Abstract

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 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

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 state-space 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 [2–6] 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 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, , denote nonnegative vector and nonnegative matrix, means that is symmetric positive definite matrix, indicates the eigenvalue spectrum of a square matrix, the symbol marks the -th order unit matrix, enters up a diagonal matrix, , refer to the set of all -dimensional real vectors and real matrices, respectively, and , refer to the set of all -dimensional real nonnegative vectors and real nonnegative matrices, respectively.

2. System Description

In the following the Metzlerian class of positive linear dynamical systems is considered where state-space description iswhere , , stand for state, control input, and measurable output, , , .

Definition 1 ([17], 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.

Definition 2. A matrix and a vector are nonnegative if all its entries are nonnegative and at least one is positive.

Definition 3 (see [18–20]). A square matrix is Metzler matrix if its off-diagonal elements are nonnegative. A Metzler matrix is called strictly Metzler if its diagonal elements are negative and its off-diagonal elements are positive. A Metzler matrix is stable if it is strictly Metzler and Hurwitz.

Proposition 4 (see [21]). A Metzler matrix is stable if and only if all principal minors of the matrix are positive.

Proposition 5 (see [22, 23]). A square matrix is said to be strictly diagonally dominant ifwhere 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.

Proposition 6 (see [17]). A solution of (1) for is asymptotically stable and positive if is stable Metzler matrix, is nonnegative matrix and for given and . The linear system (1), (2) is asymptotically stable and positive if is stable Metzler matrix, , are nonnegative matrices and for all and . The linear system (1), (2) is asymptotically stable and externally positive if is stable Metzler matrix and , are nonnegative matrices.

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 , the output variables are also nonnegative.

Proposition 7 ([24], Lyapunov inequalities). Autonomous system (1) is asymptotically stable if there exist symmetric positive definite matrices or such that

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 and setting ; then (5) implies (4).

Lemma 8. If is a symmetric positive definite matrix, it yields the following relation:where denotes the symmetric item to .

Proof. Since it yields(7) and (8) imply (6). This concludes the proof.

Definition 9 ([25], congruent modulo ). Let be a fixed positive integer. Two integers and are congruent modulo if they differ by an integral multiple of the integer (they leave the same remainder when divided by ). If and are congruent modulo , the expression is called a congruence, and the number is called the modulus of the congruence.

Note that the statement is equivalent to the statement “ is divisible by ” or to the statement “there is an integer for which ” and the word “modulo” generally means “to the modulus”.

Definition 10 (see [26]). Let be the complete set of residues for any positive integer . The addition modulo on is , where is the element of to which the result of the usual sum of integers 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 . From this reason let be the complete set of residues for any positive integer . Then the addition modulo on is in the following defined as , where is the element of to which the result of the usual sum of integers and is congruent modulo . The used shorthand symbolical notation for is so .

3. 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 aswhere is the vector of the system state variables, , are the input and output variables, is Metzler matrix, , are nonnegative vectors, the pair is controllable, and the pair is observable.

Note that if is Metzler matrix, and , are nonnegative vectors, (9) and (10) are Metzlerian system.

3.1. State Controller Design

Considering system (9), (10), the full state control law is defined aswhere is the gain vector. Thus, using (9), (11), it yieldswhereis 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.

Theorem 12. The closed-loop system (12), (13) is strictly Metzlerian if system (9), (10) is strictly Metzlerian and there exists a positive definite diagonal matrix such that for ,wherewhile , .
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:respectively, where is the diagonal matrix variable.
Moreover, (26) impliesThat 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) asit can set for the diagonal elements of (29)which leads towhere is the diagonal matrix with one circular shift of its diagonal elements. It is evident that (31) is also a symmetric matrix.
Repeating this procedure -times, it can be obtained from (23) thatand, consequently,which gives the symmetric matrix inequalitywhere is the diagonal matrix with circular shifts of its diagonal elements.
Using the permutation matrix of the structure (58) [22] it can be easily verified that for it yieldsThus, premultiplying the right side by and postmultiplying the left side by then (34) leads torespectively, and using (6), then (37) implies (16). This concludes the proof.

Theorem 13. The closed-loop system (12), (13) is stable strictly Metzlerian if system (9), (10) is strictly Metzlerian and there exist positive definite diagonal matrices such that for ,where the parameters are given in (18)–(22).
When the above conditions hold, the control law gain vector is given as

Proof. Inserting (14) into (4) givesSince with a positive diagonal matrix it also yieldsusing the notation(47) implies (38).
Multiplying the right side of (27) by givesand with the notation(49) implies (39).
Multiplying the right side of (37) by givesand with the notation (50) then (51) implies (40). This concludes the proof.

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.

3.2. Luenberger Observer Design

Considering the observable system (9), (10), the Luenberger observer is given asand using (9), (10) and (52), (53) it yieldswhere is the vector of the system state estimate, is the output variable estimate, and is the observer gain vector.

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 such that for where is defined in (18),while , .

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.
Noting down the observer matrix structure (55) as it is evident that is a strictly Metzler matrix ifWriting (64) in the diagonal matrix structurerespectively, where is the diagonal matrix variable, it is evident that such structure is symmetric and (67) can be written as (56) using notations (60), (61).
Overwriting inequality (63) asit has to be for the diagonal elements of (68)which leads to the symmetric inequalitywhere is the diagonal matrix with one circular shift of its diagonal elements.
Repeating this procedure -times, it can obtain the following:respectively. Thus, analogously, (72) results inUsing the permutation matrix of the structure (58) it yields for thatand premultiplying the left side by and postmultiplying the right side by then the symmetric structure (73)respectively, and using (6), then (76) implies (57). This concludes the proof.