System identification and self regulating systems

Discrete Equivalents - Overview controller D(s) plant G(s) r(t)u(t)y(t)e(t) + - Translation to discrete controller (emulation) Numerical Integration Forward rectangular rule Backward rectangular rule Trapeziod rule (Tustin’s method, bilinear transformation) Bilinear with prewarping Zero-Pole Matching Hold Equivalents Zero order hold (ZOH) Triangle hold (FOH) Translation to discrete plant Zero order hold (ZOH) Emulation Purpose: Find a discrete transfer function which approximately has the same characteristics over the frequency range of interest. Digital implementation: Control part constant between samples. Plant is not constant between samples.

Numerical Integration Fundamental concept –Represent H(s) as a differential equation. –Derive an approximate difference equation. We will use the following example –Notice, by partial expansion of a transfer function this example covers all real poles. Example Transfer function Differential equation

Numerical Integration

Now, three simple ways to approximate the area. –Forward rectangle approx. by looking forward from kT-T –Backward rectangle approx. by looking backward from kT –Trapezoid approx. by average kT-T kT

Numerical Integration Forward rectangular rule (Euler’s rule) (Approximation kT-T)

Numerical Integration Backward rectangular rule (app kT)

Numerical Integration Trapezoid rule (Tustin’s Method, bilinear trans.) (app ½(old value + new value))

Numerical Integration Comparison with H(s)

Numerical Integration Transform s ↔ z Comparison with respect to stability –In the s-plane, s = j  is the boundary between stability and instability.

The rest of this power point is not required in the exam Just for completeness purpose