Assessment and Future Directions of Nonlinear Model Predictive ...

344 L. BlankTheorem 3. S is a linear isomorphism and with the real values A = α, C = δ,Q = q, R −1 = r we havemin(4r, qδ 2 ) ≤ ‖S‖ H 1 →(H 1 ) ′ ≤ 2r max(1,α2 )+qδ 2 , (22){ }c(α)2qδ 2 ≤‖S −1 ‖ (H 1 ) ′ →H ≤ max 1 r , 2α2 +1qδ 2 , (23)e 2H −1for large |α|. Hence for fixed regularization pa-2(e 2H +1)√with c(α) ≈|α| 3/2rameters r and qcond(S) =‖S‖ H 1 →(H 1 ) ′‖S−1 ‖ (H 1 ) ′ →H 1to infinity with |α| →∞.is bounded but tendsThe exact value of c(α) :=‖ exp(α·)‖ H 1/‖ exp(α·)‖ (H1 ) ′isc 2 (α) =(1+α 2 )(α+1) 2 (α−1) 2 (e H −e −H )(e 2αH −1)(2α+1)(α−1) 2 (e H −e −H )(e 2αH −1)+4α 3 e −H (e αH −e H ) 2 .Considering several state functions one has to take into account observability toshow positive definiteness and herewith continuity and coercivity of a.Weshortlysketch this step while refering for the other arguments to [4]. Given a(x, x) =0then x is the solution of the system ẋ − Ax ≡ 0, x(0) = x 0 and (y ≡)Cx ≡ 0.Since the system is observable y ≡ 0 yields x 0 = 0 and consequently x ≡ 0.Hence, a(x, x) > 0forx ≢ 0. Then continuity and coercivity of a yieldTheorem 4. For any z,u ∈ L 2 the solution x of the weak formulation (20)determines the unique solution of the minimization problem (18).Hence for well-posedness only the question of stability has still to be answered.Using the Riesz representation theorem [2] and considering like for one stateonly and Lemma 1b.) the exponential function we obtainTheorem 5. S : H 1 → ( H 1) ′is bounded, has a bounded inverse and0 < 2/‖R‖≤‖S‖ H 1 →(H 1 ) ′ ≤ 2‖R‖−1 max(1, ‖A‖ 2 )+‖C T QC‖, (24)max{c(α)/‖C T QCv‖ l2 }≤‖S −1 ‖ (H 1 ) ′ →H1. (25)for all normed v ∈ IR nx eigenvectors of A with real eigenvalue α. In (24) thelower bound is valid if there exists an α 2 > 1. Observability guarantees Cv ≠0.As a consequence of Theorem 5 and Lemma 1b.) we have:Corollary 2. For any fixed regularization cond(S) is large, if there exists an inmodulo large real eigenvalue of A or if there exists a real eigenvector v of Awhich is close to the null space of C. If this is the case then the observabilitymeasure is low too.Nevertheless, the boundedness of the inverse S −1 and the compact inbeddingH 1 ↩→ C 0 yieldsCorollary 3. Linear state estimation formulated as least squares problemmin 1 2(y − z)T Q(y − z)+w T R w wdt∫ H0