lec11 - MIT OpenCourseWare http://ocw.mit.edu 16.323...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 time-varying, 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: Open-loop. Closed-loop. June 18, 2008 Spr 2008 Open-loop 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) = ?...
View Full Document

Page1 / 27

lec11 - MIT OpenCourseWare http://ocw.mit.edu 16.323...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online