𝐻_∞ Control for Linear Positive Discrete-Time Systems

Abstract

This paper is concerned with 𝐻_∞ control for linear positive discrete-time systems. Positive systems are characterized by nonnegative restriction on systems’ variables. This restriction results in some remarkable results which are available only for linear positive systems. One of them is the celebrated diagonal positive definite matrix solutions to some existed well-known results for linear systems without nonnegative restriction. We provide an alternative proof for criterion of 𝐻_∞ norm by using separating hyperplane theorem and Perron-Frobenius theorem for nonnegative matrices. We also consider 𝐻_∞ control problem for linear positive discrete-time systems via state feedback. Necessary and sufficient conditions for such problem are presented under controller gain with and without nonnegative restriction, and then the desired controller gains can be obtained from the feasible solutions.

1. Introduction

Positive systems are widespread in many practical systems, such as economic systems [1], biology systems [2], and age-structured population models [3], whose variables are required to be nonnegative and have no meaning with negative values. The explicit definition of a positive system is that its state and output are always nonnegative for any nonnegative initial state and any nonnegative input. Due to the nonnegative restriction on systems’ variables, positive systems are defined on cones rather than linear space. Hence, there are excellent and remarkable outcomes which are available only for positive systems. One of them is the existence of diagonal positive definite matrix solutions to some celebrated results for linear systems without nonnegative restriction. Therefore, the investigation of positive systems is interesting and challenging and developed a new branch in systems theory. Positive systems have been of great interest to many researchers over several decades. A great number of results have been reported in the literature; see, for instance, [3–16].

It is worth noting that convex optimization is a powerful tool for analysis of positive systems. In [9–14], some remarkable results for positive systems are studied using convex optimization. On the other hand, the problem of 𝐻_∞ control has been a topic of recurring interest for several decades. A great number of results on 𝐻_∞ control have been obtained, and different approaches have been proposed. In recent years, increasing attention has been paid to 𝐻_∞ norm analysis for positive systems. In [12], the KYP lemma for linear positive continuous-time systems is proved based on the semidefinite programming duality. The alternative proofs along the line of the rank-one separable property are given to several remarkable and peculiar results for positive systems in [13]. In [14], the KYP lemma for linear positive discrete-time systems is studied using a theorem of alternatives on the feasibility of linear matrix inequalities (LMIs).

This paper is organized as follows. Preliminaries are introduced in Section 2. Main results on 𝐻_∞ control for linear positive discrete-time systems are presented and proved in Section 3. Section 4 is devoted to illustrate the effectiveness of the obtained results by numerical examples. Section 5 concludes this paper.

2. Preliminaries

In this section, we introduce terminology, positive systems, various other definitions, and lemmas, which will be essentially used for proving our main results.

At first, the following notations will be used throughout this paper.

ℤ₊ denotes the nonnegative integers. 𝑥≥0 denotes the vector 𝑥∈ℝ^𝑛 with nonnegative entries. 𝐴≥0 denotes the matrix 𝐴∈ℝ^𝑛×𝑚 with nonnegative entries. ℝ^𝑛₊ denotes the set of all vectors 𝑥∈ℝ^𝑛 with 𝑥≥0. ℝ₊^𝑛×𝑚 denotes the set of all matrices 𝐴∈ℝ^𝑛×𝑚 with 𝐴≥0. 𝑆^𝑛 denotes the set of all symmetric matrices. 𝐷₊^𝑛×𝑛 denotes the set of all diagonal positive definite matrices. 𝐴⪰0, 𝐴≻0, and 𝐴≺0 mean that 𝐴 is a positive semidefinite, positive definite, and negative definite matrix, respectively. 𝐴_𝑖𝑗 denotes the 𝑖𝑗th entry of matrix 𝐴. 𝑥_𝑖 denotes the 𝑖th entry of vector 𝑥. For two matrices 𝐴,𝐵∈ℝ^𝑛×𝑚, 𝐴≥𝐵 means 𝐴_𝑖𝑗≥𝐵_𝑖𝑗, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚. For two vectors 𝑥,𝑦∈ℝ^𝑛, 𝑥≥𝑦 means 𝑥_𝑖≥𝑦_𝑖, 𝑖=1,2,…,𝑛. 𝜌(𝐴) denotes the spectral radius of matrix 𝐴 which is defined as 𝜌(𝐴)∶=max_{1≤𝑖≤𝑛}|𝜆_𝑖|, where 𝜆₁,𝜆₂,…,𝜆_𝑛 are the eigenvalues of 𝐴. 𝐷(𝐴) denotes the vector which is composed of the diagonal entries of 𝐴∈ℝ^𝑛×𝑛. ⟨𝑋,𝑌⟩=trace(𝑋𝑌) is the inner product on 𝑆^𝑛.

Consider the following linear discrete-time system: 𝑥()(𝑘)(𝑘),𝑦(𝑘)(𝑘)(𝑘),𝑘+1=𝐴𝑥+𝐵𝑢=𝐶𝑥+𝐷𝑢 (1) where 𝑥(𝑘)∈ℝ^𝑛 is the state, 𝑢(𝑘)∈ℝ^𝑚 is the input, and 𝑦(𝑘)∈ℝ^𝑝 is the output. 𝐴∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑚, 𝐶∈ℝ^𝑝×𝑛, and 𝐷∈ℝ^𝑝×𝑚 are known matrices.

Definition 1 (see [3]). System (1) is said to be positive if and only if 𝑥(𝑘)≥0, 𝑦(𝑘)≥0, 𝑘∈ℤ₊ for any 𝑥(0)≥0 and for any 𝑢(𝑘)≥0, 𝑘∈ℤ₊.

Definition 2 (see [3]). System (1) is said to be asymptotically stable if 𝜌(𝐴)<1.

Definition 3 (see [13]). For a given matrix 𝐻∈𝑆^𝑛 with 𝐻⪰0, we define ℎ∈ℝ^𝑛₊ by ℎ_𝑖∶=𝐻_𝑖𝑖,𝑖=1,2,…,𝑛. (2)

Lemma 4 (see [3], Perron-Frobenius theorem for nonnegative matrices). Let 𝐴∈ℝ₊^𝑛×𝑛; then 𝜌(𝐴) is an eigenvalue of 𝐴 and 𝐴 has a nonnegative eigenvector 𝑣 corresponding to 𝜌(𝐴).

A matrix 𝐴∈ℝ^𝑛×𝑛 is called a Metzler matrix if 𝐴_𝑖𝑗≥0, for all 𝑖, 𝑗 with 𝑖≠𝑗.

Lemma 5 (see [13]). For a given Metzler matrix 𝐴∈ℝ^𝑛×𝑛 and 𝐻∈𝑆^𝑛 with 𝐻⪰0, the following conditions hold:(i)(ℎℎ^𝑇)_𝑖𝑖=𝐻_𝑖𝑖, (ℎℎ^𝑇)_𝑖𝑗≥𝐻_𝑖𝑗, 𝑖≠𝑗,(ii)𝐷(ℎℎ^𝑇𝐴)≥𝐷(𝐻𝐴),
where ℎ∈ℝ^𝑛₊ is defined from 𝐻 as in Definition 3.

Lemma 6 (see [3]). System (1) is positive if and only if 𝐴≥0, 𝐵≥0, 𝐶≥0, and 𝐷≥0.

Lemma 7 (see [17], separating hyperplane theorem). Suppose that 𝐶₁ and 𝐶₂ are two convex sets that do not intersect; that is, 𝐶₁∩𝐶₂=Ø. Then, there exist 𝑎≠0 and 𝑏 such that 𝑎^𝑇𝑥≤𝑏 for all 𝑥∈𝐶₁ and 𝑎^𝑇𝑥≥𝑏 for all 𝑥∈𝐶₂.

For 𝐴∈ℝ₊^𝑛×𝑛, we consider 𝐵∶=𝑠𝐼−𝐴. If 𝑠>0 and 𝑠≥𝜌(𝐴), then 𝐵 is called an M-matrix. If 𝑠>𝜌(𝐴), then 𝐵 is a nonsingular M-matrix.

Lemma 8 (see [4]). A nonsingular matrix 𝐴∈ℝ^𝑛×𝑛 is an M-matrix if and only if 𝐴⁻¹≥0.

Lemma 9 (see [18], Schur complement). Given any real matrices 𝑄, 𝑆, and 𝑅 with 𝑄=𝑄^𝑇 and 𝑅=𝑅^𝑇, the following statement holds: 𝑆𝑄𝑆_𝑇𝑅≺0 (3) if and only if 𝑅<0,𝑄−𝑆𝑅⁻¹𝑆^𝑇<0, (4) or, equivalently, 𝑄<0,𝑅−𝑆^𝑇𝑄⁻¹𝑆<0. (5)

The transfer function matrix of system (1) is given by 𝐺(𝑧)=𝐶(𝑧𝐼−𝐴)⁻¹𝐵+𝐷, (6) and its 𝐻_∞ norm is defined as ‖𝐺‖_∞∶=sup_{(]𝜃∈−𝜋,𝜋}𝜎𝐺𝑒^𝑗𝜃, (7) where 𝜎(𝐺(𝑒^𝑗𝜃)) denotes the maximum singular value of 𝐺(𝑒^𝑗𝜃). In [14], it has been pointed out that ‖𝐺‖_∞=‖𝐺(1)‖, where ‖𝐺(1)‖=𝜎(𝐺(1)), if system (1) is positive and asymptotically stable.

3. 𝐻_∞ Control

In this section, we give an alternative proof for the existed result of 𝐻_∞ norm for positive discrete-time systems and investigate the 𝐻_∞ control under state feedback.

At first, we propose the following theorem which is helpful for the alternative proof.

Theorem 10. Suppose that system (1) is positive; the following conditions are equivalent:(i)There exists a nonzero ℎ∈ℝ^𝑛₊ such that (𝐴−𝐼)ℎ≥0.(ii)System (1) is not asymptotically stable.

Proof. (i)⇒(ii). Since 𝐴≥0, from Perron-Frobenius theorem for nonnegative matrices, it follows that 𝐴^𝑇𝑣=𝜌(𝐴)𝑣≥0, where 𝜌(𝐴)≥0 is the spectral radius of matrix 𝐴^𝑇 and 𝑣≥0 is an eigenvector corresponding to 𝜌(𝐴). Then it is obtained from condition (i) that 𝜌(𝐴)−1𝑣^𝑇ℎ≥0, (8) which implies 𝜌(𝐴)≥1; namely, system (1) is not asymptotically stable.
(ii)⇒(i). System (1) is not asymptotically stable; that is, 𝜌(𝐴)≥1. From Perron-Frobenius theorem, we immediately obtain the following result: 𝜌(𝐴)ℎ=𝐴ℎ≥ℎ, (9) where ℎ≥0 is an eigenvector corresponding to 𝜌(𝐴). This completes the proof.

The following theorem was firstly presented and proved in the light of alternatives on the feasibility of LMIs by Federico Najson in the literature [14]. Now we will give another proof using separating hyperplane theorem and Theorem 10.

Theorem 11. Suppose that system (1) is positive; the following conditions are equivalent.(i) System (1) is asymptotically stable and ‖𝐺‖_∞<1.(ii) There exists a diagonal positive definite matrix 𝑃 such that 𝐴^𝑇𝑃𝐴−𝑃+𝐶^𝑇𝐶𝐴^𝑇𝑃𝐵+𝐶^𝑇𝐷𝐵^𝑇𝑃𝐴+𝐷^𝑇𝐶𝐵^𝑇𝑃𝐵+𝐷^𝑇𝐷−𝐼≺0. (10)(iii) There exists a diagonal positive definite matrix 𝑃 such that −𝑃0𝐴^𝑇𝑃𝐶^𝑇0−𝐼𝐵^𝑇𝑃𝐷^𝑇𝑃𝐴𝑃𝐵−𝑃0𝐶𝐷0−𝐼≺0. (11)

Proof. We only prove (i)⇒(ii) since the implication (ii)⇒(i) is obvious from the existed criterion for linear discrete-time systems, and the equivalence between (ii) and (iii) is immediately obtained using Schur complement.
To the contrary, suppose that condition (10) does not hold for any diagonal positive definite matrix 𝑃. Define the following two sets: 𝐶₁𝐴∶=^𝑇𝑃𝐴−𝑃+𝐶^𝑇𝐶𝐴^𝑇𝑃𝐵+𝐶^𝑇𝐷𝐵^𝑇𝑃𝐴+𝐷^𝑇𝐶𝐵^𝑇𝑃𝐵+𝐷^𝑇𝐷−𝐼∣𝑃∈𝐷₊^𝑛×𝑛,𝐶₂∶=𝑄∣𝑄≺0,𝑄∈𝑆^𝑛+𝑚; (12) then it is easy to check that sets 𝐶₁ and 𝐶₂ are nonempty and convex. By the assumption, we have 𝐶₁∩𝐶₂=Ø. Then from the separating hyperplane theorem, there exists a nonzero 𝐻∈𝑆^𝑛+𝑚 such that ⟨𝐻,𝑆⟩≥0,∀𝑆∈𝐶₁,⟨𝐻,𝑆⟩≤0,∀𝑆∈𝐶₂. (13)(14) By condition (14), we can conclude that ⟨𝐻,𝑆⟩=trace(𝐻𝑆)≤0, for all 𝑆≺0, from which it is easy to verify that 𝐻⪰0. Thus it follows from condition (13) that there exists a nonzero 𝐻⪰0 such that trace𝐻𝐴^𝑇𝑃𝐴−𝑃+𝐶^𝑇𝐶𝐴^𝑇𝑃𝐵+𝐶^𝑇𝐷𝐵^𝑇𝑃𝐴+𝐷^𝑇𝐶𝐵^𝑇𝑃𝐵+𝐷^𝑇𝐷−𝐼≥0,∀𝑃∈𝐷₊^𝑛×𝑛, (15) which is equivalent to trace𝐻𝐴^𝑇𝑃𝐴−𝑃𝐴^𝑇𝐵𝑃𝐵_𝑇𝑃𝐴𝐵^𝑇+𝑃𝐵trace𝐻𝐶^𝑇𝐶𝐶^𝑇𝐷𝐷^𝑇𝐶𝐷^𝑇𝐷−𝐼≥0,∀𝑃∈𝐷₊^𝑛×𝑛. (16) Let 𝐻∶=^𝐻11𝐻12_{𝐻𝑇12𝐻22}, 𝐻₁₁⪰0, 𝐻₂₂⪰0; then the above condition can be rewritten as trace𝐻₁₁𝐴^𝑇𝑃𝐴−𝑃+𝐻₁₂𝐵^𝑇𝑃𝐴+𝐻^𝑇₁₂𝐴^𝑇𝑃𝐵+𝐻₂₂𝐵^𝑇+𝑃𝐵trace𝐻𝐶^𝑇𝐶𝐶^𝑇𝐷𝐷^𝑇𝐶𝐷^𝑇𝐷−𝐼≥0,∀𝑃∈𝐷₊^𝑛×𝑛, (17) or, equivalently, trace𝑃𝐴𝐻₁₁𝐴^𝑇−𝐻₁₁+𝐴𝐻₁₂𝐵^𝑇+𝐵𝐻^𝑇₁₂𝐴^𝑇+𝐵𝐻₂₂𝐵^𝑇+trace𝐻𝐶^𝑇𝐶𝐶^𝑇𝐷𝐷^𝑇𝐶𝐷^𝑇𝐷−𝐼≥0,∀𝑃∈𝐷₊^𝑛×𝑛, (18) which implies that(a)𝐷(𝐴𝐻₁₁𝐴^𝑇−𝐻₁₁+𝐴𝐻₁₂𝐵^𝑇+𝐵𝐻^𝑇₁₂𝐴^𝑇+𝐵𝐻₂₂𝐵^𝑇)≥0,(b)trace(𝐻^{𝐶𝑇𝐶𝐶𝑇𝐷}_{𝐷𝑇𝐶𝐷𝑇𝐷−𝐼})≥0.From condition (a), it follows that 𝐷𝐴𝐻₁₁𝐴^𝑇+𝐴𝐻₁₂𝐵^𝑇+𝐵𝐻^𝑇₁₂𝐴^𝑇+𝐵𝐻₂₂𝐵^𝑇𝐻=𝐷𝐴𝐵00¹¹𝐻₁₂𝐻^𝑇₁₂𝐻₂₂𝐴^𝑇0𝐵^𝑇0𝐻≥𝐷¹¹0.00 (19) Since system (1) is positive, then ,𝐴𝐵𝐶00^𝑇𝐶𝐶^𝑇𝐷𝐷^𝑇𝐶𝐷^𝑇𝐷−𝐼 (20) are nonnegative matrix and Metzler matrix, respectively. Define the nonzero ℎ∈ℝ^𝑛₊ from 𝐻 as in Definition 3 then from Lemma 5, we obtain 𝐷𝐴𝐵00ℎℎ^𝑇𝐴^𝑇0𝐵^𝑇0𝐻≥𝐷𝐴𝐵00¹¹𝐻₁₂𝐻^𝑇₁₂𝐻₂₂𝐴^𝑇0𝐵^𝑇0𝐻≥𝐷¹¹0,00traceℎℎ^𝑇𝐶^𝑇𝐶𝐶^𝑇𝐷𝐷^𝑇𝐶𝐷^𝑇≥𝐷−𝐼trace𝐻𝐶^𝑇𝐶𝐶^𝑇𝐷𝐷^𝑇𝐶𝐷^𝑇𝐷−𝐼≥0. (21)(22) Set ℎ∶=ℎ^𝑇₁ℎ^𝑇₂^𝑇, ℎ₁∈ℝ^𝑛₊, ℎ₂∈ℝ^𝑚₊; conditions (21) and (22) can be rewritten as 𝐷𝐴ℎ₁ℎ^𝑇₁𝐴^𝑇+𝐴ℎ₁ℎ^𝑇₂𝐵^𝑇+𝐵ℎ₂ℎ^𝑇₁𝐴^𝑇+𝐵ℎ₂ℎ^𝑇₂𝐵^𝑇≥𝐷ℎ₁ℎ^𝑇₁,ℎ^𝑇₁𝐶^𝑇𝐶ℎ₁+ℎ^𝑇₂𝐷^𝑇𝐶ℎ₁+ℎ^𝑇₁𝐶^𝑇𝐷ℎ₂+ℎ^𝑇₂𝐷^𝑇𝐷ℎ₂=𝐶ℎ₁+𝐷ℎ₂^𝑇𝐶ℎ₁+𝐷ℎ₂≥ℎ^𝑇₂ℎ₂. (23)(24) Since 𝐻 is nonzero, then we have the following three cases.(1) Consider that ℎ₁=0,ℎ₂≠0. By condition (24), ℎ^𝑇₂𝐷^𝑇𝐷ℎ₂≥ℎ^𝑇₂ℎ₂, which contradicts ‖𝐺‖_∞<1.(2) Consider that ℎ₁≠0,ℎ₂=0. We observe from (23) that 𝐷(𝐴ℎ₁ℎ^𝑇₁𝐴^𝑇)≥𝐷(ℎ₁ℎ^𝑇₁), which implies that 𝐴ℎ₁≥ℎ₁.  From Theorem 10, this is a contradiction. (3) Consider that ℎ₁≠0,ℎ₂≠0. From condition (24), it yields that 𝐶ℎ₁+𝐷ℎ₂≠0. Define matrix ℎΔ=²𝐶ℎ₁+𝐷ℎ₂^𝑇𝐶ℎ₁+𝐷ℎ₂^𝑇𝐶ℎ₁+𝐷ℎ₂, (25)which is well-defined and satisfies 𝜎(Δ)≤1. Note that ℎ₂=Δ(𝐶ℎ₁+𝐷ℎ₂). On the other hand, 𝜎(𝐷)<1; otherwise, this contradicts ‖𝐺‖_∞<1. It is known that for matrices 𝑀∈ℂ^𝑙×𝑚 and 𝑁∈ℂ^𝑚×𝑙 with 𝜎(𝑀)≤1 and 𝜎(𝑁)<1, det(𝐼−𝑀𝑁)≠0. Thus, 𝐼−Δ𝐷 is invertible and (𝐼−Δ𝐷)⁻¹≥0 since 𝐼−Δ𝐷 is a nonsingular 𝑀-matrix. Therefore, ℎ₂=(𝐼−Δ𝐷)⁻¹Δ𝐶ℎ₁ holds. Then from (23) we have 𝐷𝐴ℎ₁ℎ^𝑇₁𝐴^𝑇+𝐴ℎ₁ℎ^𝑇₂𝐵^𝑇+𝐵ℎ₂ℎ^𝑇₁𝐴^𝑇+𝐵ℎ₂ℎ^𝑇₂𝐵^𝑇=𝐷𝐴ℎ₁ℎ^𝑇₁𝐴^𝑇+𝐴ℎ₁ℎ^𝑇₁𝐶^𝑇Δ^𝑇(𝐼−Δ𝐷)^−𝑇𝐵^𝑇+𝐵(𝐼−Δ𝐷)⁻¹Δ𝐶ℎ₁ℎ^𝑇₁𝐴^𝑇+𝐵(𝐼−Δ𝐷)⁻¹Δ𝐶ℎ₁ℎ^𝑇₁𝐶^𝑇Δ^𝑇(𝐼−Δ𝐷)^−𝑇𝐵^𝑇=𝐷𝐴+𝐵(𝐼−Δ𝐷)⁻¹Δ𝐶ℎ₁ℎ^𝑇₁𝐴+𝐵(𝐼−Δ𝐷)⁻¹Δ𝐶^𝑇=𝐷𝐴ℎ₁ℎ^𝑇₁𝐴^𝑇≥𝐷ℎ₁ℎ^𝑇₁, (26) where 𝐴∶=𝐴+𝐵(𝐼−Δ𝐷)⁻¹Δ𝐶≥0. From Theorem 10, 𝜌(𝐴)≥1, which contradicts ‖𝐺‖_∞<1.

Corollary 12. Suppose that system (1) is positive; then the following conditions are equivalent.(i) System (1) is asymptotically stable and ‖𝐺‖_∞<𝛾.(ii) There exists a diagonal positive definite matrix 𝑃 such that 𝐴^𝑇𝑃𝐴−𝑃+𝐶^𝑇𝐶𝐴^𝑇𝑃𝐵+𝐶^𝑇𝐷𝐵^𝑇𝑃𝐴+𝐷^𝑇𝐶𝐵^𝑇𝑃𝐵+𝐷^𝑇𝐷−𝛾²𝐼≺0. (27)(iii) There exists a diagonal positive definite matrix 𝑃 such that −𝑃0𝐴^𝑇𝑃𝐶^𝑇0−𝛾²𝐼𝐵^𝑇𝑃𝐷^𝑇𝑃𝐴𝑃𝐵−𝑃0𝐶𝐷0−𝐼≺0. (28)

Now, our purpose is to design a state feedback controller given by 𝑢(𝑘)=𝐾𝑥(𝑘)+𝑣(𝑘), (29) where 𝐾∈ℝ^𝑚×𝑛 is the controller gain to be designed, and 𝑣(𝑘)∈ℝ^𝑛₊, such that the closed-loop system described as 𝑥()=()𝑥(𝑘)(𝑘),𝑦(𝑘)=()𝑥(𝑘)(𝑘),𝑘+1𝐴+𝐵𝐾+𝐵𝑣𝐶+𝐷𝐾+𝐷𝑣 (30) is positive, asymptotically stable, and ‖𝐺‖_∞<1, where 𝐺(𝑧)=(𝐶+𝐷𝐾)(𝑧𝐼−(𝐴+𝐵𝐾))⁻¹+𝐷.

At first, we focus on nonnegative control gain, as it has practical importance in many cases. For instance, for a chemical system whose variables represent concentrations of reactants and reaction speed is impacted by concentrations, in order to improve the speed of reaction, it is natural to consider such controller for increasing concentrations.

Theorem 13. For the given positive system (1), there exists a nonnegative controller of the form in (29) such that the closed-loop system (30) is asymptotically stable and ‖𝐺‖_∞<1 if and only if there exist 𝑋∈𝐷₊^𝑛×𝑛 and 𝑌≥0 satisfying −𝑋0𝑋𝐴^𝑇+𝑌^𝑇𝐵^𝑇𝑋𝐶^𝑇+𝑌^𝑇𝐷^𝑇0−𝐼𝐵^𝑇𝐷^𝑇𝐴𝑋+𝐵𝑌𝐵−𝑋0𝐶𝑋+𝐷𝑌𝐷0−𝐼≺0. (31) Under the above condition, the desired nonnegative controller gain is obtained as 𝐾=𝑌𝑋⁻¹. (32)

Proof. Necessity. From Theorem 11, there exists a diagonal positive definite matrix 𝑃 such that −𝑃0(𝐴+𝐵𝐾)^𝑇𝑃(𝐶+𝐷𝐾)^𝑇0−𝐼𝐵^𝑇𝑃𝐷^𝑇𝑃(𝐴+𝐵𝐾)𝑃𝐵−𝑃0𝐶+𝐷𝐾𝐷0−𝐼≺0. (33) Multiplying on both sides of inequality (33) by 𝑇=diag(𝑃⁻¹,𝐼,𝑃⁻¹,𝐼), it follows that −𝑃⁻¹0𝑃⁻¹𝐴^𝑇+𝑃⁻¹𝐾^𝑇𝐵^𝑇𝑃⁻¹𝐶^𝑇+𝑃⁻¹𝐾^𝑇𝐷^𝑇0−𝐼𝐵^𝑇𝐷^𝑇𝐴𝑃⁻¹+𝐵𝐾𝑃⁻¹𝐵−𝑃⁻¹0𝐶𝑃⁻¹+𝐷𝐾𝑃⁻¹𝐷0−𝐼≺0. (34) By defining 𝑋=𝑃⁻¹, 𝑌=𝐾𝑃⁻¹, inequality (31) is immediately obtained. On the other hand, since 𝑋∈𝐷₊^𝑛×𝑛 and 𝑌≥0, it is easy to see that 𝐾≥0.
Sufficiency. Positivity of the closed-loop system (30) is obvious. From (32), 𝑌=𝐾𝑋; substituting it into inequality (31) leads to −𝑋0𝑋𝐴^𝑇+𝑋𝐾^𝑇𝐵^𝑇𝑋𝐶^𝑇+𝑋𝐾^𝑇𝐷^𝑇0−𝐼𝐵^𝑇𝐷^𝑇𝐴𝑋+𝐵𝐾𝑋𝐵−𝑋0𝐶𝑋+𝐷𝐾𝑋𝐷0−𝐼≺0. (35) Multiplying on both sides of inequality (35) by 𝑇=diag(𝑋⁻¹,𝐼,𝑋⁻¹,𝐼), we have −𝑋⁻¹0(𝐴+𝐵𝐾)^𝑇𝑋⁻¹(𝐶+𝐷𝐾)^𝑇0−𝐼𝐵^𝑇𝑋⁻¹𝐷^𝑇𝑋⁻¹(𝐴+𝐵𝐾)𝑋⁻¹𝐵−𝑋⁻¹0𝐶+𝐷𝐾𝐷0−𝐼≺0. (36) Therefore, from Theorem 11, the closed-loop system (30) is asymptotically stable and ‖𝐺‖_∞<1.

Remark 14. Under the assumption that system (1) is positive, it is worth noting that there does not exist any 𝑋∈𝐷₊^𝑛×𝑛 or 𝑌≥0 satisfying condition (31) if there exists 𝐴_𝑖𝑖>1, 𝑖=1,2,…,𝑛. It is easy to verify in the light of the following facts.(1)A linear positive discrete-time system is unstable if at least one diagonal entry of matrix 𝐴 is greater than 1 which is presented in the literature [3]. (2)𝜌(𝐵)≤𝜌(𝐴) if 0≤𝐵≤𝐴 which has been pointed out in the literature [19].
On the other hand, it is known that the maximal eigenvalue 𝜌(𝐴) of 𝐴≥0 belongs to the interval maxmin𝑐_𝑖,min𝑟_𝑖≤𝜌(𝐴)≤minmax𝑐_𝑖max𝑟_𝑖, (37) where 𝑐_𝑖 and 𝑟_𝑖 denote the sum of the elements of the 𝑖th column and the 𝑖th row of matrix 𝐴, respectively. Therefore, there also does not exist 𝑋∈𝐷₊^𝑛×𝑛 or 𝑌≥0 satisfying condition (31) if max{min𝑐_𝑖,min𝑟_𝑖}≥1.

Remark 15. If 0≤𝐵≤𝐴, then 𝜎(𝐴)≥𝜎(𝐵) which is obtained immediately due to the fact (2) in Remark 14. Suppose that system (1) is positive, ‖𝐺‖_∞≥1, and there exists a nonnegative controller (29) such that system (30) is asymptotically stable; it is obvious that system (1) is also asymptotically stable. Thus, 𝐺_∞=𝐺(1=()(𝐶+𝐷𝐾)(𝐼−𝐴+𝐵𝐾))⁻¹=)𝐵+𝐷(𝐶+𝐷𝐾^∞_𝑖=0(𝐴+𝐵𝐾)^𝑖=𝐵+𝐷𝜎(𝐶+𝐷𝐾)^∞_𝑖=0(𝐴+𝐵𝐾)^𝑖≥𝐵+𝐷𝜎𝐶^∞_𝑖=0𝐴^𝑖𝐵+𝐷=‖𝐺‖_∞≥1. (38) Therefore, if system (1) is positive, asymptotically stable, but ‖𝐺‖_∞≥1, there does not exist any nonnegative state feedback (29) such that system (30) is asymptotically stable and ‖𝐺‖_∞<1.

Corollary 16. For the given positive system (1), there exists a nonnegative controller of the form in (29) such that the closed-loop system (30) is asymptotically stable and ‖𝐺‖_∞<𝛾 if and only if there exist 𝑋∈𝐷₊^𝑛×𝑛 and 𝑌≥0 satisfying −𝑋0𝑋𝐴^𝑇+𝑌^𝑇𝐵^𝑇𝑋𝐶^𝑇+𝑌^𝑇𝐷^𝑇0−𝛾²𝐼𝐵^𝑇𝐷^𝑇𝐴𝑋+𝐵𝑌𝐵−𝑋0𝐶𝑋+𝐷𝑌𝐷0−𝐼≺0. (39) Under the above condition, the desired controller gain is obtained as (32).

From Remarks 14 and 15, for some positive systems, there is no nonnegative state feedback (29) such that system (30) is asymptotically stable and ‖𝐺‖_∞<1. Hence, we are obligated to pay attention to state feedback without nonnegative restriction. It is also natural to consider such controller. For example, for an ecosystem whose variables represent population of animals in a forest; population cannot exceed the ecological capacity of the forest, otherwise, the ecosystem may be destroyed. Therefore, we must decrease the number of animals by means of harvesting or other methods when population of certain animals exceeds their ecological capacity.

Now we are looking for a state feedback without nonnegative restriction having form in (29) such that the closed-loop system (30) is positive, asymptotically stable, and ‖𝐺‖_∞<1.

Theorem 17. For the given positive system (1), there exists a controller of the form in (29) such that the closed-loop system (30) is positive, asymptotically stable, and ‖𝐺‖_∞<1 if and only if there exist 𝑋∈𝐷₊^𝑛×𝑛 and 𝑌∈ℝ^𝑚×𝑛 satisfying (39) and 𝐴𝑋+𝐵𝑌≥0,𝐶𝑋+𝐷𝑌≥0. (40)(41) Under the above conditions, the desired controller gain is given by (32).

Proof. We only prove conditions (40) and (41).
Necessity. The close-loop system (30) is positive then it follows that 𝐴+𝐵𝐾≥0,𝐶+𝐷𝐾≥0. (42) Since 𝑋 is a diagonal positive definite matrix, then it is easy to check that 𝐴𝑋+𝐵𝐾𝑋≥0,𝐶𝑋+𝐷𝐾𝑋≥0. (43) Conditions (40) and (41) are immediately obtained.
Sufficiency. The proof is similar to necessity.

Corollary 18. For the given positive system (1), there exists a controller of the form in (29) such that the closed-loop system (30) is positive, asymptotically stable, and ‖𝐺‖_∞<𝛾 if and only if there exist 𝑋∈𝐷₊^𝑛×𝑛 and 𝑌∈ℝ^𝑚×𝑛 satisfying conditions (39), (40), and (41). Under these conditions, the desired controller gain is obtained as (32).

4. Examples

In this section, we give some examples to illustrate the validity of the obtained results.

Example 1. Consider a linear positive discrete-time system with the following parameter matrices: ,.𝐴=0.604800.08610.4187,𝐵=0.095200.04570.1813𝐶=1001,𝐷=0.0921000.1286 (44) Solving the conditions in Theorem 13, one feasible solution is obtained as 𝑋=.0.3033000.4477,𝑌=0.08540.11420.07430.1150 (45) Then the desired controller gain is given by 𝐾=.0.28160.25510.25510.2551 (46)

Example 2. Consider a linear positive discrete-time system with the following parameter matrices: 𝐴=,1.214800.08610.9187,𝐵=0.17020.0457𝐶=0.8160,𝐷=0.3157. (47)
We observe that 𝐴₁₁>1 and check the condition in Theorem 13; as pointed out in Remark 14, there is no feasible solution.
Then solving the conditions in Theorem 17, one feasible solution is obtained as 𝑋=.2.174700810.9260,𝑌=−4.00625.0132 (48) The desired controller gain is 𝐾=−1.84210.0062. (49)

Example 3. Consider a linear positive discrete-time system with the following parameter matrices: 𝐴=,0.91870.12950.08610.9187,𝐵=0.17020.1570𝐶=0.57000.3850,𝐷=0.2103. (50)

By calculating, max{min𝑐_𝑖,min𝑟_𝑖}=1.0048. We check the condition in Theorem 13; there is no feasible solution.

Solving the conditions in Theorem 17, one feasible solution is obtained as 𝑋=.0.4315001.6751,𝑌=−0.2126−1.2241 (51)

Then the controller gain is given by 𝐾=−0.4926−0.7308. (52)

5. Conclusion

In this paper, we are interested in 𝐻_∞ control for linear positive discrete-time systems. We present a necessary and sufficient condition to check the stability of linear positive discrete-time systems using Perron-Frobenius theorem for nonnegative matrices, which is the key point for the alternative proof. We believe that it is useful for checking the existence of diagonal positive definite matrices to some other results given for linear discrete-time systems without nonnegative restriction. The alternative proof is along the line of separating hyperplane theorem and Theorem 10 given in this paper. In addition, we investigate the 𝐻_∞ control for such systems under state feedback. Necessary and sufficient conditions for such problem are presented under controller gain with and without nonnegative restriction, and then the desired controller gains can be obtained from the feasible solutions. However, in this paper, we have restricted our attention to the case of state feedback, but in practice, it is not always possible to have access to all of the state variables. The case of static and dynamic output feedback is left for future research. Robust 𝐻_∞ analysis and synthesis for positive uncertain systems are also open problems.

Acknowledgments

The work was supported by the National Natural Science Foundation of China under Grant 61273008 and China Scholarship Council.