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MIT16_30F10_lec14

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

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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. 596–602, Dec 1971. Estimation Theory (with noise) Kalman Reading: FPE 7.5

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Fall 2010 16.30/31 14–2 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 ˆ y Same as the proportional feedback we looked at at the beginning of the root locus work. This type of control is very diﬃcult 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 ˆ( t ) . 2. Then switch from u ( t ) = K x ( t ) to u ( t ) = K x ˆ( t ) . Two key questions: How do we find x ˆ( t ) ? Will this new plan work? October 17, 2010
Fall 2010 16.30/31 14–3 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 ˆ( t ) = x ( t ) for all time t 0 . Two primary approaches: Open-loop. Closed-loop. October 17, 2010

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Fall 2010 16.30/31 14–4 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 ˆ ˙ ( t ) = A x ˆ( t ) + B u ( t ) Then x ˆ( t ) x ( t ) t provided that x ˆ(0) = x (0) Actual System x : A, B, C Estimator ˆ x : A, B, C y ˆ y u Analysis of this case: x ˙ ( t ) = A x ( t ) + B u ( t ) x ˆ ˙ ( t ) = A x ˆ( t ) + B u ( t ) Define the estimation error x ˜( t ) = x ( t ) x ˆ( 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 14–5 Subtract to get: d dt ( x ( t ) x ˆ( t )) = A ( x ( t ) x ˆ( 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 ) 0 as t → ∞ ? (which is a more realistic goal). Response is fine if x ˜(0) = 0 . But what if x ˜(0) = 0 ? If A stable, then x ˜( t ) 0 as t → ∞ , but the dynamics of the estimation error are completely determined by the open-loop dynamics of the system (eigenvalues of A ). Could be very slow. No obvious way to modify the estimation error dynamics.

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