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jms_MAE154S_discussion_notes - 1 MAE154S Supplementary...

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Unformatted text preview: 1 MAE154S Supplementary Discussion Notes In discussion this past Thursday, I o ered the following statement without proof: Given the state-space system ˙ ~x = A~x + B~u A ∈ C nxn , B ∈ C nxm ~ y = C~x + D~u, C ∈ C pxn , D = 0 nxm If you implement a feedback control system which feeds the output ~ y (premultiplied by some gain K) back to the input ~u , the augmented A matrix (system dynamics with feedback) will be: A aug = A + BKC, K ∈ C mxp (1) This is fairly easy to show. Note that it is required that there is no input feedthrough (D = 0). I presented the statement in full generality, but for the purposes of this class, let's consider a single-input, single-output (SISO) system. If you see a problem like this on the nal, it will de nitely be like that. This assumption entails setting n equal to the state dimension, and m = p = 1 . This means A is a square matrix with n rows and columns, B is a column vector with length n , and C is a row vector with length n ....
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