Assessment and Future Directions of Nonlinear Model Predictive ...

534 M. AlamirFig. 3. Closed loop behavior of the double inverted pendulum system underthe hybrid controller given by (35) with the design parameters valuesgiven by (τ s,N,R,λ 1,λ 2,η) = (0.3, 10, 100, 100, 20, 1), L = (360, 30), and Q =diag(1, 1, 10 4 , 1, 1, 1) for two different force saturation levels: F max =20N (continuousthick line) / F max =10N (dotted thin line). The maximum number of functionevaluations parameter in the optimization code has been set to 20 in order to make thesolution real-time implementable. This may explain the behavior of the optimal costfor the lower values that is not monotonically decreasing. Initial condition: downwardequilibrium.p min (x) := 1 2[( ) ṙ ]−F max + K pre ; prmax (x) := 1 ( ṙ ]+F max + K pre .2[r)that clearly enables to meet the requirement |F (t)| ≤ F max given theparametrization (34) being used. Again, denoting by (ˆq(x), ˆp(x)) the optimalsolutions, the nonlinear receding-horizon control is given by :u(kτ s + t) =K RH (x(kτ s )) := u 1 (ˆp(x(kτ s ))) ; t ∈ [0,τ s [.Since a hybrid scheme is used here, the local controller has to be defined. This isdone by using an LQR-based method that enables a feedback gain L to be computed.Hence, the local controller is given by K L (x) =−L · (x m 1 xm 2 x ) T3 ... x 6where x m 1 and xm 2 are the minimum norm angles that are equal (modulo 2π)to θ 1 and θ 2 respectively while the gain matrix L ∈ R 1×6 satisfies the followingRiccati equation for some positive definite matrices S and Q :A T d SA d − S − (A T d SB d )(R + B T d SB T d )(B T d SA d )+Q =0.