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

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

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

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

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:

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

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