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# lec13 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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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 .

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16.323 Lecture 13 LQG Robustness Stengel Chapter 6 Question: how well do the large gain and phase margins discussed for LQR (6–29) map over to LQG?
Spr 2008 16.323 13–1 LQG When we use the combination of an optimal estimator and an optimal regulator to design the controller, the compensator is called Linear Quadratic Gaussian (LQG) Special case of the controllers that can be designed using the sep- aration principle. The great news about an LQG design is that stability of the closed-loop system is guaranteed . The designer is freed from having to perform any detailed mechanics - the entire process is fast and can be automated. So the designer can focus on the “performance” related issues, being confident that the LQG design will produce a controller that stabilizes the system. How to specify the state cost function (i.e. selecting z = C z x ) and what values of R zz , R uu to use. Determine how the process and sensor noise enter into the system and what their relative sizes are (i.e. select R ww & R vv ) This sounds great so what is the catch?? The remaining issue is that sometimes the controllers designed using these state-space tools are very sensitive to errors in the knowledge of the model. i.e., the compensator might work very well if the plant gain α = 1 , but be unstable if it is α = 0 . 9 or α = 1 . 1 . LQG is also prone to plant–pole/compensator–zero cancelation, which tends to be sensitive to modeling errors. J. Doyle, ”Guaranteed Margins for LQG Regulators”, IEEE Trans- actions on Automatic Control , Vol. 23, No. 4, pp. 756-757, 1978. June 18, 2008

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T,(s) if if and Spr 2008 16.323 13–2 June 18, 2008 Excerpt from document by John Doyle. Removed due to copyright restrictions.
Spr 2008 16.323 13–3 The good news is that the state-space techniques will give you a con- troller very easily. You should use the time saved to verify that the one you designed is a good controller. There are, of course, different definitions of what makes a controller good , but one important criterion is whether there is a reasonable chance that it would work on the real system as well as it does in Matlab. Robustness . The controller must be able to tolerate some modeling error, be- cause our models in Matlab are typically inaccurate.

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lec13 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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