Performance Analysis of Descriptor Jump Linear Systems*

Proceedings of the 4th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS 12) June 6-8, 2012. Eindhoven, The Netherlands

Performance Analysis of Descriptor Jump Linear Systems ⋆ Jorge R. Ch´ avez-Fuentes ∗ Eduardo F. Costa ∗∗ Marco H. Terra ∗∗∗ ∗

Dept. of Sciences, Pontificia Universidad Cat´ olica del Per´ u, Lima, Per´ u and Dept. of Electrical Engineering-EESC University of S˜ ao Paulo at S˜ ao Carlos, Brazil (e-mail: [email protected]). ∗∗ Department of Applied Mathematics and Statistics, University of S˜ ao Paulo, S˜ ao Carlos, Brazil, (e-mail: [email protected]). ∗∗∗ Dept. of Electrical Engineering-EESC University of S˜ ao Paulo at S˜ ao Carlos, Brazil (e-mail: [email protected]). Abstract: This paper deals with the problem of obtaining analytic expressions for the mean output energy and the steady-state output power of a closed loop control system described by a discrete-time descriptor jump linear dynamics driven by an homogeneous Markov chain. The results given in this paper are new and generalize the formulas given for the same metric of performance of a Markov jump parameter linear system. Keywords: Analysis; Stochastic systems; Switched systems; Singular systems. 1. INTRODUCTION Practical applications may feature certain phenomena that produce abrupt changes in the behavior of dynamic systems, sometimes causing degradation of performance and stability margin, which has motivated the study of Markov jump linear systems. For instance, high intensity radiated fields can produce upset in digital electronics, as has been shown in Kim et al. (2000), thereby affecting the digital flight control system. Metrics of performance for control closed loop Markov chain jump linear systems (MCJLS) have been derived in the literature before Ch´avez et al. (2010); Gray et al. (2003); Ling and Lemmon (2004). Since abrupt changes in the modes of the system, caused by the upsets, affect its performance, it is of great interest to have available analytic expressions to measure the degradation of performance of the control system. This has been shown for instance in Gray et al. (2010); Wang et al. (2011); Zhang et al. (2008). A MCJLS is a type of descriptor jump parameter linear system (DJPLS) that does not have singularity issues. This is because the lefthand-side state vector of the system is multiplied, in the worst case scenario, by a non-singular matrix, while in the singular case it is multiplied by a singular one. The theory of discrete-time descriptor jump linear system with ⋆ Research supported by FAPESP under Grants 2011/13160-0 and 10/12360-3 and CNPq under Grant 306466/2010-4.

978-3-902823-00-7/12/$20.00 © 2012 IFAC

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singularity is still being built Manfrim (2010); Manfrim et al. (2008a,b). Thus, no available formulas to measure the performance of a singular control closed loop system subject to jumping parameters exist. The main goal of this paper is to develop theoretical tools to analyze the tracking error performance of a singular system while it is operating under a harsh environment. In this paper, closed form expressions for the mean output energy, J0 , and the steady-state output power, Jω , (see Section 4) for a DJPLS are given. These results represent a generalization of the formulas given in Gray et al. (2003) and Ch´ avez et al. (2010). A standard assumption regarding descriptor systems refers to the existence and uniqueness of a solution of the system. In the classical theory, that is, for non-jumping descriptor systems this problem is handled by imposing a regularity condition, Dai (1988). This notion of regularity is extended to stochastic regularity for DJPLS in Manfrim (2010). Under this assumption it is ensured the existence of a unique solution of the system for an admissible initial condition x(0) ∈ X, where X is a subset of Rn , the ndimensional euclidian space of real numbers. Furthermore, stochastic regularity also let us define second moment matrices related with the (well-defined) random variable x(k) (see (2)). Throughout the paper it is assumed that the system (1) is stochastically regular.

10.3182/20120606-3-NL-3011.00077

ADHS 12, June 6-8, 2012 Eindhoven, The Netherlands

This paper is organized as follows. In Section 2, the main problem to solve is introduced. The notation and basic concepts are presented in Section 3. The performance analysis is developed in Section 4. Finally, the conclusions are given in Section 5. 2. PROBLEM FORMULATION Consider the following descriptor jump linear system: Sθ(k+1) x(k + 1) = Aθ(k) x(k) + Bθ(k) ω(k),

(1a)

x(0) = x0 , y(k) = Cθ(k) x(k),

(1b)

where x(k) ∈ Rn , y(k) ∈ Rp and x0 ⊂ Rn is a random vector with finite second moment. The process θ(k) is an homogeneous Markov chain taking values in the finite-state space Sθ , {1, . . . , N } with transition probability matrix P , [pij ], where pij , Pr(θ(k + 1) = j|θ(k) = i), i, j ∈ Sθ , and initial state probability ′ vector π = [π1 . . . πN ] , where πi , (Pr(θ(0) = i), i ∈ Sθ . Furthermore, π(k) = [p1 (k), . . . , pN (k)], where pi (k) , Pr(θ(k) = i), denotes the state probability vector at the time k ≥ 1. The initial random variables θ(0) and x(0) are assumed to be independent. The stochastic process ω(k) ∈ Rq is a zero mean white noise process with identity covariance matrix Iq and independent of θ(k) and x0 . For all i ∈ Sθ , the matrix Si is a square matrix of order n with rank(Si ) = ri ≤ n. When ri = n for all i ∈ Sθ the system is called non-singular, otherwise it called singular. Singular and non-singular systems are jointly called descriptor systems. The focus of this paper is on singular systems. The matrices Aθ(k) , Bθ(k) and Cθ(k) have appropriate dimension. When Si = I for all i ∈ Sθ the non-singular system is said to be in standard normal form and the system is referred to as Markov chain jump linear system, that has been widely studied by the literature before (see, e.g., Ch´avez et al. (2010), Costa et al. (2005), Costa et al. (2011), Dragan and Morozan (2006), Fang et al. (1995), Fang and Loparo (2002), Feng et al. (1992), Gray et al. (2003), Ji et al. (1991)). In Dragan and Morozan (2006); Fang and Loparo (2002); Fang et al. (1995); Feng et al. (1992) the authors are basically concerned with the problem of stability of the MCJLS, while in Gray et al. (2003) and Ch´avez et al. (2010) the attention is focussed in obtaining analytic expressions for the metric of performance J0 and Jω . These formulas are of great interest for application as they provide a mean to measure the performance, at steady-state, of the control system subject to abrupt changes. The main goal of this paper is to establish suitable analytic expressions for J0 and Jω associated with the DJPLS (1) that generalize the results given in Gray et al. (2003) and Ch´avez et al. (2010). 449

3. NOTATION The notation used in this paper is largely consistent with Costa et al. (2005). The symbol Rn denotes the ndimensional euclidian space of real numbers. The nonnegative integer numbers is denoted by Z+ . The symbol k · k denotes the 2-norm, the Euclidian norm. The normed space of matrices of order m × n defined on R is denoted by Mmn and, for short, Mn denotes the normed space of square matrices of order n. The expression B(Mn ) denotes the Banach space of matrices in Mn with the norm k·k. The ′ transpose and the trace of a matrix A are dented by A and tr(A), respectively. The space of matrices in Mn that are symmetric positive semi-definite are denoted by Mn0 . All random variables are defined in the space (Ω, F , Pr), where Ω is the sample space, F is the sigma algebra of events and Pr is the probability measure. A random variable is written in boldface, x, and its expectation is denoted by E{x}. A discrete-time process is simply denoted by x(k), k ∈ Z+ . The indicator function with respect to the event A ∈ F , is denoted, as usual, by 1A . The Kroneker product is denoted with the symbol ⊗ and the stacking column vector operator, by vec. Since in this paper the operator vec is only applied to square matrices, the notation vec−1 (A) returns back the column vector vec(A) to the matrix A. The symbol Θ is used to denote the set of all initial distribution of θ(k). For each k ∈ Z+ let us introduce the following notation. n o ′ Q(k) , E x(k)x (k) (2a) n o ′ Qi (k) , E x(k)x (k)1{θ(k)=i} , i ∈ Sθ . (2b)

The operators A, B, T , S defined below play a fundamental role   in what follows. Note that they belong to 2 B MN n  ′  A , P ⊗ In2 diag (A1 ⊗ A1 , . . . , AN ⊗ AN )  ′  B , P ⊗ In2 diag (B1 ⊗ B1 , . . . , BN ⊗ BN ) S , diag (S1 ⊗ S1 , . . . , SN ⊗ SN )

T , (S − A) . Observe that when B = 0 the System (1a) is homogeneous.

4. PERFORMANCE ANALYSIS To characterize the performance of a control descriptor closed loop jump linear system, analytic expressions for J0 and Jω , are derived. These new expressions generalize the corresponding formulas given for these metrics when the system is a MCJLS. The output performance metric for the DJPLS (1) is defined next.

ADHS 12, June 6-8, 2012 Eindhoven, The Netherlands

 ∞ n o X  2   J0 , E ky(k)k , B = 0 J= k=0 n o    Jw , lim E ky(k)k2 , B 6= 0.

which is equivalent to n o ′ ′ Sj E x(k + 1)x (k + 1)1{θ(k+1)=j} Sj

k→∞

The following definition for mean square stability (MSS) is standard in the literature (see, e.g., Costa (1994)). Definition 1. The DJPLS (1) is said to be MSS if for any admissible initial condition x0 ∈ X and any θ 0 ∈ Θ there exists a matrix Q ∈ Mn0 such that Q = lim Q(k). k→∞

(3)

When the system is homogeneous, that is, when B = 0 the matrix Q is the zero matrix. In this section, it is assumed that System (1) is MSS and the operator T , invertible.

Consider the homogeneous version of System (1). For each k ∈ Z+ and i ∈ Sθ define k k n o X X ′ M (k) , Q(n) = E x(n)x (n) (4) Mi (k) ,

n=0

n=0 k X

Qi (n) =

n=0

=

i=1

n o ′ ′ Ai E x(k)x (k)1{θ(k+1)=j} 1{θ(k)=i} Ai .

Since the system is MSS, by Lemma 2 and after some manipulations it follows ! ∞ n o X ′ ′ Sj E x(k + 1)x (k + 1)1{θ(k+1)=j} Sj k=0 N X

=

∞ X

pij Ai

i=1

n

′

E x(k)x (k)1{θ(k)=i}

k=0

! o

′

Ai .

This equation can be written as N   ′ X ′ ′ Sj Mj − E{x(0)x (0)1{θ(0)=j} } Sj = pij Ai Mi Ai . i=1

Since x(0) and θ(0) are independent then

A. The homogeneous case.

n=0 k X

N X

n o ′ E x(n)x (n)1{θ(n)=i} . (5)

The following lemma gives an equivalent characterization of MSS in terms of the matrices Mi . Lemma 2. The system (1a) is MSS if and only if there exists Mi ∈ Mn0 such that ∞ X Mi = Qi (k), i ∈ Sθ . (6) k=0

Proof. See Ch´avez et al. (2011). If the DJPLS (1) is MSS then the matrix Mi has a closed form expression. In effect, denoting by h ′ ′ i′ f= M f · · · M f M , (7) 1 N

fi = vec(Mi ), the following lemma gives a formula where M f for M .

Lemma 3. Assume that the operator T is invertible and f let the DJPLS (1) be MSS. Then the column matrix M defined in (7) is given by f = T −1 SX 0 . M (8)  n o ′ where X 0 = (π ⊗ In2 ) vec E x(0)x (0) . Proof. Fix j ∈ Sθ . From Equation (1a) it follows o n ′ ′ E Sθ(k+1) x(k + 1)x (k + 1)Sθ(k+1) 1{θ(k+1)=j} o n ′ ′ = E Aθ(k) x(k)x (k)Aθ(k) 1{θ(k+1)=j} ,

′

Sj M j Sj −

N X

′

pij Ai Mi Ai

i=1

n o ′ ′ = Sj E x(0)x (0) P (θ(0) = j)Sj .

Applying the vec operator on both sides of this equation yields (Sj ⊗ Sj ) vec Mj −

N X

pij (Ai ⊗ Ai ) vec Mi

i=1

= (Sj ⊗ Sj ) vec(X 0 )πj , n o ′ where X 0 = E x(0)x (0) . By the notation introduced in (7) this can be written as fj − (Sj ⊗ Sj )M

N X i=1

fi pij (Ai ⊗ Ai )M

= (Sj ⊗ Sj ) vec(X 0 )πj ,

j ∈ Sθ .

Collecting all values of j ∈ Sθ and writing this equation in a compact form yield  diag(S1 ⊗ S1 , . . . , SN ⊗ SN )  ′ f − (P ⊗ In2 ) diag(A1 ⊗ A1 , . . . , AN ⊗ AN ) M = diag(S1 ⊗ S1 , . . . , SN ⊗ SN )(π ⊗ In2 ) vec(X 0 ).

Denoting the column vector (π ⊗ In2 ) vec(X 0 ) by X 0 , this equation can be written as f = SX 0 . TM

From this equation, the claim follows by multiplying from the left by T −1 . f has N blocks column matrices of size n2 The matrix M each one. The matrix Mi is given by fi ), i ∈ Sθ , Mi = vec−1 (M (9) f f where Mi is the i-th block of M , which can be calculated from (8).

The following theorem gives an analytic expression for J0 . 450

ADHS 12, June 6-8, 2012 Eindhoven, The Netherlands

Theorem 4. Assume that the operator T is invertible and let the DJPLS (1) with B = 0 be MSS. Then the mean output energy J0 is finite and N   X ′ J0 = tr Ci Mi Ci , (10)

e = T −1 Beb, Q (12) ′ h ′ i ′ where eb = eb1 . . . ebN and ebi = vec(Ii )pi , where Ii is the identity matrix of the same order than Bi .

i=1

where the set of matrices Mi is given in (9). Proof. From (1b) it follows ∞ X  E ky(k)k2

=

k=0

= = =

∞ X

k=0 ∞ X k=0 ∞ X

n ′ o ′ E x (k)Cθ(k) Cθ(k)x(k)

tr

k=0

i=1

+

P∞

k=0

= tr

i=1

= tr

N X i=1

=

N X i=1

Ci

∞ X ′

Ci Mi Ci

n

′

E x(k)x (k)1{θ(k)=i}

k=0

!

!

.

n o ′ E x(k)x (k)1{θ(k)=i} con-

k=0

N X

N X i=1

n o ′ ′ Ci E x(k)x (k)1{θ(k)=i} Ci

By Lemma 2 the series verges. Thus, ∞ X  E ky(k)k2

N X i=1

o n  ′ ′ E tr Cθ(k) x(k)x (k)Cθ(k) N X

Proof. By applying the same argument as in the proof of Lemma 3, it follows n o ′ ′ Sj E x(k + 1)x (k + 1)1{θ(k+1)=j} Sj

! o

′

Ci

!

n o ′ ′ Ai E x(k)x (k)1{θ(k+1)=j} 1{θ(k)=i} Ai

 ′ Bi E 1{θ(k+1)=j} 1{θ(k)=i} Bi .

By Lemma 5 and recalling that the process θ(k) is ergodic, by tacking limits as k → ∞ on both sides of this equation it follows N N X X ′ ′ ′ Sj Q j Sj − pij Ai Qi Ai = pij Bi Bi pi . i=1

i=1

Applying the vec operator on both sides of this equation yields N X (S1 ⊗ S1 ) vec(Qj ) − pij (Ai ⊗ Ai ) vec(Qi ) =

i=1 N X

pij (Bi ⊗ Bi ) vec(Ii )pi .

i=1

  ′ tr Ci Mi Ci .

B. The non-homogeneous case. Consider now the DJPLS (1) with B 6= 0 and ω(k) 6= 0, that is, the non-homogeneous case. In this subsection an analytic form for the steady-mean output power, Jω , is derived. First, it is given a lemma that characterizes MSS in terms of the matrices Qi (k) defined in (2b). Lemma 5. The DJPLS (1) is MSS if and only if for every i ∈ Sθ there exists Qi ∈ Mn0 such that Qi = lim Qi (k). k→∞

Proof. See Ch´avez et al. (2011).

Collecting all values of j ∈ Sθ and using the notation introduced in (11), this equation becomes  diag (S1 ⊗ S1 , . . . , SN ⊗ SN )  ′   e − Π ⊗ In2 diag (A1 ⊗ A1 , . . . , AN ⊗ AN ) Q  ′  = Π ⊗ In2 diag (Bi ⊗ Bi ) eb,

which is equivalent to

e = Beb. TQ

From this equation, the claim follows by multiplying from the left by T −1 . e has N blocks columns matrices. The matrix The matrix Q Qi is given by e i ), i ∈ Sθ , Qi = vec−1 (Q (13)

e i is the i-th block of Q, e which can be calculated where Q When the system is MSS a closedn form expression foro the from (12). ′ limit limk→∞ Qi (k) = limk→∞ E x(k)x (k)1{θ(k)=i} , i ∈ The following theorem gives an analytic expression for Jω . Sθ , can be obtained. In effect, denoting by ′ h ′ i ′ Theorem 7. Let the DJPLS (1) be MSS. Then the steadye= Q e · · · Q e Q , (11) 1 N state mean output power, Jω , is finite and e N where Qj = vec Qj , the following lemma gives a formula   X ′ e Jw = tr Ci Qi Ci , (14) for Q. It is assumed that the process θ(k) is ergodic, that i=1 is, there exists pi ∈ R such that pi = lim pi (k). k→∞ where the set of matrices Qi is given in (13). Lemma 6. Let the DJPLS (1) be MSS. Then the column e defined in (11) is given by matrix Q Proof. From (1b) it follows 451

ADHS 12, June 6-8, 2012 Eindhoven, The Netherlands

n ′ o  ′ E ky(k)k2 = E x (k)Cθ(k) Cθ(k) x(k) n  o ′ ′ = E tr Cθ(k) x(k)x (k)Cθ(k) ( !) N X ′ ′ = E tr Ci x(k)x (k)1{θ(k)=i} Ci i=1

= tr

N X i=1

′

Ci Qi (k)Ci

!

.

MSS makes it possible to take limits as k → ∞ on both sides of this equation resulting in (14). 5. CONCLUSIONS Closed form expressions have been given to calculate the metrics of performance J0 and Jω of a singular control closed loop jump linear system. These formulas generalize the known one given by a Markov jump parameters linear system, which is a non-singular system. Three main operators, S, A and B, were introduced to derive these formulas, under the assumption that the systems is stochastically regular, MSS and the operator T = S − A is invertible. The formulas given here have the aim to provide analytic tools to measure the performance of a singular system that is operating under a harsh environment. REFERENCES Jorge R. Ch´avez, O. R. Gonz´alez, and W. S. Gray. Performance analysis of fault tolerant control systems with i.i.d. upsets. In Proc. 2010 American Control Conf., pages 6197–6204, Baltimore, MD, 2010. Jorge R. Ch´avez, Eduardo F. Costa, and M. H. Terra. Equivalence between Mean Square, Stochastic and Exponential Stability for Singular Jump Linear Systems. In Proc. 50th IEEE Conf. on Decision and Control and the European Control Conf. 2011, pages 2877 – 2882, Orlando, FL, 2011. E. F. Costa, A. N. Vargas, and J. B. R. do Val. Quadratic costs and second moments of jump linear systems with general Markov chain. Math. Control Signals Systems, 23(1):141 – 157, 2011. O. L. V. Costa. Linear minimum mean square error estimation for discrete-time markovian jump linear systems. IEEE Trans. Automat. Contr., 39(8):1685–1689, 1994. O. L. V. Costa, M. D. Fragoso, and R. P. Marques. Discrete-Time Markov Jump Linear Systems. Springer, New York, 2005. L. Dai. Observer for Discrete Singular Systems. IEEE Trans. Automat. Control, 32(2):187–191, 1988. V. Dragan and T. Morozan. Mean square exponential stability for some stochastic linear discrete time systems. Eur. J. Contr., 12(4):1–123, 2006. 452

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