H ∞ Control for Linear Positive Discrete-Time Systems

This paper is concernedwithH ∞ 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 ofH ∞ norm by using separating hyperplane theoremandPerron-Frobenius theorem for nonnegativematrices.We also considerH ∞ control problem for linear positive discretetime 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.


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][4][5][6][7][8][9][10][11][12][13][14][15][16].
It is worth noting that convex optimization is a powerful tool for analysis of positive systems.In [9][10][11][12][13][14], some remarkable results for positive systems are studied using convex optimization.On the other hand, the problem of  ∞ control has been a topic of recurring interest for several decades.A great number of results on  ∞ control have been obtained, and different approaches have been proposed.In recent years, increasing attention has been paid to  ∞ 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 discretetime 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  ∞ 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.

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.
Consider the following linear discrete-time system: where () ∈ R  is the state, () ∈ R  is the input, and () ∈ R  is the output. ∈ R × ,  ∈ R × ,  ∈ R × , and  ∈ R × are known matrices.
A matrix  ∈ R × is called a Metzler matrix if   ≥ 0, for all ,  with  ̸ = .
Lemma 5 (see [13]).For a given Metzler matrix  ∈ R × and  ∈   with  ⪰ 0, the following conditions hold: where ℎ ∈ R  + is defined from  as in Definition 3.

𝐻 ∞ Control
In this section, we give an alternative proof for the existed result of  ∞ norm for positive discrete-time systems and investigate the  ∞ 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: Proof.(i)⇒(ii).Since  ≥ 0, from Perron-Frobenius theorem for nonnegative matrices, it follows that   V = ()V ≥ 0, where () ≥ 0 is the spectral radius of matrix   and V ≥ 0 is an eigenvector corresponding to ().Then it is obtained from condition (i) that which implies () ≥ 1; namely, system (1) is not asymptotically stable.
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.
(ii) There exists a diagonal positive definite matrix  such that (iii) There exists a diagonal positive definite matrix  such that Proof.We only prove (i)⇒(ii) since the implication (ii)⇒(i) is obvious from the existed criterion for linear discretetime 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 .Define the following two sets: then it is easy to check that sets  1 and  2 are nonempty and convex.By the assumption, we have  1 ∩  2 = ⌀.Then from the separating hyperplane theorem, there exists a nonzero  ∈  + such that ⟨, ⟩ ≤ 0, ∀ ∈  2 .
Since  is nonzero, then we have the following three cases.
Corollary 12. Suppose that system (1) is positive; then the following conditions are equivalent.
(ii) There exists a diagonal positive definite matrix  such that (iii) There exists a diagonal positive definite matrix  such that Now, our purpose is to design a state feedback controller given by where  ∈ R × is the controller gain to be designed, and V() ∈ R  + , such that the closed-loop system described as is positive, asymptotically stable, and ‖‖ ∞ < 1, where () = ( + )( − ( + )) −1 + .
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 closedloop system (30) is asymptotically stable and ‖‖ ∞ < 1 if and only if there exist  ∈  × + and  ≥ 0 satisfying Under the above condition, the desired nonnegative controller gain is obtained as By defining  =  −1 ,  =  −1 , inequality (31) is immediately obtained.On the other hand, since  ∈  × + and  ≥ 0, it is easy to see that  ≥ 0.
Remark 14.Under the assumption that system (1) is positive, it is worth noting that there does not exist any  ∈  × + or  ≥ 0 satisfying condition (31) if there exists   > 1,  = 1, 2, . . ., .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  is greater than 1 which is presented in the literature [3].
On the other hand, it is known that the maximal eigenvalue () of  ≥ 0 belongs to the interval max {min   , min   ≤  () ≤ min {max   max   } , (37) where   and   denote the sum of the elements of the th column and the th row of matrix , respectively.Therefore, there also does not exist  ∈  × + or  ≥ 0 satisfying condition (31) if max{min   , min   } ≥ 1.
From Remarks 14 and 15, for some positive systems, there is no nonnegative state feedback (29) such that system (30) is asymptotically stable and ‖‖ ∞ < 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 ‖‖ ∞ < 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 ‖‖ ∞ < 1 if and only if there exist  ∈  × + and  ∈ R × satisfying (39) and  +  ≥ 0, (40) Under the above conditions, the desired controller gain is given by (32).