The ellipsoidal bound of reachable sets for linear neutral systems with disturbances

Journal of the Franklin Institute 348 (2011) 2570–2585 www.elsevier.com/locate/jfranklin

The ellipsoidal bound of reachable sets for linear neutral systems with disturbances$ Changchun Shena,, Shouming Zhonga,b a

School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 611731, PR China b Key Laboratory for NeuroInformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu 611731, PR China Received 20 July 2010; received in revised form 22 June 2011; accepted 27 July 2011 Available online 4 August 2011

Abstract In this paper, we consider the problem of finding an ellipsoidal bound of reachable sets for neutral systems with bounded peak disturbances. Up to now, the result related to the ellipsoidal bound of reachable sets was rarely proposed for linear neutral systems. Based on the modified augmented Lyapunov–Krasovskii type functional, we obtain some delay-dependent results expressed in the form of matrix inequalities containing only one non-convex scalar. Furthermore, a modified integral inequality is used to remove the limitation on the variation rate of the delay. Numerical examples are given to indicate significant improvements over some existing results. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction The reachable set for a dynamic system with disturbances is a set that bounds the state trajectories starting from the origin by inputs with peak value. Reachable set estimation is an important issue in the control theory [3,4,15,17]. It was first considered in the late 1960s in the context of state estimation and it has found a wide range of applications, such as peak-to-peak gain minimization problem [1], control systems with actuator saturation [1,7,8]. Generally speaking, the reachable set is not an ellipsoid but rather some closed $

This research was supported by National Basic Research Program of China (2010CB732501) and UESTC Research Fund For Youth Science and Technology (L08010701JX0809). Corresponding author. E-mail address: [email protected] (C. Shen). 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.07.017

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

2571

bounded set. But, a smaller ellipsoidal bound of a reachable set for linear time-delayed systems with saturating actuators permits a larger control gains, and this result in a better performance of the system [9]. For the ellipsoidal bound of a reachable set for linear systems without any time delay, a well-known result, which formulated in terms of linear matrix inequality (LMI), is given in [2], via Lyapunov function applying the S-procedure, can be found, and it is widely used to design control systems that have saturating actuators [8,9]. However, time delay phenomenon is frequently encountered in many practical systems, such as biological systems, chemical systems, hydraulic systems and electrical networks. It is well known that the existence of time delays in a system may cause instability or bad system performance (see [5,6,10,11,19,21]). Hence it is natural to ask what about the reachable set of systems with time delays. Up to now, only few results are available. In [5], a delay-dependent condition for an ellipsoid bounding the set of reachable states was presented by using the Lyapunov– Razumikhin function and the S-procedure. Five non-convex scalar parameters have to be treated as tuning parameters to find the ‘‘smallest’’ possible ellipsoid. Recently, Kim [10] modified the Lyapunov–Krasovskii functional used for exponential stability analysis [13,14], and, based on it, he proposed an improved ellipsoidal bound of the reachable set of reachable states. The derivative of the time delay is assumed to be less than one in [5]. This limitation may restrict the scope of application for this method. However, if the value of the derivative of the time delay is large, this method will yield a bigger ellipsoid bounding the reachable set than that in [5]. The reachable set bounding for delayed systems subject to both polytopic uncertainties and bounded peak inputs was considered in [22]. Some criteria bounding the reachable set are derived, by using a maximal Lyapunov–Krasovskii functional, which is constructed by taking pointwise maximum over a family of Lyapunov–Krasovskii functionals. But the bound of reachable set bounding for delayed systems with polytopic uncertainties is not small enough. As is well known, neutral system being a special case of time delay system exists in many dynamic systems [12,16,18,20]. However, the ellipsoidal bound of reachable sets were presented for linear neutral systems with bounded peak disturbances has not been investigated, which motivates this paper. In this paper, we consider the problem of finding an ellipsoidal bound of reachable sets for neutral systems with bounded peak disturbances. Based on the modified augmented Lyapunov–Krasovskii type functional, some delaydependent results are derived in the form of matrix inequalities containing only one nonconvex scalar. Furthermore, a modified integral inequality is used, which have been introduced for stability analysis of systems with delays in [18], to remove the limitation on the variation rate of the delay and obtain a ‘‘smaller’’ ellipsoidal bound of reachable sets. Numerical examples illustrate the effectiveness and improvement of the obtained results. 2. Problem statement Consider the following linear neutral systems with disturbances _ _ xðtÞC xðttÞ ¼ AxðtÞ þ BxðthðtÞÞ þ DwðtÞ,

xðyÞ  0, 8y 2 ½tn ,0,

ð1Þ

where xðtÞ 2 Rn is the state vector, wðtÞ 2 R‘ is the disturbance. A 2 Rnn , B 2 Rnn , C 2 Rnn and D 2 Rnl are known constant matrices, t40 is a constant neutral delay, the discrete delay h(t) is a continuous-time differentiable function and the disturbance w(t) is a

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

2572

bounded function. For discrete time delay and disturbance, we assume that _ 0rhðtÞrhM , hðtÞrh wT ðtÞwðtÞrw2m , d,

ð2Þ

n

where hM, hd are constants, t ¼ maxðt,hM Þ. We denote the set of reachable states with w(t) that satisfies Eq. (2) by Rx 9fxðtÞ 2 Rn 9xðtÞ, wðtÞ satisfies Eqs: ð1Þ and ð2Þ:

ð3Þ

We will bound Rx by an ellipsoid of the form IðP,1Þ9fxðtÞ 2 Rn : xT ðtÞPxðtÞr1; P40:

ð4Þ

Before proceeding further, we will state well-known lemmas. Lemma 1 (Kim [10]). For any positive-definite matrix F 2 Rnn , scalar g40, vector function o : ½0,g-Rn such that the integrations concerned are well defined, then Z g T Z g  Z g oðsÞ ds F oðsÞ ds rg oT ðsÞFoðsÞ ds: 0

0

0

Lemma 2 (Boyd et al. [2]). Let V ðxð0ÞÞ ¼ 0 and wT ðtÞwðtÞrw2m , if V_ ðxt Þ þ aV ðxt ÞbwðtÞT wðtÞr0, a40, b40 then we have V ðxt Þrðb=aÞw2m for 8tZ0. Remark 1. It can be easily seen that system investigated in this paper is quite different from the one studied in [5,10]. In this paper, the system (1), that its system matrix C is set to 0, is studied in [10]. 3. Main results Our aim is to find an ellipsoid set as small as possible to bound the reachable set defined in Eq. (3). Based on Lyapunov method and linear matrix inequality techniques, following theorems are derived. Theorem 1. Consider the time-delayed system (1) with constraints (2), if there exist real matrices Y, Q12, P2 and P3, symmetric matrices X, Z, Q11, Q22, P1 40, R40, S1 Z0, S2 40, G40, M40 and a scalar a40 satisfying the following matrix inequalities: 3 2 F11 F12 F13 PT2 CY 0 aY PT2 D 7 6 6 n F22 PT3 B PT3 C 0 Y þZ PT3 D 7 7 6 6 n n F33 0 0 0 0 7 7 6 7 6 at 7 6 n n n e R 0 Z 0 ð5Þ F¼6 7r0, 7 6 n ahM n n n e S 0 0 2 7 6 7 6 6 n n n n n aZeat G 0 7 6 a 7 4 n n n n n n  2 I5 wm "

Q11

Q12

QT12

Q22

# Z0,

2

X

4

YT

3 Y 5Z0, 1 Z þ eat R t

ð6Þ

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

2573

where F11 ¼ aP1 þ aX þ PT2 A þ AT P2 þ MeahM S1 , F12 ¼ P1 PT2 þ AT P3 þ X þ Y , F13 ¼ PT2 B þ Q12 þ eahM S1 , F22 ¼ R þ t2 G þ h2M ðS1 þ S2 Þ þ hM eahM Q11 P3 PT3 , F33 ¼ hM Q22 QT12 Q12 minðð1hd ÞeahM ,1hd ÞMeahM S1 : Then, the reachable sets of the system (1) having the constraints (2) is bounded by an ellipsoid IðP1 ,1Þ defined in Eq. (4). Proof of Theorem 1. Choose a new class of functional candidate for systems (1) as following: V ðxt Þ ¼ V1 ðxðtÞÞ þ V2 ðxt Þ þ V3 ðxt Þ þ V4 ðxt Þ þ V5 ðxt Þ,

ð7Þ

where V1 ðxðtÞÞ ¼ xT ðtÞP1 xðtÞ ¼ ½xT ðtÞ x_ T ðtÞ Z

t

eaðstÞ xT ðsÞMxðsÞ ds þ

V2 ðxt Þ ¼



Z

thðtÞ

Z Z

0 0

"

P1

0

P2

P3

#"

# xðtÞ , _ xðtÞ

t

_ ds, eaðstÞ x_ T ðsÞRxðsÞ

tt

t

_ ds eaðstÞ ðhM t þ sÞx_ T ðsÞ½S1 þ S2 xðsÞ

V3 ðxt Þ ¼ hM

thM t aðstÞ

e

þt

I 0

_ ds, ðtt þ sÞx_ T ðsÞG xðsÞ

tt

Z

þe

"

y

dy

V4 ðxt Þ ¼ ahM

Z

t

Z

0

e

aðytÞ

T

T

½x_ ðsÞ x ðyhðyÞÞ 

yhðyÞ

Q11

Q12

QT12

Q22

#"

_ xðsÞ xðyhðyÞÞ

# ds

t

_ ds, eaðstÞ  ðhM t þ sÞx_ T ðsÞQ11 xðsÞ

thM

" T

Z

T # 

t

_ ds xðsÞ

V5 ðxt Þ ¼ x ðtÞ tt

X

Y

YT

Z

"

# xðtÞ , _ tt xðsÞ ds

Rt

where P1 40, Q11 Z0, Q22 Z0, R40, S1 Z0, S2 40, G40, M40, Q12, X, Y, Z, P2, P3 and a scalar a40 are solutions to Eqs. (1) and (2). First, we show that V ðxt Þ in Eq. (7) is a good L–K functional candidate. For thM rsrt, we have 0oeahM reaðstÞ r1 and 0rhM t þ srhM . Furthermore, for ttrsrt, we have 0oeat reaðstÞ r1 and 0rtt þ srt, so we have V3 ðxt ÞZ0.

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

2574

By using Lemma 1, we get Z t T Z t  Z t 1 at at T V2 ðxt ÞZe y ðsÞRyðsÞ dsZ e yðsÞ ds R yðsÞ ds : t tt tt tt As a result, we further have 2 " Z t T # X 5 X T 4 T Vi Z x ðtÞ yðsÞ ds Y tt

3" # Y xðtÞ 5 Rt 1 Z0: Z þ eat R tt yðsÞ ds t

i¼2

Therefore, we get 8 5 X > > < V ðxt Þ ¼ Vi ZV1 ðxðtÞÞ ¼ xT ðtÞP1 xðtÞ,

ð8Þ

ð9Þ

i¼1 > > : V ðxt Þ ¼ 0 when xðyÞ ¼ 0, y 2 ½ttn ,t:

Next, the derivative of V ðxt Þ along the trajectory of system (1) is given by V_ ðxt Þ ¼ V_ 1 ðxðtÞÞ þ V_ 2 ðxt Þ þ V_ 3 ðxt Þ þ V_ 4 ðxt Þ þ V_ 5 ðxt Þ: From Eq. (10), we have

"

V_ 1 ðxðtÞÞ ¼ 2½xT ðtÞ x_ T ðtÞ " T

T

¼ 2½x ðtÞ x_ ðtÞ "

#

P1

PT2

0

PT3 # PT2 PT3

P1 0

_ xðtÞ 0

ð10Þ



_ xðtÞ  _ þ AxðtÞ þ BxðthðtÞÞÞ þ C xðttÞ _ xðtÞ þ DwðtÞ

#

_ ¼ xT ðtÞ½PT2 A þ AT P2 xðtÞ þ 2xT ðtÞ½P1 PT2 þ AT P3 xðtÞ T T T T T _ þ 2x ðtÞPT2 DwðtÞ þ2x ðtÞP2 BxðthðtÞÞ þ 2x ðtÞP2 C xðttÞ T T T T _ þ 2x_ ðtÞP3 BxðthðtÞÞ x_ ðtÞ½P3 þ P3 xðtÞ T T _ þ2x_ ðtÞP3 C xðttÞ þ 2x_ T ðtÞPT3 DwðtÞ,

ð11Þ

ahðtÞ T _ _ hðtÞÞe x ðthðtÞÞMxðthðtÞÞ V_ 2 ðxt Þ ¼ xT ðtÞMxðtÞ þ x_ T ðtÞRxðtÞð1 Z t _ eat x_ T ðttÞRxðttÞa eaðstÞ xT ðsÞMxðsÞ ds thðtÞ Z t aðstÞ T _ ds a e x_ ðsÞRxðsÞ tt at T _ _ rxT ðtÞMxðtÞ þ x_ T ðtÞRxðtÞe x_ ðttÞRxðttÞ ahM T ,1hd Þx ðthðtÞÞMxðthðtÞÞaV2 , minðð1hd Þe Z t T 2 2 _ _ _ ds V 3 ðxt Þ ¼ x_ ðtÞ½hM ðS1 þ S2 Þ þ t GxðtÞt eaðstÞ x_ T ðsÞGxðsÞ tt Z t Z t _ dsat hM eaðstÞ x_ T ðsÞ½S1 þ S2 xðsÞ eaðstÞ ðtt þ sÞ thM

tt

ð12Þ

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

_ dsahM x_ T ðsÞG xðsÞ

Z

2575

t

_ ds eaðstÞ ðhM t þ sÞx_ T ðsÞðS1 þ S2 ÞxðsÞ Z t at _ _ ds rx_ T ðtÞ½h2M ðS1 þ S2 Þ þ t2 GxðtÞte x_ T ðsÞG xðsÞ tt Z t _ dsaV3 hM eahM x_ T ðsÞðS1 þ S2 ÞxðsÞ thM

thðtÞ

_ reahM xT ðtÞS1 xðtÞ þ x_ T ðtÞ½h2M ðS1 þ S2 Þ þ t2 GxðtÞ þ2eahM xT ðtÞS1 xðthðtÞÞeahM xT ðthðtÞÞS1 xðthðtÞÞ Z t T Z t  ahM _ ds S2 _ ds e xðsÞ xðsÞ eat

Z

thðtÞ t

T Z _ ds G xðsÞ

tt

V_ 4 ðxt Þr

Z

thðtÞ

t

 _ ds aV3 , xðsÞ

ð13Þ

tt

t

_ ds þ 2 x_ T ðsÞQ11 xðsÞ

thðtÞ

Z

T

t

_ ds xðsÞ

Q12 xðthðtÞÞ

thðtÞ

_ þhðtÞxT ðthðtÞÞQ22 xðthðtÞÞ þ hM eahM x_ T ðtÞQ11 xðtÞ Z t _ dsaV4  x_ T ðsÞQ11 xðsÞ thM Z t _ ds þ 2½xðtÞxðthðtÞÞT Q12 xðthðtÞÞ r x_ T ðsÞQ11 xðsÞ thðtÞ

_ þhM xT ðthðtÞÞQ22 xðthðtÞÞ þ hM eahM x_ T ðtÞQ11 xðtÞ Z t _ dsaV4 x_ T ðsÞQ11 xðsÞ  thðtÞ

_ ¼ 2xT ðtÞQ12 xðthðtÞÞ þ hM eahM x_ T ðtÞQ11 xðtÞ T T þx ðthðtÞÞ½hM Q22 Q12 Q12 xðthðtÞÞaV4 , " # " Z t T #  _ xðtÞ X Y _ ds V_ 5 ðxt Þ ¼ 2 xT ðtÞ xðsÞ _ _ xðtÞ xðttÞ YT Z tt  Z t  T _ _ _ ds ðtÞY xðttÞ þ x_ T ðtÞðY þ ZÞ xðsÞ ¼ 2 xT ðtÞðX þ Y ÞxðtÞx tt Z t  T _ ds : x_ ðttÞZ xðsÞ tt

Finally, by combining Eqs. (11)–(15), we further have a V_ ðxt Þ þ aV ðxt Þ 2 wðtÞT wðtÞrxT ðtÞ½aP1 þ aX þ PT2 A þ AT P2 þ M wm _ eahM S1 xðtÞ þ 2xT ðtÞ½P1 PT2 þ AT P3 þ X þ Y xðtÞ T T ahM T þ2x ðtÞ½P2 B þ Q12 þ e S1 xðthðtÞÞ þ 2x ðtÞ½PT2 CY  Z t  T _ ds þ 2xT ðtÞPT2 DwðtÞ _ xðsÞ xðttÞ þ 2ax ðtÞY tt

_ þx_ T ðtÞ½R þ t2 G þ h2M ðS1 þ S2 Þ þ hM eahM Q11 P3 PT3 xðtÞ

ð14Þ

ð15Þ

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

2576

_ þ2x_ T ðtÞPT3 BxðthðtÞÞ þ 2x_ T ðtÞPT3 C xðttÞ þ 2x_ T ðtÞ½Y þ Z Z t  _ ds þ 2x_ T ðtÞPT3 DwðtÞ þ xT ðthðtÞÞ½hM Q22 QT12 xðsÞ  tt

Q12 minðð1hd ÞeahM ,1hd ÞMeahM S1 xðthðtÞÞ Z t  at T T _ _ ds e x_ ðttÞRxðttÞ2 x_ ðttÞZ xðsÞ tt Z t T Z t  Z t T ahM _ ds S2 _ ds  _ ds ½eat G e xðsÞ xðsÞ xðsÞ thðtÞ thðtÞ tt Z t  a _ ds  2 wT ðtÞwðtÞ ¼ X T ðtÞFX ðtÞ, aZ xðsÞ w tt m

ð16Þ

where " T

T

T

T

Z

T Z

t

_ ds xðsÞ

X ðtÞ ¼ x ðtÞ x_ ðtÞ x ðthðtÞÞ x_ ðttÞ thðtÞ

T

t

_ ds xðsÞ

#T T

w ðtÞ

tt

and F is the same as defined in Theorem 1. Thus, from matrix inequalities (5) and (6), we get a V_ ðxt Þ þ aV ðxt Þ 2 wðtÞT wðtÞr0, ð17Þ wm which means, by Lemma 2, that V ðxt Þ ¼ V1 ðxðtÞÞ þ V2 ðxt Þ þ V3 ðxt Þ þ V4 ðxt Þ þ V5 ðxt Þr1, and this results in V1 ðxðtÞÞ ¼ xT ðtÞP1 xðtÞr1 since V2 ðxt Þ þ V3 ðxt Þ þ V4 ðxt Þ þ V5 ðxt ÞZ0 from Eq. (8). This completes the proof. & Remark 2. In [23,24], the authors considered the discrete systems, and applied quadratic boundedness to deal with the design of receding-horizon estimators and the design of state estimators for discrete time linear systems with polytopic uncertainties, respectively. However, our paper considers linear neutral systems with disturbances. Under the conditions of Theorem 1, system (1) is bounded with Lyapunov matrix P1. In fact, if xT ðTÞP1 xðTÞZ1 then we have V ðxT ÞZ1: From a V_ ðxt Þ þ aV ðxt Þ 2 oT or0 om we can obtain V ðxt ÞrV ðxT ÞeaðtTÞ þ a

Z

t

eas dseat rV ðxT ÞeaðtTÞ þ 1:

T

Hence, for all tZT, there is xT ðtÞP1 xðtÞrV ðxT Þ þ 1o1: Remark 3. In [23–25], the considered systems do not include time delays, the quadratically boundedness implies the systems are reachable. But, system (1) in our paper considers linear delayed neutral systems, the V ðxt Þ in our paper includes not only V1 ðxt Þ ¼ xT ðtÞP1 xðtÞ, but

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

2577

also V2 , V3 , V4 , V5 . Therefore, by conditions of Theorem 1, it is difficult to derive dV 1 ðxt Þ=dto0 for xT ðtÞP1 xðtÞ41. However, at the absent of time delays, system (1) becomes _ ¼ AxðtÞ þ DwðtÞ xðtÞ then along the proof in [25], we can show that the above system is quadratically bounded. Next, we consider the following linear neutral system with uncertainties _ _ xðtÞC xðttÞ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞxðthðtÞÞ þ ðD þ DDðtÞÞwðtÞ, xðt0 þ yÞ ¼ 0,

8y 2 ½tn ,0,

ð18Þ

where the uncertainties are expressed as a linear convex-hull of known matrices Adi, Bdi and Ddi: DAðtÞ ¼

N X

mi ðtÞAdi ,

DBðtÞ ¼

i¼1

N X

mi ðtÞBdi

and

DDðtÞ ¼

i¼1

N X

mi ðtÞDdi

i¼1

P with mi ðtÞ 2 ½0,1 and N i ¼ 1 mi ðtÞ ¼ 1, 8t40. Corollary 1 is a result for the ellipsoidal bound of a reachable set for an uncertain time-delayed system (18) having constraints in Eq. (2). Theorem 2. Consider the time-delayed system (1) with constraints (2), if there exist real matrices Y, Q12, P2 and P3, symmetric matrices X, Z, Q11, Q22, P1 40, R40, S1 Z0, S2 40, G40, M40 and a scalar a40 satisfying the following matrix inequalities for all i ¼ 1, . . . ,N 3 2 F11i F12i F13i PT2 CY 0 aY F17i 7 6 F22i F23i PT3 C 0 Y þZ F27i 7 6 n 7 6 6 n n F33i 0 0 0 0 7 7 6 7 6 n n eat R 0 Z 0 7 6 n ð19Þ Fi ¼ 6 7r0, 6 n n n n eahM S2 0 0 7 7 6 7 6 n n n n F66i 0 7 6 n 6 a 7 5 4 n n n n n n  2 I wm "

Q11 QT12

# Q12 Z0, Q22

2

X

4

YT

3 Y 5Z0, 1 Z þ eat R t

where F11i ¼ aP1 þ aX þ PT2 ðA þ Adi Þ þ ðA þ Adi ÞT P2 þ MeahM S1 , F12i ¼ P1 PT2 þ ðA þ Adi ÞT P3 þ X þ Y , F13i ¼ PT2 ðB þ Bi Þ þ Q12 þ eahM S1 , F17i ¼ PT2 ðD þ Ddi Þ, F22i ¼ P3 PT3 þ R þ t2 G þ h2M ðS1 þ S2 Þ þ hM eahM Q11 ,

ð20Þ

2578

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

F23i ¼ PT3 ðB þ Bi Þ, F27i ¼ PT3 ðD þ Ddi Þ, F33i ¼ hM Q22 QT12 Q12 minðð1hd ÞeahM ,1hd ÞMeahM S1 , F66i ¼ aZeat G: Then, the reachable sets of the system (21) having the constraints (2) is bounded by an ellipsoid IðP1 ,1Þ defined in Eq. (4). Proof of Theorem 2. This is straightforward from the proof of Theorem 1 and the properties of a convex-hull, so we omit the details. & Remark 4. Let us remark that the derivative of discrete delay needs to satisfy constraint _ condition: hðtÞrh d , which is less conservative than the one (satisfy constraint condition: _ o1) in [10]. As we can see from Eq. (6), the derivative of time delay hd does not hðtÞrh d need to be less than one. Since the term hM Q22 QT12 Q12 minðð1hd ÞeahM ,1hd ÞM eahM S1 could be negative definite by choosing appropriate Q12, Q22, S1, M, hM and hd when hd Z1. However, the (2,2) block of matrix inequality (4) for Theorem 1 in [10] is ð1hd ÞeahM S þ h2d W with S40, W 40. To ensure ð1hd ÞeahM S þ h2d W o0, it is required that hd o1. Remark 5. The solution to Eqs. (5) and (6), or Eqs. (19) and (20), if it exists, need not be unique. It is well known [2] that the volume of I defined in Eq. (4) is proportional to detðP1 Þ1=2 , so the minimization of log detðP1 Þ1=2 is the same as minimizing the volume of I. However, the minimization of log detðP1 Þ1=2 needs additional variables, so we use a simple approximation as that in [5,10,22]. That is, maximize d subject to dIrP1 which can be equivalent to the following optimization problem:   1 minimize d d ¼ d 8 " # > > < ðaÞ dI I Z0, I P1 ð21Þ subject to > > : ðbÞ Eqs: ð5Þ and ð6Þ or Eqs: ð19Þ and ð20Þ: Remark 6. The matrix inequalities in Theorems 1 and 2 contain only one non-convex scalar a40 (for given hM and hd), and these become LMI by fixing the scalar a. The feasibility check of a matrix inequality having only one non-convex scalar parameter is numerically tractable, and a local optimum value of a can be found by fminsearch.m. 4. Numerical examples The following three examples are used to demonstrate that the proposed methods are an improvement over some previous ones.

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

2579

Example 1. Consider the following uncertain time-delayed system which has been studied in [5,10,22]: " # " #   2 0 1 0 0:5 _ ¼ xðtÞ xðtÞ þ xðthðtÞÞ þ wðtÞ, ð22Þ 0 0:9 þ r 1 1 þ 0:5r 1 where 9r9r0:2 and wT ðtÞwðtÞr1. This system can be rewritten in the form of Eq. (21) with         2 0 1 0 0 0 0:5 A¼ , B¼ , C¼ , D¼ , 0 0:9 0 0 1 1 1         0 0 0 0 0 0 0 0 , Ad2 ¼ , Bd1 ¼ Bd2 ¼ Ad1 ¼ 0 0:2 0 0:2 0 0:1 0 0:1 and t ¼ 0:1. By solving the optimization problem (21), We get the sizes of the ellipsoidal bound of a reachable set for various hd when hM ¼ 0.7 and hM ¼ 0.75. These results are respectively summarized in Tables 1 and 2, and are compared to the previous results in [10,22]. According to Tables 1 and 2, it is obvious that our result decreases the size of the ellipsoid significantly. Fig. 1 is the plot of the ellipsoidal bound reachable sets I defined in Eq. (3), obtained for hM ¼ 0.7 and hd ¼ 0.2. We let wðtÞ ¼ 0:9 sin t, m1 ðtÞ ¼ sin2 t and m2 ðtÞ ¼ cos2 t. Figs. 2 and 4 depict the time response of state variable x(t) for system (22) with hðtÞ ¼ 0:75 and hðtÞ ¼ 0:5 þ 0:2 sin t, respectively. Fig. 3 is the plot of the ellipsoidal bound reachable sets and state trajectory for hðtÞ ¼ 0:75. Fig. 5 is the plot of the ellipsoidal bound reachable sets for hðtÞ ¼ 0:5 þ 0:2 sin t. It is clear to see that the state trajectory for system (22) is inside the ellipsoidal bound as found by our method and our result decreases the size of the ellipsoid significantly compared to the previous results in [5]. As we can see in the above tables and figures, our results get significant improvements over the existing results [5,10,22]. Table 1 The sizes (d) of ellipsoidal bound for hM ¼ 0.7. hd

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.90

2

Kim [10] Zuo [22] hM in this paper

2.97 1.89 1.15

3.30 1.94 1.20

3.85 2.00 1.26

4.85 2.08 1.35

6.93 2.19 1.41

12.84 2.35 1.51

53.86 2.60 1.64

– 3.51 2.69

– 5.30 4.59

Table 2 The sizes (d) of ellipsoidal bound for hM ¼ 0.75. hd

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.90

2

Kim [10] Zuo [22] hM in this paper

3.34 2.28 1.34

3.79 2.35 1.41

4.53 2.45 1.50

5.88 2.57 1.60

8.85 2.68 1.72

18.36 2.85 1.81

127.70 4.62 2.08

– 5.57 4.07

– 13.39 12.09

2580

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585 5 4 3 2

y

1 0 −1 −2 −3 −4 −5 −2

−1.5

−1

−0.5

0 x

0.5

1

1.5

2

Fig. 1. The bounding ellipsoids I of system (22) for hM ¼ 0.7 and hd ¼ 0.2 (solid: as found by the method of this note, dotted: as in [5]).

0.2

x1(t)

0.1 0 −0.1 −0.2 0

1000

2000

3000

4000

5000

3000

4000

5000

t 1

x2(t)

0.5 0 −0.5 −1 0

1000

2000 t

Fig. 2. The time responses of state variable x(t) of system (22) for hðtÞ ¼ 0:75.

Example 2. Consider the following time-delayed system which has been studied in [22] " #     0 0:120:0264r 0:1 0:35 1 _ ¼ xðtÞ xðt0:1Þ þ wðtÞ, ð23Þ xðtÞ þ 1 0:465 þ 0:22r 1 0:3 1 where 9r9r1 and wT ðtÞwðtÞr1. This system can be rewritten in the form of Eq. (21) with         0 0:12 0:1 0:35 0 0 1 A¼ , B¼ , C¼ , D¼ , 1 0:465 1 0:3 0 0 1

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

2581

1.5 ellipsoidal bound state trajectory

1

x2(t)

0.5 0 −0.5 −1 −1.5 −1.5

−1

−0.5

0 x1(t)

0.5

1

1.5

Fig. 3. The bounding ellipsoids I and state trajectory of system (22) for hðtÞ ¼ 0:75 (solid: bounding ellipsoids, dotted: state trajectory).

0.2

x1(t)

0.1 0 −0.1 −0.2 0

1000

2000

3000

4000

5000

3000

4000

5000

t 1

x2(t)

0.5 0 −0.5 −1 0

1000

2000 t

Fig. 4. The time responses of state variable x(t) of system (22) for hðtÞ ¼ 0:5 þ 0:2 sin t.

 Ad1 ¼

0

0:0264

0

0:22



 ,

Ad2 ¼

0

0:0264

0

0:22



and t ¼ 0:1. By solving the optimization problem (21), d is computed as 8.57 with a ¼ 0:59. Fig. 6 depicts the time response of state variable x(t). Fig. 7 is the plot of the ellipsoidal bound reachable sets and state trajectory. It is clear to see that the state trajectory for system (23) is inside the ellipsoidal bound as found by our method. Using the method of Zuo [22], the size of the ellipsoidal bound of a reachable set is d ¼ 2:8686  104 . It is obvious that our

2582

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

5 4 3 2 x2(t)

1 0 −1 −2 −3 −4 −5 −2

−1.5

−1

−0.5

0 x1(t)

0.5

1

1.5

2

Fig. 5. The bounding ellipsoids I of system (22) for hðtÞ ¼ 0:5 þ 0:2 sin t (dotted: bounding ellipsoids as in [5]; solid: bounding ellipsoids as found by the method of this note).

2

x1(t)

1 0 −1 −2 −3 0

200

400

600

800

1000

600

800

1000

t 1.5

x2(t)

1 0.5 0 −0.5 −1 0

200

400 t

Fig. 6. The time responses of state variable x(t) of system (23) for hðtÞ ¼ 0:1.

result decreases the size of the ellipsoid significantly. Note that there is no feasible solution using the methods of Fridman [5] and Kim [10]. Therefore, for this example, our method leads to a wider application range comparing with those in [5,10]. Example 3. Consider the following uncertain neutral system with time delay " #   2 0 0:2 0 _ _ xðtÞ xðt0:1Þ ¼ xðtÞ 0 1 þ r 0 0:2

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585 5

2583

ellipsoidal bound state trajectory

4 3 2

x2(t)

1 0 −1 −2 −3 −4 −5 −5

0 x1(t)

5

Fig. 7. The bounding ellipsoids I and state trajectory of system (23) for hðtÞ ¼ 0:1 (solid: bounding ellipsoids, dotted: state trajectory).

Table 3 The sizes (d ) of ellipsoidal bound for hM ¼ 0.7 and hM ¼0.75. hd

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

hM ¼ 0.7 hM ¼ 0.75

1.74 2.05

1.92 2.29

2.16 2.61

2.48 3.04

2.92 3.64

3.58 4.68

4.73 6.48

6.97 10.40

12.99 22.89

44.53 130.26

0.2

x1(t)

0.1 0 −0.1 −0.2 0

1000

2000

3000

4000

5000

3000

4000

5000

t 0.4

x2(t)

0.2 0 −0.2 −0.4 0

1000

2000 t

Fig. 8. The time responses of state variable x(t) of system (24) for hðtÞ ¼ 0:5 þ 0:2 sin t.

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

2584

2 ellipsoidal bound state trajectory

1.5 1

x2(t)

0.5 0 −0.5 −1 −1.5 −2

−2

−1.5

−1

−0.5

0 x1(t)

0.5

1

1.5

2

Fig. 9. The bounding ellipsoids I and state trajectory of system (24) for hðtÞ ¼ 0:5 þ 0:2 sin t (solid: bounding ellipsoids, dotted: state trajectory).

"

#   1 0 0:6 þ xðthðtÞÞ þ wðtÞ, 1 1 þ 0:5r 0:8

ð24Þ

where 9r9r0:2 and wT ðtÞwðtÞr1. By solving the optimization problem (21), We get the sizes of the ellipsoidal bound of a reachable set for various hd when hM ¼ 0.7 and hðtÞ ¼ 0:75. These results are respectively summarized in Table 3. We let wðtÞ ¼ 0:9 sin t, m1 ðtÞ ¼ sin2 t and m2 ðtÞ ¼ cos2 t. Fig. 8 depicts the time response of state variable x(t) for hðtÞ ¼ 0:5 þ 0:2 sin t. Fig. 9 is the plot of the ellipsoidal bound reachable sets and state trajectory for hðtÞ ¼ 0:5 þ 0:2 sin t. It is clear to see that the state trajectory for system (24) is inside the ellipsoidal bound as found by our method. However, we cannot obtain the ellipsoidal bound of a reachable set for system (24) by using the results in [5,10,22]. Therefore, for this example, our method leads to a wider application range comparing with those in [5,10,22]. References [1] J. Abedor, K. Nagpal, K. Poola, A linear matrix inequality approach to peak-to-peak minimization, International Journal of Robust and Nonlinear Control 6 (1996) 899–927. [2] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, PA, 1994. [3] Al. Claudio, The reachable set of a linear endogenous switching system, Systems & Control Letters 47 (2002) 343–353. [4] P. Collins, Continuity and computability of reachable sets, Theoretical Computer Science 341 (2005) 162–195. [5] E. Fridman, On reachable sets for linear systems with delay and bounded peak inputs, Automatica 39 (2003) 2005–2010. [6] L.V. Hien, V.N. Phat, Exponential stability and stabilization of a class of uncertain linear time-delay systems, Journal of the Franklin Institute 346 (2009) 611–625.

C. Shen, S. Zhong / Journal of the Franklin Institute 348 (2011) 2570–2585

2585

[7] T. Hu, Z. Lin, Control Systems with Actuator Saturation: Analysis and Design, Birkhauser, Boston, 2001. [8] T. Hu, Z. Lin, Composite quadratic Lyapunov functions for constrained control systems, IEEE Transactions on Automatic Control 48 (2003) 440–450. [9] J.H. Kim, F. Jabbari, Scheduled controllers for buildings under seismic excitation with limited actuator capacity, Journal of Engineering Mechanics 130 (2004) 800–808. [10] J.H. Kim, Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances, Automatica 44 (2008) 2940–2943. [11] O.M. Kwon, Ju H. Park, S.M. Lee, On stability criteria for uncertain delay-differential systems of neutral type with time-varying delays, Applied Mathematics and Computation 197 (2008) 864–873. [12] M. Li, L. Liu, A delay-dependent stability criterion for linear neutral delay systems, Journal of the Franklin Institute 346 (2009) 33C37. [13] C.H. Lien, K. W Yu, Y.J. Chung, Y.F. Lin, L.Y. Chung, J.D. Chen, Exponential stability analysis for uncertain switched neutral systems with interval-time-varying state delay, Nonlinear Analysis: Hybrid Systems 3 (2009) 334–342. [14] S. Mondie, V.L. Kharitonov, Exponential estimates for retarded time delay systems: an LMI approach, IEEE Transactions on Automatic Control 50 (2005) 268–273. [15] T. Pecsvaradi, Reachable sets for linear dynamical systems, Information and Control 19 (1971) 319–344. [16] F. Qiu, B.T. Cui, Y. Ji, Further results on robust stability of neutral system with mixed time-varying delays and nonlinear perturbations, Nonlinear Analysis: Real World Applications 11 (2010) 895–906. [17] T.I. Seidman, Time-invariance of the reachable set for linear control problems, Journal of Mathematical Analysis and Applications 72 (1979) 17–20. [18] C.C. Shen, S.M. Zhong, New delay-dependent robust stability criterion for uncertain neutral systems with time-varying delay and nonlinear uncertainties, Chaos, Solitons & Fractals 40 (2009) 2277–2285. [19] B. Wang, X. Liu, S. Zhong, New stability analysis for uncertain neutral system with time-varying delay, Applied Mathematics and Computation 197 (2008) 457–465. [20] D. Zhang, L. Yu, H1 filtering for linear neutral systems with mixed time-varying delays and nonlinear perturbations, Journal of the Franklin Institute 347 (2010) 1374–1390. [21] L. Zhang, C.H. Wang, H.J. Gao, Delay-dependent stability and stabilization of a class of linear switched time-varying delay systems, Journal of Systems Engineering and Electronics 18 (2007) 320–326. [22] Z.Q. Zuo, D.W.C. Ho, Y.J. Wang, Reachable set bounding for delayed systems with polytopic uncertainties: the maximal Lyapunov–Krasovskii functional approach, Automatica 46 (2010) 949–952. [23] A. Alessandri, M. Baglietto, G. Battistelli, On estimation error bounds for receding-horizon filters using quadratic boundedness, IEEE Transactions on Automatic Control 49 (2004) 1350–1355. [24] A. Alessandri, M. Baglietto, G. Battistelli, Design of state estimators for uncertain linear systems using quadratic boundedness, Automatica 42 (2006) 497–502. [25] M.L. Brockman, M. Corless, Quadratic boundedness of nonlinear dynamical systems, in: Proceedings of the 34th Conference on Decision and Control New Orleans, LA-December, 1995, pp. 504–509.