On Control of Discrete-time LTI Positive Systems

Transcription

1 Applied Mathematical Sciences, Vol. 11, 2017, no. 50, HIKARI Ltd, On Control of Discrete-time LTI Positive Systems Dušan Krokavec and Anna Filasová Department of Cybernetics and Artificial Intelligence Faculty of Electrical Engineering and Informatics Technical University of Košice Letná 9/B, Košice, Slovakia Copyright c 2017 Dušan Krokavec and Anna Filasová. 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 systems, 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 timeinvariant (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 studied, especially for single input and single output (SISO) continuous-time linear

2 2460 D. Krokavec and A. Filasová 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 references therein). Applicable methods for stabilization of positive linear discretetime systems, maintaining its positivity when using linear state feedback, are given in [7], [18], [31]. The synthesis problem of state-feedback controllers, guaranteeing the closed-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 x T, X T denotes transpose of the vector x and matrix X, respectively, x +, X + indicates a nonnegative vector and a nonnegative matrix, X = X T 0 means that X is a symmetric positive definite matrix, ρ(x) reports the eigenvalue spectrum of a square matrix X, the symbol I n marks the n-th order unit matrix, diag[ ] enters up a diagonal matrix, IR n, IR n r refers to the set of all n-dimensional real vectors and n r real matrices, respectively, IR n, n IR + n r 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)

3 On control of discrete-time LTI positive systems 2461 q(i) IR n, u(i) IR r, and y(i) IR m are vectors of the system, input and output variables, respectively, and F IR n n, G IR n r, C IR m n. The transfer function matrix to (1), (2) is H(z) = C(zI n F ) 1 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 H 2 and H norms of H(z) are used. Definition 2.1 [12], [29] The H 2 -norm and H -norm of the transfer functions matrix (3) are defined as H d (z) = H(z) 2 2 = 1 2π tr π H(e jω )H (e jω )dω, (4) π sup σ o (H d (e jω )) = sup σ o (eig(h d (e jω )H d(e jω )), (5) ω π,π ω π,π where z = e jω, ω is the frequency variable, j := 1, H (e jω ) is the adjoint of H(e jω ) and σ o means the largest singular value of the matrix H(e jω ). Definition 2.2 [5], [20] A square matrix F is Schur (stable) if every eigenvalue 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 system (1), (2) is stable if there exist a symmetric positive definite matrix P IR n n and a positive scalar γ IR such that P = P T > 0, γ > 0, (6) P 0 γ I r CP 0 γ I m 0 < 0, (7) F P G 0 P where γ IR is the H norm of H(z). Hereafter, labels the symmetric item in a symmetric matrix. Proposition 2.2 [9] (Lyapunov inequalities) Autonomous part of the discrete-time system (1), (2) is asymptotically stable if there exist symmetric positive definite matrices P, Q IR n n such that [ ] P + Q P = P T 0, Q = Q T 0, 0. (8) F P P

4 2462 D. Krokavec and A. Filasová Lemma 2.1 If the matrix F of the system (1), (2) is Schur then F W c F T W c + GG T = 0, (9) γ 2 2 = tr(cw c C T ), (10) where W c IR n n is a positive definite symmetric matrix and γ 2 IR is H 2 norm of H(z). Proof: Since a solution of (1), (2) is n 1 q(n) = F n q(0) + A(l)u(n 1 l), A(l) = F l G, (11) l=0 as an explicit test for linear independence of A(l) can be used its Gramian n 1 W (n) = F l GG T 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 F W (n)f T = F l+1 GG T (F T ) l+1 = l=0 n F l GG T F T l (13) l=1 and, subtracting (12) from (13), it is obtained F W (n)f T W (n) = F n GG T F T n GG T. (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 n GG T 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 H(z) 2 2 = tr g l g T l = tr CF l GG T F T l C T = tr(cw c C T ) = γ2, 2 l=0 l=0 (16) where g l is the l-th impulse response function of the system with the transfer function matrix (3). Thus, (16) gives (10). This concludes the proof.

5 On control of discrete-time LTI positive systems 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 discretetime system then q(i) IR +, n u(i) IR +, r y(i) IR + m, F IR + n n, G IR + n r, C IR + m n and i Z +. Then a solution q(i) of (1) is asymptotically stable and positive, i.e., lim i q(i) = 0 while q(i) IR + n for u(i) IR + r and the initial value q(0) IR +, n if F is a positive Schur matrix and G IR + n r is a non-negative matrix. The linear system (1), (2) is asymptotically stable and positive if F is a positive Schur matrix, G IR + n r, C IR + m n are non-negative matrices and y(i) IR + m for u(i) IR + r and the initial value q(0) IR +. The linear system (1), (2) is asymptotically stable and internally positive if F is a positive Schur matrix and G IR + n r, C IR + m n are nonnegative matrices. Definition 3.2 [1] A square matrix F IR + n n is positive if its elements are nonnegative. A square matrix F IR + n n is strictly positive stable matrix if is Schur and all its elements are positive. Proposition 3.2 [10] A positive matrix F is stable if and only if is diagonally dominant. Definition 3.3 [6] (congruent modulo n) Let n be a fixed positive integer. Two integers j and h are congruent modulo n if they differ by an integral multiple of the integer n (they leave the same remainder when divided by n). If j and h are congruent modulo n, the expression (j = h) mod n is called a congruence, and the number n is called the modulus of the congruence. The statement (j = h) mod n is equivalent to the statement (j h) is divisible by n or to the statement there is an integer m for which j h = mn. Definition 3.4 [27] Let S = {0, 1, 2,..., n 1} be the complete set of residues for any positive integer n. The addition modulo n on the set S is (j + h) mod n = r, where r is the element of S to which the result of the usual sum of integers j and h is congruent modulo n. Corollary 3.1 The problem of indexing in this paper is that the rows and columns of a square matrix of dimension n n are generally denoted from 1 to n and not from 0 to n 1. From this reason let S = {0, 1, 2,..., n} be the complete set of residues for any positive integer n + 1. Then, the addition modulo n + 1 on S is in the following defined as (j + h) mod n+1 = r + 1, where r is the element of S to which the result of the usual sum of integers j and k is congruent modulo n + 1. The used shorthand symbolical notation for (j + h) mod n+1 = r + 1 is so (j + h) (1 n)/n = r + 1.

6 2464 D. Krokavec and A. Filasová 4 Control of Positive Discrete-time Systems Linear discrete-time closed-loop MIMO systems, obtained from the controllable positive system (1), (2) by using the state control law u(i) = Kq(i), K IR r n +, (17) is described by the state-space equations where q(i + 1) = (F GK)q(i) = F c q(i), (18) y(i) = Cq(i), (19) G = [ g 1 g r ], K T = [ k 1 k r ], F c = F g g k T k. (20) Naturally, if F IR + n n is a strictly positive matrix, and G IR + n r, C IR + m n are non-negative matrices, the system (1), (2) is positive system. Thus, it is necessary to render the closed-loop system matrix F c be a stable strictly positive matrix. The conditions for the stabilizing control, H control and H 2 /H control of discrete-time positive linear systems with a strictly positive system matrix F are given by the following theorems. Theorem 4.1 (H control) The state feedback control (17) stabilizes the linear discrete-time positive system (1), (2) and H(z) < γ if for given strictly positive system matrix F there exist positive definite diagonal matrices P, R k IR n n and a positive scalar γ IR such that for h = 0, 1, 2,... n 1, k = 1, 2,... r, P = P T 0, γ > 0, (21) where P 0 γ I r CP 0 γ I m 0 F P r g k r T k G 0 P T h F (j, j + h) (1 n)/n T ht P < 0, (22) T h G dk T ht R k 0, (23) T =..., T 1 = T T, (24)

7 On control of discrete-time LTI positive systems 2465 F (j, j + h) (1 n)/n = diag [ ] f 1,1+h f n h,n f n h+1,1 f n,h, (25) g 11 g 12 g 1r G = [ ] g 21 g 22 g 2r g 1 g 2 g r =., (26) g n1 g n2 g nr G dk = diag [ g 1k g 2k g nk ] = diag [ {glk }, l = 1,..., n ], (27) R k is the structured matrix variable such that R k = diag [ r k1 r k2 r kn ] 0, (28) r T k = [ r k1 r k2 r kn ] = l T R k, l = [ ] T, (29) and F (j, j + h) (1 n)/n, T, G dk IR + n n. When the above conditions hold, the control gain matrix K is given as where K IR r n +. K dk = R k P 1, k T k = l T K dk, K = k T 1. k T r, (30) Proof: Writing the closed-loop system matrix F c as follows f 11 f 12 f 1n g 1k f 21 f 22 f 2n... g 2k [ ] kk1 k k2 k kn 0, (31). f n1 f n2 f nn g nk it is evident that F c be a strictly positive matrix if all its elements satisfy the conditions f jl g kj k kl > 0 j, l = 1, 2,..., n. (32) To solve by an LMI solver, LMIs have to be symmetric and so, using the notations (25), (27), then with h = 0 the diagonal elements of (31) can be rewritten in the diagonal matrix structure F (j, j) (1 n)/n G dk K dk 0, (33) K dk = diag [ k k1 k k2 k kn ] = [ {kkj } j=1,...,n ], k = 1, 2..., r. (34)

8 2466 D. Krokavec and A. Filasová Rewriting (31) as f 12 f 13 f 1n f 11 f 22 f 23 f 2n f 21. f n2 f n3 f nn f n1 g 1k g 2k. g nk [ kk2 k k3 k kn ] k k1, (35) it can set, analogously, for the diagonal elements of (35), F (j, j + 1) (1 n)/n G d K dkc1 0, (36) where K dkc1 is the diagonal matrix K dk with one circular shift of its diagonal elements. Since it yields using the permutation matrix (24) that K dk = T K dkc1 T 1 = T K dkc1 T T, (37) premultiplying the left side by T and postmultiplying the right side by T T the inequality (36) implies T F (j, j + 1) (1 n)/n T T r T G d T T T K dkc1 T T = = T F (j, j + 1) (1 n)/n T T r T G d T T K dk 0. (38) Repeating this procedure h-times, it can be obtained from (31) that f 1,1+h f 1,n f 1,1 f 1,h g 1k f 2,i+h f 2,n f 2,1 f 2,h g 2k [ kk,1+h. f n,i+h f n,n f n,1 f n,h g nk k k,2+h ] k kh (39) and so, consequently, with K dkch representing K dk with h circular shifts of its diagonal elements it yields F (j, j + h) (1 n)/n G dk K dkch 0, (40) which can be interpreted for h = 0,1,2,... n-1 as T h F (j, j + h) (1 n)/n T ht T h G dk T ht K dk 0. (41) Multiplying the right side of (41) by a diagonal positive definite matrix P leads to T h F (j, j + h) (1 n)/n T ht P T h G dk T ht K dk P 0 (42)

9 On control of discrete-time LTI positive systems 2467 and because (42) is a symmetric matrix inequality, with the notation R k = K dk P, (43) then (42) implies (23). Inserting the closed-loop system matrix (20) into (7) gives P 0 γ I r CP 0 γ I m 0 (F r g k k T < 0 (44) k )P G 0 P and using from (43) implying notation (44) implies (22). This concludes the proof. r T k = k T k P, (45) Remark 4.1 It can be noted, the conditions (21)-(23) are all LMIs that is they are convex in the defined matrix variables. Moreover, the necessary diagonal matrix variable structure of K dk directly implies the diagonal matrix variable strictures of P, R k in Theorem 4.1. To simplify obtaining relation in A(j, j+h) (1 n)/n it is possible to construct the following matrix F = [ F F ] = f 11 f 12 f 1n f 11 f 12 f 2n f 11 f 12 f 1n f 11 f 12 f 1n. (46) Then, using the main diagonal elements and the set of n 1 upper sub-diagonals of dimension of n, the matrices A(j, j + h) (1 n)/n can be sequentially constructed for h = 0, 1, 2,... n 1 from (46). Lemma 4.1 The matrix F of the system (1), (2) is stable and H(z) 2 < γ 2 if there exists a symmetric positive definite matrix V IR n n such that V = V T > 0, (47) F V F T V + GG T < 0, tr(cv C T ) > γ 2 2. (48) Proof: Let (48) yields for a symmetric positive definite matrix V. Then subtracting (9) from (48) leads to the inequality F (V W c )F T (V W c ) < 0 (49)

10 2468 D. Krokavec and A. Filasová and with V > W c the Lyapunov inequality implies that (49) is negative definite if and only if F is stable. Moreover, the relation V > W c gives tr(cv C T ) > tr(cw c C T ) = γ 2 2 (50) and so (49), (50) imply (48). This concludes the proof. Combining the algorithms for H 2 and H control design, the H 2 /H principle with H 2 and H performance constraints on positive discrete-time systems is given by the following theorem. Theorem 4.2 (H 2 /H control) The state feedback control (17) stabilizes the linear discrete-time positive system (1), (2) and H(z) 2 < γ 2 as well as H(z) < γ if for given strictly positive system matrix F there exist positive definite diagonal matrices P, R k IR n n, U IR m m and a positive scalar γ IR + such that for h = 0, 1, 2,... n 1, k = 1, 2,... r, P = P T 0, U = U T 0, γ > 0, (51) [ ] P P C T > 0, (52) U P 0 γ I r CP 0 γ I m 0 F P r < 0, (53) g k r T k G 0 P P F P r g k r T k G P 0 < 0. (54) I r T h F (j, j + h) (1 n)/n T ht P T h G dk T ht R k 0, (55) where T is defined in (24), F (j, j + 1) (1 n)/n in (25), G dk in (27) and R k is the structured matrix variable in the same structure as is introduced in (28), (29), while F (j, j + h) (1 n)/n, T, G dk IR + n n. When the above conditions hold, the control gain matrix K IR r n can be computed using (30). Proof: Rearranging the first inequality in (48) by using the Schur complement property leads to the matrix inequality V F V G V F T V 0 < 0. (56) G T 0 I r

11 On control of discrete-time LTI positive systems 2469 Therefore, inserting the closed-loop system matrix (20) into (56) gives V (F r g k k T k )V G V (F r g k k T k ) T V 0 < 0 (57) G T 0 I r and with the notation w T k = k T k V, (58) where the diagonal matrix variable V has to be used, then (57) implies V F V r g k w T k G V 0 I r < 0. (59) By H 2 control nomination the inequality (50) could be minimized, but this form cannot directly support the set of LMIs. Introducing the inequality U > CV C T = CV V 1 V C T, (60) with U IR m m being diagonal, symmetric and positive definite, and applying appropriate the Schur complement property, then (60) implies [ ] V V C T > 0, tr(u) = η. (61) U It is evident that now η = tr(u) > tr(cv C T ) > γ 2 2. Setting down a unique solution of K in consideration in (57) and (44) that is V = P, w T k = r T k, k = 1, 2..., r, (62) then (21)-(23), (59), (61) subject to joint formulation implies (52)-(55). This concludes the proof. Theorem 4.3 (control in Lyapunov sense) The state feedback control (17) stabilizes the linear discrete-time positive system (1), (2) if for the strictly positive matrix F there exist positive definite diagonal matrices P, R k, Q IR n n such that for h = 0, 1, 2,... n 1, k = 1, 2,... r, P = P T 0, Q = Q T 0, (63) P + Q F P r g k r T 0. (64) k P

12 2470 D. Krokavec and A. Filasová T h F (j, j + h) (1 n)/n T ht P T h G dk T ht R k 0, (65) where T is defined in (24), F (j, j + 1) (1 n)/n in (25), G dk in (27) and R k is the structured matrix variable in the same structure as is introduced in (28), (29), while F (j, j + h) (1 n)/n, T, G dk IR + n n. When the above conditions hold, the control gain matrix K IR r n can be computed using (30). Proof: Inserting the closed-loop system matrix (20) into (8) means P + Q (F r g k k T 0 (66) k )P P and using (45) then (66) gives P + Q F P r g k r T k P 0. (67) Considering that P, Q are positive definite diagonal matrices then, combining (67) with (23), it can prescribe (63)-(65). This concludes proof. If a linear positive discrete-time system is stabilizable by the state control, the closed-loop discrete-time system matrix F c is a Schur strictly positive matrix. In this sense, the conditions listed in Theorem can be considered as necessarily and sufficient. 5 Illustrative Example The generating strictly Metzlerian system is represented by the continuoustime linear state-space model with the parameters A = , B = , C = [ It is possible to verify that the Metzler matrix A is not Hurwitz since its eigenvalue spectrum is ]. ρ(a) = { , , ± i }.

13 On control of discrete-time LTI positive systems 2471 Converting for the sampling period t s = 0.02 s by the MATLAB function c2d( ), the computed discrete-time system parameters are F = , G = , where the matrix F is strictly positive but, consequently, not Schur, and the matrices G, C are positive matrices. To solve the control design task, the auxiliary parameters are constructed as follows: F (i, i) (1 4) = F (i, i + 1) (1 4)/4 = F (i, i + 2) (1 4)/4 = F (i, i + 3) (1 4)/4 = T = , , Using the SeDuMi package [25] to solve in MATLAB environment the set of LMIs (51) (55), then the resulting LMI variables are P = diag [ ], R 1 = diag [ ], R 2 = diag [ ],,,. U = diag [ ], γ , γ 2 2 <

14 2472 D. Krokavec and A. Filasová q1 q2 q3 q4 2.5 q(t) t [s] Figure 1: State variables response The control law gain matrix K IR 2 4, computed by using (28), is positive matrix, since k T 1 = [ ], k T 2 = [ ], [ ] K =, which implies the Schur strictly positive closed-loop system matrix F c = , with the eigenvalue spectrum ρ(f c ) = { , , ± i }. Analyzing the numerical results, it is evident that the Schur matrix F c is diagonally dominant. Although the matrix looks only positive in the given accuracy, it is, in fact, strictly positive. The obtained results are illustrated in Fig. 1 and Fig. 2, where the state vector q(t) as well as the output vector y(t) are positive, when the input in the closed-loop system is the positive vector w T = [ ] and u(i) = Kq(i) + w. It can verify that using the solution of (21) (23) the designed closed-loop system structure is represented as [ ] K =,

15 On control of discrete-time LTI positive systems y 1 (t) y 2 (t) y(t) t [s] F c = Figure 2: Output variables response , ρ(f c ) = { , , ± i }. This structure gives the closed-loop system a slower dynamics as above given, but mainly a lower sensitivity of the control signal, because the H norm upper-bound γ is greater. Implying from (63) (65), the closed loop structure is given as follows [ ] K =, F c = , ρ(f c ) = { , ± i }. Evidently, the dynamics of the closed-loop system in this case is the worst. With respect to the n 2 boundary conditions (22) it is evident that not every linear positive system is stabilizable by the state control, while it is clear that a linear discrete-time system is also stabilizable when the generating linear continuous-time system is stabilizable. 6 Concluding Remarks A novel approach is presented in the paper to address the problem of effectively computing a state feedback control law gain that makes the positive system in

16 2474 D. Krokavec and A. Filasová closed-loop to be strictly positive and stable. Based on the matrix properties of a Schur strictly positive matrix, the algebraic constraints implying from the linear programming approach are reformulated as a set of LMIs, and replenished by the Lyapunov matrix inequality in the sense of the second Lyapunov method. It is derived that all matrix variables associated with this set of LMIs have to be positive definite and diagonal. The design conditions are formulated for all basic principle, including H 2 /H approach. The proposed approaches provide numerically reliable computational frameworks, as illustrated using a numerical example, and might be extended to other particular cases. Acknowledgements. The work presented in this paper was supported by VEGA, the Grant Agency of the Ministry of Education and the Academy of Science of Slovak Republic, under Grant No. 1/0608/17. This support is very gratefully acknowledged. References [1] M. Ait Rami and F. Tadeo, Positive observation problem for linear discrete positive systems, Proceedings of the 45th IEEE Conference on Decision & Control, San Diego, CA, USA (2006), [2] M. Ait Rami and F. Tadeo, Controller synthesis for positive linear systems with bounded controls, IEEE Transactions on Circuits and Systems II: Express Briefs, 54 (2007), [3] P.J. Antsaklis and A.N. Michel, A Linear Systems Primer, Birkhäuser, Boston, [4] A. Berman, M. Neumann and R. Stern, Nonnegative Matrices in Dynamic Systems, John Wiley & Sons, New York, [5] R. Bhatia, Matrix Analysis, Springer-Verlag, New York, [6] G. Birkhoff and S. Mac Lane, A Survey of Modern Algebra, Macmillan Publishing, New York, [7] B. Canto, R. Canto and S. Kostova, Stabilization of positive linear discrete-time systems by using a Brauer s theorem, The Scientific World Journal, 2014 (2014), Article ID ,

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