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lecture21 - 16.322 Stochastic Estimation and Control Fall...

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 8 Lecture 21 Last time: ( ) ( ) ( ) c s F s K s d s = + ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) L A c s s a s c s b s A c a c a b a B s a s a s + = + + + = + + Plugging in for the optimum compensator: ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) 0 ( ) ( ) ( ) ( ) L L L ii L n n s H s F s F s S s Bs d s a s a s K c s S b s Bs d s S K c s b s = + + = + + + + = + + This means that in the cascade of 0 ( ) H s with ( ) F s , the effect of the RHP zero on the amplitude of the product is cancelled out, but the effect on the phase is not. In fact, the phase lag due to the RHP zero is doubled in the product. Note: Cancellation of K and two LHP poles in ( ) F s Zero at s c = (RHP) is not cancelled, but a pole is placed at the symmetric point s c = − Another pole is added at s b = − which is beyond the signal cut-off by an amount which depends on n A S Also the gain n B S depends on n A S directly Also: ( ) R s has the form ( ) ( )( ) e fs C D c s b s c s b s + + =
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 8 Then ( ) ( )( )( ) ( )( )( )( )( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) R R ii R R L s F s F s S s s K c s b s e fs s d s a s c s b s K e fs s d s a s ε ε = − + = − + = − which is analytic in the LHP and varies as 2 1 s for large . s Necessary condition: ( ) L s must be analytic in LHP and go to zero for large s at least as fast as 1 s . Now find the loop compensator if the feedback transfer is unity. ( ) 0 2 0 1 n n n B s d s H KS C H FD B Bc s b c s cb S S + = = + + + + + Cancellation of the pole at the origin leaves the system with an uncontrollable mode corresponding to that pole. This is not good since that normal mode does not attenuate. As a practical matter, it might be better to move that zero away from the origin a bit. That also means the system will not have unit input-output sensitivity . Also note that the in-the-loop compensator need not be stable . It depends on the parameter values.
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