Assessment and Future Directions of Nonlinear Model Predictive ...

174 H.G. Bock et al.to yield not only an improvement of feasibility, but also of optimality for theoriginal NLP (8).The remaining online computations are slightly more expensive than for levelsB and C, as we need to recover the multipliers λ QPk,µQP kof the uncondensedQP (14) for the transition step 3, as follows:First, the inequality multipliers µ QPkare directly obtained as the multipliersµ cQPkof the condensed QP (16): µ QPk:= µ cQPk. Second, the equality multipliersλ QPkcan be computed as λ QPk:= (BS T ) −T S(A∆w k + a k − C T µ QPk)whereS isa projection matrix that maps w to its subvector s.The matrix BS T contains only those columns of B that correspond to thevariables s, cf Eq. (10), and is thus invertible. Abbreviating a := S(A∆w k +a k − C T µ QPk), a =(ax 0 ,az 0 ,...,ax N ), we can compute λQPk=(λ x 0 ,λz 0 ,...,λx N )recursively backwards: Starting with λ x N := ax N , we compute, for i = N − 1,N−2,...,0:λ z i =(Z z i ) −T ( a z i +(X z i ) T λ x i+1), λxi = a x i +(X x i ) T λ x i+1 − (Z x i ) T λ z i .where we employ the submatrix notation of Eq. (10) for the matrix B, respectivelyBS T . The proof of nominal stability of NMPC based on this variant followsthe lines of the proof for the standard scheme mentioned in section 3.3.5 A Real-Time NMPC ExampleTo demonstrate the real-time applicability of the NMPC schemes discussedabove, a simulation experiment has been set up. In this experiment, a chainof massive balls connected by springs is perturbed at one end, and the controltask is to bring the system back to steady state.An ODE Model for a Chain of Spring Connected MassesConsider the following nonlinear system of coupled ODEsẍ i + β ẋ i − 1 m (F i+ 1 − F 2 i− 1 ) − g =0, i =1, 2,...,N−1 (18a)2()F i+ 1 S L1 −(x2 i+1 − x i ), i =0, 1,...,N−1. (18b)‖x i+1 − x i ‖x 0 (t) ≡ 0, ẋ N (t) =u(t), (18c)for the ball positions x 0 (t),...,x N (t) ∈ R 3 with boundary conditions (18c) and aprescribed control function u(t) ∈ R 3 . Equations (18a)–(18c) describe the motionof a chain that consists of eleven balls (i.e. N = 10), numerated from 0 to 10,that are connected by springs. At one end, the first ball is fixed in the origin, thevelocity of the other end (the ”free” end) is prescribed by the function u. Themotion of the chain is affected by both laminar friction and gravity (gravitationalacceleration g =9.81 m/s 2 ) as an external force. The model parameters are: massm =0.03 kg, spring constant S =1N/m, rest length of a spring L =0.033 m,