Time-optimal tracking for constrained linear systems with bounded disturbance

The problem of tracking an exogenous signal in the presence of bounded disturbances is considered. A version of dynamic programming, which computes level sets of the value function rather than the value function itself, is used to design robust nonlinear controllers for linear, discrete-time, dynamical systems with hard constraints on controls and states, an external bounded disturbance, and an exogenous input to be tracked. The controller stabilizes the system and regulates to a control invariant set in minimum time, and thereafter maintains the state in this set despite the disturbance; the tracking error is bounded in this set and is small if the disturbance is small. Two nonlinear controllers which utilize the level sets of the value function, are described. The first requires the controller to solve, online, a modest linear program whose dimension is approximately the same as that of the control variable. The second decomposes each level set into a set of simplices; a piecewise linear control law, affine in each simplex, is then constructed