Numerical Integration

Starting with a CT System $x^{\prime }=f(x,u$, $y=g(x,u)$, our goal is to determine the trajectory $x(t)$ for the system. However, our challenge here, is that it is computers operate in discrete time.

The solution here, is approximating the continuous time system with a related DT system.

$x={\int }_0^{t}f(x(s),u(s))ds={\int }_0^{T}f(x(s),u(s))ds+{\int }_T^{2T}f(x(s),u(s))ds+...+{\int }_{(n-1)T}^{nT}f(x(s),u(s))ds$
Assuming $t=nT$ for some fixed $t>0$

$={\sum }_{k=0}^{n-1}{\int }_{kT}^{(k+1)T}f(x(s),u(s))ds$
Let the entire integral term be $\Delta [k+1]$… $={\sum }_{k=0}^{n-1}\Delta [k+1]$

$x(kT)=x((k-1)T)+\Delta [k]$

The idea here is to approximate $Delta[k+1]$ using the values of f at some finite number of points.

• Integration Approximation Techniques