For time invariant system, the state equation is given by

which is divided into two types as follows:

- homogeneous state equation
- non-homogeneous state equation

The equation is said to be homogeneous if **A** is a constant matrix and **U** is a zero vector. In this case no control forces are applied. The equation is said to be non-homogeneous if **A** is constant matrix and **U** is a non-zero vector. In the first case discharging a charged capacitor is an example while in the second case total response is an example.

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