Assessment and Future Directions of Nonlinear Model Predictive ...

64 J.A. Rossiter, B. Pluymers, and B. De Moor2 BackgroundThis section introduces notation, the LPV model used in this paper, basic conceptsof invariance, feasibility and performance, and some prediction equations.2.1 Model and ObjectiveDefine the LPV model (uncertain or nonlinear case) totaketheform:x(k +1)=A(k)x(k)+B(k)u(k), k =0,...,∞, (1a)[A(k) B(k)] ∈ Ω Co{[A 1 B 1 ],...,[A m B m ]},(1b)The specific values of [A(k) B(k)] are assumed to be unknown at time k. Othermethods [5, 6] can take knowledge of the current values of the system matricesor bounded rates of change of these matrices into account but these cases arenot considered in this paper. However, it is conceivable to extend the algorithmspresented in this paper to these settings as well.When dealing with LTI models (m = 1), we will talk about the nominal case.The following feedback law is implicitly assumed :u(k) =−Kx(k); ∀k. (2)For a given feedback, the constraints at each sample are summarised as:x(k) ∈X = {x : A x x ≤ 1}, ∀ku(k) ∈U= {u : A u u ≤ 1}, ∀k⇒ x(k) ∈S 0 = {x : A y x ≤ 1}, ∀k.(3)where 1 is a column vector of appropriate dimensions containing only 1’s andA y =[A x ; −A u K]. We note that the results of this paper have been proven onlyfor feedback gains giving quadratic stabilisability, that is, for feedback K, theremust exist a matrix P = P T > 0 ∈ R nx×nx such thatΦ T j PΦ j ≤ P, ∀j, Φ j = A j − B j K. (4)Problem 1 (Cost Objective). For each of the algorithms discussed, the underlyingaims are: to achieve robust stability, to optimise performance and toguarantee robust satisfaction of constraints. This paper uses a single objectivethroughout. Hence the algorithms will seek to minimise, subject to robust satisfactionof (3), an upper bound on:2.2 Invariant SetsJ =∞∑(x(k) T Qx(k)+u(k) T Ru(k)). (5)k=0Invariant sets [2] are key to this paper and hence are introduced next.