New stability criterion using a matrix-based quadratic ... - IEEE Xplore

www.ietdl.org Published in IET Control Theory and Applications Received on 15th September 2013 Revised on 11th December 2013 Accepted on 11th February 2014 doi: 10.1049/iet-cta.2013.0840

ISSN 1751-8644

New stability criterion using a matrix-based quadratic convex approach and some novel integral inequalities Xian-Ming Zhang, Qing-Long Han Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton QLD 4702, Australia E-mail: [email protected]

Abstract: This study is concerned with the stability of a linear system with an interval time-varying delay. First, t a new augmented Lyapunov–Krasovskii functional (LKF) is constructed, which includes three integral terms in the form of t−h (h − t + s)j x˙ T Rj x˙ (s) ds (j = 1, 2, 3). Second, three novel integral inequalities are established to estimate the upper bounds of the t integrals t−h (h − t + s)j x˙ T Rj x˙ (s) ds (j = 0, 1, 2) appearing in the derivative of the LKF. Third, a matrix-based quadratic convex approach is introduced to prove not only the negative definiteness of the derivative of the LKF along with the trajectory of the system, but also the positive definiteness of the LKF. Finally, a novel delay-derivative-dependent stability criterion is formulated. The effectiveness of the stability criterion is shown through two numerical examples.

1

Introduction

During the last decade, time-delay systems have received increasing attention because of their applications in control systems [1, 2]. It has been shown that networked control systems [3, 4] and event-triggered control systems [5, 6], which are two hot research topics, can be modelled as linear systems with interval time-varying delays. Since time-delays are the factor degrading system performance or even destabilising a control system, much effort has been made on time-delay systems and up to date, a number of results on delay-dependent stability have been published in the literature, see e.g. [7–16]. Consider a time-delay system described by x˙ (t) = Ax(t) + Ad x(t − d(t)),

θ ∈ [−h, 0] (1) where x(t) ∈ Rn is the system state; A ∈ Rn×n and Ad ∈ Rn×n are real matrices; φ0 is an initial condition; and the timevarying delay d(t) satisfies 0 ≤ d(t) ≤ h < ∞,

x(θ ) = φ(θ ),

˙ ≤μ 0 and R0 > 0. It is obvious 2 0 t rewritten as a double integral term −h t+θ x˙ T (s)R0 x˙ (s) ds dθ . Recently, Kim [7] has introduced an LKF, where the above two terms are replaced with ⎧ ⎤ ⎡ ⎤T ⎡ x(t) x(t) ⎪ ⎪   ⎪ t ⎪ ⎦P⎣ t ⎦ T := ⎣ ⎪ ⎪ x(s) ds x(s) ds ⎨ 3 t−h

t−h

⎪ 3 t ⎪

⎪ ⎪ ⎪ (h − t + s)j x˙ T (s)Rj x˙ (s) ds ⎪ ⎩ T4 := j=1

(4)

t−h

(2)

The delay-dependent stability analysis for the system described by (1) and (2) aims to establish a stability condition to calculate the maximum allowable upper bound hmax of h such that the system described by (1) and (2) is asymptotically stable for 0 ≤ d(t) ≤ hmax . The hmax is thus regarded as a performance index to measure the conservatism of a stability criterion. To reduce the conservatism, during the last decade, several approaches have been proposed, for example, a descriptor model transformation, a freeweighting matrix method, an integral inequality approach, a delay decomposition approach, a (reciprocally) convex 1054 © The Institution of Engineering and Technology 2014

delay analysis approach and a Wirtinger’s inequality-based technique. As is known, the Lyapunov–Krasovskii functional (LKF) plays a crucial role in deriving less conservative stability criteria. The following two terms are usually included in the LKF t (h − t + s)˙xT (s)R0 x˙ (s) ds T1 := xT (t)P0 x(t), T2 :=

The difference between (3) and (4) is that: (i) the t term T1 is augmented by T3 ; and (ii) two terms, namely, t−h (h − t + s)j x˙ T (s)Rj x˙ (s) ds (j = 2, 3), are added to T4 . As T2 represents a double integral, the term T4 can be considered as the sum of a double integral, a triple integral and a quadruple integral. Therefore one can see that the LKF [7] opens a door to investigate the stability of time-delay systems. Combining with a quadratic convex approach, some stability criteria have been derived in [7]. However, several issues should be addressed, which are listed in the following. For the first issue, by taking the time derivative of t T4 , it is found that the integral terms − t−h (h − t + IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1054–1061 doi: 10.1049/iet-cta.2013.0840

www.ietdl.org s)i x˙ T (s)Ri+1 x˙ (s) ds (i = 1, 2) appear. The main characteristic of these integral terms is that the integrands are the quadratic functions x˙ T (s)Ri+1 x˙ (s) (i = 1, 2) multiplied by a polynomial with respect to h − t + s. How to bound these integral terms is certainly a challenging issue. Although some upper bounds of the integral terms are offered in [7], they are conservative because the estimation is made just on the integrands (h − t + s)i x˙ T (s)Ri+1 x˙ (s) (i = 1, 2) rather than on the integrals. Moreover, the estimation on the integrands is based on the so-called basic inequality. As is known, the basic inequality usually leads to some conservative results. For the second issue, although T1 is augmented by T3 , the augmented matrix P is required to be positive definite such that the positive definiteness of the chosen LKF can be ensured. However, the constraint P > 0 possibly results in the conservatism of the stability criterion. For the third issue, the use of the quadratic convex approach in [7] is improper. In the proof of Theorem 1 in [7], the author claims that the function ξtT [0 + d(t)1 + ϒd ]ξt is a convex quadratic function on d(t). In fact, this function is not a quadratic function on d(t) because ξt = col{x(t), x(t − d(t)), . . .} is dependent on d(t) implicitly. For the fourth issue, when the lower bound of the timevarying delay d(t) is known to be greater than zero, the stability criteria obtained in [7] is certainly conservative because this case is not considered. However, how to address the above issues is still challenging, which motivates the current study. In this paper, we will introduce a matrix-based quadratic convex approach to study the stability of a linear system with an interval timevarying delay. Suppose that the time-varying delay is in the following general form ˙ ≤ μ2 < ∞ −∞ < μ1 ≤ d(t) (5) Correspondingly, the initial condition φ(θ ) of the system (1) is defined on [−h2 , 0]. First, a more general LKF V (t, xt , x˙ t ) than the one in [7] is constructed, where the augmented vector includes the delay x(t − h1 ), some t−hstate t−d(t) distributed delay t 1 x(s) ds and t−h2 x(s) ds. Second, terms t−h1 x(s) ds, t−d(t) some new integral inequalities for the integral terms in t−h the form of t−h21 (h2 − t + s)i x˙ T (s)Ri+1 x˙ (s) ds (i = 1, 2) are established using Jensen’s inequality and some new bounding techniques recently reported in the literature, rather than using the basic inequality as in [7]. Third, a matrix-based quadratic convex approach is introduced not only to the proof of V˙ (t, xt , x˙ t ) ≤ −2 xT (t)x(t) but also to the proof of V (t, xt , x˙ t ) ≥ 1 xT (t)x(t), where 1 and 2 are two positive constants. It is clear to see that the constraint of P > 0 is removed from the obtained stability criterion. Finally, two examples are given to demonstrate that the proposed result is less conservative than some existing ones. 0 ≤ h1 ≤ d(t) ≤ h2 < ∞,

2

Some new integral inequalities

In this section, we will establish a few new integral inequalities, which are useful in the delay-dependent stability analysis for time-delay systems. In doing so, we first introduce an inequality, which includes the so-called basic inequality as its special case. Lemma 1: Let α and β be real column vectors with dimensions of n1 and n2 , respectively. For given real positive symmetric matrices M1 ∈ Rn1 ×n1 and M2 ∈ Rn2 ×n2 , the following inequality holds for any scalar κ > 0 and matrix IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1054–1061 doi: 10.1049/iet-cta.2013.0840

S ≥ 0 is satisfied M2



S∈R

n1 ×n2

M1 provided that ST

−2α T Sβ ≤ κα T M1 α + κ −1 β T M2 β Proof: Notice that   κM1 M1 S ≥ 0 ⇐⇒ S T M2 ST

(6)

S ≥0 κ −1 M2

Then the inequality (6) can be readily obtained.



If setting κ = 1, M1 = Q > 0, M2 = Q−1 and S = I , then the inequality (6) reduces to the so-called basic inequality −2α T β ≤ α T Qα + β T Q−1 β

(7)

Now, based on Lemma 1 weestablish three novel integral t−h inequalities for the integrals t−h21 (h2 − t + s)j x˙ T (s)Rj+1 x˙ (s) ds (j = 0, 1, 2). To begin with, denote 1 ν1 (t) := h2 − d(t) ν2 (t) :=

1 d(t) − h1

 t−d(t) x(s) ds, t−h2

 t−h1 x(s) ds

(8)

t−d(t)

Notice that lim ν1 (t) = x(t − h2 ),

d(t)→h− 2

lim ν2 (t) = x(t − h1 )

d(t)→h+ 1

Thus, ν1 (t) and ν2 (t) are well defined. Lemma 2: Let d(t) be a continuous function satisfying 0 ≤ h1 ≤ d(t) ≤ h2 . For any n × n real matrix R3 > 0 and a vector x˙ : [−h2 , 0] → Rn such that the integration concerned below is well defined, the following inequality holds for any

 R S n × n real matrix S2 satisfying S2T3 R23 ≥ 0 −

 t−h1

(h2 − t + s)2 x˙ T (s)R3 x˙ (s) ds

t−h2 T T ≤ −(h2 − d(t))[φ12 R3 φ12 + 2φ22 (R3 − S2 )φ23 ]

where

⎧ ⎪ ⎨ φ12 := x(t − d(t)) − ν1 (t) φ22 := x(t − h1 ) − x(t − d(t)) ⎪ ⎩ φ23 := x(t − h1 ) − ν2 (t)

(9)

(10)

Proof: Since −

 t−h1

(h2 − t + s)2 x˙ T (s)R3 x˙ (s) ds = η1 (t) + η2 (t)

(11)

t−h2

where  t−d(t) η1 (t) := −

(h2 − t + s)2 x˙ T (s)R3 x˙ (s) ds t−h2

η2 (t) := −

 t−h1

(h2 − t + s)2 x˙ T (s)R3 x˙ (s) ds

t−d(t)

Set w(s) = (h2 − t + s)˙x(s). We consider three cases. 1055 © The Institution of Engineering and Technology 2014

www.ietdl.org Case 1: h1 < d(t) < h2 . Apply Jensen inequality to obtain 1 η1 (t) ≤ − h2 − d(t) 1 η2 (t) ≤ − d(t) − h1

 t−d(t)

T

 t−d(t) w(s)ds t−h2

R3

w(s)ds

Lemma 3: Let d(t) be a continuous function satisfying 0 ≤ h1 ≤ d(t) ≤ h2 . For any n × n real matrix R2 > 0 and a vector x˙ : [−h2 , 0] → Rn such that the integration concerned below is well defined, the following inequality holds for any q q×q q×n φ satisfying

i1 ∈ R and real matrices Zi ∈ R , Ni ∈ R Zi Ni NiT R2 ≥ 0 (i = 1, 2)

w(s) ds t−h2

 t−h1

T

 t−h1



R3

w(s) ds

t−d(t)

t−d(t)

Performing some simple algebraic manipulations yields T η1 (t) ≤ −(h2 − d(t))φ12 R3 φ12

(12)

(h2 − d(t))2 T T η2 (t) ≤ − R3 φ23 φ R3 φ22 − (d(t) − h1 )φ23 d(t) − h1 22 T − 2(h2 − d(t))φ22 R3 φ23

bounded by zero, which shows the limitation to the application of the upper bound obtained in [7].

−

 t−h1

≤

(13)

An observation of (13) is that −

(h2 − d(t))2 T T R3 φ23 φ R3 φ22 − (d(t) − h1 )φ23 d(t) − h1 22

Proof: It is clear to see that

where κ0 := [h2 − d(t)]/[d(t) − h1 ]. Apply Lemma 1 to obtain +

T κ0−1 φ23 R3 φ23

≥

T −2φ22 S2 φ23

1 T T Z1 φ11 + 2(h2 − d(t))φ11 N1 φ12 (h2 − d(t))2 φ11 2 1 T Z2 φ21 + [(h2 − h1 )2 − (h2 − d(t))2 ]φ21 2 T + 2φ21 N2 [(h2 − d(t))φ22 + (d(t) − h1 )φ23 ] (17)

where φ12 , φ22 and φ23 are defined in (10).

T T = −(h2 − d(t))[κ0 φ22 R3 φ22 + κ0−1 φ23 R3 φ23 ]

T R3 φ22 κ0 φ22

(h2 − t + s)˙xT (s)R2 x˙ (s) ds

t−h2

−

 t−h1

(h2 − t + s)˙xT (s)R2 x˙ (s) ds = φ1 (t) + φ2 (t)

(18)

t−h2

(14) where

which leads to T η2 (t) ≤ 2(h2 − d(t))φ22 (S2 − R3 )φ23

 t−d(t) (15)

Substituting (12) and (15) into (11) yields (9). Case 2: h1 = d(t) < h2 . In this case, φ22 = 0. Similar to the analysis in Case 1, we have −

 t−h1

T (h2 − t + s)2 x˙ T (s)R3 x˙ (s) ds ≤ −(h2 − h1 )φ12 R3 φ12

t−h2

(16)

which gives (9) with d(t) = h1 . Case 3: h1 < d(t) = h2 . In this case, the inequality (9) becomes  t−h1 (h2 − t + s)2 x˙ T (s)R3 x˙ (s) ds ≤ 0 − t−h2

which is true as R3 > 0. The proof is thus completed.



Remark 1: The proof of Lemma 2 gives a new way to bound the integral term in (9), which is based on Jensen’s inequality and Lemma 1. The observation of (13) leading to (14) plays a key role in this bounding technique. Nonetheless, although an estimation on the integral term is made in [7], the obtained upper bound is conservative. In fact, in [7], the upper bound of the integral term is derived to bound the integrand rather than the integral itself, and thus the characteristic of integral is not taken into account. Moreover, the integrand is bounded based on the so-called basic inequality, leading to a conservative result. On the other hand, when applying the upper bound obtained in [7] to estimate the term η2 (t) in (11), t−h1 (d(t) − η2 (t) should be first enlarged as η2 (t) ≤ − t−d(t)  t−h1 2 T t + s) x˙ (s)R3 x˙ (s) ds. In this way, the term − t−d(t) [(h2 − d(t))2 + 2(h2 − d(t))(d(t) − t + s)]˙xT (s)R3 x˙ (s)ds is overly 1056 © The Institution of Engineering and Technology 2014

φ1 (t) := −

(h2 − t + s)˙xT (s)R2 x˙ (s) ds t−h2

φ2 (t) := −

 t−h1

(h2 − t + s)˙xT (s)R2 x˙ (s) ds

t−d(t)

Now, we first make an estimation on φ1 (t). Set w(s) = (h2 − t + s)˙x(s). Then (h2 − t + s)˙xT (s)R2 x˙ (s) 1 = wT (s)R2 w(s) (h2 − t + s) T T ≥ −(h2 − t + s)φ11 Z1 φ11 − 2φ11 N1 w(s)

(19)

where the above inequality is obtained by Lemma 1. It follows that  t−d(t) T T φ1 (t) ≤ [(h2 − t + s)φ11 Z1 φ11 + 2φ11 N1 w(s)] ds t−h2

1 T T = (h2 − d(t))2 φ11 Z1 φ11 + 2(h2 − d(t))φ11 N1 φ12 2 (20) Similarly φ2 (t) ≤

1 T Z2 φ21 [(h2 − h1 )2 − (h2 − d(t))2 ]φ21 2 T + 2φ21 N2 [(h2 − d(t))φ22 + (d(t) − h1 )φ23 ]

Substituting (20) and (21) into (18) yields (17).

(21) 

Remark 2: Lemma t−h3 presents a new upper bound for the integral term − t−h21 (h2 − t + s)˙xT (s)R2 x˙ (s) ds. Similar to the discussion in Remark 1, one can see that, even though IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1054–1061 doi: 10.1049/iet-cta.2013.0840

www.ietdl.org an estimation on this term is also made in [7], the obtained upper bound is conservative and limited when the upper bound is used. Lemma 4: Let d(t) be a continuous function satisfying 0 ≤ h1 ≤ d(t) ≤ h2 . For any n × n real matrix R1 > 0 and a vector x˙ : [−h2 , 0] → Rn such that the integration concerned below is well defined, the following holds for any 

inequality 2n × 2n real matrix S1 satisfying − (h2 − h1 )

 t−h1

R˜ 1 S1 S1T R˜ 1

where R˜ 1 := diag{R1 , 3R1 }; and  ⎧ x(t − d(t)) − x(t − h2 ) ⎪ ⎪ ⎨ ψ11 := x(t − d(t)) + x(t − h2 ) − 2ν1 (t)  ⎪ x(t − h1 ) − x(t − d(t)) ⎪ ⎩ ψ21 := x(t − h1 ) + x(t − d(t)) − 2ν2 (t)

(22)

(23)

Proof: It is clear to show that the inequality (22) is true for two special cases where h1 = d(t) < h2 and h1 < d(t) = h2 , respectively. Now, suppose that h1 < d(t) < h2 . Then x˙ T (s)R1 x˙ (s) ds = ψ1 (t) + ψ2 (t)

(24)

t−h2

where  t−d(t) x˙ T (s)R1 x˙ (s) ds t−h2

ψ2 (t) := −(h2 − h1 )

 t−h1

x˙ T (s)R1 x˙ (s) ds

Applying Corollary 5 in [12], we have  T ψ1 (t) ≤ −(1 + κ)ψ11 diag{R1 , 3R1 }ψ11 T ψ2 (t) ≤ −(1 + κ −1 )ψ21 diag{R1 , 3R1 }ψ21 d(t)−h1 . h2 −d(t)

∀τ ∈ [τ1 , τ2 ]

+ τi X1 + X2 < 0 (≤0),

(i = 1, 2)

(25)

Remark 3: Lemma 4 provides an upper bound for the intet−h gral term −(h2 − h1 ) t−h21 x˙ T (s)R1 x˙ (s) ds, which just introduces one 2n × 2n slack matrix variable S1 . However, in [7], two 2n × 5n slack matrix variables are introduced to bound this term. When h1 = 0, the inequality (22) reduces to a similar one in [12].

3 Matrix-based quadratic convex approach and a novel stability criterion In this section, we first introduce a matrix-based quadratic convex approach. Then we employ this approach to establish a novel stability criterion for the system described by (1) and (5).

(26)

∀τ ∈ [τ1 , τ2 ]

+ τi X1 + X2 > 0 (≥0),

(i = 1, 2)

Since τ 2 X0 + τ X1 + X2 is a quadratic convex combination of matrices Xj (j = 0, 1, 2) with X0 ≥ 0 on τ ∈ [τ1 , τ2 ], an approach based on Lemma 5 is referred to as a matrixbased quadratic convex approach. It is worth pointing out that if taking X0 = 0, X1 = X1 − X2 and X2 = X0 + τ2 X2 − τ1 X1 , where X0 , X1 and X2 are real symmetric matrices, then (26) reduces to ∀τ ∈ [τ1 , τ2 ]

(28)

which is well used in the stability analysis for linear systems with interval time-varying delays. Clearly, the condition in (28) is the special case of (26). 3.2

Stability criterion

Now, we employ the matrix-based quadratic convex approach, together with the integral inequalities established in the previous section to formulate a novel stability criterion for the system described by (1) and (5). For this goal, let (see (29)) Then the LKF candidate is chosen as V (t, xt , x˙ t ) = V1 (t) + V2 (t) + V3 (t)

(30)

where xt denotes the function x(s) defined on the interval [t − h2 , t], and t V1 (t) := ϑ T (t)Pϑ(t) +

x˙ T (s)Q0 x˙ (s) ds t−h1

 t−h1



t

x(s) ds, t−h2

(27)

Proof: Suppose that ζ = 0 is an arbitrary vector in Rm . Let f (τ ) := ζ T (τ 2 X0 + τ X1 + X2 )ζ . Then f (τ ) is a con2 vex function on τ ∈ [τ1 , τ2 ] because dτd 2 f (τ ) = 2ζ T X0 ζ ≥ 0 for X0 ≥ 0. By the convex function property, f (τ ) < 0 (≤0), ∀τ ∈ [τ1 , τ2 ], is equivalent to f (τi ) < 0 (≤ 0) (i = 1, 2), or f (τ ) > 0 (≥0), ∀τ ∈ [τ1 , τ2 ], is equivalent to f (τi ) > 0 (≥0) (i = 1, 2). Since the arbitrariness of the vector ζ , either (26) or (27) is true. This completes the proof. 

 t−d(t))

ϑ(t) := col x(t), x(t − h1 ), IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1054–1061 doi: 10.1049/iet-cta.2013.0840

⇐⇒

τi2 X0

⇐⇒ X0 + (τ2 − τ1 )Xi < 0 (i = 1, 2)

By Lemma 1, 



τ 2 X0 + τ X1 + X2 > 0 (≥0),

X0 + (τ − τ1 )X1 + (τ2 − τ )X2 < 0,

t−d(t)

where ψj1 is defined in (23) and κ := one readily obtains (22).

τi2 X0

or

x˙ T (s)R1 x˙ (s) ds

ψ1 (t) := −(h2 − h1 )

τ 2 X0 + τ X1 + X2 < 0 (≤0), ⇐⇒

T ˜ T T ˜ ≤ 2ψ11 R1 ψ21 R1 ψ11 − ψ21 S1 ψ21 − ψ11

−(h2 − h1 )

Matrix-based quadratic convex approach

Lemma 5: Let X0 , X1 and X2 be m × m real symmetric matrices and a continuous function τ satisfy τ1 ≤ τ ≤ τ2 , where τ1 and τ2 are constants satisfying 0 ≤ τ1 ≤ τ2 . If X0 ≥ 0, then

≥0

t−h2

 t−h1

3.1

x(s) ds, t−d(t)

x(s) ds

(29)

t−h1

1057 © The Institution of Engineering and Technology 2014

www.ietdl.org t

if the following linear matrix inequalities (LMIs) are satisfied

[xT (t) xT (s)]Q1 [xT (t) xT (s)]T ds

V2 (t) := t−h1

+

 t−h1

e¯ 1 P e¯ 1T > 0,

[x (t) x (s)]Q2 [x (t) x (s)] ds T

T

T

T

T

t−d(t)

 t−d(t) + t−h2

+ (d(t) − h1 )[¯e1T e¯ 4T ]Q2 [¯e1T e¯ 4T ]T

t−h1

+ (h2 − d(t))[¯e1T e¯ 3T ]Q3 [¯e1T e¯ 3T ]T

+ (h1 − t + s)2 x˙ T (s)W2 x˙ (s)] ds t (h1 − t + s)3 x˙ T (s)W3 x˙ (s) ds + +

2 := col{¯e1 − e¯ 2 , e¯ 1 + e¯ 2 − 2¯e5 } (h2 − t + s) x˙ (s)R3 x˙ (s) ds 3 T

P0 :=

t−h2

+

 t−h1

(¯e4T e¯ 4

−

e¯ 3T e¯ 3 )P(¯e4T e¯ 4

−

e¯ 3T e¯ 3 )

(38) (39)

and e¯ i ∈ Rn×5n (i = 1, 2, . . . , 5) are the ith row-block vector of the 5n × 5n identity matrix.

[(h2 − t + s)˙xT (s)h21 R1 x˙ (s)

t−h2

+ (h2 − t + s)2 x˙ T (s)R2 x˙ (s)] ds

Proof: The proof of the first ‘≤‘ in (33). Denote

where Q0 > 0, Qj > 0, Wj > 0, Rj > 0 (j = 1, 2, 3) and P are real matrices to be determined; and h21 := h2 − h1 . Remark 4: Compared with [7], the LKF (30) has the following advantages:

• the augment vector ϑ(t) includes not only the delayed state x(t − h1 ) but also some distributed delay terms; • information about the lower bound of the delay, h1 , is taken into account in V3 (t) and • the augmented Lyapunov matrix P does not need to be positive definite, which can be seen from Lemma 6 below. It is clear to see that V (t, xt , x˙ t ) in (30) is a quadratic LKF closely depending on derivatives. With this LKF, we need the following stability theorem to establish a novel stability criterion for the system described by (1) and (5). Theorem 1 (Zhang and Han [16]): The system described by (1) and (5) is asymptotically stable if the exists a quadratic ˙ such that for some εi > 0 (i = 1, 2, 3) LKF V (t, φ, φ) ˙ ≤ ε2 φ2W ε1 φ(0)2 ≤ V (t, φ, φ) ˙ ≤ −ε3 φ(0)2 V˙ (t, φ, φ)

(36)

1 := col{¯e1 , e¯ 2 , (h2 − d(t))¯e3 , (d(t) − h1 )¯e4 , h1 e¯ 5 } (37)

t−h1

 t−h1

(34)

ϒ2 (d(t)) := h1 [¯e1T e¯ 5T ]Q1 [¯e1T e¯ 5T ]T

[(h1 − t + s)˙xT (s)h1 W1 x˙ (s)

V3 (t) :=

ϒ1 (h1 ) + ϒ2 (h1 ) ≥ 0,

where (see (35))

[xT (t) xT (s)]Q3 [xT (t) xT (s)]T ds

t

P0 ≥ 0,

ϒ1 (h2 ) + ϒ2 (h2 ) ≥ 0

(31) (32)

0 0 ˙ )2 where φ2W = φ(0)2 + −h2 φ(s)2 ds + −h2 φ(θ dθ with the vector norm  ·  denoting the Euclidean norm. In the sequel, we will employ the matrix-based quadratic convex approach to establish sufficient conditions such that the chosen LKF (30) satisfies (31) and (32).

χ (t) := col {x(t), x(t − h1 ), ν1 (t), ν2 (t), ν3 (t)} where νj (j = 1, 2) are defined in (8); and  1 t x(s) ds ν3 (t) := h1 t−h1

(40)

(41)

Then ϑ(t) = 1 χ (t), where ϑ(t) is defined in (29). Applying Corollary 5 in [12] to V1 (t) and Jensen’s inequality to V2 (t), we have V1 (t) ≥ xT (t)¯e1 P e¯ 1T x(t) + χ T (t)ϒ1 (d(t))χ (t)

(42)

V2 (t) ≥ χ (t)ϒ2 (d(t))χ (t)

(43)

T

where ϒ1 (d(t)) and ϒ2 (d(t)) are defined in (35) and (36), respectively. It follows that V (t, xt , x˙ t ) ≥ xT (t)¯e1 P e¯ 1T x(t) + χ T (t)ϒ0 (d(t))χ (t)

(44)

where ϒ0 (d(t)) := ϒ1 (d(t)) + ϒ2 (d(t)). Observe that ϒ0 (d(t)) can be rewritten as ϒ0 (d(t)) = d 2 (t)P0 + d(t)P1 + P2

(45)

where P0 is given in (39) and P1 and P2 are some corresponding constant matrices. If the LMIs in (34) hold, applying Lemma 5 yields ϒ0 (d(t)) ≥ 0 for d(t) ∈ [h1 , h2 ]. Let ε1 := λmin (¯e1 P e¯ 1T ). Then ε1 > 0 because of e¯ 1 P e¯ 1T > 0. Therefore there exists a scalar ε1 > 0 such that ε1 xT (t)x(t) ≤ V (t, xt , x˙ t ) is satisfied. The proof of the second ‘≤’ in (33). From (34), it is clear that λmax (P) > 0. After some simply algebraic manipulations, we have that V1 (t) ≤ 2 max{λmax (P), λmax (Q0 )}xt 2W V2 (t) ≤ max {λmax (Qi )} max{h2 , 1}xt 2W

Lemma 6: For the LKF (30), there exist scalars ε1 > 0 and ε2 > 0 such that ε1 xT (t)x(t) ≤ V (t, xt , x˙ t ) ≤ ε2 xt 2W  ϒ1 (d(t)) :=

(33)

T1 P1 − e¯ 1T e¯ 1 P e¯ 1T e¯ 1 , T1 P1 − e¯ 1T e¯ 1 P e¯ 1T e¯ 1 +

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i=1,2,3

V3 (t) ≤ a0 (h2 + h22 + h32 )xt 2W where a0 := max{λmax (h1 W1 ), λmax (W2 ), λmax (W3 ), λmax (h21 R1 ), λmax (R2 ), λmax (R3 )}. Thus, one can conclude

1 T2 diag{Q0 , 3Q0 }2 , h1

h1 = 0 h1 = 0

(35)

IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1054–1061 doi: 10.1049/iet-cta.2013.0840

www.ietdl.org that there exists a scalar ε2 > 0 such that V (t, xt , x˙ t ) ≤ ε2 xt 2W . 

with ei (i = 1, 2, . . . , 8) denoting the ith row-block vector of ˜ 1 = diag{W1 , 3W1 }; and the 8n × 8n identity matrix; W

Remark 5: The condition on the positive definiteness of the chosen LKF is also discussed in [17] for discrete-time singular systems. Quite different from [17], the key to the proof of Lemma 6 is the introduction of the vector χ (t) in (40). As a result, the quadratic convex combination of matrices P0 , P1 and P2 on d(t) [i.e. (45)] can be obtained so that the matrix-based quadratic convex approach can be applied. Moreover, from Lemma 6, one can see clearly that the constraint P > 0 appearing in [7] is no longer required for the positive definiteness of the chosen LKF.

20 := [e1T e3T ](Q2 − Q1 )[e1T e3T ]T + h1 [C0T 0]Q1 [e1T e7T ]T

Lemma 7: For the derivative of the LKF (30) along with the trajectory of the system described by (1) and (5), there exists a scalar ε3 > 0 such that V˙ (t, xt , x˙ t ) ≤ −ε3 xT (t)x(t)

(46)



R˜ 1 S1T

Zi NiT

− [e1T e4T ]Q3 [e1T e4T ]T + [e1T e1T ]Q1 [e1T e1T ]T 21 := [e1T e6T ]Q2 [C0T 0]T + [C0T 0]Q2 [e1T e6T ]T 22 := [e1T e5T ]Q3 [C0T 0]T + [C0T 0]Q3 [e1T e5T ]T 31 := 2N1 (e2 − e5 ) − 3(e2 − e5 )T R3 (e2 − e5 ) + 2N2 (e3 − e2 ) − 3(e3 − e2 )T (R3 − S2 )(e3 − e6 ) + 2(e3 − e2 )T N2T − 3(e3 − e6 )T (R3 − S2 )T (e3 − e2 ) + 2(e2 − e5 )T N1T 32 := 2N2 (e3 − e6 ) + 2(e3 − e6 )T N2T ψ˜ 1 := col{e2 − e4 , e2 + e4 − 2e5 } ψ˜ 2 := col{e3 − e2 , e3 + e2 − 2e6 } ψ˜ 3 := col{e1 − e3 , e1 + e3 − 2e7 }

if the following LMIs are satisfied 

+ h1 [e1T e7T ]Q1 [C0T 0]T T T T T T ˙ − (1 − d(t))[e 1 e2 ](Q2 − Q3 )[e1 e2 ]

 R3 S2 S1 ≥ 0, ≥ 0, Z1 ≥ Z2 T S 2 R3 R˜ 1  Z3 N3 Ni ≥ 0, (i = 1, 2), ≥0 R2 N3T W2

⎧  (h , μ ) + 2 (h1 , μ1 ) + 3 (h1 ) + 4 ⎪ ⎪ 1 1 1 ⎨ 1 (h1 , μ2 ) + 2 (h1 , μ2 ) + 3 (h1 ) + 4 ⎪  (h , μ ) + 2 (h2 , μ1 ) + 3 (h2 ) + 4 ⎪ ⎩ 1 2 1 1 (h2 , μ2 ) + 2 (h2 , μ2 ) + 3 (h2 ) + 4

(47)

C1 := col{e1 , e3 , (h2 − d(t))e5 , (d(t) − h1 )e6 , h1 e7 } ˙ C2 := col{C0 , e8 , (1 − d(t))e 2 − e4 , ˙ e3 − (1 − d(t))e 2 , e1 − e3 }

(48)

Proof: For simplicity of presentation, we denote

0, Ri > 0, Zi , Ni (i = 1, 2, 3), S1 , S2 and P of appropriate dimensions such that (34), (47), (48) and (49) are satisfied. Remark 7: Proposition 1 presents a novel delay-derivativedependent stability criterion by incorporating with the matrix-based quadratic convex approach and some new integral inequalities. As stated in Remarks 1, 2 and 5, using Proposition 1 one may obtain some less conservative results than using some existing approaches, which is confirmed by two examples in the next section. 1060 © The Institution of Engineering and Technology 2014

V˜ 1 (t) := xT (t)Px(t) +

t x˙ T (s)Q0 x˙ (s) ds t−h1

where P > 0, the following result is obtained. Corollary 1: For given scalars h1 ≥ 0, h2 ≥ h1 , μ1 and μ2 (μ1 ≤ μ2 ), the system described by (1) and (5) is asymptotically stable if there exist real matrices P > 0, Q0 > 0, Qi > 0, Wi > 0, Ri > 0, Zi , Ni (i = 1, 2, 3), S1 and S2 of appropriate dimensions such that (47)–(49) are satisfied, where C1 and C2 are replaced with e1 and C0 , respectively. Remark 8: Example 1 in the next section shows that using Corollary 1 one can also obtain some less conservative result than the one using Theorem 1 in [7]. It should be pointed out that the improvement of Corollary 1 over Theorem 1 in [7] benefits from the proposed integral inequalities in Lemmas 2–4. Remark 9: As classified in [1, 18], there are two kinds of LKFs: simple LKFs and complete LKFs. Stability criteria employing simple LKFs can be used to test the stability of a time-delay system which is stable when the time-delay is zero. Using complete LKFs one can derive some stability criteria which can be applied to check the stability of a time-delay system which is unstable when the time-delay is zero. Therefore, Proposition 1 and Corollary 1 can only be utilised to test stability of the time-delay system which is stable when the time-delay is zero due to the fact that they are derived based on the simple LKF. However, if one chooses a complete LKF [18, 19], the matrix-based quadratic convex approach and the integral inequalities proposed in this paper can be applied to obtain some stability criteria which can be used to check the stability of a time-delay system which is unstable when the time-delay is zero.

4

Numerical examples

Example 1: Consider the system (1), where [7]   0 0 0 1 A= , Ad = 0 −1 −1 −1 We take this example to make a comparison with the results reported in [7]. In the case of the constant delay d(t) = h, the analytical bound of h to ensure the system stability is π . For comparison, suppose that the delay d(t) is timevarying satisfying (2). For h1 = 0 and various values of μ2 = −μ1 = μ, the admissible upper bounds of h2 calculated by the approaches in [7] and in this paper are listed in Table 1. From this table, it is clear that both Proposition 1 Table 1 Admissible upper bound h2max for h1 = 0 and different values of μ Method\μ

0.0

0.05

0.1

0.5

3.0

Theorem 1 in [7] 2.52 2.17 2.02 1.62 1.60 Proposition 1 3.0278 2.5547 2.3741 1.7037 1.6514 Corollary 1 3.0176 2.5543 2.3728 1.6973 1.6503

IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1054–1061 doi: 10.1049/iet-cta.2013.0840

www.ietdl.org Table 2 Admissible upper bound h2max for h1 = 1 and μ2 = 1

6

Method \ μ1

−0.3

−0.5

−0.7

−1

Theorem 7 in [12] Theorem 1 in [8] Proposition 1 in [16] Theorem 1 in [20] Theorem 1 in [21] Theorem 2 in [22] Proposition 1 Proposition 1 with P > 0 Corollary 1

2.128 2.189 2.201 2.283 2.101 2.872 2.460 2.450 2.415

2.128 2.185 2.141 2.283 2.101 2.465 2.423 2.418 2.391

2.128 2.182 2.096 2.283 2.101 2.222 2.400 2.397 2.369

2.128 2.179 2.046 2.283 2.101 1.991 2.382 2.377 2.342

This work was supported in part by the Australian Research Council Discovery Project under Grant DP1096780, and the Research Advancement Awards Scheme Program (January 2010–December 2012) at Central Queensland University, Australia.

7 1 2

and Corollary 1 can achieve some larger upper bounds of h2 than Theorem 1 in [7]. Specifically, when μ = 0, that is ˙ = 0, the obtained hmax by Proposition 1 is improved by d(t) 2 20.15% over that in [7], where 97.99% improvement benefits from the proposed integral inequalities in Lemmas 2–4.

3

Example 2: Consider the system (1), where [8]   −2 0 −1 0 A= , Ad = 0 −0.9 −1 −1

6

4 5

7 8

This example is used to compare Proposition 1 in this paper with the methods proposed in [8, 12, 16, 20–22]. For h1 = 1 and μ2 = 1, Table 2 lists the results for various values of μ1 obtained by the methods in [8, 12, 16, 20–22] and Proposition 1 in this paper. It is clear to see that Proposition 1 in this than the methods in paper produces some larger values hmax 2 [8, 12, 16, 20, 21]. Theorem 2 in [22] is obtained using an integral quadratic constraint approach. One can see clearly from Table 2 that for μ1 = −0.3 and μ1 = −0.5, Theorem 2 in [22] gives some larger hmax than Proposition 1, while for 2 μ1 = −0.7 and μ1 = −1, Proposition 1 in this paper can obtain some better results of hmax than Theorem 2 in [22]. 2 That is, for this example, Proposition 1 in this paper can be complementary with Theorem 2 in [22]. On the other hand, from Table 2, one can see clearly that (i) if we replace the condition (34) with P > 0, then Proposition 1 gives some smaller values hmax and (ii) Proposition 1 2 outperforms Corollary 1.

5

Conclusion

The stability of a linear system with an interval time-varying delay has been analysed. A new LKF has beenconstructed t including three integral terms in the form of t−h (h − t + j T s) x˙ (s)Rj x˙ (s) ds (j= 1, 2, 3). Some new integral inequalities t for the integrals t−h (h − t + s)j x˙ T (s)Rj x˙ (s) ds (j = 0, 1, 2) appearing in the derivative of the chosen LKF have been established. The matrix-based quadratic convex approach has been introduced to prove both the negative definiteness of the derivative of the LKF and the positive definiteness of the LKF such that a novel delay-derivative-dependent stability criterion has been formulated. The effectiveness of the proposed results has been demonstrated by two numerical examples.

IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1054–1061 doi: 10.1049/iet-cta.2013.0840

Acknowledgments

9 10

11 12 13 14 15 16 17 18 19 20

21 22

References Gu, K., Kharitonov, V.L., Chen, J.: ‘Stability of time-delay systems’ (Birkhäuser, Boston, MA, USA, 2003) Yu, L., Han, Q.-L., Yu, S., Gao, J.: ‘Delay-dependent conditions for robust absolute stability of uncertain time-delay systems’. Proc. 42nd IEEE Conf. on Decision and Control, Maui, HI, USA, December 2003, pp. 6033–6037 Yue, D., Han, Q.-L., Lam, J.: ‘Network-based robust control of systems with uncertainty’, Automatica, 2005, 41, (6), pp. 999–1007 Gao, H., Chen, T., Lam, J.: ‘A new delay system approach to networkbased control’, Automatica, 2008, 44, (1), pp. 39–52 Yue, D., Tian, E., Han, Q.-L.: ‘A delay system method for designing event-triggered controllers of networked control systems’, IEEE Trans. Autom. Control, 2013, 58, (2), pp. 485–481 Peng, C., Han, Q.-L.: ‘A novel event-triggered transmission scheme and L2 control co-design for sampled-data control systems’, IEEE Trans. Autom. Control, 2013, 58, (10), pp. 2620–2626 Kim, J.H.: ‘Note on stability of linear systems with time-varying delay’, Automatica, 2011, 47, (9), pp. 2118–2121 Fridman, E., Shaked, U., Liu, K.: ‘New conditions for delayderivative-dependent stability’, Automatica, 2009, 45, (11), pp. 2723–2727 Han, Q.-L.: ‘Improved stability criteria and controller design for linear neutral systems’, Automatica, 2009, 45, (8), pp. 1948–1952 He, Y., Wang, Q.G., Xie, L.H., Lin, C.: ‘Further improvement of freeweighting matrices technique for systems with timevarying delay’, IEEE Trans. Autom. Control, 2007, 52, (2), pp. 293–299 Peng, C.: ‘Improved delay-dependent stabilisation criteria for discrete systems with a new finite sum inequality’, IET Control Theory Appl., 2012, 6, (3), pp. 448–453 Seuret, A., Gouaisbaut, F.: ‘Wirtinger-based integral inequality: application to time-delay systems’, Automatica, 2013, 49, (9), pp. 2860–2866 Souza, F.O.: ‘Further improvement in stability criteria for linear systems with interval time-varying delay’, IET Control Theory Appl., 2013, 7, (3), pp. 440–446 Tang, M., Wang, Y.-W., Wen, C.: ‘Improved delay-range-dependent stability criteria for linear systems with interval time-varying delays’, IET Control Theory Appl., 2012, 6, (6), pp. 868–873 Zhang, J., Xia, Y., Shi, P., Mahmoud, M.S.: ‘New results on stability and stabilisation of systems with interval time-varying delay’, IET Control Theory Appl., 2011, 5, (3), pp. 429–436 Zhang, X.-M., Han, Q.-L.: ‘Novel delay-derivative-dependent stability criteria using new bounding techniques’, Int. J. Robust Nonlinear Control, 2013, 23, (13), pp. 1419–1432 Weng, F., Mao, W.: ‘Robust stability and stabilization of uncertain discrete singular time-delay systems based on PNP Lyapunov functional’, IMA J. Math. Control Inf., 2013, 30, pp. 301–314 Han, Q.-L.: ‘On stability of linear neutral systems with mixed timedelays: a discretized Lyapunov functional approach’, Automatica, 2005, 41, (7), pp. 1209–1218 Han, Q.-L.: ‘New results for delay-dependent stability of linear systems with time-varying delay’, Int. J. Syst. Sci., 2002, 33, (3), pp. 213–228 Lee, W., Jeong, C., Park, P.G.: ‘Further improvement of delaydependent stability criteria for linear systems with time-varying delays’. Proc. 12th Int. Conf. on Control, Automation and Systems, Jeju Island, Korea, October 2012, pp. 1300–1304 Qian, W., Liu, J.: ‘New stability analysis for systems with interval time-varying delay’, J. Franklin Inst., 2013, 350, pp. 890–897 Ariba, Y., Gouaisbaut, F., Johansson, K.H.: ‘Robust stability of timevarying delay systems: the quadratic separation approach’, Asian J. Control, 2012, 14, pp. 1205–1214

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