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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 16.323 Lecture 11 Estimators/Observers Bryson Chapter 12 Gelb Optimal Estimation Spr 2008 16.323 111 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. And certainly there is usually error in our knowledge of x . Usually can only feedback information that is developed from the sen sors measurements. Could try output feedback u = K x u = K y But this is type of controller is hard to design. 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: called a separation principle 1. Develop estimate of x ( t ) , called x ( t ) . 2. Then switch from u = K x ( t ) to u = K x ( t ) . Two key questions: How do we find x ( t ) ? Will this new plan work? (yes, and very well) June 18, 2008 Spr 2008 Estimation Schemes 16.323 112 Assume that the system model is of the form: x = A x + B u , x (0) unknown y = C y x where A , B , and C y are known possibly timevarying, but that is sup pressed here. u ( t ) is known Measurable outputs are y ( t ) from C y = I Goal: Develop a dynamic system whose state x ( t ) = x ( t ) t Two primary approaches: Openloop. Closedloop. June 18, 2008 Spr 2008 Openloop Estimator 16.323 113 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 + B u ( t ) Then x ( t ) x ( t ) t provided that x (0) = x (0) System A , B , C y x ( t ) y ( t ) u ( t ) Observer A , B , C y x ( t ) y ( t ) To analyze this case, start with: 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) June 18, 2008 Spr 2008 16.323 114 Subtract to get: d dt ( x x ) = A ( x x ) x ( t ) = A x 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 = 0 t ? Or even that x as t ? (which is a more realistic goal). Response is fine if x (0) = . But what if x (0) = ?...
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 Spring '08
 jonathanhow

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