Approximate explicit NMPC with guaranteed stability ensured by a simple auxiliary controller

We investigate two methods for the calculation of suboptimal explicit solutions to nonlinear MPC problems and show that these two methods can be combined for guaranteed stability and good performance. The first method calculates an explicit piecewise constant (PWC) control law and a corresponding positively invariant set that is represented by a hyperrectangular partition in the state space. The explicit PWC law provides a suboptimal solution to the nonlinear MPC problem, but asymptotical stability of the closed-loop system can be guaranteed. A second explicit controller is constructed by solving the nonlinear MPC problem for a representative set of initial conditions and interpolating these pointwise solutions nonlinearly. Note that the PWC law provides feasible initial solutions to the nonlinear MPC problem and therefore can be used to speed up the construction of the second controller significantly. The PWC control law and the explicit nonlinear control law can be combined for guaranteed asymptotical stability (by virtue of the PWC control law) and good performance (from the nonlinear control law). We claim the hybrid controller is an interesting alternative, because its domain of attraction is typically larger than that of the nonlinear controller alone.