# Nonlinear Stochastic Optimal Control Using Piezoelectric Stack Inertial Actuator

Authors: Q. F. Lü; X. F. Wang; K. Lu; R. H. Huan
Journal: Shock and Vibration
 (2020)

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## Abstract

An optimal control strategy for the random vibration reduction of nonlinear structures using piezoelectric stack inertial actuator is proposed. First, the dynamic model of the nonlinear structure considering the dynamics of a piezoelectric stack inertial actuator is established, and the motion equation of the coupled system is described by a quasi-non-integrable-Hamiltonian system. Then, using the stochastic averaging method, this quasi-non-integrable-Hamiltonian system is reduced to a one-dimensional averaged system for total energy. The optimal control law is determined by establishing and solving the dynamic programming equation. The proposed control law is analytical and can be fully executed by a piezoelectric stack inertial actuator. The responses of optimally controlled and uncontrolled systems are obtained by solving the Fokker–Planck–Kolmogorov (FPK) equation to evaluate the control effectiveness of the proposed strategy. Numerical results show that our proposed control strategy is effective for random vibration reduction of the nonlinear structures using piezoelectric stack inertial actuator, and the theoretical method is verified by comparing with the simulation results.


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## Body

## 1. Introduction

Piezoelectric stack actuators have been widely used in vibration control of mechanical structures due to their fast response and high precision, such as aerospace, precision machining, biomedical engineering, and semiconductor manufacturing [1–4]. Generally, there are two basic approaches when a piezoelectric stack actuator is used as an actuator. One is the direct actuator, where one side of the piezoelectric stack is fixed and the other is bonded to the structure. Although this kind of actuator has large output force and an easily determined control law, it could bring new excitation sources to the structure. The other way is to use as an inertial actuator, where one side is combined with an inertial mass and the other side is bonded to a structure. The inertial mass can effectively isolate unnecessary interference and also can protect the pressure sensor from being damaged by excessive force [5]. This way is commonly used and has been applied by many scholars in some different areas. For example, Choi et al. [6] applied a piezoelectric stack inertial actuator to the vibration control of simply supported beam at both ends and achieved good control effectiveness. Li et al. [7] applied a piezoelectric stack actuator to an active shaft transverse vibration control system with large reduction of housing vibrations. Zhao et al. [8] used a piezoelectric rotary inertia actuator to control the vibration of the rotating structure, which effectively reduced the noise propagation of the structure. Shao et al. [9] proposed a new type of inertial piezoelectric actuator which has a miniaturization structure and dynamic performance of high precision and high load capacity. Wang et al. [10] obtained an actuator with stable linear motion performance using integrated piezoelectric vibrator and MRF control structure. The traditional piezoelectric inertia actuator can only bear the force in the axial direction. For this reason, Konstanzer et al. [11] studied and designed an actuator that can bear the bending stress, which greatly improved the control effect with the condition of large force and low frequency vibration. Introducing the modal H2 norm and change rate of natural frequencies, Lu et al. [12] proposed an optimal placement criterion for piezoelectric actuators. With this criterion, the piezoelectric smart SFM system has a better single modal controllability and observability and has a good result on the vibration suppression.Through the survey of these literatures, it can be found that most of the studies on vibration control using piezoelectric stack inertial actuator mentioned above are limited to the study of the dynamic characteristics of the actuator itself or the vibration control of linear structure under the action of deterministic load. Far less is known about the control of random vibration, especially nonlinear random vibration. Thus, the development of a control strategy for a nonlinear stochastic system using a piezoelectric stack inertial actuator is much deserving, and that is the motivation of the present paper.In the present paper, an optimal control problem for a strong nonlinear and stochastically excited structure with a piezoelectric stack inertial actuator is investigated. Our original contributions are highlighted as follows: the dynamic model of the nonlinear structure considering random excitation and the dynamics of a piezoelectric stack inertial actuator is established; the control problem is firstly investigated in the Hamiltonian frame, which makes the stochastic averaging method for the quasi-Hamiltonian system available for dimension reduction; the proposed optimal control law, which can be fully executed by a piezoelectric stack inertial actuator, is robust and effective in reduction of random vibration.


## 2. Mathematical Modeling of the Coupled System

Figure1 presents schematic configuration of the piezoelectric stack inertial actuator consisting of an inertial mass and a piezoelectric stack. The electromechanical behavior of the piezoelectric actuator can be expressed as follows [13]:(1)S3=s33ET3+d33E3,D3=d33T3+ε33TE3,where S is the strain, D is the electrical displacement, T is the stress, E is the electric field, d is the piezoelectric material constant, εE is the elastic compliance, and εT is the permittivity at a constant stress. Thus, the dynamic behavior of n piezoceramic layers can be derived as follows:(2)Fpt=AT3=As33S33−Ad33s33E3=As33δtl−nAd33s33Vptl=Kpδt−αpVpt=Kpδt−Fat,where Fpt is the load of the piezoelectric stack inertial actuator, A is the cross-sectional area of the piezoelectric stack, l and δt are the length and deformation of the piezoelectric stack, Fat is the force generated by voltage Vpt, Kp is the stiffness of the spring, and αp is the proportional constant. Considering the damping in piezoelectric stack, the motion equation of the mechanical model in Figure 1(b) can be established:(3)mx¨+cpx˙+kpx+αpVpt=0.Figure 1
Configuration and model of piezoelectric stack inertial actuator. (a) Schematic configuration. (b) Mechanical model.
(a)(b)Here, we use this inertial actuator for vibration control of a nonlinear structure. The coupled system is shown in Figure2. In this figure, kp and cp denote, respectively, the stiffness and damping of the piezoelectric stack actuator; m and M are the mass of the inertial actuator and the mass of vibration structure, respectively; x is the displacement of the vibration structure; y is the displacement of inertial mass; and F is the active control force exerted by voltage. Zt is random disturbance of the base. Then, the motion equation of the coupled system can be established:(4)Mx¨+cx˙−Z˙t+kx−Zt+k3x−Zt3−kpy−x−cpy˙−x˙=−F,my¨+cpy˙−x˙+kpy−x=F.Figure 2
Mechanical model of the coupled system.System (4) is a two-degree-of-freedom, strong nonlinear, stochastic excited, and controlled system.


## 3. Partial Averaging of the Controlled System

Introduce the transformation(5)x¯=x−Zt,y¯=y−x.Substituting (5) to (4) yields(6)Mx¯¨+cx¯˙−cpy¯˙+kx¯+k3x¯3−kpy¯=−Fu−MZ¨t,my¯¨+cp1+mMy¯˙−mMcx¯˙−mMkx¯+k3x¯3+kp1+mMy¯=1+mMFu.Introduce the generalized displacement(7)q1=x¯,p1=Mx¯˙,q2=x¯+y¯,p2=mx¯˙+y¯˙,and system (4) can be rewritten as(8)q˙1=∂H∂p1,p˙1=−∂H∂q1−c+cp∂H∂p1+cp∂H∂p2−F−MZ¨t,q˙2=∂H∂p2,p˙2=−∂H∂q2−cp∂H∂p2−∂H∂p1+−mZ¨t,where Z¨t is acceleration of the base, which is assumed to be a zero-mean Gaussian white noise with correlation function Eωkt×ωlt+τ=2Dklδτ. System (8) is called a quasi-Hamiltonian system. The Hamiltonian function associated with equation (8) is(9)H=12p12M+12p22m+12kq12+14k3q14+12kpq2−q12.Obviously,H is the only first integral, which indicates that system (5) is a quasi–non-integrable-Hamiltonian system [14]. In most engineering applications, the Hamiltonian function H denotes the total vibration energy of the structures.Theoretically, by adding Wong–Zakai terms, system (8) can be converted into the following Itô form:(10)dQ1=∂H∂P1dt,dP1=−∂H∂Q1−m11∂H∂P1−m12∂H∂P2−udt+σ1dBt,dQ2=∂H∂P2dt,dP2=−∂H∂Q2−m21∂H∂P1−m22∂H∂P2+udt+σ2dBt,where σ12=2DM2, σ22=2Dm2, m11=c+cp, m12=−cp, m21=−cp, m22=cp, u1=−F, u2=F, and Bt is the standard Wiener process. Then, the stochastic averaging method for quasi-non-integrable-Hamiltonian system is applied to system (10), and the averaged Itô equation for H can be obtained as follows:(11)dH=m¯H+u−∂H∂P1+∂H∂P2dt+σ¯HdBt,where m¯H and σ¯H are drift and diffusion coefficients, and(12)m¯H=1TH∫Ω−∑i,j=12mij∂H/∂pi∂H/∂pj+∑i,j=121/2σiσj∂2H/∂pi∂pj∂H/∂p1dq1dq2dp2=1TH∫Ω−m11∂H/∂p1∂H/∂p1−2m12∂H/∂p1∂H/∂p2−m22∂H/∂p2∂H/∂p2+1/2σ12∂2H/∂2p1+σ1σ2∂2H/∂p1∂p2+1/2σ22∂2H/∂2p2∂H/∂p1dq1dq2dp2,σ¯2H=1TH∫Ω∑i,j=12σiσj∂H/∂pi∂H/∂pj∂H/∂p1dq1dq2dp2=1TH∫Ωσ1σ1∂H/∂p1∂H/∂p1+2σ1σ2∂H/∂p1∂H/∂p2+σ2σ2∂H/∂p2∂H/∂p2∂H/∂p1dq1dq2dp2,·=1TH∫Ω·∂H/∂p1dq1dq2dp2,TH=∫Ω1∂H/∂p1dq1dq2dp2,Ω=q1,q2,p2Hq1,q2,0,p2≤H.To complete the integration above, make the following polar coordinate transformation:(13)q1=Rω1cosθ,q2=Rω2sinθ,and the coefficients m¯H, σ¯2H, and TH can be rewritten as(14)m¯H=2πTHω1ω2∫02π12λ1H+λ2H+λ3R2−λ1+λ2AH,θdθ,σ¯2H=2πTHω1ω2∫02π12λ4H+λ5HR2−λ4+λ5AH,θdθ,TH=2πMmω1ω2∫0πR2dθ,where λ1=−m11m/M, λ2=−m22M/m, λ3=1/2σ12m/M+σ22M/m, λ4=σ12m/M, λ5=σ22M/m, and(15)AH,θ=18kcos2θω12R4+124k3cos4θω14R6+18kpcosθω1−sinθω22R4.R2 in equations (14) and (15) is the solution of following equation:(16)H−12kR2ω12cos2θ−14k3R4ω14cos4θ−12kpR2sinθω2−cosθω12=0.


## 4. Optimal Control Law

Usually, the following performance of index is used for vibration control problem:(17)Ju=E∫totfX,u,tdt+gXtf.Consistent with the averaged equation (11), the averaged form for the proposed performance is obtained as [15](18)Ju=limtf⟶∞1tf∫0tffHs,usds+gHtf.Equations (11) and (18) constitute the mathematical formulation of the optimal control problem of the partially averaged quasi-non-integrable-Hamiltonian system with piezoelectric stack inertial actuator.According to the dynamic programming principle, the dynamic programming equation is established as(19)∂V∂t=−infu12σ¯2Hd2HdH2+m¯H+u−∂H∂P1+∂H∂P2dVdH+fH,u,where VH,t=gHtf is the terminal condition. The necessary condition for minimizing the right-hand side of equation (19) is(20)∂∂uu−∂H∂P1+∂H∂P2dVdH+fH,u=0.Assume the cost function(21)fH,u=f1H+uTru,f1H=s0+s1H+s2H2+s3H3,where r is a constant. Thus, the optimal control force is determined:(22)u∗=12r∂V∂H∂H∂P1−∂H∂P2.Replaceu with u∗ in equation (19) and finish the averaging to obtain final dynamical programming equation. Then, ∂V/∂H can be obtained by solving this final dynamical programming equation. Substituting ∂V/∂H into equation (22) and replacing u with optimal control force u∗ in equation (11), the fully averaged Itô equation is(23)dH=m¯¯Hdt+σ¯HdBt,where(24)m¯¯H=m¯H+u∗−∂H∂P1+∂H∂P2.According to the theory of stochastic dynamics,H is a Markov diffusion process, and the transition probability density function is satisfied by the so-called Fokker–Planck–Kolmogorov (FPK) equation. Solving the FPK equation yields the following stationary probability density of Ht:(25)pcH=Ccexp−∫0H−2m¯¯u+dσ¯2u/duσ¯2udu.The stationary joint probability densitiespcq,p and mean square value EQ12,EQ22 are then obtained as follows [14]:(26)pq,p=pcHTHH=Hq,p,EQi2=∫−∞∞qi2pq,pdq1dq2dp1dp2,i=1,2.Introduce control effectiveness to measure the performance of the proposed control strategy:(27)Ki=EuQi2−EcQi2EuQi2,i=1,2,where EuQi2 and EcQi2 denote uncontrolled and controlled mean square value of Qi, respectively.


## 5. Numerical Results

As a verification method of control strategy, Monte Carlo simulation has been widely used in many research studies, which is practical and efficient. In this paper, the Monte Carlo simulation method is used, too. With specific system parameterskp=10.702kN/m, cp=28.681N⋅s/m, k=54.684kN/m, k3=300kN/m, c=196.02N⋅s/m, D=0.1, and m=2kg, M=10kg and control parameters s0=1, s1=1, s2=1,andr=0.05, some results for uncontrolled and optimally controlled system (4) are obtained and shown in Figures 3–7.Figure 3
Stationary probability densityp(H) of controlled and uncontrolled system (10).Figure 4
Sample of processq1 of system (10).Figure 5
Mean valueEH of system (10) as function of D.Figure 6
Mean square valueEQ12 of system (10) as function of D.Figure 7
Control effectivenessK1 of system (10) as function of D.In Figure3, the stationary probability density p(H) of total energy H of system (4) is plotted. It is seen that the p(H) curve of the optimally controlled system shifts to the left and has higher peak value when the optimal control force is applied. This means that the structure has higher probability to vibrate in small amplitude, which indicates the proposed control strategy is very effective for response reduction. In Figure 3, the solid lines are analytical results obtained from solving equation (25) while the symbols are Monte Carlo simulation results directly obtained from equation (4). They agree well, which illustrates the accuracy of the proposed method. Figure 4 plots the samples of generalized displacement Q1 of optimally controlled and uncontrolled system (10), from which the response reduction of our proposed method can be observed intuitively. Figures 5 and 6 show the mean value EH of total energy and mean square value EQ12 of generalized displacement varying with the intensity of random excitation. It is seen that with the increase of the intensity of excitation, the response of the optimally controlled and uncontrolled systems increases. However, the response of the optimally controlled system is always much smaller than the uncontrolled one. Figure 7 shows the control effectiveness K1 varying with the intensity of the excitation. With different intensities of excitation, control effectiveness changes smoothly between 53%-54%, which indicates this control strategy has good robustness. Compared with existing literatures, the control effectiveness of this control strategy using the piezoelectric stack inertial actuator is much higher, for example, in ref. [7], it can be seen from the figure of vibration response for simultaneous control of multiple harmonics that the control effectiveness is about 10%–30%. In ref. [8], it can be seen from the figure of comparison of the plate vibrations in the frequency domain without control and with control that the control effectiveness is about 10%–35%.


## 6. Conclusion

An optimal control strategy for nonlinear stochastic vibration using a piezoelectric stack inertial actuator has been proposed in this paper. This proposed procedure has some advantages: the control problem is investigated in the Hamiltonian frame, which makes the stochastic averaging method for quasi-Hamiltonian system available for dimension reduction; the proposed control law is analytical and can be fully executed by a piezoelectric stack inertial actuator. Numerical results showed that the strategy is fairly robust and effective in reduction of stationary response of the controlled system by using piezoelectric stack inertial actuator; compared with those in some literatures, this proposed control strategy has higher effectiveness. The main work of our further research is to use the theoretical advantage of this method to specific experiments. Thus, it is potentially promising for practical control applications after further research.



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