Numerical Integration

Recall the general state-space model in continuous time:

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

An approximate discrete-time numerical solution is

$$\underline{x}(t_n+T_n) \eqsp \underline{x}(t_n) + T_n\,f_{t_n}[\underline{x}(t_n),\underline{u}(t_n)]$$

for $n=0,1,2,\ldots\,.\quad$ (Forward Euler)

Let $g_{t_n}[\underline{x}(t_n),\underline{u}(t_n)] \isdef \underline{x}(t_n) + T_n\,f_{t_n}[\underline{x}(t_n),\underline{u}(t_n)]$ :

• This is a simple example of numerical integration for solving the ODE

• ODE can be nonlinear and/or time-varying

• The sampling interval $T_n$ may be fixed or adaptive