## 5.4    Solution of the State-Space Equations in the z-Domain

We are given the state-space system.

$eq1127.jpg$

By taking the z-transform on the previous state equations, we will get

$eq1128.jpg$

Notice that z is a scalar and cannot be subtracted from A. Therefore, we write the previous equations as

$eq1129.jpg$

where I is the identity matrix. Solving the previous equation for the states we get

 $eq1130.jpg$ (5.11)

v(n) is the inverse transform of the previous equation. For the output y(n), we have

$eq1131.jpg$

This is a good place to try to find H(z) from the state-space equations. With v(0) = 0, we have

$eq1132.jpg$

or

 $eq1133.jpg$ (5.12)

## 5.5    General Solution ...

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