Assessment and Future Directions of Nonlinear Model Predictive ...

190 A. Grancharova, T.A. Johansen, and P. Tøndelx − space0.35u(t)0.70.30.60.250.50.2x 20.40.150.30.10.20.050.1000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9x 1−0.050 5 10 15 20 25 30tFig. 1. State space partition of the approximate explicit NMPC (left) and the controlinput for x(0) = [0.1 0.05] T (right). The solid curves are with the approximate explicitNMPC and the dotted curves are with the exact NMPC.0.6x 1(t)0.7x 2(t)0.550.60.50.450.50.40.40.350.30.30.250.20.20.10.150.10 5 10 15 20 25 30t00 5 10 15 20 25 30tFig. 2. State variables x 1 and x 2: the solid curve is with the approximate explicitNMPC and the dotted curve is with the exact NMPCpartition is performed on a 1.8 GHz AMD Athlon(tm)XP 2200+, where theCPU time needed for solving the NLP (10)–(11) at a single point in the statespace is about 3 sec and the CPU time necessary to solve the NLP (14)–(15) fora single region is about 210 sec.The performance of the closed-loop system is simulated for initial conditionx(0) = [0.1 0.05] T and with sampling time T s =0.02. Euler integration withstep size T s is applied to solve the ordinary differential equations (18)–(19).The resulting closed-loop response is depicted in the state space (Fig.1 (left)),as well as trajectories in time (Fig.1 (right) and Fig.2). In Fig.1 and Fig.2 theexact NMPC solution is also shown, which at each time step is determined bycomparing the local minima of the cost function (22) corresponding to severalinitial guesses for the optimization variables.