topic15 - Topic #15 16.31 Feedback Control Systems...

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Unformatted text preview: Topic #15 16.31 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, December 1971. Estimation Theory (with noise) Kalman Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 151 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 = Kx u = Ky Same as the proportional feedback we looked at at the begin- ning 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 measured 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 20, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 152 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 20, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 153 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) Major Problem: We do not know x (0) 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 ) ....
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This note was uploaded on 11/07/2011 for the course AERO 16.31 taught by Professor Jonathanhow during the Fall '07 term at MIT.

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topic15 - Topic #15 16.31 Feedback Control Systems...

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