Assessment and Future Directions of Nonlinear Model Predictive ...

532 M. AlamirFig. 2. Stabilization of the inverted pendulum for two different saturation levels:F max = 1.0 N (dotted thin line) / F max = 2.0 N (continuous thick line). Initialcondition: Downward equilibrium x =(π,0, 0, 0) T .Define the weighting function h(x) by:h(x)= 1 [ ˙θ2 +βr 2 +ṙ 2] +[1− cos(θ)] 2 = 1 []x 2 322 + βx2 2 + x2 4 +[1− cos(x 1 )] 2 (28)In order to explicitly handle the saturation constraint on the force, the constrainthas to be expressed in term of the new control variable u, namely:( )x2|−K pre + u| ≤Fx max . (29)4Using the expression of the control parametrization (27) this yields the followingstate dependent definition of the parameter bounds p min and p max :p min (x) =−F max + K pre1 x 2 + K pre2 x 4 (30)p max (x) =+F max + K pre1 x 2 + K pre2 x 4 (31)These bounds are used in the definition of the optimization problem P ε α(x) :P ε α(x) : min(q,p)∈{1,...,N}×[p min(x) ,p max(x)]J(x, q, p) =h(qτ s ,x,p)+ α N · min{ε, h∞ q(·,x,p)}. (32)Let ˆp(x) and ˆq(x) be optimal solutions of Pα ε (x). This defines the feedbackK RH (x) = u 1 (ˆp(x)) according to the receding horizon principle. The valuesof the system’s parameters used in the forthcoming simulations are given by :(m, M, L, k x ,k θ ,I)=(0.3, 5.0, 0.3, 0.001, 0.001, 0.009)