Mean Square Stability Analysis of Sampled-Data Supervisory Control Systems

Transcription

1 17th IEEE International Conference on Control Alications Part of 28 IEEE Multi-conference on Systems and Control San Antonio, Texas, USA, Setember 3-5, 28 WeA21 Mean Square Stability Analysis of Samled-Data Suervisory Control Systems Arturo Tejada, Heber Herencia-Zaana, Oscar R González and W Steven Gray Abstract Testable second moment stability conditions for discrete-time suervisory control systems were recently develoed using the Hybrid Jum Linear Systems (HJLS s) framework These tests are extended here to samled-data suervisory control systems using the new samled-data HJLS (SD-HJLS) framework It is shown first that the -moment stability of SD-HJLS s equied with ideal samle and hold oerators is equivalent to that of their associated HJLS s Then, the second moment stability analysis tools develoed for HJLS s are extended to SD-HJLS s Finally, the results are illustrated through a Monte Carlo simulation of a simle examle N k Suervisor k FSM State Evolution Equation z k OututMa I INTRODUCTION In the ast few years, suervisory control systems have received a great amount of attention due to their increasing use in industrial alications (eg fly-by-wire aircraft, embedded control systems, etc) In most ractical suervisory systems, discrete-time control is alied to continuous-time lants The controllers are usually designed to control a discretized version of the lant, assuming that stability and other roerties of the discrete-time closed-loo system imly the corresonding roerties for the original samled-data closed-loo system The equivalence assumtion between the stability of a deterministic samled-data system and its associated discrete-time system when the continuoustime lant is linear time-invariant (LTI), linear time-varying (LTV), and nonlinear has been roven in, for examle, 1, 2, and 3, resectively More recently, this equivalence was roved correct for deterministic, LTI, continuous-time lants controlled by stochastic jum linear controllers 4 To the best of our knowledge, it has not been roven for suervisory hybrid systems This aer rovides such a roof First, the model class of Hybrid Jum Linear Systems (HJLS s), which was recently introduced to study discretetime, stochastic, suervisory hybrid systems 5, is extended to address samled-data suervisory hybrid systems The resulting model class, called samled-data hybrid jum linear systems (SD-HJLS s), is shown in Figure 1 SD-HJLS s have a structure similar to that of iecewise affine systems with samled-data switching 6 or samled-data hybrid automata (SDHA) 7 However, unlike these models, the discrete states are not simle functions of the continuous states, the clock structure is fixed (no jitter and no samle eriod variation), and the discrete transitions are based on guards imosed on both the continuous and discrete Delft Center for Systems and Control, TU Delft, The Netherlands Deartment of Electrical and Comuter Engineering, Old Dominion University, Norfolk, Virginia , USA atejadaru@gmailcom, hhere1@oduedu, {gonzalez,gray}@eceoduedu * y k y k c y () t k JumLinear Controller Plant JumLinearClosed-looSystem u() t Fig 1 A samled-data hybrid jum linear system states Like HJLS s, SD-HJLS s are comosed of a highlevel suervisor, reresented by a finite state machine (FSM), interconnected with a jum linear closed-loo system via an analog-to-symbol oerator, Ψ However, they differ from HJLS s in two asects: (i) Their jum linear closed-loo systems are comosed of a continuous-time lant connected to a discrete-time jum linear controller The necessary A/D and D/A converters are reresented by ideal samle and hold oerators without considering quantization (ii) The FSM selects the jum linear controller s oerating mode, θk, based on both an external stochastic inut, Nk, and on information from both the controller s and the lant s outut signals, (y k) T y T ck T (the suervisor and the feeding back of closed-loo system information to the suervisor s FSM are the main innovations over 4) Next, it is shown that the -moment stability of a SD-HJLS and that of its associated HJLS are equivalent This in turn allows one to study the second moment stability of SD-HJLS with the testable sufficient conditions develoed in 8 Finally, the equivalency is illustrated via a Monte Carlo simulation of a simle examle The rest of the aer is organized as follows The remainder of Section I introduces the notation Section II resents the SD-HJLS formalism Section III resents the main result of the aer, namely, Theorem 31, which establishes the /8/$25 28 IEEE 37

2 equivalency of -moment stability of a SD-HJLS and its associated HJLS This result is illustrated in Section IV via the simulation of a scalar lant controlled by a faultrecoverable digital controller Finally, Section VI resents the conclusions In this aer, C n denotes the subsace of continuous, R n -valued functions that ma the nonnegative reals, R +, into R n, are bounded on comact subsets of R +, and are right continuous at the origin Similarly, PC n denotes the subsace of iecewise-continuous, R n -valued functions that ma R + into R n, are bounded on comact subsets of R +, are continuous from the right, and have finite limits from the left on half-oen intervals of the form t, t 1 ), t, t 1 R +, where t 1 could be finite or infinite S n denotes the sace of bounded R n -valued sequences that ma the non-negative integers, Z +, into R n Likewise, Il denotes the sace of I l -valued sequences that ma Z + into I l {,,l 1}, l < Note that for any l <, Il S 1 The arguments of functions in C n or PC n will be denoted between arentheses and those from S n or Il between square brackets In addition, if (E, d) is a metric sace, then B(E) denotes the Borel σ-algebra induced by the metric d Finally, random quantities will be shown in boldface II SAMPLED-DATA HYBRID JUMP LINEAR SYSTEMS Let (Ω, F, Pr) be the underlying robability sace, and consider the samled-data HJLS shown in Figure 1 The jum linear closed-loo system is comosed of a continuoustime lant and a jum linear controller, whose oerating mode is set by the FSM s outut, θk The continuous-time lant is reresented by ẋ (t) A x (t) + B u(t) y (t) C x (t), where A R n n, B R n mc, and C R m n The discrete-time jum linear controller is reresented by x c k + 1 A θk x c k + B θk y k y c k C θk x c k, where A θk R nc nc, B θk R nc m and C θk R mc nc (θk is defined in (3)) The samle aths of the lant s outut are samled every T seconds by the samling oerator S T : C m S m y y S T y, where the entries in the sequence are evaluated at the following limit y k lim t (kt) y (t), k Z + The samle aths of the lant s inut are a zero-order-hold transformation of the controller s outut sequence The zeroorder-hold oerator is given by H T : S mc PC mc y c u H T y c, where segments comrising the control inut are characterized by u(t) y c k for each t kt, (k + 1)T) and k Z + (1) (2) The FSM dynamics are described by two relations, the state evolution equation and the outut ma, which are driven by the FSM s inuts: the external rocess Nk, k, and the internal rocess νk, k Nk, k, is a Markov chain that takes values from I ln with initial distribution µ N and transition robability matrix Π N i,j The second inut, νk, k, is the outut of a measurable quantization oerator, Ψ, given by Ψ : S m+mc Il ν y y c ν Ψ y y c where the entries ( of the sequence are given by νk i I lν i1 Ri (y k) T yck T) T, where {R i } i Ilν denotes a artition of R m+mc such that R i B(R m+mc ) and 1 {x A} 1 A (x) equals 1 if x A and otherwise At every k, the FSM s state, zk, takes values from the finite set Σ S {e 1, e 2,, e ls }, where e j 1 T has a one in the j-th osition, j I ls + 1 {1,, l S } The FSM s state evolution equation is given by zk + 1 S Nk,νk zk, where each of the l N l ν matrices S η,ν R ls ls, η I ln, ν I lν, is a matrix with columns containing exactly a single one and l S 1 zeros The FSM s outut ma is : Σ S I lo, zk θk (zk), where I lo {,,l O 1} ( is a Moore outut ma) The formal definition of samled-data hybrid jum linear systems follows Definition 21: The system in Figure 1 described by (1)- (3) is called a samled-data HJLS For all k Z + and t kt, (k + 1)T), its state evolves according to ẋ(t) x ck+1 zk+1 A B H T C θk B θk S T C A θk S Nk,νk x(t) x ck zk with state vector y(t, k) x T (t) xt c k zt k T Y R n+nc Σ S, and initial condition y x T xt c zt T Note that for every k Z + and t kt, (k + 1)T) x (t) e A(t kt) x (kt) + t kt (3) (4) e A(t τ) dτ B C θk x c k (5) Thus, if x k lim t (kt) x (t), it follows that T x k + 1 eat x k + e Aτ dτ B C c x c k (6) and y k C x k This motivates the next definition Definition 22: The system described by (2)-(3) and (6) is called the associated HJLS of the SD-HJLS in (4) For all k Z +, its state evolution is given by xk+1 zk+1 Mθk S Nk,νk xk zk, (7) 38

3 with initial condition x T z T T x T x T c z T T, where xk (x k) T x T ck T R n+nc and M θk T e AT e Aτ dτb C θk B θk C A θk For analysis, it is better to restate (7) as follows Let yk x T k z T k T Y and F : Y I ln Y be defined as l O 1 M l x1 { (z)l} F(x, z, i) l l N 1 l ν 1, S l,j z1 {il} 1 {ψ(x)j} l j where ψ : R n+nc I lν is similar to the action of Ψ at every time k That is, ψ(x) l ν 1 i i1 R i ( C C c x) Then, it is not hard to show that (7) can be rewritten as yk + 1 F(yk, Nk), (8) with initial condition y x T zt T Note that F is a Borelmeasurable function and that yk Y R n+nc+ls Thus, it follows from 9, Theorem 21 that (yk, Nk), k, is a homogeneous Markov chain in X Y I ln if the following assumtion holds Asumtion 21: The initial states x, z, and the Markov chain Nk, k, are indeendent This assumtion is fundamental in the analysis that follows and will hold throughout the aer The Markov kernel associated with (yk, Nk), k is the function P (y,n) : X B(X), 1 defined as 2 P (y,n) ( xzi, B ) Pr l N 1 j { yk+1 Nk+1 B yk Nk 1 B (F(x, z, i) T j T ) N i,j, (9) for all ( x z, i) X and B B(X)The equivalency of the -moment stability of a SD-HJLS (4) and that of its associated HJLS (7) is shown next III STABILITY ANALYSIS OF SD-HJLS S As in 4 and 9, the hybrid system reresentation develoed in 1, 11 will be used here to describe the solutions of (4) The descrition assumes that SD-HJLS s are well-osed systems In a deterministic setting, this requires the solutions of (1) and (2) to be unique This is clearly guaranteed by the linearity of both the lant and the controller In a stochastic setting, however, one must additionally guarantee that the needed robability measures can be induced throughout the closed-loo system To do so, first observe that all the signals in a SD-HJLS belong to one of the following measurable saces for the aroriate integers n and l ν : (C n, B(C n )), (PC n, B(PC n )), (S n, B(S n )), and (Il ν, B(Il ν )), where B(C n ), B(PC n ), B(S n ), and B(Il ν ) are Borel σ-algebras induced, resectively, by the metrics d C, d PC, d S, and d S as described in 4 Next, 2 With slight abuse of notation, ( x xz z, i) X will be written as xz i i } note that these signals constitute stochastic rocesses if all the transformations and oerators in (1)-(3) are measurable mas Since the lant and the controller are linear, one must only show that the oerators S T, H T, and Ψ are measurable mas To show this, recall that if (Ω, F) and (E, E) are measurable saces, then a function X : Ω E is said to be a random element if {ω Ω : X(ω) B} F for every B E If (E, E) (R, B(R)) (or (E, E) (R n, B(R n ))) then random elements coincide with the usual concet of random variables (or random vectors) A family of random variables indexed over a subset of R is a random rocess Now, if (E, E) (PC n, B(PC n )), then it can be shown that every random element X taking values in PC n is a random rocess with index R + (see 4) Similarly, if (E, E) (S n, B(S n )) (or (E, E) (I l ν, B(I l ν ))) then every random element X taking values in S n (or I l ν ) is a random rocess with index Z + Moreover, for each given ω Ω, the samle aths or trajectories of the random rocesses in the last two cases are functions in PC n and S n (or I l ν ), resectively The oerators S T, H T, and Ψ can then be considered to be measurable maings over measurable saces as shown below Lemma 31: The samling oerator, S T ; zero-order-hold oerator, H T ; and quantization oerator, Ψ, are random elements between the following measurable saces: S T : (C m, B(C m )) (S m, B(S m )) H T : (S mc, B(S mc )) (PC mc, B(PC mc )) Ψ : (S m+mc, B(S m+mc )) (I l ν, B(I l ν )) Proof : The roof that S T and H T are random elements can be found in 4 To show that Ψ is a random element, one must show that for every B B(Il ν ), the set γ {(y )T yc TT S m+mc : Ψ(y )T yc TT B} B(S m+mc ) Note that every B B(Il ν ) is determined by imosing restrictions over the sequences ν on at most a countable set of oints (see 12, age 148) This, in turn, imlies that every sequence (y) T yc T T in γ must satisfy a set of at most countable restrictions as well, which is enough to show that γ B(S m+mc ) Thus, Ψ is a random element Clearly, all the signals in the SD-HJLS are random elements from the measurable sace (Ω, F) into their resective saces In articular, the states of the lant, the controller and the FSM are random elements, resectively, from (Ω, F) into C n, from (Ω, F) into S nc, and from (Ω, F) into S ls That is, x (t), x c k, and zk are stochastic rocesses with samle aths in, resectively, C n, S nc, and S ls Note that the hybrid state vector, y(t, k) takes values from Y for all k Z + and t kt, (k +1)T), and that (Y, d Y ) is a metric sace under the metric d Y d Y ( x z, x z ) x x + z z with x, x R n+nc ; z, z Σ S ; and and denote the Euclidean -norm on R n+nc and the discrete metric, resectively The distance between a oint y Y and a set M Y can now be defined in the usual way That is, 39

4 d Y (y, M) inf{d Y (y, ỹ) : ỹ M} In order to introduce aroriate stability notions, the following concets, which are similar to those in 11, must be defined for the SD-HJLS in (4) Definition 31: Consider the metric sace (Y, d Y ) and let A Y denote the set of initial states of (4) If y k A is the initial state corresonding to the initial time k T, k Z + then a stochastic rocess y(t, ω, y k, k T), t kt, (k + 1)T), k k, taking values in Y is called a stochastic motion if y(k T, ω, y k, k T) y k for all ω Ω Definition 32: A family S of stochastic motions of (4) taking values in Y will be called a stochastic dynamical system if S {y(,, y k, k T) : y(k T, ω, y k, k T) y k ω Ω, y k A, t kt, (k + 1)T), k k Z + } The following lemma rovides a characterization of the stochastic motions that form the stochastic dynamical systems for the SD-HJLS (4) To simlify the notation, the deendence on Ω is suressed Lemma 32: The stochastic motions of the SD-HJLS (4) are given by Nθk (t, k) y(t, yk, kt) y(t, k) yk, (1) I for all k Z +, t kt, (k + 1)T), where y(t, k) and yk are defined, resectively in (4) and (8), and N θk (t, k) is the nonsingular and bounded matrix N θk (t, k) ea(t kt) t kt I e Aτ dτ B C θk (11) Proof : Since yk x T (kt) x T ck z T k T, the result follows by exressing equation (5) and the identities x c k Ix c k and zk Izk in matrix form Lemma 32 shows the strong relation between the samle solutions of the SD-HJLS (4) and those of its associated HJLS (7) Thus, it is now ossible to formally define - moment stability and to rove that (4) and (7) have equivalent -moment stability characteristics The following additional definitions from 11 are needed Definition 33: Let S be a stochastic dynamical system A set M A is said to be invariant with resect to S if y(,, y k, k T) S, y k M imlies that Pr{ω Ω : d Y (y(t, ω, y k, k T), M), t kt, (k + 1)T), k k } 1 Definition 34: A oint y A is called an equilibrium oint of the stochastic dynamical system S if the set M {y} is invariant with resect to S An imortant invariant set is M { z T T, z Σ S } because d Y (y(t, ω, yk, kt), M) x T (t) x T ck T Hence, any stability notion defined for the invariant set M can be redefined in terms of only the equilibrium oint x e R n+nc of the lant and the controller So, without loss of generality, consider the following stability definition written in terms of x e only Definition 35: The stochastic dynamical system S is said to be -moment stable if E{ x T (t) xt c k T } as k, t kt, (k + 1)T) Similar definitions hold for the stochastic motions of HJLS s (see 9) The main result of this aer follows Theorem 31: The samled-data HJLS (4) is -moment stable if and only if its associated HJLS (7) is -moment stable Proof : First, observe from (11) that N i (t, k) is a bounded function of the difference t kt, T) for every i I lo, k Z +, and t kt, (k + 1)T) Thus, let N i su{ N i (t, k) : k Z +, t kt, (k + 1)T)}, i I lo Define N max{ N i : i I lo } and note that N θk (t, k) N Now, let the HJLS in (7) be -moment stable, and observe from (7) and (1) that x T (t) xt c k T N θk (t, k)xk Then, all the stochastic motions of (4) satisfy the following inequalities for all k Z +, t kt, (k+1)t), and ω Ω: x T (t) x T ck T N θk (k, t) xk x T (t) x T ck T N xk E{ x T (t) x T ck T } N E{ xk } Clearly, if E{ xk } then E{ x T (t) x T ck T } Thus, if the HJLS in (7) is -moment stable, so is the SD-HJLS in (4) Since N θk (t, k) is nonsingular, the converse argument can be roven with a similar rocedure, by observing that xk (N θk (t, k)) 1 x T (t) xt c k T Note that Theorem 1 in 4 can be recovered as a secial case of Theorem 31 Also observe that, with minor modifications, Theorem 31 can be extended to two more general classes of SD-HJLS s: (i) SD-HJLS s such that in (3) the outut ma,, is a function of both zk and Nk, and (ii) SD-HJLS s such that is a function of zk, Nk, and νk, and such that in (2) y c k C c x c k For 2, stability can be verified with the testable sufficient condition resented next The following setu adated from 5 and 8 is needed first Let x R n+nc ; r, ˆr I ls + 1; i, î I ln ; and set s (r, i) and ŝ (ˆr, î) Consider again the Markov kernel P (y,n) in (9) and let s,ŝ (x) be defined as ( xeri ) s,ŝ (x) P (y,n), R n+nc {eˆr } {î} 1 {eˆr }(S i,ψ(x) e r ) N i,î For every set R l in the artition of R m+mc induced by ψ, choose an arbitrary element x l R l and define ˆ s,ŝ max{ s,ŝ (x i ) : i I lν } Theorem 32: The SD-HJLS in (4) is second moment stable ( 2) if the sectral radius of A is less than 1, ie, if ρ(a) < 1, where A (Π T I (n +n c) 2)diag ( A (1,) A (1,), A (1,1) A (1,1),, A (ls,l N 1) A (ls,l N 1)), 4

5 TABLE I MODEL STRUCTURE FOR ROLLBACK AND COLD-RESTART RECOVERY Nominal model Rollback Recovery model (A RB, B RB, C RB ) (A 1RB, B 1RB, C 1RB ) ( ( ) Ac I nc, B c,i nc ) Inc I nc,, I nc Nominal model Cold-restart Recovery model (A CR, B CR, C CR ) (A 1CR, B 1CR, C 1CR ) ( ) ( ) Ac I nc, B c,i nc I nc,, I nc A (r,i) M (er) for all r I ls + 1 and i I ln (M i, i I lo, are defined in Definition 22), and Π ˆ (1,),(1,) ˆ (1,),(1,1) ˆ (1,),(lS,l N 1) ˆ (1,1),(1,) ˆ (1,1),(1,1) ˆ (1,1),(lS,l N 1) ˆ (ls,l N 1),(1,) ˆ (l S,l N 1),(1,1) ˆ (l S,l N 1),(l S,l N 1) Proof : The result follows immediately from Theorem 31 and 8, Theorem 53 Note that ρ(a) < 1 if and only if equation (9) in 4 is satisfied with π ij relaced by ˆ s,ŝ (see ref 9 in 4) Thus, Theorem 32 is the testable form of 4, Theorem 5 Theorems 31 and 32 are illustrated next IV EXAMPLE As noted in 5, many comuter-controlled, safety-critical alications, such as aircraft, are subject to harsh environments that induce faults in their comuter latforms These faults can be handled by error recovery mechanisms that usually sto the comuter latform s activity while the faults are removed The overall effect of these mechanisms is an abrut change in the closed-loo dynamics Thus, according to the absence or resence of faults, the closedloo dynamics are determined by either a nominal or a recovery state sace reresentation As shown in Figure 2, when the lant is linear, this behavior can be reresented by the jum linear system ẋ (t) A x(t) + B H T C Nk x c k x c k + 1 A Nk x c k + B Nk S T C x (t), where Nk is a two-state Markov chain that reresents the absence (Nk ) or resence (Nk 1) of faults in the comuter latform (see (1) and (2)) The secific structure of (A Nk, B Nk, C Nk ) deends on the articular recovery mechanism emloyed This structure is shown in Table I for both rollback and cold-restart recovery Rollback often rovides better stability roerties than cold-restart, while cold-restart requires simler hardware and less ower consumtion It was roosed in 5 that an advanced recovery mechanism (ARM) could combine the * y k N k JumLinear Controller y k c Plant y () t u() t Fig 2 Plant and its fault-recoverable controller 1x 1 x x e 1 x 11 e e 2 3 Fig 3 Structure of the examle s FSM benefits of rollback and cold-restart as follows: Uon the arrival of a fault, cold-restart is executed only if the norm of the closed-loo system s outut, (y k) T y T ck T, is below a certain erformance threshold, α However, if the erformance threshold is exceeded, rollback is executed Such an ARM can be modeled with the following SD-HJLS: 1x ẋ (t) A x (t) + B H T C θk x c k x c k + 1 A θk x c k + B θk S T C x (t) zk + 1 S Nk,νk zk νk 1 { y T (t) y T c kt α}, (12) where (A θk, B θk, C θk ) is given by (A RB, B RB, C RB ) (A CR, B CR, C CR ), (A 1CR, B 1CR, C 1CR ), or (A 1RB, B 1RB, C 1RB ) whenever zk is, resectively, equal to e 1, e 2, or e 3 The FSM s structure is given in Figure 3, and its state transition matrices S,,, S 1,1 in (12) are given by S, S, , S 1, 1 1 1, S 1, Nk is a Markov chain with transition robability matrix Π N, 1 N, 1 N 1,1 N 1,1 This examle was numerically tested using the following arameters: A , B , C 1, A c 98648, B c 182, C c 97, T 1 sec, and α 13 The second moment stability of the HJLS associated with (12) was analyzed with Theorem 32 ρ(a) was comuted for several values of N, and N 1,1 Theorem 32 shows that for every oint ( N 1,1, N, ) that lies in the region above the stability boundary (blue (to) line) in Figure 4, the HJLS is second moment stable Figure 4 also shows the stability boundaries for systems equied solely with rollback or cold-restart recovery (see 5) Note that the ARM s stability region is the smallest of the three, which shows that Theorem 32 is conservative 41

6 * Rollback Cold restart HJLS 2 Rollback asym Cold restart asym Simulation result N, Second moment stability region log 1 (E{ xk 2 }) N 1,1 Fig 4 HJLS s stability boundary (ρ(a) 1, blue (to) line) and stability region (ρ(a) < 1, above blue (to) line) as functions of N, and N 1,1 The stability boundaries for rollback and cold-restart are also shown A Monte Carlo simulation of (simultaneously) the SD- HJLS and its associated HJLS was erformed for N, 996 and N 1,1 8 (Figure 4) The simulation comrised 5 runs of 5 seconds (5, samle eriods) in duration In both cases, x and z were arbitrarily set to 1 T and e 1, resectively The logarithm of the estimate of E{ xk 2 } E{ (x k)t x T c kt 2 } is shown in Figure 5 This figure also shows the asymtotes for the theoretical behavior of log 1 (E{ xk 2 }) for systems equied solely with rollback or cold-restart recovery As exected, the examle s resonse lies between these two ideal cases (see 5 for details) Finally, Figure 6 shows the average behavior of the continuous-time lant in (12) (red line) and that of its associated discrete-time counterart (blue + ) Clearly, as redicted by Theorem 31, the second moment stability of (12) and that of its associated HJLS are equivalent V CONCLUSIONS An analysis framework for autonomous, stochastic, samled-data hybrid systems, called samled-data hybrid jum linear systems, has been introduced It was shown that the -moment stability of these systems can be inferred from that of their associated discrete-time counterarts A sufficient, second moment stability test was also resented Both the test and the equivalency result were illustrated via a Monte Carlo simulation of a simle scalar lant with a digital controller equied with an advanced fault recovery mechanism Future research effort should extend these results to include systems with stochastic control inuts ACKNOWLEDGEMENTS This research was suorted by the NASA Langley Research Center under grant NNX7AD52A REFERENCES 1 B A Francis and T T Georgiou, Stability theory for linear timeinvariant lants with eriodic digital controllers, IEEE Trans Automat Contr, vol 33, no 9, , Setember P A Iglesias, Inut-outut stability of samled-data linear timevarying systems, IEEE Trans Automat Contr, vol 4, no 9, , Setember Time (sec) Fig 5 Estimate of the second moment of the examle s associated HJLS E{ x (t)} Time (sec) Detail E{ x * k} E{ x (t)} Fig 6 Average behavior of the examle s continuous-time lant and its associated discrete-time counterart 3 L Hou, A N Michel, and H Ye, Some qualitative roerties of samled-data control systems, IEEE Trans Automat Contr, vol 42, no 12, , December O R González, H Herencia-Zaana, and W S Gray, Stochastic stability of nonlinear samled-data systems with a jum linear controller, in Proc 43rd IEEE Conference on Decision and Control, Nassau, Bahamas, 24, A Tejada, Stability analysis of Markov jum linear systems with Markov inuts, PhD dissertation, Old Dominion University, Norfolk, VA, 26 6 S Azuma and J Imura, Synthesis for otimal controllers for iecewise affine systems with samled-data switching, Automatica, vol 5, no 42, , 26 7 B Silva and B Krogh, Modeling and verification of samled-data hybrid systems, in Proc of the 4th International Conference on Automation of Mixed Processes: Hybrid Dynamic Systems, Setember 2, A Tejada, O R González, and W S Gray, Stability analysis of hybrid jum linear systems with Markov inuts, in Proc 46th IEEE Conference on Decision and Control, New Orleans, LA, 27, , On the Markov roerty for nonlinear discrete-time systems with Markovian inuts, in Proc of the 26 American Control Conference, Minnesota, MN, 26, H Ye and A N Michel, Stability theory for hybrid dynamical systems, IEEE Trans Automat Contr, vol 43, no 4, , Aril L Hou and A N Michel, Moment stability of discontinuous stochastic dynamical systems, IEEE Trans Automat Contr, vol 46, no 6, , June A N Shiryaev, Probability, Second Edition New York: Sringer,