Proceedings of the 36th IEEE Southeastern Symposium on System Theory Atlanta, Georgia, March 14-16, 2004
4C-5
Stochastic Stability of Sampled Data Systems with a Jump Linear Controller Oscar R. Gonz´ alez Heber Herencia-Zapana W. Steven Gray Department of Electrical and Computer Engineering Old Dominion University Norfolk, Virginia 23529-0246, U.S.A.
[email protected] [email protected] [email protected] Keywords— Sampled-data systems, stochastic jump-linear hybrid systems, stability. Abstract— In this paper an equivalence between the stochastic stability of a sampled-data system and its associated discrete-time representation is established. The sampled-data system consists of a deterministic, linear, time-invariant, continuous-time plant and a stochastic, linear, time-invariant, discrete-time, jump linear controller. The jump linear controller models computer systems and communication networks that are subject to stochastic upsets or disruptions. This sampled-data model has been used in the analysis and design of fault-tolerant systems and computer-control systems with random communication delays without taking into account the inter-sample response. This paper shows that the known equivalence between the stability of a deterministic sampled-data system and the associated discrete-time representation holds even in a stochastic framework.
in the analysis and design of fault-tolerant systems and computer-control systems with random communication delays without taking into account the inter-sample response. The results in this paper show that stochastic stability of the associated discrete-time system is equivalent to stochastic stability of the sampled-data system. Thus, the known results in the discrete-time jump linear system literature such as [2–4] are given the appropriate foundation to validate their application in the analysis and design of sampled-data systems. The paper is organized as follows. The mathematical preliminaries and notation are introduced at the end of this section. In Section 2, the sampled-data system and its corresponding discrete-time representation are defined together with representations of their stochastic motions. Section 3 presents two definitions for pth moment stability and shows the equivalence between the pth moment stability of the sampled-data and its associated discrete-time system. Finally, Section 4 gives the paper’s conclusions.
i. Introduction
The following notation is similar to that which appears in sampled-data papers such as [1, 12, 16]. Let C n denote the subspace of continuous, Rn -valued functions that map the nonnegative reals, R+ = [0, ∞), into Rn , are bounded on compact subsets of R+ , and are right continuous at the origin. Similarly, let PC n denote the subspace of piecewise-continuous, Rn -valued functions that map R+ = [0, ∞) into Rn , are bounded on compact subsets of R+ , and are continuous from the right and have finite limits from the left on half-open intervals of the form [t0 , t1 ), t0 , t1 ∈ R, where t0 ≥ 0 and t1 could be finite or infinite. Let S n denote the space of bounded Rn -valued sequences that map the non-negative integers, Z+ , into Rn . The arguments of functions taking values in C n or PC n will be denoted between parentheses and those in S n between square brackets. In addition, random quantities will be shown in boldface and 0 denotes transposition.
The equivalence between the stability of a deterministic sampled-data system and its associated discrete-time system when the continuous-time plant is linear timeinvariant (LTI), linear time-varying (LTV), or nonlinear is well known (see, for example, [5, 10, 11]). In this paper it is shown that a similar equivalence is possible when the plant is a deterministic LTI continuous-time system and the controller is a stochastic jump linear system, that is, the stochastic stability of the sampled-data system is equivalent to the stochastic stability of its associated discrete-time jump linear system. The jump linear controller models computer systems and communication networks that are subject to stochastic upsets or disruptions [6, 7, 13, 14]. Since controller parameter jumps affect the closed-loop system only at the sampleinstants, a discrete-time random process with a finite number of states is used to model their occurrence. For example, the control law could switch between a normal and a fail-safe operation algorithm. Discrete-time representations of these sampled-data models have been used
c 0-7803-8281-1/04/$20.00/°2004 IEEE
256
u(t)
where the amplitudes of the segments comprising the control input are characterized by u(t) = ψ[k] for each t ∈ [kT, (k + 1)T ) and k ∈ Z+ .
y(t)
Plant
HT
State-space representations of the closed-loop sampled data system result in hybrid representations that include both continuous-time and discrete-time dynamics. For example, in [5,12] the hybrid systems are given in terms of periodic system representations. One such representation is · ¸ · ¸· ¸ A BF HT x (t) x˙ (t) = Bθ[k] CST Aθ[k] ξ [k] ξ [k + 1] (1) · ¸ £ ¤ x (t) y (t) = C 0 , ξ [k]
ST
ψ[k] Jump Linear
η[k]
Controller
θ[k]
where k ∈ Z+ and kT ≤ t < (k + 1)T . Since the controller parameters are switched according to a discrete process, the switching sequence is taken to be {(θ [k] , k)} = {(θ [0] , 0), (θ [1] , 1), . . .}, where (θ [k] , k) ∈ Σθ × Z+ and Σθ is finite (cf. [9]). An alternate nonhybrid, closed-loop system representation where some of the states jump at the sampling instants is considered in [16].
Figure 1. Sampled-data system with ideal sample and zero-order-hold operators, ST and HT , respectively.
ii. Modeling The sampled-data system under consideration is shown in Figure 1, where the A/D and D/A conversions are performed by ideal sampling and zero-order-hold operators, respectively,. Quantization is not considered. Let (Ω, F, P ) be the underlying probability space, and let {θ(k); k ∈ Z+ } be a finite-state, discrete-time random process that drives the switching in the jump linear controller. The continuous-time plant is represented by
In this paper the hybrid system representation introduced for deterministic, discontinuous systems in [18,19] and extended to the stochastic setting in [8] will be used. A concept used throughout is that of measurable mappings or random elements [15]. Let (Ω, F) and (E, E) be measurable spaces. A function X = X(ω) defined on Ω and taking values in E is said to be a random element if {ω ∈ Ω : X(ω) ∈ B} ∈ F for every B ∈ E. If (E, E) ≡ (R, β(R)), where β(R) is the Borel algebra of subsets of R, then random elements coincide with the usual concept of random variables. A family of random variables indexed over a subset of R is a random process. Now, if (E, E) ≡ (PC n , βo (PC n )), where βo (PC n ) is the σ-algebra generated by open sets, then it can be shown that every random element X taking values in PC n is a random process with index R+ . Similarly, if (E, E) ≡ (S n , βo (S n )) then every random element X taking values in S n is a random process with index Z+ . Moreover, the sample paths or trajectories for each given ω ∈ Ω of the random processes in the last two cases are functions {X(t)} in PC n and S n , respectively. The sample and hold operators defined above can now be considered to be measurable mappings over measurable spaces as shown below.
x(t) ˙ = Ax(t) + Bu(t) y(t) = Cx(t), where A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rm×n . The jump linear discrete-time controller is represented by ξ[k + 1] = Aθ[k] ξ[k] + Bθ[k] η[k] ψ[k] = F ξ[k], where Aθ[k] ∈ Rnξ ×nξ , Bθ[k] ∈ Rnξ ×m , and F ∈ Rm×nξ . The sample paths of the plant’s output are sampled at uniformly spaced sampling instants by the sampling operator ST : C n y
−→ S n 7−→ η = ST y,
where the entries in the sequence are evaluated at the following limit η[kT ] = limt→(kT )− y(t), k ∈ Z+ . The sample paths of the plant’s input are a zero-order-hold transformation of the controller’s output sequence. The zero-order-hold operator is given by
Lemma 1: The sampling, ST , and zero-order-hold, HT , operators are measurable maps between the following measurable subspaces: ST HT
HT : S n −→ PC n ψ 7−→ u = HT ψ,
: (C n , βo (C n )) : (S n , βo (S n ))
−→ (S n , βo (S n )) −→ (PC n , βo (PC n )).
That is, ST is measurable with respect to βo (C n ) and
257
βo (S n ), and HT is measurable with respect to βo (S n ) and βo (PC n ).
Proof. In (1) the random elements X[ω], Y[ω], U[ω], Θ[ω], Ξ[ω], Ψ[ω], and η[ω] are, respectively, the random processes x(t), y(t), u(t), θ[k], ξ[k], ψ[k], and η[k] with sample paths x(t), y(t), u(t), θ[k], ξ[k], ψ[k], and η[k] in either PC n or S n . The sample path responses of the hybrid system due to any initial state z(kT, k), k ∈ Z+ and initial distribution for θ[0] result in the following usual convolution response for the random process (x(t), kT ≤ t < (k + 1)T ) Z t x(t) = eA(t−kT ) x(kT ) + eA(t−s) dsBF ξ[k].
Proof. The sampling operator is a random element if the inverse image of any event in βo (S n ) is an event in βo (C n ), that is, {y ∈ C n : ST y ∈ B} ∈ βo (C n ) for every B ∈ βo (S n ). It is known that every event in βo (C n ) is determined by restrictions imposed on the functions y on at most a countable set of points [15]. Thus, {y ∈ C n : ST y ∈ B} is an event in βo (C n ), since every element of {y ∈ C n : ST y ∈ B} has restrictions given by the sequence ST y, which is on a countable set of points. The proof for the zero-order-hold operator follows similarly. 2
kT
The desired result follows by expressing this equation and ξ[k] = Iξ[k] in matrix form. 2
For stability analysis, the stochastic process of interest is the family of random variables of the hybrid state vector in (1) that takes values in X ⊂ Rn+nξ , that is, £ ¤0 z(t, k) , x0 (t) ξ0 [k] ∈ X , where k ∈ Z+ and kT ≤ t < (k + 1)T . For stability analysis, two more concepts similar to those in [8] need to be defined.
In Section III, stability of the hybrid system will be compared to the stability of its associated discrete-time system. This closed-loop discrete-time system is found by interconnecting the jump linear discrete-time controller to the zero-order-hold equivalent model of the plant as seen from the input/output channels of the controller. The discrete-time system is given by
Definition 1 (Stochastic Motion of Hybrid Systems) Let (X , k · kp ) be a p-normed space with X ⊂ Rn+nξ . Any z(kT, k) ∈ A ⊂ X is called the initial state at the initial time (kT, k). A stochastic process z(t, ω, z(kT, k), (kT, k)), t ∈ [kT, (k + 1)T ), taking values in X is called a stochastic motion if z(kT, ω, z(kT, k), (kT, k)) = z(kT, k) for all ω ∈ Ω.
where M θ[k]
·
eAT = Bθ[k] C
RT 0
(3)
¸ · ¸ x[k] eA(T −s) dsBF and z[k] = . ξ[k] Aθ[k]
iii. Moment Stability
Definition 2 (Stochastic Dynamical System) Let S be a family of stochastic motions taking values in X given by
In this section the equivalence between p th-moment stability of the hybrid stochastic system in (1) and the p thmoment stability of the discrete stochastic system in (3) is shown. The following additional definitions from [8] are needed.
S ⊂ {z(·, ·, z(kT, k), (kT, k)) : z(kT, ω, z(kT, k), (kT, k)) = z(kT, k) ∀k ∈ Z+ , ω ∈ Ω, t ∈ [kT, (k + 1)T )}.
Definition 3 (Invariant Set) Let S be a stochastic dynamical system. A set M ⊂ A is said to be invariant with respect to S if a ∈ M implies that ¡ ¢ P z(t, ω, a, (kT, k)) ∈ M, t ∈ [kT, (k + 1)T ), ω ∈ Ω = 1 for all k ∈ Z+ .
Then S is called a stochastic dynamical system. A characterization of the stochastic motions that form the stochastic dynamical system for (1) is given in the following lemma where the dependence on Ω is omitted for simplification. Also, let z[k] = z(kT, k) = £ 0 ¤0 x (kT ) ξ 0 [k] .
Definition 4 (Equilibrium) a ∈ A is called an equilibrium of the stochastic dynamical system S if the set {a} is invariant with respect to S.
Lemma 2: The stochastic motions of the hybrid system (1) are given by z(t, z(kT, k), (kT, k)) = N (kT, t)z[k],
z[k + 1] = M θ[k] z[k],
Since the closed-loop system in (1) is autonomous, the origin of Rn+nξ is an equilibrium of the stochastic dynamical system. This is clear since z(t, ω, 0, (kT, k)) = 0, ∀k¡ ∈ Z+ , t ∈ [kT, (k + 1)T ), and ω ∈ Ω. Thus, ¢ P z(t, ω, 0, (kT, k)) = 0, t ∈ [kT, (k + 1)T ), ω ∈ Ω = 1 ∀k ∈ Z+ .
(2)
£ ¤0 where z(t, z(kT, k), (kT, k)) = x0 (t) ξ0 [k] , k ∈ Z+ , kT ≤ t < (k + 1)T , and N (kT, t) is the nonsingular and bounded matrix · ¸ R t A(t−s) eA(t−kT ) e dsBF kT N (kT, t) = . 0 I
Definition 5 (p th-Moment Stability) The equilibrium state 0 ∈ Rn+nξ of the hybrid stochastic dynamical system S is stable in the p th-moment if
258
∀² > 0, ∃ δ = δ(², k) > 0 such that kz(0, 0)k < δ implies Ekz(t, ω, z(kT, k), (kT, k))kp < ² ∀k ∈ Z+ , t ∈ [kT, (k + 1)T ), ω ∈ Ω. The origin is said to be asymptotically stable in the p th-moment if in addition to being stable in the p th-moment, Ekz(t, ω, z(kT, k), (kT, k))kp → 0 as k → ∞, t ∈ [kT, (k + 1)T ), and ω ∈ Ω.
nate stability definitions for p = 1 and p = 2 are called stability in the mean and mean square stability. Definition 6 (Alternate Notions of Stability) The equilibrium state 0 ∈ Rn+nξ of the hybrid stochastic dynamical system S is • stable in the mean if ∀² > 0, ∃ δ = δ(², k) > 0 such that kz(0, 0)k < δ implies ° ³ ¡ ¢´° ° ° °E z t, ω, z(kT, k), (kT, k) ° → ∞ as k → ∞
A similar definition holds for p th-moment stability of the origin of the discrete stochastic system in (3). The first theorem below shows that stability of the associated discrete-time system implies stability of the hybrid system. This is the condition that is needed in most applications.
t ∈ [kT, (k + 1)T ), and ω ∈ Ω, mean square stable if ∀² > 0, ∃ δ = δ(², k) > 0 such that kz(0, 0)k < δ implies ° ³ ¡ ¢´ ° °E z t, ω, z(kT, k), (kT, k) ¡ ¢´° ° z0 t, ω, z(kT, k), (kT, k) ° → ∞ as k → ∞
•
Theorem 1: If the origin of the discrete stochastic system (3) is asymptotically stable in the p th-mmoment, the origin of the hybrid system (1) is asymptotically stable in the p th-moment. Proof. It follows from the stochastic motion in (2) that all stochastic motions satisfy the following inequalities for k ∈ Z+ , t ∈ [kT, (k + 1)T ), and ω ∈ Ω kz(t, z(kT, k), (kT, k))kp Ekz(t, z(kT, k), (kT, k))kp Ekz(t, z(kT, k), (kT, k))kp
t ∈ [kT, (k + 1)T ), and ω ∈ Ω. It is known that this mean square stability definition is equivalent to the asymptotic 2nd-moment stability in Definition 5 (cf. [17]). In fact, the measures of meansquare stability and 2nd-moment stability are equivalent in the sense¡shown below. To simplify the nota¢ tion let z(·) = z t, ω, z(kT, k), (kT, k) ∀k ∈ Z+ , t ∈ [kT, (k + 1)T ), and ω ∈ Ω.
≤ kN (k, t)kp kz[k]kp ≤ kN (k, t)kp Ekz[k]kp ≤ N Ekz[k]kp
where kN (k, t)kp < N since N (·, ·) is bounded . Thus, if Ekz[k]kp → 0 then Ekz(t, ω, z[k], kT )kp → 0. 2
Theorem 4: The 2nd-moment and mean square measures of stability are equivalent in the sense that they satisfy the inequality
In general, it is easier to show that if the origin of the hybrid system is asymptotically stable in the p th-moment then the origin of the associated discrete-time system is also asymptotically stable in the p th-moment.
1 Ekz(·)k2 ≤ kEz(·)z0 (·)k ≤ Ekz(·)k2 . n + nξ
Theorem 2: If the origin of the hybrid system (1) is asymptotically stable in the p th-moment then the origin of the discrete-time system (3) is also asymptotically stable in the p th-moment.
Proof. See [17]. There is no such equivalence between the 1st-moment stability and the stability in the mean measures.
Proof. The proof is similar to Theorem 1, since N (k, t) is nonsingular ∀k ∈ Z+ , and t ∈ [kT, (k + 1)T ). 2
Theorem 5: The origin of the hybrid system in (1) is stable in the mean and mean square stable if and only if the origin of the discrete-time system in (3) is stable in the mean and mean square stable.
Combining the previous two theorems produces the main result of the paper.
Proof. There is only a need to show that stability in the mean for the hybrid and the discrete-time systems is equivalent, since mean square stability is equivalent to asymptotic 2nd-moment stability by Theorem 4. Taking expected values of (2) gives ¡ ¢ ¡ ¢ E z(t, z(kT, k), (kT, k)) = N (kT, t)E z[k] .
Theorem 3: The origin of the hybrid system (1) is asymptotically stable in the p th-moment if and only if the origin of the discrete-time system (3) is asymptotically stable in the p th-moment. Equivalence of the stability of the hybrid system and its corresponding discrete-time system can also be shown for two more popular stochastic stability definitions. These alternate notions of stability correspond roughly to a reversal in order of the expectation and the norm in the p th-moment stability when p = 1, 2. the alter-
Applying norms to both sides produces ° ¡ ¢° °E z(t, z(kT, k), (kT, k)) ° ° °° ¡ ¢° ≤ °N (kT, t)°°E z[k] °.
259
(4)
Since N (kT, t) is nonsingular it follows from (2) that ¡ ¢ ¡ ¢ E z[k] = N −1 (kT, t) E z(t, z(kT, k), (kT, k)) .
[7]
Hence, ° ¡ ¢° °E z[k] ° ° °−1 ° ¡ ¢° ≤ °N (kT, t)° °E z(t, z(kT, k), (kT, k)) °. (5)
W. S. Gray, O. R. Gonz´alez, and M. Do˘gan, “Stability analysis of digital linear flight controllers subject to electromagnetic disturbances,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, no. 4, pp. 1204– 1218, October 2000.
[8]
From (4) and (5) it follows that the hybrid system is stable in the mean iff the discrete-time system is stable in the mean. 2
L. Hou and A. N. Michel, “Moment stability of discontinuous stochastic dynamical systems,” IEEE Trans. Automat. Contr., vol. 46, no. 6, pp. 938– 943, June 2001.
[9]
L. Hou, A. N. Michel, and H. Ye, “Stability analysis of switched systems,” in Proc. 35th IEEE Conference on Decision and Control, Kobe, Japan, December 1996, pp. 1208–1212.
iv. Conclusions
[10] ——, “Some qualitative properties of sampled-data control systems,” IEEE Trans. Automat. Contr., vol. 42, no. 12, pp. 1721–1725, December 1997.
The main result of the paper is the equivalence between asymptotic stability in the p th-moment of a stochastic hybrid dynamical system and the asymptotic stability in the p th-moment of its corresponding discrete-time system. This equivalence holds for two more popular definitions: stability in the mean and mean square stability.
[11] P. A. Iglesias, “Input-output stability of sampleddata linear time-varying systems,” IEEE Trans. Automat. Contr., vol. 40, no. 9, pp. 1646–1650, September 1995. [12] P. T. Kabamba and S. H. Hara, “Worst-case analysis and design of sampled-data control systems,” IEEE Trans. Automat. Contr., vol. 38, no. 9, pp. 1337–1357, September 1993. ¨ [13] R. Krtolica, U. Ozguner, H. Chan, H. Goktas, J. Winkelman, and M. Liubakka, “Stability of linear feedback systems with random communication delays,” Int. J. of Control, vol. 59, no. 4, pp. 925– 953, 1994.
v. Acknowledgements This research was supported by the National Science Foundation under grant CCR-0209094 and by the NASA Langley Research Center under grants NCC-1-392 and NCC-1-03026. References
[14] Q. Liang and M. D. Lemmon, “Soft real-time scheduling of networked control systems with dropouts governed by a Markov chain,” in Proc. 2003 American Control Conference, Denver, CO, June 2003, pp. 4845–4850.
[1] T. Chen and B. A. Francis, “Input-output stability of sampled-data systems,” IEEE Trans. Automat. Contr., vol. 36, no. 1, pp. 50–58, January 1991. [2] H. J. Chizeck, A. S. Willsky, and D. Castanon, “Discrete-time Markovian-jump quadratic optimal control,” Inter. J. of Control, vol. 43, no. 1, pp. 213–231, 1986.
[15] A. N. Shiryaev, Probability. Verlag, 1991, 2nd ed.
Berlin: Springer-
[16] N. Sivashankar and P. P. Khargonekar, “Characterization of the L2 -induced norm for linear systems with jumps with applications to sampled-data systems,” SIAM J. Control and Optimization, vol. 32, no. 4, pp. 1128–1150, 1994.
[3] O. L. V. Costa and M. D. Fragoso, “Stability results for discrete-time linear systems with Markovian jumping parameters,” Mathematical Analysis and Applications, vol. 179, pp. 154–178, 1993.
[17] A. Tejada, Analysis of error recovery effects on digital flight control systems. M.S. Thesis, Old Dominion University, 2002.
[4] Y. Fang and K. A. Loparo, “Stochastic stability of jump linear systems,” IEEE Trans. Automat. Contr., vol. 47, pp. 1204–1208, 2002.
[18] H. Ye and A. N. Michel, “Stability theory for hybrid dynamical systems,” IEEE Trans. Automat. Contr., vol. 43, no. 4, pp. 461–474, April 1998.
[5] B. A. Francis and T. T. Georgiou, “Stability theory for linear time-invariant plants with periodic digital controllers,” IEEE Trans. Automat. Contr., vol. 33, no. 9, pp. 820–832, September 1988.
[19] H. Ye, A. N. Michel, and L. Hou, “Stability theory for hybrid dynamical systems,” in Proc. 34th IEEE Conference on Decision and Control, New Orleans, December 1995, pp. 2679–2684.
[6] O. R. Gonz´alez, W. S. Gray, A. Tejada, and S. Patilkulkarni, “Stability analysis of electromagnetic interference upset recovery methods,” in Proc. 40th IEEE Conference on Decision and Control, 2001, pp. 4134–4139.
260
