# Analysis and Design of Robust Positivity and Stability for Continuous-Time Linear Uncertain Systems

Authors: Kuo-Liang Yen; Kuo-Shong Wang; Yau-Tarng Juang
Journal: Journal of Applied Mathematics
 (2015)

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## Abstract

This paper deals with the analysis and design of positivity and stability of linear continuous conic systems. First, two robustness analysis theorems are proposed for the systems with state-feedback. Second, the state-feedback stabilization problem is solved by using linear programming (LP). Numerical examples are given for illustration. Finally, the conclusions are made.


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## Body

## 1. Introduction

Many researchers study positive systems. The design of observers and dynamic output-feedback controllers for positive linear systems with interval uncertainties was finished by Shu et al. [1]. The continuous-time case and the discrete-time case were both treated in a unified linear matrix inequality (LMI) framework. Liu et al. [2] tried to adapt the linear copositive Lyapunov function to LTI discrete-time positive systems with delays, focusing on controller design under the positivity constraint and/or with bounds on the inputs and the states. Liu [3] finished the continuous-time positive systems with delays. Roszak and Davison [4] considered the problem of stabilization via output-feedback, state-feedback, and stabilization via the sequential separation principle for positive linear time-invariant (LTI) systems. Jacquez and Simon [5] applied compartmental systems in different fields ranging from biology and chemistry, over ecology to economy and sociology. Marchetti et al. [6] proposed an improved PID control strategy for blood glucose control. This is a great contribution for type 1 diabetes. Rami and Tadeo [7] pointed out that linear programming (LP) approach is simpler than the linear matrix inequality (LMI) approach and possesses a low computational complexity. They also solved an important synthesis problem: finding bounded stabilizing state-feedback control. In this paper, we will solve the state-feedback stabilization problems for positive linear systems with conic uncertainty in terms of linear programming (LP).


## 2. Preliminary

Consider the following linear system:(1)x˙=Ax,where x(t)∈Rn is the system state and A∈Rn×n. The system in (1) is said to be positive if and only if, for any nonnegative initial condition, denoted as x(0)=x0≥0, the resulting state should be nonnegative; that is, x(t)≥0 for all t>0.Definition 1.
A
∈
R
n
×
n is said to be Metzler if all off-diagonal entries are nonnegative.


## 3. Positivity and Stability of Continuous-Time Conic Linear Systems

Lemma 2 (see [8]).
The system (1) is a continuous-time positive system if and only if the matrix A is a Metzler matrix.Lemma 3 (see [7]).
Continuous-time positive linear system is asymptotically stable if and only if there exists ad∈R+n with Ad<0.Consider an autonomous system with the system matrixA represented by a more general conic form:(2)x˙t=Axt,x0=x0∈R+n,A∈∑p=1p¯αpAp∣αp≥0,∑p=1p¯αp≠0,where Ap are given constant matrices and p¯ is a given number. Then, the following result is obtained.Theorem 4.
The linear conic system in (2) is positive and asymptotically stable if Ap are Metzler matrices and there exists a d∈R+n such that Apd<0, for all p=1,…,p¯.Proof.
Owing to the matricesAp are Metzler; let ap,ij and aij be the entries of Ap and, for 1≤i≠j≤n, p=1,…,p¯. From (2), aij=∑p=1p¯αpap,ij≥0, for 1≤i≠j≤n. This shows that the off-diagonal entries of A are nonnegative. By Lemma 2, the system (2) is positive. On the other hand, Apd<0, for p=1,…,p¯; that is, ∑j=1nap,ijdj<0, for 1≤i≤n, p=1,…,p¯. From (2), Ad=∑j=1n∑p=1p¯αpap,ijdj<0, for 1≤i≤n. By Lemma 3, system (2) is asymptotically stable.Remark 5.
The polytopic and interval representations are the special cases of the considered conic set representation.In the following, consider the continuous-time linear conic system:(3)x˙t=Axt+But,x0=x0∈R+n,A∈∑p=1p¯αpAp∣αp≥0,∑p=1p¯αp≠0,B∈∑q=1q¯βqBq∣βq≥0,∑q=1q¯βq≠0,where Ap=[ap,ij]n×n and Bq=[bq,ih]n×m are given constant matrices and p¯ and q¯ are given constants. When the state-feedback u=Kx, where K=[khj]∈Rm×n, is applied, the closed-loop system becomes(4)x˙t=A+BKxt,x0=x0∈R+n,A∈∑p=1p¯αpAp∣αp≥0,∑p=1p¯αp≠0,B∈∑q=1q¯βqBq∣βq≥0,∑q=1q¯βq≠0.Lemma 6.
System (4) is positive and asymptotically stable if Ap+BqK are Metzler matrices and there exists a d∈R+n such that (Ap+BqK)d<0, for p=1,…,p¯, q=1,…,q¯.Proof.
By Theorem4, the system in (2) is positive and asymptotically stable if Ap are Metzler matrices and there exists a d∈R+n such that Apd<0, for all p=1,…,p¯.
Now, in system (4), assume that Acl=A+BK where(5)A∈∑p=1p¯αpAp∣αp≥0,∑p=1p¯αp≠0,B∈∑q=1q¯βqBq∣βq≥0,∑q=1q¯βq≠0.Then Acl=A+BK=∑q=1q¯∑p=1p¯αpβq(Ap+BqK); that is, Acl has p¯q¯ vertices. Hence, system (4) is positive and asymptotically stable if Ap+BqK are Metzler matrices and there exists a d∈R+n such that (Ap+BqK)d<0, for p=1,…,p¯, q=1,…,q¯.Theorem 7.
Using the state-feedbacku=Kx with K=[khj]∈Rm×n, the closed-loop system in (4) is positive and asymptotically stable if there exist a [dj]∈R+n and a [zhj]∈Rm×n such that, for all p=1,…,p¯ and q=1,…,q¯,(6a)ap,ijdj+∑h=1mbq,ihzhj≥0,1≤i≠j≤n,(6b)∑j=1nap,ijdj+∑h=1mbq,ihzhj<0,1≤i≤n, where K=ZD-1, K=[khj]∈Rm×n, and (7)D=diag⁡d1,d2,…,dn.Proof.
(a) Positivity is as follows.
Due to(8)ap,ijdj+∑h=1mbq,ihzhj≥0,dj>0,owing to khj=zhjdj-1,(9)ap,ij+∑h=1mbq,ihkhj≥0for 1≤i≠j≤n, p=1,…,p¯, and q=1,…,q¯, this shows that the off-diagonal entries of the matrices Ap+BqK are nonnegative. By Lemma 6, the closed-loop system in (4) is positive.
(b) Stability is as follows.
Due to(10)∑j=1nap,ijdj+∑h=1mbq,ihzhj<0,dj>0,owing to khj=zhjdj-1,(11)∑j=1nap,ij+∑h=1mbq,ihkhjdj<0for 1≤i≤n, p=1,…,p¯, and q=1,…,q¯, this shows that there exists a [dj]∈R+n such that (Ap+BqK)d<0. By Lemma 6, the closed-loop system in (4) is asymptotically stable.Remark 8.
Since the conditions in ((6a), (6b)) are linear in the entries of [dj] and [zhj], these variables can be easily solved by using the linear programming. After [dj] and [zhj] are decided, the robust state-feedback is determined by K=ZD-1. This will accomplish the design for the positivity and stability of continuous-time conic positive linear systems.


## 4. Numerical Examples

In this section, the corresponding example is presented to show the effectiveness of our results.Example 1.
Consider a continuous-time conic system described by (3). Let p¯=4 and q¯=4; the matrices A=∑p=14αpAp and B=∑q=14βqBq with the following parameters:(12)A1=-0.5-0.40.6-0.3,A2=-0.70.60.90.5,A3=-0.30.5-0.7-0.5,A4=-0.90.40.60.3,B1=0.50.20.7-0.4,B2=0.40.50.3-0.5,B3=0.30.40.6-0.3,B4=0.60.30.4-0.2,α1≥0,α2≥0,α3≥0,α4≥0,∑p=14αp=4,β1≥0,β2≥0,β3≥0,β4≥0,∑q=14βq=4.By Theorem 7 and MATLAB to get(13)Z=-8.7788-16.7758-57.897072.5071,d=8.20474.3698,K=ZD-1=-1.0700-3.8390-7.056616.5926.For the uncertain parameters α1,α2,α3,α4,β1,β2,β3, and β4, the following cases are presented.
Case  1 (α1=0,α2=2,α3=0,α4=2,β1=2,β2=0,β3=2, and β4=0). The matrix of the closed-loop system (4) is obtained as (14)A+BK=-13.379815.768710.0972-31.6110.
Case  2 (α1=1,α2=1,α3=1,α4=1,β1=1,β2=1,β3=1, and β4=1). The matrix of the closed-loop system (4) is obtained as(15)A+BK=-14.205117.41959.1392-30.9076.


## 5. Conclusion

The proofs of the two theorems are finished. The simulations are complete by using Theorem7. All results are Hurwitz matrix. In short, the systems are positive and asymptotically stable.



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