Optimal Control Of Linear Systems Based On Dynamic Compensation

Linear quadratic optimal control theory has attracted considerable atten-tion and achieved plentiful results owing to its abundant contents and compre-hensive practical applications. Most of the results are developed based on statefeedback and static output feedback, while the result for dynamic compensa-tion is not well studied yet. Dynamic compensation plays an irreplaceable rolein analyzing and synthesizing for the systems, which possesses larger degree offreedom and makes the system achieve better performance. That representsa promising research area. This thesis studies the linear quadratic optimalproblem based on dynamic compensation for the linear system and rectan-gular system, which provides some new control algorithm to obtain the newresults. The main contents are as follows:(1) The linear quadratic optimal problem based on dynamic compensa-tion is considered for normal linear systems. According to Lyapunov stabilitytheory, the design method for dynamic compensator is given such that theclosed-loop system is asymptotically stable and its associated Lyapunov equa-tion has a positive-definite solution. Then the quadratic performance indexis derived to be an expression related to the positive-definite solution and theinitial value of the closed-loop system. In the light of the path-following al-gorithm, the optimal quadratic performance index and dynamic compensatorcan be obtained. Several numerical examples are provided to demonstrate theclosed-loop system can achieve approving performance based on compensatorwith proper dynamic order by comparing with the performance indices of theclosed-loop system under state feedback, static output feedback and dynamiccompensation with di?erent orders.(2) The linear quadratic optimal control problem for rectangular descrip-tor system by dynamic compensation is considered. First a dynamic compen-sator with a proper dynamic order is given such that the closed-loop system is regular, impulse-free and stable (it is called admissible), and its associatedmatrix inequality and Lyapunov equation have a solution. Then the quadraticperformance index is expressed as a simple form related to the public solutionand the initial value of the closed-loop system. Secondly, the correspondingsolving algorithm is designed in order to obtain the optimal quadratic perfor-mance index and the dynamic compensator. Finally, a numerical example isprovided to demonstrate the correctness of the proposed results.(3) The optimal disturbance rejection and optimal tracking control prob-lems for the linear system with disturbance signal are considered. Firstly, theoptimal disturbance rejection by dynamic compensation for linear system af-fected by external persistent disturbances with known dynamic characteristicsis studied. By means of combining the system with disturbance system, thisoptimal disturbance rejection problem can be transformed into the standardlinear quadratic optimal control problem without disturbance. The dynamiccompensator for the normal system and rectangular descriptor system are de-signed respectively. The numerical examples are provided to show that theproposed method can not only make the closed-loop system stable but rejectthe disturbance e?ectively. Secondly, the optimal tracking control by dynamiccompensation is considered. The given quadratic performance index containsthe error signal of the real output and the reference output signal. In a similarway, this optimal tracking control problem can be transformed into the stan-dard linear quadratic optimal control problem. The numerical examples areprovided to demonstrate that the proposed results can make the closed-loopsystems achieve better tracking performance.(4) The inverse linear quadratic optimal control problem based on dy-namic compensation is considered according to the positive real lemma. Firsta dynamic compensator with a proper dynamic order is given such that theclosed-loop system is asymptotically stable and Extended Strictly PositiveReal. In this case, a su?cient condition for the existence of the optimal solu-tion is presented. Then the weight matrices of the linear quadratic performance index are derived to be parameterized expressions. An algorithm to the min-imization problem with the Bilinear Matrix Inequality constraint is proposedbased on path-following algorithm, in which an optimal dynamic compensatorand the weight matrices of the linear quadratic performance index can be ob-tained. Then, the corresponding results for the rectangular descriptor systemcan be obtained in view of the Generalized Positive Real Lemma. Finally,several numerical examples are provided to demonstrate the e?ectiveness andfeasibility of the proposed results.