State-Constrained Sub-Optimal Tracking Controller for Continuous-Time Linear Time-Invariant (CT-LTI) Systems and Its Application for DC Motor Servo Systems

In this paper, we propose an analytic solution of state-constrained optimal tracking control problems for continuous-time linear time-invariant (CT-LTI) systems that are based on model-based prediction, the quadratic penalty function, and the variational approach. Model-based prediction is a concept taken from model-predictive control (MPC) and this is essential to change the direction of calculation for the solution from backward to forward. The quadratic penalty function plays an important role in deriving the analytic solution since it can transform the problem into a form that does not have inequality constraints. For computational convenience, we also propose a sub-optimal controller derived from the steady-state approximation of the analytic solution and show that the proposed controller satisfies the Lyapunov stability. The main advantage of the proposed controller is that it can be implemented in real time with a lower computational load compared to the implicit MPC. Finally, the simulation results for a DC motor servo system are shown and compared with the results of the direct multi-shooting method and the implicit MPC to verify the effectiveness of the proposed controller.