Fall 2001 6.3116-5 X() SYSTEM A,BC OBSERVER A16 y() Approach: Feedback y to improve our estimate of the state. Basic form of the estimator is (t)=Ai(t)+ Bu(t)+Lio 0(t)=Cx(t) where l is the user selectable gain matric analysi [Ac+ Bu-[A c+ Bu+L(y-y) A(a-a)-L(Cr-Cr)=Ai- LCa=(a-lc)a So the closed-loop estimation error dynamics are now =(A- LC)i with solution a(t) (A-LC)t c(O Bottom line: Can select the gain L to attempt to improve the convergence of the estimation error(and or speed it up But now must worry about observability of the system model
Fall 2001 16.31 16—5 • Approach: Feedback ˜y to improve our estimate of the state. Basic form of the estimator is: ˙ xˆ(t) = Axˆ(t) + Bu(t) + Ly˜(t) yˆ(t) = Cxˆ(t) where L is the user selectable gain matrix. • Analysis: x˜˙ = x˙ − ˙ xˆ = [Ax + Bu] − [Axˆ + Bu + L(y − yˆ)] = A(x − xˆ) − L(Cx − Cxˆ) = Ax˜ − LCx˜ = (A − LC)˜x • So the closed-loop estimation error dynamics are now ˙ x˜ = (A − LC)˜x with solution ˜x(t) = e(A−LC)t x˜(0) • Bottom line: Can select the gain L to attempt to improve the convergence of the estimation error (and/or speed it up). — But now must worry about observability of the system model
Fall 2001 16.3116-6 Note the similarity Regulator Problem: pick K for A- BK o Choose K ERIXn(SiSo) such that the closed-loop poles det(s-A+BK)=重(s are in the desired locations Estimator Problem: pick L for A-LC o Choose E RxI(SiSO)such that the closed-loop poles dt(s-A+LC)=重(s) are in the desired locations o These problems are obviously very similar-in fact they are called dual problems
Fall 2001 16.31 16—6 • Note the similarity: — Regulator Problem: pick K for A − BK 3 Choose K ∈ R1×n (SISO) such that the closed-loop poles det(sI − A + BK) = Φc(s) are in the desired locations. — Estimator Problem: pick L for A − LC 3 Choose L ∈ Rn×1 (SISO) such that the closed-loop poles det(sI − A + LC) = Φo(s) are in the desired locations. • These problems are obviously very similar — in fact they are called dual problems