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General Nonlinear ODE

In state-space form (§1.3.7) [452],8.7a general class of $ N$ th-order Ordinary Differential Equations (ODE), can be written as

$\displaystyle \dot{\underline{x}}(t) \eqsp f(t,\underline{x},\underline{u}) \protect$ (8.9)

where $ t$ denotes time in seconds, $ \underline{x}(t)$ denotes a vector of $ N$ state variables at time $ t$ ,

$\displaystyle \dot{\underline{x}}(t) \isdefs \frac{d}{dt}\underline{x}(t)

denotes the time derivative of $ \underline{x}(t)$ , and $ \underline{u}(t)$ is a vector (any length) of the system input signals, if any. Thus, Eq.(7.9) says simply that the time-derivative of the state vector is some function $ f$ depending on time $ t$ , the current state $ \underline{x}(t)$ , and the current input signals $ \underline{u}(t)$ . The basic problem is to solve for the state trajectory $ \underline{x}(t)$ given its initial condition $ \underline{x}(0)$ , the system definition function $ f$ , and the input signals $ \underline{u}(t)$ for all $ t\ge 0$ .

In the linear, time-invariant (LTI) case, Eq.(7.9) can be expressed in the usual state-space form for LTI continuous-time systems:

$\displaystyle \frac{d}{dt}\underline{x}(t) \eqsp A\,\underline{x}(t) + B\,\underline{u}(t) \protect$ (8.10)

In this case, standard methods for converting a filter from continuous to discrete time may be used, such as the FDA7.3.1) and bilinear transform7.3.2).8.8

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University