Assessment and Future Directions of Nonlinear Model Predictive ...

Hybrid MPC: Open-Minded but Not Easily Swayed 29it is in general not trivial to obtain functions V q , but it is true that a larger µwill yield more robustness to measurement error, or at least it will not degraderobustness. However, the larger µ, the longer it may take for the closed loop toconverge to the desired attractor. Also, a very large µ could make the systemincapable of adapting to large changes in conditions.5 Illustrations of Modified MPC Algorithms5.1 Pendulum Swing UpHere we consider the problem of swinging up a pendulum and stabilizing itsinverted equilibrium. The continuous-time model of the system after an inputfeedback transformation (ẋ = F (x, v) wherex ∈ R 2 , v ∈ R) and normalizationis ẋ 1 = x 2 ; ẋ 2 =sin(x 1 ) − cos(x 1 )v, wherex 1 is the angle of the pendulum (0at the upright position) and x 2 is the angular velocity. Following [20, Ex. 8.3],we design three different feedback laws v 1 (·), v 2 (·), v 3 (·) for the system. In [20],each of these control laws are activated in a different prespecified region of thestate space to perform the swing-up (the design purpose of v 1 (·), v 2 (·), v 3 (·) isto kick the system from the resting condition, to pump energy into the system,and to stabilize the inverted pendulum to the upright position, respectively).Given a sampling period T>0, for each u ∈{1, 2, 3} let x + = f(x, u) bethediscrete-time model of the closed loop ẋ = F (x, v u (x)) obtained via integrationover an interval of length T , i.e. f(x, u) =φ(T )whereφ(·) is the solution ofẋ = F (x, v u (x)) starting at φ(0) = x. WecannowuseMPCtodecidetheswing-up strategy.We construct the stage cost for standard MPC by adding the kinetic andpotential energy of the pendulum. We also include a term in the stage costthat penalizes the control law during the continuous-time horizon to avoid largecontrol efforts. The cost function is periodic in x 1 with period 2π and therefore,there exists a surface on the state space x 1 − x 2 where on one side the algorithmtries to reach the upright position rotating the pendulum clockwise and on theother side rotating the pendulum counterclockwise. For one such particular cost,the surface and two different trajectories in opposite directions starting close tothe surface are given in Fig. 3(a). As discussed in Section 2, the closed-loopsystem is vulnerable to small measurement noise in the vicinity of that surfacewhen T is small.The vulnerability to measurement noise mentioned above can be resolvedvia the approach discussed in Section 4.2. Despite the fact that x 1 = 2πk,k ∈{0, ±1, ±2, ...}, correspond to the same physical location, one can constructtwo stage costs, namely l q for q ∈ {1, 2}, that are not periodic in x 1 suchthat l 1 vanishes at x =(0, 0, 0, 0) and positive elsewhere and l 2 vanishes atx =(2π, 0, 0, 0) and positive elsewhere. By doing so we can attain a robustnessmargin that does not depend on the size of sampling period T but on µ only,which can be increased to enhance robustness. Fig. 3(b) shows the switching linesfor several values of µ for both possible switches (q =1→ 2, q =2→ 1). For aparticular value of µ, the robustness margin is related to the separation of the