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MIT6_241JS11_lec21

# MIT6_241JS11_lec21 - 6.241 Dynamic Systems and Control...

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6.241 Dynamic Systems and Control Lecture 21: Minimal Realizations Readings: DDV, Chapters 25 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology April 25, 2011 E. Frazzoli (MIT) Lecture 21: Minimal Realizations April 25, 2011 1 / 12

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�� �� Reachable/(un)observable subspaces Recall: The set of reachable states is a subspace of the state space R n , given by Ra ( R n ) := Ra A n 1 B | . . . | AB | B . The set of unobservable states is a subspace of the state space R n , given by Null ( O n ) := Null C CA . . . CA n 1 . Both the reachable space and the unobservable space are A invariant, i.e., if x is reachable (resp., unobservable) so is Ax . E. Frazzoli (MIT) Lecture 21: Minimal Realizations April 25, 2011 2 / 12
Kalman Decomposition Construct an invertible matrix in the following way: T = T r T ¯ r = T ro ¯ T ro T ¯ ro ¯ T ¯ ro , where the columns of T r = [ T ro ¯ T ro ] form a basis for the reachable space. In particular, the columns of T ro ¯ are also in the unobservable space. the columns of T ¯ r [ T ¯ ro ¯ T ¯ ro ] complement the reachable space. In particular, the columns of T ¯ ro ¯ are also in the unobservable space. Note that any of the matrices appearing in the definition of T may in fact have 0 columns, i.e., not be present in particular instances (e.g., for reachable and observable systems, one would only have T ro ) Use the matrix T for a similarity transformation: ( A , B , C , D ) ( T 1 AT , T 1 B , CT , D ) = ( A ˆ , B ˆ , C ˆ , D ˆ ); this is called the Kalman decomposition . E. Frazzoli (MIT) Lecture 21: Minimal Realizations April 25, 2011 3 / 12

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Kalman Decomposition structure of the system matrix Based on the definition of T , one can
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MIT6_241JS11_lec21 - 6.241 Dynamic Systems and Control...

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