Assessment and Future Directions of Nonlinear Model Predictive ...

342 L. Blank(G T λ)(0) = D(x(0) − x ref0 ) and (G T λ)(H) = 0 (15)ν T c(x, u) =0 c(x, u) ≤ 0 ν ≥ 0. (16)In case of ODE’s as state constraints with no regularization of the initial state,this yields w ∈ H 1 ,whereH 1 denotes the Sobolev-space of weakly differentiablefunctions in L 2 ,andw = 0 at the end points. If model error functions arepresent in all state equations, the Lagrange parameter λ can be eliminated toowith λ = R −1 (Gẋ − f(x, u)) . Then, the necessary conditions reduce to a secondorder DAE system with mixed boundary constraints for the state x only and theequations (16) for the inequality constraints.3.2 Analysis for Linear ODE’sIn the solution process for a nonlinear problem with inequality constraints nearlyalways optimization problems with linear model equations and without inequalityconstraints (or an equivalent linear equation system) appear as subproblems.Although for the linear case, in particular with regularization of the initial data,efficient methods as the Kalman filter are well established [10, 11], these methodsand its extensions cannot be applied or do not perform sufficiently in thepresence of nonlinearity and inequality constraints [9]. However, the study oflinear model equations without inequality constraints enhances already some ofthe main features we face for the treatment of the optimization formulation ofthe nonlinear state estimation. The goal of the study here is to iluminate theinfluence of the eigenvalues of the system matrix, the influence of the observabilitymeasure and the influence of the regularization parameters. As a startwe assume possible model error functions in all state equations. We consider thefollowing problem:min∫ H0(Cx − z) T Q(Cx − z)+w T R w wdt+ ‖D 1/2 (x(0) − x ref0 )‖ 2 l 2(17)s.t. ẋ − Ax − w = u.A detailed analysis of this problem and its results can be found in [4], wherethey are presented for H =1andD = 0. With a few modifications they can beextended to any H>0. In this paper we summarize some of the main results. Tostudy the properties we choose one of the following three equivalent formulation,which derive from elimination of the error function w using the state equationor from the necessary conditions eliminating the Lagrange parameter λ. Theboundary value problem and its weak formulation are not only necessary butalso sufficient condition, which is shown later (Theorem 4):Optimization problem:{}min ‖Q 1 2 (Cx − z) ‖2x∈H 1 L2+ ‖R − 1 2 (ẋ − Ax − u) ‖2L2+ ‖D 1 2 (x(0) − xref0 )‖ 2 l 2.(18)