Invertibility of discrete distributed systems: A state space approach

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1 Invertibility of discrete distributed systems: A state space approach J Karrakchou Laboratoire d étude et de Recherche en Mathématiques Appliquées Ecole Mohammadia d Ingénieurs, BP: 765, Rabat-Agdal, Maroc jkarrak@emiacma J Bouyaghroumni, A Abdelhak and M Rachik Département de Mathématiques et d Informatique Faculté des Sciences Ben M Sik Boulevard commandant Driss El Harti, BP 7955, Casablanca, Maroc Keywords: Inverse problem, optimal control, quadratic criterion, state space technique Abstract Given a desired signal y d y d i i 1,,N} we investigate, under certain hypothesis, the optimal control which applied to the system x i+1 Ax i + Bu i, gives the output function y d and that with a minimal cost A state space technique is used and numerical simulations are given the case of discrete delayed systems is also considered 1 Introduction The first systematic paper on the problem of input retrieval in finite-dimensional linear systems appears to have been the one by Brockett and Mesarovic [1], in which a condition under which the input was retrieved from the output was given in terms of the rank of matrix involving the system parameters Subsequently this condition was simplified by Sain and Massey [2] and later by Wang and Davison [3] In [4], Rachik used a state space technique to show that the problem of input retrieval can be seen as a controllability problem but in a large space, consequently they used an approach similar to Hilbert Uniqueness Method HUM see [5, 6] In the second section of this paper we restrict our attention to discrete systems described by the difference equation xi+1 Ax S i + Bu i, i,, N 1} x The output function is given by y i Cx i, i,, N} where x i X, u i U and y i Y X, U and Y are Hilbert spaces Given a desired output function y d y d i i 1,,N}, we determine under certain hypothesis, the control function u u i i,1,,n 1} which verify Cx i y d i, i 1,, N} and that with a minimal cost To solve this problem, we use a new technique based on the construction of a large infinite dimensional space in which the problem of input retrieval can be interpreted as a controllabity one In secton 3 we study the problem for discrete delayed systems 2 Statement of the problem We consider the discrete system described by xi+1 Ax S i + Bu i, i,, N 1} 1 x and the corresponding output y i Cx i, i,, N} 2 where x i X is the state of system S, u i U is the control variable and y i Y is the output function, A LX, B LU, X and C LX, Y Consider the following control problem Given a desired trajectory y d y1, d, yn d, the optimal control u u, u 1,, u N 1 which minimizes the functional cost over all controls satisfying Ju u 2 3 Cx i y d i, i 1,, N}

2 21 An adequate state space approach In this subsection, we give some technical results which will be used in the sequel For a finite subset σ s r r, r + 1,, s} of Z, with s r, let l 2 σ s r, X be the space of all sequences z i i σ s r, z i X Remark 1 l 2 σr, s X is a Hilbert space with the usual addition, scalar multiplication and with an inner product defined by s < x, y > l 2 σr s,x < x i, y i > X ir Let L and F be the operators given by L : l 2 σ 1 N ; X l2 σ 1 N ; X z N,, z 1 z N+1,, z 1, F : X l 2 σ 1 N ; X x,,, x and define the variables z i l 2 σ 1 N ; Y by z i z N i,, zi 1 zk i xi+k, if i + k x, else where x i i is the solution of system S Then the sequence z i i is the unique solution of the following difference equation z i+1 Lz i + F x i, i,, N 1} z 4 x, x,, x Let e i X l 2 σ 1 N ; X be the signals defined by e i xi z i, then we easily establish the following result Proposition 1 e i i,,n} is the unique solution of the difference equation described by e i+1 ψe i + Bu i, i,, N 1} S 1 x e x,, x 5 A B where ψ and B F L Remark 2 The equality xn e N z N xn x,, x N 1, 6 allows to assimilate the trajectory x, x 1,, x N 1, x N of system S to the final state e N of S 1 This implies that our problem of input retrieval is equivalent to a problem of optimal control with constraints on the final state e N 22 The optimal control expression let H N be the operator defined by H N : l 2 σ N 1 ; U X l 2 σ 1 u u,, u N 1 then it is easy to see that Define the operator Γ by Γ : N 1 N ; X ψ N j 1 Buj e N ψ N e + H N u 7 X l 2 σ 1 N ; X l2 σ N 1 ; Y x, ξ i N i 1 t 1,, t N where t i Cξ N+i, for i 1,, N 1 and t N Cx Then clearly we have the following proposition Proposition 2 Given a desired trajectory y d y d 1,, y d N, the problems P 1 and P 2 defined as follows are equivalent, P 1 Find the optimal control u such that Cx i y d i, i 1,, N} with a minimal cost 3 P 2 Find the optimal control u such that Γe N y d with a minimal cost 3 As a natural consequence of proposition 2, we focus our attention on the resolution of problem P 2 For this, we introduce the operator K N defined by K N : l 2 σ N 1 ; U l 2 σ N 1 ; Y u u,, u N 1 ΓoH N Definition 1 a The system S is said to be C-controllable on,, N} if x X y l 2 σ N 1 ; Y u l 2 σ N 1 ; U / Γe N y b The system S is said to be C-weakly controllable on,, N} if ɛ > x X y l 2 σ N 1 ; Y u / Γe N y l 2 σ N 1 ;Y ɛ Remark 3 Using equation 7, it is easy to establish that i The system S is C-controllable if and only if Im K N l 2 σ N 1 ; Y ii The system S is C-weakly controllable if and only if Im K N l 2 σ N 1 ; Y or equivalently Ker KN }, where K N is the adjoint operator of K N

3 It follows from equation 6 that Cx 1 Cx 2 Γe N ĊxN CAx CA 2 x + CA N x and since ΓΨ N e CB CAB CB u CA N 1 B CA N 2 B CB CAx CA 2 x CA N x, we deduce from equa- tion 7 that forall u l 2 σ N 1, U, CB CAB CB K N u CA N 1 B CA N 2 B CB u 8 Remark 4 As a consequence of equation 8 we deduce that if Ker B C } then the system S is C-weakly controllable If we define the operator Λ N by Λ N : l 2 σ N 1 ; Y l 2 σ N 1 ; Y y K N K N y On the other hand, if the system S is C-weakly controllable on, 1,, N} then 1 is a norm on l 2 σ N 1 ; Y Consequently if we denote F N the completion of l 2 σ N 1 ; Y with respect to the norm 1, then it is known that Λ N can be extended to an isomorphism denoted also by Λ N and defined from F N to its dual space F N, moreover l2 σ N 1 ; Y F N with dense and continuous embedding Now we give the fundamental result of this section Proposition 3 If the system S is C-weakly controllable on,, N} and y d Γψ N e F N then the unique optimal controle u u,, u N 1 is given by or equivalently u j N 1 j k u K Nf B A N 1 j k C f N k, j,, N 1} where f f 1,, f N is the unique solution of the equation Λ N f y d Γψ N e 11 Moreover, we have u f proof Since Λ N is an isomorphism, there exists f F N such that y d Γψ N e Λ N f, if we consider the controle u KN f, we deduce from equation 11 that y d Γψ N e + Λ N f Γψ N e + K N KN f Γψ N e + H N u ΓeN u where e u i i,,n} is the solution of 5 corresponding to the control u On the other hand, if v v,, v N 1 is a control such that Γe v N yd, then we have then Λ N is a bounded, self-adjoint operator and satisfies ker K N ker Λ N Consequently the system S is C-weakly controllable on,, N} if and only if using 7 we obtain which implies y d Γe u N Γe v N K N u v KerΛ N } Define the following inner product on l 2 σ N 1 ; Y by << x, y >> < x, Λ N y > l 2 σ N 1 ;Y < K N x, K N y > l 2 σ N 1 ;U The corresponding semi norm is given by Clearly we have 9 x N K N x l 2 σ N 1 ;U 1 x N K N x hence thus < K N u v, f > < u v, K N f > or < u v, u > u v so u is the unique optimal control Proposition 4 Assume that S is C-weakly controllable, then given an initial state x X, the set E N of all reachable output trajectories, ie, E N y i 1 i N / u Cx i y i, i 1,, N}}

4 is given by E N Γψ N e + F N Proof It follows from proposition 3 that Γψ N e + F N E N In order to establish that E N Γψ N e + F N, let y ΓeN u E N, for some u l 2 σ N ; U, and consider the linear form g : l 2 σ N 1 ; Y IR z < z, y Γψ N e > l 2 σ N 1 ;Y we have hence gz < z, y Γψ N e > < z, K N u > < KN z, u > gz u z FN then we deduce by density, since F N is the completion of l 2 σ1 N ; Y, that g is a continuous linear form on F N, ie, g F N, finally we use the Riesz theorem to conclude that 23 Example y Γψ N e F N Consider the system xi+1 Ax S i + Bu i, i,, N 1 x 12 where x i L 2 Ω, Ω ], 1[ and u i IR The operator A is defined by A S δ, where δ is a fixed real and S t t the strongly continuous semigroup generated by the Laplacien operator, ie, S t x e i2 Π 2t < x, e i > e i, x L 2 Ω i1 where <, > is the usual inner product on L 2 Ω and e i i 1 the basis of L 2 Ω given by e i s 2 sin iπs, s ], 1[ The operator B is defined by B : IR L 2 Ω v v S σ e 1 dσ Remark 5 The difference equation 12 can be interpreted as the sampling version of the following continuous diffusion system x t x gsut, s Ω, t [, T ] x, x in Ω xt, s in Ω ], T [ where g e 1 The system S is augmented with the following output function y i Cx i, i,, N where and C : L 2 Ω IR 2 x e Π2 < x, g1 > < x, g 2 > g 1 s s, g 2 s s 3 s ], 1[ Optimal control Let y d be a desired output trajectory By application of proposition 3 the optimal control is given by u K Nf where f is the unique solution of the equation Λ N f y d For every u l 2 σ N 1, U, if we denote K N u by K N u z 1,, z N T l 2 σ N 1, Y then it follows from equation 8 that i 1 z i C u j S i 1 jδ S σ e 1 dσ i 1 C u j S i 1 jδ+σ e 1 dσ i 1 C u j e Π2 i 1 jδ+σ e 1 dσ i 1 u j e Π2 i 1 jδ+σ e 1 dσ Ce 1 N 1 α i,j u j where α i,j IR 2 δ e α i,j Π2 i 1 jδ+σ e 1 dσ Ce 1 j,, i 1 else Thus the expression of the matrix K N K N i, j, 1 i 2N, 1 j N, is given by for all i 1,, 2N} and all j 1,, i} K N 2i 1, j K N 2i, j e Π2 i 1 jδ+σ dσe Π2 < e 1, g 1 > e Π2 i 1 jδ+σ dσe Π2 < e 1, g 2 > for all i 1,, 2N 1} and all j i + 1,, N} K N 2i 1, j K N 2i, j Finaly, we verify that the matrix Λ N K N KN is invertible which implies that the

5 system S is C-weakly controllable Numerical simulation If we take N 1, δ 1 and y d 1 1, 2 2,, 9 9 1, 1 then the optimal control takes the following values T i u i 22, 33, 6, 15, 3 i u i, 17, 1, 19, 17, 18 and is illustrated by figure 1 energy is J The corresponding optimal and u minimizes the functional cost 31 State and state space Ju u 2 Define the state variable e i W by x i e i z i w i, i,, N} 16 where W X l 2 σ 1 d, X l2 σ q 1, U 17 x i i is the solution of 13, z i i and w i i are described as follows z i z i k d k 1, w i w i k q k 1 z i k w i k xi+k, if i + k α i+k, if i + k < ui+k, if i + k µ i+k, if i + k < 18 Define the operators L, L, C, C, F, and F by 3 Discrete delayed systems with delays in the control variable In this section, we consider the system S d described by d q x i+1 A j x i j + B j u i j, i N 1 x x k α k, k d,, 1} u k µ k, k q,, 1} 13 Where x i X is the state variable, u i U is the input variable X and U are Hilbert spaces, x, α d,, α 1 and µ q,, µ 1 are a fixed initial conditions The output function of 13is described by r y i C j x i j, i 14 where C j LX, Y, Y a Hilbert space and r d Given a desired trajectory y d y1, d, yn d, we investigate the optimal control u u,, u N 1 such that r C j x i j yi d, i 1,, N} 15 L : L : C : C : l 2 σ 1 d, X l2 σ 1 d, X z d,, z 1 z d+1,, z 1, l 2 σ q 1, U l2 σ q, 1 U w q,, w 1 w q+1,, w 1, l 2 σ 1 d, X X d z d,, z 1 A j z j j1 l 2 σ q, 1 U U q w q,, w 1 B j w j j1 F : X l 2 σ 1 d, X x,,, x F : U l 2 σ 1 q, U w,,, w Then we have the following result Proposition 5 The sequence e i i,,n} defined by 16 is the solution of the discrete equation e i+1 φe i + Gu i, i,, N 1} S x d e α d,, α 1 µ q,, µ 1 19 where φ and G are the bounded operators given by φ A C C F L L, G B F

6 Proof From the definition of the operators L, L, C, C, F, and F we easily establish that x i+1 A x i + Cz i + Cw i + B u i z i+1 Lz i + F x i 2 w i+1 Lw i + F x i where x i i is the solution of 13 and z i, w i are given by Optimal control We investigate the optimal control which minimizes the functional cost Ju u 2 over all controls satisfying 15 The use of the state space technique shows that this is equivalent to search the control u which verify Ce i y d i, i 1,, N} 21 and minimizes the cost Ju, where e i i is the solution of the discrete system without delay 19 and C a bounded operator defined by C : W Y x, z k k, w k k C x + r C j z 22 j j1 References [1] Brocket RW and Mesarovic MD, The reproducibilty of multivariable systems, JMaths Analysis and Applications, n 11, pp , 1965 [2] Sain MK and Massey JL, Invertibility of linear time invariant dynamical systems, IEEE Trans On autamatic control, AC-14, pp , 1969 [3] Wang SH and Davison EJ, A New invertibility criterion for linear multivariable systems, IEEE Trans On autamatic control TechNotes and Corresp, AC-18, pp , 1973 [4] Rachik M, Quelques Eléments sur l analyse des Systèmes Distribuès, These de doctorat d état, Université Mohamed V, EMI-Rabat, 1995 [5] Lions JL, Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles, Dunod et Gauthiers-Villars, Paris, 1968 [6] Lions JL, Exact controllability, stabilization and perturbation for distributed systems, SIAM Rev, n 3, pp 71-86, 1988 where the space W is given by 17 Application of the results of section 2 gives the following proposition Proposition 6 If the system S d is C-weakly controllable and y d Γ Ψ N ē F N then the unique optimal control u is given by u K Nf where f is the unique solution of the equation Λ N f y d Γ ψ N ē Γ, ψ, K N, Λ N, ē and F N are defined with respect to the system 19 and the output function y i Ce i by the similar way that was defined Γ, Ψ, K N, Λ N, e and F N with respect to the system 1 and the output function 2 Moreover we have u f 4 Conclusion In this paper we have considered the problem of input retrieval for discrete systems We have shown that the problem can be reduced to a controllability one in an appropriate space For this, we have used a state space techniques and a Hilbert Uniqueness method The problem of input retrieval for disrete delayed systems was also considered The case of discrete non linear systems is under investigation