INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 12, 2018 LQ Problem in Stabilization of Linear Metzlerian Continuous-time Systems D. Krokavec and A. Filasova represented by state-space equations, the synthesis of Abstract—The paper present a consistent set of linear matrix stabilizing state-feedback controllers, guaranteeing the closed- inequalities which guaranties asymptotic stability of the closed-loop loop system is asymptotically stable and internally positive, is system, warranties strictly Metzlerian system structure, and adjusts conditionally supported by linear programming to meet the the state and output variables coincident with prescribed quadratic closed-loop system positive structure [11], [12]. In order to limits. To realize with a positive control law gain, the diagonal stabilizability of strictly Metzlerian linear continuous-time systems is reduce the number of constraints entering the solution in linear approved, and the related closed-form expression of design condi- programming methods, an alternative synthesis procedure with tions is provided. The results are illustrated using a particular LQ is proposed in [13], where the system parameter boundaries problem, for which numerical examples are given. are defined by n linear matrix inequalities (LMI), if the system is strictly Metzlerian. Because a solution of such defined base Keywords—asymptotic stability, linear matrix inequalities, linear set of LMIs only assures that the closed-loop system matrix is quadratic control, Metzlerian continuous-time systems, state feedback strictly Metzler, the design conditions are complemented by stabilization. another LMI that imposes a stable asymptotic solution. Since the applied LMI variables are of diagonal matrix structure, it I. INTRODUCTION can be refereed about diagonal stabilizability of the strictly Positive systems indicate the processes whose variables Metzlerian continuous-time linear systems. represent quantities that do not have meaning unless they are Constraining the class of controller matrix gains to be nonnegative [1]. Since, in the relevant continuous-time state- positive, it does not alleviate the complexity of the solutions space description, the system matrix of a positive system is for non strictly Metzlerian systems. Proceeding along the same Metzler, theory of Metzler matrices is naturally applied to this lines, and pursuing the formal system analogy, some appli- kind of dynamical systems [2]. Additionally limited in the way cable extensions of the above formulations for strictly positive that the system input and output matrices are at least discrete-time linear systems can be found in [14], [15]. nonnegative matrices [3], system stabilization means strictly Analyzing the challenging problem of state-feedback defined task to design a positive gain matrix of control law so stabilization of strictly Metzlerian linear continuous-time that the closed-loop system matrix is Metzler and Hurwitz [4]. systems, the main motivation of this paper are design Therefore, most of techniques applicable to ordinary linear conditions formulated for infinite-time horizon control with systems can not be straightly nominated to positive linear linear quadratic cost functions. Since, at defined constraints on systems [5], [6]. Mainly the books [7], [8] treat a considerable elements of a strictly Metzlerian system matrix structure, the number of the approaches to positive system analysis, and task cannot be formulated using a Riccati equation form, the include illustrative algorithms for many specific tasks, but matrices of cost function are used to extend that one LMI, there still remains a wide variety of related problems which reflects stability condition in overall completion of the (controllability, observability, speed of response, robustness) LMIs set in design conditions. The configuration chosen corres- which need to be solved and addressed to Metzlerian linear ponds a way exploiting the minimizing of the quadratic cost systems. A more detailed treatments of problems are given, criterion subject to a closed-loop stability constraint, the e.g. in [9], [10]. framework used is standard and convenient because other additive The trend in synthesis of feedback control of Metzlerian constraints may be included into design formulation. systems tends to simplify and disambiguate the strictly defined Used notations are conventional so that xT , X T denote design conditions. Supposing that the Metzlerian systems is transpose of the vector x , and matrix X , respectively, x+ , X+ indicate a nonnegative vector and a nonnegative matrix, ≻ means that X is a symmetric positive definite matrix, The work presented in this paper was supported by VEGA, the Grant X 0 Agency of Ministry of Education and Academy of Science of Slovak ρ()X reports the eigenvalue spectrum of the square matrix Republic, under Grant No. 1/0608/17. This support is very gratefully i X , the symbol In marks the n-th order unit matrix, diag[ ] acknowledged. ℝ n ℝ n× r The both authors are with the Department of Cybernetics and Artificial enters up a diagonal matrix, + , + signify the set of all n Intelligence, Faculty of Electrical Engineering and Informatics, Technical dimensional real non-negative vectors and n× r real non- University of Kosice, Letnna 9/B, 042 00 Kosice, Slovakia (phone: ++421 negative matrices, respectively. 55 602 4213; e-mail: [email protected], [email protected]) ISSN: 1998-0140 236 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 12, 2018 II. LINEAR CONTINUOUS-TIME POSITIVE SYSTEMS Lemma1: [15] Within the basic notations as above and To define the system positive structures, and to extend their applying the vector input variable formal stabilizability properties, it is preferred in the follo- kT 1 wing the state-space system description defined in the standard u()()()t= − Kq t = − ⋮ q t (10) way as T kr ɺq()()()t= Aq t + Bu t (1) on the strictly Metzler MIMO system (1), (2), while the y()()t= Cq t (2) × positive gain matrix K ∈ ℝ r n is prescribed to force the where the equations (1), (2) belong to the Metzlerian class of closed-loop system matrix n r m positive systems if q()t ∈ ℝ + , u()t ∈ ℝ + , y()t ∈ ℝ + (all r ≥ = − = − T variables are nonnegative) for all t 0 . Ac A BK A∑ bk k k (11) n× n In the general case, the matrix A∈ ℝ is restricted to k =1 being strictly Metzler (its diagonal elements are negative and then the matrix Ac is Metzler, if for given non-negative matrix its off-diagonal elements are positive) and the matrices ∈ ℝ n× r ∈ ℝ n× n n× r m× n B + and a strictly Metzler matrix A + there exist B ∈ ℝ , C ∈ ℝ are nonnegative (all its entries are × + + positive definite diagonal matrices PR, ∈ ℝ n n such that for nonnegative and at least one is positive). Satisfying these k + h=1,2, … n − 1, k=1,2, … r restrictions, the system (1), (2) is referred as a linear strictly T Metzlerian system. Note, a strictly Metzler matrix is stable if it PP= ≻ 0 (12) = T ≻ is Hurwitz. RRk k 0 (13) Terminating the class of admissible controllers to be linear r − ≺ and considering, for simplicity, a SISO linear strictly A(,)i i (1↔n ) ∑ BR dk k 0 (14) Metzlerian system (1), (2) controlled by the dimensionally k =1 compatible control, constrained to use a linear function of the r h+ hT− h hT ≻ state measurements, and a strictly positive real vector k such TA(,)j j h (1↔n )/ nTPTBTR∑ dk k 0 (15) that k =1 subject to the notations u( t )= −kT q ( t ), k ∈ ℝ n (3) 0 0⋯ 0 1 then the state-space enrollment of the closed-loop system is ⋯ =1 0 0 0 −1 = T given as T ⋱ , TT (16) ɺ = −T = ⋯ q()()()()t A bk qt Ac q t (4) 0 0 1 0 y()()t= Cq t (5) + = ⋯ ⋯ A(j , j h )(1↔n )/ n diag a1,1+h a n − h, n a n − h + 1,1 a n , h where (17) A= A − bkT (6) ⋯ c b11 b 1r ⋯ has to be a strictly Metzler matrix. Consequently, the closed- = [ ⋯ ] = b21 b 2r B b1 b 2 br ⋮ (18) loop system matrix structure (5) prescribes the algebraic ⋯ inequalities corresponding to the strictly Metzler matrix Ac as bn1 b nr follows = ⋯ B dk diagb1k b 2 k b nk (19) a= a − b k <0 for alli = 1,2, … , n (7) cii ii i i Then, if there are satisfied the above conditions for = − > = … ≠ acij a ij b i k j 0 for alli , j 1,2, , n , i j (8) prescribed set of variables, the control gain is = −1 TTT= = ⋯ where the detailed formats of the Metzler system matrix Kdk R k P, k k l K dk , l [ 1 1 1 ] (20) parameters, as well as the state controller gain vector structure are Remark 1: Since the rows and columns of an n× n square matrix are indexed from 1 to n, the addition modulo n +1 on a a⋯ a b k 11 12 1n 1 1 the set of residues S is considered in the following as a a⋯ a b k = 21 22 2n =2 = 2 (9) + = + A ⋮ ,,b⋮ k ⋮ (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 a a ⋯ a b k n1 n 2 nn n n n +1. The used shorthand symbolical notation is + = + Although the structure of the state feedback control law (3) (j h )(1↔n )/ n r 1. is simple, it should be noted that the positiveness constraint for Comment 1: As it is seen from Lemma 1, the resulting 2 ∈ ℝ n× n the solvability of the gain k is extended by the set of n conditions prescribe the Metzlerian structure of Ac + scalar inequalities (6), (7).