Continuous Time Control Systems

We typically assume the plant satisfies an LTI ODE of the form:

• ${\sum }_{i=0}^n{a}_i\frac{d^{i}y}{d{t}^i}\left(+\right)-{\sum }_{j=0}^m{b}_j\frac{d^{j}u}{d{t}^j}(+)=0$ (needs to sum to zero) Where $a_i$, $b_j$ are real numbers

To go from the time domain to the frequency domain, we can apply a one-sided LT with zero initial conditions $Y\left(s\right)=\frac{{\sum }_{j=0}^m{b}_j{s}^j}{{\sum }_{i=0}^m{a}_i{s}^j}U\left(s\right)$ Where $Y\left(s\right)=P\left(s\right)U\left(s\right)$

How do we discretize a continuous jawn

See Direct Design of DT Controllers but for something with no integrator(s)…
$G[z]={\sum }_{i=1}^n\frac{c_i}{{\lambda }_i}\frac{e^{{\lambda }_{i}T}-1}{z-e^{{\lambda }_{i}T}}$

Related Topics

• Continuous Time