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# 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. 596–602, 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 15–1 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. • Usually can only feedback information that is developed from the sensors measurements. – Could try “output feedback” u = Kx ⇒ u = ˆ Ky – Same as the proportional feedback we looked at at the begin- ning 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 measured output of the system. • New plan: 1. Develop estimate of x ( t ) that will be called ˆ x ( t ) . 2. Then switch from u ( t ) = − K x ( t ) to u ( t ) = − K ˆ x ( t ) . • Two key questions: – How do we find ˆ 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 15–2 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 ˆ 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 15–3 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 ˙ ˆ x ( t ) = A ˆ x ( t ) + B u ( t ) • Then ˆ x ( t ) ≡ x ( t ) ∀ t provided that ˆ x (0) = x (0) • Major Problem: We do not know x (0) • Analysis of this case: ˙ x ( t ) = A x ( t ) + B u ( t ) ˙ ˆ x ( t ) = A ˆ x ( t ) + B u ( t ) • Define the estimation error ˜ x ( t ) = x ( t ) − ˆ x ( t ) ....
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topic15 - Topic#15 16.31 Feedback Control Systems...

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