Assessment and Future Directions of Nonlinear Model Predictive ...

172 H.G. Bock et al.contractivity of the real-time iterates, but that we leave this rare updating unconsideredhere for simplicity of presentation. For mildly nonlinear systems, however,the matrices A, B, C might really be kept constant without any updates,for example evaluated once for all at a reference solution.In what follows, three different variants of the real-time iteration scheme willbe shown in detail, differing in the choice of a k , b k ,andc k . While variant A isnothing else than linear MPC, variant B converges to nonlinearly feasible (butsuboptimal) MPC solutions. Variant C will even converge to the true nonlinearMPC feedback - without the need to evaluate any derivative matrix online. Butbefore we proceed, a preliminary remark on condensing of a QP is necessary.4.1 A Prerequisite: Offline CondensingIn all approaches we use fixed approximations of the Jacobians B and C, e.g.byevaluating offline ∇ w b(¯w) T and ∇ w c(¯w) T for a reference trajectory ¯w that maybe an exact or approximate solution of an NLP P (¯x) for some state ¯x. Wealsouse a fixed approximation A of the Hessian, that may be based on the referencesolution of P (¯x) and computed as ∇ 2 w L(¯w, ¯λ, ¯µ), or be chosen otherwise. Online,we use these fixed components A, B, C to formulate a QP of the form (14),where only the vectors a k ,b k ,c k and the initial value x k are changing online.It is well known that because ∇ s b(¯w) is invertible, the online QP solution canbe prepared by a condensing of the QP [5, 8]: We divide ∆w into its state andcontrol components ∆s and ∆u, and resolve the equality constraints (14b) toobtain ∆s as a linear function of ∆u (and x k ), such that we can substitute∆w = m(b k )+ ˜Lx k + M∆u. (15)Note that the matrices ˜L and M are independent of a k ,b k ,c k ,x k and can in allvariants be precomputed offline, exploiting the structure of B in Eq. (10). InEq. (17) below, we show how m(b k ) can be computed efficiently online. We useexpression (15) to substitute ∆w wherever it appears in the QP, to yield thecondensed QP:1(min∆u 2 ∆uT A c ∆u +subject toTa c (a k ,b k k) )+Ãx ∆u. (16)(c c (c k ,b k )+ ˜Cx k)+ C c ∆u ≥ 0All matrices, A c := M T AM, à := M T A˜L, ˜C := C ˜L, C c := CM of this condensedQP are precomputed offline. Online, only the vectors a c (a k ,b k )+Ãx kand c c (c k ,b k )+ ˜Cx k need to be computed, as shown in the following.4.2 Variant A: Linear MPC Based on a Reference TrajectoryIn the first approach [3, 8], we compute offline the fixed vectors b := b 0 (¯w),c:=c(¯w)anda := ∇ w a(¯w), and set a k := a, b k := b, c k := c in all real-time iterations.We can therefore precompute m := m(b) andalsoa c := a c (a, b) =M T (Am + a)and c c := c c (c, b) =c + Cm.