Trajectory Following Methods in Control System Design

A trajectory following method for solving optimization problems is based on the idea of solving ordinary differential equations whose equilibrium solutions satisfy the necessary conditions for a minimum. The method is “trajectory following” in the sense that an initial guess for the solution is moved along a trajectory generated by the differential equations to a solution point. With the advent of fast computers and efficient integration solvers, this relatively old idea is now an attractive alternative to traditional optimization methods. One area in control theory that the trajectory following method is particularly useful is in the design of Lyapunov optimizing feedback controls. Such a controller is one in which the control at each instant in time either minimizes the “steepest decent” or “quickest decent” as determined from the system dynamics and an appropriate (Lyapunovlike) decent function. The method is particularly appealing in that it allows the Lyapunov control system design method to be used “on-line.” That is, the controller is part of a normal feedback loop with no off-line calculations required. This approach eliminates the need to solve two-point boundary value problems associated with classical optimal control approaches. We demonstrate the method with two examples. The first example is a nonlinear system with no constraints on the control and the second example is a linear system subject to bounded control.