Reachability, Observability and Minimality for a Class of 2D Continuous-Discrete Systems

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1 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 27 Reachability, Observability and Minimality for a Class of 2D Continuous-Discrete Systems VALERIU PREPELIŢĂ University Politehnica of Bucharest Department of Mathematics Splaiul Indpendentei 33 ROMANIA vprepelita@mathem.pub.ro IONEL TEVY University Politehnica of Bucharest Department of Mathematics Splaiul Indpendentei 33 ROMANIA tevy@mathem.pub.ro Abstract: Reachability and observability criteria are obtained for 2D continuous-discrete time-variable Attasi type systems by using suitable 2D reachability and observability Gramians. Necessary and sufficient conditions of reachability and observability are derived for time-invariant systems. The duality between the two concepts is emphasized as well as their connection with the minimality of these systems. Key Words: 2D continuous-discrete linear systems, controllability, reachability, observability, duality principle. Typing manuscripts, LATEX Introduction The concepts of controllability and observability, introduced by Kalman for D systems were extended to 2D systems for Roesser [9], Fornasini and Marchesini [2], and Attasi [] models; in order to keep their relationship with minimality, new concepts of modal controllability and modal observability were introduced in [6]. In this paper a class of 2D continuous-discrete time-variable linear systems is studied, which is related to Attasi s 2D discrete model and represents the extension to time-variable framework of the hybrid systems introduced in [8]. Such systems can appear in various problems as signal and image processing, seismology and geophysics, control of multipass processes, iterative learning control synthesis [5] or repetitive processes [3]. The state and output formulæ for these systems are established in Section 2 and the notions of complete reachability and complete observability are defined. These properties are characterized by means of the full ranks of suitable 2D reachability and observability Gramians. Section 4 is devoted to time-invariant 2D continuous-discrete systems and a list of criteria of reachability and observability is provided. The duality between the two concepts is emphasized. In Section 5 the relation between reachability, observability and minimality is established. WSEAS Transactions on Mathematics,... (27),... 2 The state space representation of the 2D continuous-discrete systems We consider the linear spaces X R n, U R m and Y R p, called respectively the state, input and output spaces. The time set is T R Z. Definition A two-dimensional continuousdiscrete linear system (2Dcd) is a quintuplet Σ (A (t, k), A 2 (t, k), B(t, k), C(t, k), D(t, k)) R n n R n n R n m R p n R p m with A (t, k)a 2 (t, k) A 2 (t, k)a (t, k) (t, k) T, where all matrices are continuous with respect to t R for any k Z; the state space representation of Σ is given by the state and output equations ẋ(t,k+)a (t,k+)x(t,k+)+a 2 (t,k)ẋ(t,k) A (t, k)a 2 (t, k)x(t, k) + B(t, k)u(t, k) () y(t, k) C(t, k)x(t, k) + D(t, k)u(t, k) (2) where ẋ(t, k) x t (t, k). Let us denote by Φ(t, t ; k) or Φ A (t, t ; k) the (continuous) fundamental matrix of A (t, k) with respect to t R, for any fixed k Z. Φ(t, t ; k) has the following properties, for any t, t, t R: i) d dt Φ(t, t ; k) A (t, k)φ(t, t ; k), ii) Φ(t, t ; k) I n, iii) Φ(t, t ; k)φ(t, t ; k) Φ(t, t ; k), iv) Φ(t, t ; k) Φ(t, t; k). (3)

2 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, If A is a constant matrix, then Φ(t, t ; k) e A (t t ). The discrete fundamental matrix F (t; k, k ) of the matrix A 2 (t, k) is defined by F (t; k, k ) { A2 (t,k )A 2 (t,k 2) A 2 (t,k ) for k > k I n for k k for any fixed t R. If A 2 is a constant matrix, then F (t; k, k ) A k k 2. Φ(t, t ; k) and F (s; l, l ) are commutative matrices for any t, t, s R and k, l, l Z since A (t, k) and A 2 (t, k) are commutative matrices. Definition 2 A vector x X is said to be the initial state of Σ at the moment (t, k ) T if, for any (t, k) T with (t, k) (t, k ) the following conditions hold: x(t,k )Φ(t, t ;k )x, x(t,k)f (t ; k,k )x. (4) In [7] it was proved: Proposition 3 The state of the system Σ at the moment (t, k) T determined by the control u(, ) and by the initial state x X is t t k x(t,k)φ(t,t ;k)f (t ;k,k )x + lk Φ(t, s; k)f (s; k, l + )B(s, l)u(s, l)ds. (5) By replacing the state x(t, k) given by (5) in the output equation (2) we obtain Proposition 4 The input-output map of the system Σ is given by the formula t y(t,k)c(t,k)φ(t,t ;k)f (t ;k,k )x + k C(t,k) t lk Φ(t, s; k)f (s; k,l+)b(s,l)u(s,l)ds+d(t,k)u(t,k). (6) 3 Reachability and observability of time-variable 2D continuousdiscrete systems For the concept of reachability we need only the state equation (), hence a 2Dcd system can be reduced to the triplet Σ (A (t, k), A 2 (t, k), B(t, k)). For observability the system can be reduced to the triplet Σ (A (t, k), A 2 (t, k), C(t, k)). For both notions a system will be a quadruplet Σ (A (t, k), A 2 (t, k), B(t, k), C(t, k)). By (s, l) < (t, k) we mean s t, l k and (s, l) (t, k). A triplet (t, k, x) R Z X is said to be a phase of Σ if x is the state of Σ at the moment (t, k) (i.e. x x(t, k), where x(t, k) is given by (5)). Definition 5 A phase (t, k, x) of Σ is said to be reachable if there exist (t, k ) T, (t, k ) < (t, k) and a control u(, ) which transfers the phase (t, k, ) to (t, k, x). A phase (t, k, x) is said to be controllable if there exist (t, k ) T, (t, k ) > (t, k) and a control u(, ) which transfers the phase (t, k, x) to (t, k, ). If for some fixed (τ, χ) T every phase (τ, χ, x) is reachable (controllable), the system Σ is said to be completely reachable (completely controllable) at the moment (τ, χ). Definition 6 A phase (τ, χ, x) is said to be unobservable if for any control u it provides the same output y(s, l) for (s, l) (τ, χ) as the phase (τ, χ, ). In this case the state x X is said to be unobservable at (τ, χ). The system Σ is said to be completely observable at (τ, χ) if there is no state unobservable at (τ, χ). Definition 7 The matrices R Σ (t, τ; k, χ) τ χ t lk Φ(τ, s; χ)f (s; χ, l+) B(s, l)b(s, l) T F (s; χ, l + ) T Φ(τ, s; χ) T ds, (7) O Σ (τ, t; χ, k) t τ k Φ(s, τ; l) T F (τ; l, χ) T lχ C(s, l) T C(s, l)f (τ; l, χ)φ(s, τ; l)ds. (8) are called respectively the reachability Gramian and the observability Gramian of Σ. We have proved in [8]: Theorem 8 Σ is completely reachable at (τ, χ) if and only if rankr Σ (t, τ; k, χ) n for some (t, k) (τ, χ). Theorem 9 The system Σ (A (t, k), A 2 (t, k), C(t, k)) is completely observable at (τ, χ) if and only if for some (t, k) (τ, χ). ranko Σ (τ, t; χ, k) n. (9)

3 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, Reachability and observability of time-invariant 2D continuousdiscrete systems Let us consider a time invariant system Σ (A, A 2, B, C), i.e. a system with A, A 2, B and C constant matrices. In this case we can consider the initial moment (t, k ) (, ) and the time set T R + Z +. Then the state formula (5) and the input-output map (6) become t k l t k l x(t, k) e A t A k 2x + e A (t s) A k l 2 Bu(s, l)ds () y(t, k) Ce A t A k 2x + Ce A (t s) A k l 2 Bu(s, l)ds. () Definition The system Σ d (A d, Ad 2, Bd, C) is called the dual of Σ if A d AT, Ad 2 AT 2, Bd C T, C d B T. We can prove (see [8]) the Duality Principle: Theorem The system Σ d is completely observable if and only if Σ is completely reachable. We associate to Σ the reachability matrix C Σ [B A B... A n B A 2 B A A 2 B... A n A 2 B... A n 2 B A A n 2 B A n A n 2 B] and the observability matrix O Σ [C T A T CT... (A T )n C T A T 2 CT A T AT 2 CT... (A T )n A T 2 CT... (A T 2 )n C T A T (AT 2 )n C T... (A T )n (A T 2 )n C T ] T. Theorem 8 gives (see [7],Theorem 4.2) Theorem 2 Σ (A, A 2, B) is completely reachable if and only if We can prove rankc Σ n. (2) Proposition 3 The set of all reachable states of Σ is X r ImC Σ. Proposition 4 The set of all reachable states of Σ is the smallest subspace of X which is (A, A 2 )- invariant and contains the columns of B. An immediate consequence of Proposition 4 is the following Theorem 5 Σ (A, A 2, B) is completely reachable if and only if X is the smallest subspace of X which is (A, A 2 )-invariant and contains the columns of B. By duality (Theorem ) and Theorem 2 we obtain Theorem 6 The system Σ (A, A 2, C) is completely observable if and only if ranko Σ n. (3) From () and Definition 7 we obtain Proposition 7 The set of all unobservable states of Σ is X uo KerO Σ. Proof: By using Hamilton-Cayley Theorem for A and A 2 we obtain X uo {x X CA l Ak 2 x, l, k, n } KerO Σ. Proposition 8 The set X uo of all unobservable states of Σ is the greatest subspace of X which is (A, A 2 ) invariant and is contained in KerC. We obtain from Proposition 8: Theorem 9 The system Σ (A, A 2, C) is completely observable if and only if {} is the greatest subspace of X which is (A, A 2 ) invariant and is contained in KerC. Definition 2 Two systems Σ (A, A 2, B, C) and Σ (Ã, Ã2, B, C) are said to be isomorphic if there exists a nonsingular matrix T R n n such that à i T A i T, i, 2; B T B, C CT. (4) Theorem 2 Any system Σ (A, A 2, B) is isomorphic to a system Σ (Ã, Ã2, B) of the form [ ] [ ] A A à 2 A2 A, à A 2 22, 22 A 222 [ ] B B, (5) with A, A 2 R q q, B R q m, q n. The triplet Σ (A, A 2, B ) is completely reachable. Proof: We consider the direct sum decomposition of the state space X R n as X X X 2 where X r X. The partitions of the matrices in (5) are obtained with respect to this decomposition, since by Proposition 4 X r is (A, A 2 )-invariant and contains the columns of B; q is the dimension of the subspace X r. We can derive other criteria of reachability.

4 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 27 4 Theorem 22 Σ (A, A 2, B) is completely reachable if and only if there is no common left eigenvector of matrices A and A 2, orthogonal on the columns of B. Proof:Let us assume that there exists v R n \ {} such that λ, µ C with v T A λv T, v T A 2 µv T and v T B. Then v T A i Aj 2 B λi µ j v T B i, j, hence v T C Σ, i.e. Σ (A, A 2, B) is not completely reachable. Conversely, if Σ is not completely reachable, then there exists v R n \ {} such that v T C Σ, hence the subspace S {x R n x T C Σ } contains a vector v. If x S, then x T A i Aj 2B for any i, j, n and by Hamilton-Cayley Theorem this equality is true for any i, j. Then, for any x S, (A T x)t A i Aj 2 B xt A i+ A j 2B, i, j, hence A T x S, i.e. S is A T -invariant; analogously, S is A T 2 -invariant. It results that S contains an eigenvector x of A T ; let λ be the corresponding eigenvalue. Let us consider the subspace S 2 {x X A T x λx}. If x S 2 then A T (AT 2 x) AT 2 AT x λat 2 x, hence AT 2 x S 2, that is S 2 is A T 2 -invariant and so is S 3 S S 2. Then S 3 contains an eigenvector w of A T 2 and since S 3 S 2, w is an eigenvector of A T too. Moreover, since S 3 S, we have w T C Σ and particularly w T B, hence w is a common left eigenvector of A and A 2 orthogonal on the columns of B. The following theorem is an extension to 2Dcd systems of the Popov-Hautus-Belevitch criterion of reachability. Theorem 23 Σ (A, A 2, B) is completely reachable if and only if for any λ, λ 2 C rank[ B λ I A λ 2 I A 2 ] n. Proof: Obviously, the existence of λ, λ 2 C such that rank[ B λ I A λ 2 I A 2 ] < n is equivalent to the existence of v R n \ {} such that v T [ B λ I A λ 2 I A 2 ] which means v T B, v T A λ v T, v T A 2 λ 2 v T that is, by Theorem 22, to the fact that Σ is not completely reachable. By duality we obtain the following results concerning observability: Theorem 24 The system Σ (A, A 2, C) is completely observable if and only if there is no common eigenvector of the matrices A and A 2 belonging to KerC. Theorem 25 The system Σ (A, A 2, C) is completely observable if and only if for any λ, λ 2 C C rank λ I A n. λ 2 I A 2 5 Reachability, observability and minimality Definition 26 The matrix T Σ (s, z) C(sI A ) (zi A 2 ) B (6) is called the transfer matrix of the time-invariant system Σ (A, A 2, B, C). Obviously, T Σ (s, z) is a p m rational strictly proper (in both variables s and z) matrix with separable denominator, since it has the form T Σ (s, z) det(si A )det(zi A 2 ) C(sI A ) (zi A 2 ) B. Definition 27 Given a strictly proper matrix T (s, z), a system Σ (A, A 2, B, C) is said to be a realization of Σ if T (s, z) T Σ (s, z), that is if T (s, z) C(sI A ) (zi A 2 ) B. (7) A realization Σ of T (s, z) is minimal if dimσ dim Σ for any realization Σ of T (s, z). Now let us consider the Laurent series expansion of T (s, z) about s, z T (s, z) M i,j s i z j. (8) i j The matrices M i,j R p m are called the Markov parameters of T (s, z). Proposition 28 Σ (A, A 2, B, C) is a realization of T (s, z) if and only if, for any i, j N, M i,j CA i A j 2B. (9) Proof: By (7) we have T (s, z) C(sI A ) (zi A 2 ) B C( i A i s i )( j A j 2 z j )B i j CA i Aj 2 s i z j. Since (8) holds and two equal Laurent series have equal corresponding coefficients, (9) is true. The following theorem establishes the connection between the concepts of reachability, observability and minimality.

5 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 27 5 Theorem 29 A system Σ (A, A 2, B, C) is a minimal realization of some strictly proper matrix T (s, z) if and only if Σ is completely reachable and completely observable. Proof: Necessity. By negation, let us assume that Σ is not completely reachable. Then Σ is isomorphic to a system Σ as in Theorem 2 with q < n. If we partition C as C [ C C 2 ] with C R p q, since (si n [ ] à k ) (siq A k ) A 2k (si n q A 22k ) [ (siq A k ) ]? (si n q A 22k ), k, 2 we obtain T (s, z) T Σ (s, z) T Σ(s, z) C (si q A ) (zi q A 2 ) B T Σ z), hence Σ (A, A 2, B, C ) is a realization of T (s, z) of dimension q < n, i.e. Σ is not minimal. The case Σ not completely observable is similar. Sufficiency. Let us assume that Σ is completely reachable and completely observable. If Σ is not minimal, let Σ (Â, Â2, B, Ĉ) be a realization of T (s, z) with dim Σ n < n dimσ. Let us consider the controllability and observability matrices C Σ and O Σ and let us denote by and the matrices C Σ O Σ obtained from C Σ and O Σ by replacing the matrices A, A 2, B and C by Â, Â2, B and Ĉ. By Proposition 28, since Σ and Σ are realizations of the same matrix T (s, z) we have (see (9)): CA i Aj 2 B ĈÂi B, Âj 2 i, j ; these matrices are the block elements of the product matrices O Σ C Σ and Ô ΣĈΣ hence O Σ C Σ ÔΣĈΣ. Then, by Sylvester Inequalities, we have ranko Σ + rankc Σ n ranko Σ C Σ rankôσĉσ min(rankôσ, rankĉσ). By hypothesis ranko Σ n, rankc Σ n and rankôσ n < n, rankĉσ n < n; we get n n, contradiction, hence Σ is minimal. From Theorems 23, 25 and 29 we get Theorem 3 Σ (A, A 2, B, C) is a minimal realization if and only if and rank[ B λ I A λ 2 I A 2 ] n, (2) rank for any λ, λ 2 C. C λ I A λ 2 I A 2 n (2) Since for any n n matrix A det(si A) if and only if s σ(a) (where σ(a) denotes the spectrum of A, rank(si A) n for any s / σ(a) and Theorem 3 gives Corollary 3 Σ (A, A 2, B, C) is a minimal realization if and only if equalities (2) and (2) hold for any (s, z) σ(a ) σ(a 2 ). Application Let us consider the 2Dcd system a Σ (A, A 2, B, C), where A b c, c b A 2 C [ d, B and ]. The characteristic polynomials are det(si A ) (s a)(s b c)(s b+c) and det(si A 2 ) (z a)(z )(z + ). The matrices in (2) and (2) are respectively C s a z d s b c z c s b z O T and s a z d s b c z c s b z The spectra are respectively σ(a ) {a, b c, b + c} and σ(a 2 ) {d,, } Then the 3 rd order minors of C have respectively the values, (a b) 2 c 2,d 2 and a + b + dc for (s, z) (a, d), c(b + c a), a b + c c(d ) and d for (s, z) (b + c, ), c(a b + c), 2c(a b + c), and c(d + ) for (s, z) (b c, ) and there are similar expressions for the other values of (s, z) σ(a ) σ(a 2 ). We can conclude that Σ is unreachable in the following four cases:(a b c, d ), (c, d ), (a b c, d ) and (c, d ); otherwise Σ is completely reachable. Obviously, for s a and z d ranko 2, hence Σ is unobservable. By Corollary 3 Σ is not minimal. Now, the transfer matrix of Σ is T Σ (s, z) C(sI A ) (zi A 2 ) B (s a)(s b c)(z d)( z) (s a)(s b c)(s b+c)(z d)(z )(z+) (s b+c)(z+) hence a minimal realization of Σ is à b c, à 2, B and C. 6 Conclusion Reachability and observability criteria were obtained for 2D continuous-discrete time-variable systems by using suitable 2D reachability and observability Gramians. In the case of time-invariant systems, necessary and sufficient conditions of reachability and observability were obtained by means of duality. The relationship between reachability, observability and minimality was emphasized. This research can be developed and extended to other topics of the Systems.

6 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 27 6 Theory such as stability, stabilizability, detectability, feedback and observers and optimal control for 2D continuous-discrete systems. Acknowledgements: Partially supported by the Grant CNCSIS 86/ 27 and by the 5-th Italian-Romanian Executive Programme of S&T Cooperation for 26-28, University Politehnica of Bucharest. References: [] S. Attasi, Introduction d une classe de systèmes linéaires reccurents à deux indices, C.R. Acad. Sc. Paris 277, 973, p. 35. [2] E. Fornasini and G. Marchesini, State Space Realization Theory of Two-Dimensional Filters, IEEE Trans.Aut.Control AC 2, 976, pp [3] K. Galkovski, E. Rogers and D.H. Owens, New 2D models and a transition matrix for discrete linear repetitive processes, Int.J.Control 5,999, pp [4] S. Y. Kung, B.C. Lévy, M. Morf and T. Kailath, New Results in 2-D Systems Theory, Part II: 2-D State-Space Models-Realizations and the Notions of Controllability, Observability and Minimality, Proc. of the IEEE 6, 977, pp [5] J. Kurek and M.B. Zaremba, Iterative learning control synthesis on 2D system theory, IEEE Trans.Aut.Control, 993, pp [6] V. Prepeliţă, Linear hybrid systems. Bull.Math.Soc.Sci.Math. de Roumanie 4, 979, pp [7] V. Prepeliţă, Criteria of reachability for 2D continuous-discrete systems, Rev. Roumaine Math. Pures Appl., 23, pp [8] V. Prepeliţă and Constantin Drăguşin, The duality principle in a class of 2D continuous-discrete linear systems, Math. Reports 4, 23, pp [9] R.P. Roesser, A Discrete State-Space Model for Linear Image Processing, IEEE Trans.Aut.Control, 975, pp. -.