AN ADDENDUM TO THE PROBLEM OF STOCHASTIC OBSERVABILITY

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1 AN ADDENDUM TO THE PROBLEM OF STOCHASTIC OBSERVABILITY V. Dragan and T. Morozan Institute of Mathematics of the Romanian Academy, P.O.Box , RO-77, Bucharest, Romania Abstract In this paper we introduce a concept of stochastic observability for a class of linear stochastic systems subjected both to multiplicative white noise and Markovian jumping. The definition of stochastic observability adopted here extends to this framework the definition of uniform observability of a time varying linear deterministic system. By several examples we show that the concept of stochastic observability introduced in this paper is less restrictive than those introduced in some previous papers. Also we show that the concept of stochastic observability introduced in this paper, does not imply always the stochastic detectability as it would be expected. Keywords. Linear stochastic systems, Markovian jumping, multiplicative white noise, stochastic observability. 1 Introduction Both the observability property and controlability property of a linear system plaied a crucial role in solving a wide class of control problems. Here we recall that starting with the pioneer paper of Kalman [7], the observability and controlability properties provide sufficient conditions guaranteeing the existence of the stabilizing solution of a matrix Riccati differential equation which is connected with the linear quadratic problem and filtering problem [12, 13]. In stochastic framework the concept of stochastic observability was introduced in order to provide conditions which guarantee the existence of the stabilizing solutions of the matrix Riccati differential equations of stochastic control [9]. For the linear systems subjected to Markovian jumping some definitions of stochastic observability were introduced in [8, 6]. In this paper we introduce the concept of stochastic observability for a class of linear stochastic systems subjected both to multiplicative white noise and Markovian jumping. The definition of stochastic observability adopted here extends to this framework the definition of uniform observability of a time varying linear deterministic system [7, 1]. By several examples we show that the concept of stochastic observability introduced in this paper is less restrictive than those introduced by [8] and [6]. Also we show that the concept of stochastic observability introduced in this paper, does not imply always the stochastic detectability as it would be expected. Finally we show that 1

2 the stochastic observability introduced here guarantees the positivity of the observability Gramian (if it exists) and additionaly any positive solution of corresponding Riccati equation is stabilizing as it happens in deterministic case. 2 Stochastic observability Consider the system: dx(t) = A (η(t))x(t)dt + A k (η(t))x(t)dw k (t) (2.1) y(t) = C (η(t))x(t) (2.2) x(t) R n,y R p,w(t) =(w 1 (t)... w r (t)),t isastandard Wiener process on a given probability space (Ω, F, P); η(t), t isaright continuous homogenous Markov chain, with the state space set D = {1, 2,..., d} and the probability transition matrix P (t) =e Qt,t >,Q=[q ij ] with d j=1 q ij =,i D,q ij,i j (see[2]). Throughout this paper assume that {w(t)} t and {η(t)} t are independent stochastic processes, and P{η() = i} > for all i D.Foreach t and x R n,x(t, t,x ) stands for the solution of (2.1) with initial condition x(t,t,x )=x.(for precise definition of the solution see [5, 11]). Let Φ(t, t )bethe fundamental matrix solution of (2.1). That is the j-th column of Φ(t, t ) is x(t, t,e j ), j =1, 2,..., n, where e j =( ) being vector of canonic basis of R n. Now we are able to introduce the definition of stochastic observability. Definition 2.1 We say that the system (2.1)-(2.2) is stochasticaly observable, or the triple (C,A,A 1,..., A r ; Q) isobservable if there exist β>,τ > such that +τ E[ Φ (s, t)c(η(s))c ) (η(s))φ(s, t)ds η(t) =i] βi n, (2.3) t i D, ( )t, E[ η(t) = i] being the conditional expectation with respect to the event η(t) =i. Sometimes we shall write (C, A; Q) isobservable instead of (C,A,A 1,..., A r ; Q) isobservable. Remark 2.1 a) In the particular case D = {1} and A k =1 k r the inequality (2.3) reduces to the well known definition of uniform observability for a time varying linear deterministic system (see e.g. [7, 1]). 2

3 b) If D = {1} the equations (2.1)-(2.2) become dx(t) = A x(t)dt + A k x(t)dw k (t) (2.4) y(t) = C x(t). (2.5) In this case (2.3) is +τ E[ Φ (s, t)cc Φ(s, t)ds] βi n (2.6) t for all t, Φ(s, t) being now the fundamental matrix solution of (2.4). c) In the case d 2,A k (i) =, 1 k r, i D, (2.1)-(2.2) reduce to ẋ(t) = A (η(t)) (2.7) y(t) = C (η(t))x(t) (2.8) and in this case if (2.3) holds with Φ(s, t) standing for the fundamental matrix solution of (2.7) we say that the triple (C,A ; Q) isobservable. 3 Some auxiliary results 3.1 Liapunov type operators and exponential stability Let S n R n n be the space of symmetric matrices and S d n = S n S n... S n (d factors). Let L : S d n S d n be defined by (LX)(i) =A (i)x(i)+x(i)a (i)+ A k (i)x(i)a k(i)+ d q ij X(j) (3.9) ( ) i D,X =(X(1)... X(d)) S d n. The operator L will be termed as the Liapunov type operator defined by the system (A,A 1,..., A r ; Q). It is easy to see that the adjoint operator L is given by (L X)(i) =A (i)x(i)+x(i)a (i)+ A k(i)x(i)a k (i)+ j=1 d q ij X(j) (3.1) L : S d n S d n is the linear operator obtained from (3.1) taking A k (i) =,1 k r, i D. Consider the linear differential equation on S d n d S(t) =LS(t) (3.11) dt and set e Lt the linear evolution operator defined on Sn d by (3.3). 3 j=1

4 The following result was proved in [3] in a more general setting. Proposition 3.1 a) e Lt X e L t X, e L t X e L t X, ( ) X =(X(1) X(2)...X(d)) S d n,x(i),i D,t. b) [e L (t s) X](i) = E[Φ (t, s)x(η(t))φ(t, s) η(s) = i] ( ), t s >,i cald, X = (X(1) X(2)... X(d)) S d n. Definition 3.1 We say that the zero solution of the equation (2.1) is mean square exponentially stable (MSES for shortness), or that the system (A,A 1,...A r ; Q)isstable if there exist α>,β > such that E[ x(t) 2 ] βe αt x() 2, ( )t,x() R n. The following result provides necessary and sufficient conditions for the MSES of the zero solution of (2.1). Theorem 3.1 The following are equivalent: (i) The solution x(t) =of the equation (2.1) is MSES. (ii) lim t E[ x(t) 2 ]=for any solution x(t) of (2.1). (iii) The eigenvalues of the operator L are located in the half plane Re λ <. (iv) There exists H =(H(1) H(2)... H(d)) S d n with H(i) >,i D, such that the equation LX + H = has a solution X =(X(1)... X(d)),X(i) >,i D. (v) There exists H as before such that the equation L X + H = has a solution X =(X(1)... X(d)),X(i) >,i D. (vi) There exists X =(X(1)... X(d)),X(i) > such that L X<. For detailed proof see [3]. 3.2 Stochastic detectability Definition 3.2 We say that the system (2.1)-(2.2) is stochasticaly detectable, or that the triple (C, A; Q) isdetectable if there exist L =(L(1) L(2)... L(d)), L(i) R n p such that the zero solution of the equation dx(t) =(A (η(t)) + L(η(t))C (η(t)))x(t)dt + is MSES. L will be termed stabilizing injection. 4 A k (η(t))x(t)dw k (t)

5 Proposition 3.2 The following are equivalent: (i) The triple (C, A; Q) is detectable. (ii) The system of linear equations A (i)x(i)+x(i)a (i)+c(i)γ (i)+γ(i)c (i)+ d A k(i)x(i)a k (i)+ q ij X(j)+I n = (3.12) has a solution X =(X(1)... X(d)) S d n, Γ=(Γ(1)... Γ(d)), X(i) >, Γ(i) R n p,i D. (iii) The linear matrix inequality j=1 A (i)x(i)+x(i)a (i)+c(i)γ (i)+γ(i)c (i)+ dq ij X(j) < (3.13) A k(i)x(i)a k (i)+ j=1 has a solution X =(X(1) X(2)... X(d)) S d n, Γ=(Γ(1) Γ(2)... Γ(d)), X(i) >, Γ(i) R n p,i D. Moreover, if (X, Γ),X > is a solution of (3.4), then L =(L(1) L(2)... L(d)), L(i) =X 1 (i)γ(i) provides a stabilizing injection. 4 Main results Based on Proposition 3.1 (b) we obtain: Proposition 4.1 The following are equivalent: (i) The system (2.1)-(2.2) is stochastically observable. (ii) There exists τ> such that τ e L s Cds > (4.14) C =( C(1)... C(d)), C(i) =C (i)c (i),i D. (iii) There exists τ> such that X (τ) >, where X (t) is the solution of the problem with initial value d dt X (t) =L X (t)+ C, X() =. (4.15) Proof. (i) (ii) follows from Proposition 3.1 (b). Since X (t) = e L (t s) Cds = e L s Cds, t it follows that (iii) (ii). The proof is complete. 5

6 Further from Porposition 4.1 and Proposition 3.1 (a) we obtain: Proposition 4.2 The following hold: (i) If for each i D, the pair (C (i),a (i)) is observable (in deterministic sense) then the triple (C,A ; Q) is observable. (ii) If (C,A ; Q) is observable, then (C, A; Q) is observable. Proposition 4.3 Let X (t) be the solution of the problem with initial value (4.2). If there exists τ>such that X (τ) > then X (t) > for all t>. Proof. For each t>, we write the representation X (t) =(X (t, 1), X (t, 2),..., X (t, d)) = e L (t s) Cds. Since e L (t s) : Sn d Sn d is a positive operator, we deduce that X (t), for all t. Moreover if t τ we have X (t) X (τ), therefore if X (τ) >, we have X (t) > for all t τ. It remains to show that X (t) >, <t<τ. To this end we show that detx (t, i) >, <t<τ, i D. Indeed, since detx (t, i) =det { el (t s) Cds)(i) },we deduce that t detx (t, i) isananalytic function. The set of its zeros on [,τ] has no accumulation point. In this way it will follow that there exist τ 1 > such that detx (t, i) > for all t (,τ 1 ]. Invoking again the monotonicity of the function t X (t) weconclude that X (t) > for all t τ 1, and the proof ends. Remark 4.1 From Proposition 4.1 and Proposition 4.3 it follows that the stochastic observability for the system (2.1)-(2.2) may be checked by using a numerical procedure to verify the positivity of the solution X (t) through an enough long interval of time. Proposition 4.4 The triple (C, A; Q) is observable if and only if does not exist τ>,i D and x = such that E [ y(t,,x ) 2 η() = i ] = ( ) t [,τ] where y(t,,x )=C (η(t))x(t,,x ), x(t,,x ) being the solution of (2.1) having the initial condition x(,,x )=x. The next result provide a sufficient condition assuring the stochastic observability. Its proof may be found in [1]. Proposition 4.5 The triple (C, A; Q) is observable if for every i D,rank M(i) =n, where M(i) =[C(i),A (i)c(i),...,(a (i)) n 1 C(i), q i1 C(1),...,q id C(d),A 1(i)C(i),...,A r(i)c(i)]. In the following examples, the stochastic observability used in this paper is compared with other types of stochastic observability, for example the one introduced in [6] and [8]. We also 6

7 show that the stochastic observability used in this paper doesn t imply always the stochastic detectability as we would expected. Example 1 The case of a system subjected to Markovian jumping with d =2,n=2,p=1. Take [ ] α A (1) = A (2) = α [ ] q q C (1) =[1 ],C (2) =[ 1],Q=,α R,q >. q q It is obvious that the pairs (C (1),A (1)), (C (2),A (2)) are not observable in the deterministic sense. Therefore this system is not stochastically observable, in the sense of [8, 6]. We shall show that this system is stochastically observable in the sense of Definition 2.1. To this end we use the implication (iii) = (i) inproposition 4.1. We show that there exists τ > such that X 1 (τ) >,X 2 (τ) >, where X i (t),i =1, 2isthe solution of the Cauchy problem: From the representation formula d dt X i(t) = A (i)x i (t)+x i (t)a (i)+ +C(i)C (i), X i () =, i =1, 2. (X 1 (t),x 2 (t)) = e L (t s) Cds 2 q ij X j (t) (4.16) it follows that X i (t) for all t. Therefore it is sufficient to show that there exists τ>such that detx i (τ) >. ( ) xi (t) y Set X i (t) = i (t),i=1, 2. After some simple cmputations we have detx i (t) = y i (t) z i (t) x i (t)z i (t) yi 2 (t) =x i (t)z i (t) = x(t) z(t),t where x(t) = 1 2 [e αs + e (2α q)s ]ds z(t) = 1 [e 2αs e 2(α q)s ]ds. 2 For detail see [4]. It is easy to see that for every α R,q >wehave lim t x(t) z(t) >. Remark 4.2 Let us consider the system of type (2.1)-(2.2) with n =2,d=2,p=1,r =1 and A (1) = A (2) = αi 2,C (1) = (1 ),C (2) = ( 1),A 1 (i) a2 2 arbitrary matrix, 7 j=1

8 [ q q Q = q q ],α R,q >. Combining the conclusion of Example 1 with Proposition 4.2 (ii) it follows that the system (C, (A,A 1 ); Q) isobservable. Example 2 The stochastic observability does not imply always stochastic detectability. Let us consider the system subjected to Markovian jumping with d =2,n=2,p=1, A (1) = A (2) = q [ q q 2 I 2,C (1) =[1 ],C (2) =[ 1],Q= q q ]. (4.17) From the previous example we conclude that the system (C,A ; Q) isobservable. Invoking (i) (ii) from Proposition 3.2 we deduce that if the system (4.4) would be stochastically [ ] γ1 (i) detectable, then would exist the matrices X(i) > and Γ(i) =,i =1, 2 which γ 2 (i) verify the following system of linear equations: A (i)x(i)+x(i)a (i)+γ(i)c (i)+c (i)γ (i)+ 2 q ij X(j)+I 2 =,i=1, 2 [ ] 2γ1 (1) γ which implies I (1) < which is a contradiction. γ 2 (1) Example 3 Let us consider the stochastic system (2.1)-(2.2) with n = 2, d = 2, r = 1, p =1, A (1) = A (2) = αi 2, C (1) =[1 ], C (2) =[ 1], A 1 (1) = βi 2, A 1 (2) is a [ ] q q 2 2 arbitrary matrix, Q =,α R, β R, q> which satisfy 2α q+β 2 =. q q From Remark 4.2 it follows that the above system is stochastically observable. As in the previous example based on equivalence (i) (ii) from Proposition 3.2 we may show that it is not stochastically detectable (see citepreprint1 for details). The following two results show that the stochastic observability defined in this paper guarantees the positivity of the observability gramian and the fact that all semipositive solutions of the Riccati equations are stabilizing solutions as in the deterministic case. The proofs are omitted for shortness and may be found in [1]. Proposition 4.6 Assume that (C,A,...,A r ; Q) is observable and the algebraic equation on Sn d j=1 L X + C = (4.18) has a solution X. Then : (i) The system (A,A 1,...,A r ; Q) is stable. (ii) X >. (iii) The equation (4.5) has a unique positive semidefinite solution. 8

9 Let us consider the following system of general algebraic Riccati equations: A (i)x(i)+x(i)a (i)+ dq ij X(j) (X(i)B (i)+ A K(i)X(i)A k (i)+ j=1 A k(i)x(i)b k (i))(d (i)d(i)+ Bk(i)X(i)B k (i)) 1 (B(i)X(i)+ Bk(i)X(i)A k (i)) + C(i)C (i) = (4.19) where A k (i),c (i) are as in (2.1)-(2.2) and B k (i),d(i) are n m and p m given matrices. Such algebraic equations are closly related to the linear quadratic optimization problem for a linear stochastic system subjected both to Markovian jumping and multiplicative white noise (see [4, 1] for details). The following result is proved in [1]. Proposition 4.7 If (C, A; Q) is observable then any semipositive solution of the system (4.6) is a stabilizing solution. References [1] B.D.O.Anderson, J.B.Moore, New results in linear system stability, SIAM j. Control, vol. 7, no. 3, pp , [2] J.L.Doob, Stochastic processes, Wiley, New-York, [3] V. Dragan, T. Morozan, Stability and robust stabilization to linear stochastic systems described by differential equations with markovian jumping and multiplicative white noise, Stochastic Analysis and Application, nr. 1, 22 (to appear). [4] V. Dragan, T. Morozan, The linear quadratic optimization problem and tracking problem for a class of linear stochastic systems with multiplicative white noise and Markovian jumping, Preprint nr. 3/ 21, Institute of Mathematics od Romanian Academy. [5] A. Friedman, Stochastic differential equations and applications, Academic Press, vol. I, [6] Y. Ji, H.J.Chizeck, Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control, IEEE Trans. Auto. Control, 35, no.7, pp , 199. [7] R. Kalman, Contributions on the theory of optimal control, Buletin de la Sociedad Math. Mexicana, Segunda Serie, 5, 1, pp ,

10 [8] T. Morozan, Optimal stationary control for dynamic systems with Markov perturbations, Stochastic Analysis Appl., 1, pp , [9] T. Morozan, Stochastic uniform observability and Riccati equations of stochastic control, Revue Roum. Math. Pures et Appl., 9, [1] T. Morozan, Linear quadratic control and tracking problems for time-varying stochastic differential systems perturbed by a Markov chain, Revue Roum. Math. Pure et Applique, to appear. [11] B. Oksendal, Stochastic differential equations, Springer, Fifth Edition, 2. [12] W.H. Wonham, On a matrix Riccati equation of stochastic control, SIAM J. Control Optim., 6, pp , [13] W.H. Wonham, Random diferential equations in control theory, Probabilistic Methods in Applied Math.,2, (A.T.Barucha-Reid, Ed.) Academic Press, New-York, pp ,