# Vibration-Attenuation Controller Design for Uncertain Mechanical Systems with Input Time Delay

Authors: Yuanchun Ding; Falu Weng; Xiaohua Jiang; Minkang Tang
Journal: Shock and Vibration
 (2016)

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## Abstract

The problems of vibration-attenuation controller design for uncertain mechanical systems with time-varying input delay are of concern in this paper. Firstly, based on matrix transformation, the mechanical system is described as a state-space model. Then, in terms of introducing the linear varying parameters, the uncertain system model is established. Secondly, the LMI-based sufficient conditions for the system to be stabilizable are deduced by utilizing the LMI technique. By solving the obtained LMIs, the controllers are achieved for the closed-loop system to be stable with a prescribed level of disturbance attenuation. Finally, numerical examples are given to show the effectiveness of the proposed theorems.


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## Body

## 1. Introduction

Mechanical systems play an important role in human lives, production, and industrial engineering. Although some new technologies, especially information and computer technologies, are developed recently, mechanical systems are still not replaceable in many engineering fields. Meanwhile, with the development of automatic control theories, mechanical systems become much more stable and effective than before [1–3]. In recent years, many advanced theories in systems and control have been applied to mechanical systems. For example, sliding mode control [1], H∞ control [3, 4], optimal design and control [5, 6], energy-based control [7], finite-time stabilization [8], and position feedback control [9] have been widely studied and used for control of mechanical systems.On the other hand, time delay or transportation lag is inevitable in many practical systems, such as chemical processes, long transmission lines, and pneumatic and hydraulic systems [10–12]. For mechanical systems, unavoidable time delays always appear in the controlled channel, particularly in the digital controller as it carries out the calculations associated with complex sophisticated control law and in sensors and actuators hardware such as large hydraulic actuators. Though the delay time may be short, it can nevertheless limit the control performance or even cause the instability of the system when the delay appears in the feedback loop [10]. Furthermore, most of those delays are time-varying because of the perturbance and uncertainties always existing in the practical systems [13, 14]. Recently, many control strategies have been presented to deal with the control problem for time-delay systems. Such as [15] presented a robust H∞ controller design approach to deal with the vibration-attenuation of uncertain mechanical systems with input delay. Based on LMI technique, the problem of robust active vibration control for a class of electrohydraulic actuated structural systems with time delay in the control input channel and parameter uncertainties appearing in all the mass, damping, and stiffness matrices was investigated in [16]. Based on LMI technique, the delayed H∞ control for an offshore steel jacket platforms subject to external wave force was presented in [17]. Based on Lyapunov theory, the problem of finite-time vibration control of earthquake excited linear structures with input time-delay and saturation was investigated in [12]. Moreover, uncertainty is another problem that need to be considered [18, 19]. Due to the modeling errors, material properties, or environments changing, the description of mechanical systems inevitably contains uncertainties, which can affect the stability and control performance of the mechanical systems significantly. For mechanical systems, the parameter uncertainties and time delay often appear simultaneously, thus, in our investigation, it is important to take both of them into consideration, otherwise the designed controllers will collapse in the practical situations. However, to the best of the authors’ knowledge, the uncertain mechanical systems with time-varying input delay still have not been fully investigated, which is the main motivation of this paper.In this paper, we consider the problems of vibration-attenuation controller design for uncertain mechanical systems with time-varying input delay. The objective of designing controllers is to guarantee the asymptotic stability of closed-loop systems while attenuating disturbance from external excitation. The main contribution of this paper consists in two aspects. First, in terms of introducing the linear varying parameters, the uncertain system model for mechanical systems with time-varying input delay is established. Second, based on Lyapunov stability theory and LMI technique, the LMI-based condition for the system to be stabilizable is deduced. By solving these LMI, the controller is established for the closed-loop system to be stable with the performancez2<γω2. Furthermore, example is given to demonstrate the effectiveness of the proposed theorems.


## 2. Problem Formulation and Preliminaries

The mechanical systems with time-varying input delay are described as [3, 12, 15](1)M-x-¨t+C-x-˙t+K-x-t=B-ut-dt.This model has been widely applied to the description of mechanical and structural systems. Where x-(t)∈Rn is the displacement; M-∈Rn×n,  C-∈Rn×n, and K-∈Rn×n are the mass, damping, and stiffness matrices, respectively; M- is assumed to be nonsingular as usual; B-∈Rn×m is the gain matrix for the state-feedback control input u(t),  d(t) is a time-varying delay satisfying 0≤d(t)≤d, and d is a constant. The state feedback controller consists of the displacement and velocity feedback signals given by(2)ut=Fdx-t+Fvx-˙t,where Fd, Fv are the feedback gain matrices for the displacements and velocities, respectively. By defining x(t)=x-T(t),x-˙T(t)T and considering an input disturbance signal ωt∈L20,∞ (i.e., with bounded energy) and output signal Zt, we have the following feedback control systems:(3)Ex˙t=Axt+But-dt+Bωωt,Zt=Cxt,ut=Fxt,where E=I00M-, A=0I-K--C-, B=0B-, Bω=0B-ω is the disturbance gain matrix, C∈Rq×2n is the output matrix, and F is the state-feedback gain matrix given by F=FdFv.Remark 1.
It is worthy to point out that the input time delayd(t) considered in this paper is not a constant. Thus, the results obtained in this paper should be less conservatism and more fit for using in practical systems than those achievements obtained by the systems with constant time delay (some results about constant time delay can be found in [12, 15–18] and those references therein).By considering the additive uncertainties, the uncertain description of system (3) is written by [20] (4)Eαx˙t=Aαxt+Bαut-dt+Bωωt,Zt=Cxt,ut=Fxt,where Eα is nonsingular, and Eα, A(α), B(α) are in the following convex sets:(5)Eα,Aα,Bα=∑i=1kαiEi,Ai,Biwith ∑i=1kαi=1,αi≥0,i=1,2,…,k.Lemma 2 (see [21]).
For any positive definite symmetric constant matrixΠ∈Rn×n, scalars r1, r2 satisfying r1<r2, a vector function ω:[r1,r2]→Rn such that the integrations concerned are well defined, then(6)∫r1r2ωsdsTΠ∫r1r2ωsds≤r2-r1∫r1r2ωsTΠωsds.Lemma 3 (see [22]).
For any vectorsx,y∈Rn and any positive definite matrix L∈Rn×n, then (7)-2xTy≤xTL-1x+yTLy.In this paper, the aim is to find a controller gainF such that the closed-loop system has the following properties: (i) asymptotically stable; (ii) for all non-zero ω∈L20,∞ and the prescribed constant γ>0, it has the performance z2<γω2.


## 3. Main Results

In this section, we give a solution to the stabilization and robust stabilization problems formulated previously, by using LMI technique.Theorem 4.
The system (3) is asymptotically stabilizable with time delay d(t) satisfying 0≤d(t)≤d and performance z2<γω2 for all nonzero ω∈L20,∞, and constant γ>0, if there exist positive definite symmetric matrices P, R, Q, matrices V, Y1, Y2, Y3, Y4, Z, positive scalar μ, and scalars β1, β2 satisfying the following LMI:(8)Ψ=Ξ1dY∗-dQ<0,where(9)Ξ1=Ξ11Ξ12P-VET+β2AVT+Y3TVCT+Y4T∗Ξ22β1VET+β2BZ-Y3Tβ1VCT-Y4T∗∗d2Q-β2VET-β2EVTβ2VCT∗∗∗-γ2I,Ξ11=R+BωBωT+VAT+AVT+Y1T+Y1,Ξ12=ZTBT+β1AVT+Y2T-Y1,Ξ22=-1-μR+β1ZTBT+β1BZ-Y2T-Y2,Y=Y1TY2TY3TY4TT.Moreover, a state-feedback controller is described as F=ZV-T.Proof.
Consider the system without uncertainties. Substituting the control lawu(t)=Fx(t) into the system (3) results in the following closed-loop system:(10)Ex˙t=Axt+BFxt-dt+Bωωt,Zt=Cxt.Noting that the solutions to det⁡(SE-A-BKe-sτ)=0 are the same as those of det⁡(SET-AT-KTBTe-sτ)=0, and Tzω∞=TzωT∞, as long as the H∞ performance and stability are concerned, we can consider the following system instead of system (10):(11)ETx˙t=ATxt+FTBTxt-τ+CTωt,Zt=BωTxt.Choose a Lyapunov-Krasovskii functional candidate as(12)Vt=xTtPxt+∫t-dttxTsRxsds+d∫-d0∫t+θtx˙TsQx˙sdsdθ.By Lemma 2, the time derivative of V(t) along the solution of (11) is given by(13)V˙t≤x˙TtPxt+xTtPx˙t+xTtRxt-1-μxTt-dtRxt-dt+d2x˙TtQx˙t-∫t-dtx˙TθdθQ∫t-dtx˙θdθ.For any matrix Y=Y1TY2TY3TY4TT with appropriate dimension, the following equation holds:(14)2ηTtYxt-xt-d-∫t-dtx˙θdθ=0,where η(t)=x(t)Tx(t-d(t))Tx˙(t)Tω(t)TT. For any matrix V with appropriate dimension, the following equation holds:(15)2xTt+β1xTt-dt+β2x˙TtVATxt+FTBTxt-τ+CTωt-ETx˙t=0.By Lemma 3, we have (16)-2ηTtY∫t-dtx˙θdθ≤dηTtYTQ-1Yηt+∫t-dtx˙TθdθQ∫t-dtx˙θdθ.Next, we will establish the z2<γω2 performance for the system under zero initial condition, that is, Φ(t)=0, ∀t∈-τ,0, and Vtt=0=0. Consider the following index:(17)J=∫0∞ZTtZt-γ2ωTtωtdt.Then, for any nonzero ω(t)∈L20,∞, there holds(18)J≤∫0∞ZTtZt-γ2ωTtωtdt+Vtt=∞-Vtt=0=∫0∞ZTtZt-γ2ωTtωt+V˙tdt.Choosing FVT=Z and noting (13)–(18), after some algebraic manipulations, we obtain(19)ZTtZt-γ2ωTtωt+V˙t≤ηTtΞ1+dYQYTηt.
Then, ifΞ1+dYQYT<0, we have ∫0∞ZT(t)Z(t)-γ2ωT(t)ω(t)+V˙(t)dt<0. Thus, for any nonzero ω(t)∈L20,∞, we obtain J<0 and z2<γω2. If Ξ1+dYQYT<0 and ω(t)≡0, we can easily obtain V˙(t)<0, and the asymptotic stabilizability of system (11) is established. According to the Schur compliment, we can obtain Ξ1+dYQYT<0 from (8). This completes the proof.Remark 5.
Note that the matrix inequality (8) in Theorem 4 is actually an LMI when the scalars β1 and β2 are given in advance; thus, the solutions of Theorem 4 can be easily obtained by the powerful LMI toolbox in MATLAb. Furthermore, the two scalars β1 and β2 supply an additional degree of freedom for the feasibility of LMI (8). For example, the scalars β1 and β2 can all be set to 1, initially. If LMI (8) is infeasible, a possible solution may be searched by tuning β1 and β2 or by iterating over β1 and β2.Theorem 6.
The system (4) is robustly stabilizable with time delay dt satisfying 0≤d(t)≤d and performance z2<γω2 for all nonzero ω∈L20,∞, and constant γ>0, if there exist positive definite symmetric matrices P, R, Q, matrices V, Y1, Y2, Y3, Y4, Z, positive scalar μ, and scalars β1, β2 satisfying the following LMIs:(20)Ξ~idY∗-dQ<0,i=1,2,…,k,where(21)Ξ~i=Ξ~i11Ξ~i12P-VEiT+β2AiVT+Y3TVCT+Y4T∗Ξ~i22β1VEiT+β2BiZ-Y3Tβ1VCT-Y4T∗∗d2Q-β2VEiT-β2EiVTβ2VCT∗∗∗-γ2I,Ξ~i11=R+BωBωT+VAiT+AiVT+Y1T+Y1,Ξ~i12=ZTBiT+β1AiVT+Y2T-Y1,Ξ~i22=-1-μR+β1ZTBiT+β1BiZ-Y2T-Y2.Moreover, a state-feedback controller is described as F=ZV-T.Proof.
Suppose there exist positive definite symmetric matricesP, R, Q, matrices V, Y1, Y2, Y3, Y4, Z, positive scalar μ, and scalars β1, β2 satisfying (20). Multiply the above inequalities (20) by αii=1,2,…,k, αi≥0, and ∑i=1kαi=1, and sum to get(22)Ξ~11Ξ~12P-VEαT+β2AαVT+Y3TVCT+Y4TdY1∗Ξ~22β1VEαT+β2BαZ-Y3Tβ1VCT-Y4TdY2∗∗d2Q-β2VEαT-β2EαVTβ2VCTdY3∗∗∗-γ2IdY4∗∗∗∗-dQ<0,where(23)Ξ~11=R+BωBωT+VAαT+AαVT+Y1T+Y1,Ξ~12=ZTBαT+β1AαVT+Y2T-Y1,Ξ~22=-1-μR+β1ZTBαT+β1BαZ-Y2T-Y2,Eα=∑i=1kαiEi,Aα=∑i=1kαiAi,Bα=∑i=1kαiBi.Then, by replacing E, A, and B with Eα, Aα, and Bα, respectively, we can obtain the desired result from Theorem 4, easily.


## 4. Numerical Example

We consider the system (4) with (24)M-=δ10001.80001.6,C-=δ2-0.60-0.61.2-0.60-0.60.6,K-=2-10-12-10-11,B-=100,B-ω=000.1T,C=0.10.10.50000000.10.10.5,x-t=x-1tx-2tx-3tT,where δ1 and δ2 are uncertain parameters. Firstly, consider the system without uncertainties, and δ1=1.1, δ2=1.2. This example was considered in [3, 15, 16]. Assume the control forces input time delay is 0.1 s. By [15], we can obtain the minimum γ (such that there exists an admissible controller) which is 0.278. By [3], the minimum γ is 0.119. By [16], the minimum γ is 0.031. However, by Theorem 4 in this paper, it is found that the admissible controller exists even for γ=0.025 (β1=-1,β2=-1,μ=0.1). Thus, the less conservatism of our method is obvious. The more calculated results are presented in Table 1, which show the minimum feasible γ for different input delay d. It is shown from Table 1 that the obtained results by Theorem 4 have less conservatism than those in [3, 15, 16], obviously.Table 1
Minimum feasible disturbance attenuationγ by different input time delay d.
d
0.05
0.1
0.15
0.4
[15]
0.147
0.278
0.624
—
[3]
0.082
0.119
0.156
—
[16]
0.029
0.031
0.033
0.071
Theorem4
0.025
0.025
0.026
0.04Then, let us consider vibration-attenuation performance of the closed-loop system. The corresponding disturbance input (see Figure1) is obtained by the following signal function:(25)ωt=100sin⁡t,0≤t≤1,0,t>1.Figure 1
Disturbance input signal.By assuming the parametersδ1 and δ2 satisfying δ1=1.1,δ2=1.2, and choosing β1=-0.5, β2=-0.1, μ=0.1, d=0.05, and γ=0.2, we solve LMI (8) and obtain a state feedback controller which has the following gain matrix:(26)F=0.2171-0.3699-0.1808-2.1308-0.2099-0.9866.For description in brevity, this state feedback controller is denoted as controller I thereafter. By taking the signal function (25) as the disturbance excitation, we can obtain the displacement responses of open-loop and closed-loop system which is composed of controller I, which are shown from Figures 2–4. It can be obtained from Figures 2–4 that controller I is effective to attenuate the vibration of the system, obviously.Figure 2
The responses of state variablex-1(t).Figure 3
The responses of state variablex-2(t).Figure 4
The responses of state variablex-3t.Secondly, let us consider the parameter uncertainties. By assuming the parameter uncertainties satisfyingδ1∈1.11.8, δ2∈0.81.4 and choosing β1=-0.5, β2=-0.1, μ=0.1, d=0.05, γ=0.2, we solve LMIs (20) and obtain a robust state feedback controller which has the following gain matrix:(27)F=-1.49730.24850.0501-2.3346-0.5335-0.3927.For description in brevity, this state feedback controller is denoted as controller II thereafter. By taking the signal function (25) as the disturbance excitation, we can obtain the responses of x-1t in open-loop and closed-loop system which is composed of controller II, which are shown in Figure 5. The responses of x-2t and x-3t have a similar varying trend, which is omitted here for brevity. Figure 5 illuminates that designed controller II is robust to parameter uncertainties.Figure 5
The responses of state variablex-1(t) in the uncertain case.


## 5. Conclusions

In this paper, the problems of vibration-attenuation controller design for uncertain mechanical systems with time-varying input delay are considered. First, based on matrix transformation, the mechanical system is described as a state-space model. Then, by introducing the linear varying parameters, the system model is extended to its uncertain case. Second, based on Lyapunov stability theory and LMI technique, the LMI-based condition for the system to be stabilizable is deduced. By solving these LMIs, the state-feedback control law is obtained for the closed-loop system to be stable with the performancezt2<γωt2. In the end, numerical example is given to show the effectiveness of the proposed theorems.



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