# Robust Stabilization of Controlled Positive Switched Linear Systems by Output Feedback

Authors: Jinjin Liu; Ting Zhang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This study focuses on the controller synthesis issues for constrained switched linear systems with uncertainties under mode-dependent average dwell time (MDADT) switching strategy. First, output feedback controllers ensure that the closed-loop systems are positive and asymptotically stable. Second, the bounded controllers are acquired based on system states with interval and polytopic uncertainties. Also, the proposed approach can be applied to the systems with the constrained output. Then, the presented conditions can be formulated in terms of linear programming. Finally, illustrative example is provided to show the effectiveness of the theoretical results.


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## Body

## 1. Introduction

A switched system consists of a family of subsystems and a law determining which subsystem is active during a certain time interval [1, 2]. In real world, the associated states are always nonnegative, such as the population size of species, level of liquid, and concentration of substances. Such models are described by positive systems [3, 4]. As a special switched system, the positive switched system has not only the properties of the switched system but also the nature of the positive system. The positive switched linear system receives much attention in recent years due to the widely practical applications, such as congestion control in communication networks [5], medical treatment [6], general anesthesia [7, 8], multiagent system [9], and so on.Many researchers have been concerned with the controller design problem, especially the constrained controller synthesis for the general system and positive system [10–12]. In fact, it is inevitable that the control system encounters limited situations, for example, maximum current or highest voltage in the circuits, the maximum temperature in thermal systems, and so on. Therefore, it is of great practical significance to design a controller which makes the control signal and state of the closed-loop system be bounded by the specified boundary. The study of bounded control for the positive system starts from [13, 14], where stability and stabilization conditions for the positive system by means of linear composite Lyapunov function are proposed under the constrained state feedback controller. Then, the results are extended to the positive system with delay in [15, 16], Takagi–Sugeno fuzzy system [17], fractional order system [18], and event-triggered system [19]. Meanwhile, stability and stabilization issues of positive switched systems are developing rapidly. Reference [20] considered the stability analysis problem for a class of positive switched systems with average dwell time switching; then, [21] continued this idea and further studied this issue under MDADT switching, that is, each subsystem has its own average dwell time. Reference [22] tried to adapt the constrained control problem to both continuous and discrete-time positive switched systems with delays, focusing on the nonnegative controller design under a prescribed upper bound with the arbitrary switching signal. [23, 24] investigated the controller synthesis for a class of discrete-time switched linear systems with bounds on the controls and the states, which guarantees positivity and stability of the closed-loop system. However, most of these results are based on the state feedback.On the other hand, output variables are easy to directly measure and technically implement and also have a clear physical meaning in most cases, so the output feedback is a common form of the feedback mode. Based on linear programming, [25] solved the stabilization issue by the output feedback controller having one rank gains. On the basis of the singular value decomposition approach, [26] revisited the output feedback stabilization problem with and without interval uncertainties. By using the system augmentation approach, an iterative linear matrix inequality algorithm is presented to compute the feedback gain matrix [27]. Turning to positive switched systems, [28] continued the linear programming method and addressed feedback control under system input matrix. Reference [29] involved controller synthesis via a constrained output feedback, but the rank of the controller gain is one. Furthermore, it is well known that because of modeling errors and variations of plant parameters, there always exists uncertainty due to some unpredictable factors in many real systems, and some effective results have been achieved in this area [30–32].In this context, we consider robust stabilization of enforced positive switched linear systems by the output feedback with and without boundary under the MDADT switching scheme. The so-called controlled positive systems mean that the resulting closed-loop systems we received is positive, even if the open-loop system is not positive at all. In order to overcome the incomplete controller gain, we start from the elements view of the system matrix. By introducing quantity matrices, sufficient conditions for the existence of the output feedback controller are formulated, such that the closed-loop system can not only guarantee robust asymptotic stability but also satisfy positive. Simultaneously, inputs and outputs are limited by the prescribed boundary provided that the initial condition is within the same boundary.The rest of study is structured as follows. Section2 deals with problem statement and presents some preliminaries. Robust stabilization of controlled positive switched linear systems interval uncertainties and polytopic uncertainties are solved in in Sections 3 and 4, respectively. Section 5 gives a numerical example to reveal our design. Finally, Section 6 concludes this study.


### 1.1. Notations

ℝn denotes the vector of n-tuples of real numbers, and ℝm×n means the space of m×n matrices with real entries. For a vector v∈ℝn and a matrix A∈ℝm×n, v≻0v≽0 and A≻0A≽0 imply that all its elements are positive (nonnegative), respectively. Furthermore, v⪯0v⪯¯0 and A⪯0A⪯¯0 stand for all its elements are negative (not positive), respectively. For two matrices A, B∈ℝm×n, aij and bij are the elements in the ith row and jth column of A and B for i=1,2,…,m and j=1,2,…,n. So, A≻BA≽B means that aij>bijaij≥bij, and A⪯BA⪯¯B denotes aij<bijaij≤bij for all i=1,2,…,m and j=1,2,…,n, respectively.


## 1.1. Notations

ℝn denotes the vector of n-tuples of real numbers, and ℝm×n means the space of m×n matrices with real entries. For a vector v∈ℝn and a matrix A∈ℝm×n, v≻0v≽0 and A≻0A≽0 imply that all its elements are positive (nonnegative), respectively. Furthermore, v⪯0v⪯¯0 and A⪯0A⪯¯0 stand for all its elements are negative (not positive), respectively. For two matrices A, B∈ℝm×n, aij and bij are the elements in the ith row and jth column of A and B for i=1,2,…,m and j=1,2,…,n. So, A≻BA≽B means that aij>bijaij≥bij, and A⪯BA⪯¯B denotes aij<bijaij≤bij for all i=1,2,…,m and j=1,2,…,n, respectively.


## 2. Problem Statement and Preliminaries

In this section, we present precise problem description for switched systems under consideration. Consider the following switched linear systems:(1)x˙t=Aσtxt+Bσtut,yt=Cσtxt,where xt∈ℝn, ut∈ℝm, and yt∈ℝs denote the state vector, control input, and output, respectively. The switching signal σt:0,∞⟶ℙ=1,2,…,N is a piecewise constant and right continuous function of time t, with N being the number of modes of the overall switched system. σt=p∈ℙ implies that the pth subsystem is active, and Ap=apij∈ℝn×n, Bp=bpij∈ℝn×m, and Cp=cpij∈ℝs×n are the system matrices.The main problem we studied in this article is the following: under the premise of system matrices are not precisely known, the output feedback must be set up in such a way that the resulting closed-loop system is positive and asymptotically stable. In addition, the control law we designed should be not restricted in sign and limited within certain boundary. To be specific, consider the output feedback controllerut=Gpyt, leading to the corresponding closed-loop system.(2)x˙t=Acpxtis positive and asymptotically stable under MDADT switching signal, while the open-loop system may not be positive. Here, Acp=Ap+BpGpCp,∀p∈ℙ.Before presenting the main results of this study, the following definitions and lemmas need to list.Definition 1 (see [22]).
The continuous-time linear system(3)x˙t=Apxtis said to be positive if for any switching signal and any initial condition x0≽0, and the corresponding trajectory of system xt≽0 holds for all t≥0.Definition 2 (see [4]).
MatrixA is called a Metzler matrix if the off-diagonal entries of A are nonnegative.Definition 3 (see [21]).
For a switching signalσt and any t2>t1≥0, let Nσpt1,t2 be the switching number that the pth subsystem is activated in time interval t1,t2 and Tpt1,t2 denotes the total running of the pth subsystem in time interval t1,t2. If there exists a nonnegative constant N0p, such that(4)Nσpt1,t2≤N0p+Tpt1,t2ταp,then ταp is a MDADT of σt, and N0p is the mode-dependent chattering bound.Lemma 1 (see [22]).
System (3) is positive if and only if Ap is a Metzler matrix for each p∈ℙ.
Here, it is worth mentioning that the closed-loop system (2) is positive only if system matrix Acp is Metzler for each p∈ℙ.Lemma 2 (see [21]).
If there exist vectorλp≻0 and constant γp>0, such that Ap+γpIλp⪯0, a SPLS x˙t=Apxt is asymptotically stable under the MDADT switching signal ταp>ταp∗=lnμp/γp, where μp=maxλp/λq for p,q∈ℙ×ℙ.Lemma 3 (see [22]).
Consider a SPLSx˙t=Apxt; if there exist vector λp≻0 and constant γ>0, such that Ap+γpIλp⪯0 sets up for 0⪯¯x0⪯¯λp, then the state trajectory of SPLS satisfies that 0⪯¯xt⪯¯λp for p∈ℙ.


## 3. Robust Stabilization with Interval Uncertainties

In this section, we consider the controller synthesis issue if switched systems with interval uncertainties. Here, the subsystem matricesAp and Bp are not precisely known but belong to the following interval uncertainty domain: A¯p⪯¯Ap⪯¯A¯p and B¯p⪯¯Bp⪯¯B¯p. In addition, we divide this section into three cases.


### 3.1. Sign-Restricted Controls

First of all, a fundamental result for robust stabilization of switched linear systems under MDADT switching strategy with nonnegative or negative controls is now put forward. Let us begin by the conditionut≽0.Theorem 1.
Consider the switched linear system (1) with interval uncertainties. For prescribed constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s subjected to(5)a¯pijdp+∑l=1s∑k=1mb¯pikzpklcplj≥0,∀1≤i≠j≤n,(6)∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mb¯pikzpklcplj+γpdp<0,∀1≤i≤n,(7)∑l=1szpilcplj≥0,∀1≤i,≤m,1≤j,≤n,then the closed-loop system (2) with the output feedback gain Gp=ZpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. Furthermore, ut≽0 holds for initial condition 0⪯¯x0⪯¯dvp, where dvp=dp,dp,…,dpT∈ℝn.Proof.
Using the output feedback gain matrix, it follows that(8)acpijdp=apijdp+∑l=1s∑k=1mbpikgpklcpljdp,∀1≤i,j≤n.
Note the fact thatzpkj=gpkjdp, and it is easy to give(9)acpijdp=apijdp+∑l=1s∑k=1mbpikzpklcplj≥a¯pijdp+∑l=1s∑k=1mb¯pikzpklcplj.
Using the fact (5) and recalling the condition dp>0, one can derive acpij≥0 for 1≤i≠j≤n. In consequence, Acp is the Metzler matrix and the closed-loop (2) system is positive by Lemma 1.
With the vectordvp and matrix operation rule in mind, we have(10)Acpdvpi=∑j=1napij+∑l=1s∑k=1mbpikgpklcpljdp=∑j=1napijdp+∑j=1n∑l=1n∑k=1mbpikzpklcplj,≤∑j=1na¯pijdp+∑j=1n∑l=1n∑k=1mb¯pikzpklcplj.
By (6), we can get Acp+γpIdvp=Ap+BpGpCp+γpIdvp⪯0, that is, the closed-loop system (2) is asymptotically stable by Lemma 2.
Finally, it derives thatZpCp=GpCpdp≽0 from (7). Since dp>0, it yields GpCp≽0. Thus, under the output feedback controller and (7), ut=Gpyt=GpCpxt≽0 sets up.Remark 1.
We do not restrict the state of the original system because our goal is to design the output feedback controller, so that the closed-loop system is positive. This is so-called the controlled positive system.
Next, we turn our attention on negative control, that is,ut⪯¯0.Theorem 2.
Consider the switched linear system (1) with interval uncertainties. For predetermined constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s, such that(11)a¯pijdp+∑l=1s∑k=1mb¯pikzpklcplj≥0,∀1≤i≠j≤n,(12)∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mb¯pikzpklcplj+γpdp<0,∀1≤i≤n,(13)∑l=1szpilcplj≤0,∀1≤i≤m,1≤j≤n,then the closed-loop system (2) is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. Moreover, the output feedback gain is given by Gp=ZpDp−1. Also, ut⪯¯0 sets up, and assume that the initial condition is 0⪯¯x0⪯¯dvp for dvp∈ℝn.Proof.
By (11) and (12), it verifies that the closed-loop system (2) is positive and asymptotically stable using a similar proof to Theorem 1. Taking dp>0 and (13) into account, we can obtain GpCp⪯¯0. So, it follows that ut=Gpyt=GpCpxt⪯¯0.Remark 2.
For the model of a positive system, system control is also required to be nonnegative. On the contrary, negative controllers are also very common in real life. So, it is important to talk about sign-constrained controllers.


### 3.2. Bounded Controls

In this subsection, the bounded control problems are investigated for the switched system with interval uncertainties under MDADT switching. That is, the designed output feedback controllers allow positivity and asymptotic stability of the closed-loop system with the initial condition0⪯¯x0⪯¯dvp, and the controllers should be limited to boundaries decided in advance. In this sequel, we consider the condition of 0⪯¯ut⪯¯u¯ for fixed u¯≽0.Theorem 3.
Consider the switched linear system (1) with interval uncertainties. For fixed constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s, such that(14)a¯pijdp+∑l=1s∑k=1mb¯pikzpklcplj≥0,∀1≤i≠j≤n,(15)∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mb¯pikzpklcplj+γpdp<0,∀1≤i≤n,(16)∑l=1szpilcplj≥0,∀1≤i≤m,1≤j≤n,(17)∑j=1n∑l=1szpilcplj≤u¯i,∀1≤i≤m,then the closed-loop system (2) with the output feedback gain Gp=ZpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq and ∀p,q∈ℙ×ℙ. Furthermore, the restricted condition 0⪯¯ut⪯¯u¯ meets if initial condition is 0⪯¯x0⪯¯dvp, where dvp∈ℝn.Proof.
By Theorem1, the closed-loop system is positive and asymptotically stable, and ut≽0 also sets up. Furthermore, it is clear that 0⪯¯xt⪯¯dvp holds for the initial condition meets 0⪯¯x0⪯¯dvp by Lemma 3. Then, we have(18)ut=Gpyt=GpCpxt⪯¯GpCpdvp.
Lete=1,1…,1T∈ℝs. We have dvp=dpe. Recalling the output feedback gain matrix, we can obtain(19)GpCpdvp=GpdpCpe=ZpCpe.
Then, the inequalityut⪯¯u¯ holds by (17).
In the following, we aim to the condition−u˜⪯¯ut⪯¯0 of controlled positive switched systems for prescribed u˜≻0.Theorem 4.
Consider the switched linear system (1) with interval uncertainties. For given constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s, such that(20)a¯pijdp+∑l=1s∑k=1mb¯pikzpklcplj≥0,∀1≤i≠j≤n,∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mb¯pikzpklcplj+γpdp<0,∀1≤i≤n,∑l=1szpilcplj≤0,∀1≤i≤m,1≤j≤n,−u˜i≤∑j=1n∑l=1szpilcplj,∀1≤i≤m,then the closed-loop system (2) with the output feedback gain Gp=ZpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. Moreover, −u˜⪯¯ut⪯¯0 fulfills suppose that initial condition satisfies 0⪯¯x0⪯¯dvp, where dvp∈ℝn.Proof.
The proof is similar to Theorem3 and therefore omitted here.
At the end of this subsection, asymmetrically bounded control−u˜⪯¯ut⪯¯u¯ is addressed to stabilize switched systems with interval uncertainties for given u˜ and u¯.Theorem 5.
Consider the switched linear system (1) with interval uncertainties. For given constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s, Wp=wpij∈ℝm×s, such that(21)a¯pijdp+∑l=1s∑k=1mminb¯pikzpkl−wpklcplj,b¯pikzpkl−wpklcplj≥0,∀1≤i≠j≤n,(22)∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mmaxb¯pikzpkl−wpklcplj,b¯pikzpkl−wpklcplj+γpdp<0,∀1≤i≤n,(23)∑l=1szpilcplj≥0,∀1≤i≤m,1≤j≤n,(24)∑k=1swpilcplj≥0,∀1≤i≤m,1≤j≤n,(25)∑j=1n∑l=1szpilcplj≤u¯i,∀1≤i≤m,(26)∑j=1n∑l=1swpilcplj≤u˜i,∀1≤i≤m,then the closed-loop system (2) with the output feedback gain Gp=Zp−WpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. Moreover, −u˜⪯¯ut⪯¯u¯ holds if the initial condition is 0⪯¯x0⪯¯dvp, where dvp∈ℝn.Proof.
According to the proof of Theorem1, the closed-loop system is positive and asymptotically stable for the output feedback gain matrix Gp=ZpDp−1−WpDp−1. Since, inequalities (21) and (22) are together with Lemma 3, 0⪯¯xt⪯¯dvp, sets up for the initial condition 0⪯¯x0⪯¯dvp. Note that vector e and combining with (23)–(26), we can get(27)0⪯¯ZpDp−1yt=ZpDp−1Cpxt⪯¯ZpDp−1Cpdvp=ZpCpe,(28)0⪯¯WpDp−1yt=WpDp−1Cpxt⪯¯WpDp−1Cpdvp=WpCpe.
Using the properties of the inequality, (27) can be rewritten as(29)−WpCpe⪯¯−WpDp−1yt⪯¯0.
So, one has(30)−u˜⪯¯utGpyk=Zp−WpDp−1yt⪯¯u¯.
This completes the proof.


### 3.3. Constrained Output

The matrixCp is nonnegative. Under this general premise, this subsection handles the constrained output issue for controlled positive switched systems. The objective is to design the output feedback controller, such that the corresponding closed-loop system is positive and asymptotically stable via MDADT switching signals. Here, the output satisfies the bound 0⪯¯yt⪯¯y¯ for fixed y¯≻0.Theorem 6.
Consider the switched linear system (1) with interval uncertainties. Assume that Cp≽0. For given constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s satisfying(31)a¯pijdp+∑l=1s∑k=1mminb¯pikzpklcplj,b¯pikzpklcplj≥0,∀1≤i≠j≤n,(32)∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mmaxb¯pikzpklcplj,b¯pikzpklcplj+γpdp<0,∀1≤i≤n,(33)∑j=1ncpijdp≤y¯i,∀1≤i≤s,then the closed-loop system (2) with the output feedback gain Gp=ZpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. In addition, 0⪯¯yt⪯¯y¯ satisfies if initial condition is 0⪯¯x0⪯¯dvp, where dvp∈ℝn.Proof.
It is similar to the previous proof of Theorem1, and we know that the closed-loop system (2) is positive and asymptotically stable from (31) and (32). By using Cp≽0 and Lemma 3, it follows yt=Cpxt⪯¯Cpdvp. Together with (33), we immediately arrive at constraints 0⪯¯yt⪯¯y¯.


## 3.1. Sign-Restricted Controls

First of all, a fundamental result for robust stabilization of switched linear systems under MDADT switching strategy with nonnegative or negative controls is now put forward. Let us begin by the conditionut≽0.Theorem 1.
Consider the switched linear system (1) with interval uncertainties. For prescribed constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s subjected to(5)a¯pijdp+∑l=1s∑k=1mb¯pikzpklcplj≥0,∀1≤i≠j≤n,(6)∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mb¯pikzpklcplj+γpdp<0,∀1≤i≤n,(7)∑l=1szpilcplj≥0,∀1≤i,≤m,1≤j,≤n,then the closed-loop system (2) with the output feedback gain Gp=ZpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. Furthermore, ut≽0 holds for initial condition 0⪯¯x0⪯¯dvp, where dvp=dp,dp,…,dpT∈ℝn.Proof.
Using the output feedback gain matrix, it follows that(8)acpijdp=apijdp+∑l=1s∑k=1mbpikgpklcpljdp,∀1≤i,j≤n.
Note the fact thatzpkj=gpkjdp, and it is easy to give(9)acpijdp=apijdp+∑l=1s∑k=1mbpikzpklcplj≥a¯pijdp+∑l=1s∑k=1mb¯pikzpklcplj.
Using the fact (5) and recalling the condition dp>0, one can derive acpij≥0 for 1≤i≠j≤n. In consequence, Acp is the Metzler matrix and the closed-loop (2) system is positive by Lemma 1.
With the vectordvp and matrix operation rule in mind, we have(10)Acpdvpi=∑j=1napij+∑l=1s∑k=1mbpikgpklcpljdp=∑j=1napijdp+∑j=1n∑l=1n∑k=1mbpikzpklcplj,≤∑j=1na¯pijdp+∑j=1n∑l=1n∑k=1mb¯pikzpklcplj.
By (6), we can get Acp+γpIdvp=Ap+BpGpCp+γpIdvp⪯0, that is, the closed-loop system (2) is asymptotically stable by Lemma 2.
Finally, it derives thatZpCp=GpCpdp≽0 from (7). Since dp>0, it yields GpCp≽0. Thus, under the output feedback controller and (7), ut=Gpyt=GpCpxt≽0 sets up.Remark 1.
We do not restrict the state of the original system because our goal is to design the output feedback controller, so that the closed-loop system is positive. This is so-called the controlled positive system.
Next, we turn our attention on negative control, that is,ut⪯¯0.Theorem 2.
Consider the switched linear system (1) with interval uncertainties. For predetermined constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s, such that(11)a¯pijdp+∑l=1s∑k=1mb¯pikzpklcplj≥0,∀1≤i≠j≤n,(12)∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mb¯pikzpklcplj+γpdp<0,∀1≤i≤n,(13)∑l=1szpilcplj≤0,∀1≤i≤m,1≤j≤n,then the closed-loop system (2) is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. Moreover, the output feedback gain is given by Gp=ZpDp−1. Also, ut⪯¯0 sets up, and assume that the initial condition is 0⪯¯x0⪯¯dvp for dvp∈ℝn.Proof.
By (11) and (12), it verifies that the closed-loop system (2) is positive and asymptotically stable using a similar proof to Theorem 1. Taking dp>0 and (13) into account, we can obtain GpCp⪯¯0. So, it follows that ut=Gpyt=GpCpxt⪯¯0.Remark 2.
For the model of a positive system, system control is also required to be nonnegative. On the contrary, negative controllers are also very common in real life. So, it is important to talk about sign-constrained controllers.


## 3.2. Bounded Controls

In this subsection, the bounded control problems are investigated for the switched system with interval uncertainties under MDADT switching. That is, the designed output feedback controllers allow positivity and asymptotic stability of the closed-loop system with the initial condition0⪯¯x0⪯¯dvp, and the controllers should be limited to boundaries decided in advance. In this sequel, we consider the condition of 0⪯¯ut⪯¯u¯ for fixed u¯≽0.Theorem 3.
Consider the switched linear system (1) with interval uncertainties. For fixed constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s, such that(14)a¯pijdp+∑l=1s∑k=1mb¯pikzpklcplj≥0,∀1≤i≠j≤n,(15)∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mb¯pikzpklcplj+γpdp<0,∀1≤i≤n,(16)∑l=1szpilcplj≥0,∀1≤i≤m,1≤j≤n,(17)∑j=1n∑l=1szpilcplj≤u¯i,∀1≤i≤m,then the closed-loop system (2) with the output feedback gain Gp=ZpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq and ∀p,q∈ℙ×ℙ. Furthermore, the restricted condition 0⪯¯ut⪯¯u¯ meets if initial condition is 0⪯¯x0⪯¯dvp, where dvp∈ℝn.Proof.
By Theorem1, the closed-loop system is positive and asymptotically stable, and ut≽0 also sets up. Furthermore, it is clear that 0⪯¯xt⪯¯dvp holds for the initial condition meets 0⪯¯x0⪯¯dvp by Lemma 3. Then, we have(18)ut=Gpyt=GpCpxt⪯¯GpCpdvp.
Lete=1,1…,1T∈ℝs. We have dvp=dpe. Recalling the output feedback gain matrix, we can obtain(19)GpCpdvp=GpdpCpe=ZpCpe.
Then, the inequalityut⪯¯u¯ holds by (17).
In the following, we aim to the condition−u˜⪯¯ut⪯¯0 of controlled positive switched systems for prescribed u˜≻0.Theorem 4.
Consider the switched linear system (1) with interval uncertainties. For given constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s, such that(20)a¯pijdp+∑l=1s∑k=1mb¯pikzpklcplj≥0,∀1≤i≠j≤n,∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mb¯pikzpklcplj+γpdp<0,∀1≤i≤n,∑l=1szpilcplj≤0,∀1≤i≤m,1≤j≤n,−u˜i≤∑j=1n∑l=1szpilcplj,∀1≤i≤m,then the closed-loop system (2) with the output feedback gain Gp=ZpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. Moreover, −u˜⪯¯ut⪯¯0 fulfills suppose that initial condition satisfies 0⪯¯x0⪯¯dvp, where dvp∈ℝn.Proof.
The proof is similar to Theorem3 and therefore omitted here.
At the end of this subsection, asymmetrically bounded control−u˜⪯¯ut⪯¯u¯ is addressed to stabilize switched systems with interval uncertainties for given u˜ and u¯.Theorem 5.
Consider the switched linear system (1) with interval uncertainties. For given constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s, Wp=wpij∈ℝm×s, such that(21)a¯pijdp+∑l=1s∑k=1mminb¯pikzpkl−wpklcplj,b¯pikzpkl−wpklcplj≥0,∀1≤i≠j≤n,(22)∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mmaxb¯pikzpkl−wpklcplj,b¯pikzpkl−wpklcplj+γpdp<0,∀1≤i≤n,(23)∑l=1szpilcplj≥0,∀1≤i≤m,1≤j≤n,(24)∑k=1swpilcplj≥0,∀1≤i≤m,1≤j≤n,(25)∑j=1n∑l=1szpilcplj≤u¯i,∀1≤i≤m,(26)∑j=1n∑l=1swpilcplj≤u˜i,∀1≤i≤m,then the closed-loop system (2) with the output feedback gain Gp=Zp−WpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. Moreover, −u˜⪯¯ut⪯¯u¯ holds if the initial condition is 0⪯¯x0⪯¯dvp, where dvp∈ℝn.Proof.
According to the proof of Theorem1, the closed-loop system is positive and asymptotically stable for the output feedback gain matrix Gp=ZpDp−1−WpDp−1. Since, inequalities (21) and (22) are together with Lemma 3, 0⪯¯xt⪯¯dvp, sets up for the initial condition 0⪯¯x0⪯¯dvp. Note that vector e and combining with (23)–(26), we can get(27)0⪯¯ZpDp−1yt=ZpDp−1Cpxt⪯¯ZpDp−1Cpdvp=ZpCpe,(28)0⪯¯WpDp−1yt=WpDp−1Cpxt⪯¯WpDp−1Cpdvp=WpCpe.
Using the properties of the inequality, (27) can be rewritten as(29)−WpCpe⪯¯−WpDp−1yt⪯¯0.
So, one has(30)−u˜⪯¯utGpyk=Zp−WpDp−1yt⪯¯u¯.
This completes the proof.


## 3.3. Constrained Output

The matrixCp is nonnegative. Under this general premise, this subsection handles the constrained output issue for controlled positive switched systems. The objective is to design the output feedback controller, such that the corresponding closed-loop system is positive and asymptotically stable via MDADT switching signals. Here, the output satisfies the bound 0⪯¯yt⪯¯y¯ for fixed y¯≻0.Theorem 6.
Consider the switched linear system (1) with interval uncertainties. Assume that Cp≽0. For given constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s satisfying(31)a¯pijdp+∑l=1s∑k=1mminb¯pikzpklcplj,b¯pikzpklcplj≥0,∀1≤i≠j≤n,(32)∑j=1na¯pijdp+∑j=1n∑l=1s∑k=1mmaxb¯pikzpklcplj,b¯pikzpklcplj+γpdp<0,∀1≤i≤n,(33)∑j=1ncpijdp≤y¯i,∀1≤i≤s,then the closed-loop system (2) with the output feedback gain Gp=ZpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. In addition, 0⪯¯yt⪯¯y¯ satisfies if initial condition is 0⪯¯x0⪯¯dvp, where dvp∈ℝn.Proof.
It is similar to the previous proof of Theorem1, and we know that the closed-loop system (2) is positive and asymptotically stable from (31) and (32). By using Cp≽0 and Lemma 3, it follows yt=Cpxt⪯¯Cpdvp. Together with (33), we immediately arrive at constraints 0⪯¯yt⪯¯y¯.


## 4. Robust Stabilization with Polytopic Uncertainties

This section is focused on robust stabilization for controlled positive switched systems with polytopic uncertainties. Suppose that the subsystem matricesAp and Bp are not precisely known but belong to the following polytopic uncertainty set:(34)Θp=∑s=1MγsAps,Bps∑s=1Mγs=1,γs≥0,where Aps and Bps are the certain matrices and stand for extreme points of the pth subsystem, and M denotes the total number of extreme points. Except, we are talking about asymmetrically constrained control and constrained output cases. Others can be discussed similarly.Theorem 7.
Consider the switched linear system (1) with polytopic uncertainties. For given constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s, Wp=wpij∈ℝm×s, such that(35)apijsdp+∑l=1s∑k=1mbpikszpkl−wpklcplj≥0,∀1≤i≠j≤n,(36)∑j=1napijsdp+∑j=1n∑l=1s∑k=1mbpikszpkl−wpklcplj+γpdp<0,∀1≤i≤n,(37)∑l=1szpilcplj≥0,∀1≤i≤m,1≤j≤n,(38)∑k=1swpilcplj≥0,∀1≤i≤m,1≤j≤n,(39)∑j=1n∑l=1szpilcplj≤u¯i,∀1≤i≤m,(40)∑j=1n∑l=1swpilcplj≤u˜i,∀1≤i≤m,then the closed-loop system (2) with the output feedback gain Gp=Zp−WpDp−1 is positive and asymptotically stable under MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. Moreover, −u˜⪯¯ut⪯¯u¯ can be achieved if the initial condition 0⪯¯x0⪯¯dvp, where dvp∈ℝn.Proof.
First, from the closed-loop system matrix, one has(41)acpijs=apijs+∑l=1s∑k=1mbpiksgpklcplj,∀1≤i,j≤n.
Then, it is not hard to see that(42)acpijsdp=apijsdp+∑l=1s∑k=1mbpiksgpklcpljdp,
Notice the matrix element of the output feedback gain, and it follows thatgpkjdp=zpkj−wpkj. Hence,(43)acpijsdp=apijsdp+∑l=1s∑k=1mbpikszpkl−wpkjcplj.
By virtue of (35), acpijsdp≥0 holds for 1≤i≠j≤n. Thus, Acps is the Metzler matrix due to dp>0. This means that Acp is also the Metzler matrix by a convexity of argument, so the closed-loop (2) system is positive by Lemma 1.
Next, consider vectordvp, and we have(44)Acpsdvpi=∑j=1napijs+∑l=1s∑k=1mbpiksgpklcpljdp=∑j=1napijsdp+∑j=1n∑l=1n∑k=1mbpikszpkl−wpkjcplj.
The above together with (36) means that Acps+γpIdvp⪯0, and it follows Acp+γpIdvp=Ap+BpGpCp+γpIdvp⪯0. Thus, the closed-loop system is asymptotically stable by Lemma 2.
The rest of the proof is similar to Theorem5.Remark 3.
In fact, the processes of interval uncertainties and polytopic uncertainties can convert each other. IfAp¯⪯¯Aps⪯¯Ap¯ and Bp¯⪯¯Bps⪯¯Bp¯, Theorem 7 can be treated in the same way as Theorem 5.Theorem 8.
Consider the switched linear system (1) with polytopic uncertainties. Assume that Cp≽0. For given constants γp>0, if there exist diagonal matrices Dp=diagdp,dp,…,dp∈ℝs×s with positive diagonal entries and matrices Zp=zpij∈ℝm×s subjected to(45)apijsdp+∑l=1s∑k=1mbpikszpklcplj≥0,∀1≤i≠j≤n,(46)∑j=1napijsdp+∑j=1n∑l=1s∑k=1mbpikszpklcplj+γpdp<0,∀1≤i≤n,(47)∑j=1ncpijdp≤y¯i,∀1≤i≤s,then the closed-loop system (2) with the output feedback gain Gp=ZpDp−1 is positive and asymptotically stable under the MDADT switching strategy ταp>ταp∗=lnμp/γp, where μp=maxdp/dq, ∀p,q∈ℙ×ℙ. And then, 0⪯¯yt⪯¯y¯ holds if 0⪯¯x0⪯¯dvp, where dvp∈ℝn.Proof.
It is similar to the previous proof of Theorem7, and we know that the closed-loop system (2) is positive and asymptotically stable from (45) and (46). The rest section can be deduced by Theorem 6.Remark 4.
It should be pointed out that we discuss the general case of switching signals in this study. When we chooseμp=μ and γp=γ for ∀p∈ℙ, MDADT switching strategy degenerates into the simple case of average dwell time. Furthermore, if we make μp=μ, γp=γ, and dp=d, the presentation of our conclusion becomes that under the arbitrary switching signal.


## 5. Numerical Examples

Consider system (1) with the interval uncertainties, where the bounds of the system matrices are given by(48)A¯1=−2.1−0.9−0.6−1.3,A¯1=−1.9−0.7−0.4−1.1,B¯1=1.7−1.3−0.70.9,B¯1=1.9−1.1−0.51.1,A¯2=−1.3−0.60.4−1.1,A¯2=−1.1−0.40.6−0.9,B¯2=−2.30.7−0.9−1.9,B¯2=−2.10.9−0.7−1.7,C1=10.80.51,C2=0.6110.5.In this part, we take the constrained output feedback controller0⪯¯ut⪯¯u¯, for example, take u¯=6,8T. Address inequalities (14)–(17) in Theorem 3 by utilizing the linear programming algorithm in MATLAB, and it can readily obtain that D1=diag6.5164,6.5164, D2=diag6.7756,6.7756, and matrices(49)Z1=−2.67546.64349.5755−7.2464,Z2=−1.24363.02809.5813−5.5690.Then, the output feedback gain matrix can be calculated by(50)G1=−0.41061.01951.4694−1.1120,G2=−0.18350.44691.4141−0.8219.It is easy to see that the gain matricesG1 and G2 are nonsingular, and the rank of the matrix is full. Furthermore, the extreme closed-loop system matrices are(51)A¯c1=A¯1+B¯1G1C1=−3.11890.19220.1527−1.7265,A¯c2=A¯2+B¯2G2C2=−2.05600.01040.0465−3.0419,A¯c1=A¯1+B¯1G1C1=−2.71630.54310.5552−1.3756,A¯c2=A¯2+B¯2G2C2=−1.78330.41900.3192−2.6333.We can observe thatA¯c1 and A¯c2 are the Metzler matrices. Select Ai=1/2A¯i+A¯i and Bi=1/2B¯i+B¯i for i=1,2; hence, the close-loop system is positive. Let γ1=0.4, γ2=0.6 and μ1=1.2, μ2=1.25; we get the minimum MDADT τ1∗=0.1990 and τ2∗=0.4368. The state trajectories of the closed-loop system together with the extreme plants under initial x0=6,6T are given in Figures 1 and 2. Figures 3 and 4 plot the control input, which illustrates that the controls meet the fixed bound 0⪯¯uk⪯¯6,8T when the initial value is limited to 0⪯¯x0⪯¯dv1=6.5164,6.5164T. The switching signal σt with MDADT is shown in Figure 5.Figure 1
Simulation of closed-loop system state trajectories.Figure 2
Simulation of control input.Figure 3
Simulation of closed-loop system state trajectories.Figure 4
Simulation of control input.Figure 5
Simulation of switching signal.


## 6. Conclusion

The problem of robust stabilization of controlled positive switched linear systems by the output feedback is addressed in this article. First, we pay attention to find constrained output feedback controllers rendering the positiveness and asymptotic stability of the closed-loop systems, eventhough the open-loop system is not positive. Second, by utilizing the MDADT switching scheme, the robust stabilization of the controlled positive switched system is solved under interval and polytopic uncertainties. It is worth pointing out that all the controllers we designed satisfy certain limiting conditions, including bounded control inputs and constrained outputs. All the conditions can be solvable in terms of linear programming, instead of LMI. Along the approach of this study, further works may refer to delay problems and nonfragile reliable control.



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