An Improvement of Delay-Derivative-Dependent Asymptotic Stability
Criterion for Interconnected Switched Systems with Time-Varying Delays
Ming-Sheng Yang
Department of Electrical Engineering
Chienkuo Technology University
Changhua, 500, Taiwan, R.O.C.
E-mail:
[email protected]
ABSTRACT
This paper discusses the problem of delay-dependent asymptotic stability for interconnected switched neutral-type systems with time-varying delays. By applying weighting-delay approach, introducing both singular model transformation technique and Finsler’s lemma, and constructing an augmented Lyapunov-Krasovskii functional combined with slack matrices, an improved delay-derivative-dependent stability criterion is derived to guarantee the asymptotic stability of above systems. The obtained criterion is formulated in terms of matrix inequalities, which can be efficiently solved via standard numerical software. Two numerical examples are included to show that the proposed method is effective and can provide less conservative results.
Keywords: Interconnected switched neutral-type systems, time-varying delays, weighting-delay approach,
singular model transformation, delay-derivative-dependent stability criterion.
1. Introduction
It is well known that a wide class of physical systems in power systems, chemical procedure control systems, navigation systems, automobile speed change system, etc. may be appropriately described by the switched model. Switched systems are a special class of hybrid dynamical systems, which consist of a family of subsystems and a switching law specifying the switching between the subsystems. Recently, there has been increasing interest in the stability problem of switched systems with time delay due to their significance both in theory and applications. To the best of our knowledge, it seems that few people have studied the asymptotic stability problem for continuous-time interconnected switched neutral-type systems with time-varying delays. This has motivated our research.
In this paper, we will give preliminary knowledge for our main result. First of all, consider the following interconnected switched neutral-type system with time-varying delays
)) ( ( )) ( ( ) ( { ) ( ) ( 1 t d t x C t d t x A t x A t t x i k i i k i i k i r k k i =
∑
+ − + − = α [ () ( ())]} 1∑
≠ = − + + N i j j j k ij j k ijx t B x t d t B (1a), 1 ) ( 1
∑
= = r k k t α xi(t)=ϕi(t), t∈[−h, 0] (1b) = otherwise , 0 mode th by the described is system switched the when , 1 ) (t k k α (1c) where ni i t Rx ()∈ is the state vector of the ith subsystem, Aik,Aik, , k ij
B Bijk,Cikare known constant matrices
with appropriate dimensions, i=1 ,2, ,N, k=1,2,,r. The delay d(t)is a time-varying continuous function satisfying 0≤d(t)≤h andd(t)≤µ.ϕi(t) is a given continuous vector-valued initial function.
The following notations will be used throughout this paper. The notation F>G(F≥G) means that the matrix F−G is positive definite (positive semi-definite) for two symmetric matrices F, G. Ii is an identity matrix of appropriate dimensions.
Assumption 1[1]: All the eigenvalues of matricesCik, i=1 ,2, ,N, are inside the unit circle.
Lemma 1[2]: For any real vectors κ1,κ2 and any matrixM>0with appropriate dimensions, it follows that
2 T 2 1 1 T 1 2 T 1 2κ κ ≤κ M−κ +κ Mκ (2)
Lemma 2[3]: For any symmetric positive definite matrixP and scalarsλ>0, δ >1, the following inequality holds ds s e P s e ds s e P s e( ) ( ) ( ) ( ) 0 T 0 T
∫
∫
≤− − λδ λ T 0 ( ) ) ( ) 1 ( s d s e∫
− − λ λ δ ) ) ( ( 0e s ds P∫
λ (3)Lemma 3[4]: For any symmetric positive definite matrix Q and scalars 0≤b1<b2, the following inequality
holds Q b t x b t x b b d x Q x b t b t T 2 1 1 2 T )] ( ) ( [ 1 ) ( ) ( 1 2 − − − − − ≤ −
∫
− − θ θ θ [x(t−b1)−x(t−b2)] (4)Lemma 4(Finsler’s lemma)[5]: Consider a vector ζ ∈Rn, a symmetric positive definite matrix S∈Rn×n
and a matrixD∈Rm×n, such that rank(D)<n. The following conditions are equivalent:
(i)ζ ST ζ <0, ∀ such that ζ Dζ =0, ζ ≠0 (5a)
(ii)(D⊥)TSD⊥<0 (5b)
2. Main Result
In the following theorem, an improved delay-derivative-dependent criterion for asymptotic stability of interconnected switched neutral-type system (1) is proposed in terms of matrix inequalities.
Theorem 1: Under Assumption 1, the interconnected switched neutral-type system (1) is asymptotically stable
for i=1 ,2, ,N and k=1,2,,r, if there exist positive definite matrices J11i,J22i,J33i,J44i,Y1i, Y2i, ,
11i
Z Z22i, Z33i, Z44i, Z55i, Z66i, Z77, Pi,Qi, Ri,W1i,W2i,W3i,W4i, , ~ , ˆ , , , , 2 1i X i Mij Mij Mij Mij X real matrices , i
H J12i, J13i, J14i, J23i, J24i, J34i, Z12i,Z13i, Z14i, Z15i, Z16i, Z17i, Z23i, Z24i, Z25i, Z26i,Z27i,Z34i, Z35i, ,
36i
Z Z37i, Z45i,Z46i,Z47i, Z56i, Z57i, Z67i,and scalars 0<ρ<1,δi>1 such that the following conditions
hold 0 2 T 1 > i i i i Y H H Y (6a)
0 44 T 34 T 24 T 14 34 33 T 23 T 13 24 23 22 T 12 14 13 12 11 > i i i i i i i i i i i i i i i i J J J J J J J J J J J J J J J J (6b) 0 77 T 67 T 57 T 47 T 37 T 27 T 17 67 66 T 56 T 46 T 36 T 26 T 16 57 56 55 T 45 T 35 T 25 T 15 47 46 45 44 T 34 T 24 T 14 37 36 35 34 33 T 23 T 13 27 26 25 24 23 22 T 12 17 16 15 14 13 12 11 > = i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z (6c) 0 ) 1 ( 1 77 2i+ − X i−Z i > X µ (6d) 0 ) (Di⊥ TΠiD⊥i < (6e) where = ⊥ i i i i i i i i i i i I I I I I I I I I I D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (7a) Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π Π= i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 77 T 67 T 57 T 47 T 37 T 27 T 17 67 66 T 56 T 46 T 36 T 26 T 16 57 56 55 T 45 T 35 T 25 T 15 47 46 45 44 T 34 T 24 T 14 37 36 35 34 33 T 23 T 13 27 26 25 24 23 22 T 12 17 16 15 14 13 12 11 (7b) i i i i i i k i k i i i X X Z W W h R A A R T 2 1 66 1 2 11 [ (1 ) ] 1 ) ( − + − − + + + = Π µ ρ +J11i+W4i+Y1i+ρhZ11i { [ ( ) ( ) ] 1 ˆ 1} 1 T T − − ≠ = + + + +
∑
N ji ji i j j i k ij ij k ij k ij ij k ij i B M B B M B R M M R (7c) i i k i i i i i i k i i i i P R A Q J12 hZ12 13 RA J13 hZ13 T 12= − +( ) + +ρ ,Π = + +ρ Π (7d) ] ) 1 ( [ 1 77 1 2 14 14i i X i Xi Z i h Z h + + − − = Π µ ρ ρ (7e) i i i i i i k i i i RC hZ15 16 hZ16 17 J14 Z17 15 = + ,Π = ,Π = + Π ρ ρ (7f)∑
≠ = + − + + + = Π N i j j k ij ij k ij i i i i i i i W Y J hZ Q Q B M B 1 T 22 22 2 3 22 ρ 2 [ ˆ ( ) i k ij ij k ijM B Q B ~ ( )T] + (7g) i i i i k i i i QA J23 hZ23 24 hZ24 23= + +ρ ,Π =ρ Π ,Π25i=QiCik+ρhZ25i,Π26i=ρhZ26i,Π27i=J24i+Z27i (7h)∑
≠ = − −+ + + − − + = Π N i j j ji ji i i i i i J hZ W Y M M 1 1 1 1 1 33 33 33 ) ~ ( ) ( ) 1 ( µ ρ (7i)i i i i i hZ34 35 H hZ35 34 =ρ ,Π =−(1−µ) +ρ Π ,Π36i =ρhZ36i, Π37i=J34i+Z37i (7j) i i i i i i X h Z X X h 2 1 77 2 44 [ (1 ) ] 1 µ δ ρ + − − − − = Π −(1−ρµ)W2i+ρhZ44i (7k) , 45 45i=ρhZ i Π 46i 46i iX2i, h Z h δ ρ + = Π Π47i=Z47i (7l) ) )( 1 ( 3 2 55 55i= hZ i− − Wi+Yi Π ρ µ ,Π56i=ρhZ56i,Π57i=Z57i (7m) i i i i i i i i i i X h J Z X h Z h 66 2 67 67 77 44 2 66 ) 1 ( , , ρ δ δ ρ − Π = Π = − − = Π (7n)
Proof: Based on singular model transformation [6], system (1) can be written as
) ( ) (t y t xi = i (8a) )) ( ( ) ( { ) ( 0 1 t d t y C t y t i k i i r k k − + − =
∑
= α A x(t) A xi(t d(t)) k i i k i + − + [ () ( ( ))]} 1∑
≠ = − + + N i j j j k ij j k ijx t B x t d t B (8b)By means of the idea of [7] and [8], we use the following Lyapunov-Krasovskii functional to derive the stability criterion ) ( ) ( ) ( ) ( ) ( [ ) ( 4 5 1 3 2 1 t V t V t V t V t V t V i i N i i i i + + + + =
∑
= )] ( ) ( ) ( ) ( ) ( 7 8 9 10 6 t V t V t V t V t Vi + i + i + i + i + (9) where = ) ( ) ( 0 0 0 0 )] ( ) ( [ ) ( T T 1 t y t x Q R P I t y t x t V i i i i i i i i i (10a) ds s x W s x t V i i t t d t i i() ( ) 1 ( ) ) ( T 2 =∫
− (10b) ds s x W s x t V i i t t d t i i() ( ) 2 ( ) ) ( T 3 =∫
−ρ (10c) ds s y W s y t V i i t t d t i i() ( ) 3 ( ) ) ( T 4 =∫
− (10d) ds s x W s x t V t i i h t i i() ( ) 4 ( ) T 5 =∫
− (10e) ds s y s x Y H H Y s y s x t V i i i i i t t d t i i i =∫
− ( ) ) ( ) ( ) ( ) ( 2i T 1 T ) ( 6 (10f) θ θ y s X y s dsd t V i i t d t t i i() ( ) 1 ( ) 0 ) ( T 7 =∫
−∫
+ (10g) θ δ θ y s X y s dsd t V i i h t t i i i() ( ) 2 ( ) 0 T 8 =∫ ∫
− + (10h) θ θ θ θ θ ρ θ e s Ze s dsd t V i i t d i i() ( , ) ( , ) 0 ( ) T 9 =∫ ∫
− (10i)T 0 ) ( 10 ) ( )) ( ( ) ( ) ( ) (
∫
∫
− = − t d i i i i i ds s y d x y x t V θ θ ρ θ θ θ θ θ i i i i i i i i i i i i i i i i J J J J J J J J J J J J J J J J 44 T 34 T 24 T 14 34 33 T 23 T 13 24 23 22 T 12 14 13 12 11 −∫
− y s ds d x y x d i i i i θ θ ρ θ θ θ θ θ ) ( ( ) )) ( ( ) ( ) ( θ d (10j) whereei(θ,s)=[xiT(θ)yiT(θ)xiT(θ−d(θ))xiT(θ−ρd(θ)) T T T T )] ( ) ( )) ( ( d x h y syi θ− θ i θ− i and matrixZiis defined in
(6c).
Taking the time derivative of V(t)along the trajectories of system (1) and noting that 0≤d(t)≤h
andd(t)≤µ,it yields ) ( ) ( ) ( ) ( ) ( [ ) ( 4 5 1 3 2 1 t V t V t V t V t V t V i i N i i i i =
∑
+ + + + = )] ( ) ( ) ( ) ( ) ( 7 8 9 10 6 t V t V t V t V t Vi + i +i + i + i + (11) where = i i i i i i Q R P t y t x t V 0 )] ( ) ( [ 2 ) ( T T 1 − + + − + + − + − ×∑
∑
≠ = = N i j j j k ij j k ij i k i i k i i k i i r k k i t d t x B t x B t d t x A t x A t d t y C t y t t y 1 1 } ))] ( ( ) ( [ )) ( ( ) ( )) ( ( ) ( { ) ( ) ( α (12a) ) ( ) ( ) ( T 1 2 t x tW x t Vi ≤ i i i −(1−µ)xiT(t−d(t))W1ixi(t−d(t)) (12b) ) ( ) ( ) ( 2 T 3 t x tW x t Vi ≤ i i i (1 ) ( ( )) 2 ( ()) T t d t x W t d t xi ρ i i ρ µ ρ − − − − (12c) ) ( ) ( ) ( T 3 4 t y t W y t Vi ≤ i i i −(1−µ)yiT(t−d(t))W3iyi(t−d(t)) (12d) ) ( ) ( ) ( ) ( ) ( 4 T 4 T 5 t x t W x t x t h W x t h Vi ≤ i i i − i − i i − (12e) ≤ ) ( ) ( ) ( ) ( ) ( 2i T 1 T 6 y t t x Y H H Y t y t x t V i i i i i i i i − − − − − − )) ( ( )) ( ( )) ( ( )) ( ( ) 1 ( 2i T 1 T t d t y t d t x Y H H Y t d t y t d t x i i i i i i i µ (12f) ) ( ) ( ) ( 1 T 7 t hy t X y t Vi ≤ i i i y s Xiyi s ds t t d t i( ) ( ) ) 1 ( 1 ) ( T∫
− − − µ (12g) ) ( ) ( ) ( 2 T 8 t h y t X y t Vi ≤ δi i i i y s X iyi s ds t t d t () i i( ) 2 ( ) T∫
− − ρ δ y s X iyi s ds t d t h t i i( ) 2 ( ) ) ( T∫
−− − ρ δ (12h) T 9 ) ( )) ( ( )) ( ( )) ( ( ) ( ) ( ) ( ) ( − − −− = h t x t d t y t d t x t d t x t y t x t d t V i i i i i i i ρ ρ − − −− ) ( )) ( ( )) ( ( )) ( ( ) ( ) ( 66 T 56 T 46 T 36 T 26 T 16 56 55 T 45 T 35 T 25 T 15 46 45 44 T 34 T 24 T 14 36 35 34 33 T 23 T 13 26 25 24 23 22 T 12 16 15 14 13 12 11 h t x t d t y t d t x t d t x t y t x Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i iρ
ds s y Z t y Z t xi i i i i t t d t 2[ () () 27] ( ) T 17 T ) ( + +∫
−ρ xi t d t Z i xi t d t Z i yi sds t t d t 2[ ( ()) ( ()) 47] ( ) T 7 3 T ) ( ρ ρ − + − +∫
− ds s y Z h t x Z t d t y i i i t t d t 2[ i ( ()) i ( ) 67] ( ) T ) ( 57 T − + − +∫
−ρ y s Z iyi s ds t t d t () i( ) 77 ( ) T∫
− + ρ (12i)T ) ( 10 ) ( )) ( ( ) ( ) ( ) ( − =
∫
− y s ds t d t x t y t x t V t t d t i i i i i ρ i i i i i i i i i i i i i i i i J J J J J J J J J J J J J J J J 44 T 34 T 24 T 14 34 33 T 23 T 13 24 23 22 T 12 14 13 12 11 −∫
− y s ds t d t x t y t x t t d t i i i i ) ( ( ) )) ( ( ) ( ) ( ρ (12j)Applying Lemma 1, we have
∑∑
= ≠ = N i N i j j j k ij i i t RB x t x 1 1 T ) ( ) ( 2∑∑
= ≠ = − + ≤ N i N i j j j ij j i i k ij ij k ij i i t RB M B Rx t x t M x t x 1 1 1 T T T )] ( ) ( ) ( ) ( ) ( [ ) ( ] ) ( )[ ( 1 1 1 T T t x M R B M B R t x i N i N i j j ji i k ij ij k ij i i∑∑
= ≠ = − + = (13a)∑∑
= ≠ = N i N i j j j k ij i i t QB x t y 1 1 T ) ( ) ( 2∑∑
= ≠ = − + ≤ N i N i j j j ij j i i k ij ij k ij i i t QB M B Qy t x t M x t y 1 1 1 T T T() ˆ ( ) () () ˆ ()] [∑∑
= ≠ = = N i N i j j i i k ij ij k ij i i t QB M B Qy t y 1 1 T T ) ( ) ( ˆ ) ( [ +xiT(t)Mˆ−ji1xi(t)] (13b)∑∑
= ≠ = − N i N i j j j k ij i i t RB x t d t x 1 1 T )) ( ( ) ( 2∑∑
= ≠ = ≤ N i N i j j i i k ij ij k ij i i t RB M B Rx t x 1 1 T T ) ( ) ( ) ( [ xT(t d(t))M 1x (t d(t))] j ij j − − + −∑∑
= ≠ = = N i N i j j i i k ij ij k ij i i t RB M B Rx t x 1 1 T T ) ( ) ( ) ( [ xT(t d(t))M 1x(t d(t))] i ji i − − + − (13c)∑∑
= ≠ = − N i N i j j j k ij i i t QB x t d t y 1 1 T )) ( ( ) ( 2∑∑
= ≠ = ≤ N i N i j j i i k ij ij k ij i i t QB M B Qy t y 1 1 T T ) ( ) ( ~ ) ( [ xT(t d(t))M 1x (t d(t))] j ij j − − + −∑∑
= ≠ = = N i N i j j i i k ij ij k ij i i t QB M B Qy t y 1 1 T T ) ( ) ( ~ ) ( [ xT(t d(t))M~ 1x(t d(t))] i ji i − − + − (13d)According to Lemma 2 and using the idea of [9], we get
ds s y X s y i i t t d t () i i ( ) 2 ( ) T
∫
− − ρ δ − − ≤∫
∫
− − y s ds X y sds h i t t d t i i t t d t i 1) ( ) ( ) ( ) ( 2 T ) ( ρ ρ ρ δ ds s y X s y i i t t d t () i ( ) 2 ( ) T∫
− − ρ (14)From (6d), (12g), (12i), (14) and Lemma 3, we have
ds s y Z X X s y i i i i t t d t () i ( )[ 2 (1 ) 1 77] ( ) T + − − −
∫
−ρ µ[
() ( ())]
[ (1 ) ] 1 77 1 2 T i i i i it x t d t X X Z x h − − + − − − ≤ ρ µ ρ[
xi(t)−xi(t−ρd(t))]
(15a)∫
−− − () T 2 ) ( ) ( t d t h t iyi s X iyi sds ρ δ[
i i]
i i X h t x t d t x h 2 T ) ( ) ) ( ( − − − − ≤ δ ρ[
xi(t−ρd(t))−xi(t−h)]
(15b) From(11)−(15),we obtain∑∑
= = Π ≤ N i i i i r k k t t t t V 1 T 1 ) ( ) ( ) ( ) ( α ω ω (16) where ωi(t)=[xiT(t) yiT(t) xiT(t−d(t)) xTi(t−ρd(t)) − −∫
− t t d t i i i t d t x t h y sds y ) ( T T T T ] ) ) ( ( ) ( )) ( ( ρ and matrix i Π is defined in (7b).Based on Leibniz-Newton formula, we get
∫
− = − − − t t d t i i i t x t d t y sds x ) ( ( ) 0 )) ( ( ) ( ρ ρ (17) This means 0 ) (t = Diωi (18) whereDi=[
Ii 0 0 −Ii 0 0 −Ii]
.From Lemma 4, it is seen that ωiT(t)Πiωi(t)<0is equivalent to inequality (6e). Obviously, if inequality (6e)
holds, then V(t)<0,which ensures that system (8) is asymptotically stable [1]. It means that system (1) is asymptotically stable, too. The proof is completed.
3. Numerical Examples
In this section, two examples are given to show the benefits of our result.
Example 1: Consider the following interconnected switched time-varying-delay system composed of two
individual switched systems:
Switched system 1 (k = 1): )) ( ( 3 . 0 1 . 0 4 . 0 5 . 0 ) ( 3 . 3 0 0 5 . 5 ) ( 1 1 1 t x t x t d t x − − − + − − = ( ( )) 0 1 . 0 0 5 . 0 ) ( 2 . 0 5 . 0 1 2 . 0 2 2 t x t d t x − + + )) ( ( 0 1 . 0 0 1 ) ( 0 2 . 0 1 . 0 0 3 3 t x t d t x − − + − − + )) ( ( 1 5 . 0 0 7 . 0 ) ( 3 . 6 0 0 3 . 8 ) ( 2 2 2 t x t x t d t x − − − + − − = ( ()) 0 3 . 0 2 . 0 1 . 1 ) ( 7 . 0 1 . 0 3 . 0 2 . 0 1 1 t x t d t x − + + )) ( ( 1 . 0 7 . 0 5 . 0 1 ) ( 2 . 0 3 . 0 1 . 0 1 . 1 3 3 t x t d t x − − + + ( ()) 3 5 . 0 1 1 ) ( 2 . 7 0 0 2 . 9 ) ( 3 3 3 t x t x t d t x − − − + − − = ( ()) 1 2 . 0 0 1 . 0 ) ( 0 1 4 . 0 0 2 2 t x t d t x − + + )) ( ( 1 1 1 0 ) ( 1 . 0 0 5 . 0 1 . 0 1 1 t x t d t x − + + (19a) Switched system 2 (k = 2): )) ( ( 1 . 0 5 . 0 0 1 . 0 ) ( 5 . 3 0 0 5 . 2 ) ( 1 1 1 t x t x t d t x =− − +− − − ( ()) 1 . 0 2 . 0 0 0 ) ( 0 1 . 0 2 . 0 1 . 0 2 2 t x t d t x + − + )) ( ( 0 2 . 0 1 . 0 0 ) ( 1 . 0 0 0 1 . 0 3 3 t x t d t x + − + )) ( ( 5 . 0 0 0 5 . 0 ) ( 5 0 0 6 . 3 ) ( 2 2 2 t x t x t d t x =− − +− − − ( ( )) 0 1 . 0 5 . 0 0 ) ( 0 5 . 0 1 . 0 0 1 1 t x t d t x − + + )) ( ( 0 1 . 0 1 . 0 0 ) ( 2 . 0 0 0 2 . 0 3 3 t x t d t x + − + )) ( ( 5 . 0 0 0 5 . 0 ) ( 6 . 2 0 0 7 ) ( 3 3 3 t x t x t d t x =− − +− − − ( ()) 0 0 1 . 0 0 ) ( 1 . 0 0 0 1 . 0 2 2 t x t d t x − + + )) ( ( 2 . 0 0 3 . 0 0 ) ( 0 2 . 0 0 0 1 1 t x t d t x − + + (19b)
Our purpose in example 1 is to find the maximum allowed delay h ofd(t)satisfying d(t)≤µ,such that the switching system (19) is asymptotically stable. A comparison between our Theorem 1 and the method of [10] is shown in Table 1, which also displays the maximum allowed delay h and its time derivative µfor guaranteeing the asymptotic stability of system (19). Obviously, it can be seen that the weighting-delay-dependent stability criterion in this paper is less conservative than one given by [10].
Table 1. Allowable delay bound h for differentμ µ h ([10]) h (Our Theorem 1) 0.5 Fail 6.5631 1.0 Fail 5.7382 1.5 Fail 4.6297 2.0 Fail 3.9156 2.5 Fail 2.8353
Example 2: Consider the following switched systems with time-varying delay
Switched system 1: )) ( ( 5 . 0 1 . 0 2 . 0 1 . 0 ) ( 5 . 1 0 0 5 . 5 ) (t xt xt d t x =− − +− − (20a) Switched system 2: )) ( ( 1 . 0 5 . 0 5 . 0 1 . 0 ) ( 7 . 7 0 0 2 . 2 ) (t xt xt d t x =− − + − (20b)
Our purpose in example 2 is to find the maximum allowed delay h ofd(t)satisfyingd(t)≤µ,such that the switching system (20) is asymptotically stable. A comparison between our Theorem 1 and the methods of [11], [12] and [13] is shown in Table 2, which also displays the maximum allowed delay h and its time derivative
µ
for guaranteeing the asymptotic stability of system (20). It is clear that our new method produces better results than those in [11], [12] and [13].Table 2. Allowable delay bound h for differentμ
µ h ([11]) h ([12]) h ([13]) h (Our Theorem 1) 0.1 1.3519 2.5381 3.3215 9.3129 0.3 0.6287 1.9236 2.6738 8.9153 0.7 0.4093 1.0153 1.3596 7.5816 0.9 0.3182 0.6928 0.9361 6.6187 1.1 0.1016 0.3527 0.5329 5.8652 4. Conclusion
A class of interconnected switched neutral-type system with time-varying delays has been investigated in this paper. By means of an augmented Lyapunov-Krasovskii functional form combined with slack matrices, singular model transformation technique, Finsler’s lemma and weighting-delay approach, an improved delay-derivative-dependent stability criterion is derived in terms of matrix inequalities. Two numerical examples are given to show the effectiveness and benefits of the proposed criterion.
References
[1] Kolmanovskii, V., & Myshkis, A. (1992). Applied Theory of Functional Differential Equations. Boston, MA: Kluwer.
[2] Cao, Y. Y., Sun, Y. X., & Cheng, C. (1998). Delay-dependent robust stabilization of uncertain systems with multiple state delays. IEEE Trans. Automat. Control, 43(11), 1608-1612.
[3] Kwon, O. M., & Park, J. H. (2004). On improved delay-dependent robust control for uncertain time-delay systems. IEEE Trans. Automat. Control, 49(11), 1991-1995.
[4] Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of Time-Delay Systems. Boston, MA: Birkhauser.
[5] De Oliveira, M. C., & Skelton, R. E. (2001). Stability tests for constrained linear systems. in Perspectives
in Robust Control, S. O. Reza Moheimani (Ed.), series Lecture Notes in Control and Information Sciences
(pp. 241-257). New York: Springer-Verlag.
[6] Fridman, E., & Shaked, U. (2002). A descriptor system approach to H∞ control of linear time-delay systems. IEEE Trans. Automat. Control, 47(2), 253-270.
[7] Zhu, X. L., & Yang, G. H. (2008). Jensen integral inequality approach to stability analysis of continuous-time systems with time-varying delay. IET Control Theory Appl., 2(6), 524-534.
[8] Yang, Z., & Yang, Y. P. (2010). New delay-dependent stability analysis and synthesis of T-S fuzzy systems with time-varying delay. Int. J. Robust Nonlinear Control, 20(3), 313-322.
[9] Zhu, X. L., & Yang, G. H. (2010). New results of stability analysis for systems with time-varying delay.
Int. J. Robust Nonlinear Control, 20(5), 596-606.
[10] Chiou, J. S. (2006). Stability analysis for a class of switched large-scale time-delay systems via time-switched method. IEE Proc.-Control Theory Appl., 153(6), 684-688.
[11] Sun, X. M., Wang, W., Liu, G. P., & Zhao, J. (2008). Stability analysis for linear switched systems with time-varying delay. IEEE Trans. Syst., Man, Cybern. B, Cybern., 38(2), 528-533.
[12] Sun, Y. G., Wang, L., & Xie, G. (2009). Exponential stability of switched systems with interval time-varying delay. IET Control Theory Appl., 3(8), 1033-1040.
[13] Tissir, E. H. (2011). Exponential stability of switched linear systems with mixed time delays. ICIC
Express Letters, 5(7), 2101-2109.
![Table 1. Allowable delay bound h for different μ µ h ([10]) h (Our Theorem 1) 0.5 Fail 6.5631 1.0 Fail 5.7382 1.5 Fail 4.6297 2.0 Fail 3.9156 2.5 Fail 2.8353](https://thumb-ap.123doks.com/thumbv2/9libinfo/8964736.278015/9.892.275.611.131.319/table-allowable-delay-bound-different-theorem-fail-fail.webp)
