Generalized 2D Continuous-Discrete Systems with Boundary Conditions

Generalized 2D Continuous-Discrete Systems with Boundary Conditions VALERIU PREPELIŢĂ University Politehnica of Bucharest Department of Mathematics-Informatics I Splaiul Independentei 313, 060042 Bucharest ROMANIA vprepelita@mathem.pub.ro Abstract: 2D generalized hybrid systems are considered, having the coefficient matrices of bounded variation or regulated matrix functions, the controls being regulated vector functions. The formulæ of the state and of the input-output map of these systems are obtained. Some 2D separable kernels are associated to these systems and a 2D continuous-discrete Wiener-Hopf equation is studied in this framework. The realization problem is discussed for 2D separable kernels and necessary and sufficient conditions for the minimality of a realization are provided. Key Words: 2D hybrid systems, input-output map, regulated functions, functions of bounded variation, semiseparable kernels, realizations, minimality. 1 Introduction An important class of 2D systems is represented by the hybrid models, their behavior depending on two variables, one continuous and the other one discrete [5], [12]. They were used as models in the study of linear repetitive processes [1], [14] with practical applications in iterative learning control synthesis [8] or in long-wall coal cutting and metal rolling. One aim of the paper is to extend the study of the hybrid systems to the general framework represented by the space of regulated functions. The topic of regulated functions is studied in a series of monographs or papers (e.g. [4], [16], [17]). We use Tvrdý s results [16], [17], concerning the properties of the Perron-Stieltjes integral with respect to regulated functions and the differential equation in this space. A class of 2D generalized hybrid linear systems is considered, having the controls over the space of regulated functions, the drift matrix with respect to the continuous variable being of bounded variation and the other coefficient matrices being regulated matrix functions. This class is the 2D hybrid counterpart of 1D continuous-time acausal systems introduced by Krener [6], [7] and developed by Gohberg and Kaashoek [2], [3]. Some extended models were studied in [10], [13]. The formulæ of the state and of the input-output map of these systems are provided. Generalized 2D separable kernels are associated to these systems and a 2D continuous-discrete Wiener- Hopf equation is studied in this framework. The realization problem is discussed for 2D separable kernels and necessary and sufficient conditions for minimality are obtained. We shall use the following definitions and notations. A function f : [a, b] R which posseses finite one sided limits f(t ) and f(t+) for any t [a, b] (where by definition f(a ) = f(a) and f(b+) = f(b)) is said to be regulated on [a, b]. The set of all regulated functions denoted by G(a, b), endowed with the supremal norm, is a Banach space; the set BV (a, b) of functions of bounded variation on [a, b] with the norm f = f(a) + var b af is also a Banach space; the Banach space of n-vector valued functions belonging to G(a, b) and BV (a, b) respectively are denoted by G n (a, b) and BV n (a, b) (or simply G n and BV n ); BV n m denotes the space of n m matrices with entries in BV (a, b). The set of functions f : [a, b] Z R such that k Z, f(, k) G(a, b) (BV (a, b)) will be denoted G 1 (a, b) (BV 1 (a, b)) and similar significances will have the above mentioned spaces with subscript 1 (G n 1, BV 1 n n m, BV1 ). A pair D = (d, s) where d = {t 0, t 1,..., t m } is a division of [a, b] (i.e. a = t 0 < t 1 <... < t m = b) and s = {s 1,..., s m } verifies t j 1 s j t j, j = 1,..., m is called a partition of [a, b]. A function δ : [a, b] (0, + ) is called a gauge on [a, b]. Given a gauge δ, the partition (d, s) is said to be δ-fine if [t j 1, t j ] (s j δ(s j ), s j + δ(s j )), j = 1,..., m. ISSN: 1790-2769 67 ISBN: 978-960-474-131-1

Given the functions f, g : [a, b] R and a partition D = (d, s) of [a, b] let us associate the integral sum m S D (f g) = f(s j )(g(t j ) g(t j 1 )). j=1 Definition 1 The number I R is said to be the Perron-Stieltjes (Kurzweil) integral of f with respect to g from a to b and it is denoted as or b a b a fdg f(t)dg(t) if for any ε > 0 there exists a gauge δ on [a, b] such that I S D (f g) < ε for all δ-fine partitions D of [a, b]. Given f G(a, b) and g G([a, b] [a, b]) we define the differences +,, and + s, s, s by + f(t) = f(t+) f(t), f(t) = f(t) f(t ), f(t) = f(t+) f(t ), + s g(t, s) = g(t, s+) g(t, s), s g(t, s) = g(t, s) g(t, s ); D (f), D + (f) denote respectively the set of the left and right discontinuities of f in [a, b] and similarly for g we can define D t (g), D+ t (g) with respect to the argument t. We denote by the sum where t t D D = D (f) D + (f) D (g) D + (g). 2 Generalized Differential Equations The symbol dx = d[a]x + dg (1) where A BV n n and g G n (a, b) is said to be a generalized linear differential equation (GLDE) in the space of regulated functions. Definition 2 A function x : [a, b] R n is said to be a solution of GLDE (1) if for any t, t 0 [a, b] it verifies the equality x(t) = x(t 0 ) + d[a(s)]x(s) + g(t) g(t 0 ). (2) t 0 Theorem 3 ([10, Theorem III.2.10]) Assume that for any t [a, b] the matrix A BV n n verifies the condition det[i + + A(t)] det[i A(t)] 0. (3) Then there exists a unique matrix valued function U : [a, b] [a, b] R n n such that, for any (t, s) [a, b] [a, b] U(t, s) = I + s d[a(τ)]u(τ, s). (4) U(t, s) is called the fundamental matrix solution of the homogeneous equation dx = d[a]x Some methods for the calculus of the fundamental matrix U(t, s) were provided in [9]. 3 Input-output maps of 2D generalized systems The linear spaces X = G n 1, U = Gm 1 and Y = Gp 1 are called respectively the state, input and output spaces. The time set is T = [, b 1 ] {a 2, a 2 + 1,..., b 2 }, where [, b 1 ] R and a 2, b 2 Z. Definition 4 A 2D generalized continuous-discrete system (2Dgcd) is an ensemble Σ = (A 1 (t, k), A 2 (t, k), B(t, k), C(t, k), D(t, k), N 1, N 2, M 1, M 2 ) BV n n 1 G n n 1 G n m 1 G p n 1 G p m 1 R n n R n n R n n R n n where A 1 (t, k)a 2 (t, k) = A 2 (t, k)a 1 (t, k), (t, k) T, with the following state equation, output equation, boundary condition and output vector equation: dx(t, k + 1) = d[a 1 (t, k + 1)]x(t, k + 1)+ +A 2 (t, k)dx(t, k) d[a 1 (t, k)]a 2 (t, k)x(t, k)+ +B(t, k)du(t, k), y(t, k) = C(t, k)x(t, k) + D(t, k)u(t, k), (6) N 1 x(, a 2 ) + N 2 x(b 1, b 2 ) = v, (7) z = M 1 x(, a 2 ) + M 2 x(b 1, b 2 ). (8) n is called the dimension of the system Σ and it is denoted dimσ. Let U(t, t 0 ; k) be the fundamental matrix of A 1 (t, k), k {a 2, a 2 + 1,..., b 2 } and F (t; k, k 0 ) the discrete fundamental matrix of A 2 (t, k), t [a, b]. Definition 5 A vector x 0 X is called the initial state of Σ at the moment (t 0, k 0 ) T if (t, k) T with (t, k) (t 0, k 0 ) From [13] we have Proposition 6 If x(t, k 0 ) = U(t, t 0 ; k 0 )x 0, x(t 0, k) = F (t 0 ; k, k 0 )x 0. det[(i A i (t, k))(i + + A i (t, k))] 0, i = 1, 2, t [a, b], k Z, (9) (10) (5) ISSN: 1790-2769 68 ISBN: 978-960-474-131-1

then the state of the system at the moment (t, k) T determined by the initial state x 0 at the initial moment (t 0, k 0 ) T and by the control u : [t 0, t] {k 0, k 0 + 1,..., k 1} R m is x(t, k) = U(t, t 0 ; k)f (t 0 ; k, k 0 )x 0 + + t 0 k 1 l=k 0 U(t, s; k)f (s; k, l + 1)B(s, l)du(s, l)+ k 1 + + s U(t, s; k) F (s; k, l + 1) t 0 s<t l=k 0 B(s, l) + u(s, l) s U(t, s; k) k 1 t 0 <s t F (s; k, l + 1)B(s, l) u(s, l). l=k 0 (11) Definition 7 The boundary condition (7) is said to be well-posed if the homogeneous problem corresponding to (5) and (7) (i.e. with u 0 and v = 0) has the unique solution x = 0. Proposition 8 The boundary condition (7) is wellposed if and only if the matrix R = N 1 + N 2 U(b 1, ; b 2 )F ( ; b 2, a 2 ) is nonsingular. Proof: By (11) with u 0 we get x(b 1, b 2 ) = U(b 1, ; b 2 )F ( ; b 2, a 2 )x(, a 2 ); we replace x(b 1, b 2 ) and v = 0 in (7). It results that (7) is well-posed if and only if the equation [N 1 + N 2 U(b 1, ; b 2 )F ( ; b 2, a 2 )]x(, a 2 ) = 0 has the unique solution x(, a 2 ) = 0, condition which is equivalent to R nonsingular. In the sequel we shall consider systems with wellposed boundary condition (7) and which verify (11). Moreover, the discrete-time character of Σ with respect to the variable k imposes the following assumption: the matrices A 2 depend only on k and A 2 (k) are nonsingular for any k {a 2, a 2 + 1,..., b 2 }. Definition 9 The matrix P = P Σ = R 1 N 2 U (b 1, ; b 2 )F (b 2, a 2 ) is called the canonical boundary value operator of the system Σ with well-posed boundary condition. Theorem 10 The state of the system Σ determined by the control u : T R m and by the input vector v R n is x(t, k) = U(t, ; k)f (k, a 2 )R 1 v b U(t, ; k)f (k, a 2 ) P U(, s; b 2 )F (a 2, l + 1)B(s, l)du(s, l)+ k 1 + U(t, s; k)f (k, l + 1)B(s, l)du(s, l)+ +U(t, ; k)f (k, a 2 )(I P ) b + s U(, s; b 2 ) F (a 2, l + 1) s<b 1 B(s, l) + u(s, l) b <s t s U(, s; b 2 ) F (a 2, l + 1)B(s, l) u(s, l). (12) Proof: We replace x(b 1, b 2 ) given by (11) in the boundary condition (7). We get [N 1 + N 2 U(b 1, ; b 2 )F (b 2, a 2 )]x 0 + +N 2 b U(b 1, s; b 2 )F (b 2, l + 1)B(s, l)du(s, l)+ + b + s U(b 1, s; b 2 ) F (b 2, l + 1) s<b 1 B(s, l) + u(s, l) s U(b 1, s; b 2 ) <s b 1 b F (b 2, l + 1)B(s, l) u(s, l) = v hence, by the semigroup properties of the fundamental matrices U and F, we obtain b x 0 = R 1 v P U(, s; b 2 )F (a 2, l + 1) B(s, l)du(s, l) P + s U(, s; b 2 ) s<b 1 b F (a 2, l + 1)B(s, l) + u(s, l)+ +P s U(, s; b 2 ) (13) <s b 1 b F (a 2, l + 1)B(s, l) u(s, l). ISSN: 1790-2769 69 ISBN: 978-960-474-131-1

We replace x 0 = x(, a 2 ) in (11); then (12) results by using the semigroup property, i.e. U(b 1, s; b 2 ) = U(b 1, ; b 2 )U(, s; b 2 ) and F (b 2, l + 1) = F (b 2, a 2 )F (a 2, l + 1). Theorem 11 The input-output map of the 2Dgcd system Σ is H : G m 1 Rn G p 1 Rn, H(u, v) = (y, z) where y(t, k) is obtained by replacing the state x(t, k) (12) in the output equation (6) and, by denoting Q = M 1 + M 2 U(b 1, ; b 2 ) F (b 2, a 2 ), S = Q(I P ) M 1, b z = QR 1 v + S U(, s; b 2 )F (a 2, l + 1) B(s, l)du(s, l) + S s<b 1 + s U(, s; b 2 ) b F (a 2, l + 1)B(s, l) + u(s, l) s U(, s; b 2 ) <s t b F (a 2, l + 1)B(s, l) u(s, l). (14) Proof: By replacing x(, a 2 ) = x 0 (13) and x(b 1, b 2 ) given by (12) in (8) and by a long calculus which uses the semigroup property and which is omitted, we get (14). Corollary 12 If u G m 1 then x Gn 1 and y Gp 1. If A 2 BV1 n n, B BV1 n m, C BV p n 1, D BV1 n m and u BV1 m then x BV1 n p and y BV1. Proof: We apply [15, Theorem III 3.1] and [17, Proposition 2.5] Definition 13 The space of admissible controls is the set U = {u G m 1 (a, b) D+ t (A i(, k) D + t (u(, k)) =, D t (A i(, k)) D t (u(, k)) =, i = 1, 2, k Z}. Corollary 14 If u U, then the output of the system Σ is given by the following formula: y(t, k) = C(t, k)u(t, ; k)f (k, a 2 )R 1 v b C(t, k)u(t, ; k)f (k, a 2 ) P U(, s; b 2 )F (a 2, l + 1)B(s, l)du(s, l)+ k 1 + C(t, k)u(t, s; k)f (k, l + 1) B(s, l)du(s, l) + D(t, k)u(t, k). (15) 4 Realizations of 2D semiseparable kernels Let us consider the case A 1 : [, b 1 ] R n n and A 2 : {a 2, a 2 + 1,..., b 2 } R n n. Therefore the fundamental matrices of A 1 and A 2 are respectively U(t, s) and F (k, l). We denote by T (t, k) and T (t, k) the sets T (t, k) = [, t) {a 2, a 2 + 1,..., k 1} and T (t, k) = T \ T (t, k) respectively. We shall consider the notation T b def = b k 1. Assume v = 0. Then, by the semigroup properties U(t, s) = U(t, )U(, s), F (k, l + 1) = F (k, a 2 )F (a 2, l + 1), (15) can be written as y(t, k) = k 1 F (k, a 2 )(I P ) C(t, k)u(t, ) U(, s)f (a 2, l + 1)B(s, l)du(s, l) T b C(t, k)u(t, )F (k, a 2 ) P U(, s)f (a 2, l + 1)B(s, l)du(s, l)+ +D(t, k)u(t, k), hence (16) can be written in the form y(t, k) = b K(t, s; k, l)du(s, l)+ + D(t, k)u(t, k). (16) (17) Definition 15 The function K(t, s; k, l) is called the kernel of the equation (17). A kernel is said to be 2D semiseparable if it has the form = K(t, s; k, l) = { E1 (t, k)g 1 (s, l) if (s, l) T (t, k) E 2 (t, k)g 2 (s, l) if (s, l) T (t, k). (18) where E 1, E 2 BV p n 1, G 1, G 2 BV1 n m. The kernel K obtained from the input-output equation (16) of a system Σ is denoted by K Σ and it is called the kernel of the system Σ. Proposition 16 The kernel K Σ of any 2Dgcd system with well-posed boundary condition is 2D semiseparable. ISSN: 1790-2769 70 ISBN: 978-960-474-131-1

Proof: From (16) and (17) we get (18) with E 1 (t, k) = C(t, k)u(t, )F (k, a 2 )(I P ), G 1 (s, l) = U(, s)f (a 2, l + 1)B(s, l), E 2 (t, k) = C(t, k)u(t, )F (k, a 2 )P, G 2 (s, l) = U(, s)f (a 2, l + 1)B(s, l), hence K Σ is 2D semiseparable. Definition 17 Given a 2D semiseparable kernel K, a 2Dgcd system Σ is said to be a realization of K if K = K Σ. Proposition 18 For any 2D semiseparable kernel K there exists a realization of K. Proof: If K has the form (18), a realization of K is the system Σ given by A 1 = O n a.e. on [, b 1 ], [ ] G1 (t, k) A 2 = I n, B(t, k) =, C(t, k) = G 2 (t, k) [E 1 (t, k) E 2 (t, k)], D = Op m R p m, N 1 = [ ] [ ] In1 0 0 0, N 2 = with n 1, n 2 > 0, 0 0 0 I n2 n 1 + n 2 = n; M 1 and M 2 are arbitrary. Obviously U(t, s) = I n a.e. on [, b 1 ], F (k, l) = I n, hence R = N 1 + N 2 = I n ; it results that this system has well-posed boundary condition and its canonical operator is P = R 1 N 2 = N 2. Example Let us consider the 2D continuousdiscrete Wiener-Hopf equation b y(t, k) K(t s, k l)du(s, l) = = Du(t, s), (t, s) T (19) where K(t, k) BV p m 1 for any (t, k) T. Assume that K(t, k) can be extended to a function K(t, k) defined on R Z which admits a proper rational 2D continuous-discrete Laplace transform of θ(s, z) the form T (s, z) = with θ(s, z) π 1 (s)π 2 (z) R p n [s, z], π 1 (s) R[s], π 2 (z) R[z]. Then, using the algorithm of minimal realization described in [12], we can determine the constant matrices A 1, A 2, B, C, D with A 1 A 2 = A 2 A 1 such that T Σ (s, z) = C(s A 1 ) 1 (z A 2 ) 1 B + D. By considering the matrices N 1 = I P and N 2 = P where P is a suitable spectral projection and M 1, M 2 arbitrary matrices we obtain a system Σ whose inputoutput map (15) coincides with the 2Dgcd Wiener- Hopf equation (19). Definition 19 A realization Σ of a 2D semiseparable kernel K is said to be minimal if dim Σ dim ˆΣ for any realization ˆΣ of K. We introduce the controllability and the observability Gramians of the system Σ: C(Σ) = b U(, s)f (a 2, l) B(s, l)b(s, l) T F (a 2, l) T U(, s) T ds, O(Σ) = b U(s, ) T F (l, a 2 ) T C(s, l) T C(s, l)f (l, a 2 )U(s, )ds. The canonical boundary value operator of Σ is P = [N 1 + N 2 U(b 1, )F (b 2, a 2 )] 1 N 2 U(b 1, )F (b 2, a 2 ). Now we shall extend [3, Theorem 3.1] to the case of 2Dgcd acausal systems. Theorem 20 A realization Σ of the 2D semiseparable kernel K is minimal if and only if the following conditions hold: Im [C(Σ) P C(Σ)] = R n, (20) Ker [ O(Σ) O(Σ)P ] = {0}, (21) Ker O(Σ) Im C(Σ). (22) Proof: Necessity. Let us consider an arbitrary direct sum decomposition R n = X 1 X 2 with n 1 = dim X 1, 0 < n 1 < n and the corresponding partitions of the following operators: U(, t)f (a 2, k)b(t, k) =, [ ] B1 (t, k) B 2 (t, k) C(t, k)u(t, )F (k, a 2 ) = [C 1 (t, k) C 2 (t, k)], [ ] [ ] D1 (t, k) P11 P 12 D(t, k) =, P =. Let D 2 (t, k) P 21 P 22 us denote by Σ 1 the 2Dgcd system with well-posed boundary conditions determined by the matrices A 1 1 = O n1, A 1 2 = I n 1, B 1 (t, k), C 1 (t, k), D 1 (t, k), N1 1 = I P 11, N2 1 = P 11. If the condition (20) is not fulfilled we take X 1 = Im[C(Σ) P C(Σ)] and X 2 = X1 ; if (21) is not fulfilled we take X 2 = Ker and X 1 = X2 [ ] O(Σ) O(Σ)P ; if (22) is not true we consider X 2 the subspace of KerC(Σ) such that ImC(Σ)+KerO(Σ) = ImC(Σ) X 2 and X 1 is the complement of X 2 in R n which includes ImC(Σ). As in [3, lemmas 3.2-3.4] we can ISSN: 1790-2769 71 ISBN: 978-960-474-131-1

prove that in all these cases K Σ1 = K, hence Σ 1 is a realization of K and dim Σ 1 = n 1 < n = dim Σ, i.e. Σ is not minimal. Sufficiency. If the conditions (4.1)-(4.3) hold for some realization Σ of K, we consider the direct sum decomposition of the state space R n given by X 2 = Ker O(Σ), X 1 X 2 = Im C(Σ) and X 1 X 2 X 3 = R n. Following the lines of [9, Theorem 3.1] we can prove that dimσ dimˆσ for any realization ˆΣ of K. 5 Conclusion The state space representation was studied for a class of time-varying 2D hybrid systems in the general framework of spaces over the set of regulated functions. The formulæ of the state and of the input-output map of these systems were obtained. The realization problem was analysed and a minimality criterion was provided. This study can be continued by analysing for this class other important concepts as stability, controllability, observability, adjoint systems etc. References: [1] M. Dymkov, F. Gaishun, K. Gałkovski, E. Rogers, D.H. Owens, Exponential stability of discrete linear repetitive processes, Int. J. Control, 75, 2002, pp. 861-869. [2] I. Gohberg and M.A. Kaashoek, Time varying linear systems with boundary conditions and integral operators, Integral Equations and Operator Theory, 7, 1984, pp. 325-391. [3] I. Gohberg and M.A. Kaashoek, Minimality and irreducibility of time invariant linear boundaryvalue systems, Int. J. Control, 44, 1986, pp. 363-379. [4] Ch. S. Hönig, The adjoint equation of a linear Volterra-Stieltjes integral equation with a linear constraint, Lecture Notes in Mathematics 957, Springer-Verlag, Berlin Heidelberg New York 1982. [5] T. Kaczorek, Controllability and minimum energy control of 2D continuous-discrete linear systems, Appl. Math. and Comp. Sci., 5, 1995, pp. 5-21. [6] [13] A.J. Krener, Acausal linear systems, Proc. 18th IEEE Conference on Decision and Control, Ft. Lauderdale, Fl., 1979. [7] A.J. Krener, Boundary value linear systems, Astérisque, 75/76, 1980, pp. 149-165. [8] J. Kurek and M.B. Zaremba, Iterative learning control synthesis on 2D system theory, IEEE Trans. Aut. Control, AC-38, 1993, pp. 121-125. [9] V. Prepeliţă, Calculus of the Fundamental Matrix for Generalized Linear Differential Equations, Ann. Sci. Math. Quebec, 23, 1999, pp. 87-96. [10] V. Prepeliţă, 16 A Class of qd Continuous-Time Time-Varying Acausal Systems, Proceedings of the 7th WSEAS Int. Conf. on INSTRUMENTA- TION, MEASUREMENT, CIRCUITS and SYS- TEMS (IMCAS 08), Hangzhou, China, April 6-8, 2008, pp 90-97. [11] V. Prepeliţă, Structural properties of Linear Generalized Systems, WSEAS Transactions on Systems and Control, 3, 2008, pp. 701-711. [12] V. Prepeliţă, Minimal realization algorithm for (q,r)-d hybrid systems, WSEAS Transactions on Systems and Control, 8, 2009, pp. 22-33. [13] V. Prepeliţă, Linear 2D Hybrid Systems over Spaces of Regulated Functions, Mathematical Reports, 11(61), 3 (2009). [14] E. Rogers, D.H. Owens, Stability Analysis for Linear Repetitive Processes. Lecture Notes in Control and Information Sciences, 175, Ed. Thoma H., Wyner W., Springer Verlag, Berlin Heidelberg New York 1999. [15] S. Schwabik, M. Tvrdý, O. Vejvoda, Differential and Integral Equations. Boundary Value Problems and Adjoints, D. Reidel Publishing Co., Dordrecht Boston London 1979. [16] M. Tvrdý, Regulated Functions and the Perron- Stieltjes Integral, Časopis pěst. mat., 114, 1989, pp. 187-209. [17] M. Tvrdý, Generalized Differential Equations in the Space of Regulated Functions, Mathematica Bohemica, 116, 1991, pp. 225-244. ISSN: 1790-2769 72 ISBN: 978-960-474-131-1