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System identification and self regulating systems

System identification and self regulating systems. Discrete Equivalents - Overview. r(t). e(t). controller D(s). u(t). plant G(s). y(t). +. -. Translation to discrete plant Zero order hold (ZOH). Translation to discrete controller (emulation) Numerical Integration

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System identification and self regulating systems

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  1. System identification and self regulating systems

  2. Discrete Equivalents - Overview r(t) e(t) controller D(s) u(t) plant G(s) y(t) + - Translation to discrete plant Zero order hold (ZOH) 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) 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.

  3. 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

  4. Numerical Integration

  5. 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 kT-T kT kT-T kT

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

  7. Numerical Integration • Backward rectangular rule (app kT)

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

  9. Numerical Integration • Comparison with H(s)

  10. Numerical Integration • Transform s ↔ z • Comparison with respect to stability • In the s-plane, s = jw is the boundary between stability and instability.

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

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