Efficient on-line computation of constrained optimal control

For discrete-time linear time-invariant systems with constraints on inputs and outputs, the constrained finite-time optimal controller can be obtained explicitly as a piecewise-affine function of the initial state via multi-parametric programming. By exploiting the properties of the value function, we present two algorithms that efficiently perform the online evaluation of the explicit optimal control law both in terms of storage demands and computational complexity. The algorithms are particularly effective when used for model-predictive control (MPC) where an open-loop constrained finite-time optimal control problem has to be solved at each sampling time.