Assessment and Future Directions of Nonlinear Model Predictive ...

30 S.E. Tuna et al.21.521.5µ=51 → 2µ=101 → 21(x1 (0),x 2(0)) 0(x 1(T),x 2(T)) 2π1µ=201 → 20.50.5µ=31 → 2x 20−0.5−1−1.5−2(x 1(0),x 2(0)) 2π(x 1(T),x 2(T)) 0x 1−1 0 1 2 3 4 5 6 7x 20−0.5−1−1.5µ=202 → 1µ=32 → 1µ=102 → 1µ=52 → 1µ=31 → 2µ=32 → 1µ=52 → 1−20 1 2 3 4 5 6x 1(a) Vulnerability of standard MPC tomeasurement noise: trajectories, denotedby ∗, starting close to the thick line approachdifferent equilibrium points.(b) MPC with logic: switching linesfrom q =1→ 2andq =2→ 1 for variousµ.Fig. 3. Swing-up with standard MPC and MPC with memorylines. The margin is independent of the sampling time T as long as NT remainsconstant, N being the horizon for MPC with logic. The design of an MPC withmemory controller and the extension to the case of swinging up the pendulumon a cart follows directly, but due to space limitations we do not include themhere.5.2 Obstacle Avoidance with Constant Horizontal VelocityConsider a vehicle moving on the plane x + = x + δ, y + = y + uδ where δ>0and u ∈ {−1, 1} (note that this system can be thought of as sampling thesystem ẋ =1,ẏ = u). Suppose that the goal for the vehicle is to avoid hittingan obstacle defined by a block of unit height centered about the horizontalaxis at x = 0 (i.e. the vehicle must leave the region y ∈ [−0.5, 0.5]before x = 0). We design a controller using MPC with logic. Let q ∈{1, 2},l 1 ([x, y] T ,u)=l 2 ([x, −y] T ,u)=exp(y), and g(·) = 0. Since the costs are invarianton x and symmetric about the x axis, the decision lines defined by µturn out to be horizontal lines. Let the spacing between these lines be s(µ). Inthis case, s(µ) =ln(µ), since VN 1 ([x, y]T )=µVN 2 ([x, y]T )wheny = ln(µ)2andVN 2 ([x, y]T )=µVN 1 ([x, y]T )wheny = − ln(µ)2for any N.Note that when µ = 1 (or s(µ) = 0) MPC with logic is equivalent tothe standard MPC algorithm implemented using the stage cost l([x, y] T ,u)=min{l 1 ([x, y] T ,u),l 2 ([x, y] T ,u)}. Asµ is increased, the spacing s(µ) increases.Table 1 shows the average number of switches and the total number of crashes for50, 000 runs of the system. The initial conditions are set to be x(0) = −1.5 andy(0) normally distributed (though kept within (−1, 1)) around y = 0. The noiseis uniformly distributed in [−0.8, 0.8]. The key variables of comparison are thespacing of the decision lines s(µ) and the sampling time δ. With the increased