Abstract

This paper is concerned with the design of a robust observer for the switched positive linear system with uncertainties. Sufficient conditions of building a robust observer are established by using the multiple copositive Lyapunov-krasovskii function and the average dwell time approach. By introducing an auxiliary slack variable, these sufficient conditions are transformed into LMI (linear matrix inequality). A numerical example is given to illustrate the validities of obtained results.

1. Introduction

The switched system is a type of hybrid dynamical system, which is composed of several subsystems and a switching law [1]. The switching law governs the switches between subsystems. The switched positive system is a special kind of switched system, whose state and output are nonnegative whenever the initial state and input are nonnegative. In practice, many systems can be modeled as switched positive systems, such as communication system [2], formation flying [3], and viral mutation [4].

Recently, the switched positive system has attracted a lot of attention. As the stability and stabilization problems are basic problems for control systems, the obtained results mainly focus on them [5–11]. Most of the obtained results are sufficient conditions. However, it should be pointed out that Benzaouia and Tadeo proposed the necessary and sufficient condition for the existence of a stabilization controller [12]. In practice, system state may not be measurable. In this case, the problem of building a state observer for switched system is very significant. Considerable attention has been devoted to this problem. In [13], the observers were designed by using the common Lyapunov function and the multiple Lyapunov function, respectively. In [14], an effective method was used to build an observer for the switched linear system with state jumps. Taking uncertainties into account, Xiang et al. designed a robust observer for the switched nonlinear system [15]. However, since the state of switched positive system is positive, the state observer must also be positive. The straightforward application of the above methods to the switched positive system may result in meaningless results [16]. Thus, the state of observer should be restricted to be positive. Rami et al. designed positive observers for the linear continuous positive system [17] and the linear discrete positive system [18]. In [19, 20], a positive observer was built for the positive system with time delays. For the positive linear system with interval uncertainties, Shu et al. proposed necessary and sufficient conditions for the existence of a positive observer. Furthermore, these conditions were described by system matrices. Hence complex matrix decomposition was avoided [21]. Although these results are concerned with the positive system observers, they also contribute to the design of switched positive system observers.

In practice, switched systems are commonly subjected to time delays which have great impacts on the performances of systems. Some published papers have discussed time delay in detail [22–25]. Besides, model uncertainties universally exist in systems and may deteriorate the performances of systems. Thus, the state observer should be robust to model uncertainties. In [26, 27], two different methods were proposed to deal with the polytypic uncertainty. In [28], a robust observer was built for the switching discrete system with uncertainties. Furthermore, by introducing slack variables, the obtained results were presented in form of LMI.

This paper focuses on the robust state observer of switched positive system with uncertainties and time-varying delay. The main contributions of this paper are summarized as follows. Taking model uncertainties into account, the robust observer is obtained; the sufficient conditions of building a robust observer are proposed in form of LMI; the designed state observer is positive.

The rest of this paper is organized as follows. Some necessary definitions and lemmas are introduced in Section 2. In Section 3, a robust positive observer is designed for the switched positive linear system. In Section 4, a numerical example is given. The conclusions are presented in Section 5.

Notations.    stands for -dimensional real (positive) vector space; denotes the space of matrices with real entries; represents Metzler matrices whose off diagonal entries are nonnegative;    implies that all elements of matrix are positive (nonnegative and negative); define , where is the th element of vector ; define and , where and are arbitrary positive integers.

2. Problem Statements and Preliminaries

Consider the following switched linear system: where is the system state; denotes time-varying delay; is the output of system; is the control input; is the switching law which is a piecewise continuous function; is the continuous vector-valued initial function; and are known positive constants; the model uncertainties and are norm bounded, described by ; is unknown matrix satisfying ; , , and are known matrices; , , , and are known system matrices with appropriate dimensions; besides, , .

Next, some necessary definitions and lemmas are introduced.

Definition 1. For any initial state and , if the corresponding trajectories and hold for , then the system (1) is called switched positive linear system [5].

Definition 2. If there exist positive constants and such that then the system (1) is globally uniformly exponentially stable (GUES) under switching law [16].

Definition 3. For , let denote the switching number of over . If holds for and an integer , then is called the average dwell time (ADT) [6].

Assumption 4. The state trajectory of system (1) is continuous everywhere. In other words, state variable does not jump at switching instants.

Lemma 5. System (1) is positive if and only if  , , , , , and hold for [16].

Lemma 6 (see [16]). For matrices and and symmetric matrix , holds for if and only if there exists a positive constant such that

3. Robust Observer Design

In this section, we focus on the design of a robust observer for system (1). According to the structure of system (1), the desired observer is written as or, equivalently, where and denote the state and output of observer, respectively, and is the gain matrix to be determined.

According to Lemma 5 and the given conditions, system (1) is a switched positive linear system. Since the state of system (1) is positive, the desired observer is required to be positive. Thus, should satisfies the following conditions:

Define and . From (1) and (6), error system (8) is obtained as follows:

Define

From (1) and (8), the augmented system (10) is obtained as follows:

Let , , , and .

Consequently,

Remark 7. According to Lemma 5, if conditions (7a) and (7b) hold, then system (6) is positive. Furthermore, if system (10) is stable, then is converged to zero. This fact implies that the state of system (6) is also converged to state of system (1). Then, system (6) is a positive observer of system (1). Therefore we should choose appropriate such that (a) the conditions (7a) and (7b) are satisfied and (b) the system (10) is stable.
Next, we propose two lemmas which are utilized to build an observer for system (1).

Lemma 8. For given constants and , if there exist symmetric positive definition matrices , , and , matrix , and positive scalar such that then where

Proof. Introduce a new matrix which is written as
According to Schur complement, (12) is equivalent to
By Lemma 6, (16) is equivalent to
Consequently,
Therefore (13) holds. This completes the proof.

Note the following problems in Lemma 8. (a) Inequality (12) includes which involves product terms between and ; (b) also exists in inequality (12). Thus, inequality (12) is not a LMI. We propose Lemma 9 to deal with these problems.

Lemma 9. For given constants and , inequality (12) holds for if there exist symmetric positive definition matrices , , and , matrix , and positive scalars and such that where

Proof. Since ,
It follows that
Applying (22) to (19), we have
Premultiplying diag and postmultiplying diag to (23) yield
Let and let . Then, (24) is equivalent to (12). This completes the proof.

Remark 10. In the proof of Lemma 9, the matrix variable is employed to replace the term of which involves . The slack scalar b is used to decouple the product terms brought in by . By this way, inequality (12) is transformed into a standard LMI.

Now, Theorem 11 is proposed for building a robust positive observer.

Theorem 11. For given constants , , and , if there exist symmetric positive definition matrices , , and , matrix , and positive scalars and such thatand the ADT satisfies then system (10) is GUES with an ADT (26) and system (6) is the desired robust positive observer, where and represent the elements in th row and th column of and , respectively.

Proof. The proof of Theorem 11 is divided into two parts. We prove that (I) system (10) is stable and (II) system (6) is a positive observer of system (1).
(I) System (10) is stable.
Construct multiple copositive Lyapunov-Krasovskii function for system (10) as follows: where
Assume that the th subsystem is activated over . Let represent the instant of the th switching and let denote the instant just before .
Take the derivatives of , , and with respect to along the trajectory of system (10) on .
According to Jensen inequality and (31),
Noting (29), (30), and (32), hence,
We derive from Lemma 8, Lemma 9, (33), (25a), and (25c) that
By iterative calculation, we have
Define , , , and , where represents all eigenvalues of :
Note that and ; hence,
It is derived from (35), (37), and the definition of that
Consequently,
Let and . Obviously, . Since , . According to Definition 2, system (10) is GUES with an ADT (26). Hence the estimated state converges to state of system (1). In other words, system (6) is an observer for system (1).
(II) System (6) is a positive observer of system (1).
Since for , . Furthermore, for implies. Hence system (6) is a positive system.
Synthesizing (I) and (II), system (6) is the desired robust positive observer for system (1). This completes the proof of Theorem 11.