MIT16_30F10_lec15

MIT16_30F10_lec15 - Topic #15 16.30/31 Feedback Control...

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Unformatted text preview: Topic #15 16.30/31 Feedback Control Systems State-Space Systems Closed-loop 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 closed-loop estimator with gain L x ( t ) = ( A LC ) x ( t ) + B u ( t ) + L y ( t ) y ( t ) = C x ( t ) The closed-loop 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 closed-system 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 closed-loop 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 closed-loop 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: 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 com- bine them at the end. Called the Separation Principle . Keep in mind that the pole locations you are picking for these two sub-problems will also be the closed-loop pole locations. Note: 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!!! 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 )...
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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.

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MIT16_30F10_lec15 - Topic #15 16.30/31 Feedback Control...

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