Assessment and Future Directions of Nonlinear Model Predictive ...

y i (k) =∑n um=1MPC for Stochastic Systems 257g T im(k)ũ m (k − 1), ũ m (k − 1) = [u m (k − n) ...u m (k − 1)] T (1)where u m (k), m =1,...,n u , y i (k), i =1,...,n y , are input and output variablesrespectively, and the plant parameters g im (k) are Gaussian random variables.For convenience we consider the case of two outputs (n y =2):y 1 is taken to beprimary (in that a probabilistic measure of performance on it is to be optimized)whereas y 2 is subject to probabilistic performance constraints and is referred toas secondary.As a result of the linear dependence of the model (1) on uncertain plant parameters,the prediction of y i (k + j) madeattimek (denoted y i (k + j|k)) isnormally distributed. Therefore bounds on y i (k + j|k) thataresatisfiedwithaspecified probability p can be formulated as convex (second-order conic) constraintson the predicted future input sequence. Bounds of this kind are usedin [9] to derive a probabilistic objective function and constraints for MPC. Theseare combined with a terminal constraint that forces predictions to reach a precomputedsteady-state at the end of an N-step prediction horizon to define a stablereceding horizon control law. Subsequent work has applied this methodologyto a sustainable development problem using linear time-varying MA models [13].Though often convenient in practice, MA models are non-parsimonious, andan alternative considered in [12] is given by the state space model:x(k +1)=Ax(k)+Bu(k), y i (k) =c T i (k) x(k), i =1, 2 (2)where x(k) ∈ R n is the state (assumed to be measured at time k), u(k) ∈ R nuis the input, and A, B are known constant matrices. The output maps c i (k) ∈R n , i =1, 2 are assumed to be normally distributed: c i (k) ∼N(¯c i ,Θ c,i ), with{c i (k),c i (j)} independent for k ≠ j. The stability constraints of [9] are relaxedin [12], which employs less restrictive inequality constraints on the N step-aheadpredicted state.This paper considers a generalization of the model class in order to handle thecase that the future plant state is a random variable. For simplicity we restrictattention to the case of uncertainty in the input map:x(k+1) = Ax(k)+B(k)u(k), B(k) = ¯B+L∑q r (k)B r , y i (k) =c T i x(k), i=1, 2(3)where A, ¯B, Bi , c i are known and q(k) = [q 1 (k) ··· q L (k)] T are Gaussianparameters. We assume that q(k) ∼N(0,I) since it is always possible to definethe model realization (A, B, C) so that the elements of q(k) are uncorrelated,and that {q(k),q(j)} are independent for k ≠ j. Correlation between modelparameters at different times could be handled by the paper’s approach, butthe latter assumption simplifies the expressions for the predicted covariances insection 5 below. The state x(k) is assumed to be measured at time k. The paperfocuses on the design of the MPC cost and terminal constraints so as to ensureclosed-loop stability (for the case of soft constraints) and recursive feasibilitywith a pre-specified confidence level.r=1