Dynamic Output Feedback Stabilization of Controlled Positive Discrete-Time Systems with Delays

Abstract

The problem of stabilization by means of dynamic output feedback is studied for discrete-time delayed systems with possible interval uncertainties. The control is under positivity constraint, which means that the resultant closed-loop system must be stable and positive. The robust resilient controller is respect to additive controller gain variation which also belongs to an interval. Necessary and sufficient/sufficient conditions are established for the existence of the dynamic output feedback controller. The desired controller gain matrices can be determined effectively via the cone complementarity linearization techniques.

1. Introduction

A dynamical system is called positive if any trajectory of the system starting from nonnegative initial states remains forever nonnegative. Such systems abound in almost all fields, for instance, engineering, ecology, economics, biomedicine, and social science [1–3]. Since the states of positive systems are confined within a “cone" located in the positive quadrant rather than in linear spaces, many well-established results for general linear systems cannot be readily applied to positive systems. This feature makes the analysis and synthesis of positive systems a challenging and interesting job, and many results have been obtained, see [4–12]. It should be pointed out that in [9–12], the governed system is not necessarily positive, while a control strategy can be designed such that the closed-loop system is positive. We call systems in this class controlled positive.

The reaction of real world systems to exogenous signals is never instantaneous and always infected by certain time delays. The delay presence may induce complex behaviors, such as oscillations, instability, and poor performance [13]. Recently, the study on delayed positive systems has drawn increasing attention and many important results have been obtained, see [14–17] for stability and [18–20] for control. It has been shown that the stability of delayed positive systems has nothing to do with the amplitude of delays.

It should be noted that in most literature aforementioned for delayed systems, it is assumed that the parameters of systems are exactly known, and the controller takes the form of state feedback. However, in practical applications, it is inevitable that uncertainties enter the system parameters and it is often impossible to obtain the full information on the state variables. Hence, it is necessary to investigate the output feedback stabilization problem of uncertain positive system with delays. On the other hand, in practice, instead of being precise or exactly implemented, many controllers do have a certain degree of errors and may be sensitive to these errors. Such controllers are often termed “fragile". Therefore, it is considered beneficial to design a “resilient" controller being capable of tolerating some level of controller gain variations [21, 22]. All of the above motivate our research.

This paper deals with the dynamic output feedback stabilization problem for discrete-time delayed systems (not necessarily positive) under the positivity constraint, which means that the closed-loop systems are not only stable, but also positive. First, a new necessary and sufficient condition is given for the stability of discrete-time positive systems with delays, which is more useful for designing output feedback controllers. Then for systems with/without uncertainties, necessary and sufficient/sufficient conditions for the existence of the dynamic output feedback controllers are established in terms of linear matrix inequalities (LMIs) together with a matrix equality constraint. The controller gain matrices can be determined via the cone complementarity linearization techniques.

Notations
ℝ, ℝ^𝑛, and ℝ^𝑛_0,+ denote the reals, the 𝑛-dimensional linear vector space over the reals the nonnegative quadrant of ℝ^𝑛, respectively. ℝ^𝑛×𝑚 denotes the set of all 𝑛×𝑚 real matrices. 𝐴≽0(⪯0) means that the elements of 𝐴 are nonnegative (nonpositive). For matrices 𝐴,𝐵∈ℝ^𝑛×𝑚, the notation 𝐴≽𝐵 or 𝐵⪯𝐴 means that 𝐴−𝐵≽0. 𝐴>0(<0) stands for a symmetric positive (negative) definite matrix 𝐴. The symbol 𝜌(𝐴) denotes the spectral radius of matrix 𝐴, that is, 𝜌(𝐴)=max{|𝜆|∶𝜆∈𝜎(𝐴)} with 𝜎(𝐴) being the spectrum of 𝐴. The superscript 𝑇 represents the transpose. The symbol ∗ will be used in some matrix expressions to induce a symmetric structure.

2. Mathematical Preliminaries

In this section, we will give some definitions and lemmas about positive discrete-time delayed systems.

Consider the discrete-time system with delay𝑥(𝑡+1)=𝐴𝑥(𝑡)+𝐴_𝜏𝑥(𝑡−𝜏),𝑦(𝑡)=𝐶𝑥(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡=−𝜏,−(𝜏−1),…,0, (2.1) where 𝑥(𝑡)∈ℝ^𝑛 is the state, 𝑦(𝑡)∈ℝ^𝑚 is the measurable output, 𝐴,𝐴_𝜏, and 𝐶 are known real constant matrices, 𝜏∈ℕ is a constant delay and 𝜙(𝑡)∶[−𝜏,−(𝜏−1),…,0]→ℝ^𝑛_0,+ is the vector valued initial function.

First, some definitions and lemmas are given.

Definition 2.1 (see [17]). System (2.1) is said to be positive if for any 𝜙(𝑡)∶[−𝜏,−(𝜏−1),…,0]→ℝ^𝑛_0,+, one has 𝑥(𝑡)≽0 and 𝑦(𝑡)≽0 for all 𝑡∈ℕ.

Lemma 2.2 (see [17]). System (2.1) is positive if and only if 𝐴≽0, 𝐴_𝜏≽0 and 𝐶≽0.

Definition 2.3 (see [15]). A square matrix 𝐴 is called a Schur matrix if 𝜌(𝐴)<1.

Lemma 2.4 (see [15]). Positive system (2.1) is asymptotically stable if and only if (𝐴+𝐴_𝜏) is a Schur matrix.

Lemma 2.5 (see [3]). A matrix 𝐴≽0 is a Schur matrix if and only if there exists a diagonal matrix 𝑃>0 such that 𝐴^𝑇𝑃𝐴−𝑃<0.

Combining the above lemmas, we have

Lemma 2.6. Positive system (2.1) is asymptotically stable if and only if there exists a diagonal matrix 𝑃>0 such that (𝐴+𝐴_𝜏)^𝑇𝑃(𝐴+𝐴_𝜏)−𝑃<0. (2.2)

Lemma 2.7 (see [2]). For two matrices 𝐴,𝐵∈ℝ^𝑛×𝑛, if 𝐴≽𝐵≽0, then 𝜌(𝐴)≽𝜌(𝐵).

3. Dynamic Output Feedback Stabilization

Now consider the discrete-time system with delay and control𝑥(𝑡+1)=𝐴𝑥(𝑡)+𝐴_𝜏𝑥(𝑡−𝜏)+𝐵𝑢(𝑡),𝑦(𝑡)=𝐶𝑥(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡=−𝜏,−(𝜏−1),…,0, (3.1) where 𝑥(𝑡)∈ℝ^𝑛, 𝑢(𝑡)∈ℝ^𝑝, 𝑦(𝑡)∈ℝ^𝑚 are, respectively, the state, the control input and the measurable output. 𝐴,𝐴_𝜏,𝐵 and 𝐶 are known constant matrices, 𝜏∈ℕ is a constant delay and 𝜙(𝑡)∶[−𝜏,−(𝜏−1),…,0]→ℝ^𝑛_0,+ is the vector valued initial function.

The purpose of this section is to design a dynamic output feedback controller𝛿(𝑡+1)=𝐴_𝑘𝛿(𝑡)+𝐵_𝑘𝑦(𝑡),𝑢(𝑡)=𝐶_𝑘𝛿(𝑡)+𝐷_𝑘𝑦(𝑡),𝛿(0)=𝛿₀ (3.2) such that the resultant closed-loop system=𝑥(𝑡+1)𝛿(𝑡+1)𝐴+𝐵𝐷_𝑘𝐶𝐵𝐶_𝑘𝐵_𝑘𝐶𝐴_𝑘+𝑥(𝑡)𝐴𝛿(𝑡)^𝜏0,,00𝑥(𝑡−𝜏)𝛿(𝑡−𝜏)𝑦(𝑡)=𝐶0𝑥(𝑡)𝛿(𝑡)𝑥(𝑡)=𝜙(𝑡),𝑡=−𝜏,−(𝜏−1),…,0,𝛿(0)=𝛿₀ (3.3) is positive and asymptotically stable. Where 𝛿(𝑡)∈ℝ^𝑟 is the state of the controller, 𝛿₀∈ℝ^𝑟_0,+, 𝐴_𝑘,𝐵_𝑘,𝐶_𝑘 and 𝐷_𝑘 are the controller gain matrices to be determined. The above stabilization problem will be called Problem DOFS (Dynamic Output Feedback Stabilization).

Remark 3.1. 𝑟 may be either equal to or less than 𝑛. In the case of 𝑟=𝑛 or 𝑟<𝑛, controller (3.2) is called the full-order or reduced-order dynamic output feedback controller for system (3.1), respectively.

First, similar to [8], in order to design dynamic output feedback controller for system (3.1), we give an equivalent form of Lemma 2.6.

Theorem 3.2. Positive system (2.1) is asymptotically stable if and only if there exist diagonal matrices 𝑃>0 and 𝑄>0 satisfying the LMI −𝑃(𝐴+𝐴_𝜏)^𝑇∗−𝑄<0, (3.4) and the matrix equality constraint 𝑃𝑄=𝐼. (3.5)

Proof. By Schur complement, it is easy to see that (2.2) holds if and only if the following inequality holds: −𝑃(𝐴+𝐴_𝜏)^𝑇∗−𝑃⁻¹<0. (3.6) Taking 𝑃⁻¹=𝑄, we have that there exist a diagonal matrix 𝑃>0 satisfying (2.2) if and only if there exist diagonal matrices 𝑃>0 and 𝑄>0 satisfying (3.4) and (3.5).

Remark 3.3. Comparing with Lemma 2.6, the conditions in Theorem 3.2 are a little more complicated since a matrix equality constraint is introduced. However, it can be seen that in Theorem 3.2, the Lyapunov matrix 𝑃 and the system parametric matrices have been decoupled. Hence, Theorem 3.2 is more useful when designing output feedback controllers for system (3.1).

Based on Theorem 3.2, we will establish the necessary and sufficient conditions for the solvability of Problem DOFS.

Theorem 3.4. For discrete-time delayed system (3.1) with 𝐴_𝜏≽0, 𝐶≽0, there exists a solution to Problem DOFS if and only if there exist matrices 𝐿_𝑖, 𝑖=1,2,3,4, and diagonal matrices 𝑃_𝑗>0, 𝑄_𝑗>0, 𝑗=1,2, satisfying the LMIs 𝐿₁𝐿≽0,₂𝐶≽0,𝐵𝐿₃≽0,𝐴+𝐵𝐿₄𝐶≽0,−𝑃₁0𝐴^𝑇+𝐶^𝑇𝐿^𝑇₄𝐵^𝑇+𝐴^𝑇_𝜏𝐶^𝑇𝐿^𝑇₂∗−𝑃₂𝐿^𝑇₃𝐵^𝑇𝐿^𝑇₁∗∗−𝑄₁0∗∗∗−𝑄₂<0 (3.7) and the matrix equality constraints 𝑃₁𝑄₁𝑃=𝐼,₂𝑄₂=𝐼. (3.8) In this case, the controller gain matrices in (3.2) are designed as 𝐴_𝑘=𝐿₁,𝐵_𝑘=𝐿₂,𝐶_𝑘=𝐿₃,𝐷_𝑘=𝐿₄. (3.9)

4. Robust Resilient Stabilization of Interval Systems

In this section, we consider the discrete-time interval uncertain system (3.1), where the system parametric matrices are all uncertain with𝐴∈[𝐴_𝑚,𝐴_𝑀],𝐴_𝜏∈[𝐴_𝜏𝑚,𝐴_𝜏𝑀],𝐵∈[𝐵_𝑚,𝐵_𝑀],𝐶∈[𝐶_𝑚,𝐶_𝑀𝐴],_𝜏𝑚≽0,𝐵_𝑚≽0,𝐶_𝑚≽0 (4.1) and 𝐴_𝑚,𝐴_𝑀,𝐴_𝜏𝑚,𝐴_𝜏𝑀,𝐵_𝑚,𝐵_𝑀,𝐶_𝑚,𝐶_𝑀 are known constant matrices.

For uncertain system (3.1), we will design a resilient dynamic output feedback controller𝛿(𝑡+1)=(𝐴_𝑘+Δ𝐴_𝑘)𝛿(𝑡)+(𝐵_𝑘+Δ𝐵_𝑘)𝑦(𝑡),𝑢(𝑡)=(𝐶_𝑘+Δ𝐶_𝑘)𝛿(𝑡)+(𝐷_𝑘+Δ𝐷_𝑘)𝑦(𝑡),𝛿(0)=𝛿₀, (4.2) such that the resultant closed-loop system 𝜂(𝑡+1)=𝐴_𝑐𝜂(𝑡)+𝐴_𝑐𝜏𝜂(𝑡−𝜏),𝑦(𝑡)=𝐶_𝑐𝜂(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡=−𝜏,−(𝜏−1),…,0,𝛿(0)=𝛿₀, (4.3) with𝜂(𝑡)=𝑥(𝑡)𝛿(𝑡),𝐴_𝑐=𝐴+𝐵𝐷_𝑘𝐶+𝐵Δ𝐷_𝑘𝐶𝐵𝐶_𝑘+𝐵Δ𝐶_𝑘𝐵_𝑘𝐶+Δ𝐵_𝑘𝐶𝐴_𝑘+Δ𝐴_𝑘,𝐴_𝑐𝜏=𝐴_𝜏000,𝐶_𝑐=𝐶0, (4.4) is positive and robustly stable. In (4.2), 𝛿(𝑡)∈ℝ^𝑟 is the state of the controller, 𝐴_𝑘, 𝐵_𝑘, 𝐶_𝑘, and 𝐷_𝑘 are the controller gain matrices to be determined and Δ𝐴_𝑘, Δ𝐵_𝑘, Δ𝐶_𝑘, and Δ𝐷_𝑘 are the controller gain variations which are assumed to satisfyΔ𝐴_𝑘∈[𝐴_𝑘𝑚,𝐴_𝑘𝑀],Δ𝐵_𝑘∈[𝐵_𝑘𝑚,𝐵_𝑘𝑀],Δ𝐶_𝑘∈[𝐶_𝑘𝑚,𝐶_𝑘𝑀],Δ𝐷_𝑘∈[𝐷_𝑘𝑚,𝐷_𝑘𝑀] (4.5) with 𝐴_𝑘𝑚, 𝐴_𝑘𝑀, 𝐵_𝑘𝑚, 𝐵_𝑘𝑀, 𝐶_𝑘𝑚, 𝐶_𝑘𝑀, 𝐷_𝑘𝑚, 𝐷_𝑘𝑀 being known constant matrices.

In the sequel, the above stabilization problem will be stated as Problem RRDOFS (Robust Resilient Dynamic Output Feedback Stabilization).

Assumption. Assume that 𝐴_𝑘𝑀≽0,𝐵_𝑘𝑀≽0,𝐶_𝑘𝑀≽0,𝐷_𝑘𝑀≽0,𝐴_𝑘𝑚=−𝐴_𝑘𝑀,𝐵_𝑘𝑚=−𝐵_𝑘𝑀,𝐶_𝑘𝑚=−𝐶_𝑘𝑀,𝐷_𝑘𝑚=−𝐷_𝑘𝑀. (4.6)

Remark 4.2. In fact, Assumption 4.1 is without loss of generality. For example, for any matrices 𝐴_𝑘𝑚 and 𝐴_𝑘𝑀 with 𝐴_𝑘𝑚⪯𝐴_𝑘𝑀, let 𝐴_𝑘=(𝐴_𝑘𝑚+𝐴_𝑘𝑀)/2, 𝐴_𝑘=(𝐴_𝑘𝑀−𝐴_𝑘𝑚)/2≽0, then 𝐴_𝑘+Δ𝐴_𝑘 can be rewritten as 𝐴_𝑘+Δ𝐴_𝑘=𝐴_𝑘+𝐴_𝑘+Δ𝐴_𝑘−𝐴_𝑘=𝐴_𝑘+Δ𝐴_𝑘, (4.7) where 𝐴_𝑘=𝐴_𝑘+𝐴_𝑘 is the new controller gain matrix to be determined and Δ𝐴_𝑘=Δ𝐴_𝑘−𝐴_𝑘 is the new controller gain variation which satisfies Δ𝐴_𝑘=∈[−𝐴_𝑘,𝐴_𝑘]. Similarly, we can prove the generality of the assumption on Δ𝐵_𝑘,Δ𝐶_𝑘, and Δ𝐷_𝑘.

Next, we will establish the sufficient conditions for the solvability of Problem RRDOFS.

Theorem 4.3. For the interval uncertain delayed system (3.1) with the parametric matrices satisfying (4.1), there exists a solution to Problem RRDOFS if there exist matrices 𝐿₁, 𝐿_𝑖≽0, 𝐻_𝑖⪯0, 𝑖=2,3,4 and diagonal matrices 𝑃_𝑗>0, 𝑄_𝑗>0, 𝑗=1,2, satisfying the following LMIs: 𝐿₁+𝐴_𝑘𝑚𝐿≽0,₂𝐶_𝑚+𝐻₂𝐶_𝑀+𝐵_𝑘𝑚𝐶_𝑀𝐵≽0,_𝑚𝐿₃+𝐵_𝑀𝐻₃+𝐵_𝑀𝐶_𝑘𝑚𝐴≽0,_𝑚+𝐵_𝑚𝐿₄𝐶_𝑚+𝐵_𝑀𝐻₄𝐶_𝑀+𝐵_𝑀𝐷_𝑘𝑚𝐶_𝑀≽0,−𝑃₁0𝐴^𝑇_𝑀+𝐶^𝑇_𝑀𝐿^𝑇₄𝐵^𝑇_𝑀+𝐶^𝑇_𝑚𝐻^𝑇₄𝐵^𝑇_𝑚+𝐶^𝑇_𝑀𝐷^𝑇_𝑘𝑀𝐵^𝑇_𝑀+𝐴^𝑇_𝜏𝑀𝐶^𝑇_𝑀𝐿^𝑇₂+𝐶^𝑇_𝑚𝐻^𝑇₂+𝐶^𝑇_𝑀𝐵^𝑇_𝑘𝑀∗−𝑃₂𝐿^𝑇₃𝐵^𝑇_𝑀+𝐻^𝑇₃𝐵^𝑇_𝑚+𝐶^𝑇_𝑘𝑀𝐵^𝑇_𝑀𝐿^𝑇₁+𝐴^𝑇_𝑘𝑀∗∗−𝑄₁0∗∗∗−𝑄₂<0 (4.8a)(4.8b)(4.8c)(4.8d)(4.8e)and the matrix equality constraints (3.8). In this case, the controller gain matrices in (4.2) are designed as 𝐴_𝑘=𝐿₁,𝐵_𝑘=𝐿₂+𝐻₂,𝐶_𝑘=𝐿₃+𝐻₃,𝐷_𝑘=𝐿₄+𝐻₄. (4.9)

Proof. Letting 𝐵_𝑘=𝐵_𝑘1+𝐵_𝑘2,𝐶_𝑘=𝐶_𝑘1+𝐶_𝑘2,𝐷_𝑘=𝐷_𝑘1+𝐷_𝑘2 (4.10) with 𝐵_𝑘1≽0, 𝐵_𝑘2⪯0, 𝐶_𝑘1≽0, 𝐶_𝑘2⪯0, 𝐷_𝑘1≽0, 𝐷_𝑘2⪯0, and noting that (4.1) and (4.5)-(4.6), we get𝐴_𝑚+𝐵_𝑚𝐷_𝑘1𝐶_𝑚+𝐵_𝑀𝐷_𝑘2𝐶_𝑀+𝐵_𝑀𝐷_𝑘𝑚𝐶_𝑀≼𝐴+𝐵𝐷_𝑘𝐶+𝐵Δ𝐷_𝑘𝐶=𝐴+𝐵𝐷_𝑘1𝐶+𝐵𝐷_𝑘2𝐶+𝐵Δ𝐷_𝑘𝐶≼𝐴_𝑀+𝐵_𝑀𝐷_𝑘1𝐶_𝑀+𝐵_𝑚𝐷_𝑘2𝐶_𝑚+𝐵_𝑀𝐷_𝑘𝑀𝐶_𝑀,𝐵_𝑚𝐶_𝑘1+𝐵_𝑀𝐶_𝑘2+𝐵_𝑀𝐶_𝑘𝑚⪯𝐵𝐶_𝑘+𝐵Δ𝐶_𝑘=𝐵𝐶_𝑘1+𝐵𝐶_𝑘2+𝐵Δ𝐶_𝑘⪯𝐵_𝑀𝐶_𝑘1+𝐵_𝑚𝐶_𝑘2+𝐵_𝑀𝐶_𝑘𝑀,𝐵_𝑘1𝐶_𝑚+𝐵_𝑘2𝐶_𝑀+𝐵_𝑘𝑚𝐶_𝑀⪯𝐵_𝑘𝐶+Δ𝐵_𝑘𝐶=𝐵_𝑘1𝐶+𝐵_𝑘2𝐶+Δ𝐵_𝑘𝐶⪯B_𝑘1𝐶_𝑀+𝐵_𝑘2𝐶_𝑚+𝐵_𝑘𝑀𝐶_𝑀,𝐴_𝑘+𝐴_𝑘𝑚⪯𝐴_𝑘+Δ𝐴_𝑘≼𝐴_𝑘+𝐴_𝑘𝑀𝐴_𝑐,+𝐴_𝑐𝜏𝐴⪯𝑀∶=^𝑀+𝐵_𝑀𝐷_𝑘1𝐶_𝑀+𝐵_𝑚𝐷_𝑘2𝐶_𝑚+𝐵_𝑀𝐷_𝑘𝑀𝐶_𝑀+𝐴_𝜏𝑀𝐵_𝑀𝐶_𝑘1+𝐵_𝑚𝐶_𝑘2+𝐵_𝑀𝐶_𝑘𝑀𝐵_𝑘1𝐶_𝑀+𝐵_𝑘2𝐶_𝑚+𝐵_𝑘𝑀𝐶_𝑀𝐴_𝑘+𝐴_𝑘𝑀. (4.11a)(4.11b)(4.11c)(4.11d)(4.11e) From (4.8a)–(4.8d), (4.9)–(4.11a), (4.11b), (4.11c), (4.11d), and (4.11e) and 𝐵_𝑚≽0, 𝐶_𝑚≽0, 𝐿_𝑖≽0, 𝐻_𝑖⪯0, 𝑖=2,3,4, we obtain that 𝐴_𝑐≽0, 𝐴_𝑐𝜏≽0, 𝐶_𝑐≽0 for all uncertainties and 𝑀≽0. By using Lemma 2.2, we conclude that the closed-loop system (4.3) is positive.
Noting (4.8e) and by using Lemma 2.4 and Theorem 3.2, we have that 𝑀≽0 is a Schur matrix considering (4.9). From (4.11e) and Lemma 2.7, we get 𝐴_𝑐+𝐴_𝑐𝜏≽0 is also a Schur matrix for all uncertainties. Hence, the positive system (4.3) is robustly stable.

Remark 4.4. From the proof of Theorem 4.3, we can see that the condition in (4.1) that 𝐵_𝑚≽0, 𝐶_𝑚≽0 is given for the purpose to find the upper bound and the lower bound about the parametric matrices of the closed-loop system (4.3).

If in system (3.2), there are no uncertainties in the parametric matrices 𝐵≽0 and 𝐶≽0, that is, 𝐵≽0 and 𝐶≽0 are known constant matrices, we will obtain the the necessary and sufficient conditions for the solvability of Problem RRDOFS, which will be given as follows.

Theorem 4.5. For the interval uncertain delayed system (3.1) with (4.1) and 𝐵≽0 and 𝐶≽0 being known constant matrices, there exists a solution to Problem RRDOFS if there exist matrices 𝐿_𝑖, 𝑖=1,2,3,4 and diagonal matrices 𝑃_𝑗>0, 𝑄_𝑗>0, 𝑗=1,2, satisfying the following LMIs: 𝐿₁+𝐴_𝑘𝑚𝐿≽0,₂𝐶+𝐵_𝑘𝑚𝐶≽0,𝐵𝐿₃+𝐵𝐶_𝑘𝑚𝐴≽0,_𝑚+𝐵𝐿₄𝐶+𝐵𝐷_𝑘𝑚𝐶≽0,−𝑃₁0𝐴^𝑇_𝑀+𝐶^𝑇𝐿^𝑇₄𝐵^𝑇+𝐶^𝑇𝐷^𝑇_𝑘𝑀𝐵^𝑇+𝐴^𝑇_𝜏𝑀𝐶^𝑇𝐿^𝑇₂+𝐶^𝑇𝐵^𝑇_𝑘𝑀∗−𝑃₂𝐿^𝑇₃𝐵^𝑇+𝐶^𝑇_𝑘𝑀𝐵^𝑇𝐿^𝑇₁+𝐴^𝑇_𝑘𝑀∗∗−𝑄₁0∗∗∗−𝑄₂<0 (4.12a)(4.12b)(4.12c)(4.12d)(4.12e)and the matrix equality constraints (3.8). In this case, the controller gain matrices in (4.2) are designed as 𝐴_𝑘=𝐿₁,𝐵_𝑘=𝐿₂,𝐶_𝑘=𝐿₃,𝐷_𝑘=𝐿₄. (4.13)

Proof. The sufficiency can be easily obtained from Theorem 4.3 by letting 𝐻_𝑖=0, 𝑖=2,3,4 and 𝐵_𝑚=𝐵_𝑀=𝐵, 𝐶_𝑚=𝐶_𝑀=𝐶.
Now we will prove the necessity. Suppose that for the interval uncertain delayed system (3.1) with (4.1) and 𝐵≽0 and 𝐶≽0 being known constant matrices, Problem RRDOFS is solvable, that is, there exists matrices 𝐴_𝑘,𝐵_𝑘,𝐶_𝑘 and 𝐷_𝑘 such that the closed-loop (4.3) is positive and asymptotically stable for any 𝐴∈[𝐴_𝑚,𝐴_𝑀], 𝐴_𝜏∈[𝐴_𝜏𝑚,𝐴_𝜏𝑀] and Δ𝐴_𝑘∈[𝐴_𝑘𝑚,𝐴_𝑘𝑀],Δ𝐵_𝑘∈[𝐵_𝑘𝑚,𝐵_𝑘𝑀],Δ𝐶_𝑘∈[𝐶_𝑘𝑚,𝐶_𝑘𝑀],Δ𝐷_𝑘∈[𝐷_𝑘𝑚,𝐷_𝑘𝑀], then we have that both the systems=𝑥(𝑡+1)𝐴𝛿(𝑡+1)^𝑚+𝐵𝐷_𝑘𝐶+𝐵𝐷_𝑘𝑚𝐶𝐵𝐶_𝑘+𝐵𝐶_𝑘𝑚𝐵_𝑘𝐶+𝐵_𝑘𝑚𝐶𝐴_𝑘+𝐴_𝑘𝑚+𝑥(𝑡)𝐴𝛿(𝑡)^𝜏𝑚0=𝐴00𝑥(𝑡−𝜏)𝛿(𝑡−𝜏),𝑥(𝑡+1)𝛿(𝑡+1)^𝑀+𝐵𝐷_𝑘𝐶+𝐵𝐷_𝑘𝑀𝐶𝐵𝐶_𝑘+𝐵𝐶_𝑘𝑀𝐵_𝑘𝐶+𝐵_𝑘𝑀𝐶𝐴_𝑘+𝐴_𝑘𝑀+𝑥(𝑡)𝐴𝛿(𝑡)^𝜏𝑀000𝑥(𝑡−𝜏)𝛿(𝑡−𝜏) (4.14a)(4.14b)are positive and asymptotically stable.
From 𝐴_𝜏𝑚≽0, we obtain that system (4.14a) is positive if and only if𝐴_𝑚+𝐵𝐷_𝑘𝐶+𝐵𝐷_𝑘𝑚𝐶𝐵𝐶_𝑘+𝐵𝐶_𝑘𝑚𝐵_𝑘𝐶+𝐵_𝑘𝑚𝐶𝐴_𝑘+𝐴_𝑘𝑚≽0. (4.15) Thus (4.12a)–(4.12d) hold considering (4.13).
From the positivity and stability of system (4.14b) and using Theorem 3.2 again, we conclude that there exist matrices 𝐿_𝑖, 𝑖=1,2,3,4 and diagonal matrices 𝑃_𝑗>0, 𝑄_𝑗>0, 𝑗=1,2, satisfying (3.8) and (4.12e). The necessity is proved.

Remark 4.6. We stress out that the conditions in above theorems do not impose the restriction on the governed system that the system matrix 𝐴≽0. That is, the free system is not necessarily positive. Therefore, the governed system considered in this paper is called controlled positive system.

Remark 4.7. The matrix equality constraint in the above theorems can be solved via the cone complementarity linearization techniques [8].

5. Numerical Examples

Example 5.1. Consider the discrete-time delayed system (3.1) with 𝐴=0.2−0.10.40.6,𝐴_𝜏=0.6000.6,𝐵=−0.2000.2,𝐶=01. (5.1)
It is easy to see that 𝐴 is not nonnegative, which implies that the unforced system (3.1) is not positive. By solving the conditions in Theorem 3.4, after 1 iteration, we obtain the full-order DOFS controller gain matrices𝐴_𝑘=0.26380.26380.26380.2638,𝐵_𝑘=,𝐶0.10920.1092_𝑘=−0.3517−0.35170.37630.3763,𝐷_𝑘=−0.6241−2.8714 (5.2) and the reduced-order DOFS controller gain matrices 𝐴_𝑘=0.2460,𝐵_𝑘=0.2079,𝐶_𝑘=−0.59110.6413,𝐷_𝑘=−0.6268−2.8685. (5.3)

Example 5.2. Consider the uncertain discrete-time delayed system (3.1) with 𝐴_𝑚=0.1−0.080.30.5,𝐴_𝑀=0.20.330.40.6,𝐴_𝜏𝑚=0.1000.1,𝐴_𝜏𝑀=,𝐵0.2000.2_𝑚=0.15000.15,𝐵_𝑀=0.2000.2,𝐶_𝑚=01,𝐶_𝑀=,𝐴01.2_𝑘𝑀=0.10.10.10.1,𝐵_𝑘𝑀=0.10.1,𝐶_𝑘𝑀=0.10.10.10.1,𝐷_𝑘𝑀=.0.10.1 (5.4)
It is easy to see that 𝐴_𝑀 is nonnegative while 𝐴_𝑚 is not, which implies that the unforced system (3.1) is not always positive within the set of uncertain system matrices. And computation shows that the eigenvalues of 𝐴_𝑀+𝐴_𝜏𝑀 are 0.1853, 1.0147. From Lemma 2.4, we know that the unforced system (3.1) is not always asymptotically stable.
Solving the conditions in Theorem 4.3 gives the RRDOFS controller gain matrices𝐴_𝑘=0.10190.10190.10190.1019,𝐵_𝑘=0.12060.1206,𝐶_𝑘=0.13900.13900.13890.1389,𝐷_𝑘=0.6945−1.9818 (5.5) after 1 iteration.

6. Conclusions and Future Works

In this paper, we have studied the dynamic output feedback stabilization problem for delayed systems with/wihtout interval uncertainties. The controller/resilient controller which has additive controller gain variation belonging to an interval, is designed to guarantee that the resulting closed-loop systems are not only stable, but also positive. Necessary and sufficient/sufficient conditions for the existence of such controllers are established in terms of linear matrix inequalities together with a matrix equality constraint. And the controller gain matrices can be determined via the cone complementarity linearization techniques. The approach presented in this paper can also solve the corresponding problems for continuous-time delayed systems.

Acknowledgments

This work was supported by National Natural Science Foundation of P. R. China (50977054, 61004011) and Science Research Foundation of Shandong Economic University.