Research On Some Problem Of Optimal Control For Generalized System

The optimal control problem for the generalized system is a significant branch of modern control theory.The research on this problem has important theoretical and practical significance,and has been widely concerned by scholars at home and abroad.In the research of the traditional optimal control theory,the model-based methods have obtained a large number of research results.But in practice,due to the complexity and uncertainty of the system,the traditional optimal control method cannot solve practical problems or cannot be solved.Therefore,in view of two aspects,i.e.,the complexity of system which causes the system to be unable to solve and the unknown system,this paper studies the non-traditional optimal control problem for the generalized system.Specifically,we discussed the discrete-time generalized linear system,discrete-time generalized nonlinear system and continuous-time generalized nonlinear system,respectively.According to the actual situation,the non traditional optimal control methods are studied in the context of the double oligopoly market advertising competition as the background,and for different optimal control problem,the optimal control strategies are given.The main results are summarized as follows.(1)A new data-based method to testify system is investigated,and the controllability of the system is analyzed in the view of different systems.Firstly,we describe a linear discrete-time system in a general form,then the system type can be testified via the measured state data and the control input satisfying certain condition,including the normal system and the generalized system.Next,for the system in a general form,we derive a necessary and sufficient condition for the equivalent condition of complete controllability.To analyze the complete controllability,the controllability matrices are constructed by only the input and measured data without system identification.And data-based generalized Lyapunov stability criterion is given as well.Furthermore,for the generalized linear system,we also discuss the causality,causal controllability and R-controllability separately.Finally,a numerical example and a stock price example are used to show the effectiveness and feasibility of the proposed data-based method.(2)A new data-based linear quadratic regulator(LQR)optimal control method for linear generalized discrete-time system is discussed.In the case of model free,using only the measured input/output data,its optimal control is designed to minimize the given performance index by using the approximate dynamic programming(ADP)technique.We develop the value iteration(VI)algorithm by transforming the generalized system into a restricted equivalent form in order to obtain the equivalent value function which can be expressed by measured data.In consideration of two different cases,two improved VI algorithms are presented respectively so that fewer numbers of measured data are required at every iteration step.Therefore,the calculation cost can be reduced significantly in the process of solving the improved VI algorithms.Finally,some illustrative examples are used to show the effectiveness of the proposed data-based optimal control method.(3)A data-based method for the optimal control problem of advertising competition model of duopoly is discussed.We will use the ADP method mentioned aforementioned to deal with this optimal control problem for the discrete-time generalized nonlinear system.A discrete-time generalized nonlinear model of a double oligopoly advertising competition is presented,and the optimal control problem of the minimization of advertising investment and controlling cost is investigated.And the corresponding data-based VI algorithm is given.The optimal control strategies can be obtained by solving VI algorithms,which give the optimal control strategies for the enterprises in the competition.Finally,the illustrative examples are used to show the effectiveness of the proposed method.(4)Optimal investment strategy on advertisement in duopoly is discussed.With the double oligopoly market as the background,we propose a new market model characterized by nonlinear generalized nonlinear system with continuous inequality constraints.The aim is to present a new optimal advertising strategy of market share for new product in the commencing period of horizontal expansion.We apply a constraint transcription technique to covert these constraints into some canonical forms.In order to solve the investment problem numerically based on proposed model,the control parameterization technique together with the constraint transcription method is used by transforming the proposed problem into a sequence of optimal parameter selection problems.Finally,a practical example on beer sales is used to show the effectiveness of proposed model and we present the optimal advertising strategies corresponding to different competition situations.(5)The problem of the shortest time and optimal investment control in duopoly is investigated.We use a generalized nonlinear system with continuous inequality constraints and terminal state constraint to formulate this optimal duopoly competition problem,and this is a free terminal time optimal control problem.In order to obtain a feasible advertising strategy,it maps the varying partition time points into fixed time points in a new time horizon via introducing an additional control variable,so that the free terminal time problem is transformed into the fixed terminal time problem.By the control parameterization technique and the constraint transcription method,the optimal control problem is transformed into an optimal parameter selection problem.Particularly,in order to optimize the switching time for reducing the investment cost,a time scaling transform is developed with the partition time points regarded as decision variables,so that the better efficiency investment strategy can be obtained.Finally,a beer sales problem is studied to show the effectiveness of the proposed method,and we also compare the investment strategies with applying the time scaling transform and the coarsely equal partition over the time horizon.