Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 125343, 10 pages

http://dx.doi.org/10.1155/2015/125343

## Robust Stabilization and Disturbance Rejection of Positive Systems with Time-Varying Delays and Actuator Saturation

^{1}School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China^{2}School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, China

Received 11 June 2014; Accepted 17 August 2014

Academic Editor: Mohammed Chadli

Copyright © 2015 Xi Ding 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 focuses on the problems of robust stabilization and disturbance rejection for a class of positive systems with time-varying delays and actuator saturation. First, a convex hull representation is used to describe the saturation characteristics. By constructing an appropriate copositive type Lyapunov functional, we give sufficient conditions for the existence of a state feedback controller such that the closed-loop system is positive and asymptotically stable at the origin of the state space with a domain of attraction. Then, the disturbance rejection performance analysis in the presence of actuator saturation is developed via -gain. The design method is also extended to investigate the problem of -gain analysis for uncertain positive systems with time-varying delays and actuator saturation. Finally, three examples are provided to demonstrate the effectiveness of the proposed method.

#### 1. Introduction

Positive systems, whose states and outputs are nonnegative whenever the initial conditions and inputs are nonnegative, are of fundamental importance to multitudinous applications in areas such as economics, biology, sociology, and communications [1–4]. Recently, positive systems have been investigated by many researchers [5–10]. The linear copositive Lyapunov functional approach has been used to study the stability of positive systems due to the fact that it is less conservative than the traditional quadratic Lyapunov functional method [11]. It is well known that, in real engineering, time-delays are involved in many subjects and fields, such as mechanics, medicine, chemistry, physics, engineering, and control theory [12]. The existence of time-delay may lead to the deterioration of system performance and instability. Many results have been reported for time-delay systems [13–18], and a few results on positive systems with time-delay have appeared in [19–21].

On the other hand, in practice, the reaction to exogenous signals is not instantaneous, and the outputs will be inevitably affected. Because of the peculiar nonnegative property of positive systems, it is natural to evaluate the size of such systems via the -gain in terms of the ratio of input and output signals [19]. Some results on -gain analysis and control of positive systems have been reported in the literature [19, 22].

Recently, several works on positive systems have been done [12, 19, 23–28]. It should be pointed out that, in almost all available results on positive systems, it has been assumed that the actuator provides unlimited amplitude signal. However, actuator saturation is commonly unavoidable in almost all practical control systems because of the existence of physical, technological, or even safety constraints [29, 30]. Actuator saturation can lead to performance degradation of the closed-loop system; even more, it will make the additional stable closed-loop system unstable for large perturbations. Thus, more and more attention has been focused on the analysis and control synthesis for dynamic systems with actuator saturation for a long time and many methods have been developed to deal with actuator saturation [31–42]. To the best of our knowledge, few results on positive systems with actuator saturation have been proposed [43, 44]. In addition, because of the phenomena of actuator saturation nonlinearities and the peculiar nonnegative property of positive systems, the research of positive systems with actuator saturation becomes more difficult for both analysis and synthesis tasks.

In this paper, we focus our attention on the investigation of robust stabilization and disturbance rejection for a class of positive systems with time-varying delays and actuator saturation. The main contributions of this paper lie in three aspects. First, a convex hull representation is used to describe the saturation behavior, and a domain of attraction, which is different from the ellipsoid, is for the first time proposed for positive systems. Secondly, by constructing a copositive type Lyapunov functional, a state-feedback controller design scheme is developed to guarantee the stability with -gain performance of the resulting closed-loop systems. Thirdly, the proposed controller design method is further extended to the case of uncertain positive systems.

The remainder of this paper is organized as follows. In Section 2, the necessary definitions and lemmas are reviewed. In Section 3, sufficient conditions for the existence of -gain controller are presented. An extension of the obtained results to uncertain positive systems with time-varying delays and actuator saturation is given in Section 4. Three examples are provided to illustrate the feasibility of the proposed method in Section 5. Concluding remarks are given in Section 6.

*Notation.* In this paper, means that all the entries of matrix are nonnegative (nonpositive); means that all the entries of are positive (negative); means that ; means the transpose of matrix ; is the set of all real (positive real) numbers; is the -dimensional real (positive) vector space; is the set of all real matrices of dimension ; refers to the set of all positive integers. For the vector , -norm is denoted by , where is the th element of ; denotes a column vector with rows containing only entry; is the space of absolute integrable vector-valued functions on ; that is, we say is in if ; , , stands for the th column of . For a vector and a positive scalar , denotes a domain of attractive regions, and for a matrix , denotes a linear region.

#### 2. Problem Statements and Preliminaries

Consider the following system with time-varying delays: where is the state vector, is the controlled output vector and is the disturbance input which belongs to , is the initial condition on , , is the initial time, , , , , and are constant matrices of appropriate dimensions, and denotes the time-varying delay satisfying where and are known constants.

*Definition 1 (see [24]). *System (1) is said to be positive if for any initial condition , , and any inputs , it satisfies and , .

*Definition 2 (see [45]). * is called a Metzler matrix if its off-diagonal entries are nonnegative.

Lemma 3 (see [24]). *System (1) is positive if and only if is a Metzler matrix, and , , and .*

*Definition 4 (see [19]). *For a given positive scalar , system (1) is said to have an -gain performance level if the following conditions hold.(a)System (1) is asymptotically stable when .(b)Under the zero initial condition, that is, , , system (1) satisfies

Now let us consider the following system subject to actuator saturation:
where is the control input vector.

The function is the saturation function which is defined as
where

Let be the set of all diagonal matrices in with diagonal elements that are either 1 or 0; then there are elements in , and for each , is also an element in .

Lemma 5 (see [46]). *Given and in , then
**
for all satisfying , , and represents the convex hell. Consequently, can be expressed as
**
where with .**Consider the following state-feedback control law:
**
where is a gain matrix to be determined.*

From Lemma 5, it is clear that satisfies (8) when . Applying controller (9) to system (4) yields the closed-loop system:

By Lemma 3, should be Metzler matrices to ensure the positivity of system (10).

The aim of the paper is to determine the controller gain matrix such that the resulting closed-loop system (10) is positive and asymptotically stable with an -gain performance.

#### 3. Main Results

##### 3.1. Stability Analysis

In this section, we firstly consider the stability of the closed-loop system (10) with .

Theorem 6. *Given a matrix , if there exist vectors , and a matrix , such that, for ,
**
then the closed-loop system (10) with is positive and asymptotically stable for any initial states satisfying
**
where () is the th element of ().*

*Proof. *For any , that is, for any satisfying , we have from (14) that , , that is, ; therefore, . When , system (10) can be written by the following representation:

By Lemma 3, it is easy to get from (11) that system (10) is positive. Choose the following copositive type Lyapunov functional candidate for system (10):
where
and , are positive vectors to be determined.

When , along the trajectory of system (10), we have, ,
It follows that
Combining (12)–(14), we obtain
Therefore, system (10) with is locally asymptotically stable.

Furthermore, it can be obtained from (17) that

Therefore, all the trajectories of that start from will remain inside of .

The proof is completed.

##### 3.2. -Gain Analysis

The following theorem establishes a condition under which the closed-loop system (10) possesses positivity and has an -gain performance.

Theorem 7. *Given positive constants , , and a matrix , if there exist , and a matrix , such that, for , (11), (13), and the following conditions hold,
**
then the closed-loop system (10) is positive and asymptotically stable with an -gain performance level for any initial states satisfying (15).*

*Proof. *By Lemma 3, it is easy to get from (11) that system (10) is positive. Choose the Lyapunov functional candidate (17). By Theorem 6, the stability of system (10) with is ensured if (13), (23)–(25) hold. To establish the -gain performance, we define
Under zero initial condition, we have
From (13), (23)–(25), we obtain
Under zero initial condition, it gives rise to

Considering that , there exists a constant satisfying
Then from (28) and (30) we can get
It means that if , that is, , then and hence .

Therefore, system (10) has an -gain performance level , and all trajectories will remain inside of .

The proof is completed.

In what follows, we will give a method for the controller design based on Theorem 7.

From the definition of the Metzler matrix, (11) can be converted into where

*Remark 8. *It should be noted that both and are variables to be determined in Theorem 7, and we cannot directly compute by using the LMI (linear matrix inequality) method. In order to obtain , we introduce vectors satisfying , ; then (23) holds if the following inequality is satisfied:

Thus, we can firstly obtain , , and by solving (13), (24)-(25), and (34). Then from (32) and , , we can get .

*Remark 9. *There are several results on the stabilization of positive systems with time-varying delays [19, 22, 26]; however, the controllers proposed in these papers may fail to work when the actuator is subject to saturation. In this paper, the actuator saturation, which brings difficulties for the controller design, is taken into account, and the convex hull technique is used to deal with it. The controller proposed in Theorem 7 can guarantee the positivity and the -gain performance of the closed-loop system despite the existence of actuator saturation.

#### 4. Extension to Uncertain Case

In this section, we will extend the results proposed in previous section to the following uncertain positive system: where is the state vector, is the controlled output vector, is the disturbance input which belongs to , is the initial condition on , , and denotes the time-varying delay satisfying (2).

and are uncertain matrices satisfying where , , , and are known constant matrices.

Lemma 10. *If is a Metzler matrix and , , and , then system (35) is positive.*

*Proof. *Because is a Metzler matrix and , , and , it is easy to obtain that is a Metzler matrix and ; then system (35) is positive.

The proof is completed.

Consider the following system with actuator saturation: where is the control input vector.

Similarly, the system (37) can be rewritten as the following closed-loop system:

By Lemma 10, should be Metzler matrices to ensure the positivity of system (38).

The following theorem gives sufficient conditions which ensure the positivity and -gain property of the closed-loop system (38).

Theorem 11. *Given positive constants , , and a matrix , if there exist , and a matrix , such that, for , (24)-(25) and the following conditions hold,
**
then the closed-loop system (38) is positive and asymptotically stable with an -gain performance level for any initial states satisfying (15).*

*Proof. *By Theorem 7, it is easy to get from (14) and (39) that and system (38) is positive.

Choose the Lyapunov functional candidate (17). Along the trajectory of system (38), we have

When , we obtain

Therefore, system (38) with is locally asymptotically stable.

When , similar to the proof line of Theorem 7, the -gain performance can be obtained.

The proof is completed.

From the definition of the Metzler matrix, (39) can be converted into where

*Remark 12. *It should be noted that condition (40) is not expressed in the form of LMI. We can adopt the method proposed in Remark 8 to find the gain matrix ; that is to say, we can firstly get , , and by solving (13), (24)-(25), and
Then from (44) and , , we can obtain the gain matrix .

#### 5. Examples

In this section, three examples are presented to check the validity of the proposed results.

*Example 1. *Consider system (4) with the following parameters:
Let , , , , and

Solving the matrix inequalities (13), (24)-(25), and (34) in Theorem 7 gives rise to

Then from (32) and , , we can get
and , , are Metzler matrices.

The simulation results are shown in Figures 1–3, where the initial condition of the systems is , , , and the disturbance input is . Figure 1 shows the domain of attraction. Figure 2 plots the state responses of the closed-loop system. Figure 3 shows the control signals and . It is not hard to find that the feedback controller can guarantee the positivity and the asymptotical stability of the closed-loop system.