SYMPLECTIC TRANSFORMATION BASED ANALYTICAL AND NUMERICAL METHODS FOR LINEAR QUADRATIC CONTROL WITH HARD TERMINAL CONSTRAINTS

Feedback gain and feedforward input of conventional linear quadratic (LQ) hard terminal controllers tend to infinity at terminal time, so that the conventional controllers have to go open-loop for a short interval before the final time. This short interval is called blind time. To avoid the terminal infinite feedback control gain and feedforward input, new optimal control laws are proposed for hard terminal control of linear time-varying systems. Furthermore, a structure-preserving numerical method is also presented to evaluate the time-varying optimal feedback and feedforward control gains and corresponding optimal trajectory. The analytical and numerical methods being developed here are based on the application of symplectic (canonical) transformation and generating functions of Hamiltonian systems. Different from the existing generating function method for optimal control, the first type of generating function plays a key role in solving the associated Hamiltonian two-point boundary-value problem (TPBVP), while the second generating function is employed to recover the first type. This note uses the second and third types of generating functions to find novel optimal control laws by solving the Hamiltonian TPBVP, which eliminates the infinite control gains of conventional optimal control laws near terminal time. Since the optimal trajectory of the closed-loop system is a solution of the Hamiltonian TPBVP, by using symplecticity of the solution operator of the linear Hamiltonian system, this paper also derives a structure-preserving matrix recursive algorithm for the computation of time-varying optimal control gains and the systems optimal trajectories. Numerical simulations show that this structure-preserving algorithm gives accurate results for relative large discrete steps and keeps geometric properties of the solutions.