L. Giarr´e 2017-2018
Systems and Control Theory Lecture Notes
Laura Giarr´e
Lesson 18: Minimal State-Space Realization
I Kalman Decomposition
I Interconnections and minimality
The Kalman Decomposition
I Suppose we construct a transformation matrix T = [Tr ¯o Tro T¯r ¯o T¯r o] such that
1. The columns of Tr ¯o form a basis for RTO, the subspace that¯ is both reachable and unobservable
2. Tro complements Tr ¯o in the reachable subspace, so that Ra[Tr ¯o Tro] = R
3. T¯r ¯o complements Tr ¯o in the unobservable subspace, so that Ra[Tr ¯o T¯r ¯o] = O
4. T¯r o complements the rests of the column to span Rn, so that T is invertible.
I Perform a similarity transformation using T
I The system (ˆA, ˆB, ˆC , D) is said to be in Kalman Decomposed form
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Kalman Decomposition
Minimality
I Theorem
A state-space realization of a SISO transfer function H is minimal iff it is reachable and observable.
I Theorem
All minimal realizations of a given transfer function are similar to each other.
I Note the relation with Interconnected systems and pole/zero cancellation.
I Note the relation among internal and external stability.
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Thanks
DIEF- laura.giarre@unimore.it Tel: 059 2056322