Assessment and Future Directions of Nonlinear Model Predictive ...

L i (s i ,u i ):=Numerical Methods for Efficient and Fast NMPC 167t∫i+1t iL(x i (t),z i (t),φ i (t, u i )) dt. (6)Note that up to now the multiple shooting parameterization does not involveany discretization of differential operators f,g, but is exact.Structured Nonlinear Programming ProblemThe parameterization of problem (3a)–(3f) using multiple shooting and a suitablecontrol representation leads to the following structured nonlinear programming(NLP) problem :N−1∑minu,si=0L i (s i ,u i ) + E(s x N ) (7a)subject to s x 0 = x 0, (7b)0=g(s x i ,s z i ,φ i (t i ,u i )), i =0, 1,...N−1, (7c)s x i+1 = x i(t i+1 ; s i ,u i ), i =0, 1,...N−1, (7d)r(s x N ) ≥ 0,(7e)with initial condition (7b), consistency conditions (7c), continuity conditions(7d), and terminal constraint (7e). Additional control and path constraints aresupposed to be imposed pointwise for a suitable discretization (at n i points τ ijon each interval, τ ij ∈ [t i ,t i+1 ), j =0,...,n i − 1)h(x i (τ ij ; s i ,u i ),z i (τ ij ; s i ,u i ),u i ) ≥ 0, j =0,...,n i −1, i =0,...N−1. (7f)The NLP (7a)–(7e) can be summarized asP (x 0 ):{bx0 (w)=0min a(w) subject towc(w)≥0,(8)where w contains all the multiple shooting state variables and controls:w =(s x 0,s z 0,u 0 ,s x 1,s z 1,u 1 ,...,u N−1 ,s x N ) ∈ R nw .The function a(w) is the objective (7a), the vector valued equation b x0 (w) =0summarizes all equalities from (7b)-(7d), and the vector valued c(w) ≥ 0containsthe inequality constraints (7f) and (7e).It is important to note that the initial condition (7b) is a linear constraintamong the equality constraints, with the varying parameter x 0 entering linearlyonly in this constraint, so that⎡s x 0 − x ⎤⎡ ⎤0−I nxg(s x 0,s z 0,φ 0 (t 0 ,u 0 ))b x0 (w) =⎢ s⎣1 − x 0(t 1 ; s x 0 ,sz 0 ,u 0) ⎥⎦ = b 00(w)+Lx 0 with L :=⎢ 0 ⎥⎣ ⎦ . (9)..