Adaptive iterative learning control for vibration of flexural rectangular plate

The adaptive vibration control (AVC) problem of flexible plate structures has attracted considerable attention during the last two decades. Many researchers proposed different control strategies for the purpose of AVC of flexible plate structures. Hu et al [1] applied LMI (Linear Matrix Inequality)-based H∞ robust control for AVC of a flexible plate structure. They used specific transformations of Lyapunov variable with appropriate linearizing transformations of the controller variables, which give rise to a tractable and practical LMI formulation of the vibration control problem. Based on LMI, a H∞ output feedback controller was designed to suppress the low-frequency vibrations caused by external disturbances. The simulation results showed that the proposed robust active control method is efficient for active vibration suppression. Other research on the effectiveness of the robust H∞ control for AVC of the flexible structures has been addressed in [2 4]. Based on the previously outlined literature, there is no published report in which the adaptive iterative learning MIMO control is used for the purpose of intelligent AVC of a flexible rectangular plate system. In this research, an adaptive iterative learning MIMO control strategy is applied to the problem of AVC of a rectangular flexible rectangular plate. First, the flexible rectangular plate system is modeled using the FEM method and new modeling method. Then, the validity of the obtained new model is investigated by comparing the plate natural frequencies, mode shape, static analysis and forced vibration response analysis predicted by the finite element model with the calculated values obtained from new model. After validating the model, adaptive iterative learning MIMO controller is applied to the plate dynamics via the MATLAB/Simulink platform. The algorithms were then coded in MATLAB to evaluate the performance of the control system. Disturbances were employed to excite the plate system at different excitation points and the controller ability to attenuate the vibration of observation point was investigated. The simulation results clearly demonstrate an effective vibration suppression capability that can be achieved using adaptive iterative learning MIMO controller.


Introduction
The adaptive vibration control (AVC) problem of flexible plate structures has attracted considerable attention during the last two decades.Many researchers proposed different control strategies for the purpose of AVC of flexible plate structures.Hu et al [1] applied LMI (Linear Matrix Inequality)-based H ∞ robust control for AVC of a flexible plate structure.They used specific transformations of Lyapunov variable with appropriate linearizing transformations of the controller variables, which give rise to a tractable and practical LMI formulation of the vibration control problem.Based on LMI, a H ∞ output feedback controller was designed to suppress the low-frequency vibrations caused by external disturbances.The simulation results showed that the proposed robust active control method is efficient for active vibration suppression.Other research on the effectiveness of the robust H ∞ control for AVC of the flexible structures has been addressed in [2 -4].
Based on the previously outlined literature, there is no published report in which the adaptive iterative learning MIMO control is used for the purpose of intelligent AVC of a flexible rectangular plate system.In this research, an adaptive iterative learning MIMO control strategy is applied to the problem of AVC of a rectangular flexible rectangular plate.First, the flexible rectangular plate system is modeled using the FEM method and new modeling method.Then, the validity of the obtained new model is investigated by comparing the plate natural frequencies, mode shape, static analysis and forced vibration response analysis predicted by the finite element model with the calculated values obtained from new model.After validating the model, adaptive iterative learning MIMO controller is applied to the plate dynamics via the MATLAB/Simulink platform.The algorithms were then coded in MATLAB to evaluate the performance of the control system.Disturbances were employed to excite the plate system at different excitation points and the controller ability to attenuate the vibration of observation point was investigated.The simulation results clearly demonstrate an effective vibration suppression capability that can be achieved using adaptive iterative learning MIMO controller.

Modelling of flexible rectangular plate system
Cartesian coordinate system (x, y, z) is introduced, consider a thin flexible rectangular plate of length a along x-axis, width b along y-axis and thickness h along z-axis.This condition is illustrated in Fig. 1.

Fig. 2 A discrete flexible rectangular plate
The quality and flexibility of plate structure is a continuous distribution, the system has an infinite number of degrees of freedom.To simplify the research and facilitate the calculation, construct spring-mass system and make the system discrete, the system is simplified as multi DOF vibration system.After the process of discrete, the flexible rectangular plate is shown in Fig. 2.Where ( ) damping coefficients.

Fig. 3 A flexible cantilever plate
Considering the boundary conditions, take the modeling of cantilever flexible rectangular plate as an example so as to elaborate new flexible rectangular plate modeling method.Consider a thin cantilever flexible rec-http://dx.doi.org/10.5755/j01.mech.17.5.724 tangular plate of length a along x-axis, width b along yaxis and thickness h along z-axis.This condition is illustrated in Fig. 3.After the process of discrete, the cantilever flexible rectangular plate is shown in Fig. 4.Where ij m ( 1 2 3 i , , p; = 1 2 3 j , , = ) are masses; ( ) are damping coefficients.
Letting As a convention, we denote a dot as a first derivative with respect to time (i.e., y dx / dt = ), and a double dot as a second derivative with respect to time (i.e., 2 2 y d y / dt = ).Let d n be a number of degrees of freedom of the system (linearly independent coordinates describing the finite-dimensional structure), let r be a number of outputs, and let s be a number of inputs.A flexible structure in nodal coordinates is represented by the following secondorder matrix differential equation In this equation X is the (all its eigenvalues are positive), and the stiffness and damping matrices are positive semidefinite (all their eigenvalues are nonnegative).

Modelling of flexible cantilever plate system
Finite element analysis for 10×10 m plate, ρ = 7800 kg/m 3 .Thickness is 0.001 m.This condition is illustrated in Fig. 6.

Forced vibration response analysis
When system is excited by a harmonic force, the vibration response of 39th node is shown by Fig. 8.

Adaptive iterative learning control design
Using the Lagrangian formulation, the equations of motion of a n degrees-of-freedom rectangular plate system may be expressed by M q t q t C q t q t q t K q t t d t where t denotes the time and the nonnegative integer, k Z + ∈ denotes the operation or iteration number.The sig- ∈ are the node position, node velocity and node acceleration vectors, respectively, at the iteration k .
is the inertia matrix, ( ) is the control input vector containing the forces to be applied at each node.( ) ∈ is the vector containing the unmodeled dynamics and other unknown external disturbances.
Assuming that the node positions and the node velocities are available for feedback, our objective is to design a control law ( ) guaranteeing the boundedness of ( ) and t Z + ∀ ∈ , and the convergence of ( ) k q t to the desired reference trajectory ( ) when k tends to infinity.
Throughout this paper, we will use the pe norm defined as follows where ( ) x t denotes any norm of x , and t belongs to the finite interval [ ] 0,T .We say that pe x ∈ when ( ) pe x t exists (i.e., when ( ) is finite).
We assume that all the system parameters are unknown and we make the following reasonable assumptions.
(A1) The reference trajectory and its first and second time-derivative, namely ( ) d q t , ( ) d q t and ( ) d q t , as well as the disturbance ( ) (A2) The resetting condition is satisfied, i.e., ( ) ( ) ( ) ( ) We will also need the following properties, which are common to rectangular plate system. (P1) is symmetric, bounded, and positive definite.(P2) The matrix where [5] ( ) ∈ is a known matrix and ( ) ( ) where q t q t q t = − and ( ) q q q q q sgn q φ Ψ ⎡ ⎤ ⎣ ⎦ , where ( ) k sgn q is the vector obtained by applying the signum function to all elements of k q .The matrices are symmetric positive definite.Let assumptions (A1-A2) be satisfied, then ( ) The proof of this theorem is in three parts.The first part consists of taking a positive definite Lyapunovlike composite energy function, namely k W , and show that this sequence is nonincreasing with respect to k and hence bounded if 0 W is bounded.In the second part, we show that ( ) . Finally, in the third part, we show that Proof.Part 1: Let us consider the following Lyapunov-like composite energy function where ( ) ( ) and ( ) ( ) ( ) ⎦ is the estimated value of ( ) t θ .The unknown vector ( ) t ξ is defined in (P3) and the unknown parameter β is obtained according to (P1) and (A1) such that ( ) The term The difference of k W is given by ( ) ( ) where . On the other hand, one can rewrite k V as follows V q t q t V q q q Mq q Mq q K q dτ Now, using (33) and (P2, P3) we have ( ) ( ) ( ) From known conditions we can have q q sgn q q q sgn q q q q Ψ ξ β Ψ Eq. ( 40) leads to Using Eqs.(36), ( 46) and (A2), Eq. (39) leads to Hence k W is a nonincreasing sequence.Thus if 0 W is bounded one can conclude that k W is bounded.In Part 2 of the Proof we will show that 0 W is bounded for Part 2: Now, we will show that ( ) 0 W t is bounded over the time interval [ ] 0,T .In fact, considering (37) with 0 k = , the time-derivative of 0 W can be bounded as follows ( ) ( ) Since ( ) Using Young's inequality, we have for any 0 .Hence, using (37), one has ( ) Which implies that ( ) ( ) Note that under properties (P1-P3) the control law (35)-(36) involves m iterative parameters, where m is generally larger than the number of degrees-of-freedom n .
It also requires the knowledge of the matrix ( ) However, by using (P4) instead of (P3), the knowledge of the matrix ( ) Ψ is not required anymore and the number of iterative parameters is reduced to two as stated in the following theorem.

Conclusions
Adaptive iterative learning MIMO control strategy for the active vibration control of a flexible rectangular plate structure was developed.It was shown that the new modeling method is a kind of development with respect to the plant modeling theory of current control theory.It provides theoretical basis for low order controller design of high order plant with unknown parameters, adaptive controller design and intelligent controller design.It also brings about great convenience for engineering design.The first nine natural frequencies, mode shapes, static analysis and forced vibration response analysis of the flexible rectangular plate structure considered in this study were predicted accurately and compared by the FEM method and new modeling method and thus, the validity of the proposed new model was confirmed.An adaptive iterative learning MIMO controller was then employed to attenuate the unwanted vibration of a rectangular flexible plate system simulated using the MATLAB/Simulink platform.The simulation results demonstrate the effectiveness of the proposed control technique.Future works will be directed towards the development of an experimental rig to validate the theoretical results obtained in the study.In this paper, we developed an approach for active vibration control of flexible rectangular plate structures using control theory.The flexible rectangular plate system is firstly modeled and simulated via a finite element method; and secondly, a new type of modeling method, and the state-space model are involved in the development of the equation of motion in state-space, which is efficiently used for the analysis of the system and design of control laws with a modern control framework.Then, the validity of the obtained new model is investigated by comparing the plate natural frequencies, mode shapes, statical analysis and forced vibration response analysis predicted by the finite element model with the calculated values obtained from new model.After validating the model, adaptive iterative learning MIMO controller is applied to the plate dynamics via the MATLAB/Simulink platform.The simulation results clearly demonstrate an effective vibration suppression capability that can be achieved using adaptive iterative learning MIMO controller.

Fig. 4 AF
Fig. 4 A discrete flexible cantilever plate

FigFig. 7 4 .
Fig. 6 Finite element model response of flexible rectangular plate system (acting point of force is 39th node): a -result of FEM model; b -result of new model the minimal (maximal) eigenvalue of ( ) * .Since θ is continuous over [ ] 0,T , it is clear that it is bounded over [ ] 0,T , i.e., .