Assessment and Future Directions of Nonlinear Model Predictive ...

66 J.A. Rossiter, B. Pluymers, and B. De MoorDefinition 4 (Feasibility). Let Φ i (k) =A(k) − B(k)K i , then [1] the followinginput sequence and the corresponding state predictions are recursively feasiblewithin S:u(k) =− ∑ ni=1 K ∏ k−1i j=0 Φ i(k − 1 − j)ˆx i ,if one ensures thatx(0) =x(k) = ∑ ni=1n∑ˆx i ,i=1∏ k−1j=0 Φ i(k − 1 − j)ˆx i ,with⎧⎪⎨ ˆx i = λ i x i ,∑ ni=1⎪⎩λ i =1, λ i ≥ 0,x i ∈S i .(12)(13)Definition 5 (Cost). With ˜x =[ˆx T 1 ... ˆx T n ] T , Lyapunov theory gives an upperbound ˜x T P ˜x on the infinite-horizon cost J for predictions (12) using:P ≥ Γ T u RΓ u + Ψ T i Γ T x QΓ x Ψ i + Ψ T i PΨ i , i =1,...,m, (14)with Ψ i =diag(A i − B i K 1 ,...,A i − B i K n ), Γ x =[I,...,I], Γ u =[K 1 ,...,K n ].These considerations show that by on-line optimizing over ˜x, one implicitly optimizesover a class of input and state sequences given by (12). Due to recursivefeasibility of these input sequences, this can be implemented in a receding horizonfashion.3 Interpolation Schemes for LTI SystemsInterpolation is a different form of methodology to the more usual MPCparadigms in that one assumes knowledge of different feedback strategies withsignificantly different properties. For instance one may be tuned for optimal performanceand another to maximise feasibility. One then interpolates betweenthe predictions (12) associated with these strategies to get the best performancesubject to feasibility. The underlying aim is to achieve large feasible regions withfewer optimisation variables, at some small loss to performance, and hence facilitatefast sampling. This section gives a brief overview and critique of someLTI interpolation schemes; the next section considers possible extensions to theLPV case.3.1 One Degree of Freedom Interpolations [17]ONEDOF uses trivial colinear interpolation, hence in (12) use:x =ˆx 1 +ˆx 2 ; ˆx 1 =(1− α)x; ˆx 2 = αx; 0 ≤ α ≤ 1. (15)Such a restriction implies that α is the only d.o.f., hence optimisation is trivial.Moreover, if K 1 is the optimal feedback, minimising J of (5) over predictions(15,12) is equivalent to minimising α, α ≥ 0. Feasibility is guaranteed only in⋃i S i.