Assessment and Future Directions of Nonlinear Model Predictive ...

A Low Dimensional Contractive NMPC Scheme 533while the values of the parameters used in the controller definition are thefollowing :(τ s ,N,t r ,α,β)=(0.4, 8, 0.2, 0.01, 10) ; K pre =(2.5, 10) ; F max ∈{1, 2}The behavior of the closed loop systems under the contractive receding horizoncontrol is depicted on figure 2. Two scenarios are presented for different valuesof the input saturation levels F max =1andF max = 2. The computation timesare also given vs the sampling period (the computations have been performed ona 1.3 GHz PC-Pentium III). Note that these computation times never exceeded0.1 s. This has to be compared to the sampling period τ s =0.4 s. This suggeststhat the proposed receding horizon feedback can be implementable in real timecontext.4.2 Swing Up and Stabilization of a Double Inverted Pendulum onCart: A Hybrid SchemeThe system is depicted on figure 1 together with the definition of some auxiliaryvariables. The numerical values are given by :(m 1 ,m 2 ,m,l 1 ,l 2 ,J 1 ,J 2 )=(0.3, 0.2, 5.0, 0.3, 0.2, 1.3 × 10 −2 , 4 × 10 −3 ).The system equations are given by [1] :h 1¨r + h 2 ¨θ1 cos θ 1 + h 3 ¨θ2 cos θ 2 = h 2 ˙θ2 1 sin θ 1 + h 3 ˙θ2 2 sin θ 2 + Fh 2¨r cos θ 1 + h 4 ¨θ1 + h 5 ¨θ2 cos(θ 1 − θ 2 )=h 7 sin θ 1 − h 5 ˙θ2 2 sin(θ 1 − θ 2 )h 3¨r cos θ 2 + h 5 ¨θ1 cos(θ 1 − θ 2 )+h 6 ¨θ2 = h 5 ˙θ2 1 sin(θ 1 − θ 2 )+h 8 sin θ 2Again, a pre-compensation is done using the change in control variable givenby :(ṙF = −K pre · + u, (33)r)while a two-dimensional control parametrization is needed this time :P =[p min ,p max ] 2 ⊂ R 2 ; u i (p) =p 1 · e λ1ti + p 2 e −λ2ti ; t i = (i − 1)τ sN (34)The weighting function h(·) invoked in the general formulation (24) is here takenas followsh(x) = h 42 ˙θ 2 1 + h 62 ˙θ 2 2 + h 5 ˙θ1 ˙θ2 cos(θ 1 − θ 2 )+h 7[1 − cos(θ1 ) ] + h 8[1 − cos(θ2 ) ] ++ h 1[r 2 +ṙ 2] .This is inspired by the expression of the total energy given in [1]. The constrainedopen-loop optimal control problem is then given by (24) in which the admissibledomain of the parameter vector is [p min (x),p max (x)] 2 where :