Selecting and Simulation of Weighting Matrices for LQR Control System with Input Delays

In this paper, how to select weighting matrices for linear quadratic regulation (LQR)control system with input delays is researched. By a number of simulation examples, some unprecedented and useful conclusions are obtained. The simulation results indicate that the two weighting matrixes Q and R for LQR time-delay systems are interactional. But the effect of the state weighting matrix Q on the system dynamic performance is more significant than the control weighting matrix R; and the control effect is good if R is selected for a unit matrix. So R is usually taken as a unit matrix fixedly, then the element values of Q are changed according to the control requirements. Both control effect and energy consumption should be considered carefully. In addition, the element values of two weighting matrices Q and R should not be changed simultaneously in proportion, because if so, both the optimal controller and the state response will remain unchanged. When two weighting matrices Q,R are fixed and delays of the control system increase, the energy consumption of the system will certainly increase. So in order to achieve the same good control effect, more energy must be added when delays of the control system increase. These comprehensive conclusions greatly enrich theories and methods of selecting the two weighting matrices for LQR control system with time-delays, and then give guidance in practical applications.