Assessment and Future Directions of Nonlinear Model Predictive ...

274 J.M. Maciejowski, A. Lecchini Visintini, and J. LygerosIfLet h(ω, x 1 ,x 2 ,...,x J ) denote the joint distribution of (Ω,X 1 ,X 2 ,X 3 ,...,X J ).h(ω, x 1 ,x 2 ,...,x J ) ∝J∏u(ω, x j )p Ω (x j ) (10)then the marginal distribution of Ω, also denoted by h(ω) for simplicity, satisfies[∫Jh(ω) ∝ u(ω, x)p Ω (x)dx]=[E X u(ω, x)] J (11)which is the property we specified as property 3 above.The random accept/reject mechanism, governed by the probability ρ, isaparticular formulation of the Metropolis-Hastings algorithm, which is a generalalgorithm for generating a homogeneous Markov chain [19] . For a desired (target)distribution given by h(ω, x 1 ,x 2 ,...,x J ) and proposal distribution givenbyg(ω) ∏ p Ω (x j )jthe acceptance probability for the standard Metropolis-Hastings algorithm is{min 1, h(˜ω, ˜x 1, ˜x 2 ,...,˜x J ) g(ω) ∏ j p }Ω(x j )h(ω, x 1 ,x 2 ,...,x J ) g(˜ω) ∏ j p ˜Ω(˜x j )By inserting (10) in this expression one obtains the probability ρ as definedin the algorithm. Under minimal assumptions, the Markov Chain generated bythe Ω(k) is uniformly ergodic with equilibrium distribution h(ω) givenby(11).Therefore, after a burn in period, the extractions Ω(k) accepted by the algorithmwill concentrate around the modes of h(ω), which, by (11) coincide with theoptimal points of U(ω). Results that characterize the convergence rate to theequilibrium distribution can be found, for example, in [19].j=15 Using MCMC for NMPCThe original successful idea of MPC was to use general-purpose optimisationalgorithms to solve a sequence of open-loop problems, with feedback being introducedby resetting the initial conditions for each problem on the basis ofupdated measurements. Since the algorithm which we have described is an optimisationalgorithm, we can use it in exactly this way to implement nonlinearMPC. But it is desirable to modify it in some ways, so as to make it more suitablefor solving MPC problems. Increased suitability has at least two aspects:1. Addressing the theoretical concerns that have been raised for MPC, in particularstability, feasibility, and robust versions of these.2. Reducing the computational complexity, to allow a wider range of problemsto be tackled by the MCMC approach.