Real-time dynamic optimization of controllable linear systems

A real-time optimization controller is developed to steer a linear time-varying control system to closed-loop trajectories that optimize a cost functional of interest. When advantage is taken of the differential flatness of the linear systems, a basis function approach is used to parameterize the open-loop trajectories of the system and to approximate the optimal solution of the finite time optimal control problem. An adaptive optimization method is used to formulate the real-time optimization scheme. The problem is posed as a real-time optimal trajectory generation problem in which the approximate optimal trajectories are computed using an extremum-seeking approach. The control algorithm provides tracking of the approximate optimal trajectories. Two optimal control problems are considered to demonstrate the application of the technique. It is shown that the technique can be successfully implemented in real time.