Assessment and Future Directions of Nonlinear Model Predictive ...

Computational Aspects of Approximate Explicit NMPC 189The following compressor and valve characteristics are used:( ( x1) (Ψ e (x 1 )=ψ c0 + H 1+1.5W − 1 x1) ) 3− 0.5W − 1 (20)Φ(x 2 )=γsign(x 2 ) √ |x 2 | (21)with γ =0.5, B =1,H =0.18, ψ c0 =0.3 andW =0.25. Like in [10], the controlobjective is to avoid surge. This is formulated as [10]:J(U, x(t)) =N−1∑k=0[α(xt+k|t − x ∗ ) T (x t+k|t − x ∗ )+ku 2 t+k]+ Rv2+β(x t+N|t − x ∗ ) T (x t+N|t − x ∗ ) (22)with α, β, k, R ≥ 0andtheset-pointx ∗ 1 =0.4, x ∗ 2 =0.6 corresponds to anunstable equilibrium point. We have chosen α =1,β =0andk =0.08. Thehorizon is chosen as T = 12, which is split into N = 15 equal-sized intervals,leading to a piecewise constant control input parameterization. Valve capacityrequires the following constraint to hold:0 ≤ u(t) ≤ 0.3 (23)The pressure constraint:x 2 (t) ≥ 0.4 − v (24)avoids operation too far left of the operating point. The variable v ≥ 0isaslackvariable introduced in order to avoid infeasibility and R = 8 is a large weight.Numerical analysis of the cost function shows that it is non-convex [10]. It canbe seen that this NMPC problem formulation differs from that in section 2 inthe absence of a terminal constraint and in the use of a slack variable.The NLP (10)–(11) has 16 free variables and 46 constraints, while the NLP(14)–(15) has 46 free variables and 811 constraints. One internal region X0 1 ⊂ X 0is used in Procedures 1, 2 and 3. In (14), it is chosen µ = 10 and the control inputonly at the first sample is considered. The approximation tolerance is chosen todepend on X 0 such that:ε(X 0 )=max(ε a , ε r V ∗ min) (25)where ε a =0.0001 and ε r =0.02 can be interpreted as absolute and relativetolerances, respectively, and V ∗ min =minx∈X 0V ∗ (x). Here, V ∗ (x) denotes a close-toglobalsolution.The partition of the approximate explicit NMPC controller is shown in Fig.1.It has 595 regions and 12 levels of search. With one scalar comparison required ateach level of the k−d tree, 12 arithmetic operations are required in the worst caseto determine which region the state belongs to. Totally, 16 arithmetic operationsare needed in real-time to compute the control input and 1368 numbers needsto be stored in real-time computer memory. The off-line computation of the