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Abstract

The problem of finite-time vibration-attenuation controller design for buildings structural systems with parameter uncertainties is the concern of this paper. The objective of designing controllers is to guarantee the finite-time stability of closed-loop systems with a prescribed level of disturbance attenuation. First, based on matrix transformation, the structural system is described as state-space model, which contains parameter uncertainties. Then, based on finite-time stability analysis method, some sufficient conditions for the existence of finite-time vibration-attenuation controllers are obtained. By solving these conditions, the desired controllers can be obtained for the closed-loop system to be finite-time stable with the performance ∥z∥₂ < γ∥ω∥₂. It is shown by the simulation results, that compared with some Lyapunov asymptotic stability results, finite-time stability control can obtain better state responses, especially while the system is under nonzero initial states.

1. Introduction

In recent years, more and more high-rising buildings are built up due to the limitation of land. Such kinds of buildings not only help save the source of land, but also present a better scene to people. However, some strong earthquakes and wind, such as the 2008 Wenchuan Earthquake, the 2010 Yushu Earthquake, and the 2011 Japanese tsunami, happened frequently in past several years. These excitations inevitably induce large vibrations which make those high-rising buildings deformation, fatigue damage, and even falling down. Thus, vibration control for high-rising buildings has received considerable attention, and some vibration control methods, such as passive control, semiactive control, and active control, were proposed for attenuating seismic or wind excitation. Passive and semiactive controls have some virtues, such as low energy consumption and low cost. However, as buildings become higher and higher, structural stability and solidity are challenged and cannot be guaranteed only by those passive and semiactive control methods. Therefore, the status of active control for seismic-excited or wind-excited buildings becomes more and more significant. Recently, many scholars have applied themselves to the research of active vibration control and many control strategies have been utilized, such as classical H_∞ theories [1 –4], energy-to-peak control [5 –7], finite frequency control [8], sliding mode control [9, 10], adaptive control [11], fuzzy control [12, 13], neural networks [14, 15], and optimal control [16], and have been developed to attenuate the vibration excited by earthquake or winds. Accompanied with the development of active vibration control techniques, some active control devices were designed for applying those control algorithms; for example, magnetorheological dampers [17], active mass damper (AMD) [18], active brace system (ABS) [19], electrohydraulic servo system [20], and so forth have been widely studied and used for vibration attenuation.

It is worth to point out that most of the existing results are obtained based on Lyapunov stability analysis method, which cares about the asymptotic convergence of the system, while t→∞. However, most of those earthquakes or strong winds happen in a very short time, and it is often the peak values of displacements or accelerations that make the buildings damaged. Thus, a better performance can be expected if finite-time stability analysis method is taken into consideration. The concept of finite-time stability was first introduced in the Russian literature [21]. A system is said to be finite-time stable if, given a bound on the initial condition, its state does not exceed a certain domain during a specified time interval. Recently, the problem of finite-time stability of regular or singular systems has received considerable attention. For example, based on Lyapunov functional, [22] discussed the finite-time stability of a class of stochastic nonlinear systems. In terms of LMI technique, some sufficient conditions were given in [23] for the uncertain discrete singular systems to be finite-time stable and stabilizable. For more results about the finite-time stability, the readers can refer to [24 –26] and the references therein. However, to the best of the authors’ knowledge, although most of the stability analysis theories have been extended to the stability analysis of structural systems, we still have not found any published paper concerning the finite-time stability of buildings structural system. That is to say, the finite-time vibration-attenuation controller design for buildings structural system is still not fully investigated.

This paper is concerned with the problem of finite-time vibration control of earthquake excited buildings structure with parameter uncertainties. Firstly, based on matrix transformation, the linear state-space model of buildings with parameter uncertainties is established. Secondly, based on a Lyapunov functional and finite-time stability analysis method, the LMIs-based conditions for the structural system to be finite-time stabilizable are established. By solving these LMIs, the desired controller is obtained for the closed-loop system to be finite-time stable with the performance ${∥ z ∥}_{2} < γ {∥ ω ∥}_{2}$ . In the end, simulation results are given to show the effectiveness of the proposed theorems.

The organization of this paper is as follows. Section 2 formulates the problem and presents the dynamic models. The main results are given in Section 3. The illustrative examples are given in Section 4 to show the applicability and improvement of the presented approaches. Finally, the paper is concluded in Section 5.

Notation. Throughout this paper, for real matrices X and Y, the notation X≥Y (resp., X>Y) means that the matrix X-Y is semipositive definite (resp., positive definite). I is the identity matrix with appropriate dimension, and a superscript “T” represents transpose. ${∥ x ∥}_{2}$ expresses the 2-norm of x. For a symmetric matrix, * denotes the symmetric terms. The symbol Rⁿ stands for the n-dimensional Euclidean space, and R^{n × m} is the set of n × m real matrices.

2. Problem Formulation and Dynamic Models

Consider an n degree-of-freedom structural system. The system under consideration is depicted in Figure 1. The linear structural model equation can be written with [1 –8]

M {\ddot{x}}_{m} (t) + C {\dot{x}}_{m} (t) + K x_{m} (t) = H_{0} u (t) + H_{ω} {\ddot{x}}_{g} (t),

(1)

where $x_{m} (t) = {[{x_{m}}_{1} (t), {x_{m}}_{2} (t), \dots, {x_{m}}_{n} (t)]}^{T}$ , x_mn(t) is the interstorey relative drift of the nth floor, u(t) is the control force input, ${\ddot{x}}_{g} (t)$ is the input disturbance that belongs to L₂[0,∞), H₀∈R^{n × m} gives the locations of these controllers, H_ω ∈ R^{n × 1} is a vector denoting the influence of disturbance excitation, and M,C,K∈R^{n × n} are the mass, damping, and stiffness matrices of the system, respectively. From Figure 1, we can obtain

\begin{matrix} M = [\begin{bmatrix} m_{1} & 0 & \dots & 0 \\ m_{2} & m_{2} & 0 \\ ⋮ & ⋱ & ⋮ \\ m_{n} & m_{n} & \dots & m_{n} \end{bmatrix}], C = [\begin{bmatrix} c_{1} & - c_{2} & \dots & 0 \\ 0 & c_{2} & ⋱ & ⋮ \\ ⋮ & ⋮ & ⋱ & - c_{n} \\ 0 & 0 & \dots & c_{n} \end{bmatrix}], \\ K = [\begin{bmatrix} k_{1} & - k_{2} & \dots & 0 \\ 0 & k_{2} & ⋱ & ⋮ \\ ⋮ & ⋮ & ⋱ & - k_{n} \\ 0 & 0 & \dots & k_{n} \end{bmatrix}], H_{ω} = [\begin{bmatrix} - m_{1} \\ - m_{2} \\ ⋮ \\ - m_{n} \end{bmatrix}] . \end{matrix}

(2)

Figure 1:

n-degree-of-freedom structural system.

Defining the state variables as $x (t) = {[x_{m} {(t)}^{T}, {\dot{x}}_{m} {(t)}^{T}]}^{T}$ , (1) can be written in the following state-space form:

\begin{matrix} \dot{x} (t) = A x (t) + B u (t) + B_{ω} ω (t), \\ z (t) = C_{z} x (t), \end{matrix}

(3)

where C_z is real constant matrix with appropriate dimensions, $ω (t) = {\ddot{x}}_{g} (t)$ satisfies $\int_{0}^{T} ω^{T} (t) ω (t) d t \leq d$ , d≥0, and

\begin{matrix} A = [\begin{bmatrix} 0 & I \\ - M^{- 1} K & - M^{- 1} C \end{bmatrix}], B = [\begin{bmatrix} 0 \\ M^{- 1} H_{0} \end{bmatrix}], \\ B_{ω} = [\begin{bmatrix} 0 \\ M^{- 1} H_{ω} \end{bmatrix}] . \end{matrix}

(4)

Consider that the system has parameter uncertainties, and, in particular, the parameter uncertainties are induced by the variations of stiffness and damping coefficients [27]. By assuming that the uncertain $k_{j} \in [{\underline{k}}_{j}, {\bar{k}}_{j}]$ , $c_{j} \in [{\underline{c}}_{j}, {\bar{c}}_{j}]$ , j = 1,2,…, n, where ${\underline{k}}_{j}$ , ${\underline{c}}_{j} ({\bar{k}}_{j}, {\bar{c}}_{j})$ are the lower (upper) bounds of the stiffness and damping, respectively, and denoting

\begin{matrix} {\hat{k}}_{j} = \frac{({\underline{k}}_{j} + {\bar{k}}_{j})}{2}, Δ {\hat{k}}_{j} = θ_{j} {\hat{k}}_{j}, \\ | θ_{j} | \leq {\bar{θ}}_{j} < 1, {\bar{θ}}_{j} = \frac{({\bar{k}}_{j} - {\underline{k}}_{j})}{({\bar{k}}_{j} + {\underline{k}}_{j})}, \\ j = 1,2, \dots, n, \\ {\hat{c}}_{j} = \frac{({\underline{c}}_{j} + {\bar{c}}_{j})}{2}, Δ {\hat{c}}_{j} = θ_{n + j} {\hat{c}}_{j}, \\ | θ_{n + j} | \leq {\bar{θ}}_{n + j} < 1, {\bar{θ}}_{n + j} = \frac{({\bar{c}}_{j} - {\underline{c}}_{j})}{({\bar{c}}_{j} + {\underline{c}}_{j})}, \\ j = 1,2, \dots, n, \end{matrix}

(5)

we can describe the uncertain system by state-space equation of the following form:

\begin{matrix} \dot{x} (t) = A (θ) x (t) + B u (t) + B_{ω} ω (t), \\ u (t) = F x (t) \\ z (t) = C_{z} x (t), \end{matrix}

(6)

where F is the controller gain to be designed later, and the uncertain matrix A(θ) satisfies $A (θ) = A_{0} + \sum_{j = 1}^{2 n} θ_{j} A_{j}$ , where

\begin{matrix} A_{0} = [\begin{bmatrix} 0 & I \\ - M^{- 1} \hat{K} & - M^{- 1} \hat{C} \end{bmatrix}], \hat{K} = [\begin{bmatrix} {\hat{k}}_{1} & - {\hat{k}}_{2} & \dots & 0 \\ 0 & {\hat{k}}_{2} & \dots & ⋮ \\ ⋮ & ⋮ & \dots & - {\hat{k}}_{n} \\ 0 & 0 & \dots & {\hat{k}}_{n} \end{bmatrix}], \\ \hat{C} = [\begin{bmatrix} {\hat{c}}_{1} & - {\hat{c}}_{2} & \dots & 0 \\ 0 & {\hat{c}}_{2} & \dots & ⋮ \\ ⋮ & ⋮ & \dots & - {\hat{c}}_{n} \\ 0 & 0 & \dots & {\hat{c}}_{n} \end{bmatrix}], A_{j} = [\begin{bmatrix} 0 & 0 \\ 0 & M^{- 1} \end{bmatrix}] {\hat{k}}_{j} e_{j} f_{j}^{T}, \\ A_{n + j} = [\begin{bmatrix} 0 & 0 \\ 0 & M^{- 1} \end{bmatrix}] {\hat{c}}_{j} e_{n + j} f_{n + j}^{T}, j = 1,2, \dots, n . \end{matrix}

(7)

e_j ∈ R²ⁿ, f_j ∈ R²ⁿ, e_{n + j} ∈ R²ⁿ, and f_{n + j} ∈ R²ⁿ are all column vectors.

Definition 1 (see [23]). The system (6) is said to be finite-time H_∞ stabilizable with respect to $(κ_{1}, κ_{2}, R, T, γ, d)$ , if there exists a controller gain F such that the closed-loop system has

x^{T} (0) R x (0) \leq κ_{1}^{2} ⟹ x {(t)}^{T} R x (t) \leq κ_{2}^{2}, {∥ z ∥}_{2} < γ {∥ ω ∥}_{2},

(8)

for any $t \in [0, T]$ , $\int_{0}^{T} ω^{T} (t) ω (t) d t \leq d$ , where 0<κ₁<κ₂, R>0, T>0, γ>0, and d≥0.

Definition 2. The system (6) is said to be robustly finite-time H_∞ stabilizable with respect to $(c_{1}, c_{2}, R, T, γ, d)$ , if the system (6) is finite-time H_∞ stabilizable for all admissible uncertainties.

Lemma 3 (see [28]). Let ϖ(t) be a nonnegative function such that ϖ(t) ≤ a $+ b \int_{0}^{t} ϖ (s) d s$ , 0 ≤ t ≤ T, for some constants a, b>0; then, one has ϖ(t) ≤ a exp⁡(bt), 0 ≤ t ≤ T.

Lemma 4 (see [29]). Given matrices χ, μ, and ν with appropriate dimensions and with χ symmetrical, then χ + μF(t)ν + ν^TF(t)^Tμ^T<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 + δμμ^T + δ⁻¹ν^Tν<0.

3. Main Results

Theorem 5. The system (6) without uncertainties is finite-time H_∞ stabilizable with respect to $(κ_{1}, κ_{2}, R, T, γ, d)$ for constant α≥0, if there exist positive definite symmetric matrix P and matrix G satisfying the following matrix inequalities:

Ψ = [\begin{bmatrix} {P A}_{0}^{T} + G^{T} B^{T} + A_{0} P + B G - α P & B_{ω} & P C_{z}^{T} \\ * & - γ^{2} & 0 \\ * & * & - I \end{bmatrix}] < 0,

(9)

Furthermore, a state-feedback controller is described as F = GP⁻¹.

Proof. By substituting the control law u(t) = Fx(t) into system (6) with θ_j = 0 $(j = 1,2, \dots, 2 n)$ , we can obtain the following closed-loop system:

\frac{1}{λ_{\min} (R^{1 / 2} P R^{1 / 2})} κ_{1}^{2} + γ^{2} d < \frac{1}{λ_{\max} (R^{1 / 2} P R^{1 / 2})} κ_{2}^{2} e^{- α T} .

(10)

Choose a Lyapunov-Krasovskii functional candidate as

\begin{matrix} \dot{x} (t) = (A_{0} + B F) x (t) + B_{ω} ω (t), \\ z (t) = C_{z} x (t) . \end{matrix}

(11)

where $\bar{P} = P^{- 1}$ . The derivative of V(t) along the solution of system (11) is given by

V (t) = x {(t)}^{T} \bar{P} x (t),

(12)

where $η (t) = {[x {(t)}^{T}, ω {(t)}^{T}]}^{T}$ and $Π = [\begin{smallmatrix} \bar{P} A_{0} + \bar{P} B F + A_{0}^{T} \bar{P} + F^{T} B^{T} \bar{P} & \bar{P} B_{ω} \\ * & 0 \end{smallmatrix}]$ .

By pre- and postmultiplying (9) with diag⁡{P⁻¹ I I} and its transpose, considering G = FP, and according to the Schur Compliment, we can obtain

\dot{V} (t) = 2 x {(t)}^{T} \bar{P} \dot{x} (t) = 2 x {(t)}^{T} \bar{P} ((A_{0} + B F) x (t) + B_{ω} ω (t)) = η {(t)}^{T} Π η (t),

(13)

From (13) and (14), it is easy to obtain

η {(t)}^{T} Π η (t) + x^{T} (t) C_{z}^{T} C_{z} x (t) - α x^{T} (t) \bar{P} x (t) - γ^{2} ω^{T} (t) ω (t) < 0 .

(14)

Integrating both sides of (15) from 0 to t with $t \in [0, T]$ , it follows that

\dot{V} (t) < γ^{2} ω^{T} (t) ω (t) - z^{T} (t) z (t) + α V (t) .

(15)

By Lemma 3, it has

V (t) < V (0) + α \int_{0}^{t} V (s) d s + \int_{0}^{t} ω^{T} (s) γ^{2} ω (s) d s .

(16)

According to (12), we have

V (t) < V (0) e^{α t} + e^{α t} \int_{0}^{t} ω^{T} (s) γ^{2} ω (s) d s .

(17)

that is,

V (t) = x^{T} (t) \bar{P} x (t) \geq \frac{1}{λ_{\max} (R^{1 / 2} P R^{1 / 2})} x^{T} (t) R x (t);

(18)

Furthermore, it holds

x^{T} (t) R x (t) \leq λ_{\max} (R^{1 / 2} P R^{1 / 2}) V (t) .

(19)

In view of (17)–(21), it yields

V (0) = x {(0)}^{T} \bar{P} x (0) \leq \frac{1}{λ_{\min} (R^{1 / 2} P R^{1 / 2})} κ_{1}^{2},

(20)

By considering the conditions (10), we can obtain x^T(t)Rx(t)<κ₂².

Next, we will establish the ${∥ z (t) ∥}_{2} < γ {∥ ω (t) ∥}_{2}$ performance of the system under zero initial condition; that is, ${V (t) |}_{t = 0} = 0$ . Integrating both sides of (15) from 0 to t with $t \in [0, \infty)$ , by Lemma 3, it yields

\int_{0}^{t} ω^{T} (s) γ^{2} ω (s) d s < γ^{2} d .

(21)

Obviously, (23) implies $\int_{0}^{t} z^{T} (s) z (s) d s < γ^{2} \int_{0}^{t} ω^{T} (s) ω (s) d s$ . From Definition 1, we know the system is finite-time H_∞ stabilizable with respect to $(κ_{1}, κ_{2}, R, T, γ, d)$ . This completes the proof.

Remark 6. By solving matrix inequalities (9) and (10) of Theorem 5, we can obtain a finite-time H_∞ stabilization controller, with which the closed-loop system has not only the H_∞ performance ${∥ z ∥}_{2} < γ {∥ ω ∥}_{2}$ for $t \in [0, \infty)$ , but also the state-constraint x(t)^TRx(t) ≤ κ₂² for $t \in [0, T]$ . Compared with those H_∞ stabilization controllers, the improvement of finite-time H_∞ stabilization controller is obvious, and it will also be demonstrated later through the numerical example in Section 4. Furthermore, by selecting α = 0, we can obtain that the LMI (9) will be reduced to Corollary 8 (a H_∞ controller design condition) in [30]; that is to say, Corollary 8 in [30] is just a special case of LMI (9) in this paper.

Remark 7. It is worth to point out that (10) is not a linear inequality; however, from a computational point of view, obtaining a set of LMIs is more desirable. Fortunately, based on some matrix transforms, we obtain the following LMIs-based corollary, which guarantees finite-time H_∞ stability of the closed-loop systems.

Corollary 8. The system (3) without uncertainties is finite-time H_∞ stabilizable with respect to $(κ_{1}, κ_{2}, R, T, γ, d)$ for constant α≥0, if there exist positive definite symmetric matrix P, matrix G, positive scalars ε₁, ε₂ satisfying (9), and the following LMIs:

x^{T} (t) R x (t) \leq e^{α T} λ_{\max} (R^{1 / 2} P R^{1 / 2}) \times (\frac{1}{λ_{\min} (R^{1 / 2} P R^{1 / 2})} κ_{1}^{2} + γ^{2} d) .

(22)

Furthermore, a state-feedback controller is described as F = GP⁻¹.

Theorem 9. The system (3) is robustly finite-time H_∞ stabilizable with respect to $(κ_{1}, κ_{2}, R, T, γ, d)$ for constant α≥0, if there exist positive definite symmetric matrix P, matrix G, positive scalars r₁,r₂,…, r_2n, ε₁, ε₂ satisfying (24)–(25), and the following LMI:

V (t) < e^{α t} (\int_{0}^{t} γ^{2} ω^{T} (s) ω (s) d s - \int_{0}^{t} z^{T} (s) z (s) d s) .

(23)

where

ε_{2} < R^{1 / 2} P R^{1 / 2} < ε_{1},

(24)

Furthermore, a state-feedback controller is described as F = GP⁻¹.

Proof. Replacing A0 with $A_{0} + \sum_{j = 1}^{2 n} θ_{j} A_{j}$ , (9) can be expressed as

[\begin{bmatrix} - ε_{1} κ_{2}^{2} e^{- α t} & ε_{1} κ_{1} & ε_{1} γ d \\ * & - ε_{2} & 0 \\ * & * & - d \end{bmatrix}] < 0 .

(25)

By Lemma 4, (28) holds if and only if there exist positive scalars r₁,r₂,…, r_2n, such that

[\begin{bmatrix} Γ & M_{12} \\ * & M_{22} \end{bmatrix}] < 0,

(26)

Applying the Schur complement, LMI (29) is equivalent to LMI (26). This completes the proof.

4. Illustrative Example

Consider the structural system with n = 3. The structural parameters are m_i = 1000 kg, ${\hat{k}}_{i} = 980$ kN/m, and ${\hat{c}}_{i} = 1.407$ kN's/m $(i = 1,2, 3)$ [3]. Then, the state-space equation (6) has the following parameters:

\begin{matrix} Γ = Ψ + \sum_{j = 1}^{n} (r_{j} {\hat{k}}_{j}^{2} {\bar{θ}}_{j}^{2} Γ_{j} Γ_{j}^{T} + r_{n + j} {\hat{c}}_{j}^{2} {\bar{θ}}_{n + j}^{2} Γ_{n + j} Γ_{n + j}^{T}), \\ Γ_{j} = {[\begin{bmatrix} {(M^{- 1} e_{j})}^{T} & 0 & 0 & 0 \end{bmatrix}]}^{T}, \\ M_{12} = [Λ_{1}, Λ_{2}, \dots, Λ_{2 n}], \\ Λ_{j} = {[\begin{bmatrix} f_{j}^{T} P & 0 & 0 & 0 \end{bmatrix}]}^{T}, \\ M_{22} = diag {- r_{1}, - r_{2}, \dots - r_{2 n}} . \end{matrix}

(27)

where

Ψ + \sum_{j = 1}^{n} (θ_{j} {\hat{k}}_{j} Γ_{j} Λ_{j}^{T} + θ_{n + j} {\hat{c}}_{j} Γ_{n + j} Λ_{n + j}^{T} + θ_{j} {\hat{k}}_{j} Λ_{j} Γ_{j}^{T} + θ_{n + j} {\hat{c}}_{j} Λ_{n + j} Γ_{n + j}^{T}) < 0 .

(28)

Assume that the displacements and velocities of the three storeys are all measurable for feedback in this case. The controlled output is chosen to be the interstorey relative drift; that is, z(t) = [x_m1(t) x_m2(t) x_m3(t)]^T. In order to verify the dynamics of the closed-loop system, a time history of acceleration (see Figure 2) from EI Centro 1940 earthquake excitation is applied to this system. The excitation has $\int_{0}^{15} ω^{T} (t) ω (t) d t = 8.5257$ . Thus, we can choose d = 9.

Figure 2:

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

Firstly, consider the system without uncertainties; that is, θ_j = 0 $(j = 1,2, \dots, 6)$ . By choosing γ = 0.2, κ₁² = 0.1, κ₂² = 0.2, α = 0.1, d = 9, R = I, and T = 15's, we solve the LMIs (9), (24), and (25) and obtain a state feedback controller which has the following gain matrix:

Ψ + \sum_{j = 1}^{n} (r_{j} {\bar{θ}}_{j}^{2} {\hat{k}}_{j}^{2} Γ_{j} Γ_{j}^{T} + r_{j}^{- 1} Λ_{j} Λ_{j}^{T} + r_{n + j} {\bar{θ}}_{n + j}^{2} {\hat{c}}_{j}^{2} Γ_{n + j} Γ_{n + j}^{T} + r_{n + j}^{- 1} Λ_{n + j} Λ_{n + j}^{T}) < 0 .

(29)

For description in brevity, we denote this designed controller as controller I thereafter. In order to facilitate the comparison, we obtain another H_∞ state feedback controller, which does not involve the finite-time stability, by solving Theorem 9 in [30] with γ = 0.2, and this controller has the following gain:

\begin{matrix} H_{0} = diag {1,1, 1}, H_{ω} = {[\begin{bmatrix} - m_{1} & - m_{2} & - m_{3} \end{bmatrix}]}^{T}, \\ A (θ) = A_{0} + \sum_{j = 1}^{3} (θ_{j} {\hat{k}}_{j} [\begin{bmatrix} 0_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & M^{- 1} \end{bmatrix}] e_{j} {f_{j}}^{T} + θ_{3 + j} {\hat{c}}_{j} \times [\begin{bmatrix} 0_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & M^{- 1} \end{bmatrix}] e_{3 + j} f_{3 + j}^{T}), \end{matrix}

(30)

We denote this state feedback controller as controller II thereafter. Consider that the initial condition of the system satisfies $x (0) = 0$ . The displacement and acceleration responses of the open-loop and closed-loop systems which are composed of controllers I and II are compared in Figures 3 and 4, and the maximum displacements and accelerations of the open-loop and closed-loop systems are shown in Table 1. Obviously, the two controllers are all effective in attenuating the structural vibrations, and the maximum responses obtained by controller I are less than those obtained by controller II.

Table 1:

The maximum displacements and accelerations of the open-loop and closed-loop systems, which are composed of controllers I and II, respectively (under zero initial states).

	x_1max (cm)	x_2max (cm)	x_3max (cm)	${\ddot{x}}_{1 \max}$ (m/s²)	${\ddot{x}}_{2 \max}$ (m/s²)	${\ddot{x}}_{3 \max}$ (m/s²)

Open-loop	4.57	3.50	2.01	9.99	7.53	5.63
Controller I	0.33	4.10 × 10⁻⁴	1.53 × 10⁻⁴	1.53	1.55 × 10⁻³	5.79 × 10⁻⁴
Controller II	4.26	2.31 × 10⁻³	1.15 × 10⁻³	7.20	2.60 × 10⁻³	1.44 × 10⁻³

Figure 3:

Displacement responses of the open-loop and closed-loop systems, which are composed of controllers I and II, respectively (under zero initial states).

Figure 4:

Acceleration responses of the open-loop and closed-loop systems, which are composed of controllers I and II, respectively (under zero initial states).

Remark 10. From this example, we can obtain the finite-time H_∞ controller that can obtain much better displacements’ and accelerations’ responses than the classical H_∞ controller, and the state constraint x(t)^TRx(t) ≤ κ₂² is reached by the finite-time H_∞ controller, obviously.

Then, we consider the initial condition satisfying $x (0) = {[0.2 0.2 0.12 0 0 0]}^{T}$ . Obviously, it has $x^{T} (0) R x (0) < κ_{1}^{2} = 0.1$ . The displacement and acceleration responses of the open-loop and closed-loop systems which are composed of controllers I and II are compared in Figures 5 and 6, and the maximum displacements, accelerations, and $\sup_{t \in [0,15] s} {x {(t)}^{T} R x (t)}$ of the open-loop and closed-loop systems are shown in Table 2. Obviously, the vibration-attenuation results obtained by controller I are much better than those obtained by controller II; furthermore, the maximum displacement and acceleration responses obtained by controller I are also less than those obtained by controller II. Moreover, it can be obtained from Table 2 that the $\sup_{t \in [1, 15] s} {x^{T} (t) \bar{R} x (t)}$ obtained by controller I is 0.157, which is less than $\sup_{t \in [1, 15] s} {x^{T} (t) \bar{R} x (t)}$ obtained by controller II is 5.364, which is much higher than 0.2. That is to say, only the finite-time sTability controller I satisfies the finite-time stability condition, and the effectiveness of Corollary 8 is obvious.

Now, let us come to see the uncertain case. The uncertainties are applied to the damping coefficients of the structural system and the parameter uncertainties satisfy $| θ_{4} | \leq 0.3$ , $| θ_{5} | \leq 0.3$ , $| θ_{6} | \leq 0.3$ . By choosing γ = 0.2, κ₁² = 0.1, κ₂² = 0.15, α = 0.1, d = 9, R = I, and T = 15's, we solve the LMIs (24)–(26) and obtain a robust state feedback controller which has the following gain matrix:

\begin{matrix} A_{0} = [\begin{bmatrix} 0 & I_{3} \\ - M^{- 1} \hat{K} & - M^{- 1} \hat{C} \end{bmatrix}], M = [\begin{bmatrix} m_{1} & 0 & 0 \\ m_{2} & m_{2} & 0 \\ m_{3} & m_{3} & m_{3} \end{bmatrix}], \\ \hat{K} = [\begin{bmatrix} {\hat{k}}_{1} & - {\hat{k}}_{2} & 0 \\ 0 & {\hat{k}}_{2} & - {\hat{k}}_{3} \\ 0 & 0 & {\hat{k}}_{3} \end{bmatrix}], \hat{C} = [\begin{bmatrix} {\hat{c}}_{1} & - {\hat{c}}_{2} & 0 \\ 0 & {\hat{c}}_{2} & - {\hat{c}}_{3} \\ 0 & 0 & {\hat{c}}_{3} \end{bmatrix}], \\ e_{1} = e_{4} = {[\begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 \end{bmatrix}]}^{T}, \\ f_{1} = {[\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}]}^{T}, \\ f_{4} = {[\begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 \end{bmatrix}]}^{T}, \\ e_{2} = e_{5} = {[\begin{bmatrix} 0 & 0 & 0 & - 1 & 1 & 0 \end{bmatrix}]}^{T}, \\ f_{2} = {[\begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix}]}^{T}, \\ f_{5} = {[\begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}]}^{T}, \\ e_{3} = e_{6} = {[\begin{bmatrix} 0 & 0 & 0 & 0 & - 1 & 1 \end{bmatrix}]}^{T}, \\ f_{3} = [\begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \end{bmatrix}], \\ f_{6} = [\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}] . \end{matrix}

(31)

F = 1 0^{4} \times [\begin{bmatrix} 88.9096 & - 98.0113 & - 4.1259 \times 1 0^{- 3} & - 10.1966 & - 0.1539 & - 4.7985 \times 1 0^{- 3} \\ - 9.0790 & 88.9286 & - 98.0051 & - 10.3241 & - 10.1749 & - 0.1467 \\ - 9.0749 & - 9.0702 & 88.9348 & - 10.3193 & - 10.3144 & - 10.1677 \end{bmatrix}] .

(32)

F = 1 0^{4} \times [\begin{bmatrix} 87.1534 & - 98.0047 & - 6.65 \times 1 0^{- 4} & - 0.3423 & - 0.1412 & - 1.61 \times 1 0^{- 4} \\ - 10.8478 & 87.1442 & - 98.0038 & - 0.4825 & - 0.3403 & - 0.1409 \\ - 10.8472 & - 10.8585 & 87.1451 & - 0.4824 & - 0.4810 & - 0.3401 \end{bmatrix}] .

(33)

F = 1 0^{4} \times [\begin{bmatrix} 21.3084 & - 23.8491 & - 0.0184 & - 107.2508 & 103.9222 & - 0.0259 \\ - 2.4493 & 21.3779 & - 23.8218 & - 3.1917 & - 107.1404 & 103.9605 \\ - 2.4308 & - 2.4164 & 21.4051 & - 3.1657 & - 3.1411 & - 107.1021 \end{bmatrix}] .

(34)

For description in brevity, we denote this designed controller as controller III thereafter. Consider a vertex case: $c_{1} = 0.7 {\hat{c}}_{1}$ , $c_{2} = 1.3 {\hat{c}}_{2}$ , and $c_{3} = 1.3 {\hat{c}}_{3}$ , and the initial condition has, $x (0) = {[0.02 0.02 0.02 0.2 0.19 0.15]}^{T}$ . Obviously, it has $x^{T} (0) R x (0) < κ_{1}^{2} = 0.1$ . The displacement and acceleration responses of the open-loop and closed-loop system which is composed of controller III are compared in Figures 7 and 8. Obviously, controller III is effective in attenuating the structural vibrations, while there exist some parameter uncertainties. Moreover, the $\sup_{t \in [1, 15] s} {x^{T} (t) R x (t)}$ obtained by controller III is 0.0998, which is less than κ₂² = 0.15. That is to say, the results obtained by the uncertain finite-time stabilization controller III satisfy the finite-time stability condition, and the effectiveness of Theorem 9 is obvious.

Table 2:

The maximum displacements and accelerations of the open-loop and closed-loop systems, which are composed of controllers I and II, respectively (under nonzero initial states).

	x_1max (cm)	x_2max (cm)	x_3max (cm)	${\ddot{x}}_{1 \max}$ (m/s²)	${\ddot{x}}_{2 \max}$ (m/s²)	${\ddot{x}}_{3 \max}$ (m/s²)	$\sup_{t \in [0,15] s} {x {(t)}^{T} R x (t)}$

Open-loop	20.9	20.0	12.0	60.1	78.4	58.4	18.88
Controller I	20.0	20.0	12.0	18.2	18.1	10.9	0.157
Controller II	20.0	20.0	12.0	21.7	21.7	13.0	5.364

Figure 5:

Displacement responses of the open-loop and closed-loop systems, which are composed of controllers I and II, respectively (under nonzero initial states).

Figure 6:

Acceleration responses of the open-loop and closed-loop systems, which are composed of controllers I and II, respectively (under nonzero initial states).

Figure 7:

Displacement responses of the open-loop and closed-loop system, which is composed of controller III.

Figure 8:

Acceleration responses of the open-loop and closed-loop system, which is composed of controller III.

5. Conclusion

The finite-time vibration control of earthquake excited buildings structure with parameter uncertainties has been investigated in this paper. First, based on matrix transform, the buildings structural system is described as a state-space model, which contains parameter uncertainties. Secondly, based on a Lyapunov functional and finite-time stability analysis method, some sufficient conditions for the existence of finite-time vibration-attenuation controllers are obtained. If the feasibility problem of these conditions is solvable, the desired controller can be obtained for the closed-loop system to be finite-time stable with a prescribed level of disturbance attenuation. The condition is also extended to the uncertain case. Finally, the simulation results are given to show the effectiveness of the designed controllers.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgments

This work was partially supported by the National Natural Science Foundation (nos. 61305019 and 51365017), Jiangxi Provincial Natural Science Foundations (nos. GJJ13430 and GJJ13385), and the Science Foundation of Jiangxi University of Science and Technology of China.

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Finite-Time Vibration-Attenuation Controller Design for Structural Systems with Parameter Uncertainties

Abstract

1. Introduction

2. Problem Formulation and Dynamic Models

3. Main Results

4. Illustrative Example

5. Conclusion

Conflict of Interests

Footnotes

Acknowledgments

References