This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Topic #15 16.30/31 Feedback Control Systems StateSpace Systems Closedloop control using estimators and regulators. Dynamics output feedback Back to reality Reading: FPE 7.6 Fall 2010 16.30/31 152 Combined Estimators and Regulators Now evaluate stability and/or performance of a controller when K designed assuming u ( t ) = K x ( t ) but implemented as u ( t ) = K x ( t ) Assume we have designed a closedloop estimator with gain L x ( t ) = ( A LC ) x ( t ) + B u ( t ) + L y ( t ) y ( t ) = C x ( t ) The closedloop system dynamics are given by: x ( t ) = A x ( t ) + B u ( t ) x ( t ) = ( A LC ) x ( t ) + B u ( t ) + L y ( t ) y ( t ) = C x ( t ) y ( t ) = C x ( t ) u ( t ) = K x ( t ) Which can be compactly written as: x ( t ) = A BK x ( t ) x cl ( t ) = A cl x cl ( t ) x ( t ) LC A BK LC x ( t ) This does not look too good at this point not even obvious that the closedsystem is stable. i ( A cl ) =?? November 5, 2010 Fall 2010 16.30/31 153 Can fix this problem by introducing a new variable x = x x and then converting the closedloop system dynamics using the similarity transformation T x cl ( t ) x ( t ) = I x ( t ) = T x cl ( t ) x ( t ) I I x ( t ) Note that T = T 1 Now rewrite the system dynamics in terms of the state x cl ( t ) A cl T A cl T 1 A cl Since similarity transformations preserve the eigenvalues we are guaranteed that i ( A cl ) i ( A cl ) Work through the math: A cl = I A BK I I I LC A BK LC I I = A BK I A LC A + LC I I = A BK BK A LC Because A cl is block upper triangular, we know that the closedloop poles of the system are given by det( sI A cl ) det( sI ( A BK )) det( sI ( A LC )) = November 5, 2010 Fall 2010 16.30/31 154 Observation: Closedloop 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 com bine them at the end. Called the Separation Principle . Keep in mind that the pole locations you are picking for these two subproblems will also be the closedloop pole locations. Note: Separation principle means that there will be no ambiguity or uncertainty about the stability and/or performance of the closedloop system. The closedloop poles will be exactly where you put them!! And we have not even said what compensator does this amazing accomplishment!!! November 5, 2010 Fall 2010 16.30/31 155 The Compensator Dynamic Output Feedback Compensator is the combination of the regulator and estimator using u ( t ) = K x ( t ) x ( t ) = ( A LC ) x ( t ) + B u ( t ) + L y ( t ) x ( t ) = ( A BK LC ) x ( t ) + L y ( t ) u ( t ) = K x ( t ) Rewrite with new state x c ( t )...
View
Full
Document
This note was uploaded on 02/03/2012 for the course AERO 16.30 taught by Professor Ericferon during the Fall '04 term at MIT.
 Fall '04
 EricFeron
 Dynamics

Click to edit the document details