Topic #17 16.31 Feedback Control tate-Space Systems Closed-loop control using estimators and regulators Dynamics output feedback “ Back to reality' Copyright[2001by JOnathan dHow.D
Topic #17 16.31 Feedback Control State-Space Systems • Closed-loop control using estimators and regulators. • Dynamics output feedback • “Back to reality” Copyright 2001 by Jonathan How. 1
Fall 2001 16.3117-1 Combined Estimators and Regulators Can now evaluate the stability and or performance of a controller when we design K assuming that u =-Ka, but we implement Assume that we have designed a closed-loop estimator with gain L a (t)=Ai(t)+ Bu(t)+l(y-y u(t)=Ci( Then we have that the closed-loop system dynamics are given by i(t)=Ax(t)+ Bu(t) (t)= Ac(t)+ Bu(t)+L(y-g t)=Ca(t) cal Which can be compactly written as BK LC A-BK-LCIi cl clcl This does not look too good at this point- not even obvious that the closed-system is stable 入(Aa)
Fall 2001 16.31 17—1 Combined Estimators and Regulators • Can now evaluate the stability and/or performance of a controller when we design K assuming that u = −Kx, but we implement u = −Kxˆ • Assume that we have designed a closed-loop estimator with gain L ˙ xˆ(t) = Axˆ(t) + Bu(t) + L(y − yˆ) yˆ(t) = Cxˆ(t) • Then we have that the closed-loop system dynamics are given by: x˙(t) = Ax(t) + Bu(t) ˙ xˆ(t) = Axˆ(t) + Bu(t) + L(y − yˆ) y(t) = Cx(t) yˆ(t) = Cxˆ(t) u = −Kxˆ • Which can be compactly written as: ∙ x˙ ˙ xˆ ¸ = ∙ A −BK LC A − BK − LC ¸ ∙ x xˆ ¸ ⇒ x˙ cl = Aclxcl • This does not look too good at this point — not even obvious that the closed-system is stable. λi(Acl) =??
Fall 2001 16.3117-2 Can fix this problem by introducing a new variable a a and then converting the closed-loop system dynamics using the similarity transformation T △ I 0 I-1| Note that T=T-1 Now rewrite the system dynamics in terms of the state cl TAct=A Note that similarity transformations preserve the eigenvalues, so we are guaranteed that 入(A4)≡A(Aa . Work through the math BK Acl I 0 A Ⅰ-1LCA-BK-LCI-I A BK I 0 A-LC-A+LCII-I A-BK BK 0 A-LC Because Ac is block upper triangular, we know that the closed DOD poles of the system are given by det(sI -Ac)= det(sI -(A- BK). det(sI-(A-LC))=0
Fall 2001 16.31 17—2 • Can fix this problem by introducing a new variable ˜x = x − xˆ and then converting the closed-loop system dynamics using the similarity transformation T x˜cl , ∙ x x˜ ¸ = ∙ I 0 I −I ¸ ∙ x xˆ ¸ = T xcl — Note that T = T −1 • Now rewrite the system dynamics in terms of the state ˜xcl Acl ⇒ T AclT −1 , A¯cl — Note that similarity transformations preserve the eigenvalues, so we are guaranteed that λi(Acl) ≡ λi(A¯cl) • Work through the math: A¯cl = ∙ I 0 I −I ¸ ∙ A −BK LC A − BK − LC ¸ ∙ I 0 I −I ¸ = ∙ A −BK A − LC −A + LC ¸ ∙ I 0 I −I ¸ = ∙ A − BK BK 0 A − LC ¸ • Because A¯cl is block upper triangular, we know that the closed-loop poles of the system are given by det(sI − A¯cl) , det(sI − (A − BK)) · det(sI − (A − LC)) = 0
Fall 2001 163117-3 Observation: The closed-loop poles for this system con- sist of the union of the regulator poles and estimator poles So we can just design the estimator/regulator separately and com- bine them at the end Called the Separation Principle Just keep in mind that the pole locations you are picking for these two sub-problems will also be the closed-loop pole locations Note: the separation principle means that there will be no ambi- guity or uncertainty about the stability and or performance of the close d-loop system The closed-loop poles will be exactly where you put them! And we have not even said what compensator does this amazin accomplishment!!!
Fall 2001 16.31 17—3 • Observation: The closed-loop poles for this system consist of the union of the regulator poles and estimator poles. • So we can just design the estimator/regulator separately and combine them at the end. — Called the Separation Principle. — Just keep in mind that the pole locations you are picking for these two sub-problems will also be the closed-loop pole locations. • Note: the separation principle means that there will be no ambiguity or uncertainty about the stability and/or performance of the closed-loop system. — The closed-loop poles will be exactly where you put them!! — And we have not even said what compensator does this amazing accomplishment!!!
Fall 2001 6.3117-4 The Compensator e Dynamic Output Feedback Compensator is the combina- tion of the regulator and estimator using u=-K3 i(t)=Ai(t)+ Bu(t)+L(y-y) A i(t)-BK2+L(y-Ca =a(t)=(A-BK-LCi(t)+Ly F Rewrite with new state c= S c= Acrc+ Bcy C where the compensator dynamics are given by A≌A-BK-LC,B全L,C≌K Note that the compensator maps sensor measurements to ac tutor commands, as expected Closed-loop system stable if regulator/estimator poles placed in the LHP, but compensator dynamics do not need to be stable Ai(A- BK-LC)=??
Fall 2001 16.31 17—4 The Compensator • Dynamic Output Feedback Compensator is the combination of the regulator and estimator using u = −Kxˆ ˙ xˆ(t) = Axˆ(t) + Bu(t) + L(y − yˆ) = Axˆ(t) − BKxˆ + L(y − Cxˆ) ⇒ ˙ xˆ(t)=(A − BK − LC)ˆx(t) + Ly u = −Kxˆ • Rewrite with new state xc ≡ xˆ x˙ c = Acxc + Bcy u = −Ccxc where the compensator dynamics are given by: Ac , A − BK − LC , Bc , L, Cc , K — Note that the compensator maps sensor measurements to actuator commands, as expected. • Closed-loop system stable if regulator/estimator poles placed in the LHP, but compensator dynamics do not need to be stable. λi(A − BK − LC) =??