Study On Uncertainty Of Piezothermoelasticity Intelligent Plate And Its Vibration Control Using H_∞ Control Law

The paper studies on robust control of uncertain piezothermoelasticity intelligent laminated Plates by using H_∞control law, the main works finished in this paper include the following:1. For composite plates with materials of each layer different, considering the heat conduction, convection and radiation boundary conditions, a 4-node rectangle element model is adopted and its temperature differential equations are deduced by means of Galerkin method, at the same time, the general formulation about the sensitivity of the temperature with respect to each design parameter is deduced, and the common part in the general formulation having an obvious inertial effect is founded in computer simulation.2. For linear transient temperature field, when all parameters are random, on the basis of decoupling method, randomicity analysis of eigenvalue and eigenvector, and second-moment method, the numeric characteristics of the temperature field are obtained. A 3-layer plates was taken as an example, the numerical results are compared with those obtained by Monte Carlo method, by which the practicability of the presented method is verified.3. For nonlinear transient temperature field, when all parameters are random, in order to decrease the operating time of Monte Carlo method, this paper introduce random factor and presents an approximate treatment of heat conduction matrix and heat capacity matrix, at the same time, the errors of temperature's numerical characteristics caused by that approximate treatment method are computed approximately by computing sensitivity and the linearization technique of first order Taylor series expansion. Computer simulation shows the time is saved largely, and the errors computed approximately are precision in a certain extent.4. The method presented in 3 is used in 2-D truss structure, two methods to deal with approximatively the randomicities of the mass matrix and stiffness matrix are presented, the first method can obtain an upper limit of standard deviation of dynamic characteristic, the second method can obtain an approximation of it, both of them can greatly decrease the works of Monte Carlo numerical simulation. Finally, the frugality and precision of the two methods proposed in this paper are proved by computer simulation.5. For random factor mentioned in 3 and 4, although it has been used for many years, it has not been proven form the angle of mathematics, here, it is proven for the first time, namely, after random variables are described by random factor method, when all random factors can be extracted from mass matrix and stiffness matrix, through analyzing the QR method process of solving random matrix's eigenvalue, the natural frequencies' random factors are proved equally, based on it, all elements of the normal model shape matrix having the same random factor are proven too. In computer simulation, all conclusions are proven by Monte Carlo method.6. For nonlinear transient temperature field, when all parameters are random, this paper presents a method to obtain the approximate solution of the temperature's main interval by using synthetically the following techniques: difference method, Monte Carlo method, Quantile-quantile plot, Box-Cox transformation, hypothesis testing and parameter interval estimation. In computer simulation, the temperature's main interval obtained is compared with lots of temperature's random samples, which shows that the method presented is reasonable.7. For piezothermoelasticity intelligent thin plate, a cube finite element model including 4 displacement nodes, 2 electric potential nodes and 8 temperature nodes is presented, the displacement field is defined by means of plane shell element model, its electric potential field and temperature field are all defined by means of linear interpolation; the finite element equations are deduced by using virtual work principle, its correctness is proved by compared with other papers; finally, an intelligent cantilever plate is taken as an example , its responses of temperature field, displacement field and output voltages are simulated, which show that the temperature having big change amplitude value has large affection on output voltages.8. For element finite differential equations in 7, when all parameters are random, through the analytic solutions of sensitivities with respect to random parameters and first order second moment method, the numerical characteristics of natural frequencies, mode shapes, temperature field, displacement field and output voltages are solved in turn; finally , an intelligent cantilever plate is taken as an example , the numerical results are compared with those of Monte Carlo Method, the results show that the computing process presented has good precision.9. For the problem of computing speed slowness existed in 8, the response's numerical characteristics are computed by using synthetically deference method and moment method, the computer simulation shows that the computing process presented is practicability and has great efficiency.10. In order to realize the robust control of uncertain piezothermoelasticity intelligent laminated plates with small interval parameters, it is studied firstly how to derive the reduced system's uncertainty from the original uncertain system with small interval parameters. On the basis of sensitivities of matrix exponential and the sensitivities of real symmetric matrixes' eigenvalue and eigenvector in orthogonal transform, the sensitivities of state transformation matrix are derived, then the reduced system's uncertainty are obtained by the first order Taylor series expansion and interval mathematics, in computer simulation, a same input is applied respectively on the original system with parametric uncertainty and the reduced system with obtained uncertainty, their output's intervals are almost the same shows the method presented is corrected.11. For the reduced uncertain system obtained in 10, a low dimension method is presented to decompose uncertain matrix; the robust controller of the reduced generalized plants is solved by using H_∞control theory, and a process is given to verify the robustness of the controller on original uncertain system. A simple example's computer simulation shows the giving method is feasible.12. In order to realize the robust control of uncertain piezothermoelasticity intelligent laminated plate with small interval parameters, except the works in 10 and 11, it is also need to be studied beforehand that the vibration control of the certain piezothermoelasticity intelligent laminated plate. Here, without treating of thermal radiation, the linear system is reduced by using Balancing reduction; it is discussed how to derive the generalized plants and using H_∞control theory to restrain vibration; an intelligent cantilever plate is taken as an example , its control effect shows that the method presented is valid; simultaneity, the example shows for this kind of structure, when both the bending displacement and tensile deformation are wanted to be controlled, the actuator has the two faces of control action and interference action in control process.13. By using synthetically the wors presented in 10-12, the robust H_∞control of uncertain piezothermoelasticity intelligent laminated plate with small interval parameters is realized.