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Unformatted text preview: Topic #15 16.31 Feedback Control Systems StateSpace Systems Openloop Estimators Closedloop 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: Openloop. Closedloop. 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 Openloop 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.
 Fall '07
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