Sampled-data-based vibration control for structural systems with finite-time state constraint and sensor outage

ISA Transactions xxx (xxxx) xxx–xxx

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Research article

Sampled-data-based vibration control for structural systems with finite-time state constraint and sensor outage Falu Wenga,∗, Mingxin Liub, Weijie Maoc, Yuanchun Dingd, Feifei Liua a

Faculty of Electrical Engineering and Automation, Jiangxi University of Science and Technology, Ganzhou, Jiangxi, 341000, China School of Computer Science and Technology, Hangzhou Dianzi University, Hangzhou 310018, China c State Key Lab of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, China d School of Resources and Environmental Engineering, Jiangxi University of Science and Technology, Ganzhou, Jiangxi, 341000, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Finite-time stability Sensor outage Structural systems Vibration Sampled-data

The problem of sampled-data-based vibration control for structural systems with finite-time state constraint and sensor outage is investigated in this paper. The objective of designing controllers is to guarantee the stability and anti-disturbance performance of the closed-loop systems while some sensor outages happen. Firstly, based on matrix transformation, the state-space model of structural systems with sensor outages and uncertainties appearing in the mass, damping and stiffness matrices is established. Secondly, by considering most of those earthquakes or strong winds happen in a very short time, and it is often the peak values make the structures damaged, the finite-time stability analysis method is introduced to constrain the state responses in a given time interval, and the H-infinity stability is adopted in the controller design to make sure that the closed-loop system has a prescribed level of disturbance attenuation performance during the whole control process. Furthermore, all stabilization conditions are expressed in the forms of linear matrix inequalities (LMIs), whose feasibility can be easily checked by using the LMI Toolbox. Finally, numerical examples are given to demonstrate the effectiveness of the proposed theorems.

1. Introduction In recent years, because strong earthquakes and hurricanes happen frequently, vibration control for structural systems has received considerable attention, and many control methods were achieved for attenuating those vibrations resulted from seismic or wind excitations. Normally, those methods can be classified into three types: passive control, semi-active control and active control. Due to the virtues of low energy consumption and low cost, the passive and semi-active controls were ever researched heatedly [1,2]. However, with the structural systems built higher and higher, the stability and solidity of structural systems are challenged and cannot be guaranteed only by those passive and semi-active control methods. Thus, the active vibration control for structural systems has been heatedly discussed recently, and many control strategies, such as, classical H∞ theories [3–5], energy-to-peak control [6–8], finite frequency control [9], sliding mode control [10,11], adaptive control [12], fuzzy control [13,14], model predictive control [15], optimal control [16], etc., have been utilized for protecting structures subjected to seismic or wind excitations. Moreover, many active control devices, such as, active mass damper (AMD) [17,18], active brace system (ABS) [19,20], etc., were also designed for applying those control algorithms. ∗

However, most of the existing results are obtained on the basis of the assumption that the sensors can provide uninterrupted signal measurement. In practice, contingent failures are possible for all sensors in a system, which may result in substantial damage, and can even be hazardous to human and environmental security. Thus, sensor failure is an inevitable problem which needs to be considered in the active control devices and algorithms design. At least, the sensor failures include two major types: sensor outage and sensor performance degradation. Sensor outage means the sensor is completely broke down; and sensor performance degradation means the sensor can still work with lower performances, such as lower degree of precision, higher error rate, etc. During the last decades, some efforts have been made by scholars to obtain the results about sensor failures, and some achievements were reached. For example, based on linear matrix inequality (LMI) technique, the problem of sensor fault-tolerant vibration attenuation controller design for uncertain buildings structural systems was investigated in Ref. [21]. In terms of H∞ theory, Ref. [22] discussed the problem of simultaneous design of reliable filter and fault detector for a class of linear continuous-time systems with bounded disturbances and nonzero constant reference inputs, and numerical example was given to illustrate the effectiveness of the proposed methods. The

Corresponding author. E-mail addresses: [email protected], [email protected] (F. Weng).

https://doi.org/10.1016/j.isatra.2018.04.021 Received 21 December 2015; Received in revised form 7 February 2018; Accepted 30 April 2018 0019-0578/ © 2018 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Weng, F., ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.04.021

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existing results show robust analysis can solve some sensor performance degradations greatly, however, the results about sensor outage are still few, and it is not fully investigated, obviously. On the other hand, with the advances in computer measurement and control technique, the analog signals are often replaced by digital signals to provide better performances [23]. Thus, sampled-data systems have attracted great attention, and many achievements have been reached during the last several decades. The corresponding results can be found in Refs. [24–27] and those references therein. Moreover, with recent focus on wireless monitoring and control of structural systems [28–31], research on sampled-data-based control for structural system is becoming significant. Classical solutions to this type of problem can be found in the literatures [8,32–34], where sampled-data control algorithms taking into account external excitations were given for structural systems, and numerical examples were given to show the validation of those methods. However, most of those existing results were obtained by using Lyapunov stability theory, which cases about asymptotic convergence of structural systems. It is well known that the structural systems are often damaged by the peak responses of displacements or accelerations, thus, obtaining some results with a constraint on the peak responses of displacements or accelerations will be much more practical, obviously. Very recently, the problem of finitetime stability of systems has received considerable attention. For example, by employing the Lyapunov-like function method, Ref. [35] addressed the problems of input-output finite-time stability analysis for linear time-delay systems and applied it to active vibration control for structural systems with input delay. In terms of a special Lyapunov functional, the finite-time vibration control of earthquake excited linear structures with input time-delay and saturation was concerned in Ref. [36], and numerical examples were given to illustrate the effectiveness of the developed theory. More achievements about this issue can also be found in Refs. [37,38] and the references therein. This paper concerns the problem of sampled-data-based vibration controller design for structural systems with finite-time state constraint and sensor outage. Based on matrix transformation, the state-space model of structural systems, which contain sampled-data signals, sensor outage and uncertainties appearing in the mass, damping and stiffness matrices, is established. Then, in terms of the obtained model and finite-time stability technique, the LMIs-based conditions are established for the structural systems to be stabilizable with finite-time state constraint and sensor outage. By solving these LMIs, the desired controller can be obtained such that the state responses of the closed-loop system constrained byxT (t) Rx (t) < c32 (R > 0) during the time interval[0,T], and the influence of the external disturbances is constrained by z 2 < γ ω 2 (γ> 0) during the whole control process. Furthermore, when sensor outages happen, the control system can reconfigure the controllers according to the signals come from the sensor outage detector. In the end, numerical examples are given to show the effectiveness of the proposed theorems.

Fig. 1. n degree-of-freedom structural system.

nth storey to ground; u(t) is the control force input; x¨ g (t) is the ground acceleration, H 0 ∈ Rn×m gives the locations of these controllers, H ω ∈ Rn×1 is an vector denoting the influence of disturbance excitation, and M, C, K ∈ Rn×n are the mass, damping and stiffness matrices of the system, respectively. From Fig. 1, it is obtained that M = diag{m1, m2, ⋯, mn}, H ω = −[m1, m2, ⋯, mn]T , ⎡ c1 + c2 − c2 ⎢ − c 2 c 2 + c3 C=⎢ ⋮ ⋮ ⎢ 0 ⎣ 0

(2)

where Cz is real constant matrix with appropriate dimensions, ω (t) = x¨ g (t) , and

I 0 I 0 0 I ⎤ 0 A = ⎡ ∼⎤ A 0 , B = ⎡ ∼⎤ ⎡ ⎤ , A 0 = ⎡ , ⎢0 M⎥ ⎢ 0 M⎥ ⎢ H0 ⎥ ⎢− K − C ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ T

⎡ ⎤ ∼ M = diag{1/m1, 1/m2, ⋯, 1/mn}, Bω = ⎢ 0, , − 1⎥ , ⋯,0 ,  −1, ⋯  ⎢ ⎥ n ⎣ n ⎦ Remark 1. Some historical earthquake records are listed in Table 1 [40,41]. It is obvious that the ground accelerations and durations are all limited in some special bounds, that is, earthquake excitations can be described as an energy-bounded disturbance, thus, the ω(t) shown in the paper satisfiesω(t) ∈ L2 [0, +∞], and for a given time interval [0,T], Table 1 Fundamental information of some earthquakes. Year

Observation site

Peak of ground acceleration

Duration (s)

(m/s2 ) 1940 1940 1952 1966 1971 1971 1979 1989 1994

Consider an n degree-of-freedom structural system, which is depicted in Fig. 1. The structural model equation can be written as [5–9,21,35,36,38,39].

wherexm (t) = [xm1(t),x m2 (t), ⋯,xmn

⋯ 0 ⎤ ⋯ ⋮ ⎥ , ⋯ − kn ⎥ ⎥ ⋯ kn ⎦

x˙ (t) = Ax (t) + Bu (t) + Bω ω(t), z (t) = Cz x (t),

2. Problem formulation and dynamic models

(t)]T

⎡ k1 + k2 − k2 ⎢ − k2 k2 + k3 K=⎢ ⋮ ⎢ ⋮ 0 ⎣ 0

Defining the state variables asx (t) = [xm (t) T , x˙ m (t) T]T , equation (1) can be written in the following state-space form:

Notation. Throughout this paper, for real matrices X and Y , the notation X ≥ Y (respectivelyX > Y ) means that the matrix X − Y is semi-positive definite (respectively, positive definite). I is the identity matrix with appropriate dimension, and a superscript “T” represents transpose. We define MH = M T + M. For a symmetric matrix, ∗ denotes the symmetric terms. The symbol Rn stands for the n -dimensional Euclidean space, and Rn×m is the set of n× m real matrices.

Mx¨ m (t) + Cx˙ m (t) + Kxm (t) = H 0 u (t) + H ω x¨ g (t),

⋯ 0 ⎤ ⋯ ⋮ ⎥ , ⋯ − cn ⎥ ⎥ ⋯ cn ⎦

(1)

, xmn (t) is the relative drift of the 2

EI Centro, 270 Deg EI Centro, 180 Deg Taft Lincoln School Parkfield Cholame, Shandon San Fernando, 69 Deg San Fernando, 159 Deg James RD., 220 Deg Loma Prieta, 270 Deg Northridge, 90 Deg

3.498 2.099 1.526 2.323 3.091 2.652 3.600 2.704 5.926

53.72 53.46 54.38 26.18 61.84 61.88 37.68 39.98 59.98

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the uncertain system can be described by state space equation of the form:

x˙ (t) = A (β,θ) x (t) + B (β) Fo x (t−τ(t)) + Bω ω (t), tk ≤ t < tk+1, z (t) = Cz x (t),

(7)

where A(β,θ) and B(β) satisfy

A (β,θ) = B (β) =

2n

I

0 ⎛ 2n θ A ⎟⎞, ∼⎤ ⎜A + ∑ i= 1 i i Mj ⎥ 0 ⎣ ⎦⎝ ⎠ I 0 0 βj ⎡ ∼ ⎤ ⎡ ⎤, ⎢ 0 Mj ⎥ ⎢ H 0 ⎥ ⎣ ⎦⎣ ⎦

∑j=1 βj ⎡⎢ 0 2n

∑ j= 1

Fig. 2. Block diagram of the structural control system with controller reconfiguration.

0 I ⎤ ˆ A0 = ⎡ , K= ˆ⎥ ⎢− K ˆ −C ⎣ ⎦

T

there is a constantd≥ 0 , such that ∫0 ωT (t) ω (t)dt ≤ d . More results in this issue can also be found in Refs. [40,41]. When considering possible sensor outage, a state-feedback controller is introduced which has the form of

u (t) = Fo x (t),

⎡ cˆ1+cˆ2 − cˆ2 ⋯ − cˆ2 cˆ2+cˆ3 ⋯ ˆ =⎢ C ⎢ ⋮ ⋮ ⋯ ⎢ 0 ⋯ ⎢ 0 ⎣

(3)

where Fo is the controller gain to be designed later, and its items in the columns corresponding to the failed sensors are zeros, for example, the items in column 2 are all zeros while the sensor 2 is in outage. In this paper, it is assumed that the state variables are measured at time instants 0= t 0 < t1<⋯ < tk < tk+1<⋯+∞ , that is, only x(tk) is available in the time interval [tk ,tk+1]. The block diagram of the sampled-data control for structural systems is shown in Fig. 2. The control signal is assumed to be generated by using a Zero-Order-Hold (ZOH), and suppose that updating signal experiences a constant signal transmission delayh (the delay time h includes transmitting the state signals from the sampler to the controller, reconfiguring the controller, calculating control forces and transmitting the control force signals from the controller to the ZOH). Assume that the sampling intervals satisfy

tk+1 − tk ≤ τ, k= 0, 1, 2, …,

⎡ kˆ1+kˆ2 − kˆ2 ⋯ 0 ⎤ ⎥ ⎢ ˆ ˆ ˆ ⎢ − k2 k2+k3 ⋯ ⋮ ⎥, ˆ ⎢ ⋮ ⋮ ⋯ − kn ⎥ ⎥ ⎢ 0 ⋯ kˆn ⎦ ⎣ 0 0 ⎤ ⋮ ⎥ , − cˆ n ⎥ ⎥ cˆn ⎦ ⎥

Ai = kˆ i ei f iT, An+i = cˆi en+i f nT+i, i= 1, 2, ⋯, n.

ei ∈ R2n , en+i ∈ R2n , fi ∈ R2n , and fn+i ∈ R2n are all column vectors. Definition 1. The system (6) is said to be finite-time state-constraint H∞ stabilizable with respect to (c1, c2, c3, R , T, γ, d) , if there exists a controller gain Fo such that the closed-loop system has

{ΦT (s) RΦ (s)} ≤ c12,

sup

{Φ˙ T (s) RΦ˙ (s)} ≤ c 22 ⇒ x (t) TRx (t)

sup

s∈ [−(h+τ),0]

s∈ [−(h+τ),0]

≤ c32 and z

2

<γ ω

(8)

2

T

for anyt∈ [0,T], ∫0 ωT (t) ω (t)dt ≤ d , where 0 < c1 < c3 , c2 > 0 , R > 0 , T > 0 , γ > 0 , d≥ 0 .

(4)

(5)

Definition 2. The system (7) is said to be robustly finite-time stateconstraint H∞ stabilizable with respect to(c1,c2 ,c3, R,T,γ,d) , if system (7) is finite-time state-constraint H∞ stabilizable for all admissible uncertainties.

By definingτ(t) = t− tk + h,tk ≤ t < tk+1, and substituting the controller (5) into the structural system (2), the closed-loop system becomes

Lemma 1. [42]: For any matrixW>0 , scalars h1 and h2 satisfying h2 > h1, a vector function φ:[h1,h2] → Rn such that the integrations concerned are well defined, then

x˙ (t) = Ax (t) + BFo x (t−τ(t)) + Bω ω (t), z (t) = Cz x (t),

(h2 − h1)

where τ > 0 is the maximum sampling interval. Therefore the state feedback controller takes the following form:

u (t) = Fo x (tk−h),

tk ≤ t < tk+1,

tk ≤ t < tk+1, (6)

∫a

∫h

h2

1

∫h

φT (s)dsW

h2

φ (s)ds.

1

b

φT (s) Qφ (s)ds ≤

4(b− a)2 π2

∫a

b

φ˙ T (s) Qφ˙ (s)ds.

Lemma 4. [45]: Let ϖ (t) be a nonnegative function such that t ϖ (t) ≤ a+ b∫0 ϖ (s)ds , 0≤ t≤ T for some constants a , b > 0 , then, we haveϖ (t) ≤ aebt , 0≤ t≤ T . Lemma 5. [46,47]: given matrices χ , μ and ν with appropriate dimensions and with χ symmetrical, then χ + μF (t) ν + νT F (t) TμT<0 holds for any F(t) satisfying F (t) TF (t) ≤ I , if and only if there exists a scalar δ > 0 such that χ + δμμT + δ−1 νT ν<0 .

θi ≤ θi <1, θi = (ki − k i)/(ki + k i),

i= 1, 2, ⋯, n,

cˆ i = (c i + ci)/2, Δcˆ i = θn+i cˆ i ,

φT (s) Wφ (s)ds ≥

Lemma 3. [44]: Given any matricesX , V and U with appropriate dimensions such that U>0 . Then, we have− XUX ≤ XVT + VXT + VU−1VT .

(mi − mi)m i 1 1 + (1 − βi1) , βi1 = , i= 1, 2, ⋯, n, mi mi (mi − m i)mi

kˆi = (k i +k i)/2, Δkˆi = θi kˆi,

h2

1

Lemma 2. [43]: Let φ (t) ∈ W [a,b] and φ(a) = 0 . Then for any matrixQ>0 , the following inequality holds:

It can be gotten that h≤ τ(t) < tk+1 − tk + h . Thus, system (6) is a continuous-time system with an uncertain and bounded delay in state. Furthermore, the mass, damping and stiffness are usually subjected to possible perturbations, such as measurement error, the changes in environmental temperature and plastic deformation, etc. Assume that the uncertainties satisfyingmi ∈ [m i,m i], ki ∈ [k i,k i], c i ∈ [c i, c i] i= 1, 2, ⋯, n , where m i, k i, c i (mi, k i, c i ) are the lower (upper) bounds of the mass, stiffness and damping, respectively, and denote

1/mi = βi1

∫h

θn+i ≤ θn+i <1,

θn+i = (c i − c i)/(c i + c i), i= 1, 2, ⋯, n,

3. Stabilization criteria

∼ ∼ 2n Then, it getsM (β) = diag{1/m1, 1/m2, ⋯, 1/mn} = ∑ j= 1 βj Mj, where n ∼ ∼ 2 Mj(j=1, 2, ⋯, 2n) are the vertices ofM(β) , βj ≥ 0 and∑ j= 1 βj = 1. Thus,

Theorem 1. The system (6) is finite-time state-constraint H∞ stabilizable with respect to (c1, c2, c3, R , T, γ, d) for constants 3

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symmetric matricesP , Qi(i=1, 2, 3, 4) , Ui(i=1, 2) , nonsingular matrixS , matrices Go Yi(i=1, 2, 3, 4, 5, 6) , positive scalars εi κ4 ri (i=1, 2, ⋯, 2n) , (i=1, 2, ⋯, 9) , scalarsκ1, κ2, κ3 and satisfying (10)–(11) and the following LMIs

h > 0, τ≥ 0 and α≥ 0 , if there exist positive definite symmetric matrices P , Qi(i=1, 2, 3, 4) , Ui(i=1, 2) , nonsingular matrixS , matricesGo , Yi (i=1, 2, 3, 4, 5, 6) , positive scalarsεi(i=1, 2, ⋯, 9) , scalars κ1, κ2, κ3 and κ 4 satisfying the following LMIs Ξ ⎤ ⎡ Ψ hY ⎢ ∗ − hQ2 0 ⎥ <0, ⎢ ∗ − I⎥ ⎣∗ ⎦

Ξ ⎡ Ψj1 hY ⎢ ∗ − hQ2 0 Ψj = ⎢ ∗ −I ⎢ ∗ ⎢ ∗ ∗ ∗ ⎣

(9)

ε1R

Ξ2 ⎤ 0 ⎥ <0, j= 1, 2, ⋯, 2n , ⎥ 0⎥ Ξ3 ⎥ ⎦

(12)

(10) where

h2 ε c2 2 5 1 τ 2ε9 hc 22 +

ε2 c12 + ε1hc12 + + h2ε8c12 +

h3 1 ε c 2 + 12 h5ε7 c 22 2 6 2 γ2d < e−αT (ε1 + hε 4)c32 ,

+

⎡ Ψj11 Ψj12 Ψj13 ⎢ * Ψ Ψ 22 j23 ⎢ ⎢ * * Ψj33 Ψj1 = ⎢ * * ⎢ * ⎢ * * * ⎢ * * * ⎣

(11)

where

⎡ Ψ11 Ψ12 Ψ13 ⎢ ∗ Ψ22 Ψ23 ⎢ ∗ ∗ Ψ33 Ψ=⎢ ∗ ∗ ⎢ ∗ ⎢ ∗ ∗ ∗ ⎢ ∗ ∗ ⎢ ⎣ ∗

Ψ14 Ψ15 Ψ24 − κ1S Ψ34 Ψ35 Ψ44 ∗ ∗

Ψ45 Ψ55 ∗

Bω κ1Bω κ2 Bω

⎤ ⎥ ⎥ ⎥, Y6T + κ3 Bω ⎥ κ 4 Bω ⎥ ⎥ − γ 2I ⎥ ⎦

Ψ33 =

Ψ45 Ψ55 *

∑ (riθ2i kˆ 2i ϒjiϒTji + rn+i θ2n+i cˆi2 ϒj(n+i) ϒTj(n+i) ), T

T

I 0 Ψj12 = Q3 + κ1STA 0T ⎡ ∼ ⎤ , ⎢ 0 Mj⎥ ⎣ ⎦ T

π2 U2, 4

I 0 I 0 0 Ψj13 = ⎡ ∼ ⎤ ⎡ ⎤ Go + κ2STA 0T ⎡ ∼ ⎤ , ⎢ 0 Mj⎥ ⎢ H0 ⎥ ⎢ 0 Mj⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

Ψ13 = BGo + κ2 STAT , π2 − 4 U2

Ψj35

Ψ44 * *

⎤ ⎥ ⎥ κ2 Bω ⎥ ⎥ Y6T + κ3 Bω ⎥ κ 4 Bω ⎥ ⎥ − γ2I ⎦

I 0 I 0 Ψj11 = Q1 + hQ2 + (h2 − 1) Q3 + hU1 − h2Q4 + ⎡ ∼ ⎤ A 0 S + STA 0T ⎡ ∼ ⎤ − αP, ⎢ 0 Mj⎥ ⎢ 0 Mj⎥ ⎣ ⎦ ⎣ ⎦

Ψ11 = Q1 + hQ2 + (h2 − 1) Q3 + hU1 − h2Q4 + ΑS + STAT − αP, Ψ12 = Q3 + κ1STAT ,

Ψ23 =

Ψj34

Bω

κ1Bω

i= 1

T

π2 U2 4

Ψj15

Ψ24 − κ1S

n

+

Ξ = [Cz S 0 0 0 0 0]T , Y = [ Y1T Y2T Y3T YT4 Y5T Y6T ] ,

Ψ22 = −Q1 − Q3 − hU1 −

Ψj14

T

I 0 Ψj14 = −U1 + Y1 + hQ4 + κ3STA 0T ⎡ ∼ ⎤ , ⎢ 0 Mj⎥ ⎣ ⎦

+ κ1BGo ,

T

Ψ14 = −U1 + Y1 + hQ4 + κ3STAT , Ψ24 = U1 + Y2,

I 0 Ψj15 = P − S + κ 4 STA 0T ⎡ ∼ ⎤ , ⎢ 0 Mj⎥ ⎣ ⎦ I 0 ⎤⎡ 0 ⎤ π2 ⎡ Ψj23 = U2 + κ1 G , ⎢0 ∼ 4 Mj⎥ ⎢ H0 ⎥ o ⎣ ⎦⎣ ⎦

Ψ34 = Y3 + κ3GoT BT ,

Ψj33 = −

+ κ2 BGo + κ2 GoT BT ,

T I 0 0 0 ⎞ π2 ⎛ I 0 U2 + κ2 ⎡ ∼ ⎤ ⎡ ⎤ Go + κ2GoT ⎜ ⎡ ∼ ⎤ ⎡ ⎤ ⎟ , ⎢ 0 Mj⎥ ⎢ H0 ⎥ ⎢ 0 Mj⎥ ⎢ H0 ⎥ 4 ⎦⎣ ⎦⎠ ⎦⎣ ⎦ ⎣ ⎝⎣ T ⎛⎡ I 0 ⎤⎡ 0 ⎤⎞ T Ψj34 = Y3 + κ3Go ⎜ , ⎢0 ∼ Mj⎥ ⎢ H0 ⎥ ⎟

YT4

− Q4 , Ψ44 = Y4 + Ψ15 = P − S + κ 4 STAT ,

⎝⎣

Ψ35 = −κ2 S + κ 4 GoT BT ,

⎦⎣

⎦⎠

T

0 ⎞ ⎛ I 0 Ψj35 = −β2 S + κ 4 GoT ⎜ ⎡ ∼ ⎤ ⎡ ⎤ ⎟ , ⎢ 0 Mj⎥ ⎢ H0 ⎥ ⎦⎣ ⎦⎠ ⎝⎣

Ψ45 = Y5T − κ3S , 1

Ψ55 = 4 h4Q4 + τ 2U2 − κ 4 S − κ 4 ST. Furthermore, a state-feedback controller is described asFo = Go S−1.

T

I 0 ⎡ ϒji = ⎢ eiT ⎡ ∼ ⎤ ⎢ 0 Mj⎥ ⎢ ⎦ ⎣ ⎣

Proof. See the Appendix. Remark 2. From Theorem 1, it is obtained that the state responses of the structural system are constrained byxT (t) Rx (t) < c32 during the time interval[0, T], and the energy gain from ω(t) to z(t) is constrained by z 2 < γ ω 2 during the whole control process. That is, Theorem 1 gives a method for designing an H∞ stabilization controller with finitetime state-constraint performance for structural systems.

T

I 0 κ1eiT ⎡ ∼ ⎤ ⎢ 0 Mj⎥ ⎣ ⎦

T

I 0 κ2eiT ⎡ ∼ ⎤ ⎢ 0 Mj⎥ ⎣ ⎦

T

I 0 κ3eiT ⎡ ∼ ⎤ ⎢ 0 Mj⎥ ⎣ ⎦

T

I 0 κ 4 eiT ⎡ ∼ ⎤ ⎢ 0 Mj⎥ ⎣ ⎦

T

⎤ 0 0 0⎥ , ⎥ ⎦

T

Ξ2 = [Λ1, Λ2, ⋯, Λ2n], Λi = [ fiTS 0 0 0 0 0 0 0] , Ξ3 = diag{ −r1, −r2, ⋯, −r2n}. Furthermore, a state-feedback controller is described asFo = Go S−1. Proof. Multiply the above inequalities (12) byβj(j=1, 2, ⋯, 2n) ,

Remark 3. Fo is a sensor-outage-tolerant controller, which is adjusted according to the signals coming from the sensor outage detector shown in Fig. 2 and Theorem 1 (for some results about detecting sensor outage, the readers can refer to [22,48,49] and the references therein). While some sensors are in outage, the outage signals will be transmitted from the sensor outage detector to the controller reconfiguration. In terms of Theorem 1, the controller reconfiguration reconstructs the controller according to the sensor outage signals and the past state response signals of the system. It is worth mentioning that the finite-time stateconstraint is also important to guarantee controller switching safely.

2n

βj > 0 , and ∑ j= 1 βj = 1, sum to get Ξ ⎡ Ψ hY ⎢ ∗ − hQ2 0 Ψ (β) = ⎢ ∗ −I ⎢∗ ∗ ∗ ⎣∗

Ξ2 ⎤ 0⎥ <0, 0⎥ ⎥ Ξ3 ⎦

where

Ψ = Ψˆ +

Theorem 2. The system (7) is robustly finite-time state-constraint H∞ stabilizable with respect to (c1, c2, c3, R , T, γ, d) for constants h > 0, τ≥ 0 and α≥ 0 , if there exist positive definite

n i= 1

4

2

2 T ˆ 2i ϒn+iϒnT+i), ∑ (riθ2i kˆ i ϒϒ i i + rn+i θn+i c

(13)

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⎡ Ψ11 Ψ12 Ψ13 ⎢ ∗ Ψ22 Ψ23 ⎢ ∗ ∗ Ψ33 Ψˆ = ⎢ ⎢ ∗ ∗ ∗ ⎢ ∗ ∗ ⎢ ∗ ⎢ ∗ ∗ ∗ ⎣

Ψ14 Ψ15 Ψ24 − κ1S Ψ34 Ψ35 Ψ44 ∗ ∗

This example was considered by Refs. [39,50–52]. It is assumed that the maximum sampling interval τ= 0.05s . By Ref. [50], The minimum γ is 0.147 such that an admissible controller is existing. By Refs. [51,52], and [39], the corresponding minimum values of γ are 0.082, 0.029, and 0.0289, respectively. Then, by solving LMI (9) in this paper with h= 0.01s , τ= 0.05s , α = 0 , κ1 = 1, κ2 = κ3= κ 4 = 0.1, it is obtained the minimumγ= 0.0289 such that an admissible controller is existing, that is, while only the H∞ performance is considered, the LMI (9) in this paper has the same minimum γ with [39]. Then, the finite-time stateconstraint performance is considered. By solving Theorem 1 with α = 0.01, c12 = 0.001, c 22 = 0.01, c32 = 0.4 , d= 8, T= 10s , it is obtained the minimum γ= 0.0288, and the minimum γ= 0.0281 and 0.0074 with α = 0.05 and 0.08, respectively. Some more results are given in Table 2, which shows the effectiveness of the finite-time state-constraint H∞ stabilizability in obtaining less conservative controllers. Fig. 3 shows the variation of minimum γ for τ= 0.05, 0.1, and0.15s . As α is varied from zero (non-finite-time state-constraint performance), the minimum γ decreases till α = 0.2 and then increases, that is, the minimum γ such that an admissible controller is existing is not a monotonically decreasing function ofα . From this example, it is concluded that the finite-time state-constraint H∞ stabilizability can achieve less conservative results than those Lyapunov asymptotic stability conditions, and there exist a optimal α such that the controlled system has the minimumγ .

Bω κ1Bω κ2 Bω

⎤ ⎥ ⎥ ⎥, Y6T + κ3 Bω ⎥ ⎥ κ 4 Bω ⎥ − γ 2I ⎥ ⎦

Ψ45 Ψ55 ∗

T

I 0 I 0 Ψ11 = Q1 + hQ2 + (h2 − 1) Q3 + hU1 − h2Q4 + ⎡ ∼ ⎤ A 0 S + STA 0T ⎡ ∼ ⎤ − αP, ⎢ 0 M (β) ⎥ ⎢ 0 M (β) ⎥ ⎣ ⎦ ⎣ ⎦ T

I 0 Ψ12 = Q3 + κ1STA 0T ⎡ ∼ ⎤ , ⎢ ⎥ ⎣ 0 M (β) ⎦ T

I 0 I 0 0 Ψ13 = ⎡ ∼ ⎤ ⎡ ⎤ Go + κ2STA 0T ⎡ ∼ ⎤ , ⎢ 0 M (β) ⎥ ⎢ H0 ⎥ ⎢ 0 M (β) ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ T

I 0 Ψ14 = −U1 + Y1 + hQ4 + κ3STA 0T ⎡ ∼ ⎤ , ⎢ 0 M (β) ⎥ ⎣ ⎦ T

I 0 Ψ15 = P − S + κ 4 STA 0T ⎡ ∼ ⎤ , ⎢ 0 M (β) ⎥ ⎣ ⎦ I 0 ⎤⎡ 0 ⎤ π2 ⎡ Ψ23 = U2 + κ1 G , ⎢0 ∼ ⎥ ⎢H ⎥ o 4 ⎣ M (β) ⎦ ⎣ 0 ⎦ Ψ33 = −

T I 0 0 0 0 ⎞ π2 ⎛ I U2 + κ2 ⎡ ∼ ⎤ ⎡ ⎤ Go + κ2GoT ⎜ ⎡ ∼ ⎤ ⎡ ⎤ ⎟ , ⎢ 0 M (β) ⎥ ⎢ H0 ⎥ ⎢ 0 M (β) ⎥ ⎢ H0 ⎥ 4

⎣

⎦⎣

⎦

⎝⎣

⎦⎣

⎦⎠

T

0 0 ⎞ ⎛ I Ψ34 = Y3 + κ3GoT ⎜ ⎡ ∼ ⎤ ⎡ ⎤ ⎟ , ⎢ 0 M (β) ⎥ ⎢ H0 ⎥ ⎦⎣ ⎦⎠ ⎝⎣ T

0 0 ⎞ ⎛ I Ψ35 = −κ2S + κ 4 GoT ⎜ ⎡ ∼ ⎤ ⎡ ⎤ ⎟ , ⎢ 0 M (β) ⎥ ⎢ H0 ⎥ ⎦⎣ ⎦⎠ ⎝⎣ T

I 0 ⎡ ϒi = ⎢ e iT ⎡ ∼ ⎤ ⎢ 0 M (β) ⎥ ⎦ ⎣ ⎣

T

T

I 0 κ1e iT ⎡ ∼ ⎤ ⎢ 0 M (β) ⎥ ⎣ ⎦

I 0 κ2 e iT ⎡ ∼ ⎤ ⎢ 0 M (β) ⎥ ⎣ ⎦

Example 2. The system is taken from the existing literatures [6,47], ˆ i = 1000kg , kˆi = 980kN/m , which has the following parameters: m andcˆ i = 1.407kNs/m(i=1, 2, 3) . Then, the state space equation (2) has the following system matrices:

T

I 0 κ3e iT ⎡ ∼ ⎤ ⎢ 0 M (β) ⎥ ⎣ ⎦

T

T

I 0 κ 4 e iT ⎡ ∼ ⎤ ⎢ 0 M (β) ⎥ ⎣ ⎦

0 I 0 I ⎤ 0 A = ⎡ ∼⎤ A 0, B = ⎡∼⎤, A 0 = ⎡ , ˆ⎥ ⎢0 M⎥ ˆ −C ⎢ M⎥ ⎢− K ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ H 0 = diag{1 1 1}, Bω = [0, 0, 0, −1, −1, −1]T ,

⎤ 0 0 0⎥ . ⎦

Then, replacingA , B withA(β,θ) , B(β) , equation (9) can be expressed

ˆ1 0 1/m 0 ⎤ ∼ ⎡ ˆ2 M = ⎢ 0 1/m 0 ⎥, ⎢ 0 ˆ 3⎥ 0 1/m ⎣ ⎦ kˆ1 + kˆ2 −kˆ2 ⎡ ⎢ ˆ ˆ ˆ K= k2 + kˆ3 −k2 ⎢ ⎢ 0 −kˆ3 ⎣

as

Ψˆ +

n

∑ (θi kˆiϒiΛ iT + θn+i cˆi Λn+iϒnT+i)

H

<0. (14)

i= 1

By Lemma 5, equation (14) holds if there exist positive scalarsr1, r2, ⋯, r2n , such that n

Ψˆ +

2 T ˆ 2i ϒn+iϒnT+i) ∑ (riθ2i kˆ 2i ϒϒ i i + rn+i θn+i c

∑ (r−i 1ΛiΛ iT + r−n+1 i Λn+iΛ nT+i) <0.

(15)

i= 1

In order to verify the dynamics of the closed-loop system, a time history of acceleration (see Fig. 4) from EI Centro 1940 earthquake excitation is applied to this system, and the displacements of three storeys are selected as the controlled output, that is, Cz = [I3 , 03]. From Fig. 4, it can be obtained that the high amplitude excitations are concentrated in the time interval (0, 6) s, thus, the T defined in Definition 1 is chosen as T= 10 . By doing an integral operation, the 10 excitation satisfies∫0 ωT (t) ω (t)dt = 7.5017 , thus, the d can be chosen

By applying the Schur complement, LMI (15) is equivalent to LMI (13). This completes the Proof. 4. Illustrative example Numerical simulations are conducted in this section to show the effectiveness of the proposed theoretical methods. Example 1. Consider system (1), which has the following system matrices:

⎡ 1.2 ⎡1.1 0 0 ⎤ M = ⎢ 0 1.8 0 ⎥, C = ⎢− 0.6 ⎣ 0 0 1.6 ⎦ ⎣ 0 2 1 0 − ⎡ ⎤ ⎡1⎤ K = ⎢ − 1 2 − 1⎥ , H 0 = ⎢ 0 ⎥ , ⎣0⎦ ⎣ 0 −1 1 ⎦

⎤ ⎥, ⎥ ⎥ ⎦

0 ⎤ ⎡ cˆ1 + cˆ2 −cˆ2 ˆ = ⎢ −cˆ2 cˆ2 + cˆ3 −cˆ3 ⎥, C ⎢ 0 cˆ3 ⎥ −cˆ3 ⎣ ⎦

i= 1 n

+

0 −kˆ3 kˆ3

Table 2 Minimum feasible disturbance attenuation γ by τ .

0 ⎤ − 0.6 1.2 − 0.6 ⎥, − 0.6 0.6 ⎦ T

⎡0⎤ Hω = ⎢ 0 ⎥ , ⎣ 0.1⎦

and the controlled output matrix is chosen as

0.1 0.1 0.5 0 0 0 ⎤ Cz = ⎡ . ⎢ 0 0 0 0.1 0.1 0.5 ⎥ ⎦ ⎣ 5

τ

0.05

0.1

0.15

[50] [51] [52] [39] Theorem Theorem Theorem Theorem

0.147 0.082 0.029 0.0289 0.0289 0.0288 0.0261 0.0074

0.278 0.119 0.031 0.0291 0.0291 0.0290 0.0277 0.0118

0.624 0.156 0.033 0.0294 0.0294 0.0292 0.0279 0.0197

1(α= 1(α= 1(α= 1(α=

0) 0.01) 0.06 ) 0.08 )

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which is denoted as controller IV thereafter. To investigate the effectiveness of the proposed controllers, the earthquake excitations mentioned above (see Fig. 4) is taken as the external input signal. When there is no sensor outage in this system, the first storey displacements of the open-loop and closed-loop systems which are composed with the controller I, II, III and IV, respectively, are shown in Fig. 5, and the displacements of the other two storeys and the accelerations of three storeys have a similar vary trend, which is omitted here for brevity. Fig. 5 shows that controller I, II and III are all effective to attenuate the displacements and accelerations of the system, however, the closedloop system, which is composed with controller IV (without considering the time delay), becomes unstable. The maximum displacements and accelerations of the open-loop and closed-loop systems are given in Table 3, which shows the values of the maximum displacement and acceleration responses obtained by controller I are less than those of the other controllers. Furthermore, the value of sup{xT (t) Rx (t)} is 9.55 × 10−5 for controller I. Thus, the finite-time stability condition sup{xT (t) Rx (t)} < c32 is satisfied. Then, consider the sensor-outage-tolerance performance of controller I, II and III. By assuming the sensors of x3 and x˙3 are in outage from the 1th second to the end, the first storey displacements of the closed-loop systems which are composed with the controller I, II and III, respectively, are shown in Fig. 6. The displacements of the other two storeys and the accelerations of three storeys have a similar vary trend, which is omitted here for brevity. Form Fig. 6, it is obtained that the closed-loop systems, which are composed with controller I, II and III, respectively, are all unstable. That is to say, the sensor outage can influents the control performance significantly. Fortunately, the sensoroutage-tolerant performances of the control systems can be improved by reconfiguring the controllers, and in the following content, an example, which concerns the controller reconfiguration, is considered. Assume the sensors of x3 and x˙3 are in outage from the 1th second to the end. By setting

Fig. 3. The minimum feasible disturbance attenuation γ with α for τ= 0.05 , 0.1, and 0.15s .

asd= 8. By choosing c12 = 0.001, c22 = 0.01, c32 = 0.4 , α= 0.35, γ= 0.3, h= 10ms , τ= 15ms , κ1 = 1, κ2 = κ3= κ 4 = 0.1, R = I , T= 10s , the LMIs (9)–(11) are solved with

12.146 1.4577 6.6188 − 57.359 − 70.022 ⎤ ⎡ 23.091 24.545 13.594 − 57.341 − 63.497 − 127.42 ⎥ ⎢ 12.142 1.4550 13.601 36.690 − 70.026 − 127.46 − 120.73 ⎥ , S=⎢ ⎢− 474.85 − 239.58 − 39.380 7798.5 2501.7 1247.4 ⎥ ⎥ ⎢− 239.44 − 514.26 − 278.78 2500.5 9048.6 3748.2 ⎢− 39.313 − 279.00 − 753.74 1247.0 3751.5 11546 ⎥ ⎦ ⎣ and a state feedback controller which has the following gain matrix ⎡ 85.977 − 90.217 − 6.0882 − 5.5556 0.5058 − 0.2380 ⎤ Fo1 = 10 4 × ⎢− 90.193 79.878 − 96.291 0.5065 − 5.7941 0.2680 ⎥. ⎣− 6.0805 − 96.311 − 10.319 − 2379.7 0.2676 − 5.2876 ⎦

For description in brevity, this designed controller is denoted as controller I thereafter. Furthermore, by choosing τ= 25ms , β1 = β2 = 0.1, β3 = β4 = 3, γ= 0.1, ψ0 = I, Corollary 2 in literature [36] is solvable, and a time-delayed H∞ state-feedback controller is obtained, which has the following gain matrix

⎡ s11 ⎢ s21 g g 0 g g 0 14 15 ⎡ 11 12 ⎤ ⎢ s31 Go = ⎢ g 21 g 22 0 g 24 g 25 0 ⎥, S = ⎢ s ⎢ ⎥ ⎢ 41 ⎣ g31 g32 0 g34 g35 0 ⎦ ⎢ s51 ⎢ ⎣ s61

⎡ 82.503 − 41.416 − 1.1726 0.3376 − 0.2275 − 0.1580 ⎤ Fo2 = 10 4 × ⎢− 68.549 109.01 − 50.420 − 0.2227 0.3325 − 0.2365 ⎥. ⎣− 21.243 − 48.791 57.533 − 0.0908 − 0.1869 0.1568 ⎦

For description in brevity, this designed controller is denoted as controller II thereafter. Moreover, by choosing τ= 25ms , γ= 0.2, Theorem 1 in literature [8] is solvable, and a sampled-data-based energy-to-peak controller is obtained, which has the following gain matrix

s12 0 s22 0 s32 s33 s42 0 s52 0 s62 s63

s14 s24 s34 s44 s54 s64

s15 0 ⎤ s25 0 ⎥ s35 s36 ⎥ , s45 0 ⎥ ⎥ s55 0 ⎥ s65 s66 ⎥ ⎦

and choosingγ= 0.2 , c12 = 0.001, c 22 = 0.01, c32 = 0.4 , α= 0.05, h= 10ms , τ= 15ms , κ1 = κ2= κ3 = 0.1, κ 4 = 2 , R = I, T= 10s , the LMIs (9)–(11) are solvable, and a state feedback controller with the sensor outages in x3 and x˙3 is obtained, which has the following gain matrix

⎡ 150.30 − 95.746 − 11.755 − 2.4703 − 0.2109 − 0.1956 ⎤ Fo3 = 10 4 × ⎢− 95.684 138.63 − 107.39 − 0.2079 − 2.6589 − 0.3984 ⎥. ⎣− 11.668 − 107.36 42.964 − 0.1920 − 0.3974 − 2.8626 ⎦

⎡ 37.623 − 27.426 0 − 4.1087 0.5786 0 ⎤ Fo5 = 10 4 × ⎢− 99.829 53.835 0 − 0.4065 − 1.3313 0 ⎥, 0.1237 0 ⎦ ⎣ 7.9593 − 4.4745 0 0.0767

This designed controller is denoted as controller III thereafter. In order to facilitate the comparison, a controller without considering the time delay was gotten in Ref. [38], which has the following gain matrix

This designed controller is denoted as controller V thereafter. It has been shown that the controller I (without considering the sensor

⎡ 41.1699 − 38.3653 3.6646 − 11.0343 − 0.0244 − 0.4810 ⎤ Fo4 = 10 4 × ⎢− 36.6996 44.4073 − 33.9185 − 0.0207 − 11.2981 − 0.2718 ⎥, − 0.4782 − 0.2776 − 11.1335 ⎦ ⎣ 6.1340 − 35.4211 8.3497

Fig. 5. Displacement responses of the open-loop and closed-loop systems, which are composed with controller I, II, III and IV, respectively (h= 10ms, τ= 15ms , and no sensor outage).

Fig. 4. The time history of acceleration from EI Centro 1940 earthquake excitation. 6

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Table 3 The maximum displacements and accelerations of the closed-loop systems which are composed with controller I, II and III, respectively.

Open-loop Controller II [36] Controller III [8] Controller I (Theorem 1)

x1max (cm)

x2max (cm)

x3max (cm)

x¨1max (m/s2)

x¨2max (m/s2)

x¨3max (m/s2)

4.25 2.86 0.48 0.28

7.48 5.34 0.53 0.31

9.20 6.77 0.56 0.30

9.36 5.62 4.02 3.57

13.88 9.00 4.96 4.34

18.27 10.39 5.30 4.46

Fig. 7. The controller switching in Case A (All of the sensors are work normally from the 0th to 1th second, and then the sensors of x3 and x˙3 are in outage from the 1th second to the end).

Fig. 6. Displacements of the closed-loop systems which are composed with controller I, II and III, respectively (the sensors of x3 and x˙3 are in outage).

outage) can't stabilize the system while the sensors of x3 and x˙3 are in outage. Now, a case (corresponds to Case A) is considered: the system is controlled by controller I while the sensors work normally, and then switched to controller V when the sensors of x3 and x˙3 are in outage. The controller switching in Case A is shown in Fig. 7. Under the earthquake excitations mentioned in Fig. 4, the first storey displacements in Case A are plotted in Fig. 8. The displacements of the other two storeys and the accelerations of three storeys have a similar vary trend, which is omitted here for brevity. Fig. 8 shows the effectiveness of Case A in attenuating the vibration of the structural system, that is, the controller reconfiguration is effective in improving the system's performance. The maximum displacements and accelerations of the open-loop and closed-loop systems are given in Table 4. Compared with the open loop, the mean value of the PDVSR obtained in Case A is 75.7% for the displacements and 62.7% for the accelerations. PDVSR is the percentage of the decreased values of the state responses, for example, the PDVSR of the first-storey's displacement is (4.25 − 1.01)/4.25 = 76.2%. Furthermore, the values of sup{xT (t) Rx (t)} sup {x˙ T (t) Rx˙ (t)} obtained in Case A are 1.89 × 10−4 and and

Fig. 8. Displacement responses of the open-loop system and Case A.

controller. Moreover, some optimal methods can also be used to solve LMIs (9)–(11), and obtain some controllers, which have the minimum γ , or the maximum time-delay-tolerance performance, etc.. Furthermore, the variablesκ1, κ2 , κ3 and κ 4 supply additional degrees of freedom for the feasibility of LMIs (9)–(11). If LMIs (9)–(11) are infeasible, a possible solution may be searched by tuning κ1, κ2 , κ3 and κ 4 , or by iterating overκ1, κ2 , κ3 and κ 4 . In order to further illuminate the effectiveness of the controller reconfiguration given in this paper, it is assumed that the sensors have some self-recovery abilities (This assumption is reasonable because more and more intelligent sensors are commonly used in various fields in recent years). The sensors of x3 and x˙3 are in outage from the 1th to 14th second, the sensors of x2 and x˙2 are in outage from the 20th second to the end, and at other times, all the sensors work normally. By setting

t∈ [1s− 25ms,1s]

1.038 × 10−3 , respectively. Thus, Case A satisfies the finite-time stability condition sup{xT (t) Rx (t)} < c32 and controller switching initial conditions sup {xT (t) Rx (t)} < c12 and sup {x˙ T (t) Rx˙ (t)} < c 22 . t∈ [1s− 25ms,1s]

t∈ [1s− 25ms,1s]

⎡ g11 0 g13 g14 Go = ⎢ g 21 0 g 23 g 24 ⎢ ⎣ g31 0 g33 g34

Remark 4. In this example, the normal system is controlled by controller 1 at the beginning. Then, the outages happen to the sensors of x3 and x˙3 at the time instant 1s. After the happening of these outages, the sensor outage detector will capture the outage signals, and send them to the controller reconfiguration. As soon as the sensor outage signals are received by the controller reconfiguration, the structures of Go and S will be chosen by the controller reconfiguration in terms of the received outage signals. According to the giving upper bounds ofc12 , c 22 , c32 , h and τ , the LMIs (9)–(11) are solved with the giving α , γ , κ1, κ2 , κ3 and κ 4 , and the corresponding sensor-outage-tolerant controller gain (Controller V) is obtained. Then, the controller gain is updated to the

⎡ s11 0 ⎢ s21 s22 0 g16 ⎤ ⎢s 0 0 g 26 ⎥, S = ⎢ 31 s41 0 ⎥ ⎢ 0 g36 ⎦ ⎢ s51 s52 ⎢ ⎣ s61 0

s13 s23 s33 s43 s53 s63

s14 0 s24 s25 s34 0 s44 0 s54 s55 s64 0

s16 ⎤ s26 ⎥ s36 ⎥ , s46 ⎥ ⎥ s56 ⎥ s66 ⎥ ⎦

and choosing γ= 0.2 , c12 = 0.001, c 22 = 0.01, c32 = 0.4 , α= 0.05, h= 10ms , τ= 15ms , κ1 = κ2= κ3= κ 4 = 0.1, R = I , T= 10s , the LMIs (9)–(11) are solvable, and a state feedback controller with the sensor outages in x2 and x˙2 is obtained, which has the following gain matrix

7

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Table 4 The maximum displacements and accelerations in the open-loop system and Case A.

Open-loop Case A PDVSR

x1max (cm)

x2max (cm)

x3max (cm)

x¨1max (m/s2)

x¨2max (m/s2)

x¨3max (m/s2)

4.25 1.01 76.2%

7.48 1.82 75.7%

9.20 2.29 75.1%

9.36 3.49 62.7%

13.88 5.49 60.4%

18.27 6.41 64.9%

⎡ 82.003 0 − 60.297 − 4.5222 0 − 0.6313 ⎤ 0.4428 0, 0.4916 ⎥ Fo6 = 10 4 × ⎢− 6.7816 0 3.4577 0.4577 0 − 3.4099 ⎦ ⎣− 32.004 0 7.6978 This designed controller is denoted as controller VI thereafter. Now, a case (corresponds to Case B) is considered: the system is controlled by controller I while the sensors work normally, and the controller is switched to controller V when the sensors of x3 and x˙3 are in outage, and switched to controller VI when the sensors of x2 and x˙2 are in outage. The controller switching in Case B is shown in Fig. 9. Under the earthquake excitations mentioned in Fig. 4, the first storey displacements in Case B are shown in Fig. 10. The displacements of the other two storeys and the accelerations of three storeys have a similar vary trend, which is omitted here for brevity. Form Fig. 10, it is obtained that the effectiveness of Case B in attenuating the vibration of the structural system, and the controller reconfiguration is effective in improving the system's performance is verified again. The maximum responses in Case B are the same as those in Case A, which are shown in Table 4. Obviously, in Case B, the controllers can attenuate the vibration significantly. Furthermore, the value ofsup{xT (t) Rx (t)} obtained in Case B is1.32 × 10−3 . Thus, Case B satisfies the finite-time state-constraint condition sup{xT (t) Rx (t)} < c32 . Moreover, sup{xT (λ) Rx (λ)} = 8.85 × 10−6 andsup{x˙ T (λ) Rx˙ (λ)} = 5.13 × 10−3, λ∈ [1 s− 25ms, 1s]∪[14 s− 25ms, 14s] ∪ [20 s− 25ms, 20s], and the controller switching initial conditionssup{xT (λ) Rx (λ)} < c12 and sup{x˙ T (λ) Rx˙ (λ)} < c 22 are satisfied, obviously. Now, the uncertain case is considered, and the uncertainties are applied to the mass, damping and stiffness coefficients of the first ˆ 1, 1.2m ˆ 1], storey. The parameter uncertainties satisfym1 ∈ [0.8m k1 ∈ [0.7kˆ1 , 1.3kˆ1], c1 ∈ [0.7ˆc1, 1.3ˆc1]. By choosingγ= 0.2 , c12 = 0.001, c 22 = 0.01, c32 = 0.2, α= 0.1, h= 10ms , τ= 15ms , κ1 = κ2= κ3= κ 4 = 0.1, R = I , T= 10s , Theorem 2 is solvable, and a state feedback controller is obtained, which has the following gain matrix

Fig. 10. Displacement responses of the open-loop system and Case B.

⎡ s11 ⎢ s21 g g 0 g g 0 14 15 ⎡ 11 12 ⎤ ⎢ s31 Go = ⎢ g 21 g 22 0 g 24 g 25 0 ⎥, S = ⎢ ⎢ ⎥ ⎢ s41 ⎣ g31 g32 0 g34 g35 0 ⎦ ⎢ s51 ⎢ ⎣ s61

s12 0 s22 0 s32 s33 s42 0 s52 0 s62 s63

s14 s24 s34 s44 s54 s64

s15 0 ⎤ s25 0 ⎥ s35 s36 ⎥ , s45 0 ⎥ ⎥ s55 0 ⎥ s65 s66 ⎥ ⎦

and choosingγ= 0.2, c12 = 1 × 10−4 , c22 = 0.1, c32 = 0.2 , α= 0.08, h= 10ms , τ= 15ms , R = I , T= 10s , κ1 = κ2= κ3= κ 4 = 0.1, Theorem 2 is solvable, and a state feedback controller with the sensor outages in x3 and x˙3 is obtained, which has the following gain matrix

⎡ 95.423 − 66.901 0 − 4.3580 0.4022 0 ⎤ Fo8 = 10 4 × ⎢− 91.213 5.4660 0 0.0870 − 5.7029 0 ⎥, 0.6327 0 ⎦ ⎣ 8.5995 − 2.7253 0 0.0632 This designed controller is denoted as controller VIII thereafter. For brevity, four cases are considered: the nominal case corresponding to Case C1 and three-vertex cases where the mass, stiffness and damping coefficients are given as their vertex values, respectively. Case C2 corresponds tom1 = 0.8mˆ1 , ˆ 1, k1 = 1.3kˆ1 and k1 = 0.7kˆ1 and c1 = 1.3ˆc1; Case C3 corresponds to m1 = 0.8m ˆ 1, k1 = 0.7kˆ1 and c1 = 0.7ˆc1. c1 = 0.7ˆc1; Case C4 corresponds to m1 = 1.2m Furthermore, the sensors of x3 and x˙3 are in outage from the 10th second to the end, and work normally at other times. Under the earthquake excitations mentioned in Fig. 4, the first storey displacements of open-loop and closedloop system in Case C1 are plotted in Fig. 11. The displacements of the other storeys and the accelerations of the three storeys in the four cases have a similar varying trend, which is omitted here for brevity. The maximum responses of the open-loop and closed-loop systems in the four cases are shown in Table 5, where OL means Open-loop system and CL means Closed-loop system. From Fig. 11 and Table 5, it is obtained that the better responses are achieved in the closed-loop systems no matter the parameter uncertainties exist or not. Thus, it is validated that the obtained controller reconfiguration is robust to parameter uncertainties. Furthermore, the value ofsup{xT (t) Rx (t)} obtained in the four cases is3.90 × 10−3 , which is less thanc32 , obviously. Thus the finite-time state-constraint condition is sasup sup {x (λ) TRx (λ)} = 6.80 × 10−5 < c12 and tisfied. Moreover,

Fo7 = 10 4 0.1495 ⎤ ⎡ 41.346 − 93.260 0.7273 − 4.2576 0.2960 × ⎢− 8.5683 75.130 − 78.547 1.7692 − 5.2613 0.3246 ⎥, ⎣− 25.306 − 59.212 13.222 − 0.5457 0.9964 − 3.7524 ⎦ This designed controller is denoted as controller VII thereafter. Then, assume the sensors of x3 and x˙3 are in outage from the 10th second to the end. By setting

Fig. 9. The controller switching in Case B.

λ∈ [10s− 25ms,10s]

8

λ∈ [10s− 25ms,10s]

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structural systems subject to sensor outage is investigated. The sampled-data-based stabilization condition is established in terms of finitetime stability technique and LMIs, which guarantees the structural system to be stable with finite-time state-constraint and has a prescribed level of vibration attenuation performance. If the feasibility problem of these conditions is solvable, the desired controller can be obtained such that the state responses of the structural system are constrained byxT (t) Rx (t) < c32 during the time interval[0, T], and the influence of the external disturbances is constrained by z 2 < γ ω 2 during the whole control process. Furthermore, when sensor outages happen, the control system can reconfigure the controllers according to the signals come from the sensor outage detector. Moreover, based on a combination of matrices and some rank-1 vectors, the uncertain system model is established, and the obtained stabilization result is also extended to its uncertain case. In the end, numerical examples show that the control scheme obtained in this paper can get a better displacement and acceleration responses than Refs. [8,36,38] did. Moreover, while some sensor outage happening in the system, the control scheme obtained in this paper can still stabilize the system significantly, and yet the systems composed with those controllers obtained in Refs. [8,36,38] become unstable. Furthermore, it should be highlighted that the proposed methodology can be of great interest to a wide variety of structural and engineering areas, where the vibration attenuation are encountered.

Fig. 11. The first storey's displacement of the open-loop and closed-loop system in Case C1.

Table 5 The maximum responses of the open-loop and closed-loop systems in the four cases. Case C1

Case C2

Case C3

Case C4

OL

CL

OL

CL

OL

CL

OL

CL

x¨1max (m/s2)

4.25 7.48 9.20 9.39

0.28 0.53 0.74 3.78

4.06 6.38 7.59 8.14

0.31 0.53 0.73 3.82

1.87 3.84 4.79 5.85

0.20 0.47 0.73 3.82

4.94 7.52 8.94 9.11

0.41 0.61 0.76 4.04

x¨2max (m/s2)

13.9

4.51

12.23

4.46

9.85

4.36

13.48

4.52

x¨3max (m/s2)

18.3

4.50

13.14

4.55

10.80

4.62

15.59

4.43

x1max (cm) x2max (cm) x3max (cm)

Acknowledgment

{x˙ (λ) TRx˙ (λ)} = 6.28 × 10−2 < c 22 , and the controller switching initial conditions is satisfied, obviously.

The work described in this paper was fully supported by grants from the National Natural Science Foundation (nos. 61463018 and 61763015), Jiangxi Provincial Natural Science Foundation (nos. 20151BAB207046, 20161BAB206146, GJJ160607 and GJJ170502) of China.

5. Conclusion In this paper, the sampled-data-based vibration controller design for Appendix

Proof of Theorem 1. According to Schur complement and equation (9), it is easy to obtain

Ψ + hYQ−2 1 Y T + ΞΞT <0,

(16)

Choose an energy function as

V(t)= V1 (t)+ V2 (t)+ V3 (t)+ V4 (t),

(17)

where

V1 (t) = xT (t) Px (t) +

t

0

t

∫t−h xT (s) Q1x (s)ds + ∫−h ∫t+ε xT (s) Q2 x (s)dsdε,

∫−h ∫t+ε x˙ T (s) Q3 x˙ (s)dsdε + 12 h2∫−h ∫θ ∫t+ε x˙ T (s) Q4 x˙ (s)dsdεdθ, 0

V2 (t) = h

t

0

t

t

0

t

t

∫t−h xT (s) U1x (s)ds − ∫t−h xT (s)dsU1∫t−h x (s)ds,

V3 (t) = h

2

∫t −h x˙ T (s) U2x˙ (s) ds − π4 ∫t −h (xT (s) − xT (tk−h)) U2 (x (s) − x (tk−h))ds.

V4 (t)=τ 2

t

k

t−h

k

P = S−1PS−T > 0 , Qi = S−1Qi S−T > 0(i=1, 2, 3, 4) , Ui = S−1UiS−T > 0(i=1, 2) . According to Lemma 1 and 2, the equality (17) has V3 (t) ≥ 0 , V4 (t) ≥ 0 , thus V(t) is a positive definite function. Then, the derivative of V(t) along the solution of system (6) is given by V˙1 (t) = 2xT (t) Px˙ (t) + xT (t) Q1 x (t) − xT ( t− h) Q1 x ( t− h) t

+ hxT (t) Q2 x (t) − ∫t−h xT (s) Q2 x (s)ds t t−h

V˙2 (t) = h2x˙ T (t) Q3 x˙ (t) − h∫

x˙ T (s) Q3 x˙ (s)ds +

(18) 0 1 1 4 T h x˙ (t) Q4 x˙ (t) − 2 h2 −h 4 T ⎤ ⎡−Q3 Q3 ⎤ ⎡ x (t)

∫ ∫ x˙ T (s) Q4 x˙ (s)dsdθ

x (t) ≤ h2x˙ T (t) Q3 x˙ (t) + ⎡ ⎢ x (t− h) ⎥ ⎢ ∗ ⎣ ⎦ ⎣ 1

1

t t+θ

0

⎤ ⎢ ⎥ − Q3 ⎥ ⎦ ⎣ x (t− h) ⎦

t

+ 4 h4x˙ T (t) Q4 x˙ (t) − 2 h2∫−h ∫t+θ x˙ T (s) Q4 x˙ (s)dsdθ

(19)

9

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V˙3 (t) = h(xT (t) U1x (t) − xT ( t− h) U1x ( t− h)) t

− 2(xT (t) − xT ( t− h)) U1∫t−h x (s)ds

(20)

T

π2 ⎡ x (t− h) ⎤ ⎡− U2 U2 ⎤ ⎡ x (t− h) ⎤ V˙4 (t)=τ 2 x˙ T (t) U2x˙ (t) + ⎥ ⎥ ⎢ ⎥⎢ 4 ⎢ ⎣ x (t−τ(t)) ⎦ ⎣ ∗ − U2 ⎦ ⎣ x (t−τ(t)) ⎦

(21)

Furthermore, t

(xT (t)+κ1 xT ( t− h)+κ 2 xT (t−τ(t))+ κ3∫t−h xT (s)ds+κ4 x˙ T (t)) S × (Ax (t) + BFo x (t−τ(t)) + Bω ω (t) − x˙ (t)) = 0, whereS = holds

−

S−1

. According to Lemma 3, for any matrices Y =

t

(22)

[ Y1T

Y2

T

Y3 T

Y4

T

Y5

T

Y6

T ]T ,

Yi =

−T , S−1YS i

(i=1, 2, 3, 4, 5) , Y6 = Y6

S−T ,

there

t

∫t−h xT (s) Q2 x (s)ds ≤ 2ξ T (t) Y∫t−h x (s)ds + hξ T (t) YQ−2 1Y Tξ (t),

(23)

t whereξ (t) = ⎡ xT (t) xT ( t− h) xT (t−τ(t)) ∫t−h xT (s)ds x˙ T (t) ωT (t) ⎤. Additionally, it can be easily verified that ⎥ ⎢ ⎦ ⎣ t t 1 2 0 t T ⎛ ⎞ ⎛ T T − h x˙ (s) Q4 x˙ (s)dsdθ ≤ −⎜hx (t) − x (s)ds⎟ Q4 ⎜hx (t) − x (s)ds⎟⎞. t−h t−h −h t+θ 2 ⎝ ⎠ ⎝ ⎠

(24)

Then, by noting (18)–(24), it follows that ∼ −1 V˙ (t) ≤ ξ (t)(Ψ + hYQ2 Y T) ξ T (t),

(25)

∫ ∫

∫

∫

and

∼ ⎡ Ψ11 ⎢ ⎢ ∗ ∼ ⎢ ∗ Ψ=⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗

∼ Ψ12 ∼ Ψ22

∼ Ψ13 ∼ Ψ23 ∼ Ψ33

∗ ∗

∗

∗ ∗

∗ ∗

∼ ∼ Ψ14 Ψ15 SBω ⎤ ∼ ⎥ Ψ24 − κ1S κ1SBω ⎥ ∼ ∼ Ψ34 Ψ35 κ2 SBω ⎥, ⎥ ∼ ∼ T Ψ44 Ψ45 Y 6 + κ3 SBω ⎥ ⎥ ∼ ∗ Ψ55 κ 4 SBω ⎥ ⎥ ∗ ∗ 0 ⎦

∼ Ψ11 = Q1 + hQ2 + (h2 − 1) Q3 + hU1 − h2Q4 + SA + ATS T , ∼ Ψ12 = Q3 + κ1ATS T , ∼ π2 Ψ22 = −Q1 − Q3 − hU1 − 4 U2, ∼ Ψ13 = SBFo + κ2 ATS T , ∼ π2 Ψ23 = 4 U2 + κ1SBFo, ∼ π2 Ψ33 = − 4 U2 + κ2 SBFo + κ2 FoT BTS T , ∼ Ψ14 = −U1 + Y1 + hQ4 + κ3ATS T , ∼ Ψ24 = U1 + Y2, ∼ Ψ34 = Y3 + κ3FoT BTS T , ∼ Ψ44 = Y4 + Y T4 − Q4 , ∼ Ψ15 = P − S + κ 4 ATS T , ∼ Ψ35 = −κ2 S + κ 4 FoT BTS T , ∼ Ψ45 = Y 5T − κ3S , ∼ 1 Ψ55 = 4 h4Q4 + τ 2U2 − κ 4 S − κ 4 S T. By pre and post-multiplying (16) withdiag{ S−1 S−1 S−1 S−1 S−1 I } and its transpose, and considering Go = Fo ST , the following inequality can be obtained ∼ −1 T Ψ + hYQ2 Y T + Ξ͠ Ξ͠ + diag{− αP 0 0 0 0 − γ2I } <0, (26) whereΞ͠ = [Cz 0 0 0 0 0]. From (25) and (26), it is easy to obtain T ͠ T (t) V˙ (t) < αxT (t) Px (t) + ωT (t)γ2ω (t) − ξ (t) Ξ͠ Ξξ T 2 T < αV(t) + ω (t)γ ω (t) − z (t) z (t),

(27)

Integrating both sides of (27) from 0 to t witht∈ [0, T], it follows

∫0

V(t) < V(0) + α

t

V(s)ds +

∫0

t

ωT (s)γ2ω (s)ds.

(28)

By Lemma 4, it has

10

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F. Weng et al.

V(t) < V(0)eαt + eαt

∫0

t

ωT (s)γ2ω (s)ds.

(29)

According to (17), V(t) satisfies

V(t) ≥ xT (t)(P + hQ2) x (t) ≥ λ min (R−1/2PR−1/2 + R−1/2hQ2 R−1/2) xT (t) Rx (t),

(30)

whereR = S−1RS−T > 0 , thus

1 V(t), λ min (R−1/2PR−1/2 + R−1/2hQ2 R−1/2)

xT (t) Rx (t) ≤

∫

≤

(31)

0 0 0 T V1 (0) = x (0) TPx (0) + −h xT (s) Q1 x (s)ds + −h ε x (s) Q2 x (s) dsdε 0 0 0 T λ max (P) x (0) TRx (0) + λ max (Q1) −h xT (s) Rx (s)ds + λ max (Q2) −h ε x (s) Rx (s) dsdε

∫ ∫

∫

< λ max (P)c12 0

∫ ∫

λ max (Q1)hc12

+

0

+ 0

1

h2 λ max (Q2) 2 c12. 0

(32)

0

V2 (0) = h∫−h ∫ε x˙ T (s) Q3 x˙ (s)dsdε + 2 h2∫−h ∫θ ∫ε x˙ T (s) Q4 x˙ (s)dsdεdθ 0

0

0

1

0

0

≤ λ max (Q3)h∫−h ∫ε x˙ T (s) Rx˙ (s)dsdε + λ max (Q4 ) 2 h2∫−h ∫θ ∫ε x˙ T (s) Rx˙ (s)dsdεdθ h3

1

< λ max (Q3) 2 c 22 + λ max (Q4 ) 12 h5c 22. 0

0

0

(33)

V3 (0) = h∫−h xT (s) U1x (s)ds − ∫−h xT (s)dsU1∫−h xT (s)ds 0

≤ λ max (U1)h∫−h xT (s) Rx (s)ds < λ max (U1)h2c12 0

π2 −h (xT (s) − xT ( −h)) U2 (x (s) 4 −h 0 λ max (U2)τ 2 −h x˙ T (s) Rx˙ (s)ds < λ max (U2)τ 2hc 22

V4 (0) = τ 2∫−h x˙ T (s) U2x˙ (s)ds − ≤

(34)

∫

− x ( −h))ds

∫

(35)

Furthermore, it holds

∫0

T

ωT (s)γ2ω (s)ds < γ2d.

(36)

In view of (29)–(36), it yieds

xT (t) Rx (t) ≤

eαT λmin (R−1/2PR−1/2 + R−1/2hQ2 R−1/2)

(λ max (P)c12

h2

h3

+ λ max (Q1)hc12 + λ max (Q2) 2 c12 + λ max (Q3) 2 c 22 h5

+ λ max (Q4 ) 12 c 22 + λ max (U1)h2c12 + λ max (U2)τ 2hc 22 + γ2d).

(37)

By considering the condition (10) and (11), it is obtained thatxT (t) Rx (t) ≤ c32 , 0≤ t≤ T . Next, the z 2 < γ ω 2 performance is established for the system under zero initial condition, that is, Φ(t) = 0 , ∀ t∈ [−(h+ τ), 0], and V(t) t=0 = 0 . Integrating both side of (27) from 0 to t with t∈ [0, ∞) , by Lemma 4, it yields

V(t) < eαt ⎜⎛ ⎝

∫0

t

γ2ωT (s) ω (s)ds −

∫0

t

zT (s) z (s)ds⎞⎟ ⎠ t T z (s) z (s)ds < γ2

(38) t

Obviously, equation (38) implies∫0 From Definition 1, it is known that the system is finite-time state-constraint ∫0 H∞ stabilizable with respect to(c1,c2 ,c3, R,T,γ,d) . This completes the Proof.

ωT (s) ω (s)ds .

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