Solving the Linearly Quadratic Optimal Control Problems of Elastic Structure by the Symplectic Group

The research and application of active structure control have been studied for almost thirty years. For the civil engineering, the active control can reduce the damage from the dynamical response of vehicles, wind, earthquake, and increase the resistance of the earthquake and disaster. It is an effective way for the structure to resist from the earthquake and decrease the damage from the disaster. Moreover, the active control of civil engineering can be divided into three parts: base isolation systems, passive energy dissipation systems, active, semi-active and intelligent control systems. In the classical linear quadratic optimal control (LQR), it is unavoidable to solve the Riccati differential equations. Therefore, the main purpose of this paper is to use the numerical method with the properties of symplectic group and avoid solving the Riccati differential equations and investigate the classical linear quadratic optimal control problems. We employ the group preserving schemes (GPS) and symplectic group shooting method in this paper, which are on the basis of the Lie group. Using these two methods, we can solve the active control problems quickly, economically and accurately. Besides, we design several examples and utilize the programming language, FORTRAN, to analyze them. Then, we will show the numerical results. Finally, the conclusions and the future work are addressed.