General Nonlinear ODE

In state-space form (§1.3.7) [449],^8.7a general class of th-order Ordinary Differential Equations (ODE), can be written as

where denotes time in seconds, denotes a vector of state variables at time ,

denotes the time derivative of , and is a vector (any length) of the system input signals, if any. Thus, Eq.(7.8) says simply that the time-derivative of the state vector is some function depending on time , the current state , and the current input signals . The basic problem is to solve for the state trajectory given its initial condition , the system definition function , and the input signals for all .

In the linear, time-invariant (LTI) case, Eq.(7.8) can be expressed in the usual state-space form for LTI continuous-time systems:

In this case, standard methods for converting a filter from continuous to discrete time may be used, such as the FDA (§7.3.1) and bilinear transform (§7.3.2).^8.8

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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.