State Space Models

Equations of motion for any physical system may be conveniently formulated in terms of the state of the system [330]:

Here, denotes the state of the system at time , is a vector of external inputs (typically forces), and the general vector function specifies how the current state and inputs cause a change in the state at time by affecting its time derivative . Note that the function may itself be time varying in general. The model of Eq.(1.6) is extremely general for causal physical systems. Even the functionality of the human brain is well cast in such a form.

Equation (1.6) is diagrammed in Fig.1.4.

The key property of the state vector in this formulation is that it completely determines the system at time , so that future states depend only on the current state and on any inputs at time and beyond.^2.8 In particular, all past states and the entire input history are ``summarized'' by the current state . Thus, must include all ``memory'' of the system.

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Subsections

• Forming Outputs
• State-Space Model of a Force-Driven Mass
• Numerical Integration of General State-Space Models
• State Definition

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About the Author: 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.