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State Space Models

Equations of motion for any physical system may be conveniently formulated in terms of the state of the system [333]:

$\displaystyle \underline{{\dot x}}(t) = f_t[\underline{x}(t),\underline{u}(t)] \protect$ (2.6)

Here, $ \underline{x}(t)$ denotes the state of the system at time $ t$ , $ \underline{u}(t)$ is a vector of external inputs (typically forces), and the general vector function $ f_t$ specifies how the current state $ \underline{x}(t)$ and inputs $ \underline{u}(t)$ cause a change in the state at time $ t$ by affecting its time derivative $ \underline{{\dot x}}(t)$ . Note that the function $ f_t$ 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.

Figure 1.4: Continuous-time state-space model $ \underline {{\dot x}}(t) = f_t[\underline {x}(t),\underline {u}(t)]$ .

The key property of the state vector $ \underline{x}(t)$ in this formulation is that it completely determines the system at time $ t$ , so that future states depend only on the current state and on any inputs at time $ t$ and beyond.2.8 In particular, all past states and the entire input history are ``summarized'' by the current state $ \underline{x}(t)$ . Thus, $ \underline{x}(t)$ must include all ``memory'' of the system.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2022-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University