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Control Systems ILecture 9: The Nyquist condition

Readings: Guzzella, Chapter 9.4–6Astrom and Murray, Chapter 9.1–4

www.cds.caltech.edu/~murray/amwiki/index.php/First_Edition

Emilio Frazzoli

Institute for Dynamic Systems and ControlD-MAVT

ETH Zurich

November 17, 2017

E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 1 / 31

www.cds.caltech.edu/~murray/amwiki/index.php/First_Edition

Tentative schedule

# Date Topic

1 Sept. 22 Introduction, Signals and Systems2 Sept. 29 Modeling, Linearization

3 Oct. 6 Analysis 1: Time response, Stability4 Oct. 13 Analysis 2: Diagonalization, Modal coordi-

nates.5 Oct. 20 Transfer functions 1: Definition and properties6 Oct. 27 Transfer functions 2: Poles and Zeros7 Nov. 3 Analysis of feedback systems: internal stability,

root locus8 Nov. 10 Frequency response9 Nov. 17 Analysis of feedback systems 2: the Nyquist

condition

10 Nov. 24 Specifications for feedback systems11 Dec. 1 Loop Shaping12 Dec. 8 PID control13 Dec. 15 Implementation issues14 Dec. 22 Robustness


Putting it all together: Bode plots for complicated transfer functions


Example

Sketch the Bode plots of

G (s) = 100s + 4

s(s2 + 10s + 100)

First thing: write the transfer function in the “Bode” form:

G (s) = 4s/4 + 1

s(s2/100 + s/10 + 1)

Second: draw the Bode plot for each factor in the transfer function.

Third: add all of the above together to get the final Bode plot.


Example

G (s) = 4(s/4 + 1)

s(s2/100 + s/10 + 1)

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Bode Diagram

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Example

G (s) = 4(s/4 + 1)

s(s2/100 + s/10 + 1)

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Bode’s Law

In the Bode plot, the magnitude slope and the phase are not independent.

In particular, if the system is open-loop stable and minimum-phase, then ifthe slope of the Bode magnitude plot is κ db/decade over a range of morethan ≈ 1 decade, the phase in that range will be approximately κ · 90◦.


The polar plot


The polar plot

In the polar plot, the frequency response G (jω) is plotted on the complexplane as a parametric function of ω.

No special rules for drawing it, but the same principles we used in the Bodeplot apply.

In fact, it is convenient to sketch a Bode plot first, so that we can have agood idea of what the polar plot looks like, especially in view of the following.

The only things that really matter in the polar plot are:

Where the plot intersects the unit circle (|G(jω)| = 1)

Where the plot crosses the real axis (∠G(jω) = l · 180◦).


Polar plot — Integrator

Re

Im


Polar plot — single real, stable pole

Re

Im


Polar plot — complex-conjugate, stable poles

Re

Im


Polar plot — complicated transfer function

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Towards Nyquist’s theorem


The principle of variation of the argument

Let D ⊂ C be a bounded, simply-connected region of the complex plane, andlet Γ be its boundary.

As s moves along the closed curve Γ, G (s) describes another closed curve.

D

G(s)

Re

Im Im

Re

Remarkable fact: The number of times G (s) encircles the origin, or,equivalently, the total variation in its argument ∠G (s), as s moves along Γ,counts the number of zeros and poles of G (s) in D.


No poles/zeros in D

Remember that if G (s) = (s− z)/(s−p), then ∠G (s) = ∠(s− z)−∠(s−p).If D contains no poles/zeros, the net variation of the argument of ∠G (s)across one complete cycle of Γ is zero.

Re

Im


A zero in D

Remember that if G (s) = (s− z)/(s−p), then ∠G (s) = ∠(s− z)−∠(s−p).If D contains one zero, the net variation of the argument of ∠G (s) acrossone complete cycle of Γ is 2π.

Re

Im


A pole in D

Remember that if G (s) = (s− z)/(s−p), then ∠G (s) = ∠(s− z)−∠(s−p).If D contains one pole, the net variation of the argument of ∠G (s) acrossone complete cycle of Γ is −2π.

Re

Im


The general case

Re

Im

N = Z - PTheorem (Variation of the argument [Proof in A&M, pp. 277–278])

The number N of times that G (s) encircles the origin of the complex plane as smoves along the boundary Γ of a bounded simply-connected region of the planesatisfies

N = Z − P,

where Z and P are the numbers of zeros and poles of G (s) in D, respectively.Note that the encirclements are counted positive if in the same direction as smoves along Γ, and negative otherwise.


How do we use these results for feedback control?


The Nyquist or D contour

For closed-loop stability, the closed-loop poles, which corresponds to theroots (i.e., zeros!) of the characteristic polynomial 1 + kL(s), must havenegative real part.

The poles of 1 + kL(s) are also the poles of L(s).

Construct the region D as a D-shaped region containing an arbitrarily large(but finite) part of the complex right-half plane.

As s moves along the boundary of this region, 1 + kL(s) encircles the originN = Z − P times, where

Z is the number of unstable closed-loop poles (zeros of 1 + kL(s) in the rhp);

P is the number of unstable open-loop poles (poles of 1 + kL(s) in the rhp);


The Nyquist plot

The previous statement can be rephrased:

As s moves along the boundary of this region, L(s) encircles the −1/k pointN = Z − P times, where

Z is the number of unstable closed-loop poles (zeros of 1 + kL(s) in theNyquist contour);

P is the number of unstable open-loop poles (poles of 1 + kL(s) in theNyquist contour);

Symmetry of poles/zeros about the real axis implies that

∠L(−jω) = −∠L(jω),

hence the plot of L(s) when s moves on the boundary of the Nyquist contouris just the polar plot + its symmetric plot about the real axis. This is what iscalled the Nyquist plot.

The key feature of the Nyquist plot is the number of encirclements of the−1/k point.


The Nyquist condition

Theorem

Consider a closed-loop system with loop transfer function kL(s), which has Ppoles in the region enclosed by the Nyquist contour. Let N be the net number ofclockwise encirclements of −1/k by L(s) when s moves along the Nyquist contourin the clockwise direction. The closed loop system has Z = N + P poles in theNyquist contour.

In particular:

If the open-loop system is stable, the closed-loop system is stable as long asthe Nyquist plot of L(s) does NOT encircle the −1/k point.

If the open-loop system has P poles, the closed-loop system is stable as longas the Nyquist plot of L(s) encircles the −1/k point P times in the negative(counter-clockwise) direction.


Nyquist condition — single real, stable pole

L(s) =2

s + 1

Re

Im

Re

Im


Nyquist condition — open-loop unstable system

L(s) =s + 2

s2 − 1= − s + 2

−s2 + 1

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Im

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Im


Dealing with open-loop poles on the imaginary axis

If there are open-loop poles on theimaginary axis, make small“indentations” in the Nyquistcontour, e.g., leaving the imaginarypoles on the left.

Be careful on how you close theNyquist plot “at infinity:” If movingCCW around the poles, then closethe plot CW.

Re

Im


Nyquist — poles on the imaginary axis

L(s) =2

(s2 + 1)(s + 1)

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Im

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Im


The Nyquist condition on Bode plots

If the open-loop is stable, then we know that in order for the closed-loop tobe stable the Nyquist plot of L(s) should NOT encircle the −1 point.

In other words, |L(jω)| < 1 whenever ∠L(jω) = 180◦.

On the Bode plot, this means that the magnitude plot should be below the 0dB line if/when the phase plot crosses the −180◦ line.

Remember that this condition is valid only if the open loop is stable. In allother cases (including non-minimum phase zeros) it is strongly recommendedto double check any conclusion on closed-loop stability using other methods(Nyquist, root locus).


Gain and Phase Margin

The “distance” from the Nyquist plot to the −1 point is a measure of robustness.On the bode plot, it is easy to measure this distance in terms of gain and phasemargin.

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Frequency (rad/s)E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 30 / 31

Summary

In this lecture, we learned:

How to sketch a polar plot (and hence a Nyquist plot), based on Bode plots

The Nyquist condition to determine closed-loop stability using a Nyquist plot.

How to check the Nyquist condition on a Bode plot.

How to quickly assess the “robustness” of a feedback control system.

Now we have three graphical methods to study closed-loop stability given the(open-)loop transfer function.

1 Root locus: always correct if applicable (assumes finite-dimensional system)2 Nyquist: always correct, always appplicable;3 Bode: very useful for control system design, however may be misleading in

determining closed-loop stability (e.g., for open-loop unstable systems).

So far we have only looked at analysis issues, i.e., how to determineclosed-loop stability; from now on we will concentrate on control synthesis,i.e., how to design a feedback control system that makes a system behave asdesired.


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