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Journal of Applied Mathematics
Volume 2013, Article ID 576379, 7 pages
http://dx.doi.org/10.1155/2013/576379
Research Article

Control for Linear Positive Discrete-Time Systems

1Institute of Systems Science, Northeastern University, Shenyang 110819, China
2Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan

Received 7 June 2013; Accepted 11 August 2013

Academic Editor: Wei-Shih Du

Copyright © 2013 Youmei Zhang et al. This 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.

Abstract

This paper is concerned with 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 norm by using separating hyperplane theorem and Perron-Frobenius theorem for nonnegative matrices. We also consider 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, [316].

It is worth noting that convex optimization is a powerful tool for analysis of positive systems. In [914], 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 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 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.

denotes the nonnegative integers. denotes the vector with nonnegative entries. denotes the matrix with nonnegative entries. denotes the set of all vectors with . denotes the set of all matrices with . denotes the set of all symmetric matrices. denotes the set of all diagonal positive definite matrices. , , and mean that is a positive semidefinite, positive definite, and negative definite matrix, respectively. denotes the th entry of matrix . denotes the th entry of vector . For two matrices , means , , . For two vectors , means , . denotes the spectral radius of matrix which is defined as , where are the eigenvalues of . denotes the vector which is composed of the diagonal entries of . is the inner product on .

Consider the following linear discrete-time system: where is the state, is the input, and is the output. , , , and are known matrices.

Definition 1 (see [3]). System (1) is said to be positive if and only if , , for any and for any , .

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

Definition 3 (see [13]). For a given matrix with , we define by

Lemma 4 (see [3], Perron-Frobenius theorem for nonnegative matrices). Let ; then is an eigenvalue of and has a nonnegative eigenvector corresponding to .

A matrix is called a Metzler matrix if , for all , with .

Lemma 5 (see [13]). For a given Metzler matrix and with , the following conditions hold:(i), , ,(ii),
where is defined from as in Definition 3.

Lemma 6 (see [3]). System (1) is positive if and only if , , , and .

Lemma 7 (see [17], separating hyperplane theorem). Suppose that and are two convex sets that do not intersect; that is, . Then, there exist and such that for all and for all .

For , we consider . If and , then is called an M-matrix. If , then is a nonsingular M-matrix.

Lemma 8 (see [4]). A nonsingular matrix is an M-matrix if and only if .

Lemma 9 (see [18], Schur complement). Given any real matrices , , and with and , the following statement holds: if and only if or, equivalently,

The transfer function matrix of system (1) is given by and its norm is defined as where denotes the maximum singular value of . In [14], it has been pointed out that , where , if system (1) is positive and asymptotically stable.

3. 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:(i)There exists a nonzero such that .(ii)System (1) is not asymptotically stable.

Proof. (i)(ii). Since , from Perron-Frobenius theorem for nonnegative matrices, it follows that , where is the spectral radius of matrix and is an eigenvector corresponding to . Then it is obtained from condition (i) that which implies ; namely, system (1) is not asymptotically stable.
(ii)(i). System (1) is not asymptotically stable; that is, . From Perron-Frobenius theorem, we immediately obtain the following result: where is an eigenvector corresponding to . 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 .(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 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 . Define the following two sets: then it is easy to check that sets and are nonempty and convex. By the assumption, we have . Then from the separating hyperplane theorem, there exists a nonzero such that By condition (14), we can conclude that , for all , from which it is easy to verify that . Thus it follows from condition (13) that there exists a nonzero such that which is equivalent to Let , , ; then the above condition can be rewritten as or, equivalently, which implies that(a),(b).From condition (a), it follows that Since system (1) is positive, then are nonnegative matrix and Metzler matrix, respectively. Define the nonzero from as in Definition 3 then from Lemma 5, we obtain Set , , ; conditions (21) and (22) can be rewritten as Since is nonzero, then we have the following three cases.(1) Consider that . By condition (24), , which contradicts .(2) Consider that . We observe from (23) that , which implies that . From Theorem 10, this is a contradiction. (3) Consider that . From condition (24), it yields that . Define matrix which is well-defined and satisfies . Note that . On the other hand, ; otherwise, this contradicts . It is known that for matrices and with and , . Thus, is invertible and since is a nonsingular -matrix. Therefore, holds. Then from (23) we have where . From Theorem 10, , which contradicts .

Corollary 12. Suppose that system (1) is positive; then the following conditions are equivalent.(i) System (1) is asymptotically stable and .(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 is the controller gain to be designed, and , such that the closed-loop system described as is positive, asymptotically stable, and , where .

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 if and only if there exist and satisfying Under the above condition, the desired nonnegative controller gain is obtained as

Proof. Necessity. From Theorem 11, there exists a diagonal positive definite matrix such that Multiplying on both sides of inequality (33) by , it follows that By defining , , inequality (31) is immediately obtained. On the other hand, since and , it is easy to see that .
Sufficiency. Positivity of the closed-loop system (30) is obvious. From (32), ; substituting it into inequality (31) leads to Multiplying on both sides of inequality (35) by , we have Therefore, from Theorem 11, the closed-loop system (30) is asymptotically stable and .

Remark 14. Under the assumption that system (1) is positive, it is worth noting that there does not exist any or satisfying condition (31) if there exists , . 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 which is presented in the literature [3]. (2) if which has been pointed out in the literature [19].
On the other hand, it is known that the maximal eigenvalue of belongs to the interval 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 satisfying condition (31) if .

Remark 15. If , then which is obtained immediately due to the fact in Remark 14. Suppose that system (1) is positive, , 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, Therefore, if system (1) is positive, asymptotically stable, but , there does not exist any nonnegative state feedback (29) such that system (30) is asymptotically stable and .

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 if and only if there exist and satisfying 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 . 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 .

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 if and only if there exist and satisfying (39) and 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 Since is a diagonal positive definite matrix, then it is easy to check that 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 if and only if there exist and 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: Solving the conditions in Theorem 13, one feasible solution is obtained as Then the desired controller gain is given by

Example 2. Consider a linear positive discrete-time system with the following parameter matrices:
We observe that 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 The desired controller gain is

Example 3. Consider a linear positive discrete-time system with the following parameter matrices:

By calculating, . We check the condition in Theorem 13; there is no feasible solution.

Solving the conditions in Theorem 17, one feasible solution is obtained as

Then the controller gain is given by

5. Conclusion

In this paper, we are interested in 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 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 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.

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