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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 11–1 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 ˆ 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 ˆ( t ) . 2. Then switch from u = − K x ( t ) to u = − K x ˆ( t ) . • Two key questions: – How do we find x ˆ( t ) ? – Will this new plan work? (yes, and very well) June 18, 2008 Spr 2008 Estimation Schemes 16.323 11–2 • 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 ˆ( t ) = x ( t ) ∀ t ≥ Two primary approaches: – Openloop. – Closedloop. June 18, 2008 Spr 2008 Openloop Estimator 16.323 11–3 • 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 + B u ( t ) – Then x ˆ( t ) ≡ x ( t ) ∀ t provided that x ˆ(0) = x (0) System A , B , C y ⇒ x ( t ) y ( t ) u ( t ) Observer A , B , C y ⇒ ˆ x ( t ) ˆ y ( t ) • To analyze this case, start with: 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) June 18, 2008 Spr 2008 16.323 11–4 • Subtract to get: d dt ( x − x ˆ) = A ( x − x ˆ) ⇒ 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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This note was uploaded on 11/07/2011 for the course AERO 16.323 taught by Professor Jonathanhow during the Spring '08 term at MIT.
 Spring '08
 jonathanhow

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