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

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

- Converting to State-Space Form by Hand
- Signal Flow Graph to State Space Model
- Controllability and Observability
- A Short-Cut to Controller Canonical Form
- Matlab Direct-Form to State-Space Conversion
- State Space Simulation in Matlab
- Other Relevant Matlab Functions
- Matlab State-Space Filter Conversion Example

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