Researches On The Optimal Control Problems For Classes Of Linear Singular Systems

In this paper, the optimal tracking control problems of a class of both lineardiscrete systems with time-delays and discrete singular systems with time-delays areresearched. In addition, the variable structure control problems under variableswitched surfaces are discussed. The theory of least-square solutions based on thegeneralized inverse matrix and the variable structure control method with variableswitched surfaces are the main research methods. The corresponding simulationexamples are given in order to illustrate the feasibility and effectiveness of someresults in the paper.The main contents summed up as follows.1. The definition and development of singular systems are reviewed, variablestructure controls and tracking controls are reduced and their practical application ofresearch and development are also reviewed. And then the main research works aresummarized.2. Theory of least squares solution is used in this chapter to discuss the optimaltracking control of linear discrete time-delay systems. The method of solvingdifferential equations according to generalized inverse matrix is utilized. The optimalcontrol sequence and the minimum-energy optimal control sequence are given for thesystems, and the minimum errors are estimated. Lastly, an example is given toillustrate the applicability and feasibility for some results in the chapter2.3. Theory of least squares solution based on the generalized inverse is used inthis chapter to discuss the optimal tracking problems of the discrete singular systemswith time-delays. According to quality of {1,3} and M-P generalized inverse, theoptimal and minimum-energy control sequence are given for the system. And then thetracking errors are estimated under the optimal control. At last, an example is given to illustrate the applicability and feasibility of the researched results.4. The problem of minimum time sliding mode variable structure control isesearched for the continuous singular linear systems under variable reaching surfaces.Firstly, a sufficient condition for asymptotic stability of the two switching subsystems’sliding movements is deduced under integral-type sliding surfaces. Secondly, anew-type reaching law is designed, that is, surfaces (reaching surfaces) where thestate trajectories of subsystems are designed to tend to the sliding surfaces at a certainrate and coincide with the sliding surfaces at last. Thirdly, an estimated value of theminimum time is calculated for subsystem reaching surface tending to the sliding one,so that the time is guaranteed as less as possible for state trajectories reached to thesliding mode. The switching sequence is designed to make the two subsystems reachhierarchically to the sliding mode and then converge in a neighborhood of the origin.Finally, the effectiveness and the feasibility are illustrated for the proposedapproached in this chapter by a simulation example.5. All the results of this paper are summed up. And the further study plan and theoutlook are given.