Output Tracking Control Of Linear Singularly Perturbed Systems

In the process of solving practical engineering problems,fast and slow changes usually exist in some variables.The difference in rate of this change always leads to high-dimensional and ill-conditioned solutions problems in the system modeling analysis and controller design,which is called singular perturbation problem and singularly perturbed system.Furthermore,with the rapid development of science and technology,some factors,such as time delay and uncertain signals,will inevitably occur in actual engineering systems.Great achievements about output tracking control in linear systems have been obtained,,but the research results of singularly perturbed system was lacking.The theoretical results of linear system can't be used directly,and the new methods should be studied.,Therefore,in this paper,the output tracking control of linear singularly perturbed systems has been investigated.The contents are as follows:(1)The optimized control of output tracking for singularly perturbed system is studied.First,select an infinite time domain quadratic performance index and link original system with external system to gain a new standard LQR(linear quadratic regulator)problem.The necessary and sufficient condition for the existence of the optimized control of output tracking controller is given in terms of matrix inequalities,such that the tracking system achieves the asymptotic tracking and the minimum value of quadratic performance index can be obtained.Furthermore,the controller design method is presented in the form of LMI for the case that the controller has an approximate solution of the generalized Riccati equation.Finally,one numerical example is given to illustrate the correctness and feasibility of the proposed results.(2)The problem of unified optimization control of output tracking for singularly perturbed systems is considered.First,the original system is transformed a linearsystem and link with external system to get an optimal adjustment problem.Based on the optimal regulation theory,the performance index is given to obtain the form of the optimal controller.Then,an optimal output tracking controller is given based on the method of minimum value principle,such that the original system achieves asymptotically stable and asymptotic tracking of the output system and the minimum value of quadratic performance index can be obtained.Furthermore,based on Nash game theory,an algorithm to optimize simultaneously which transfers multi-objective problem into a single objective problem as well we determine the objective weights.Finally,comparing simulation by output tracking optimization,one numerical example is given to illustrate the correctness and feasibility of the proposed results.(3)The optimized control of output tracking control for linear singularly perturbed systems with uncertain time-delay is proposed.First,link original system with external system and give the range of uncertain signals and time delays,to list three necessary equations and derive those into integral form with upper bound based on Lyapunov stability theorem.The integrals are split into matrix inequalities and incorporated into performance indicators in order to transform the optimal output tracking problem into a convex optimization problem.Then,derive the inequality condition of the memory state feedback output tracking controller,which makes the system meet the asymptotic stability and the performance index.What's more,mathematical inequalities and contractual transformations are performed on the inequality.,and solve the feasible solution based on the(LMI)toolbox.Finally,one numerical example is given to illustrate the correctness and feasibility of the proposed results.