Assessment and Future Directions of Nonlinear Model Predictive ...

State Estimation Analysed as Inverse Problem 339For linear model equations the corresponding operator K is affine. Let us firstconsider linear K, i.e. the model equations are linear and p and u = 0. Withoutloss of generality we consider in the rest of the paper only the case t 0 =0.Thuswe have withẋ − Ax =0, x(0) = x 0 , y − Cx =0 (3)⇒ Kx 0 = Ce At x 0 (4)Choosing now as norm on Y the L 2 ([0,H]) ny -norm we obtain∫‖Kx 0 ‖ 2 L 2= x T H0 0 (eAt ) T C T Ce At dt x 0 = x T 0 G(H)x 0where the matrix G(H) ∈ IR nx×nx is the known finite time observability Gramian(e.g. [6, 11, 12]).Lemma 1. Let the system be observable, then:a.) The observability Gramian G(H) = ∫ H0 (eAt ) T C T Ce At dt ∈ IR nx×nx is symmetricpositive definite, and therefore invertible.b.) Let v be a normed real eigenvalue of A to an eigenvalue α ∈ IR.Then‖G(H)‖ 2is large for large α and for a long horizon [0,H], while‖G(H) −1 ‖ 2 is large if −αis large or ‖Cv‖ l2 is small or if the horizon is short.Proof: a.) Symmetry is obvious. Given v ≠0theny(t) =Ce At v ≢ 0sincethesystem is observable. Hence, v T Gv = ∫ H0 yT (t)y(t) dt = ‖y‖ 2 L 2> 0.b.) Let v and α fulfill the assumption, thenv T Gv = ‖e αt ‖ 2 L 2(0,H) ‖Cv‖2 l 2= e2αH − 1‖Cv‖ 2 l2α2. (5)With ||G|| 2 =max v∈IR nx (v T Gv)/(v T v), ‖G −1 ‖ 2 =max v∈IR nx (v T v)/(v T Gv) followsthe assertion.Using the l 2 -norm for X =IR nx it follows for K:‖K‖ 2 l 2→L 2=supx 0∈IR nx ‖Kx 0 ‖ 2 L 2‖x 0 ‖ 2 l 2= supx 0∈IR nx x T 0 Gx 0x T 0 x 0= ||G|| 2 (6)‖(K ∣ ∣ R(K) ) −1 ‖ 2 L 2→l 2= supx 0∈IR nx ‖x 0 ‖ 2 l 2‖Kx 0 ‖ 2 L 2= ||G −1 || 2 (7)For linear systems with not necessarily p and u = 0 we need to consider forthe condition numbers ‖K ˜x 0 − Kx 0 ‖ 2 L 2=(˜x 0 − x 0 ) T G(˜x 0 − x 0 ). Hence, having√‖G−1‖ 2 ‖y − ỹ‖ L2 ≥‖x 0 − ˜x 0 ‖ l2 for all x 0 , we can define for linear systems-like condition numbers for matrices- a observability measure independent of thestate x 0 :Definition 4. The absolute, respectively the relative observability measure withrespect to the l 2 and L 2 -norms for linear systems is given by1/ √ ||G −1 || 2 respectively 1/ √ cond(G).