Assessment and Future Directions of Nonlinear Model Predictive ...

MPC for Stochastic Systems 265Fig. 1. Feasible initial condition sets for stochastic MPC (p 2 =0.85) with varying N.Dashed line: feasible set of robust MPC based on 85% confidence level for N =4.First consider the plant model withA =1.04 −0.620.62 1.04¯B = 0 2B 1 = −0.120.02B 2 = 0.04−0.06c T 1 =0−4.4c T 2 = 32.3U =1,Y 2 =1,andp 1 = p 2 =0.85. The offline computation for stochastic MPCinvolves maximizing a low-complexity polytopic set Ω subject to Pr(Φx ∈ Ω) ≥p 2 for all x ∈ Ω; for this example the maximal Ω has an area of 0.055.An alternative approach to MPC is to determine bounds on plant parameterscorresponding to a confidence level of, say, p 2 by settingB(k) = ¯B +L∑q i (k)B i , |q(k)| ≤N −1 (p 2 ) (27)i=1in (3), and then to implement a robust MPC law based on this approximateplant model. For a confidence level of p 2 =0.85, the maximal low-complexityset Ω ′ , which is robustly invariant for bounded parameter variations (27), issimilar in size (area = 0.048) to Ω. The similarity is to be expected since theassumption of bounded uncertainty implies that the probability that Φx ∈ Ω ′under the actual plant dynamics for any x ∈ Ω ′ is p 2 .A robust (min-max) MPC law employing open-loop predictions based on theparameter bounds of (27) is, however, significantly more conservative than thestochastic MPC law of (22) for the same confidence level. This can be seen inFig. 1, which compares the feasible sets for the two control laws for p 2 =0.85(the feasible set for robust MPC decreases with increasing N for N>4 since theplant is open-loop unstable). Closed-loop performance is also significantly worse:the closed-loop cost for robust MPC (based on the parameter bounds (27) withp 2 =0.85), averaged over 10 initial conditions and 200 uncertainty realizations,is 63% greater than that for stochastic MPC (with p 1 = p 2 =0.85). Figures 2and 3 compare the closed-loop responses for a single initial condition and 20