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

where denotes time in seconds, denotes a vector of

denotes the time derivative of , and 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 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.(7.9) 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).

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