all iden tical and the linearized state equation for the th plan is mg = where is the cart mass, is the mass of the endulum ob, is the length of the endulum and is gra vitational accleration. The initial state is zero. or these sim ulations let 10, 3, and 10. The system state where is the cart’s osition and is the endulum ob’s angle with resp ect to the ertical. The external

disturbance, is time-v arying and tak es the form, Rect( where is band-limited white noise with noise er of 100 and sampling erio of 001 sec. Rect( is unit rectangle function of duration 25 seconds starting at time represen ts the strength of the rectangular disturbance. In these sim ulations, all three plan ts are hit with the same rectangular disturbance at the same time 2) and the disturbance lev el, as set to 1000. In these sim ulations required the state magnitude to lie elo sp ecied lev el, In other ords, require for some sp ecied Immediately after the rectangular pulse,

the system state hanges and assume that the next released task can measure this hange. Once the sampled system state exceeds the task reduces its gain to 100 to more aggressiv ely reject the rectangular pulse. as set to for these sim ulations. Once the system state is sucien tly small, the gain is reset to large alue 500) consisten with less aggressiv disturbance rejection ob jectiv e. By adjusting the gain in this manner ept the system output relativ ely small without ha ving to use the more aggressiv gain throughout the en tire system’s history sim ulated the real-time system using

Matlab stateo w/sim ulink mo del. The mo del as built to accurately capture task timing that migh seen in actual real-time systems. The real-time computer as mo deled as stateo hart that consisted of the parallel comp osition of three con trol tasks, six in- terrupt handlers, and pro cesses for the sc heduler. The con trol tasks sample the plan state up on their release, compute the requested next release time Up on nishing their execution, these tasks output the con trol signal. This sim- ulation, therefore, forces the con trol, ), to constan et een consecutiv

nishing times rather than consecutiv release times as assumed in theorem 1. The sc heduler as implemen ted as pro cesses. One pro cess assigned priorities
Page 11
to the released con trol tasks and the other pro cess used the requested sampling erio to compute the actual release times. sim ulated three dieren cases. The rst \baseline" case sim ulated the resp onse of single in erted endulum an \idealized" real-time system in whic sc heduler erformance as not an issue. The second case sim ulated the resp onse of self-triggered real-time system under \rigid" and

\elastic" sc hedul- ing. The third case sim ulated the resp onse of the erio dically-triggered real-time system under dieren task erio ds. These three cases are discussed elo w. Baseline Case: The baseline case sim ulated the resp onse of single in erted endulum in whic the con trol, ), as constan et een consecutiv release times. asks ere erio dically released ev ery 25 seconds. The con troller’s state feedbac gain as hosen to ensure the closed lo op system’s gain as less than 100. This case is therefore an \idealized" real-time system in whic missed deadlines and pro cessor con ten

tion are abstracted The lefthand side of gure sho ws the time histories for the four system states: cart osition, cart elo cit endulum ob angle and angular elo cit This plot serv es as baseline against whic the other cases will compared. The plot sho ws that when the rectangular disturbances hits the system, the cart mo es quic kly to ensure the endulum ob angle, remains small. This correctiv action results in large displacemen t, of the cart that tak es ab out 10 seconds to return to the home osition. 10 15 20 25 30 −20 −10 10 20 10 15 20 25 30 −20 −10 10 20

Fig. 1. ransien Resp onse of Baseline System Self-T riggered Case: The self-triggered case consisted of sim ulations. One sim ulation used rigidly sc heduled and the other used elastically sc heduled task sets. The system clo ran at 001 seconds. All con trol tasks had iden tical computation times, 50 clo tic ks, and deadlines, 100 clo tic ks. ask erio ds ere selected based on the results in theorem 1. In general, this resulted in probabilistic distribution of task erio ds that ere dep enden on the system state at the release time. or system gain 100 and 500 the requested erio eraged 200 and

500 clo tic ks, resp ectiv ely In the \rigidly"
Page 12
self-triggered system, these requested task erio ds ere gran ted the sc heduler. The \elastically" self-triggered system had the requested task erio ds adjusted the Buttazzo algorithm using the sc hedulabilit condition of theorem 2. 10 15 20 25 30 −20 −10 10 20 10 15 20 25 30 −20 −10 10 20 10 15 20 25 30 −20 −10 10 20 10 15 20 25 30 −20 −10 10 20 10 15 20 25 30 −20 −10 10 20 10 15 20 25 30 −20 −10 10 20 Fig. 2. State histories for rigidly (left) and

elastically (righ t) self-triggered con trollers The state histories for the rigid and elastic self-triggered systems are sho wn in gure 2. The lefthand graphs are state histories for the three plan ts in the rigidly self-triggered system and the righ thand graphs are for the elastically self- triggered system. What is erhaps most in teresting here is that all systems ha the same transien eha vior regardless of whether task erio ds ere assigned in an elastic or rigid manner. Comparing the state tra jectories in gure against the baseline tra jectories in gure 1, see

little dierence; thereb suggesting that the self-triggered system as main taining the baseline transien eha vior regardless of whic sc heduling sc heme as used. erio dically-T riggered Case: As oin of comparison sim ulated e- rio dically triggered system. The task computation times and deadlines ere 50 and 100 resp ectiv ely for all tasks with clo tic of 001 seconds. or one set of sim ulations, set the task erio to 250. The state tra jectories for the three plan ts are sho wn in the lefthand plots of gure 3. task erio of 250 as the erage task erio for the rigidly sc heduled

self-triggered sim ulations. The lefthand plots in gure are therefore comparable to the \rigidly sc heduled" sim ulations in gure 2. In the other set of sim ulations, set the task erio to 500. This sim ulation’s state histories are sho wn in the righ thand plots of gure 3. task erio of 500 as the erage task erio for the \elastically sc heduled" self-triggered sim ulations. The righ thand plots are therefore comparable to the \elastically sc heduled" sim ulations in gure 2. These sim ulations sho that at the shorter task erio (lefthand side of gure 3),

the systems app ears to ha transien resp onse similar to that for the baseline system. the longer task e- rio (righ thand side of gure 3), some of the plan ts ecome extremely oscillatory with one of the systems actually ecoming unstable.
Page 13
10 15 20 25 30 −20 −10 10 20 10 15 20 25 30 −20 −10 10 20 10 15 20 25 30 −20 −10 10 20 TASK1 TASK2 TASK3 10 15 20 25 30 −20 −10 10 20 10 15 20 25 30 −20 −10 10 20 10 15 20 25 30 −20 −10 10 20 TASK1 TASK2 TASK3 Fig. 3. State tra jectories for erio dically triggered

system. (Left) ask erio 250, (Righ t) ask erio 500 Final Remarks In comparing the sim ulation results et een the baseline, erio dically triggered, and self-triggered systems it should apparen that the self-triggered system as able to main tain an acceptable lev el of transien erformance regardless of whether rigid or elastic sc heduler as used. This as somewhat surprising at rst glance. But in hindsigh it as conjectured that this migh due to the inheren feedbac nature of self-triggering. In selecting the next release based on the curren state, self-triggered system is using state

feedbac to adjust task erio ds in that assures erall system \p erformance". The feedbac nature of this in teraction suggests that the erformance of self- triggered systems should ery robust to pro cessor erloads and late (dela ed) jobs. Our sim ulations sho ed that the rigidly sc heduled system as erloaded whereas the elastically sc heduled system as not erloaded. Inspite of the fact that the rigidly sc heduled system as erloaded, gure clearly sho ws that the transien resp onse is ery similar to the baseline resp onse. The task erio ds in the elastically sc heduled self-triggered

system ere long enough to destabilize the erio dically triggered system (gure 3). The results in gure clearly sho that ev en with this longer task erio d, the self-triggered system as able to preserv con trol erformance. In other ords, self-triggered systems app ear to main tain acceptable lev els of application erformance in the face of signican pro cessor erloading. This pap er’s results therefore demonstrate that self-triggering can main tain acceptable lev els of system erformance regardless of whether or not sc hedule in an elastic manner. In particular, these

results seem to suggest practical for relaxing the need for \hard" real-time supp ort in computer con trolled systems. These results, ho ev er, are only preliminary and future ork will need to more rigorously erify them.
Page 14
References 1. Arzen, K.: simple ev en t-based pid con troller. In: Pro ceedings of the 14th IF orld Congress. (1999) 2. elasco, M., Marti, ., uertes, J.: The self triggered task mo del for real-time con trol systems. In: ork-in-Progress Session of the 24th IEEE Real-Time Systems Symp osium (R TSS03). (2003) 3. Hristu-V arsak elis, D., Kumar, .: In

terrupt-based feedbac con trol er shared comm unication medium. In: Pro ceedings of the IEEE Conference on Decision and Con trol. (2002) 4. Astrom, K., Bernhardsson, B.: Comparison of riemann and leb esgue sampling for rst order sto hastic systems. In: Pro ceedings of the IEEE Conference on Decision and Con trol. (1999) 5. oulgaris, .: Con trol of async hronous sampled data systems. IEEE ransactions on Automatic Con trol 39 (7) (1994) 1451{1455 6. abuada, ., ang, X.: Preliminary results on state-triggered sc heduling of stabi- lizing con trol tasks. In: IEEE Conference on Decision and

Con trol. (2006) 7. Buttazzo, G., Lipari, G., Ab eni, L.: Elastic task mo del for adaptiv rate con trol. In: IEEE Real-Time Systems Symp osium (R TSS). (1998) 8. Chan tem, T., Hu, X., Lemmon, M.: Generalized elastic sc heduling. In: Real-Time Systems Symp osium (R TSS). (2006) 9. Zheng, Y., Ow ens, D., Billings, S.: ast sampling and stabilit of nonlinear sampled-data systems: art 2. sampling rate estimations. IMA Journal of Mathe- matical Con trol and Information (1990) 13{33 10. Nesic, D., eel, A., Son tag, E.: orm ulas relating kl stabilit estimates of discrete- time and sampled-data

nonlinear systems. Systems and Con trol Letters 38 (1999) 49{60 11. Seto, D., Leho czky J., Sha, L., Shin, K.: On task sc hedulabilit in real-time con trol systems. In: IEEE Real-time ec hnology and Applications Symp osium. (1996) 13{ 21 12. Cervin, A., Ek er, J., Bernhardsson, B., Arzen, K.E.: eedbac k-feedforw ard sc hedul- ing of con trol tasks. Real-time Systems 23(1-2) (2002) 25{53 13. Marti, ., Lin, C., Brandt, S., elasco, M., uertes, J.: Optimal state feedbac resource allo cation for resource-constrained con trol tasks. In: IEEE Real-Time Systems Symp osium (R TSS 2004). (2004) 161172

14. Caccamo, M., Buttazzo, G., Sha, L.: Elastic feedbac con trol. In: IEEE Euromicro Conference on Real-Time Systems (ECR TS). (2000) 15. Buttazzo, G., Lipari, G., Caccamo, M., Ab eni, L.: Elastic sc heduling for exible orkload managemen t. IEEE ransactions on Computers 51(3) (2002) 289{302 16. Hu, X., Chan tem, T., Lemmon, M.: Optimal elastic sc heduling. In: IEEE Real- time and em edded tec hnology and applications symp osium orks in progress trac k. (2006) 17. Liu, C., La yland, J.: Sc heduling for ultiprogramming in hard-real-time en vi- ronmen t. Journal of the Asso ciation for

Computing Mac hinery 20 (1) (1973) 46{61 18. Baruah, S., Rosier, L., Ho ell, R.: Algorithms and complexit concerning the preemptiv sc heduling of erio dic real-time tasks on one pro cessor. Journal of Real-Time Systems (1990) 301{324