Robust Observer Design for Switched Positive Linear System with Uncertainties

and Applied Analysis 3 Define x(t) = x(t) − x(t) and y(t) = y(t) − y(t). From (1) and (6), error system (8) is obtained as follows: ̇̃ x (t) = (A σ(t) − H σ(t) C σ(t) ) x (t) + B σ(t) x (t − d (t)) + ΔA σ(t) x (t) + ΔB σ(t) x (t − d (t)) x (t) = φ (t) , t ∈ [−τ, 0] y (t) = C σ(t) x (t) . (8)


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][6][7][8][9][10][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][23][24][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.
(1) Taking model uncertainties into account, the robust observer is obtained; (2) the sufficient conditions of building a robust observer are proposed in form of LMI; (3) 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.
Assumption 4. The state trajectory of system ( 1) is continuous everywhere.In other words, state variable does not jump at switching instants.
Lemma 6 (see [16]).For matrices  and  and symmetric matrix ,  +  +       < 0 holds for    ≤  if and only if there exists a positive constant  such that

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 ẋ () =  () x () +  () x ( −  ()) +  ()  () or, equivalently, where x() and ŷ() 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 x() = () − x() and ỹ() = () − ŷ().From ( 1) and (6), error system (8) is obtained as follows: From ( 1) and ( 8), the augmented system ( 10) is obtained as follows: 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 x() 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  > 0 and  ≥ 0, 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.
Synthesizing (I) and (II), system ( 6) is the desired robust positive observer for system (1).This completes the proof of Theorem 11.
Remark 12.   and   can be obtained by solving (25a).Since   =     ,   can be obtained.If   satisfies (25b), then   meets all requirements of design.By this way, the desired robust positive observer is built for system (1).
Remark 13.Note the following problems.(a) Since the conditions (7a) and (7b) are not strictly positive, they are not strict LMI; (b) under the conditions (7a) and (7b), the state and output of system may be always zero; in this case, the obtained result may be useless for engineering practice.Considering these problems, the conditions are replaced by (25b) which is a strict LMI.

Numerical Example
A numerical example is given to illustrate the validity of the obtained result in this section.
The simulation results are shown in Figures 1-3.From them, we can find the following facts.
(2) In Figure 2, the state of observer approximates the state of original system.This fact is also revealed by Figure 3 in which the state of error system exponentially converges to zero.
(3) The state of observer is positive all the time.Therefore, the numerical example illustrates the validity of the proposed method.

Conclusions
The design of a robust observer for the switched positive linear system has been investigated in this paper.
(1) In the presence of model uncertainties, the sufficient conditions for the existence of a positive observer are proposed in form of LMI.
(2) The state of observer is positive and converges to the state of original system.
(3) In the future study, the significant task is to investigate fault detection for switched positive linear system with uncertainties based on state observer.

Figure 1 :X 1 ,
Figure 1: The figure of switching law.

Figure 2 :
Figure 2: States of observer and original system.

Ψ
For given constants  > 0,  ≥ 0, and  ≥ 1, if there exist symmetric positive definition matrices   ,   , and   , matrix   , and positive scalars  and  such that 11 Ψ 12      is the desired robust positive observer, where (  ) , and (    ) , represent the elements in th row and th column of   and (    ), respectively.