Difference Equations to State Space

Any explicit LTI difference equation (§5.1) can be converted to state-space form. In state-space form, many properties of the system are readily obtained. For example, using standard utilities (such as in Matlab), there are functions for computing the modes of the system (its poles), an equivalent transfer-function description, stability information, and whether or not modes are ``observable'' and/or ``controllable'' from any given input/output point.

Every $n$th order scalar (ordinary) difference equation may be reformulated as a first order vector difference equation. For example, consider the second-order difference equation

$$y(n) = u(n) + 2u(n-1) + 3u(n-2) - \frac{1}{2}y(n-1) - \frac{1}{3}y(n-2). \protect$$ (E.7)

We may define a vector first-order difference equation--the ``state space representation''--as discussed in the following sections.