Robust FiniteTime Stabilization of Linear Systems with Multiple Delays in State and Control

This paper is concerned with the problem of robust finitetime stabilization for a class of linear systems with multiple delays in state and control and disturbance.The disturbance under consideration are norm bounded. We first present delaydependent sufficient conditions for robust finitetime stabilization of the system via memoryless static feedback controllers based on Lyapunov functional and LMI method.Then, memory state feedback controllers are designed to finitetime stabilize the closedloop timedelay system, and the conditions are formulated in terms of delaydependent linear matrix inequalities (LMIs). Finally, two numerical examples are provided to show the effectiveness of the proposed result

Robust Finite-Time Stabilization of Linear Systems with Multiple Delays in State and

NGUYEN T THANHa and VU N PHATb, ∗

a Department of Mathematics University of Mining and Geology, Hanoi, Vietnam

bInstitute of Mathematics, VAST

18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam

∗ Corresponding author: vnphat@math.ac.vn

Abstract

This paper is concerned with the problem of robust finite-time stabilization for a class of linear systems with multiple delays in state and control and disturbance.The disturbance under consideration are norm bounded We first present delay-dependent sufficient conditions for robust finite-time stabilization of the system via memoryless static feedback controllers based on Lyapunov functional and LMI method.Then, mem-ory state feedback controllers are designed to finite-time stabilize the closed-loop time-delay system, and the conditions are formulated in terms of time-delay-dependent linear matrix inequalities (LMIs) Finally, two numerical examples are provided to show the effectiveness of the proposed results

Key words Finite-time stabilization, time delay, Lyapunov functions, linear matrix inequal-ities

1 Introduction

The problem of finite-time stabilization of linear control systems was considered in [1], which

is to design a static feedback controller to finite-time stabilize the closed-loop system

————————–

1 This paper was completed when the authors were visiting the Vietnam Institute for Advance Study in Mathematics (VIASM) We would like to thank the VIASM for support and hospitality

During the past decades, the finite-time stabilization of linear control systems becomes

a very important topic and has been studied extensively (see, e.g [2-8], and the references therein) The features among these results are the use of quadratic Lyapunov functionals and the design of static feedback controllers via solving LMIs This approach has the advantage

of providing an upper bound on a given performance index and thus the system performance degradation incurred by the uncertainties is guaranteed to be less than this bound Based on this idea, the finite-time stabilization problem was developed in [9-12] for linear control sys-tems with delays, and its solutions provide sufficient conditions for designing state feedback controller via LMIs

It should be noticed that one way to solve the stabilization problem of linear system

with delay is to design memoryless controllers u(t) = Kx(t) or of more general controllers with memory that include, nevertheless, an instantaneous feedback term u(t) = Kx(t) +

m

∑

i=1

K i x(t − h i) Although the memoryless are easy to implement, it was pointed out in [13, 14] that they tend to be more conservative when the time delay is small In fact, information on the size of the delay is often available in many processes Hence, by using this information and employing a feedback of the past control history as well as the current state, we may expect to achieve an improved performance Therefore, in this paper, we investigate the memory controller design for the exponential stabilization of linear singular time delay systems with control delay

To the best of our knowledge, finite-time stabilization problem for linear control systems with multiple state and control delays has not fully investigated Some recently published results given in [15, 16] have shown efficiency in implementing LMI approach for designing memory feedback controllers in stabilization problem in Lyapunov sence of linear control state-delay systems without control delays In this paper, we propose a new design tool to solve the finite-time stabilization for linear systems with time delays via memoryless and memory static feedback controllers

The problem studied in this paper is technically challenging due to two reasons: (a) The time delays are involved in in both the state and control; (b) The disturbance is norm bounded; (c) There has not been an effective method to design memoryless and memory static feedback controllers for linear systems with delays in both state and control to ensure that the closed-loop system is robustly finite-time stable We propose a simple set of Lyapunov-like functionals and apply LMI technique in analysing the finite-time stabilization of the control time-delay systems The conditions are obtained in terms of LMIs, which can be determined by utilizing MATLABs LMI Control Toolbox [17]

The structure of the paper is as follows In Section 2, we present definitions and some auxiliary results which will be used in the proof of our main result In Section 3, the design

of memoryless and memory feedback controllers for robust finite-time stabilization in terms

of LMIs is presented together with illustrative examples Section 4 gives some conclusions

Notation R n ×r denotes the space of all (n × r)- matrices The notation i = 1, N means

i = 1, 2, , N ; λ(A) denotes the set of all eigenvalues of A; λ max (A) = max {Reλ : λ ∈ λ(A)};

λ min (A) = min {Reλ : λ ∈ λ(A)}; λ A = λ max (A T A); the matrix norm ||A|| =√λ max (A T A);

C1([a, b], R n ) denotes the set of all R n -valued differentiable functions on [a, b]; The symmetric

terms in a matrix are denoted by∗ Matrix A is semi-positive definite (A ≥ 0) if (Ax, x) ≥ 0,

for all x ∈ R n ; A is positive definite (A > 0) if (Ax, x) > 0 for all x ̸= 0; A ≥ B means

A − B ≥ 0 The segment of the trajectory x(t) is denotes by x t ={x(t + s) : s ∈ [−τ, 0]}.

2 Preliminaries

Consider a class of linear systems described by the following equation:







˙x(t) = Ax(t) +

p

∑

i=1

A i x(t − h i ) + Bu(t) +

q

∑

j=1

B j u(t − m j ) + Dω(t), t ≥ 0, x(θ) = φ(θ), θ ∈ [−2h, 0],

(2.1)

where x(t) ∈ R n is the state vector; A, A i ∈ R n ×n , B, B

j ∈ R n ×m , D ∈ n × r, p is the number

of state delay, q is the number of control delay; the delays satisfy the following conditions

0 < h i ≤ h, 0 < m j ≤ h, ∀i = 1, p, j = 1, q;

the system matrices A, A i , B, B j , D are of appropriate dimensions; the function φ(.) ∈ C([ −2h, 0], R n ); the disturbance ω(t) is a continuous function satisfying

∃d > 0 :

T

∫ 0

ω(t) T ω(t)dt ≤ d. (2.2)

Once the above assumption on φ(.) are given, the solution of system (2.1) is well defined on [0, T ].

Let us now recall the following definitions and propositions that will be used to derive the main results of the paper

Definition 2.1 (Finite-time stability) For given positive numbers T, c1, c2 and a symmetric

positive definite matrix Q ∈ R n ×n , the system (2.1) is robustly finite-time stable w.r.t

(c1, c2, T, Q) if

sup

s ∈[−h,0] {φ(s) T Qφ(s) } ≤ c1 =⇒ x(t) T Qx(t) < c2, ∀t ∈ [0, T ],

for all disturbances ω( ·) satisfying (2.2).

Definition 2.2.(Robust finite-time stabilization) For given positive numbers T, c1, c2 and

a symmetric positive definite matrix Q ∈ R n×n , the system (2.1) is robustly finite-time

stabilizable with respect to (c1, c2, T, Q) if there exists a memoryless feedback controller u(t) = Kx(t) (or memory feedback controller u(t) = Kx(t) +

p

∑

i=1

K i x(t − h i)) such that the

closed-loop system is robustly finite-time stable w.r.t.(c1, c2, T, Q).

Proposition 2.1 (Schur Complement Lemma [18]) Given matrices X, Y, Z, where Y =

Y T > 0, X = X T Then X + Z T Y −1 Z < 0 if and only if

[

X Z T

Z −Y

]

< 0.

3 Main result

The existing methods developed so far for Lyapunov stability are mainly for linear systems with state delay In this section we give delay-dependent sufficient conditions for designing memoryless and memory state feedback controllers that enable closed-loop system trajectory

to stay within the priori given interval finite time

3.1 Memoryless feedback control

In this subsection, we give a sufficient condition for the robustly finite-time stabilization of

the system (2.1) by using the memoryless feedback controller u(t) = Kx(t) Before

intro-ducing the main result, the following notations of several matrix variables are defined for simplicity

P1 = P −1 , R1 = P −1 RP −1 ,

H 1,1 = AP + P A T + BY + Y T B T + qR +

p

∑

i=1

A i A T

i + DD T ,

H 2,2 =−I, H 1,2 =√

pP, H 2+j,2+j =−R, H 1,2+j = B j Y, j = 1, q.

α1 = λ min (P1)

λ max (Q) , α2 =

λ max (P1)

λ min (Q) + ph

1

λ min (Q) + qh

λ max (R1)

λ min (Q) .

Theorem 3.1 For given positive numbers T, c1, c2, c2 > c1, and a symmetric positive definite matrices Q ∈ R n ×n , the system (2.1) is robustly finite-time stabilizable with respect

to (c1, c2, T, Q) if there exist symmetric positive definite matrices P, R, a free weight matrix

Y, and a number β > 0 satisfying the following conditions









H11 H12 H 1(q+2)

∗ H22 H 2(q+2)







α2c1+ d

α1 e

The memoryless state feedback controller is defined by

u(t) = Y P −1 x(t).

Proof Consider the following non-negative quadratic function: V (t) = V1(t) + V2(t),

where

V1(t) = e βt x(t) T P1x(t),

V2(t) = e βt(∑p

i=1

t

∫

t −h i

x(s) T x(s)ds +

q

∑

j=1

t

∫

t−m j

x(s) T R1x(s)ds

)

.

Taking the derivative of V (t) in t along the solution of the closed-loop system, we have

˙

V1(t) = βV1(t) + e βt 2x(t) T P1˙x(t)

= βV1(t) + e βt 2x(t) T P1

[

Ax(t) +

p

∑

i=1

A i x(t − h i ) + BY P1x(t) (3.3)

+

q

∑

j=1

B j Y P1x(t − m j ) + Dω(t)

]

,

˙

V2(t) = βV2(t) + e βt

(

p x(t) T x(t) + q x(t) T R1x(t)

)

(3.4)

− e βt(∑p

i=1

x(t − h i)T x(t − h i) +

q

∑

j=1

x(t − m j)T R1x(t − m j)

)

.

We first estimate ˙V1(.) as follows Using Cauchy matrix inequality gives

2x(t) T P1

[∑p i=1

A i x(t − h i)

]

≤

p

∑

i=1

x(t) T P1A i A T i P1x(t) +

p

∑

i=1

x(t − h i))T x(t − h i ),

2x(t) T P1

[∑q

j=1

B j Y P1x(t − m j)

]

≤

q

∑

j=1

x(t) T P1B j Y R −1 Y T B j T P1x(t)

+

q

∑

j=1

x(t − m j)T R1x(t − m j ),

2x(t) T P1Dω(t) ≤x(t) T P1DD T P1x(t) + ω(t) T ω(t),

and hence from (3.3) it follows that

˙

V1(t) ≤βV1(t) + e βt x(t) T

[

P1A + A T P1+ P1(BY + Y T B T )P1

]

x(t)

+ e βt

p

∑

i=1

x(t) T P1A i A T i P1x(t) + e βt

p

∑

i=1

x(t − h i)T x(t − h i)

+ e βt

q

∑

j=1

x(t) T P1B j Y R −1 Y T B j T P1x(t) + e βt

q

∑

j=1

x(t − m j)T R1x(t − m j)

+ e βt x(t) T P1DD T P1x(t) + e βt ω(t) T ω(t),

(3.5)

Therefore, taking into account the inequalities (3.4)-(3.5), we get

˙

V (t) − βV (t) ≤e βt(

x(t) T [P1A + A T P1+ P1(BY + Y T B T )P1]x(t) +

p

∑

i=1

x(t) T P1A i A T i P1x(t)

+

q

∑

j=1

x(t) T P1B j Y R −1 Y T B j T P1x(t) + px(t) T x(t) + qx(t) T R1x(t)

+ x(t) T P1DD T P1x(t) + ω(t) T ω(t)

)

.

Setting y(t) = P1x(t), we obtain

˙

V (t) − βV (t) ≤ e βt [y(t) T M y(t) + ω(t) T ω(t)], (3.6) where

M = AP + P A T + BY + Y T B T + qR +

p

∑

i=1

A i A T i + DD T + pP2+

p

∑

j=1

B j Y R −1 Y T B j T

Using the Schur complement lemma, Proposition 2.1, the condition (3.1) leads to M < 0,

and from the inequality (3.6), it follows that

˙

V (t) − βV (t) ≤ e βt ω(t) T ω(t), ∀t ≥ 0. (3.7)

Multiplying both sides of (3.7) by e −βt , and noting that dt d (e −βt V (t)) = e −βt V (t)˙ −βe −βt V (t),

we have

d

dt (e

−βt V (t)) ≤ ω(t) T

ω(t), t ∈ [0, T ].

Integrating the above inequality from 0 to t, we obtain

e −βt V (t) − V (0) ≤

t

∫ 0

ω(s) T ω(s)ds ≤

T

∫ 0

ω(s) T ω(s)ds ≤ d, ∀t ∈ [0, T ],

and hence

V (t) ≤ [V (0) + d]e βT

, ∀t ∈ [0, T ]. (3.8)

On the other hand, it is easy to verify that

V (t) ≥ x(t) T P1x(t) ≥ λ min (P1)x(t) T x(t)

≥ λ min (P1)

λ max (Q) x(t)

T Qx(t) = α1x(t) T Qx(t), t ≥ 0, (3.9)

and

V (0) ≤ +

q

∑

j=1

0

∫

−m j

x(s) T Qx i (s) λ max (R1)

λ min (Q) ds

≤ λ max (P1)

λ min (Q) x(0)

T Qx(0) +

p

∑

i=1

0

∫

−h

x(s) T Qx(s) 1

λ min (Q) ds

+

q

∑

j=1

0

∫

−h

x(s) T Qx(s) λ max (R1)

λ min (Q) ds

≤α2 sup

s ∈[−h,0] {x(s) T

Qx(s) } = α2 sup

s ∈[−h,0] {φ(s) T

Qφ(s) } ≤ α2c1.

(3.10)

From (3.8)-(3.10), we finally obtain that

x(t) T Qx(t) ≤ 1

α1[V (0) + d]e

βT ≤ α2c1+ d

α1 e

βT ≤ c2, ∀t ∈ [0, T ].

This completes the proof of the theorem.

Remark 3.1 We note that the condition (3.2) is not LMI with respect to β Since β

does not include in (3.1), we can first find the solutions P, R, Y from LMI (3.1) and then determine β from (3.2).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time(sec) x(t)TQx(t)

x(t)TQx(t) c

1 =0.01, c

2 =4.6

Figure 1: The trajectories of x(t) T Qx(t) of the system (2.1)

Example 3.1 Consider system (2.1), where

p = q = 2, A =

[

−1 1

]

, A1 =

[

0.1 0.01 0.01 0.1

]

, A2 =

[

0.1 0.02 0.02 0.1

]

,

B =

[

0.1 0.2

0.3 0.4

]

, B1 =

[

0.1 0.2 0.2 0.1

]

, B2 =

[

0.2 0.1 0.1 0.2

]

, D =

[

0.1 0.1 0.1 0.1

]

, d = 1.

By using LMI Toolbox in MATLAB [11], the LMI (3.1) is feasible with

β = 0.01, h1 = 1, h2 = 0.9, m1 = 0.6, m2 = 0.8, h = 1,

P =

[

0.5243 −0.4792

−0.4792 1.0309

]

, R =

[

0.4747 −0.5269

−0.5269 0.7043

]

, Y =

[

2.5654 −2.4635

−2.4635 2.4176

]

,

Besides, the condition (3.2) holds with

c1 = 0.01, c2 = 4.6, T = 10, Q =

[

1 1

1 2

]

.

The feedback control can be obtained as

u(t) =

[

4.7104 −0.2000

−4.4434 0.2796

]

x(t).

Moreover, the system is robustly finite-time stable with respect to (0.01, 4.6, 10, Q).

Fig 1 shows the trajectories of x(t) T Qx(t) of the closed loop system with the initial

conditions φ(t) = [ −0.09, 0].

3.2 Memory feedback control

In this subsection, we give a sufficient condition for the robustly finite-time stabilization of

the system (2.1) by using the memory feedback controller u(t) = Kx(t) +

p

∑

i=1

K i x(t − h i ).

Let us denote

P1 = P −1 , R1 = P −1 RP −1 , U1 = P −1 U P −1 ,

H 1,1 = AP + P A T + BY0+ Y T

0 B T + qR + pqU + 2

p

∑

i=1

A i A T

i + DD T ,

H 2,2 =−I, H 1,2 =√

pP, H 2+j,2+j =−R, H 1,2+j = B j Y0, j = 1, q,

H 2+q+i,2+q+i=−I, H 1,2+q+i =√

2BY i , i = 1, p,

H 2+q+p+(j −1)p+i,2+q+p+(j−1)p+i =−U, H 1,2+q+p+(j −1)p+i = B j Y i , i = 1, p, j = 1, q,

α1 = λ min (P1)

λ max (Q) , α2 =

λ max (P1)

λ min (Q) + ph

1

λ min (Q) + qh

λ max (R1)

λ min (Q) + 2pqh

λ max (U1)

λ min (Q) .

Theorem 3.2 For given positive numbers T, c1, c2, c2 > c1, and a symmetric positive definite matrices Q ∈ R n ×n , the system (2.1) is robustly finite-time stabilizable with respect to

(c1, c2, T, Q) if there exist symmetric positive definite matrices P, R, U, free-weight matrices

Y0, Y1, , Y p and a number β > 0 satisfying the following conditions









H11 H12 H 1(2+q+p+pq)

∗ H22 H 2(2+q+p+pq)







α2c1+ d

α1 e

The memoryless state feedback controller is defined by

u(t) = Y0P −1 x(t) +

p

∑

i=1

Y i P −1 x(t − h i ).

Proof Consider the following non-negative quadratic function: V (t) = V1(t) + V2(t),

where

V1(t) = e βt x(t) T P1x(t),

V2(t) = e βt(∑p

i=1

t

∫

t −h i

x(s) T x(s)ds+

q

∑

j=1

t

∫

t −m j

x(s) T R1x(s)ds+

q

∑

j=1

p

∑

i=1

t

∫

t −m j −h i

x(s) T U1x(s)ds

)

.

Using the same method of the proof of Theorem 3.1, taking the derivative of V (t) in t along

the solution of the closed-loop system and applying the following derived estimations

2x(t) T P1

[∑p

i=1

A i x(t − h i)

]

≤ 2∑p

i=1

x(t) T P1A i A T i P1x(t) + 0.5

p

∑

i=1

x(t − h i))T x(t − h i ),

2x(t) T P1

[∑p

i=1

BY i P1x(t −h i)

]

≤ 2∑p

i=1

x(t) T P1BY i [BY i]T P1x(t)+0.5

p

∑

i=1

x(t −h i))T x(t −h i ),

2x(t) T P1

[∑q

j=1

B j Y0P1x(t − m j)

]

≤ ∑q

j=1

x(t) T P1B j Y0 R −1 Y0T B j T P1x(t)

+

q

∑

j=1

x(t − m j)T R1x(t − m j ),

2x(t) T P1

[∑q

j=1

p

∑

i=1

B j Y i P1x(t − m j − h i)

]

≤ ∑q

j=1

p

∑

i=1

x(t) T P1B j Y i U −1 Y T

i B T

j P1x(t)

+

q

∑

j=1

p

∑

i=1

x(t − m j − h i)T U1x(t − m j − h i ), 2x(t) T P1Dω(t) ≤ x(t) T P1DD T P1x(t) + ω(t) T ω(t),

we obtain that

˙

V (t) − βV (t) ≤e βt(

x(t) T [P1A + A T P1+ P1(BY0+ Y0T B T )P1]x(t)

+ 2

p

∑

i=1

x(t) T P1A i A T i P1x(t) + 2

p

∑

i=1

x(t) T P1BY i [BY i]T P1x(t)

+

q

∑

j=1

x(t) T P1B j Y0 R −1 Y0T B T j P1x(t) +

q

∑

j=1

p

∑

i=1

x(t) T P1B j Y i U −1 Y i T B j T P1x(t)

+ p x(t) T x(t) + qx(t) T R1x(t) + pqx(t) T U1x(t)

+ x(t) T P1DD T P1x(t) + ω(t) T ω(t)

)

.

Setting y(t) = P1x(t), we obtain

˙

V (t) − βV (t) ≤ e βt [y(t) T M y(t) + ω(t) T ω(t)], (3.13) where

M =AP + P A T + BY0 + Y0T B T + qR + pqU + 2

p

∑

i=1

A i A T i + DD T

+ 2

p

∑

i=1

BY i [BY i]T +

q

∑

j=1

B j Y0R −1 Y0T B T j +

q

∑

j=1

p

∑

i=1

B j Y i U −1 Y i T B j T + pP2.

Using the Schur complement lemma, Proposition 2.1, the condition (3.11) leads to M < 0,

and from the inequality (3.13), it follows that

˙

V (t) − βV (t) ≤ e βt ω(t) T ω(t), ∀t ≥ 0, (3.14) and hence

V (t) ≤ [V (0) + d]e βT

, ∀t ∈ [0, T ]. (3.15)

On the other hand, it is easy to verify that

V (t) ≥ λ min (P1)

λ max (Q) x(t)

T Qx(t) = α1x(t) T Qx(t), t ≥ 0, (3.16)

and

V (0) ≤ α2 sup

s ∈[−2h,0] {x(s) T

Qx(s) } = α2 sup

s ∈[−2h,0] {φ(s) T

Qφ(s) } ≤ α2c1. (3.17) Therefore, from (3.15)-(3.17) it follows that

x(t) T Qx(t) ≤ 1

α1[V (0) + d]e

βT ≤ α2c1+ d

α1 e

βT ≤ c2, ∀t ∈ [0, T ].

This completes the proof of the theorem

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time(sec)

x(t)TQx(t) c

1 =0.1, c

2 =4.1

Figure 2: The trajectories of x(t) T Qx(t) of the system (2.1)

Example 3.2 Consider system (2.1), where

p = q = 2, A =

[

−1 1

]

, A1 =

[

0.1 0.1 0.1 0.1

]

, A2 =

[

0.1 0.2 0.2 0.1

]

,

B =

[

0.1 0.3

0.5 0.7

]

, B1 =

[

0.2 0.4 0.4 0.2

]

, B2 =

[

0.3 0.1 0.1 0.3

]

, D =

[

−0.1 0.1

]

, d = 2.