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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 15–2 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 ˆ( t ) • Assume we have designed a closed-loop estimator with gain L x ˆ ˙ ( t ) = ( A − LC ) x ˆ( t ) + B u ( t ) + L y ( t ) y ˆ( t ) = C x ˆ( t ) • The closed-loop system dynamics are given by: x ˙ ( t ) = A x ( t ) + B u ( t ) x ˆ ˙ ( t ) = ( A − LC ) x ˆ( t ) + B u ( t ) + L y ( t ) y ( t ) = C x ( t ) y ˆ( t ) = C x ˆ( t ) u ( t ) = − K x ˆ( t ) • Which can be compactly written as: x ˙ ˙ ( t ) = A − BK x ( t ) x ˙ cl ( t ) = A cl x cl ( t ) x ˆ( t ) LC A − BK − LC x ˆ( t ) ⇒ • This does not look too good at this point – not even obvious that the closed-system is stable. λ i ( A cl ) =?? November 5, 2010 Fall 2010 16.30/31 15–3 • Can fix this problem by introducing a new variable x ˜ = x − x ˆ 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 ˆ( 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 λ i ( A cl ) ≡ λ 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 15–4 • 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 15–5 The Compensator • Dynamic Output Feedback Compensator is the combination of the regulator and estimator using u ( t ) = − K x ˆ( t ) x ˆ ˙ ( t ) = ( A − LC ) x ˆ( t ) + B u ( t ) + L y ( t ) x ˆ ˙ ( t ) = ( A − BK − LC ) x ˆ( t ) + L y ( t ) ⇒ u ( t ) = − K x ˆ( t ) • Rewrite with new state x c ( t )...
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MIT16_30F10_lec15 - Topic#15 16.30/31 Feedback Control...

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