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On Control of Discrete-Time LTI Positive Systems

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On Control of Discrete-Time LTI Positive Systems Applied Mathematical Sciences, Vol. 11, 2017, no. 50, 2459 - 2476 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.78246 On Control of Discrete-time LTI Positive Systems DuˇsanKrokavec and Anna Filasov´a Department of Cybernetics and Artificial Intelligence Faculty of Electrical Engineering and Informatics Technical University of Koˇsice Letn´a9/B, 042 00 Koˇsice,Slovakia Copyright c 2017 DuˇsanKrokavec and Anna Filasov´a.This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Incorporating an associated structure of constraints in the form of linear matrix inequalities, combined with the Lyapunov inequality guaranteing asymptotic stability of discrete-time positive linear system structures, new conditions are presented with which the state-feedback controllers can be designed. Associated solutions of the proposed design conditions are illustrated by numerical illustrative examples. Keywords: state feedback stabilization, linear discrete-time positive sys- tems, Schur matrices, linear matrix inequalities, asymptotic stability 1 Introduction Positive systems are often found in the modeling and control of engineering and industrial processes, whose state variables represent quantities that do not have meaning unless they are nonnegative [26]. The mathematical theory of Metzler matrices has a close relationship to the theory of positive linear time- invariant (LTI) dynamical systems, since in the state-space description form the system dynamics matrix of a positive systems is Metzler and the system input and output matrices are nonnegative matrices. Other references can find, e.g., in [8], [16], [19], [28]. The problem of Metzlerian system stabilization has been previously stud- ied, especially for single input and single output (SISO) continuous-time linear 2460 D. Krokavec and A. Filasov´a systems, as well as discrete-time linear systems, which have minimal degree of freedom to ensure that a solution exists (see [17], [22], [24], [30] and the refer- ences therein). Applicable methods for stabilization of positive linear discrete- time systems, maintaining its positivity when using linear state feedback, are given in [7], [18], [31]. The synthesis problem of state-feedback controllers, guaranteeing the clo- sed-loop system to be asymptotically stable and positive, has been investigated by a linear matrix inequality (LMI) and the linear programming approach in [2], [13], but as far as the authors know, there is no literature on design of controllers for positive continuous-time or discrete-time linear systems, in which the design conditions are built only on LMIs. The main motivation issue of this paper is to reformulate design conditions for stabilization of linear positive discrete-time systems with the state-feedback. Considering the stable strictly positive matrix structure, algebraic constraints implying from linear programming approach are reformulated as a set of LMIs, which is extended by an LMI, reflecting the Lyapunov stability condition. The paper is organized as follows. Within the frame of preliminaries, the standard declaration for discrete-time linear systems is presented in Sec. 2 and the basic characteristics of positive discrete-time linear systems are given in Sec. 3. A newly introduced set of LMIs, describing the design conditions of the state control law parameters for positive discrete-time LTI systems, is theoretically substantiated in Sec. 4. An example is provided to demonstrate the proposed approach in Sec. 5, while Sec. 6 draws some conclusions. Used notations are conventional so that xT , XT denotes transpose of the vector x and matrix X, respectively, x+, X+ indicates a nonnegative vector and a nonnegative matrix, X = XT 0 means that X is a symmetric positive definite matrix, ρ(X) reports the eigenvalue spectrum of a square matrix X, the symbol In marks the n-th order unit matrix, diag[ · ] enters up a diagonal matrix, IRn, IRn×r refers to the set of all n-dimensional real vectors and n × r n n×r real matrices, respectively, IRn, IR+ signifies the set of all n-dimensional real non-negative vectors and n × r real non-negative matrices, respectively, and Z+ is the set of all positive integers. 2 Basic Preliminaries This section present some basic preliminaries which are concerned with the discrete-time linear MIMO systems. To support the following parts of the paper, the state-space form of the system description is preferred, where q(i + 1) = F q(i) + Gu(i) ; (1) y(i) = Cq(i) ; (2) On control of discrete-time LTI positive systems 2461 q(i) 2 IR n, u(i) 2 IR r, and y(i) 2 IR m are vectors of the system, input and output variables, respectively, and F 2 IR n×n, G 2 IR n×r, C 2 IR m×n. The transfer function matrix to (1), (2) is −1 H(z) = C(zIn − F ) G ; (3) where a complex number z is the transform variable of the transform Z [23]. Quantifying the effect of the input onto the output of the system, the so-called H2 and H1 norms of H(z) are used. Definition 2.1 [12], [29] The H2-norm and H1-norm of the transfer func- tions matrix (3) are defined as π 2 1 j! ∗ j! kH(z)k2 = tr s H(e )H (e )d!; (4) 2π −π j! j! ∗ j! kHd(z)k1 = sup σo(Hd(e )) = sup σo(eig(Hd(e )Hd(e )) ; (5) !∈h−π,πi !∈h−π,πi p where z = ej!, ! is the frequency variable, j := −1, H∗(ej!) is the adjoint j! j! of H(e ) and σo means the largest singular value of the matrix H(e ). Definition 2.2 [5], [20] A square matrix F is Schur (stable) if every eigen- value of F lies in the unit circle in the plain of the complex variable z. If F is stable, the dynamical system (1), (2) has the stable transfer function matrix (3), i.e., the poles of all elements of H(z) lie in the unit circle in the plain of the complex variable z. Proposition 2.1 [21] (bounded real lemma) The discrete-time linear sys- tem (1), (2) is stable if there exist a symmetric positive definite matrix P 2 n×n IR and a positive scalar γ1 2 IR such that T P = P > 0; γ1 > 0 ; (6) 2 −P ∗ ∗ ∗ 3 6 0 −γ1Ir ∗ ∗ 7 6 7 < 0 ; (7) 4 CP 0 −γ1Im 0 5 FPG 0 −P where γ1 2 IR is the H1 norm of H(z). Hereafter, ∗ labels the symmetric item in a symmetric matrix. Proposition 2.2 [9] (Lyapunov inequalities) Autonomous part of the di- screte-time system (1), (2) is asymptotically stable if there exist symmetric positive definite matrices P ; Q 2 IRn×n such that −P + Q ∗ P = P T 0 ; Q = QT 0 ; ≺ 0 : (8) FP −P 2462 D. Krokavec and A. Filasov´a Lemma 2.1 If the matrix F of the system (1), (2) is Schur then T T FW cF − W c + GG = 0 ; (9) 2 T γ2 = tr(CW cC ) ; (10) n×n where W c 2 IR is a positive definite symmetric matrix and γ2 2 IR is H2 norm of H(z). Proof: Since a solution of (1), (2) is n−1 X q(n) = F nq(0) + A(l)u(n − 1 − l) ; A(l) = F lG ; (11) l=0 as an explicit test for linear independence of A(l) can be used its Gramian n−1 X W (n) = F lGGT F T l : (12) l=0 Pre-multiplying the left side by F and post-multiplying the right side by F T then (12) implies n−1 n X X FW (n)F T = F l+1GGT (F T ) l+1 = F lGGT F T l (13) l=0 l=1 and, subtracting (12) from (13), it is obtained FW (n)F T − W (n) = F nGGT F T n − GGT : (14) On the ground of that (11) for l = n inserts the input variable value u(−1), which is identically equal zero, it has to be F nGGT F T n = 0 (15) and defining the stationary solution W (n) = W c, then (14) implies (10). Utilizing the Parseval's theorem property [3] then (4), (12) gives n−1 n−1 2 X T X l T T l T T 2 kH(z)k2 = tr g l g l = tr CF GG F C = tr(CW cC ) = γ2 ; l=0 l=0 (16) where gl is the l-th impulse response function of the system with the transfer function matrix (3). Thus, (16) gives (10). This concludes the proof. On control of discrete-time LTI positive systems 2463 3 Positive Linear Discrete-time Systems Definition 3.1 [4] (positive linear system) The linear system (1), (2) is said to be positive if and only if for every nonnegative initial state and for every nonnegative input its state and output are nonnegative. Proposition 3.1 [11] If the system (1), (2) is a positive linear discrete- n r m n×n n×r time system then q(i) 2 IR+, u(i) 2 IR+, y(i) 2 IR+ , F 2 IR+ , G 2 IR+ , m×n C 2 IR+ and i 2 Z+. Then a solution q(i) of (1) is asymptotically stable n r and positive, i.e., limi!1 q(i) = 0 while q(i) 2 IR+ for u(i) 2 IR+ and the n n×r initial value q(0) 2 IR+, if F is a positive Schur matrix and G 2 IR+ is a non-negative matrix. The linear system (1), (2) is asymptotically stable n×r m×n and positive if F is a positive Schur matrix, G 2 IR+ , C 2 IR+ are m r non-negative matrices and y(i) 2 IR+ for u(i) 2 IR+ and the initial value q(0) 2 IR+.
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