Input-output finite-time stability of time-delay systems and itsapplication to active vibration control*

Wenping Xue and Kangji Li

Abstract— This paper addresses the problems of input-outputfinite-time stability (IO-FTS) analysis for linear time-delaysystems and its application to active vibration control for struc-tural systems with input delay. By employing the Lyapunov-like function method, a sufficient condition for the IO-FTS ofthe time-delay system is proposed. Then, based on the IO-FTS analysis result, a state feedback controller is designed forthe structural system to attenuate the output response due tothe exogenous disturbance. The controller design condition ispresented in terms of a set of linear matrix inequalities (LMIs).Considering a practical earthquake excitation, a vibration-attenuation example is given to illustrate the effectiveness ofthe developed theory.

I. INTRODUCTION

Active vibration control for structural systems has receivedconsiderable research interest since many countries and re-gions (such as Japan, Taiwan, etc.) have often suffered fromearthquakes and strong winds. Many control strategies on thebasis of H2 and H∞, neural networks, fuzzy logic, adaptivecontrol, etc., have been adopted to attenuate the effects ofstructural vibration [1]. Moreover, time-delays are commonlyencountered in the control channels of structural systems,and frequently a major source of instability and poor perfor-mance. Hence, active vibration control with consideration ofinput delay has been investigated by some researchers [2]–[6].

It is noticed that in [2]–[5], the authors considered tra-ditional asymptotic stability for the active vibration controlsystems. Compared with the asymptotic stability, finite-timestability (FTS) can better describe the transient behavior ofa system over a certain time interval. A system is said tobe finite-time stable if, once we fix a time interval and givea bound on the initial condition, the system state does notexceed a certain domain during this time interval. Therefore,for systems that are known to operate only over a short timeperiod or whenever, from practical considerations, the systemstate is required to remain within a prescribed bound, FTScan be used. In recent years, FTS analysis and related controlproblems for many types of systems have been extensivelystudied. For example, linear systems [7], [8], time-delaysystems [9], [10], switched systems [11], [12], stochasticsystems [13], nonlinear systems [14], etc..

*This work was supported by the National Natural Science Foundation ofChina (No. 61304075), the Jiangsu Provincial Natural Science Foundation ofChina (No. BK20130538), and the Jiangsu University Research Foundationfor Talented Scholars (No. 13JDG112).

W. Xue and K. Li are with the School of Electrical and InformationEngineering, Jiangsu University, Zhenjiang, 212013, China {xwping,likangji}@ujs.edu.cn

Amato et al. [15] proposed a new concept which is calledinput-output finite-time stability (IO-FTS). Consistently withthe FTS concept, IO-FTS focuses on the input-output be-havior of the dynamics during a specified time interval. Inlater years, some results are available on IO-FTS analysisand synthesis problems. For instance, in [16], [17], theissue of input-output finite-time stabilization was discussedfor linear continuous-time or discrete-time systems. Theauthors investigated IO-FTS synthesis problem for impulsivedynamical linear systems in [18]. Both static output andstate feedback controllers were designed for the IO-FTSof the closed loop system. Amato et al. [19] establishednecessary and sufficient conditions for the IO-FTS of linearsystems concerning the class of L2 input signals. The authorsstudied the problems of IO-FTS analysis and synthesis forstochastic Markovian jump systems in [20]. The literature[21] addressed the problem of IO-FTS analysis for discrete-time impulsive switched systems with time-invariant statedelay. Sufficient conditions are obtained for IO-FTS of thesystem under the different cases of the switching signal.To the best of our knowledge, IO-FTS analysis problemfor time-delay systems has not been fully discussed yet.Moreover, little attention has been paid to the applicationof IO-FTS to practical systems. These points motivate themain purpose of our research.

In this paper, we aim to analyze the IO-FTS of thelinear time-delay system and then, based on the IO-FTSanalysis result, design a state feedback vibration-attenuationcontroller for the structural system with input delay.

The rest of this paper is organized as follows. Somepreliminaries are introduced in Section II. In Section III,a sufficient condition for the IO-FTS of the linear time-delay system is established. Our method to design the statefeedback vibration-attenuation controller is developed in IV.An illustrative example is presented in Section V to showthe effectiveness of the proposed method. Some conclusionsare given in Section VI.

Notation: Rn denotes the n-dimensional Euclidean space.Rn×m is the set of all n×m real matrices. The notation X >Y(respectively, X ≥ Y ), where X and Y are real symmetricmatrices, means that the matrix X−Y is positive definite (re-spectively, positive semi-definite). I and 0 denote the identityand zero matrices with appropriate dimensions, respectively.λmax(P) (λmin(P)) denotes the maximum (minimum) ofeigenvalues of a real symmetric matrix P. L2[0,∞) denotesthe set of the real vector-valued functions which are squareintegrable on [0,∞). The superscript T denotes the transposefor vectors or matrices. The symbol ∗ denotes the term that

2014 IEEE International Conference onAutomation Science and Engineering (CASE)Taipei, Taiwan, August 18-22, 2014

978-1-4799-5283-0/14/$31.00 ©2014 IEEE 878

is induced by symmetry. Matrices, if not explicitly stated,are assumed to have compatible dimensions for algebraicoperations.

II. PRELIMINARIES

Consider the linear time-delay system described byx(t) = Ax(t)+Adx(t−d)+Bω ω(t),

z(t) =Cx(t),

x(t) = φ(t), t ∈ [−d,0].(1)

where x(t) ∈ Rn is the state vector, d > 0 is the state delay,ω(t)∈Rp is the exogenous input, z(t)∈Rq is the controlledoutput, φ(t) is the initial condition. A, Ad , Bω and C areknown real constant matrices.

In the sequel, the definition of IO-FTS in [16] is extendedto the linear time-delay system (1).

Definition 1: The linear time-delay system (1) is said tobe input-output finite-time stable with respect to (w.r.t.)(c1,c2,Γ,T ), where scalars c1 > 0, c2 > 0, T > 0 and theweighting matrix Γ > 0, if under the zero initial condition,i.e., φ(t) = 0, ∀t ∈ [−d,0], system (1) satisfies

∀ω(t) ∈W ⇒ zT (t)Γz(t)< c22, ∀t ∈ [0,T ], (2)

where W ,{

ω(t)|∫ T

0 ωT (t)ω(t)dt ≤ c21

}Remark 1: It is noted that IO-FTS and BIBO (bounded

input bounded output) stability are two independent concepts.The former focuses on the input-output behavior of thesystem during a fixed time interval and requires that theoutput signal remain within a prescribed bound. However,the latter considers a sufficiently long time interval and justrequires that the output signal be bounded (the value of thebound is not prescribed).

Remark 2: If c1 = c2 = 1, Γ = Q(·), Definition 1 isconsistent with the IO-FTS definition w.r.t. (W2,Q(·),T ) in[16]. However, in some applications, c2 < c1 or only c2 > c1can be satisfied. Therefore, it is more general and practicalthat c1 = c2 is not required in Definition 1.

Remark 3: The set W in Definition 1 can also be definedas W ,

{ω(t)|ωT (t)ω(t)≤ c2

1,∀t ∈ [0,T ]}

, which corre-sponds to the IO-FTS definition w.r.t. (W2,Q(·),T ). It shouldbe pointed out that the analysis process of IO-FTS is similarin spite of two different sets W mentioned above (seeRemark 5 for details).

III. IO-FTS ANALYSIS

Theorem 1: System (1) is input-output finite-time stablew.r.t. (c1,c2,Γ,T ), if there exist matrices P> 0, Q> 0, R> 0,S > 0, M, N and a scalar γ ≥ 0, satisfying the followingconditions:

Φ11 Φ12 dN PBω dAT R∗ Φ22 dM 0 dAT

d R∗ ∗ −dR 0 0∗ ∗ ∗ −S dBT

ω R∗ ∗ ∗ ∗ −dR

< 0, (3a)

CTΓC−P < 0, (3b)

S <c2

2

eγT c21

I, (3c)

where

Φ11 = AT P+PA− γP+Q−N−NT ,

Φ12 = PAd−MT +N,

Φ22 =−eγdQ+M+MT .Proof: For system (1), construct the following energy

function:

V (t) = xT (t)Px(t)+∫ t

t−deγ(t−s)xT (s)Qx(s)ds

+∫ 0

−d

∫ t

t+θ

xT (s)Rx(s)dsdθ .

Then, it is easily obtained that

V (t) = xT (t)(AT P+PA)x(t)+2xT (t−d)ATd Px(t)

+2ωT (t)BT

ω Px(t)+ γ

∫ t

t−deγ(t−s)xT (s)Qx(s)ds

+ xT (t)Qx(t)− eγdxT (t−d)Qx(t−d)

+dxT (t)Rx(t)−∫ t

t−dxT (s)Rx(s)ds. (4)

Letting ξ (t,s)=[xT (t) xT (t−d) xT (s) ωT (t)

]T , it is obviousthat

dxT (t)Rx(t) = dξT (t,s)

AT

ATd

0BT

ω

R

AT

ATd

0BT

ω

T

ξ (t,s). (5)

For any matrices M and N, we have

1d

∫ t

t−d2xT (t−d)dMx(s)ds−2xT (t−d)M[x(t)−x(t−d)]= 0,

(6)1d

∫ t

t−d2xT (t)dNx(s)ds−2xT (t)N[x(t)− x(t−d)] = 0. (7)

It follows from (4) ∼ (7) that

V (t)−γV (t)−ωT (t)Sω(t)

≤ V (t)− γ

[xT (t)Px(t)+

∫ t

t−deγ(t−s)xT (s)Qx(s)ds

]−ω

T (t)Sω(t)

=1d

∫ t

t−dξ

T (t,s)Φξ (t,s)ds, (8)

where

Φ =

Φ11 Φ12 dN PBω

∗ Φ22 dM 0∗ ∗ −dR 0∗ ∗ ∗ −S

+dξT (t,s)

AT

ATd

0BT

ω

R

AT

ATd

0BT

ω

T

ξ (t,s).

879

By Schur complement, (3a) is equivalent to Φ < 0. Based on(8), it is easily seen that

V (t)− γV (t)≤ ωT (t)Sω(t). (9)

Further, (9) can be rewritten as

ddt

[e−γtV (t)

]≤ e−γt

ωT (t)Sω(t). (10)

Integrating (10) from 0 to t, we can obtain that

V (t)≤ eγt∫ t

0e−γs

ωT (s)Sω(s)ds. (11)

Based on (3b), (3c) and (11), we have

zT (t)Γz(t) = xT (t)CTΓCx(t)≤ xT (t)Px(t)≤V (t)

≤ eγtλmax(S)

∫ t

0ω

T (s)ω(s)ds < eγT c22

eγT c21

c21

= c22, ∀t ∈ [0,T ]. (12)

From Definition 1, we know that system (1) is input-outputfinite-time stable w.r.t. (c1,c2,Γ,T ). The proof is completed.

Remark 4: If the conditions (3) hold with γ = 0, it is easilyobtained from (8) that system (1) is asymptotically stablewithout consideration of ω(t). Moreover, if the conditions (3)hold with γ = 0, it is straightforward from (12) that system(1) is input-output finite-time stable w.r.t. (c1,c2,Γ,T ) for allT > 0.

Remark 5: If the set W is defined as W ,{ω(t)|ωT (t)ω(t)≤ c2

1,∀t ∈ [0,T ]}

, just replace (3c) in

Theorem 1 with S <c2

2γ

(eγT−1)c21I (if γ = 0, taking the limit

value for the right side of this inequality, which turns intoS <

c22

T c21I). In this case, (12) can be expressed as

zT (t)Γz(t) = xT (t)CTΓCx(t)≤ xT (t)Px(t)≤V (t)

≤ λmax(S)∫ t

0eγ(t−s)

ωT (s)ω(s)ds

<c2

2γ

(eγT −1)c21

c21

eγT −1γ

= c22, ∀t ∈ [0,T ].

IV. VIBRATION-ATTENUATION CONTROLLERDESIGN

Consider the following n degree-of-freedom structuralsystem with an input delay:

Mx(t)+Cx(t)+Kx(t) = Hu(t− τ)+Hω ω(t), (13)

where x(t) = [x1(t) x2(t) · · · xn(t)]T , xi(t) (i = 1,2, . . . ,n)is the interstorey drift (relative displacement between twostoreys) of the ith floor, u(t)∈Rm is the control input vector,τ > 0 is the time-delay in the control force, ω(t) ∈ Rp isthe disturbance input which belongs to L2[0,∞). H ∈ Rn×m

denotes the location of the controllers, Hω ∈ Rn×p denotesthe influence of disturbance input. M ∈Rn×n, C ∈Rn×n andK ∈ Rn×n are the mass, damping and stiffness matrices ofthe system, respectively.

By letting q(t) =[xT (t) xT (t)

]T and considering thecontrolled output z(t), system (13) can be expressed in thefollowing state-space form:

q(t) = Aq(t)+Bu(t− τ)+Bω ω(t),

z(t) =Coq(t),

q(t) = φ(t), t ∈ [−τ,0],(14)

where Co is a known real constant matrix, φ(t) is the initialcondition,

A =

[0 I

−M−1K −M−1C

], B =

[0

M−1H

], Bω =

[0

M−1Hω

].

In this section, an active state feedback controller will bedesigned for the structural system (14) such that the closed-loop system has the following peak performance: for a givenperformance index µ > 0, a known weighting matrix Γ > 0,all t > 0, and all non-zero ω(t) ∈ L2[0,∞), the closed-loopsystem satisfies

zT (t)Γz(t)< µ2∫

∞

0ω

T (s)ω(s)ds (15)

under zero initial condition. The closed-loop system of (14)with the state feedback controller u(t) =Fq(t) can be writtenby:

q(t) = Aq(t)+BFq(t− τ)+Bω ω(t),

z(t) =Coq(t),

q(t) = φ(t), t ∈ [−τ,0].(16)

According to Definition 1, it is known that the peak per-formance mentioned above is satisfied if and only if theclosed-loop system (16) is input-output finite-time stablew.r.t. (c,µ · c,Γ,T ), where T → ∞. Therefore, our maingoal in the rest of this section is to design a controlleru(t) =Fq(t), which guarantees the IO-FTS ofthe closed-loopsystem (16).

Theorem 2: Consider the structural system (14), thereexists a state feedback controller u(t) = Fq(t) such thatthe closed-loop system (16) is input-output finite-time stablew.r.t. (c,µ · c,Γ,T ), where T → ∞, if there exist matricesZ > 0, G > 0, W > 0, S > 0, L, X , Y , satisfying the followingconditions:

Ξ11 BL−XT +Y τY Bω τZAT

∗ −G+X +XT τX 0 τLT BT

∗ ∗ τ(W −2Z) 0 0∗ ∗ ∗ −S τBT

ω

∗ ∗ ∗ ∗ −τW

< 0,

(17a)[−Z ZCT

o∗ −Γ−1

]< 0, (17b)

S < µ2I, (17c)

where Ξ11 = ZAT +AZ +G−Y −Y T .Moreover, the controller gain is given by F = LZ−1.

Proof: Based on Theorem 1 and Remark 4, it is knownthat system (16) is input-output finite-time stable w.r.t. (c,µ ·

880

c,Γ,T ), where T → ∞, if there exist matrices P > 0, Q > 0,R > 0, S > 0, M, N, satisfying (17c),

Ξ12 PBF−MT +N τN PBω τAT R∗ −Q+M+MT τM 0 τFT BT R∗ ∗ −τR 0 0∗ ∗ ∗ −S τBT

ω R∗ ∗ ∗ ∗ −τR

< 0,

(18a)CT

o ΓCo−P < 0, (18b)

where Ξ12 = AT P+PA+Q−N−NT .Pre-multiplying and post-multiplying (18a) by

diag(P−1,P−1,P−1, I,R−1) and its transpose respectively,letting Z , P−1, G , ZQZ, W , R−1, L , FZ, X , ZMZ,Y , ZNZ, and noticing that −τZW−1Z ≤ τ(W − 2Z)since (W − Z)W−1(W − Z) ≥ 0, we can obtain that (17a)guarantees (18a). Pre- and post-multiplying (18b) by Z andits transpose respectively, and using Schur complement,we obtain that (17b) is equivalent to (18b). The proof iscompleted.

Remark 6: Theorem 2 provides a sufficient condition interms of LMIs for the controller design. In fact, by perform-ing an optimization over µ , the controller design problem canbe turned into the following convex optimization problem:

min µ

s. t. LMIs (17a),(17b),(17c). (19)

V. ILLUSTRATIVE EXAMPLE

Example 1: Consider the structural system (14) with thefollowing parameters [2], [4]

M =

1.1 0 00 1.8 00 0 1.6

, C =

1.2 −0.6 0−0.6 1.2 −0.6

0 −0.6 0.6

,K =

2 −1 0−1 2 −10 −1 1

, Hω =

00

0.1

.In this example, it is assumed that only the first storey ofthe structural system has a controller, that is, H = [1 0 0].The controlled output is chosen to be the displacement ofthe third floor relative to the ground. Therefore,

Co =[1 1 1 0 0 0

].

If τ = 0.05 s, Γ = I, by solving the optimization problem(19), we obtain that µ = 0.0027 and the correspondingcontroller gain (denoted as F1)

F1 =[−91.6817 −93.5253 −92.9923

−14.6846 −15.8164 −15.4876]

In order to show the effectiveness of the designed controller,we use the 1940 El Centro earthquake excitation as thedisturbance input ω(t), which is shown in Fig. 1. The outputresponses of the open-loop and closed-loop (the controllergain F1) systems are depicted in Fig. 2.

0 5 10 15 20 25 30−4

−3

−2

−1

0

1

2

3

Time (s)

Accele

ration (

m/s

2)

Fig. 1. 1940 El Centro earthquake excitation

0 5 10 15 20 25 30−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Time (s)

Dis

pla

cem

ent (m

)

Open−loopF

1

Fig. 2. Displacement of the third floor relative to the ground (τ = 0.05 s)

For the open-loop system, it is calculated that

supt∈[0,∞)

√zT (t)Γz(t)√∫

∞

0 ωT (s)ω(s)ds=

0.01313.5802

≈ 0.0037 > 0.0027.

For the closed-loop system, it is calculated that

supt∈[0,∞)

√zT (t)Γz(t)√∫

∞

0 ωT (s)ω(s)ds=

0.00163.5802

≈ 0.00045 < 0.0027.

From above calculations, it is known that the designedcontroller guarantees the desired IO-FTS of the closed-loop system. Moreover, from Fig. 2, it is easily seen thatthe closed-loop system has clearly decreased output peakcompared with the open-loop system.

Similarly, when τ is respectively given as 0.1 s and 0.15s, the minimum value of µ can be obtained as 0.0075 and0.0135, respectively. The obtained controller gain can berespectively expressed as

F2 =[−23.8592 −25.6818 −25.1666

−7.2425 −8.4126 −8.0779]

F3 =[−10.8322 −12.6169 −12.1315

−4.7618 −5.9780 −5.6407]

In addition, the output responses of the open-loop and closed-loop systems are respectively shown in Fig. 3 and Fig. 4,

881

which further demonstrate the effectiveness of the designedcontrollers.

0 5 10 15 20 25 30−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Time (s)

Dis

pla

cem

ent (m

)

Open−loopF

2

Fig. 3. Displacement of the third floor relative to the ground (τ = 0.1 s)

0 5 10 15 20 25 30−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Time (s)

Dis

pla

cem

ent (m

)

Open−loopF

3

Fig. 4. Displacement of the third floor relative to the ground (τ = 0.15 s)

VI. CONCLUSIONSIn this paper, the problems of IO-FTS analysis for linear

time-delay systems and its application to active vibrationcontrol for structural systems are investigated. A sufficientcondition is firstly established for the linear time-delaysystem to be input-output finite-time stable. Then, based onthe obtained analysis result, a state feedback controller isdesigned for vibration attenuation of the structural systemwith time-delay in control input. Finally, considering the1940 El Centro earthquake excitation, a vibration-attenuationexample for the linear structural system is provided toshow the effectiveness of the proposed methodology. Aninteresting topic, which concerns the comparison of IO-FTS control strategy with some previously presented activevibration control strategies, will be studied in the future.

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