LMI Based Principles in Strictly Metzlerian Systems Control Design

The 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 linearmatrix inequalities principle. Incorporating an associated structure of linearmatrix inequalities, combinedwith 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.


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 [1]. The mathematical theory of Metzler matrices has a close relationship to the theory of positive linear continuous-time dynamical systems, since in the statespace description form the system dynamics matrix of a positive systems is Metzler and the system input and output matrices are nonnegative matrices. In abbreviated terms, such continuous-time linear systems are denoted in the following as Metzlerian systems. Stability and stabilization of such systems are reported, e.g., in [2][3][4][5][6] and various methods based on linear programming (LP) and linear matrix inequalities (LMI) are considered for positive stabilization and observer designs, as it is further elaborated in subsequent paragraphs.
The task of the state observer design for positive linear system is solved in [7] using the coordinates transformation. If the coordinate transform matrix is a solution of the specified Silvester equation and its inverse is nonnegative, the positive observer is constructed only for transformed state coordinates. Another proposed way is to transform system into controllable and uncontrollable parts to obtain a reduced order realization, suitable for positive observer design. The second group of methods represents [8], where LMI conditions to state observer design for positive linear systems are presented, but the observer is not of Luenberger type since the standard observer gain matrix does not specify the observer dynamics and an additive matrix is used to define the observer stability.
The most similar idea to the one presented in this paper is the approach given in [9,10], using formulation based on the set of LMIs. These approaches are considered for both positive and nonpositive systems. However, since nondiagonal matrix variables are exploited, they do not in general guarantee diagonally dominant Metzler structure of the closed-loop system matrix. The approach of the present paper makes it possible to achieve this additional constraint, if required.
The linear programming approach was effectively used for static output feedback design of positive systems in [11,12], and [13,14] consider both positive observer and control design with bounded control constraint.
The main motivation of this paper is to give design conditions for stabilization of linear positive continuous-time systems by the state-feedback, as well as the conditions for Lunberger state observers design in the same systems structures, using the formulation based on strict LMIs. Analyzing structures of the closed-loop system matrix, and the observer 2 Mathematical Problems in Engineering system matrix, the resulting algebraic constraints, which lead to strictly Metzler matrices, are formulated as a set of LMIs with diagonal matrix variables. This set is extended by an LMI, reflecting the Lyapunov stability condition [15]. Since the stability is posed as an LMI, constraints implying from the strictly Metzler matrix structure in the form of LMIs results also in the LMI formulation [16]. Advantages may also be added to the fact that these conditions can be adapted to the synthesis of nonnegative gain matrices of the regulator or estimator, as well as for the synthesis resulting in a Metzler matrix structure, if the conditions for its zero nondiagonal elements are satisfied. Preferring LMI structure, the proofs are of standard way in the sense of Lyapunov principle and reflect the facts when optimization in positive systems has LMIs representation.
The paper is organized as follows. After the Introduction, Section 2 presents some preliminaries, including the characterization of positive systems. A newly introduced set of LMIs, describing the design conditions for strictly Metzlerian SISO systems, is theoretically substantiated and proven in Section 3 and, subsequently, the design conditions for strictly Metzlerian MIMO systems are generalized in Section 4. An example is provided to demonstrate the proposed approach in Section 5, while Section 6 draws some conclusions.
Used notations are conventional so that , denote transpose of the vector and matrix , respectively, ∈ R + , ∈ R × + denote nonnegative vector and nonnegative matrix, = ≻ 0 means that is symmetric positive definite matrix, ( * ) indicates the eigenvalue spectrum of a square matrix, the symbol marks the -th order unit matrix, diag[⋅] enters up a diagonal matrix, R , R × refer to the set of all -dimensional real vectors and × real matrices, respectively, and R , R × + refer to the set of all -dimensional real nonnegative vectors and × real nonnegative matrices, respectively.

System Description
In the following the Metzlerian class of positive linear dynamical systems is considered where state-space description iṡ where ( ) ∈ R , ( ) ∈ R , ( ) ∈ R stand for state, control input, and measurable output, and a vector ∈ R + are nonnegative if all its entries are nonnegative and at least one is positive.
Proposition 5 (see [22,23] where denotes the entry in the th row and th column. If the matrix is strictly diagonally dominant and all its diagonal elements are negative, then the real parts of all its eigenvalues are negative. A strictly Metzler matrix is stable if it is strictly diagonally dominant.
To highlight some features of positive systems, presented exposition is placed within the framework introduced by the monograph [17], p. 8 and 41.
In process control, (2) defines the measurement subsystem on the plant and ( ) defines the output variables, whose values must be complied. If the matrix would be not nonnegative, some of the output variables could be negative. Thus, considering the nonnegative state variables and ∈ R × + , the output variables are also nonnegative.
= ≻ 0, Note that the equivalent statements (4) and (5) are given to circumvent formulations with inverse matrices in the following sections, since direct use (4) leads to bilinear matrix inequality in the design of the observer, and direct use (5) leads to bilinear matrix inequality in the synthesis of the regulator. It is easily verifiable that premultiplying the left side and postmultiplying the right side by the matrix = −1 and setting = −1 −1 ; then (5) implies (4).
Definition 9 ([25], congruent modulo ). Let be a fixed positive integer. Two integers and ℎ are congruent modulo if they differ by an integral multiple of the integer (they leave the same remainder when divided by ). If and ℎ are congruent modulo , the expression ( = ℎ) mod is called a congruence, and the number is called the modulus of the congruence.
Note that the statement ( = ℎ) mod is equivalent to the statement "( − ℎ) is divisible by " or to the statement "there is an integer for which − ℎ = " and the word "modulo" generally means "to the modulus".
Definition 10 (see [26]). Let = {0, 1, 2, . . . , − 1} be the complete set of residues for any positive integer . The addition modulo on is (ℎ + ) mod = , where is the element of to which the result of the usual sum of integers ℎ and is congruent modulo . Corollary 11. The problem of indexing in this paper is that the rows and columns of a square matrix of dimension × are generally denoted from 1 to and not from 0 to − 1. From this reason let = {0, 1, 2, . . . , } be the complete set of residues for any positive integer + 1. Then the addition modulo + 1 on is in the following defined as ( +ℎ) mod +1 = +1, where is the element of to which the result of the usual sum of integers ℎ and is congruent modulo +1. The used shorthand symbolical notation for (1 + ℎ) mod +1 = + 1 is so ( + ℎ) (1←→ )/ = + 1.

SISO Strictly Metzlerian Systems
The systems under consideration in this section are linear single-input, single-output (SISO) continuous-time dynamic systems represented in state-space form aṡ where ( ) ∈ R is the vector of the system state variables, ( ) ∈ R, ( ) ∈ R are the input and output variables, ∈ R × is Metzler matrix, ∈ R + , ∈ R are nonnegative vectors, the pair ( , ) is controllable, and the pair ( , ) is observable.

State Controller Design.
Considering system (9), (10), the full state control law is defined as where ∈ R + is the gain vector. Thus, using (9), (11), it yieldṡ is the closed-loop system dynamics matrix. Note that the controller has to be designed not only to stabilize the system, but also to render the stable closed-loop system matrix strictly Metzlerian.
The following theorem defines the LMI conditions to obtain a strictly Metzler matrix but does not guarantee that is stable strictly Metzler matrix.

Mathematical Problems in Engineering
Hereafter, * denotes the symmetric item in a symmetric matrix.
Proof. Writing the closed-loop dynamics matrix as follows: it is evident that is a strictly Metzler matrix if all of its main diagonal elements are negative, i.e., and all its of diagonal elements are positive, i.e., To solve by an LMI solver, LMIs have to be symmetric, and so (24) can be rewritten in the following diagonal matrix structure: Moreover, (26) implies That is, (27) is a symmetric matrix. Thus, exploiting (6), then (27) can be written as (15) using the notations (17) and (20), (21).
Rewriting (23) as it can set for the diagonal elements of (29) which leads to where 1 is the diagonal matrix with one circular shift of its diagonal elements. It is evident that (31) is also a symmetric matrix.
Mathematical Problems in Engineering 5 Repeating this procedure ℎ-times, it can be obtained from (23) that and, consequently, which gives the symmetric matrix inequality where ℎ is the diagonal matrix with ℎ circular shifts of its diagonal elements.
Note that conditions (38)-(43) are all LMIs; that is, they are convex in the defined matrix variables.

Remark 14.
It can be noted that the necessary diagonal matrix variable structure of , presented in Theorem 12, directly implies the diagonal matrix variable structures of , in Theorem 13. Since, in the terms of the Krasovskii theorem [27], the matrices , in Proposition 7 can be zero matrices, this way can be also applied in (38), and in the inequalities we exploited the proposition properties in the following parts of the paper.
Note that the observer has to be designed not only to be stable, but also to render the observer system matrix stable strictly Metzlerian.

Theorem 15. The Luenberger observer (52), (53) is strictly
Metzlerian if system (9), (10) is strictly Metzlerian and there exists a positive definite diagonal matrix ∈ R × such that for ℎ = 1, 2, . . . − 1 where is defined in (18), Proof. Since the duality principle has to be applied on the structure (23), to avoid some misinterpretation the proof is given in partly full dual sense.

Mathematical Problems in
Repeating this procedure ℎ-times, it can obtain the following: respectively. Thus, analogously, (72) results in Using the permutation matrix of the structure (58) it yields for ℎ = 1, 2, . . . − 1 that and premultiplying the left side by ℎ and postmultiplying the right side by ℎ then the symmetric structure (73) respectively, and using (6), then (76) implies (57). This concludes the proof.
where the parameters are given in (18), (59)-(62). When the above conditions hold, the observer gain vector is given as Proof. Inserting (55) into (5) gives for a positive definite diagonal matrix that respectively. Therefore, using the notation (86) implies (77).

MIMO Strictly Metzlerian Systems
Linear multiple-input, multiple-output (MIMO) continuoustime closed-loop dynamical systems obtained from the controllable system (1), (2) by using the state control law are described by the state-space equationṡ The MIMO Luenberger observer, associated with the observable system (1), (2), is given aṡ where Naturally, if ∈ R × is Metzler matrix, and ∈ R × + , ∈ R × + are nonnegative matrices, system (1), (2) is Metzlerian system. Thus, it is necessary to render the closedloop system matrix and observer system matrix as stable strictly Metzler ones.
Mathematical Problems in Engineering 9 In the same way, (107) can be interpreted as and (109) implies (102). This concludes the proof.
Remark 19. The point was finding conditions for the synthesis of the state control law to stabilize positive systems that would be sufficiently general. The stabilizability is bound to a particular system pair ( , ) and the strictly Metzler matrix does not a priori generate any boundaries on elements of the nonnegative matrix to obtain that the matrix of closedloop dynamics = − is (strictly) Metzler matrix if is positive matrix.
If the Metzler matrix element = 0, ̸ = it follows for given from condition (107) that all elements of row of the matrix must be zero. With such a boundary on the structure of the matrix , the positive gain matrix , if exists, can be designed according to Theorem 18. The second way is to design as nonnegative matrix. In this case, if = 0, ̸ = , it is necessary to define the diagonal matrices in (115) as structured matrix variables, whose elements are zero for given , and then the nonnegative gain matrix , if exists, can be designed according to Theorem 18. Potentially, both ways can be combined.
Analogously the above-mentioned approaches can be applied in Luenberger observer synthesis for Metzlerian systems.
When the above conditions hold, the observer gain matrix is given as Proof. Inserting (100) into (5) gives respectively. Therefore, using the notation

Illustrative Example
The considered strictly Metzlerian system is represented by the model (1)     Analyzing the numerical results, it is evident that in this case the stabilization of the unstable strictly Metzlerian system principally changes the value of the unstable mode, while stable modes are modified very little. It is also obvious that the strictly Metzler matrix is stable but not diagonally dominant.
The obtained results are illustrated in Figures 1 and 2 One can verify that these results are perfectly comparable with the ones obtained by the methodology constructed for the discrete-time positive LTI systems for the associated continuous-time and discrete-time system models [29].
To design the strictly Metzlerian observer, the auxiliary parameters are constructed as The observer gain matrix is computed by using (140)   Analyzing the results, it is evident that the stable strictly Metzlerian observer is designed to the unstable strictly Metzlerian system that is the observer system matrix is stable strictly Metzler, but not diagonally dominant.
In Figure 3, the response of the strictly Metzlerian observer in estimating the closed-loop system state variable is presented, acting under the same simulation conditions as the above presented for the closed-loop system. Since the state variable estimation errors can be practically neglected, no other responses of this stable Metzlerian observer are presented.
In addition, the example illustrates that at the large positive values of some elements outside the diagonal of the strictly Metzler matrix of the system , neither the closedloop system matrix nor the observer system matrix is diagonally dominant, although they are stable strictly Metzler matrices. However, the results of our other simulations show that if the system matrix is diagonally dominant strictly Metzler matrix, even the matrices and are diagonally dominant to the strictly Metzler matrix.

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 strictly Metzlerian system in closedloop to be positive and stable, as well as of recounting the observer gain that establish the stable strictly Metzlerian structures of Luenberger observers. Based on a stable strictly Metzler matrix, algebraic constraints implying from 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 LMIs ensemble have to be positive definite and diagonal. The proposed approach provides a numerically reliable computational framework, as illustrated using the numerical example, and might be extended to other particular cases.
The example also illustrates that at the large positive values of some elements outside the diagonal of the strictly Metzler matrix of the system , neither the closed-loop system matrix nor the observer system matrix is diagonally dominant, although they are stable strictly Metzler matrices. The results of our other simulations show that if the system matrix is diagonally dominant strictly Metzler matrix, the matrices and are diagonally dominant strictly Metzler matrix. Further research topics include this problem as well as the problems of forced mode in positive systems, discrete-time linear positive structures, and nonlinear positive systems.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.