Assessment and Future Directions of Nonlinear Model Predictive ...

A New Real-Time Method for Nonlinear Model Predictive Control 539L : X × U → R ≥0 is assumed to satisfy γ L (‖x, u‖ Σ) ≤ L(x, u) ≤ γ U (‖x, u‖ Σ)for some γ L ,γ H ∈K ∞ , although this could be relaxed to an appropriate detectabilitycondition. The mapping W : X f → R ≥0 is assumed to be positivesemi-definite, and identically zero on the set Σ X ⊂ X f . For the purposes ofthis paper, the functions L(·, ·), W (·) andf(·, ·) are all assumed to be C 1+ ontheir respective domains of definition, although this could be relaxed to locallyLipschitz with relative ease.3 Finite-Dimensional Input ParameterizationsIncreasing horizon length has definite benefits in terms of optimality and stabilityof the closed loop process. However, while a longer horizon obviously increasesthe computation time for model predictions, of significantly greater computationalconcern are the additional degrees of freedom introduced into the minimizationin (2a). This implies that instead of enforcing a constant horizon length,it may be more beneficial to instead maintain a constant number of input parameterswhose distribution across the prediction interval can be varied accordingto how “active” or “tame” the dynamics may be in different regions.Towards this end, it is assumed that the prediction horizon is partitionedinto N intervals of the form [t θ i−1 ,tθ i ], i =1...N, with t ∈ [tθ 0 ,tθ 1 ]. The inputtrajectory u :[t θ 0,t θ N ] → Rm is then defined in the following piecewise manneru(τ) =u φ (τ,t θ ,θ,φ) {φ(τ −tθ0 ,θ 1 ) τ ∈ [t θ 0,t θ 1](3)φ(τ −t θ i−1 ,θ i) τ ∈ (t θ i−1 ,tθ i ], i ∈{2 ...N}with individual parameter vectors θ i ∈ Θ ⊂ R n θ, n θ ≥ m, for each interval, andθ = {θ i | i ∈{1,...N}} ∈ Θ N . The function φ : R ≥0 × Θ → R m may consist ofany smoothly parameterized (vector-valued) basis in time, including such choicesas constants, polynomials, exponentials, radial bases, etc. In the remainder, a(control- or input-) parameterization shall refer to a triple P (φ, R N+1 ,Θ N )with specified N, although this definition may be abused at times to refer to thefamily of input trajectories spanned by this triple (i.e. the set-valued range ofφ(R N+1 ,Θ N )).Assumption 1. The C 1+ mapping φ : R ≥0 × Θ → R m and the set Θ aresuch that 1) Θ is compact and convex, and 2) the image of Θ under φ satisfiesU ⊆ φ(0,Θ).Let (t 0 ,x 0 ) ∈ R × ˚X represent an arbitrary initial condition for system (1), andlet (t θ ,θ) be an arbitrary choice of parameters corresponding to some parameterizationP. We denote the resulting solution to the prediction model in (2b),defined on some maximal subinterval of [t 0 ,t θ N ], by xp (·,t 0 ,x 0 ,t θ ,θ,φ). At timeswe will condense this notation, and that of (3), to x p (τ), u φ (τ).A particular choice of control parameters (t θ ,θ) corresponding to some parameterizationP will be called feasible with respect to (t 0 ,x 0 ) if, for every