22.11.2019

RTI Vorlesung 10

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Type and relative degree

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Corresponding Nyquist Diagram

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Re

Im

-1

Bode’s law

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The gradient of the magnitude plot (! ⋅ 20 ⁄&' &()) determines the phase shift (! ⋅ ⁄* +).

FD system identification

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Complete model description: Σ " ,$2(")

FD I/O measurement

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Measure the frequency response ! times at each frequency "#:

magnitude $ and phase %at frequencies "# (& = 1,2, … , ,)and measurement . (. = 1,2, … , !)

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FD I/O nominal system identification

FD I/O uncertainty identification

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Here:decimal values, not in dB

Model uncertainty is ≥ 100% as of from %&

Plot the absolute value of the relative error and find an upper bound '& (% for it:

20.09. Lektion 1 – Einführung

27.09. Lektion 2 – Modellbildung4.10. Lektion 3 – Systemdarstellung, Normierung, Linearisierung

11.10. Lektion 4 – Analyse I, allg. Lösung, Systeme erster Ordnung, Stabilität18.10. Lektion 5 – Analyse II, Zustandsraum, Steuerbarkeit/Beobachtbarkeit

25.10. Lektion 6 – Laplace I, Übertragungsfunktionen1.11. Lektion 7 – Laplace II, Lösung, Pole/Nullstellen, BIBO-Stabilität8.11. Lektion 8 – Frequenzgänge (RH hält VL)

15.11. Lektion 9 – Systemidentifikation, Modellunsicherheiten 22.11. Lektion 10 – Analyse geschlossener Regelkreise 29.11. Lektion 11 – Randbedingungen

6.12. Lektion 12 – Spezifikationen geregelter Systeme13.12. Lektion 13 – Reglerentwurf I, PID (RH hält VL)20.12. Lektion 14 – Reglerentwurf II, „loop shaping“

Modellierung

Systemanalyse im Zeitbereich

Systemanalyse im Frequenzbereich

Reglerauslegung

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Closed-loop system

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Loop gain L(s)

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The loop gain is the open-loop transfer function from ! → #.

Sensitivity S(s)

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The sensitivity is the closed-loop transfer function from ! → #(and from $ → %).

Complementary sensitivity T(s)

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The complementary sensitivity is the closed-loop transfer function from ! → # (and from $ → #).

Graphical interpretation

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General properties

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For very large and very small !(#), the following approximations hold:

Moreover, the two vectors (in the complex plane) % # and &(#)always add up to 1. Therefore, at a specific frequency, only one of them can be substantially smaller than 1:

Recap: Disturbance and Noise

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The disturbance !(#) is an external signal, which causes a deviation of the actual plant output % # .

The noise '(#) is an external signal, which causes a deviation of the measurement of the plant output % # .

Typically, the disturbance is a low-frequency signal, whereas the noise is a high-frequency signal.

If this frequency separation is not present then one cannot guarantee simultaneously disturbance rejection and noise attenuation.

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QC: What is a “reasonable” loop gain L(s)?

Controller design rule # 1: Don’t cancel unstable poles with non-minimum phase zeros

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Transfer function from ! → #?

Transfer function from ! → $?

% & = 1& + * + 1

Example:

Internal closed-loop stability

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For «internal» (complete) stability, all nine transfer functions

must be asymptotically stable.

Alternative: closed-loop completely asymptotically stable iff1 + # $ has only zeros with negative real parts (all poles of % $ or &($) have negative real parts), and no cancellations of unstable poles with non-minimum phase zeros occur in the loop gain.

From now on this is always assumed.

Nyquist theorem

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Nyquist Theorem: Let !" be the number of poles with positive real part and !# the number of poles with zero real part of the open-loop transfer function $(&), and let !( be the number of encirclements of the point −1 by $()*) counted positively in the counter-clockwise direction when * is changed from −∞ to +∞.

Then the closed-loop system is asymptotically stable iffthe number !( = ⁄!# 0 + !".

Wow!

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1. We now can test a time-domain property (asymptotic stability) by working only in the frequency domain with transfer functions.

2. We can decide if the closed-loop system ! " (nonlinear function of #(")) is asymptotically stable by analyzing the open-loop transfer function & " (linear function of #(")).

3. Even better: we don’t need & " , knowing the frequency response of & () is sufficient.

4. The Nyquist theorem provides quantitative information about “how stable” a closed-loop system is and with that one can design (“optimize”) controllers #(").

Number of encirclements

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0"

0#∞−∞

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−∞∞

0$0%

Number of encirclements

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Number of encirclements

0"

0#

+∞−∞

Example: Stability

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! " = $ " ⋅ & " = 100") + 12" + 100 ⋅ 0.1

0.1" + 10.1"

Is the corresponding closed-loop system asymptotically stable?

+∞−∞

0/

00

Proof of Nyquist theorem

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Using the “Principle of Arguments”, details: see Appendix B.3

A few remarks:

• The Nyquist theorem is valid for all SISO LTI systems, even those with non-rational transfer functions (delays).

• The proof of the Nyquist theorem is based on fundamental results of complex analysis; it does not rely on the frequency response interpretation (which works only for BIBO stable systems).

• Accordingly, even if L(s) is not BIBO stable, its frequency response L(jw) and the Nyquist theorem can be used.

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The phase margin ! is the distance from -180° where " #$ enters the unit circle in the Nyquist diagram (magnitude 1).

The gain margin % is the the inverse of the magnitude of " #$ at -180°

The minimum return difference &min is the minimum distance from -1.

Robustness measures

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Robustness measuresUncertainty model 1:

Closed loop system remains asymptotically stable as long as

!nom =0

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Robustness measures

!nom = 1

Uncertainty model 2:

Closed loop system remains asymptotically stable as long as

Example

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! =

# =

Robust Nyquist theorem

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Robust Nyquist Theorem: The uncertain closed-loop system is asymptotically stable if the nominal closed-loop system is asymptotically stable and the following inequality is satisfied