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

This paper focuses on the problems of robust stabilization and disturbance rejection for a class of positive systemswith 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 L 1 -gain. The design method is also extended to investigate the problem of L 1 -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.


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][2][3][4].Recently, positive systems have been investigated by many researchers [5][6][7][8][9][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, timedelays 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][14][15][16][17][18], and a few results on positive systems with time-delay have appeared in [19][20][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  1 -gain in terms of the ratio of input and output signals [19].Some results on  1 -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][24][25][26][27][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][32][33][34][35][36][37][38][39][40][41][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 Mathematical Problems in Engineering 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  1 -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  1 -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.

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  1 [ 0 , ∞), () is the initial condition on [−, 0],  > 0,  0 is the initial time, ,   , , , and  are constant matrices of appropriate dimensions, and () denotes the time-varying delay satisfying where  and  are known constants.
Definition 2 (see [45]). is called a Metzler matrix if its offdiagonal entries are nonnegative.
Definition 4 (see [19]).For a given positive scalar , system (1) is said to have an  1 -gain performance level  if the following conditions hold.
Now let us consider the following system subject to actuator saturation: where () ∈   is the control input vector.The function sat(⋅) :   →   is the saturation function which is defined as where Let Ω be the set of all diagonal matrices in  × with diagonal elements that are either 1 or 0; then there are 2  elements   in Ω, and for each  = 1, 2, . . ., 2  ,  −  =  −   is also an element in Ω.
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  1 -gain performance.

Stability Analysis.
In this section, we firstly consider the stability of the closed-loop system (10) then the closed-loop system (10) with () = 0 is positive and asymptotically stable for any initial states satisfying where V  (  ) is the th element of V().
When () = 0, 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 V,  are positive vectors to be determined.When () = 0, along the trajectory of system (10), we have, Mathematical Problems in Engineering It follows that Combining ( 12)-( 14), we obtain Therefore, system (10) with () = 0 is locally asymptotically stable.Furthermore, it can be obtained from ( 17) that ()     = Γ () .
The proof is completed.

𝐿 1 -Gain Analysis.
The following theorem establishes a condition under which the closed-loop system (10) possesses positivity and has an  1 -gain performance.
Therefore, system (10) has an  1 -gain performance level , and all trajectories will remain inside of (V, 1 +   ).
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 V are variables to be determined in Theorem 7, and we cannot directly compute  by using the LMI (linear matrix inequality) method.
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  1 -gain performance of the closed-loop system despite the existence of actuator saturation.
Â and Â are uncertain matrices satisfying where , ,   , and   are known constant matrices.
Proof.Because  is a Metzler matrix and   ⪰ 0,  ⪰ 0, and  ⪰ 0, it is easy to obtain that Â is a Metzler matrix and Â ⪰ 0; then system ( 35) is positive.
The proof is completed.
where () ∈   + is the control input vector.Similarly, the system (37) can be rewritten as the following closed-loop system: By Lemma 10, ∑ 2  =1   ( +    +  −  ) should be Metzler matrices to ensure the positivity of system (38).
The following theorem gives sufficient conditions which ensure the positivity and  1 -gain property of the closed-loop system (38).
Choose the Lyapunov functional candidate (17).Along the trajectory of system (38), we have
When () ̸ = 0, similar to the proof line of Theorem 7, the  1 -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 V, , and   by solving ( 13), ( 24)-( 25), and Then from (44) and   ⪰        V,  = 1, 2, . . ., 2  , we can obtain the gain matrix .

Examples
In this section, three examples are presented to check the validity of the proposed results.
Figure 5 shows the state responses of the closed-loop system.Figure 6 shows the control signals () and sat(()).It can be seen from Figures 4-6 that the closed-loop system is positive and asymptotically stable.This demonstrates the effectiveness of the proposed approach.
Example 3. Consider a model of virus treatment which can be described as system (4) [6].The system parameters are as follows: where  1 () and

Conclusions
In this paper, we have investigated the problems of robust stabilization and disturbance rejection for a class of positive systems with time-varying delays and actuator saturation.An appropriate copositive type Lyapunov functional is employed   to ensure the stability and  1 -gain performance of the positive systems.Both the existence conditions and the explicit characterization of the desired controller are derived in terms of LMIs.Finally, three examples are provided to illustrate the validity of the theoretical results.

Example 1 .
Consider system (4) with the following parameters:

Figure 4 :
Figure 4: Attractive regions and state trajectory in Example 2.

Figure 5 :
Figure 5: State responses of the closed-loop system in Example 2.

Figure 7 :
Figure 7: Attractive regions and state trajectory in Example 3.

Figure 8 :
Figure 8: State responses of the closed-loop system in Example 3.