Fall 2001 16.3116-10 Estimators Example Simple system 11.5 0.5 A B x(0 0 D=0 assume that the initial conditions are not well known System stable, but Amax(A)=-018 Test observability rank rank CA 11.5 Use open and closed-loop estimators Since the initial conditions are not well known. use 0 (0) pen-loop estimato A+ Bu y= Ca Closed-loop estimator A i+ Bu+ Ly=A.i+ Bu+l(y-y (A- LC)i+ Bu+ Ly C Which is a dynamic system with poles given by Ai(A-LC) and which e measure ed plant outputs as an input and generates an estimate of aFall 2001 16.31 16—10 Estimators Example • Simple system A = ∙ −1 1.5 1 −2 ¸ , B = ∙ 1 0 ¸ , x(0) = ∙ −0.5 −1 ¸ C = £ 1 0 ¤ , D = 0 — Assume that the initial conditions are not well known. — System stable, but λmax(A) = −0.18 — Test observability: rank ∙ C CA ¸ = rank ∙ 1 0 −1 1.5 ¸ • Use open and closed-loop estimators — Since the initial conditions are not well known, use xˆ(0) = ∙ 0 0 ¸ • Open-loop estimator: ˙ xˆ = Axˆ + Bu yˆ = Cxˆ • Closed-loop estimator: ˙ xˆ = Axˆ + Bu + Ly˜ = Axˆ + Bu + L(y − yˆ) = (A − LC)ˆx + Bu + Ly yˆ = Cxˆ — Which is a dynamic system with poles given by λi(A − LC) and which takes the measured plant outputs as an input and generates an estimate of x