Assessment and Future Directions of Nonlinear Model Predictive ...

NMPC for Complex Stochastic Systems 271N∑ (u(ω, x) =− ξTk+1 Qξ k+1 + ηk T )Rη kk=0(2)with x =(ξ 1 ,...,ξ N+1 ) being the state sequence and ω =(η 0 ,...,η N )beingthe control sequence. 1 Since we will abandon the luxury of linear models andGaussian distributions, there will be no analytical advantage in retaining thequadratic structure, and we will use more general forms of utility function. Inmost applications the distribution P ω (x) will be very complicated, and the onlyway to elicit it will be to draw samples from a simulator based on the model.The only way to estimate the integral in (1) will be by Monte Carlo methods.Thus the only requirement on the form of u(ω, x) will be sufficient regularity forsuch estimation to be possible.This allows many possibilities. For example, if a single outcome x 0 is desired,then setting u(ω, x) =δ(x − x 0 ) (the Dirac impulse) gives E X u(ω, x) =P ω (x 0 ),and hence ω ∗ is a decision which maximises the likelihood of outcome x 0 .In the sequel, and in our related publication [13], we use the term performanceinstead of utility.3 Satisfying ConstraintsAn important consideration for us is that we wish to satisfy constraints, with ahigher priority than maximising the expected performance. That is, we wish tosolve problems of the formω ∗ =argmax E Xperf(ω, x) subject to Pr{x ∗ ∈ X f } > 1 − ɛ, (0