Research On PDF Shape Control In Non-linear Stochastic Systems

When there are random inputs,random interferences or random characteristics(parameters)in a stochastic system,the system state,the output and the control are all random processes,which make the development of stochastic control theory inevitable.In during several decades,the objective of designing controller for stochastic dynamic systems is functional mean and variance.In this method,the stability of the closed-loop systems and optimality of a certain objective function are to be as starting point.But there are some limitations in both theory and application,thus restricting the performance of the controller greatly.So controlling the PDF shape of index performance has aroused researcher's close attention,and has been gradually becoming one present researching focus in control areas.Aiming at several hot point problems on stochastic systems in current domestic and international,we have a deep research in this paper.The research contents mainly conclude design methods of PDF shape non-linear controller and piecewise linear controller for polynomial non-linear stochastic systems,PDF shape control technique based on the approximate solution to FPK equation,and application of advanced intelligent algorithm to PDF shape control for non-linear stochastic systems.For key technology for above studied problems,this paper provides more novel and practical algorithm,and has achieved research results which are of a certain reference value.The specific research work includes the following aspects:1.Aiming at a class of non-linear stochastic systems which is of additive White Gauss noise and polynomial form,we study a method of designing a nonlinear controller.The goal of the optimal control for the studied non-linear stochastic systems is that the designed controller approximates the expected shape,thus the control problem of the PDF shape is converted the optimal problem of controller parameters.We design a nonlinear controller with polynomial type,and obtain the stationary PDF expression of the state variable by solving the FPK equation.When PDF shape of state variable approximates the expected PDF shape,the optimal controller gains are acquired using linear least squares method.Simulation results show that devised non-linear controller can control effectively the PDF shape of the state variable.2.We propose a designing method of linear piecewise PDF shape controller.Aiming at PDF shape control problem for polynomial non-linear stochastic systems,polynomial non-linear controller is more favorite and adopted.However,the polynomial control law will increase the computation requirements and complexity of the overall algorithm,so we present a linear piecewise controller which is composed of two proportional coefficients and a piecewise point,including three parameters,and the number of the parameters is relatively less.Besides,linear controller shows its linearity within a specific range,so it is mathematically simpler to deal with.When the PDF shape of state variable approximates the goal PDF shape in optimal way,the control problem of PDF shape is transformed into a mathematical programming problem,for which existing toolkits are very abundant.The characteristics of the problem we study is linear or of polynomial form.When solving such optimization problems,using conjugate gradient method can find optimal infinite steps,and the efficiency is very high.So we use non-linear conjugate gradient method to obtain the parameters in the controller.The comparison experiments are carried out among the piecewise linear controller,second-order non-linear controller and cubic-order non-linear controller,reflecting the advantages and characteristics of the linear controller.3.The structure model of the stochastic system determines the structure of the controller.For stochastic systems of different structures,the control laws of the controller are different.The nonlinearity of a system is not only expressed as polynomial form,but also perhaps reflects in exponential function and trigonometric function.Therefore,we need to develop a technique to deal with all kinds of nonlinear systems.As is known to all,the stationary PDF of the state variable corresponds to the solution to the FPK equation of the stochastic dynamical equation.However,it is hard to work out the accurate solution to the FPK equation.In this paper an approximate method of solving FPK equation is proposed.Find a special function firstly,and this function has the properties of PDF.So the function can be regarded as the approximate solution,then the PDF controller expression including parameter is derived.The approximate solution of the FPK equation is the PDF of the state variable,which follows the desired PDF.The involved parameters in the function can be worked out through optimization algorithm,and the approximate solution of the FPK equation can be obtained,the PDF shape controller for different desired distributions is also acquired.Meanwhile,the PDF shape control method is applied to the grining system to control the ore particle distribution.Optimizing the amount of newly added aggregate of grinding mill makes the distribution of particles from the overflow ore in the hydro-cyclone meet the requirements of the subsequent separation process.Controlling result demonstrates the effectiveness of the PDF shape control method.4.For non-linear stochastic dynamical systems,it is more complicated and even impossible to obtain the probability density function of the system state through solving the FPK equation.According the relation among the PDF,statistical moments of each order and cumulants of each order,the PDF of the system state variable can be expressed by Edgeworth expansion approximately,thus solving the proximate PDF.Firstly,the statistical moment differential equation of the response PDF is obtained in the nonlinear stochastic dynamic system,and then the PDF shape control problem is studied.Swarm intelligence algorithm has been applied to many fields for its strong ability of solving problems.In this paper,a new intelligent algorithm-fireworks algorithm is proposed.Its algorithm principle is introduced in detail,and then it is applied to the PDF shape control of stochastic systems,optimizing the parameters of the controller.In the end,the performance comparison examples in PDF shape control are carried out among fireworks algorithm,genetic algorithm and PSO algorithm,fully demonstrating the superiority of the fireworks algorithm.Finally,some concluding remarks are given,and the future research works are pointed out.