MIT16_30F10_lec14

MIT16_30F10_lec14 - Topic #14 16.30/31 Feedback Control...

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Unformatted text preview: Topic #14 16.30/31 Feedback Control Systems State-Space Systems Open-loop Estimators Closed-loop Estimators Observer Theory (no noise) Luenberger IEEE TAC Vol 16, No. 6, pp. 596602, Dec 1971. Estimation Theory (with noise) Kalman Reading: FPE 7.5 Fall 2010 16.30/31 142 Estimators/Observers Problem: So far we have assumed that we have full access to the state x ( t ) when we designed our controllers. Most often all of this information is not available. Usually can only feedback information that is developed from the sensors measurements. Could try output feedback u = K x u = K y Same as the proportional feedback we looked at at the beginning of the root locus work. This type of control is very dicult to design in general. Alternative approach: Develop a replica of the dynamic system that provides an estimate of the system states based on the mea- sured output of the system. New plan: 1. Develop estimate of x ( t ) that will be called x ( t ) . 2. Then switch from u ( t ) = K x ( t ) to u ( t ) = K x ( t ) . Two key questions: How do we find x ( t ) ? Will this new plan work? October 17, 2010 Fall 2010 16.30/31 143 Estimation Schemes Assume that the system model is of the form: x ( t ) = A x ( t ) + B u ( t ) , x (0) unknown y ( t ) = C x ( t ) where 1. A , B , and C are known. 2. u ( t ) is known 3. Measurable outputs are y ( t ) from C = I Goal: Develop a dynamic system whose state x ( t ) = x ( t ) for all time t . Two primary approaches: Open-loop. Closed-loop. October 17, 2010 Fall 2010 16.30/31 144 Open-loop Estimator Given that we know the plant matrices and the inputs, we can just perform a simulation that runs in parallel with the system x ( t ) = A x ( t ) + B u ( t ) Then x ( t ) x ( t ) t provided that x (0) = x (0) Actual System x : A, B, C Estimator x : A, B, C y y u Analysis of this case: x ( t ) = A x ( t ) + B u ( t ) x ( t ) = A x ( t ) + B u ( t ) Define the estimation error x ( t ) = x ( t ) x ( t ) . Now want x ( t ) = 0 t . (But is this realistic?) Major Problem: We do not know x (0) October 17, 2010 Fall 2010 16.30/31 145 Subtract to get: d dt ( x ( t ) x ( t )) = A ( x ( t ) x ( t )) x ( t ) = A x ( t ) which has the solution x ( t ) = e At x (0) Gives the estimation error in terms of the initial error. Does this guarantee that x ( t ) = 0 t ? Or even that x ( t ) as t ? (which is a more realistic goal). Response is fine if x (0) = . But what if x (0) = ?...
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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_lec14 - Topic #14 16.30/31 Feedback Control...

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