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Systems and Control Theory Lecture Notes

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L. Giarr´e 2017-2018

Systems and Control Theory Lecture Notes

Laura Giarr´e

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Lesson 18: Minimal State-Space Realization

I Kalman Decomposition

I Interconnections and minimality

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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

L. Giarr´e- Systems and Control Theory 2017-2018 - 3

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Kalman Decomposition

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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.

L. Giarr´e- Systems and Control Theory 2017-2018 - 5

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Thanks

DIEF- laura.giarre@unimore.it Tel: 059 2056322

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