# Improvements on Robust Stability of Sampled-Data System with Long Time Delay

Authors: Shigang Wang; Yongli Bi; Yingsong Li
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper mainly studies the problem of the robust stability analysis for sampled-data system with long time delay. By constructing an improved Lyapunov-Krasovskii functional and employing some free weighting matrices, some new robust stability criteria can be established in terms of linear matrix inequalities. Furthermore, the proposed equivalent criterion eliminates the effect of free weighing matrices such that numbers of decision variables and computational burden are less than some existing results. A numerical example is also presented and compared with previously proposed algorithm to illustrate the feasibility and effectiveness of the developed results.


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## Body

## 1. Introduction

In the last few decades, there has been much interest in long time delay system. This is due to its key role in theory research and practical application, such as welding industry, communication networks, and electrical power system. Lots of relevant research results to long time delay systems have been reported in the literature. To mention a few, modeling of long time delay system was considered in [1]; analysis and synthesis results of such system were reported in [2–8]; the fault detection and filtering problem were solved in [9, 10]. However, relationship between sampling period and time delay in the plant is not fully considered [11, 12], which has inevitably limited the applicability of the aforementioned resulted.In the recent years, increasing attention has been devoted to the problem of sampled-data system with long time delay, in which time delay of the plant is usually longer than a sampling period. When such problem is researched in the discrete-time framework, concentrated augmentation approach and direct distribution approach can be chosen. However, in concentrated augmentation approach, time delay is not taken into consideration in process of deriving stability criterion and designing desired controller [5, 6, 13]. Therefore, this approach is usually regarded as being more numbers of decision variable and computational burden than direct distribution approach.In this paper, we make an attempt to solve the robust stability problem of sampled-data systems with long time delay. Some free weighting matrices and an equivalent criterion are introduced, in order to reduce numbers of decide variable and computation burden. Numerical examples are also presented to illustrate the feasibility and effectiveness of the developed results.


## 2. Problem Formulation

Consider a continuous plant of sampled-data system with long time delay:(1)𝒮:x˙(t)=A0x(t)+A1x(t-τ),x(t)=x0,t∈[-τ,0],
where x(t)∈ℝn is the state vector and A0 and A1 are the constant matrices of appropriate dimensions. x0 is the initial state vector.Assumption 1.
Sampler is time-driven with a constant sampling periodh(h>0).Assumption 2.
The time delayτ is uncertain but is evaluated between two adjacent sampling periods; namely, (m-1)h≤τ≤mh, where m>1 is a known constant.Lemma 3 (see [14]).
For given matrices𝒬=𝒬T, ℋ, and ℰ, with appropriate dimensions,
(2)𝒬+ℋℱ(k)ℰ+ℰTℱT(k)ℋT<0
holds for all ℱ(k) satisfying ℱT(k)ℱ(k)≤I if and only if there exists ε>0:
(3)𝒬+εℋℋT+ε-1ℰTℰ<0.


## 3. Main Results

Discretizing system (1) in one period, we can obtain the discrete state equation of sampled-data system:
(4)𝒰:x(k+1)=G0x(k)+G1x(k-m+1)+G2x(k-m),x(k)=x0,k=-m,-m+1,…,0,
where
(5)G0=eA0h,G1=∫0mh-τ‍eA0tdtA1,G2=∫mh-τh‍eA0tdtA1.
Let
(6)G¯1=-Ldiag⁡{1λ1,…,1λn}L-1A1,G¯2=Ldiag⁡{1λ1eλ1h,…,1λneλnh}L-1A1,D=Ldiag⁡{1λ1eλ1β1,…,1λneλnβ1},F(τ)=diag⁡{eλ1(mh-τ-β1),…,eλn(mh-τ-βn)},E=L-1A1.
Then
(7)G1(τ)=G¯1+DF(τ)E.
In the same way, we can get
(8)G2(τ)=G¯2-DF(τ)E.Choosing appropriate variablesβ1,β2,…,βn,  so that they satisfy eλi(mh-τ-βi)≤1, then
(9)FT(τ)F(τ)≤I.Remark 4.
Modeling of sampled-data system with long time delay has been reported in [13].


### 3.1. Stability Condition

The stability condition presented in this section is based on system (4); we will proceed with the following theorem.Theorem 5.
System (4) is asymptotically stable if there exist matrices P1>0, S1>0, S2>0, Q1>0, Q2>0, R1>0, R2>0, M, L, and N, satisfying
(10)[Π1+Π2+Π2T+Π3Π4*Π5]<0,
where
(11)Π1=[G0TP1G0-P1+S1+S2+(m-1)Q1+mQ2G0TP1G1G0TP1G2*G1TP1G1-S1G1TP1G2**G2TP1G2-S2],(12)Π2=[M+NL-M-L-N],Π3=m[G0-IG1G2]T(R1+R2)[G0-IG1G2],Π4=[m-1MLmN],Π5=diag⁡{-R1-R1-R2}.Proof.
Letting(13)η(k)=x(k+1)-x(k),
then
(14)x(k+1)=x(k)+η(k),x(k)-x(k-m+1)=∑i=k-m+1k-1‍η(i),x(k)-x(k-m)=∑i=k-mk-1‍η(i).
Choosing a Lyapunov function candidate as
(15)V=V1+V2+V3+V4+V5+V6,V1=xT(k)P1x(k),V2=∑i=k-m+1k-1‍xT(i)S1x(i),V3=∑i=k-mk-1‍xT(i)S2x(i),V4=∑j=-m+1-1‍∑i=k+jk-1‍xT(i)Q1x(i),V5=∑j=-m-1‍∑i=k+jk-1‍xT(i)Q2x(i),V6=∑j=-m-1‍∑i=k+jk-1‍ηT(i)(R1+R2)η(i),
where P1=P1T>0, Si=SiT≥0, i=1,2, and Ri=RiT≥0, i=1,2, are undetermined.
DefiningΔV=V(k+1)-V(k), and then along the solution of (4), we have
(16)ΔV1=xT(k+1)P1x(k+1)-xT(k)P1x(k)ΔV2≤xT(k)S1x(k)-xT(k-m+1)S1x(k-m+1)ΔV2≤+∑j=k-m+1k-1‍xT(j)Q1x(j),ΔV3≤xT(k)S2x(k)-xT(k-m)S2x(k-m)ΔV3≤+∑j=k-mk-1‍xT(j)Q2x(j),ΔV4=(m-1)xT(k)Q1x(k)-∑j=k-m+1k-1‍xT(j)Q1x(j),ΔV5=mxT(k)Q2x(k)-∑j=k-mk-1‍xT(j)Q2x(j),ΔV6=mηT(k)(R1+R2)η(k)-∑l=k-mk-1‍ηT(l)(R1+R2)η(l)ΔV6=mηT(k)(R1+R2)η(k)-∑l=k-m+1k-1‍ηT(l)R1η(l)ΔV6=-∑l=k-mk-m‍ηT(l)R1η(l)-∑l=k-mk-1‍ηT(l)R2η(l).
Defining ξ(k)=[xT(k)xT(k-m+1)xT(k-m)]T, then we have
(17)ΔV≤[G0x(k)+G1x(k-m+1)+G2x(k-m)]TP1×[G0x(k)+G1x(k-m+1)+G2x(k-m)]-xT(k)P1x(k)+xT(k)S1x(k)-xT(k-m+1)S1x(k-m+1)+xT(k)S2x(k)-xT(k-m)S2x(k-m)+(m-1)xT(k)Q1x(k)+mxT(k)Q2x(k)+m[(G0-I)x(k)+G1x(k-m+1)+G2x(k-m)]T×(R1+R2)×[(G0-I)x(k)+G1x(k-m+1)+G2x(k-m)]-∑l=k-m+1k-1‍ηT(l)R1η(l)-∑l=k-mk-m‍ηT(l)R1η(l)-∑l=k-mk-1‍ηT(l)R2η(l)+2ξT(k)M[x(k)-x(k-m+1)-∑l=k-m+1k-1‍η(l)]+2ξT(k)L[x(k-m+1)-x(k-m)-∑l=k-mk-m‍η(l)]+2ξT(k)N[x(k)-x(k-m)-∑l=k-mk-1‍η(l)]≤ξT(k)[Π1+Π2+Π2T+Π3+(m-1)MR1-1MT000000000+LR1-1LT+mNR2-1NT]ξ(k)-∑l=k-m+1k-1‍[ξT(k)M+ηT(l)R1]×R1-1[MTξ(k)+R1η(l)]-∑l=k-mk-m‍[ξT(k)L+ηT(l)R1]×R1-1[LTξ(k)+R1η(l)]-∑l=k-mk-1‍[ξT(k)N+ηT(l)R2]×R2-1[NTξ(k)+R2η(l)].
SinceRi>0, i=1,2, the last three terms are all nonpositive. By the Schur complement [15], inequality (10) guarantees Π1+Π2+Π2T+Π3+(m-1)MR1-1MT+SR1-1ST+mNR2-1NT<0. From (17), we can attain ΔV<-ε∥x(k)∥2, for a sufficiently small ε, and x(k)≠0, and the asymptotic stability is established.


### 3.2.  Robust Stability Condition

There exist uncertainties in the stability condition of Theorem5. In this section, we will cope with uncertainties of stability condition (10), such that robust stability condition is derived.Theorem 6.
System (4) is asymptotically stable if there exist matrices P1>0, S1>0, S2>0, Q1>0, Q2>0, R1>0, R2>0, M, L, and N, satisfying
(18)[Π~1+Π~2***Π~4TΠ~5**G~03×3Π~6*Π~702×3Π~8Π~9]<0,
where
(19)Π~1=[-X+W1+W2+(m-1)W3+mW4**0-S1*00-S2],Π~2=diag⁡{P1-1,I,I}×{Π2+Π2T}×diag⁡{P1-1,I,I},Π~4T=diag⁡{R1-1,R1-1,R2-1}×Π4T×diag⁡{P1-1,I,I},G~=[G0XG¯1G¯2m(G0X-X)mG¯1mG¯2m(G0X-X)mG¯1mG¯2],Π~5=[-R~1**0-R~1*00-R~2],Π~6=[000-X**0000-R~1*00000-R~2],Π~7=[0000E1-E1],Π~8=[000ε1DTε1DTε1DT000000],Π~9=[-ε1I*0-ε1I],X=P-1,R~1=R1-1,R~2=R2-1,W1=XS1X,W2=XS2X,W3=XQ1X,W4=XQ2X.Proof.
With Theorem5, the system 𝒰 in (4) is asymptotically stable, if there exist matrices P1, Si, Qi, and Ri, i=1,2, satisfying
(20)[Π⌣1+Π2+Π2T**Π4TΠ⌣5*G⌣03×3Π~6]<0,
where
(21)Π⌣1=[-P1+S1+S2+(m-1)Q1+mQ2000-S1000-S2],Π⌣5=[-R1**0-R1*00-R2],G⌣=[G0G1G2G0-IG1G2G0-IG1G2].
Substitution of the matricesG1 and G2 defined in (7)~(8) leads to
(22)[Π⌣1+Π2+Π2T**Π4TΠ⌣5*G⌢03×3Π~6]+θ1F(τ)θ2+θ2TFT(τ)θ1T<0,
where(23)G⌢=[G0G¯1G¯2m(G0-I)mG¯1mG¯2m(G0-I)mG¯1mG¯2],θ1=[000000DTDTDT]T,θ2=[0E-E000000].
By Lemma 3, we readily obtain
(24)[Π⌣1+Π2+Π2T***Π4TΠ⌣5**G⌢03×3Π~6*Π⌣702×3Π⌣8Π⌣9]<0,
where
(25)Π⌣7=[0000E-E],Π⌣8=[ε1DTε1DTε1DT000],Π⌣9=[-ε1I*0-ε1I].
A congruence transformation is applied to (24), by diag⁡{P1-1,I,I,R1-1,R1-1,R2-1,I,I,I}, we can attain LMI (18).Fact 1.
Definiteness of a matrix is invariant under congruent transformation by a full rank matrix. For instance, ifQ=QT∈ℝn×n, and V∈ℝn×n is of full rank, then Q>0→VQVT>0.Theorem 7.
System (4) is asymptotically stable if there exist real symmetric positive definite matrices P1>0, Si>0, Qi>0, Ri>0, i=1,2, and matrices L, M, and N, such that the following LMI holds:
(26)Π1+Γ1+Π3<0,
where
(27)Γ1=(e1-e2)T(-R1m-1)(e1-e2)+(e2-e3)T(-R1)(e2-e3)+(e1-e3)T(-R2m)(e1-e3)
and ei, i=1 to 3, are block entry matrices of appropriate dimensions; namely, e1=[I00], e2=[0I0], and e3=[00I].Proof.
Consider a nonsingular transformation matrixΛ. Congruent transformation of the LMI given in (10) with Λ yields the following equivalent criterion:
(28)Λ[Π1+Π2+Π2T+Π3Π4*Π5]ΛT=[Π1+Γ1+Π3Γ2*Γ3]<0,(29)Λ=[I00-Im-10-Im0I0Im-1-I000I0IIm000Im-1000000I000000Im],Γ2=[M1+R1m-1S1N1+R2mM2-R1m-1S2+R1N2M3S3-R1N3-R2m],Γ3=diag⁡{-R1m-1-R1-R2m}.
Using Schur complement, we can express the right-side LMI of (28) as follows:
(30)Π1+Γ1+Π3-Γ2Γ3-1Γ2T<0.
Since Γ3<0 holds, we can write the equivalent stability criterion as Π1+Γ1+Π3<0, if we let the slack variable matrices M, L, and N be as follows:
(31)M=[-R1m-1R1m-10],L=[0-R1R1],N=[-R2m0R2m].
Hence,Π1+Γ1+Π3<0 is equivalent to (10). In other words, Theorem 6 is equivalent to Theorem 5.Remark 8.
Theorem7 presents an equivalent condition for the solvability of the stability condition problem for sampled-data system with long time delay. We note that Π1 and Π3 include norm-bounded uncertain parameters, such that (26) cannot be solved directly. In order to solve this problem, Theorem 9 is derived as follows.Theorem 9.
System (4) is asymptotically stable if there exist real symmetric positive definite matrices P1>0, Si>0, Qi>0, Ri>0, i=1,2, such that the following LMI holds:
(32)[Π⌣1+Γ1**G⌢Π~6*Π⌣7Π⌣8Π⌣10]<0,
where
(33)Π⌣10=[-ε2I*0-ε2I].Proof.
The process of proof is similar to Theorem6, so it is omitted.Remark 10.
The equivalent stability criterion presented in Theorems7 and 9 has no free weighting matrices, and they have only the matrix variables that are associated in the Lyapunov-Krasovskii function. Therefore, as the total number of matrix variables involved with the proposed equivalent criterion in minimum, it has less offline computational burden compared to Theorem 5. To highlight this aspect, we compare the number of decision variables involved in both the stability criterion in Table 1.Table 1
Comparison of decision variables.
Revelent theorem
Number of decision variables
By [6]
18
n
2
+
3
n
By Theorem5
12.5
n
2
+
3.5
n
By Theorem7
3.5
n
2
+
3.5
n


## 3.1. Stability Condition

The stability condition presented in this section is based on system (4); we will proceed with the following theorem.Theorem 5.
System (4) is asymptotically stable if there exist matrices P1>0, S1>0, S2>0, Q1>0, Q2>0, R1>0, R2>0, M, L, and N, satisfying
(10)[Π1+Π2+Π2T+Π3Π4*Π5]<0,
where
(11)Π1=[G0TP1G0-P1+S1+S2+(m-1)Q1+mQ2G0TP1G1G0TP1G2*G1TP1G1-S1G1TP1G2**G2TP1G2-S2],(12)Π2=[M+NL-M-L-N],Π3=m[G0-IG1G2]T(R1+R2)[G0-IG1G2],Π4=[m-1MLmN],Π5=diag⁡{-R1-R1-R2}.Proof.
Letting(13)η(k)=x(k+1)-x(k),
then
(14)x(k+1)=x(k)+η(k),x(k)-x(k-m+1)=∑i=k-m+1k-1‍η(i),x(k)-x(k-m)=∑i=k-mk-1‍η(i).
Choosing a Lyapunov function candidate as
(15)V=V1+V2+V3+V4+V5+V6,V1=xT(k)P1x(k),V2=∑i=k-m+1k-1‍xT(i)S1x(i),V3=∑i=k-mk-1‍xT(i)S2x(i),V4=∑j=-m+1-1‍∑i=k+jk-1‍xT(i)Q1x(i),V5=∑j=-m-1‍∑i=k+jk-1‍xT(i)Q2x(i),V6=∑j=-m-1‍∑i=k+jk-1‍ηT(i)(R1+R2)η(i),
where P1=P1T>0, Si=SiT≥0, i=1,2, and Ri=RiT≥0, i=1,2, are undetermined.
DefiningΔV=V(k+1)-V(k), and then along the solution of (4), we have
(16)ΔV1=xT(k+1)P1x(k+1)-xT(k)P1x(k)ΔV2≤xT(k)S1x(k)-xT(k-m+1)S1x(k-m+1)ΔV2≤+∑j=k-m+1k-1‍xT(j)Q1x(j),ΔV3≤xT(k)S2x(k)-xT(k-m)S2x(k-m)ΔV3≤+∑j=k-mk-1‍xT(j)Q2x(j),ΔV4=(m-1)xT(k)Q1x(k)-∑j=k-m+1k-1‍xT(j)Q1x(j),ΔV5=mxT(k)Q2x(k)-∑j=k-mk-1‍xT(j)Q2x(j),ΔV6=mηT(k)(R1+R2)η(k)-∑l=k-mk-1‍ηT(l)(R1+R2)η(l)ΔV6=mηT(k)(R1+R2)η(k)-∑l=k-m+1k-1‍ηT(l)R1η(l)ΔV6=-∑l=k-mk-m‍ηT(l)R1η(l)-∑l=k-mk-1‍ηT(l)R2η(l).
Defining ξ(k)=[xT(k)xT(k-m+1)xT(k-m)]T, then we have
(17)ΔV≤[G0x(k)+G1x(k-m+1)+G2x(k-m)]TP1×[G0x(k)+G1x(k-m+1)+G2x(k-m)]-xT(k)P1x(k)+xT(k)S1x(k)-xT(k-m+1)S1x(k-m+1)+xT(k)S2x(k)-xT(k-m)S2x(k-m)+(m-1)xT(k)Q1x(k)+mxT(k)Q2x(k)+m[(G0-I)x(k)+G1x(k-m+1)+G2x(k-m)]T×(R1+R2)×[(G0-I)x(k)+G1x(k-m+1)+G2x(k-m)]-∑l=k-m+1k-1‍ηT(l)R1η(l)-∑l=k-mk-m‍ηT(l)R1η(l)-∑l=k-mk-1‍ηT(l)R2η(l)+2ξT(k)M[x(k)-x(k-m+1)-∑l=k-m+1k-1‍η(l)]+2ξT(k)L[x(k-m+1)-x(k-m)-∑l=k-mk-m‍η(l)]+2ξT(k)N[x(k)-x(k-m)-∑l=k-mk-1‍η(l)]≤ξT(k)[Π1+Π2+Π2T+Π3+(m-1)MR1-1MT000000000+LR1-1LT+mNR2-1NT]ξ(k)-∑l=k-m+1k-1‍[ξT(k)M+ηT(l)R1]×R1-1[MTξ(k)+R1η(l)]-∑l=k-mk-m‍[ξT(k)L+ηT(l)R1]×R1-1[LTξ(k)+R1η(l)]-∑l=k-mk-1‍[ξT(k)N+ηT(l)R2]×R2-1[NTξ(k)+R2η(l)].
SinceRi>0, i=1,2, the last three terms are all nonpositive. By the Schur complement [15], inequality (10) guarantees Π1+Π2+Π2T+Π3+(m-1)MR1-1MT+SR1-1ST+mNR2-1NT<0. From (17), we can attain ΔV<-ε∥x(k)∥2, for a sufficiently small ε, and x(k)≠0, and the asymptotic stability is established.


## 3.2.  Robust Stability Condition

There exist uncertainties in the stability condition of Theorem5. In this section, we will cope with uncertainties of stability condition (10), such that robust stability condition is derived.Theorem 6.
System (4) is asymptotically stable if there exist matrices P1>0, S1>0, S2>0, Q1>0, Q2>0, R1>0, R2>0, M, L, and N, satisfying
(18)[Π~1+Π~2***Π~4TΠ~5**G~03×3Π~6*Π~702×3Π~8Π~9]<0,
where
(19)Π~1=[-X+W1+W2+(m-1)W3+mW4**0-S1*00-S2],Π~2=diag⁡{P1-1,I,I}×{Π2+Π2T}×diag⁡{P1-1,I,I},Π~4T=diag⁡{R1-1,R1-1,R2-1}×Π4T×diag⁡{P1-1,I,I},G~=[G0XG¯1G¯2m(G0X-X)mG¯1mG¯2m(G0X-X)mG¯1mG¯2],Π~5=[-R~1**0-R~1*00-R~2],Π~6=[000-X**0000-R~1*00000-R~2],Π~7=[0000E1-E1],Π~8=[000ε1DTε1DTε1DT000000],Π~9=[-ε1I*0-ε1I],X=P-1,R~1=R1-1,R~2=R2-1,W1=XS1X,W2=XS2X,W3=XQ1X,W4=XQ2X.Proof.
With Theorem5, the system 𝒰 in (4) is asymptotically stable, if there exist matrices P1, Si, Qi, and Ri, i=1,2, satisfying
(20)[Π⌣1+Π2+Π2T**Π4TΠ⌣5*G⌣03×3Π~6]<0,
where
(21)Π⌣1=[-P1+S1+S2+(m-1)Q1+mQ2000-S1000-S2],Π⌣5=[-R1**0-R1*00-R2],G⌣=[G0G1G2G0-IG1G2G0-IG1G2].
Substitution of the matricesG1 and G2 defined in (7)~(8) leads to
(22)[Π⌣1+Π2+Π2T**Π4TΠ⌣5*G⌢03×3Π~6]+θ1F(τ)θ2+θ2TFT(τ)θ1T<0,
where(23)G⌢=[G0G¯1G¯2m(G0-I)mG¯1mG¯2m(G0-I)mG¯1mG¯2],θ1=[000000DTDTDT]T,θ2=[0E-E000000].
By Lemma 3, we readily obtain
(24)[Π⌣1+Π2+Π2T***Π4TΠ⌣5**G⌢03×3Π~6*Π⌣702×3Π⌣8Π⌣9]<0,
where
(25)Π⌣7=[0000E-E],Π⌣8=[ε1DTε1DTε1DT000],Π⌣9=[-ε1I*0-ε1I].
A congruence transformation is applied to (24), by diag⁡{P1-1,I,I,R1-1,R1-1,R2-1,I,I,I}, we can attain LMI (18).Fact 1.
Definiteness of a matrix is invariant under congruent transformation by a full rank matrix. For instance, ifQ=QT∈ℝn×n, and V∈ℝn×n is of full rank, then Q>0→VQVT>0.Theorem 7.
System (4) is asymptotically stable if there exist real symmetric positive definite matrices P1>0, Si>0, Qi>0, Ri>0, i=1,2, and matrices L, M, and N, such that the following LMI holds:
(26)Π1+Γ1+Π3<0,
where
(27)Γ1=(e1-e2)T(-R1m-1)(e1-e2)+(e2-e3)T(-R1)(e2-e3)+(e1-e3)T(-R2m)(e1-e3)
and ei, i=1 to 3, are block entry matrices of appropriate dimensions; namely, e1=[I00], e2=[0I0], and e3=[00I].Proof.
Consider a nonsingular transformation matrixΛ. Congruent transformation of the LMI given in (10) with Λ yields the following equivalent criterion:
(28)Λ[Π1+Π2+Π2T+Π3Π4*Π5]ΛT=[Π1+Γ1+Π3Γ2*Γ3]<0,(29)Λ=[I00-Im-10-Im0I0Im-1-I000I0IIm000Im-1000000I000000Im],Γ2=[M1+R1m-1S1N1+R2mM2-R1m-1S2+R1N2M3S3-R1N3-R2m],Γ3=diag⁡{-R1m-1-R1-R2m}.
Using Schur complement, we can express the right-side LMI of (28) as follows:
(30)Π1+Γ1+Π3-Γ2Γ3-1Γ2T<0.
Since Γ3<0 holds, we can write the equivalent stability criterion as Π1+Γ1+Π3<0, if we let the slack variable matrices M, L, and N be as follows:
(31)M=[-R1m-1R1m-10],L=[0-R1R1],N=[-R2m0R2m].
Hence,Π1+Γ1+Π3<0 is equivalent to (10). In other words, Theorem 6 is equivalent to Theorem 5.Remark 8.
Theorem7 presents an equivalent condition for the solvability of the stability condition problem for sampled-data system with long time delay. We note that Π1 and Π3 include norm-bounded uncertain parameters, such that (26) cannot be solved directly. In order to solve this problem, Theorem 9 is derived as follows.Theorem 9.
System (4) is asymptotically stable if there exist real symmetric positive definite matrices P1>0, Si>0, Qi>0, Ri>0, i=1,2, such that the following LMI holds:
(32)[Π⌣1+Γ1**G⌢Π~6*Π⌣7Π⌣8Π⌣10]<0,
where
(33)Π⌣10=[-ε2I*0-ε2I].Proof.
The process of proof is similar to Theorem6, so it is omitted.Remark 10.
The equivalent stability criterion presented in Theorems7 and 9 has no free weighting matrices, and they have only the matrix variables that are associated in the Lyapunov-Krasovskii function. Therefore, as the total number of matrix variables involved with the proposed equivalent criterion in minimum, it has less offline computational burden compared to Theorem 5. To highlight this aspect, we compare the number of decision variables involved in both the stability criterion in Table 1.Table 1
Comparison of decision variables.
Revelent theorem
Number of decision variables
By [6]
18
n
2
+
3
n
By Theorem5
12.5
n
2
+
3.5
n
By Theorem7
3.5
n
2
+
3.5
n


## 4. Numerical Example

Considering the system𝒮 in (1)
(34)A0=[-2.65.0-3.6-2.6],A1=[-10.510.1].
Letting  h=0.1, m=2, and discretizing system (1), new parameter of discrete-time system is attained:
(35)G0=[0.03450.4726-0.34030.0345],G¯1=[2.923-0.302-0.150-0.083],G¯2=[-2.9050.3170.5020.047],D=[0.345000.345],E=[-0.0200.015-1.0500.033].
Let
(36)x(0)=[0.030.02]T.Remark 11.
The problem of relevant norm-bounded uncertain parameter is further solved in Theorem9. Compared with [16], the number of decision variables is 3.5n2+3.5n, which is less than 20.5n2+4.5n. Moreover, state response curve of Theorem 9’s method has smaller overshoot and quicker convergence speed, which is shown in Figure 1.State response curve ofx1,x2.
(a)
x
1
(b)
x
2


## 5. Conclusions

In this correspondence paper, we have presented an equivalent stability criterion with less number of LMI variables for a robust stability criterion reported in Theorem6. By employing congruent transformation with nonsingular matrix, the free weighting matrices in the existing criterion are eliminated without sacrificing the conservation of stability criterion; this, in turn, yields an equivalent criterion that is mathematically less complex and computationally less expensive.



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*Source: 580768-2014-04-07.xml*