Assessment and Future Directions of Nonlinear Model Predictive ...

58 M.V. Kothare and Z. Wan120100#0#180h 2(cm)60#240#32010*00 1 5 10 15 20 25 30h (cm) 1Fig.(1. Phase plots of the regulations from x(0) = (7.5, 15) T to the equilibrium(19.753, 100) T , 1.7710 ) . First formulation: solid line - state; dotted line - estimatedstate. Second formulation: dashed line - state; dashed dotted line - estimated state.where ρ = 0.001kg/cm 3 ,g= 980cm/s 2 ,S 1 = 2500cm 2 ,A 1 = 9cm 2 , S 2 =1600cm 2 ,A 2 =4cm 2 ,1cm≤ h 1 ≤ 50cm, 10 cm≤ h 2 ≤ 120cm, and 0 ≤ u ≤2.5kg/s. The sampling time is 0.5 sec. Let Q = diag(0, 1), R=0.01 and α =0.998 for all the controller designs, and ρ =0.99 for all the observer designs. Letθ =0.9, T=10andδ =10 −5 for all switches. [ ]7.5Consider the regulation from an initial state h(0) = to the equilibrium(h (0) ,u (0) )=, 1.7710 . Initial estimated state is15([ ] )19.753100ĥ(0) = h(3) .Figure 1 shows four regions of stability defined by the optimization (7) for fourequilibrium points. There are two controller formulations implemented withineach stability region. The first formulation is to represent the local nonlinearityas a LTV model and parameterize the infinite control ([ horizon ] ) in terms ofheq1a linear feedback law. Consider an equilibrium point ,u eq . The localnonlinearity within a neighborhood (Π h ,Π u ) is expressed as a polytopich eq2