Finitetime stability and H control of linear discretetime delay systems with normbounded disturbances

This paper deals with the finitetime stability and H∞ control of linear discretetime delay systems. The system under consideration is subject to interval timevarying delay and normbounded disturbances. Linear matrix inequality approach is used to solve the finitetime stability problem. First, new sufficient conditions are established for robust finitetime stability of the linear discretetime delay system subjected to normbounded disturbances, then the state feedback controller is designed to robustly finitetime stabilize the system and guarantee an adequate level of system performance. The delaydependent sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed results.

Finite-time stability and H ∞ control of linear

discrete-time delay systems with norm-bounded disturbances

Le A Tuana and Vu N Phatb, ∗

aDepartment of Mathematics College of Sciences, Hue University, Hue, Vietnam

bInstitute of Mathematics, VAST

18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam

∗Corresponding author: vnphat@math.ac.vn

Abstract

This paper deals with the finite-time stability and H ∞ control of linear discrete-time delay systems The system under consideration is subject to interval discrete-time-varying delay and norm-bounded disturbances Linear matrix inequality approach is used to solve the finite-time stability problem First, new sufficient conditions are established for robust finite-time stability of the linear discrete-time delay system subjected to norm-bounded disturbances, then the state feedback controller is designed to robustly finite-time stabilize the system and guarantee an adequate level of system performance The delay-dependent sufficient conditions are formulated in terms of linear matrix inequalities (LMIs) Numerical examples are given to illustrate the effectiveness of the proposed results

Key words. Finite-time stability, H ∞ control, time-varying delay, disturbances, linear matrix

inequalities

2000 Mathematics Subject Classifications: 34D10, 34K20, 49M7

Finite-time stability (FTS) introduced by Dorato in [1] involves dynamical systems whose solutions converge to an equilibrium state in finite time Compared with the Lyapunov stability, FTS is a more practical property, useful to study the behavior of the system within

a finite interval time, and therefore it finds many applications A lot of interesting results

on finite-time stability and stabilization in the context of linear discrete-time delay systems have been obtained (see, e.g [2-5] and the references therein)

On the other hand, problem of finite-time H ∞ control has attracted much attention due

to its both practical and theoretical importance Various approaches have been developed and a great number of results for linear continuous and discrete-time systems have been reported in the literatures (see, e.g [6-10] and the references therein) Note that these papers were limited either to Lyapunov stability or to the system with constant delays

Based on the solution to some LMIs, the finite-time H ∞ control for discrete-time systems without delay was proposed in [11, 12] Recently, the authors in [13] have considered the

finite-time H ∞ control for discrete-time system with constant delays To the best of our

knowledge, finite-time H ∞ control problem for linear discrete-time systems with interval time-varying delay has not fully investigated The problem is important and challenging in many practice applications, which motivates the main purpose of our research

In this paper, we extend further the results of finite-time stability and H ∞ control for linear discrete-time delay systems with norm-bounded disturbances Our main propose is

to design a state feedback controller which guarantees FTS of the closed-loop system and reduces the effect of the disturbance input on the controlled output to a prescribed level The novel features of this paper are: (i) The system under consideration subjected to interval time-varying delays in both the state input and observation output; (ii) Using new bounding LMI estimation technique, a set of improved Lyapunov-like functionals is constructed to

design the H ∞ feedback controller in terms of LMIs, which can be determined by utilizing MATLAB’s LMI Control Toolbox [14]

The paper is organized as follows In Section 2 some preliminary definitions are provided and the problem we deal with is precisely stated Section 3 presents the main results of the

paper: sufficient conditions for robust finite-time H ∞ boundedness and control in terms of LMIs Numerical examples showing the effectiveness of the proposed method are given

2 Preliminaries

The following notations will be used throughout this paper Z+ denotes the set of all non-negative integers; Rn denotes the n −dimensional space with the scalar product x ⊤ y; Rn×r

denotes the space of all matrices of (n × r)−dimensions; A ⊤denotes the transpose of matrix

A; A is symmetric if A = A ⊤ ; I denotes the identity matrix of appropriate dimension Matrix

A is called semi-positive definite (A > 0) if x ⊤ Ax > 0 for all x ∈ R n ; A is positive definite (A > 0) if x ⊤ Ax > 0 for all x ̸= 0; A > B means A − B > 0 The notation diag{ .} stands

for a block-diagonal matrix The symmetric term in a matrix is denoted by ∗.

Consider the following linear discrete-time systems with time-varying delay

x(k + 1) = Ax(k) + A d x(k − d(k)) + Bu(k) + Gω(k),

x(k) = φ(k), k ∈ {−d2, −d2+ 1, , 0 }, where x(k) ∈ R n is the state; u(k) ∈ R m is the control input; z(k) ∈ R p is the observation

output; A, A d ∈ R n ×n , B ∈ R n ×m , G ∈ R n ×q , C, C

d ∈ R p ×n are given real constant matrices;

d(k) is delay function satisfying the condition

0 < d1 6 d(k) 6 d2 ∀k ∈ Z+, (2)

where d1, d2 are known positive integers; φ(k) is the initial function; ω(k) ∈ R q satisfying the condition

∃d > 0 :

N

∑

k=0

Definition 2.1 (Finite-time stability) Given positive numbers N, c1, c2, c1 < c2, and a symmetric positive-definite matrix R, the discrete-time delay system (1) with u(k) = 0 is said to be robustly finite-time stable w.r.t (c1, c2, R, N ) if

max

k ∈{−d2, −d2+1, ,0 } x

⊤ (k)Rx(k) 6 c1 =⇒ x ⊤ (k)Rx(k) < c

2, ∀k = 1, 2, , N, for all disturbances ω(k) satisfying (3).

Definition 2.2 (Finite-time H ∞ boundedness) Given positive numbers γ, N, c1, c2, c1 < c2, and a symmetric positive-definite matrix R, the system (1) with u(k) = 0 is said to be robustly finite-time H ∞ bounded w.r.t (c1, c2, R, N ) if the following two conditions hold:

(i) The system (1) with u(k) = 0 is robustly finite-time stable w.r.t (c1, c2, R, N ).

(ii) Under the zero initial condition (i.e., φ(k) = 0 ∀k ∈ {−d2, −d2 + 1, , 0 }), the output z(k) satisfies

N

∑

k=0

z ⊤ (k)z(k) 6 γ

N

∑

k=0

for all disturbances ω(k) satisfying (3).

Definition 2.3 (Finite-time H ∞ control) Given positive numbers γ, N, c1, c2, c1 < c2, and

a symmetric positive-definite matrix R, the finite-time H ∞ control problem for the system

(1) has a solution if there exists a state feedback controller u(k) = Kx(k) such that the resulting closed-loop system is robustly finite-time H ∞ bounded w.r.t (c1, c2, R, N ).

Proposition 2.1 (Schur Complement Lemma, [15]) Given constant matrices X, Y, Z with

appropriate dimensions satisfying X = X ⊤ , Y = Y ⊤ > 0, then

X + Z ⊤ Y −1 Z < 0 ⇐⇒

[

]

< 0.

3 Main results

This section provides sufficient conditions for finite-time H ∞ boundedness and control for the system (1) with interval time-varying delay and norm-bounded disturbances

Theorem 3.1 Given positive constants γ, N, c1, c2, c1 < c2, a symmetric positive-definite matrix R, the system (1) with u(k) = 0 is robustly finite-time H ∞ bounded w.r.t (c1, c2, R, N )

if for scalar δ > 1, there exist symmetric positive-definite matrices P, Q, positive scalars

λ1, λ2, λ3 such that the following LMIs hold:













δ N I G ⊤ P 0













γd − c2δλ1 c1δ

N +1 λ2 ρλ3

∗ −c1δ N +1 λ2 0



where ρ := c1δ N +d2−1[

d2δ + d2(d2−1)−d1(d1−1)

2

]

Proof Consider the following non-negative quadratic functions:

V (k) = V1(k) + V2(k) + V3(k),

where

V1(k) = x ⊤ (k)P x(k),

V2(k) =

k −1

∑

s=k −d(k)

δ k −1−s x ⊤ (s)Qx(s)

V3(k) =

−d∑1 +1

s= −d2 +2

k −1

∑

t=k −1+s

δ k −1−t x ⊤ (t)Qx(t).

Taking the difference variation of V i (k), i = 1, 2, 3, we have

V1(k + 1) − δV1(k) = x ⊤ (k + 1)P x(k + 1) − δx ⊤ (k)P x(k)

=



x(k x(k) − d(k)) ω(k)





⊤

A

⊤

A ⊤ d

G ⊤



 P [A A d G] x(k)

x(k − d(k)) ω(k)



 − δx ⊤ (k)P x(k),

(V2+ V3)(k + 1) − δ(V2+ V3)(k) =

k

∑

s=k+1 −d(k+1)

δ k −s x ⊤ (s)Qx(s) −

k −1

∑

s=k −d(k)

δ k −s x ⊤ (s)Qx(s)

+

−d∑1 +1

s= −d2 +2

k

∑

t=k+s

δ k −t x ⊤ (t)Qx(t) −

−d∑1 +1

s= −d2 +2

k −1

∑

t=k −1+s

δ k −t x ⊤ (t)Qx(t)

= x ⊤ (k)Qx(k) +

k −1

∑

s=k −d1 +1

δ k −s x ⊤ (s)Qx(s) +

k∑−d1

s=k+1 −d(k+1)

δ k −s x ⊤ (s)Qx(s)

−

k −1

∑

s=k −d(k)+1

δ k −s x ⊤ (s)Qx(s) − δ d(k) x (k − d(k))Qx(k − d(k))

+

−d∑1 +1

s= −d2 +2

[

x ⊤ (k)Qx(k) +

k −1

∑

t=k+s

δ k −t x ⊤ (t)Qx(t) −

k −1

∑

t=k+s

δ k −t x ⊤ (t)Qx(t)

− δ1−s x ⊤ (k − 1 + s)Qx(k − 1 + s)

]

= x ⊤ (k)Qx(k) +

k −1

∑

s=k −d1 +1

δ k −s x ⊤ (s)Qx(s) −

k −1

∑

s=k −d(k)+1

δ k −s x ⊤ (s)Qx(s)

+

k∑−d1

s=k+1 −d(k+1)

δ k −s x ⊤ (s)Qx(s) − δ d(k)

x ⊤ (k − d(k))Qx(k − d(k))

+

−d∑1 +1

s= −d2 +2

[

x ⊤ (k)Qx(k) − δ1−s x ⊤ (k − 1 + s)Qx(k − 1 + s)]

6 x ⊤ (k)Qx(k) +

k∑−d1

s=k+1 −d(k+1)

δ k −s x ⊤ (s)Qx(s)

− δ d(k) x ⊤ (k − d(k))Qx(k − d(k)) + (d2− d1)x ⊤ (k)Qx(k)

− −d

1 +1

∑

s= −d2 +2

δ1−s x ⊤ (k − 1 + s)Qx(k − 1 + s)

6 (d2− d1+ 1)x ⊤ (k)Qx(k) − δ d1x ⊤ (k − d(k))Qx(k − d(k))

+

k∑−d1

s=k+1 −d(k+1)

δ k −s x ⊤ (s)Qx(s) −

k∑−d1

s=k+1 −d2

δ k −s x ⊤ (s)Qx(s)

6 (d2− d1+ 1)x ⊤ (k)Qx(k) − δ d1x ⊤ (k − d(k))Qx(k − d(k)).

Thus we have

V (k + 1) − δV (k) 6



x(k x(k) − d(k)) ω(k)





⊤

A

⊤

A ⊤ d

G ⊤



 P [A A d G] x(k)

x(k − d(k)) ω(k)





+ x ⊤ (k)[

−δP + (d2− d1+ 1)Q]

x(k) − δ d1x ⊤ (k − d(k))Qx(k − d(k)) + z ⊤ (k)z(k) − γ

δ N ω ⊤ (k)ω(k) + γ

δ N ω ⊤ (k)ω(k) − z ⊤ (k)z(k).

Note that by setting

ξ(k) := [

x ⊤ (k) x ⊤ (k − d(k)) ω ⊤ (k)]⊤

,

Υ :=[

P A P A d P G]

,

Φ :=



−δP + (d2− d1+ 1)Q + C

⊤ C C ⊤ C

δ N I



 ,

we can see that



x(k x(k) − d(k))

ω(k)





⊤

A

⊤

A ⊤ d

G ⊤



 P [A A d G] x(k)

x(k − d(k)) ω(k)



 = ξ ⊤ (k)Υ ⊤ P −1 Υξ(k)

and

x ⊤ (k)[

−δP +(d2 − d1+ 1)Q]

x(k) − δ d1x ⊤ (k − d(k))Qx(k − d(k)) + z ⊤ (k)z(k) − γ

δ N ω ⊤ (k)ω(k)

= x ⊤ (k)[

−δP + (d2− d1+ 1)Q + C ⊤ C]

x(k) + 2x ⊤ (k)C ⊤ C d x(k − d(k)) + x ⊤ (k − d(k))[−δ d1Q + C d ⊤ C d]

x(k − d(k)) − γ

δ N ω ⊤ (k)ω(k)

= ξ ⊤ (k)Φξ(k).

Therefore, we get

V (k + 1) − δV (k) 6 ξ ⊤ (k)[

Φ + Υ⊤ P −1Υ]

ξ(k) + γ

δ N ω ⊤ (k)ω(k) − z ⊤ (k)z(k). (8)

Next, by applying the Proposition 2.1, we have

Φ+Υ⊤ P −1 Υ < 0 ⇐⇒









−δP + (d2− d1+ 1)Q + C ⊤ C C ⊤ C d 0 A ⊤ P

δ N I G ⊤ P







 < 0

and hence









δ N I G ⊤ P







 +









C ⊤

C d ⊤

0 0







[C C d 0 0]

< 0,

which is evidently equivalent to the LMI (6) For this reason, from (8) it follows that

V (k + 1) − δV (k) 6 γ

δ N ω ⊤ (k)ω(k), ∀k ∈ Z+.

This estimation can be rewritten as

V (k) 6 δV (k − 1) + γ

δ N ω ⊤ (k − 1)ω(k − 1), ∀k = 1, 2, (9)

By iteration, and take the assumption (3) into account, the inequality (9) gives

V (k) 6 δ k V (0) + γ

δ N

k −1

∑

s=0

δ k −1−s ω ⊤ (s)ω(s)

6 δ N V (0) + γ

δ N δ N −1

N∑−1 s=0

ω ⊤ (s)ω(s)

< δ N V (0) + γ

Now, using the condition (5) and x(k) = φ(k) ∀k ∈ {−d2, −d2+ 1, , 0 }, it’s obvious that

V (0) = x ⊤ (0)P x(0) +

−1

∑

s= −d(0)

δ −1−s x ⊤ (s)Qx(s) +

−d∑1 +1

s= −d2 +2

−1

∑

t= −1+s

δ −1−t x ⊤ (t)Qx(t)

< λ2x ⊤ (0)Rx(0) + λ3δ d2−1 ∑−1

s= −d2

x ⊤ (s)Rx(s) + λ3δ d2−2 −d1

+1

∑

s= −d2 +2

−1

∑

t= −1+s

x ⊤ (t)Rx(t)

6[λ2+ λ3d2δ d2−1 + λ

3

d2(d2− 1) − d1(d1 − 1)

d2−2]

Taking (10), (11) into account, we obtain

V (k) < δ N σ + γ

where σ :=

[

λ2 + λ3d2δ d2−1 + λ

3

d2(d2− 1) − d1(d1− 1)

d2−2]

c1 On the other hand, also

from (5) it follows that

V (k) > x ⊤ (k)P x(k) > λ

1x ⊤ (k)Rx(k), ∀k ∈ Z+. (13) Note that by the Proposition 2.1, the LMI (7) is equivalent to

γd − c2δλ1+[

c1δ N +1 λ2 ρλ3] [c1δ N +1 λ2 0

0 ρλ3

]−1[

c1δ N +1 λ2

ρλ3

]

< 0

⇐⇒ γd − c2δλ1+[

c1δ N +1 λ2 ρλ3] [(c1δ N +1 λ2)−1 0

0 (ρλ3)−1

] [

c1δ N +1 λ2

ρλ3

]

< 0

⇐⇒ γd − c2δλ1+ c1δ N +1 λ2+ ρλ3 < 0

⇐⇒ γd − c2δλ1+ c1δ N +1 λ2+ c1δ N +d2−1[

d2δ + d2(d2− 1) − d1(d1− 1)

2

]

λ3 < 0

Consequently, we get from (12), (13) and (14) that:

x ⊤ (k)Rx(k) < 1

λ1δ

[

δ N +1 σ + γd

]

< c2,

which implies that the system is robustly finite-time stable w.r.t (c1, c2, R, N ) To complete the proof of the theorem, it remains to show the finite-time γ-level condition (4) For this,

from (8) it follows that

V (k + 1) 6 δV (k) + γ

δ N ω ⊤ (k)ω(k) − z ⊤ (k)z(k), ∀k ∈ Z+,

and by iteration, we have

06 V (k) 6 δ k V (0) +

k−1

∑

s=0

δ k−1−s

[ γ

δ N ω ⊤ (s)ω(s) − z ⊤ (s)z(s)]

Under the zero initial condition: V (0) = 0, as a result, inequality (15) gives

k −1

∑

s=0

δ k −1−s z ⊤ (s)z(s) 6

k −1

∑

s=0

δ k −1−s γ

δ N ω ⊤ (s)ω(s).

For k = N + 1, we have

N

∑

s=0

δ N −s z ⊤ (s)z(s) 6 γ

N

∑

s=0

δ N −s

δ N ω ⊤ (s)ω(s). (16)

Since 16 δ N −s 6 δ N ∀s ∈ {0, 1, , N}, (16) immediately yields

N

∑

s=0

z ⊤ (s)z(s) 6 γ

N

∑

s=0

ω ⊤ (s)ω(s),

which proves the condition (4) This completes the proof of the theorem

In the sequel, we will solve the problem of finite-time H ∞ control for system (1), i.e.,

we will design a state feedback controller u(k) = Kx(k) such that the resulting closed-loop

system

x(k + 1) = (A + BK)x(k) + A d x(k − d(k)) + Gω(k),

x(k) = φ(k), k ∈ {−d2, −d2+ 1, , 0 },

is robustly finite-time H ∞ bounded

Theorem 3.2 The finite-time H ∞ control of system (1) has a solution if for scalar δ > 1, there exist symmetric positive definite matrices U, V, W1, W2, W3 and any matrix Y such that the following LMIs hold:













−δU + (d2− d1+ 1)V 0 0 U A ⊤ + Y ⊤ B ⊤ U C ⊤

δ N I G ⊤ 0











< 0, (19)



−W1 c1δ

N +1 W2 ρW3

∗ −c1δ N +1 W2 0



[

W1− c2δU γdU R

]

Moreover, the state feedback controller is given by

u(k) = Y U −1 x(k), k ∈ Z+.

Proof Evidently, from the Theorem 3.1, the system (17) is robustly finite-time H ∞ bounded

if for scalar δ > 1, there exists symmetric positive definite matrices P, Q, positive scalars

λ1, λ2, λ3 such that the LMIs (5)-(7) hold, therein matrix A + BK will in place of the matrix

A In other words, in proportion to the LMI (6), we have













δ N I G ⊤ P 0











< 0. (22)

Pre- and post-multipling (22) by matrix diag{P −1 , P −1 , I, P −1 , I } > 0 yields













−δP −1 + (d

2− d1+ 1)P −1 QP −1 0 0 P −1 (A + BK) ⊤ P −1 C ⊤

δ N I G ⊤ 0











< 0.

(23)

Let’s define new matrix variables as follows: U = P −1 , V = P −1 QP −1 Then, (23) becomes













−δU + (d2 − d1+ 1)V 0 0 U (A + BK) ⊤ U C ⊤

δ N I G ⊤ 0











< 0.

Letting Y ⊤ = U K ⊤ , K = Y U −1 , we get the LMI (19) For getting LMI (20), we note that

the inequality (7) can be regarded as



(γd − c2δλ1)I c1δ

N +1 λ2I ρλ3I

∗ −c1δ N +1 λ2I 0



Post-multipling (24) by matrix diag{R, R, R} > 0 gives



γdR − c2δλ1R c1δ

N +1 λ2R ρλ3R

∗ −c1δ N +1 λ2R 0



Again pre- and post-multipling (25) by matrix diag{P −1 , P −1 , P −1 } > 0, we get



γdP

−1 RP −1 − c2δP −1 (λ1R)P −1 c1δ N +1 P −1 (λ2R)P −1 ρP −1 (λ3R)P −1

3R)P −1



 < 0 (26)

Setting new variables

W1 =−γdP −1 RP −1 + c

2δP −1 (λ1R)P −1 , W2 = P −1 (λ2R)P −1 , W3 = P −1 (λ3R)P −1 ,

the LMI (26) reduces to the LMI (20) as desired To obtain the LMI (18), we just pre- and

post-multipling (5) by the matrix P −1 Finally, note that

W1 =−γdP −1 RP −1 + c

2δP −1 (λ1R)P −1 < −γdURU + c2δU,

we obtain

W1 − c2δU + γdU R[γdR] −1 γdRU < 0,

which is obviously equivalent to the LMI (21) by the Schur Complement Lemma The proof

of the theorem is completed

Remark 3.1 Different from the previous results [8, 11-13], the Lyapunov function method

is not used for the proof of Theorem 3.1 All the sufficient conditions of Theorem 3.1, Theorem 3.2 are given in terms of LMIs, which can be easily calculated by the LMI Toolbox

in MATLAB

Remark 3.2 In the papers [6, 8, 11-13], additional unknowns and free-weighting matrices

are introduced to make the flexibility to solve the resulting LMIs However, too many unknowns and free-weighting matrices employed in the existing methods complicate the system analysis and significantly increase the computational demand Compared with the free matrix method used in these papers, our simpler uncorrelated augmented matrix method uses fewer variables, e.g., the LMI (6) has no free-weighting matrices, the LMI (19) has one free-weighting matrix Consequently, our criterions are less conservative in comparison with others This effectiveness of the results will be illustrated by the following examples

Example 3.1 Consider the linear discrete-time delay system (1) with u(k) = 0 and its

parameters are described by

A =

[

−0.25 0.1

0.2 0.3

]

, A d=

[

−0.12 0.1 0.15 0.1

]

[

0.2 0.1 0.2 0.25

]

,

C =

[

0.1 −0.2

−0.15 0.15

]

, C d=

[

−0.1 0.25 0.2 −0.15

]

, R = I, h(k) = 2 + 8 sin 2 kπ2 , k ∈ Z+.

Note that the delay function h(k) is interval time-varying and d1 = 2, d2 = 10 For given

N = 200, d = 1, c1 = 1, c2 = 7, and γ = 1, by using LMI Control Toolbox in MATLAB, the LMIs (5)-(7) are feasible with δ = 1.0001 and

P =

[

2.1225 0.0261 0.0261 2.0246

]

[

0.1901 −0.0194

−0.0194 0.1548

]

,

λ1 = 2.0180, λ2 = 2.1292, λ3 = 0.1987.

By the Theorem 3.1, the system is robustly finite-time H ∞ bounded w.r.t (1, 7, I, 200).

Example 3.2 Consider the system (1) where: