Assessment and Future Directions of Nonlinear Model Predictive ...

On the Computation of Robust Control Invariant Sets 133Based on these definitions and properties, the maximal robust control invariantset can be obtained by means of the following algorithm [8]:Algorithm 1(i) Set the initial region C 0 equal to X.(ii) C k+1 = Q(C k ).(iii) If C k+1 = C k then C k = C ∞ .Stop.Else,setk = k +1andreturntostep(ii).Note that the evolution of any initial condition belonging to set C k can be robustlymaintained in X at least k sample times. Therefore, C ∞ = lim C k constitutesthe set of initial condition for which the system is robustly controllable ink→∞an admissible way. That is, C ∞ is the maximal robust control invariant set.Suppose that algorithm 1 converges to C ∞ in k d steps. Then, applying property2 in a recursive way it is possible to state that C kd = C ∞ can be representedby means of the union of r k dconvex polyhedra. This worst-case estimation ofthe number of convex polyhedra required to represent C ∞ clearly shows that theexact computation of C ∞ for a piecewise-affine system might not be possible inthe general case: the number of sets required to represent the maximal robustcontrol invariant set grows in an exponential way with the number of iterationsof algorithm 1. Even in the case that algorithm 1 obtains (theoretically) themaximal robust control invariant set in a finite number of steps, the complexityof the representation might make it impossible to run the algorithm beyond areduced number of steps (normally insufficient to attain the greatest domain ofattraction). Therefore, it is compulsory to consider approximated approaches tothe computation of C ∞ .In this paper we propose an algorithm (based on convex outer (and inner)approximations of the one step set) that can be used to compute a convex robustcontrol invariant set for the piecewise affine system.3 Outer Bound of the Maximal Robust Control InvariantSetOne of the objectives of this paper consists in providing a procedure to obtaina convex outer approximation of C ∞ for a piecewise affine system. This outerbound has a number of practical and relevant applications:(i) It captures the geometry of C ∞ and makes the computation of a robustcontrol invariant set for the system easier (this use is explored in section4). Moreover, it can be used as the initial set in algorithm 1. If the outerbound is a good approximation, then algorithm 1 might require an (implementable)reduced number of iterations.(ii) The constraints that define the outer bound can be included as hard constraintsin a hybrid MPC scheme. Moreover, the inclusion of the aforementionedconstraints can be used to improve the convex relaxations of thenonlinear optimization problems that appear in the context of hybrid MPC.