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.

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Subsections

• Converting to State-Space Form by Hand
• Signal Flow Graph to State Space Filter
• 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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.