Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 9590253, 14 pages https://doi.org/10.1155/2018/9590253 Research Article LMI Based Principles in Strictly Metzlerian Systems Control Design Dušan Krokavec and Anna Filasová Department of Cybernetics and Artifcial Intelligence, Faculty of Electrical Engineering and Informatics, Technical University of Koˇsice, Letna9,04200Ko´ ˇsice, Slovakia Correspondence should be addressed to Duˇsan Krokavec; [email protected] Received 5 August 2017; Revised 15 January 2018; Accepted 22 April 2018; Published 17 July 2018 Academic Editor: Marzio Pennisi Copyright © 2018 Duˇsan Krokavec and Anna Filasova.´ Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te paper is concerned with the design requirements that relax the existing conditions reported in the previous literature for continuous-time linear positive systems, reformulating the linear programming approach by the linear matrix inequalities principle. Incorporating an associated structure of linear matrix inequalities, combined with the Lyapunov inequality guaranteeing asymptotic stability of positive system structures, the conditions are presented, with which the state-feedback controllers and the system state observers can be designed. A numerical example illustrates the proposed conditions. 1. Introduction design. Te second group of methods represents [8], where LMI conditions to state observer design for positive linear Positive systems are ofen found in the modeling and control systems are presented, but the observer is not of Luenberger of engineering and industrial processes, whose state variables type since the standard observer gain matrix does not specify represent quantities that do not have meaning unless they the observer dynamics and an additive matrix is used to are nonnegative [1]. Te mathematical theory of Metzler defne the observer stability. matrices has a close relationship to the theory of positive Te most similar idea to the one presented in this linear continuous-time dynamical systems, since in the state- paper is the approach given in [9, 10], using formulation space description form the system dynamics matrix of a based on the set of LMIs. Tese approaches are considered positive systems is Metzler and the system input and output for both positive and nonpositive systems. However, since matrices are nonnegative matrices. In abbreviated terms, such nondiagonal matrix variables are exploited, they do not in continuous-time linear systems are denoted in the following general guarantee diagonally dominant Metzler structure of as Metzlerian systems. Stability and stabilization of such sys- the closed-loop system matrix. Te approach of the present temsarereported,e.g.,in[2–6]andvariousmethodsbased paper makes it possible to achieve this additional constraint, on linear programming (LP) and linear matrix inequalities if required. (LMI) are considered for positive stabilization and observer Te linear programming approach was efectively used designs, as it is further elaborated in subsequent paragraphs. forstaticoutputfeedbackdesignofpositivesystemsin[11, Te task of the state observer design for positive linear 12], and [13, 14] consider both positive observer and control system is solved in [7] using the coordinates transformation. design with bounded control constraint. If the coordinate transform matrix is a solution of the Te main motivation of this paper is to give design specifed Silvester equation and its inverse is nonnegative, conditions for stabilization of linear positive continuous-time the positive observer is constructed only for transformed systems by the state-feedback, as well as the conditions for state coordinates. Another proposed way is to transform Lunberger state observers design in the same systems struc- system into controllable and uncontrollable parts to obtain tures, using the formulation based on strict LMIs. Analyzing a reduced order realization, suitable for positive observer structures of the closed-loop system matrix, and the observer 2 Mathematical Problems in Engineering system matrix, the resulting algebraic constraints, which lead A Metzler matrix � is called strictly Metzler if its diagonal to strictly Metzler matrices, are formulated as a set of LMIs elements are negative and its of-diagonal elements are with diagonal matrix variables. Tis set is extended by an positive.AMetzlermatrixisstableifitisstrictlyMetzlerand LMI, refecting the Lyapunov stability condition [15]. Since Hurwitz. the stability is posed as an LMI, constraints implying from the �×� strictly Metzler matrix structure in the form of LMIs results Proposition 4 (see [21]). AMetzlermatrix� ∈ R is stable also in the LMI formulation [16]. Advantages may also be ifandonlyifallprincipalminorsofthematrix−� are positive. added to the fact that these conditions can be adapted to the Proposition 5 � ∈ R�×� synthesis of nonnegative gain matrices of the regulator or (see [22, 23]). Asquarematrix is estimator, as well as for the synthesis resulting in a Metzler said to be strictly diagonally dominant if matrix structure, if the conditions for its zero nondiagonal � � � � �� � > ∑ �� � ∀� ∈ ⟨1, �⟩ , elements are satisfed. Preferring LMI structure, the proofs � ��� � �� � �=� ̸ (3) areofstandardwayinthesenseofLyapunovprincipleand refect the facts when optimization in positive systems has where ��� denotes the entry in the �th row and �th column. LMIs representation. If the matrix is strictly diagonally dominant and all its Te paper is organized as follows. Afer the Introduction, diagonal elements are negative, then the real parts of all its Section 2 presents some preliminaries, including the char- eigenvalues are negative. A strictly Metzler matrix is stable if acterization of positive systems. A newly introduced set of it is strictly diagonally dominant. LMIs, describing the design conditions for strictly Metzlerian SISO systems, is theoretically substantiated and proven in To highlight some features of positive systems, presented Section 3 and, subsequently, the design conditions for strictly exposition is placed within the framework introduced by the Metzlerian MIMO systems are generalized in Section 4. An monograph [17], p. 8 and 41. example is provided to demonstrate the proposed approach in Section 5, while Section 6 draws some conclusions. Proposition 6 �(�) �≥0 � � (see [17]). Asolution of (1) for is Used notations are conventional so that � , � denote asymptotically stable and positive if � is stable Metzler matrix, � � � ∈ �×� � transpose of the vector and matrix ,respectively, � ∈ R+ is nonnegative matrix and �(�) ∈ R+ for given �(�) ∈ � �×� � R+, � ∈ R+ denote nonnegative vector and nonnegative R+ and �(0) ∈ R+.Telinearsystem(1),(2)isasymptotically � �×� � = � ≻0 � stable and positive if � is stable Metzler matrix, � ∈ R+ , � ∈ matrix, means that is symmetric positive �×� � defnite matrix, �(∗) indicates the eigenvalue spectrum of R+ are nonnegative matrices and �(�) ∈ R+ for all �(�) ∈ R� �(0) ∈ R asquarematrix,thesymbol�� marks the �-th order unit + and +.Telinearsystem(1),(2)isasymptotically [⋅] R� R�×� stableandexternallypositiveif� is stable Metzler matrix and matrix, diag enters up a diagonal matrix, , refer to �×� �×� the set of all �-dimensional real vectors and �×�real matrices, � ∈ R+ , � ∈ R+ are nonnegative matrices. � �×� respectively, and R�, R+ refertothesetofall�-dimensional real nonnegative vectors and �×�real nonnegative matrices, In process control, (2) defnes the measurement sub- �(�) respectively. system on the plant and defnes the output variables, whose values must be complied. If the � matrix would be not nonnegative, some of the output variables could be negative. 2. System Description Tus, considering the nonnegative state variables and � ∈ R�×� In the following the Metzlerian class of positive linear dynam- + ,theoutputvariablesarealsononnegative. ical systems is considered where state-space description is Proposition 7 ([24], Lyapunov inequalities). Autonomous �̇ (�) = �� (�) + �� (�) , (1) system (1) is asymptotically stable if there exist symmetric �×� �×� positive defnite matrices �, � ∈ R or �, � ∈ R such � (�) = �� (�) , (2) that � � � where �(�) ∈ R , �(�) ∈ R , �(�) ∈ R stand for state, � �×� �×� � = � ≻0, control input, and measurable output, � ∈ R , � ∈ R , �×� � � ∈ R . � = � ≻0, (4) Defnition 1 ([17], positive linear system). Te linear system �� + ��� + � ≺0, (1), (2) is said to be positive if and only if for every nonnegative initial state and for every nonnegative input its state and � = �� ≻0, output are nonnegative. � = �� ≻0, �×� � (5) Defnition 2. Amatrix� ∈ R+ and a vector � ∈ R+ are � nonnegative if all its entries are nonnegative and at least one �� + � � + � ≺0. is positive. Note that the equivalent statements (4) and (5) are given �×� Defnition 3 (see [18–20]). A square matrix � ∈ R is to circumvent formulations with inverse matrices in the Metzler matrix if its of-diagonal elements are nonnegative. following sections, since direct use (4) leads to bilinear matrix Mathematical Problems in Engineering 3 inequality in the design of the observer, and direct use (5) 3. SISO Strictly Metzlerian Systems leads to bilinear matrix inequality in the synthesis of the regulator. It is easily verifable that premultiplying the lef side Te systems under consideration in this section are linear