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Signals and Systems by Smarajit Ghosh

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10.7 SOLUTION OF STATE EQUATION

The two methods to solve the state equations (i) z –transform method and (ii) series expansion method will be discussed in this article.

10.7.1 z-Transform Method

The solution of state equation of a discrete system is given by

 

image

 

where ϕ (k) = Inverse z–transform of [zIA]z

 

= z−1[zIA]−1 = state transition matrix.

 

The expression z –1[zIA] –1 can also be compared with resolvent matrix with the linear case where it is [sIA] –1. In this case it is multiplied by z and we then take the inverse to get state transition matrix.

Proof:

 

x (k + 1) = Ax (k) + Bu(k)     (10.18)

 

The z–transform of Eq. (10.18) ...

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