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Linear Control Systems Lecture #3 - Frequency Domain Analysis
Guillaume Drion Academic year 2018-2019
1
Goal and Outline
Goal:
To be able to analyze the stability and robustness of a closed-loop system
Outline:
The loop transfer function
The Nyquist plot
The simplified and general Nyquist criteria
Stability margins
2
Design in frequency domain: transfer functions
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We work in frequency domain for the analysis and design of the stability and performance of control systems ( ).
Design in frequency domain: transfer functions
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In frequency domain, systems dynamics are described by a transfer function.
LTI system LTI system
Design in frequency domain: transfer functions
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The dynamical behavior of LTI systems depends on the shape of the transfer function (mostly its poles and zeros).
Poles
Design in frequency domain: transfer functions
The frequency response of a system can be analyzed using the Bode plots.
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Example: .
(http://lpsa.swarthmore.edu/Bode/Bode.html)
H(s) =100s+ 100
s2 + 110s+ 1000
Design in frequency domain: transfer functions
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Transfer functions are ideal for the study of system interconnections:
Series interconnection:
Feedback interconnection:
H1 H2
H=H1H2
H1
H2
H=H1/(1+H1H2)
Goal and Outline
Goal:
To be able to analyze the stability and robustness of a closed-loop system
Outline:
The loop transfer function
The Nyquist plot
The simplified and general Nyquist criteria
Stability margins
8
Stability of closed-loop systems
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Can we easily assess the stability of the closed-loop system (a) while designing the control system transfer function?
Nyquist: we can assess the stability of a closed-loop system by looking at the loop transfer function: (b).
This approach is very convenient for the design of control systems.
L(s) = P (s)C(s)
Stability of closed-loop systems
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Can we easily assess the stability of the closed-loop system (a) while designing the control system transfer function?
Nyquist: we can assess the stability of a closed-loop system by looking at the loop transfer function: (b).
This approach is very convenient for the design of control systems.
L(s) = P (s)C(s)
Condition of stability and the loop transfer function
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What are the conditions under which oscillations occur?Let’s first break the loop!
The limit of stability is when an oscillation is maintained over time, i.e. if then .
Knowing that, at that frequency, , the closed-loop system oscillate if .
B = !0A = !0
L(j!0) = �1B = �L(j!0)A
Condition of stability and the loop transfer function
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Nyquist approach: we look at the stability and robustness of a feedback system by looking at the properties of the loop transfer function.
Example: we need to tune the control system transfer function in order to avoid the value .
For this, Nyquist developed a specific tool: the Nyquist plot.
L(j!0) = �1
L(s) = P (s)C(s)
Goal and Outline
Goal:
To be able to analyze the stability and robustness of a closed-loop system
Outline:
The loop transfer function
The Nyquist plot
The simplified and general Nyquist criteria
Stability margins
13
The Nyquist plot
14
Nyquist plot: plot of the loop transfer function for different values of the complex frequency (i.e. mapping in the complex plane).
For this, we introduce the Nyquist D contour, which defines the path in the complex plane containing the values of for which we plot .
s = � + j!L(s)
s = � + j! L(s)
Mapping in the complex plane: examples
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To give intuition on how to sketch and interpret a Nyquist plot, we will first see some examples of mapping of functions in the complex plane.
Examples and matlab GUI are taken from “http://lpsa.swarthmore.edu/Nyquist/Nyquist.html”(the whole website is a great source of information!!!).
Mapping in the complex plane: examples
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Illustration: let’s take the contour (clockwise).
Effect of a zero: .
Same rotation as contour, shifted of to the right.
Effect of a single pole: .
Rotates in the opposite direction as the contour, and radius varies in the opposite direction.
Illustrations using the matlab function.
s = rej✓, ✓ = 0 ! �2⇡
L(s) = s+ a = rej✓ + a, ✓ = 0 ! �2⇡
a
L(s) =1
s=
1
rej✓=
1
re�j✓, ✓ = 0 ! �2⇡
The Nyquist D contour
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In general, goes to 0 as gets big. i.e. the semicircle at infinity maps to the origin in the Nyquist plot.
L(s)
The D contour does not include poles on the imaginary axis!
s
Examples of Nyquist plots using the matlab function.
The Nyquist D contour and Nyquist plot
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Example: Nyquist plot for . L(s) =1.4e�s
(s+ 1)2
Goal and Outline
Goal:
To be able to analyze the stability and robustness of a closed-loop system
Outline:
The loop transfer function
The Nyquist plot
The simplified and general Nyquist criteria
Stability margins
19
The simplified Nyquist criterion
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Let be the loop transfer function for a negative feedback system and assume that has no poles in the open right half-plane (except for single poles on the imaginary axis). Then the closed loop system is stable if and only if the closed contour given by has no net encirclements of the critical point .
L(s)L(s)
� = {L(j!) : �1 < ! < 1} ⇢ C
s = �1
The simplified Nyquist criterion
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Let be the loop transfer function for a negative feedback system and assume that has no poles in the open right half-plane (except for single poles on the imaginary axis). Then the closed loop system is stable if and only if the closed contour given by has no net encirclements of the critical point .
L(s)L(s)
� = {L(j!) : �1 < ! < 1} ⇢ C
s = �1
The Nyquist criterion does not require that !!! |L(j!)| < 1 8!
The Nyquist plot/criterion shows how system stability is influenced by changes in the controller parameters (see stability margins).
Correspondance between Nyquist plot and Bode plots
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The frequency at which the Nyquist curve crosses the real axis at negative values corresponds to the frequency at which the phase crosses -180° in the corresponding Bode plot.
Relationships between Nyquist plot and Bode plots
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Closed-loop systems can often be destabilized by an increase in feedback gain (= “radial dilation” of the Nyquist plot, in red).
Conditional stability
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There are, however, situations where a system can be stabilized by increasing the gain.
In other words, where the loop transfer function is unstable (poles in the right half-plane), but the closed-loop system is stable.
The general Nyquist criterion
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Consider a closed loop system with the loop transfer function that has P poles in the region enclosed by the Nyquist contour.
Let N be the net number of clockwise encirclements of −1 by when encircles the Nyquist contour in the clockwise direction. Counterclockwise encirclement: N=−1.
The closed loop system then has Z = N +P poles in the right half-plane.
L(s)
L(s)s �
The general Nyquist criterion
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Example: stabilizing the inverted pendulum using a PD controller.
L(s) = P (s)C(s) =1
s2 � 1k(s+ 2) (Pole at )s = 1
�2k
Goal and Outline
Goal:
To be able to analyze the stability and robustness of a closed-loop system
Outline:
The loop transfer function
The Nyquist plot
The simplified and general Nyquist criteria
Stability margins
27
Stability margins on the Nyquist plot
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: Gain margin : phase margin : stability margingm 'm sm
Gain crossover frequency
Phase crossover frequency
Gain margin
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Gain margin ( ): smallest increase of the open-loop gain at which the closed-loop system becomes unstable
gm
Gain margin is infinite if the phase of never crosses -180°!L(s)
Phase margin
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Phase margin ( ): 180° + phase at unit gain. Any time delay in the system rotates the Nyquist curve, hence reduces the phase margin.
Phase margin is infinite if the gain of is always smaller than1.L(s)
'm
Sensitivity margin
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Sensitivity margin ( ): Shortest distance from the Nyquist curve to the critical point (-1).
Sensitivity margin is related to disturbance attenuation.
sm
Stability margins on the Nyquist plot
32
: Gain margin : phase margin : stability margingm 'm sm
Gain crossover frequency
Phase crossover frequency
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