Assessment and Future Directions of Nonlinear Model Predictive ...

270 J.M. Maciejowski, A. Lecchini Visintini, and J. Lygeros3. Constraints are respected in the sense that the probability of constraintviolations is kept below some user-defined threshold. This threshold can bearbitrarily small (at the expense of computational complexity) but hardconstraints are not enforced, strictly speaking.4. Computational complexity is very high, so that the time required to finda solution at each step may be of the order of several hours. In section 6we shall identify some applications for which this is not a problem. Theseapplications arise in diverse areas such as batch process control, financialinvestment policy, and environment management.Our method relies on stochastic optimisation. It makes use of a Monte CarloMarkov Chain technique for optimal decision-taking which was pioneered byMüller [16] and rediscovered independently by Doucet et al [8]. Our contributionto the method is a way of incorporating constraints into the problem formulation,which is presented in section 3. There is no requirement for convexity of theobjective function, convergence (in probability) resulting from the very mildconditions that are required for the convergence of a homogeneous Markov chain.The technical details of the method are given in section 4.Kouvaritakis et al. have previously proposed a stochastic MPC formulationfor solving a problem in the area of sustainable development policy [12]. Theirmethod involves the solution of a convex optimisation problem, with constraintsin the form of thresholds for the probabilities of obtaining certain outcomes — asin our approach. The convexity requirement imposes limitations on the model,objective function, and constraints. Our approach is free of these limitations,but at the expense of greater computational complexity.2 Maximising Expected Utility or PerformanceThe notion of rational decision-making under uncertainty as the maximisationof the statistical expectation of a utility function has a long history, dating backto Bernoulli [4] and Bentham [3]. Although it has gone into and out of fashionas a foundational axiom for economics, its place has been firmly establishedsince its use (and formalisation) by von Neumann and Morgenstern in GameTheory [21], and extended to other applications of decision theory by variousauthors [1, 11, 18, 20].Suppose that a set of possible decisions Ω is given, from which one decisionω ∈ Ω must be chosen. For each such ω, let the outcome be a random variableX, and let its probability distribution be P ω (x). Suppose that to each decisionoutcomepair (ω, x) we can attach a real-valued utility function u(ω, x), suchthat the pair (ω 1 ,x 1 ) is preferred to the pair (ω 2 ,x 2 )ifu(ω 1 ,x 1 ) >u(ω 2 ,x 2 ).Then the principle of maximising expected utility states that one should choosethe decision optimally as follows:∫ω ∗ =argmaxωE Xu(ω, x) = arg maxωu(ω, x)dP ω (x) (1)In familiar finite-horizon LQG control theory, the utility is the negative cost