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) ∈ 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

−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 H∞ norms of H(z) are used.

Definition 2.1 [12], [29] The H2-norm and H∞-norm of the transfer func- tions matrix (3) are defined as

π 2 1 jω ∗ jω kH(z)k2 = tr ∫ H(e )H (e )dω , (4) 2π −π

jω jω ∗ jω kHd(z)k∞ = sup σo(Hd(e )) = sup σo(eig(Hd(e )Hd(e )) , (5) ω∈h−π,πi ω∈h−π,πi √ where z = ejω, ω is the frequency variable, j := −1, H∗(ejω) is the adjoint jω jω 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 ∈ n×n IR and a positive scalar γ∞ ∈ IR such that

T P = P > 0, γ∞ > 0 , (6)  −P ∗ ∗ ∗   0 −γ∞Ir ∗ ∗    < 0 , (7)  CP 0 −γ∞Im 0  FPG 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 di- screte-time system (1), (2) is asymptotically stable if there exist symmetric positive definite matrices P , Q ∈ 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 ∈ IR is a positive definite symmetric matrix and γ2 ∈ 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) ∈ IR+, u(i) ∈ IR+, y(i) ∈ IR+ , F ∈ IR+ , G ∈ IR+ , m×n C ∈ IR+ and i ∈ Z+. Then a solution q(i) of (1) is asymptotically stable n r and positive, i.e., limi→∞ q(i) = 0 while q(i) ∈ IR+ for u(i) ∈ IR+ and the n n×r initial value q(0) ∈ IR+, if F is a positive Schur matrix and G ∈ 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 ∈ IR+ , C ∈ IR+ are m r non-negative matrices and y(i) ∈ IR+ for u(i) ∈ IR+ and the initial value q(0) ∈ IR+. The linear system (1), (2) is asymptotically stable and internally n×r m×n positive if F is a positive Schur matrix and G ∈ IR+ , C ∈ IR+ are non- negative matrices. n×n Definition 3.2 [1] A square matrix F ∈ IR+ is positive if its elements n×n are nonnegative. A square matrix F ∈ IR+ 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 diago- nally 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. 2464 D. Krokavec and A. Filasov´a

4 Control of Positive Discrete-time Systems

Linear discrete-time closed-loop MIMO systems, obtained from the control- lable positive system (1), (2) by using the state control law

r×n u(i) = −Kq(i) , K ∈ IR+ , (17) is described by the state-space equations

q(i + 1) = (F − GK)q(i) = F cq(i) , (18)

y(i) = Cq(i) , (19) where r   T   X T G = g1 ··· gr , K = k1 ··· kr , F c = F − ggkk . (20) k=1

n×n n×r Naturally, if F ∈ IR+ is a strictly positive matrix, and G ∈ IR+ , m×n C ∈ IR+ 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 H2/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 kH(z)k∞ < γ∞ if for given strictly positive system matrix F there exist positive definite diagonal matrices n×n P , Rk ∈ IR and a positive scalar γ∞ ∈ IR such that for h = 0, 1, 2, . . . n−1, k = 1, 2, . . . r, T P = P 0 , γ∞ > 0 , (21)  −P ∗ ∗ ∗   0 −γ∞Ir ∗ ∗     CP 0 −γ∞Im 0  < 0 , (22)  r   P T  FP − gkrk G 0 −P k=1 r h hT X h hT T F (j, j + h)(1↔n)/nT P − T GdkT Rk 0 , (23) k=1 where  0 0 ··· 0 1   1 0 ··· 0 0    −1 T T =  ..  , T = T , (24)  .  0 0 ··· 1 0 On control of discrete-time LTI positive systems 2465

  F (j, j + h)(1↔n)/n = diag f1,1+h ··· fn−h,n fn−h+1,1 ··· fn,h , (25)   g11 g12 ··· g1r  g g ··· g     21 22 2r  G = g1 g2 ··· gr =  .  , (26)  .  gn1 gn2 ··· gnr   Gdk = diag g1k g2k ··· gnk = diag [ {glk}, l = 1, . . . , n ] , (27)

Rk is the structured matrix variable such that   Rk = diag rk1 rk2 ··· rkn 0 , (28)

T   T T rk = rk1 rk2 ··· rkn = l Rk, l = [ 1 1 ··· 1 ] , (29) n×n and F (j, j + h)(1↔n)/n, T , Gdk ∈ IR+ . When the above conditions hold, the control gain matrix K is given as

 T  k1 K = R P −1, kT = lT K , K =  .  , (30) dk k k dk  .  T kr

r×n where K ∈ IR+ .

Proof: Writing the closed-loop system matrix F c as follows     f11 f12 ··· f1n g1k r  f21 f22 ··· f2n  X  g2k     .  −  .  kk1 kk2 ··· kkn ≺ 0 , (31)  ..   .    k=1  .  fn1 fn2 ··· fnn gnk it is evident that F c be a strictly positive matrix if all its elements satisfy the conditions r X fjl − gkjkkl > 0 ∀ j, l = 1, 2, . . . , n. (32) k=1 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

r X F (j, j)(1↔n)/n − GdkKdk 0 , (33) k=1   Kdk = diag kk1 kk2 ··· kkn = [ {kkj}j=1,...,n ] , k = 1, 2 . . . , r . (34) 2466 D. Krokavec and A. Filasov´a

Rewriting (31) as     f12 f13 ··· f1n f11 g1k  f f ··· f f  r  g   22 23 2n 21  X  2k     . −  .  kk2 kk3 ··· kkn kk1 , (35)  .  k=1  .  fn2 fn3 ··· fnn fn1 gnk it can set, analogously, for the diagonal elements of (35),

F (j, j + 1)(1↔n)/n − GdKdkc1 0 , (36) where Kdkc1 is the diagonal matrix Kdk with one circular shift of its diagonal elements. Since it yields using the permutation matrix (24) that

−1 T Kdk = TKdkc1T = TKdkc1T , (37) premultiplying the left side by T and postmultiplying the right side by T T the inequality (36) implies

r T P T T TF (j, j + 1)(1↔n)/nT − TGdT TKdkc1T = k=1 r (38) T P T = TF (j, j + 1)(1↔n)/nT − TGdT Kdk 0 . k=1 Repeating this procedure h-times, it can be obtained from (31) that     f1,1+h ··· f1,n f1,1 ··· f1,h g1k r  f2,i+h ··· f2,n f2,1 ··· f2,h  X g2k     ......  −  .  kk,1+h kk,2+h ··· kkh  ......   .   . . . .  k=1  .  fn,i+h ··· fn,n fn,1 ··· fn,h gnk (39) and so, consequently, with Kdkch representing Kdk with h circular shifts of its diagonal elements it yields

F (j, j + h)(1↔n)/n − GdkKdkch 0 , (40) which can be interpreted for h = 0,1,2,. . . n-1 as

r h hT X h hT T F (j, j + h)(1↔n)/nT − T GdkT Kdk 0 . (41) k=1 Multiplying the right side of (41) by a diagonal positive definite matrix P leads to r h hT X h hT T F (j, j + h)(1↔n)/nT P − T GdkT KdkP 0 (42) k=1 On control of discrete-time LTI positive systems 2467 and because (42) is a symmetric matrix inequality, with the notation

Rk = KdkP , (43) then (42) implies (23). Inserting the closed-loop system matrix (20) into (7) gives

 −P ∗ ∗ ∗   0 −γ∞Ir ∗ ∗     CP 0 −γ∞Im 0  < 0 (44)  r   P T  (F − gkkk )PG 0 −P k=1 and using from (43) implying notation

T T rk = kk P , (45) (44) implies (22). This concludes the proof.

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 Kdk directly implies the diagonal matrix variable strictures of P , Rk in Theorem 4.1. To simplify obtaining relation in A(j, j+h)(1↔n)/n it is possible to construct the following matrix   f11 f12 ··· f1n f11 f12 ··· f2n ◦  FF   ......  F = =  . . . .  . (46) f11 f12 ··· f1n f11 f12 ··· f1n

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 con- structed for h = 0, 1, 2, . . . n − 1 from (46).

Lemma 4.1 The matrix F of the system (1), (2) is stable and kH(z)k2 < n×n γ2 if there exists a symmetric positive definite matrix V ∈ IR such that

V = V T > 0 , (47)

T T T 2 FVF − V + GG < 0 , tr(CVC ) > γ2 . (48)

Proof: Let (48) yields for a symmetric positive definite matrix V . Then subtracting (9) from (48) leads to the inequality

T F (V − W c)F − (V − Wc) < 0 (49) 2468 D. Krokavec and A. Filasov´a

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

T T 2 tr(CVC ) > tr(CW cC ) = γ2 (50) and so (49), (50) imply (48). This concludes the proof. Combining the algorithms for H2 and H∞ control design, the H2/H∞ princi- ple with H2 and H∞ performance constraints on positive discrete-time systems is given by the following theorem.

Theorem 4.2 (H2/H∞ control) The state feedback control (17) stabilizes the linear discrete-time positive system (1), (2) and kH(z)k2 < γ2 as well as kH(z)k∞ < γ∞ if for given strictly positive system matrix F there exist n×n m×m positive definite diagonal matrices P , Rk ∈ IR , U ∈ IR and a positive scalar γ∞ ∈ IR+ such that for h = 0, 1, 2, . . . n − 1, k = 1, 2, . . . r,

T T P = P 0 , U = U 0 , γ∞ > 0 , (51)  PPCT  > 0 , (52) ∗ U  −P ∗ ∗ ∗   0 −γ∞Ir ∗ ∗     CP 0 −γ∞Im 0  < 0 , (53)  r   P T  FP − gkrk G 0 −P k=1  r  P T −PFP − gkrk G  k=1    < 0 . (54)  ∗ −P 0  ∗ ∗ −Ir r h hT X h hT T F (j, j + h)(1↔n)/nT P − T GdkT Rk 0 , (55) k=1 where T is defined in (24), F (j, j + 1)(1↔n)/n in (25), Gdk in (27) and Rk is the structured matrix variable in the same structure as is introduced in (28), n×n (29), while F (j, j + h)(1↔n)/n, T , Gdk ∈ IR+ . When the above conditions hold, the control gain matrix K ∈ IRr×n can be computed using (30).

Proof: Rearranging the first inequality in (48) by using the Schur comple- ment property leads to the matrix inequality  −VFVG  T  VF −V 0  < 0 . (56) T G 0 −Ir On control of discrete-time LTI positive systems 2469

Therefore, inserting the closed-loop system matrix (20) into (56) gives

 r  P T −V (F − gkkk )VG  k=1   r   P T T  < 0 (57)  V (F − gkkk ) −V 0   k=1  T G 0 −Ir and with the notation T T wk = kk V , (58) where the diagonal matrix variable V has to be used, then (57) implies

 r  P T −VFV − gkwk G  k=1    < 0 . (59)  ∗ −V 0  ∗ ∗ −Ir

By H2 control nomination the inequality (50) could be minimized, but this form cannot directly support the set of LMIs. Introducing the inequality

U > CVCT = CVV −1VCT , (60) with U ∈ IRm×m being diagonal, symmetric and positive definite, and applying appropriate the Schur complement property, then (60) implies

 VVCT  > 0, tr(U) = η . (61) ∗ U

T 2 It is evident that now η = tr(U) > tr(CVC ) > γ2 . Setting down a unique solution of K in consideration in (57) and (44) that is T T V = P , wk = rk , 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 , Rk, Q ∈ IRn×n such that for h = 0, 1, 2, . . . n − 1, k = 1, 2, . . . r,

P = P T 0 , Q = QT 0 , (63)

 −P + Q ∗  r  P T  ≺ 0 . (64) FP − gkrk −P k=1 2470 D. Krokavec and A. Filasov´a

r h hT X h hT T F (j, j + h)(1↔n)/nT P − T GdkT Rk 0 , (65) k=1 where T is defined in (24), F (j, j + 1)(1↔n)/n in (25), Gdk in (27) and Rk is the structured matrix variable in the same structure as is introduced in (28), n×n (29), while F (j, j + h)(1↔n)/n, T , Gdk ∈ IR+ . When the above conditions hold, the control gain matrix K ∈ IRr×n can be computed using (30).

Proof: Inserting the closed-loop system matrix (20) into (8) means

 −P + Q ∗  r  P T  ≺ 0 (66) (F − gkkk )P −P k=1 and using (45) then (66) gives

 −P + Q ∗  r  P T  ≺ 0 . (67) FP − gkrk −P k=1 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 ma- trix. In this sense, the conditions listed in Theorem 4.1 - 4.3 can be considered as necessarily and sufficient.

5 Illustrative Example

The generating strictly Metzlerian system is represented by the continuous- time linear state-space model with the parameters

 −3.3800 0.2080 6.7150 5.6760   0.4000 0.1888   0.5810 −4.2900 2.0500 0.6750   0.5679 0.2030  A =   , B =   ,  1.0670 4.2730 −6.6540 5.8930   0.1136 0.3146  0.0480 2.2730 0.1430 −2.1040 0.1136 0.1701

 4 0 1 0  C = . 0 0 0 1 It is possible to verify that the Metzler matrix A is not Hurwitz since its eigenvalue spectrum is

ρ(A) =  1.2887, −8.8079, −4.4544 ± 1.2253 i . On control of discrete-time LTI positive systems 2471

Converting for the sampling period ts = 0.02 s by the MATLAB function c2d(∗), the computed discrete-time system parameters are

 0.9361 0.0116 0.1219 0.1149   0.0081 0.0043   0.0112 0.9197 0.0375 0.0156   0.0110 0.0041  F =   , G =   ,  0.0198 0.0792 0.8784 0.1098   0.0028 0.0063  0.0012 0.0428 0.0035 0.9593 0.0025 0.0034 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:  0 0 0 1   1 0 0 0  T =   ,  0 1 0 0  0 0 1 0  0.9361   0.9197  F (i, i)(1↔4) =   ,  0.8784  0.9593  0.0116   0.0375  F (i, i + 1)(1↔4)/4 =   ,  0.1098  0.0012  0.1219   0.0156  F (i, i + 2)(1↔4)/4 =   ,  0.0198  0.0428  0.1149   0.0112  F (i, i + 3)(1↔4)/4 =   .  0.0792  0.0035 Using the SeDuMi package [25] to solve in MATLAB environment the set of LMIs (51)–(55), then the resulting LMI variables are

P = diag  0.1033 0.0111 0.0347 0.0049  ,   R1 = diag 0.0480 0.0004 0.0485 0.0001 ,   R2 = diag 0.0008 0.0291 0.0005 0.0185 ,   2 U = diag 7.2091 6.3863 , γ∞ ≤ 39.5382 , γ2 < 1.6931 . 2472 D. Krokavec and A. Filasov´a

4 q1 q2 3.5 q3 q4 3

2.5

2 q(t)

1.5

1

0.5

0 0 1 2 3 4 5 t [s]

Figure 1: State variables response

The control law gain matrix K ∈ IR2×4, computed by using (28), is positive matrix, since T   k1 = 0.4644 0.0405 1.3988 0.0218 , T   k2 = 0.0079 2.6249 0.0156 3.7404 ,  0.4644 0.0405 1.3988 0.0218  K = , 0.0079 2.6249 0.0156 3.7404 which implies the Schur strictly positive closed-loop system matrix  0.9323 0.0001 0.1106 0.0988   0.0060 0.9086 0.0221 0.0001  F c =   ,  0.0185 0.0626 0.8744 0.0862  0.0000 0.0337 0.0000 0.9464 with the eigenvalue spectrum  ρ(F c) = 0.8427, 0.9880, 0.9155 ± 0.0189 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 wT = [ 1.67 0.76 ] 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  0.4417 0.1232 1.3335 0.0656  K = , 0.0172 2.4070 0.0465 3.5486 On control of discrete-time LTI positive systems 2473

16 y (t) 1 y (t) 14 2

12

10

8 y(t)

6

4

2

0 0 1 2 3 4 5 t [s]

Figure 2: Output variables response

 0.9324 0.0004 0.1109 0.0992   0.0063 0.9086 0.0227 0.0004  F c =   ,  0.0185 0.0637 0.8744 0.0873  0.0000 0.0342 0.0001 0.9470  ρ(F c) = 0.8424, 0.9893, 0.9153 ± 0.0191 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∞ ≤ 46.5709 is greater. Implying from (63)–(65), the closed loop structure is given as follows

 0.3030 0.4014 1.0625 0.1106  K = , 0.0946 1.7006 0.2131 3.2637

 0.9332 0.0011 0.1124 0.1001   0.0075 0.9084 0.0250 0.0011  F c =   ,  0.0184 0.0674 0.8741 0.0890  0.0001 0.0360 0.0002 0.9479  ρ(F c) = 0.8415 0.9937, 0.9142 ± 0.0203 i . Evidently, the dynamics of the closed-loop system in this case is the worst. With respect to the n2 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 2474 D. Krokavec and A. Filasov´a 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 replen- ished 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 H2/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.

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Received: August 16, 2017; Published: September 26, 2017