Time Delay Systems Part 6 ppt

Conclusion In this paper, we have studied the exponential stability of uncertain switched system with time varying delay and nonlinear perturbations.. On stability of linear and weakly n

Applying Lemma 2.1 and from (2) and (3), we get

2x T(t)ΔA T

i(t)P i x(t) ≤ε−14i x T(t)H 4i T H 4i x(t) +ε 4i x T(t)P i E 4i T E 4i P i x(t),

2x T(t−h(t))ΔB T

i(t)P i x(t) ≤ε−15i x T(t−h(t))H 5i T H 5i x(t−h(t)) +ε 5i x T(t)P i E 5i T E 5i P i x(t)

Next, by taking derivative of V 2,i(x t), V 3,i(x t) and V 4,i(x t), respectively, along the system trajectories yields

˙

V 2,i(x t) ≤x T(t)Q i x(t) − (1−μ)e −2βh(t) x T(t−h(t))Q i x(t−h(t)) −2βV2,i(x t),

˙

V 3,i(x t) ≤h M x T(t)R i x(t) −t

t −h(t) e

2β (s−t) x T(s)R i x(s)ds−2βV3,i(x t),

˙

V 4,i(x t) ≤h M



x(t)

x(t−h(t))

T

S 11,i S 12,i

S T 12,i S 22,i

 

x(t)

x(t−h(t))



−t

t −h(t) e

2β (s−t)

x(s)

x(s−h(s))

T

S 11,i S 12,i

S 12,i T S 22,i

 

x(s)

x(s−h(s))



ds

−2βV4,i(x t)

Then, the derivative of V i(x t)along any trajectory of solution of (1) is estimated by

˙

V i(x t) ≤ ∑N

i=1λ i(t)



x(t)

x(t−h(t))

T

Θi



x(t)

x(t−h(t))



−2βV2,i(x t)

−t

t −h(t) e

2β (s−t) x T(s)R i x(s)ds−2βV3,i(x t)

−t

t −h(t) e

2β (s−t)

x(s)

x(s−h(s))

T

S 11,i S 12,i

S T 12,i S 22,i

 

x(s)

x(s−h(s))



ds

For i∈S u, it follows from (40) that

˙

V i(x t) ≤ ∑N

i=1λ i(t)



x(t)

x(t−h(t))

T

Θi



x(t)

x(t−h(t))



Similar to Theorem 3.1, from (33) and (41), we get

V i(x t) ≤∑N

i=1λ i(t) V i(x t0) e i (t−t0 ), t≥t0 (42)

whereξ i= 2maxi {λ M(Θi)}

min

i {λ m(P i)} .

89

Exponential Stability of Uncertain Switched System with Time-Varying Delay

For i∈S s, from (13), (14) and (40), we have

˙

V i(x t) ≤ ∑N

i=1λ i(t)



x(t)

x(t−h(t))

T

Θi



x(t)

x(t−h(t))



−2βV2,i(x t)

−(2β+ 1

h M)(V 3,i(x t) +V 4,i(x t)) (43) Similar to Theorem 3.1, from (34) and (43), we get

V i(x t) ≤∑N

i=1λ i(t) V i(x t0) e −ζ i (t−t0 ), t≥t0 (44)

whereζ i=min{mini {λ m(−Θi)}

max

i {λ M(P i)} , 2β}.

In general, from (39), (42) and (44), with the same argument as in the proof of Theorem 3.1, we get

V i(x t) ≤ l∏(t)

m=1ψe λ+(t m −t m−1 )× N (t)−1∏

n =l(t)+1

ψe ζ in hM e −λ−(t n −t n−1 )× V i0(x t0) e −λ−(t−t N (t)−1),

t≥t0 Using (35), we have

V i(x t) ≤ ∏l (t)

m=1ψ× N (t)−1∏

n =l(t)+1

ψe ζin hM× V i0(x t0) e −λ∗(t−t0 ), t≥t0

By (36) and (37), we get

V i(x t) ≤V i0(x t0) e −(λ∗−ν)(t−t0 ), t≥t0 Thus, by (38), we have

x(t) ≤

α 3

α1 x t0 e−1(λ∗−ν)(t−t0 ), t≥t0,

4 Numerical examples

Example 4.1Consider linear switched system (1) with time-varying delay but without matrix

uncertainties and without nonlinear perturbations Let N = 2, S u = {1}, S s = {2} Let

the delay function be h(t) = 0.51 sin2t We have h M = 0.51, μ = 1.02,λ(A1+B1) =

0.0046,−0.0399,λ(A2) = −0.2156, 0.0007 Letβ=0.5

Since one of the eigenvalues of A1+B1is negative and one of eigenvalues of A2is positive,

we can’t use results in (Alan & Lib, 2008) to consider stability of switched system (1) By using the LMI toolbox in Matlab, we have matrix solutions of (5) for unstable subsystems and (6) for stable subsystems as the following:

For unstable subsystems, we get



41.6819 0.0001

0.0001 41.5691



, Q1=

 24.7813 −0.0002

−0.0002 24.7848



, R1=

 33.1027 −0.0001

−0.0001 33.1044

 ,

S11,1=



33.1027 −0.0001

−0.0001 33.1044



, S12,1=



−0.0372−0.0023

−0.0023 0.7075



, S22,1=

 50.0412 0.0001 0.0001 50.0115

 ,

T1=



41.7637 −0.0001

−0.0001 41.7920

 For stable subsystems, we get

P2=



71.8776 2.3932

2.3932 110.8889



, Q2=

 7.2590 −0.3265

−0.3265 0.8745



, R2=

 10.4001 −0.4667

−0.4667 1.2806

 ,

S11,2=



12.7990 −0.4854

−0.4854 3.5031



, S12,2=



−3.1787 0.0240 0.0240 −2.8307



, S22,2=

 4.6346 −0.0289

−0.0289 4.0835

 ,

T2=



16.9964 0.0394

0.0394 17.7152



, X11,2=

 17.2639 −0.1536

−0.1536 14.2310



, X12,2=



−9.6485 −0.1466

−0.1466−12.5573

 ,

X22,2=



16.9716 −0.1635

−0.1635 13.8095



, Y2=



−3.4666−0.1525

−0.1525−6.3485



, Z2=

 6.8776 −0.0574

−0.0574 5.7924



By straight forward calculation, the growth rate isλ+ =ξ =2.8291, the decay rate isλ− =

ζ=0.0063,λ(Ω1,1) =25.8187, 25.8188, 58.7463, 58.8011,λ(Ω2,2) = −10.1108,−3.7678,

−2.0403,−0.7032 andλ(Ω3,2) =1.4217, 4.2448, 5.4006, 9.1514, 29.3526, 30.0607 Thus, we may takeλ∗ =0.0001 andν=0.00001 Thus, from inequality(7), we have T− ≥456.3226 T+ By

choosing T+=0.1, we get T−≥45.63226 We choose the following switching rules:

(i)for t∈ [0, 0.1) ∪ [50, 50.1) ∪ [100, 100.1) ∪ [150, 150.1) ∪ , subsystem i=1 is activated

(ii)for t∈ [0.1, 50) ∪ [50.1, 100) ∪ [100.1, 150) ∪ [150.1, 200) ∪ , subsystem i=2 is activated Then, by Theorem 3.1, the switching system (1) is exponentially stable Moreover, the solution

x(t)of the system satisfies

x(t) ≤11.8915e −0.000045t , t∈ [0,∞)

The trajectories of solution of switched system switching between the subsystems i=1 and

i=2 are shown in Figure 1, Figure 2 and Figure 3, respectively

0 0.2 0.4 0.6 0.8 1 1.2 1.4

time

x1 x2

Fig 1 The trajectories of solution of linear switched system

91

Exponential Stability of Uncertain Switched System with Time-Varying Delay

0 10 20 30 40 50 0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

time

x1 x2

Fig 2 The trajectories of solution of subsystem i=1

−0.05 0 0.05 0.1 0.15 0.2

time

x1 x2

Fig 3 The trajectories of solution of subsystem i=2

Example 4.2Consider uncertain switched system(1)with time-varying delay and nonlinear

perturbation Let N=2, S u= {1}, S s= {2}where

A1=



0.1130 0.00013

0.00015−0.0033



, B1=

 0.0002 0.0012 0.0014−0.5002

 ,

A2=



−5.5200 1.0002

1.0003 −6.5500



, B2=

 0.0245 0.0001 0.0001 0.0237

 ,

E 1i=E 2i=



0.2000 0.0000

0.0000 0.2000



, H 1i=H 2i=

 0.1000 0.0000 0.0000 0.1000



, i=1, 2,

F 1i=F 2i=



sin t 0

0 sin t



, i=1, 2,

f1(t, x(t), x(t−h(t))) =



0.1x1(t)sin(x1(t))

0.1x2(t−h(t))cos(x2(t))

 ,

f2(t, x(t), x(t−h(t))) =



0.5x1(t)sin(x1(t))

0.5x2(t−h(t))cos(x2(t))

 From

 f1(t, x(t), x(t−h(t))) 2 = [0.1x1(t)sin(x1(t))]2+ [0.1x2(t−h(t))cos(x2(t))]2

≤0.01x2(t) +0.01x2(t−h(t))

≤0.01x(t) 2+0.01x(t−h(t)) 2

≤0.01[x(t)  + x(t−h(t)) ]2,

we obtain

 f1(t, x(t), x(t−h(t))) ≤0.1x(t)  +0.1x(t−h(t)) 

The delay function is chosen as h(t) =0.25 sin2t From

 f2(t, x(t), x(t−h(t))) 2 = [0.5x1(t)sin(x1(t))]2+ [0.5x2(t−h(t))cos(x2(t))]2

≤0.25x2(t) +0.25x2(t−h(t))

≤0.25x(t) 2+0.25x(t−h(t)) 2

≤0.25[x(t)  + x(t−h(t)) ]2,

we obtain

 f2(t, x(t), x(t−h(t))) ≤0.5x(t)  +0.5x(t−h(t)) 

We may take h M=0.25, and from (4), we takeγ1=0.1,δ1 =0.1,γ2=0.5,δ2=0.5 Note that

λ(A1) =0.11300016,−0.00330016 Letβ=0.5,μ=0.5 Since one of the eigenvalues of A1is negative, we can’t use results in (Alan & Lib, 2008) to consider stability of switched system (1) From Lemma 2.4 , we have the matrix solutions of (33) for unstable subsystems and of (34) for stable subsystems by using the LMI toolbox in Matlab as the following:

For unstable subsystems, we get

ε31=0.8901, ε41=0.8901, ε51=0.8901,

P1=



0.2745 −0.0000

−0.0000 0.2818



, Q1=

 0.4818 −0.0000

−0.0000 0.5097



, R1=

 0.8649 −0.0000

−0.0000 0.8729

 ,

S11,1=



0.8649 −0.0000

−0.0000 0.8729



, S12,1=10−4×



−0.1291−0.8517

−0.8517 0.1326

 ,

S22,1=



1.0877 −0.0000

−0.0000 1.0902

 For stable subsystems, we get

ε32=2.0180, ε42=2.0180, ε52=2.0180,

P2=



0.2741 0.0407

0.0407 0.2323



, Q2=

 1.3330 −0.0069

−0.0069 1.3330



, R2=

 1.0210 −0.0002

−0.0002 1.0210

 ,

S11,2=



1.0210 −0.0002

−0.0002 1.0210



, S12,2=



−0.0016−0.0002

−0.0002−0.0016

 ,

S22,2=



0.8236 −0.0006

−0.0006 0.8236



By straight forward calculation, the growth rate is λ+ = ξ = 8.5413, the decay

93

Exponential Stability of Uncertain Switched System with Time-Varying Delay

rate is λ− = ζ = 0.1967, λ(Θ1) = 0.1976, 0.2079, 1.1443, 1.1723 and λ(Θ2) =

−0.7682,−0.6494,−0.0646,−0.0588 Thus, we may takeλ∗=0.0001 andν=0.00001

Thus, from inequality (35), we have T− ≥ 43.4456 T+ By choosing T+ = 0.1, we get

T ≥4.34456 We choose the following switching rules:

(i)for t∈ [0, 0.1) ∪ [5.0, 5.1) ∪ [10.0, 10.1) ∪ [15.0, 15.1) ∪ , system i=1 is activated

(ii)for t∈ [0.1, 5.0) ∪ [5.1, 10.0) ∪ [10.1, 15.0) ∪ [15.1, 20.0) ∪ , system i=2 is activated Then, by theorem 3.3.1, the switched system (1) is exponentially stable Moreover, the solution

x(t)of the system satisfies

x(t) ≤1.8770e −0.000045t , t∈ [0,∞)

The trajectories of solution of switched system switching between the subsystems i=1 and

i=2 are shown in Figure 4, Figure 5 and Figure 6, respectively

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

time

x1 x2

Fig 4 The trajectories of solution of switched system with nonlinear perturbations

5 Conclusion

In this paper, we have studied the exponential stability of uncertain switched system with time varying delay and nonlinear perturbations We allow switched system to contain stable and unstable subsystems By using a new Lyapunov functional, we obtain the conditions for robust exponential stability for switched system in terms of linear matrix inequalities (LMIs) which may be solved by various algorithms Numerical examples are given to illustrate the effectiveness of our theoretical results

6 Acknowledgments

This work is supported by Center of Excellence in Mathematics and the Commission on Higher Education, Thailand

We also wish to thank the National Research University Project under Thailand’s Office of the Higher Education Commission for financial support

0 5 10 15 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5

time

x1 x2

Fig 5 The trajectories of solution of system i=1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

time

x1 x2

Fig 6 The trajectories of solution of system i=2

7 References

Alan, M.S & Lib, X (2008) On stability of linear and weakly nonlinear switched systems with

time delay, Math Comput Modelling, 48, 1150-1157.

Alan, M.S & Lib, X (2009) Stability of singularly perturbed switched system with time delay

and impulsive effects, Nonlinear Anal., 71, 4297-4308.

Alan, M.S., Lib, X & Ingalls, B (2008) Exponential stability of singularly

perturbed switched systems with time delay, Nonlinear Analysis:Hybrid systems, 2,

913-921

Boyd, S et al (1994) Linear Matrix Inequalities in System and Control Theory, SIAM,

Philadelphia

95

Exponential Stability of Uncertain Switched System with Time-Varying Delay

Hien, L.V., Ha, Q.P & Phat, V.N (2009) Stability and stabilization of switched linear dynamic

systems with delay and uncertainties, Appl Math Comput., 210, 223- 231.

Hien, L.V & Phat, V.N (2009) Exponential stabilization for a class of hybrid systems with

mixed delays in state and control, Nonlinear Analysis:Hybrid systems, 3,

259-265

Huang, H., Qu, Y & Li, H.X (2005) Robust stability analysis of switched Hopfield

neural networks with time-varying delay under uncertainty, Phys Lett A, 345,

345-354

Kim, S., Campbell, S.A & Lib, X (2006) Stability of a class of linear switching systems with

time delay, IEEE Trans Circuits Syst I Regul Pap., 53, 384-393.

Kwon, O.M & Park, J.H (2006) Exponential stability of uncertain dynamic systems including

states delay, Appl Math Lett., 19, 901-907.

Li, T et al (2009) Exponential stability of recurrent neural networks with time-varying discrete

and distributed delays, Nonlinear Analysis Real World Appl., 10,

2581-2589

Li, P., Zhong, S.M & Cui, J.Z (2009) Stability analysis of linear switching systems with time

delays, Chaos Solitons Fractals, 40, 474-480.

Lien C.H et al (2009) Exponential stability analysis for uncertain switched neutral

systems with interval-time-varying state delay, Nonlinear Analysis:Hybrid sys- tems, 3,

334-342

Lib, J., Lib, X & Xie, W.C (2008) Delay-dependent robust control for uncertain switched

systems with time delay, Nonlinear Analysis:Hybrid systems, 2, 81-95.

Niamsup, P (2008) Controllability approach to H∞control problem of linear time- varying

switched systems, Nonlinear Analysis:Hybrid systems, 2, 875-886.

Phat, V.N., Botmart, T & Niamsup, P (2009) Switching design for exponential stability of a

class of nonlinear hybrid time-delay systems, Nonlinear Analysis:Hybrid

systems, 3, 1-10.

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systems, Automatica J IFAC, 40, 1435-1439.

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systems with state delay, Proceedings of American Control Conf., Minnesota, USA,

pp.1539- 1543

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Alexander Stepanov

Synopsys GmbH, St.-Petersburg representative office

Russia

1 Introduction

Problems of stabilization and determining of stablility characteristics of steady-state regimes are among the central in a control theory Especial difficulties can be met when dealing with the systems containing nonlinearities which are nonanalytic function of phase Different models describing nonlinear effects in real control systems (e.g servomechanisms, such as servo drives, autopilots, stabilizers etc.) are just concern this type, numerous works are devoted to the analysis of problem of stable oscillations presence in such systems

Time delays appear in control systems frequently and are important due to significant impact

on them They affect substantially on stability properties and configuration of steady state solutions An accurate simultaneous account of nonlinear effects and time delays allows to receive adequate models of real control systems

This work contains some results concerning to a stability problem for periodic solutions

of nonlinear controlled system containing time delay It corresponds further development

of an article: Kamachkin & Stepanov (2009) Main results obtained below might generally

be put in connection with classical results of V.I Zubov’s control theory school (see Zubov (1999), Zubov & Zubov (1996)) and based generally on work Zubov & Zubov (1996)

Note that all examples presented here are purely illustrative; some examples concerning to similar systems can be found in Petrov & Gordeev (1979), Varigonda & Georgiou (2001)

2 Models under consideration

Consider a system

here x=x(t ) ∈En , t ≥ t0 ≥ τ, A is real n × n matrix, c ∈En , vector x(t), t ∈ [ t0− τ, t0], is considered to be known Quantityτ >0 describes time delay of actuator or observer Control

statement u is defined in the following way:

u(t − τ) = f(σ(t − τ)), σ(t − τ) =γ  x(t − τ), γ ∈En,  γ  =0;

nonlinearity f can, for example, describe a nonideal two-position relay with hysteresis:

f(σ) =



m1, σ < l2,

On Stable Periodic Solutions of One Time Delay

System Containing Some Nonideal Relay

Nonlinearities

5

here l1< l2, m1< m2; and f(σ(t)) = f − =f(σ(t −0))ifσ ∈ [ l1; l2].

In addition to the nonlinearity (2) a three-position relay with hysteresis will be considered:

f(σ) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

0,



| σ | ≤ l0,

| σ | ∈ ( l0; l], f −=0;

m1,



σ ∈ [− l; − l0), f −=m1,

σ < − l;

m2,



σ ∈ ( l0; l], f −=m2,

σ > l;

(3)

(here m1< m < m2, 0< l0< l);

Suppose that hysteresis loops for the nonlinearities are walked around in counterclockwise direction

3 Stability of periodic solutions

Denote x(t − t0, x0, u) solution of the system (1) for unchanging control law u and initial

conditions(t0, x0)

Let the system (1), (3) has a periodic solution with four switching points ˆs isuch as

ˆs1=x(T4, ˆs4, m2), ˆs2=x(T1, ˆs1, 0), ˆs3=x(T2, ˆs2, m1), ˆs4=x(T3, ˆs3, 0)

Let s i , i=1, 4 are points of this solution (preceding to the corresponding ˆs i) such as

γ  s1=l0, γ  s2= − l, γ  s3= − l0, γ  s4=l,

(let us name them ˇTpre-switching points ˇT, for example), and

ˆs1=x(τ, s1, m2), ˆs2=x(τ, s2, 0), ˆs3=x(τ, s3, m1), ˆs4=x(τ, s4, 0),

or

ˆs i+1=x(T i , ˆs i , u i), ˆs i=x(τ, s i , u i−1), where

u1=0, u2=m1, u3=0, u4=m2 (hereafter suppose that indices are cyclic, i.e for i = 1, m one have i+1= 1 if i = m and

i −1=m if i=1)

Denote

v i=As i+1+cu i, k i=γ  v i

Theorem 1. Let k i = 0 and  M  < 1, where

M=∏1

i=4

M i, M i=I − k −1 i v i γ e ATi,

then the periodic solution under consideration is orbitally asymptotically stable.