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Unformatted text preview: Topic #21 16.31 Feedback Control Systems Robustness Analysis Model Uncertainty Robust Stability (RS) tests RS visualizations 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 16.31 211 Model Uncertainty Want to consider what happens if our model of the system is incorrect actual system dynamics G A ( s ) are in one of the sets Multiplicative model G p ( s ) = G N ( s )(1 + ( s )) Additive model G p ( s ) = G N ( s ) + ( s ) where 1. G N ( s ) is the nominal dynamics ( known ) 2. ( s ) is the modeling error not known directly , but bound E ( s ) known (assumed stable) where  ( j )   E ( j )  If E ( j ) small, our confidence in the model is high nominal model is a good representation of the actual dynamics If E ( j ) large, our confidence in the model is low nominal model is not a good representation of the actual dynamics 101 10 10 1 10 2 106 105 104 103 102 101 10 Freq (rad/sec) G G N multiplicative uncertainty Figure 1: Typical system TF with multiplicative uncertainty November 18, 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 16.31 212 Simple example: Assume we know that the actual dynamics are G A ( s ) = 2 n s 2 ( s 2 + 2 n s + 2 n ) but we take the nominal model to be G N = 1 /s 2 . Can explicitly calculate the error E ( s ) , and it is shown in the plot. 101 10 10 1 104 103 102 101 10 10 1 10 2 10 3 Freq (rad/sec) G G N E=G A /G N1 E G A G A G N E E Figure 2: Various TFs for the example system Can also calculate an LTI overbound E ( s ) of the error. Since E ( s ) is not normally known, it is the bound E ( s ) that is used in our analysis tests. November 18, 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 16.31 213 101 10 10 1 106 104 102 10 10 2 10 4 Freq (rad/sec) G G N Possible Gs given E G A G N Figure 3: G N with one partial bound. Can add many others to develop the overall bound that would completely include G A . Usually E ( j ) not known, so we would have to develop it from our approximate knowledge of the system dynamics. Want to demonstrate that the system is stable for any possible perturbed dynamics in the set G p ( s ) Robust Stability November 18, 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...
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This note was uploaded on 11/07/2011 for the course AERO 16.31 taught by Professor Jonathanhow during the Fall '07 term at MIT.
 Fall '07
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