CHEE434 notes nyquiststability 2011 · Nyquist Stability Criterion “If N is the number of times that the Nyquist plot encircles the point (-1,0) in the complex plane in the clockwise

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Nyquist Plot

Plot of in the complex plane as is varied on

Relation to Bode plot

AR is distance of from the origin Phase angle, , is the angle from the Real positive axis

Nyquist path

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Nyquist Nyquist PlotPlot

First order systemFirst order system

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Nyquist Nyquist plotplot

Second order processSecond order process

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Third order process:Third order process:

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Nyquist Nyquist PlotPlot

Delayed systemDelayed system

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System withSystem with pole at zeropole at zero Pole atPole at is on the is on the Nyquist Nyquist path - no finite dc gainpath - no finite dc gain

Strategy is to go around the singularity aboutStrategy is to go around the singularity about in the rightin the righthalf planehalf plane ( a small number)( a small number)

x

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Pole at zeroPole at zero

xx

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System with integral action:System with integral action:

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Che 446: Process Dynamics andControl

Frequency DomainController Design

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PI Controller

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PD Controller

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PID Controller

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Bode Stability Criterion

Consider open-loop control system

1. Introduce sinusoidal input in setpoint ( ) and observe sinusoidaloutput

2. Fix gain such and input frequency such that3. At same time, connect close the loop and set

Q: What happens if ?

+-++

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“A closed-loop system is unstable if the frequency of theresponse of the open-loop GOL has an amplitude ratiogreater than one at the critical frequency. Otherwise it isstable. “

Strategy:

1. Solve for ω in

2. Calculate AR

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To check for stability:

1. Compute open-loop transfer function2. Solve for ω in φ=-π3. Evaluate AR at ω4. If AR>1 then process is unstable

Find ultimate gain:

1. Compute open-loop transfer function without controller gain2. Solve for ω in φ=-π3. Evaluate AR at ω4. Let K

ARcu =1

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Bode Criterion

Consider the transfer function and controller

- Open-loop transfer function

- Amplitude ratio and phase shift

- At ω=1.4128, φ=-π, AR=6.746

G s es s

s( )

( )( . )

.=

+ +

!51 05 1

01

G s es s sOL

s( )

( )( . ).

.

.=

+ ++!

"#$%&

'51 05 1

0 4 1 101

01

AR =+ +

+

= ! ! ! ! "#$

%&'

! ! !

5

1

1

1 0 250 4 1 1

0 01

01 05 101

2 2 2

1 1 1

( ( (

) ( ( ((

..

.

. tan ( ) tan ( . ) tan.

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Bode CriterionBode Criterion

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Ziegler-Nichols Tuning

Closed-loop tuning relation

With P-only, vary controller gain until system (initially stable) starts tooscillate.

Frequency of oscillation is ωc,

Ultimate gain, Ku, is 1/M where M is the amplitude of the open-loopsystem

Ultimate Period

Ziegler-Nichols Tunings

P Ku/2PI Ku/2.2 Pu/1.2PID Ku/1.7 Pu/2 Pu/8

Puc

= 2!"

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Bode Stability CriterionBode Stability Criterion

Using Bode plots,Using Bode plots, easy criterion to verify for closed-loopeasy criterion to verify for closed-loopstabilitystability More general than polynomial criterion such as More general than polynomial criterion such as Routh Routh Array,Array,

DirectDirect Substitution,Substitution, Root locusRoot locus

Applies to delay systems without approximationApplies to delay systems without approximation

Does not require explicit computation of closed-loop polesDoes not require explicit computation of closed-loop poles

Requires that a unique frequency yield phase shift of -180 degrees Requires that a unique frequency yield phase shift of -180 degrees

Requires monotonically decreasingRequires monotonically decreasing phase shiftphase shift

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Nyquist Nyquist Stability CriterionStability Criterion

ObservationObservation Consider the transfer functionConsider the transfer function

Travel on closed path containingTravel on closed path containing in thein the domain in the domain in theclockwise directionclockwise direction

inin domain,domain, travel on closed path encircling the origin in thetravel on closed path encircling the origin in theclockwise directionclockwise direction

Every encirclement of Every encirclement of in s domain leads to one encirclement ofin s domain leads to one encirclement ofthe origin in domainthe origin in domain

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ObservationObservation Consider the transfer functionConsider the transfer function

Travel on closed path containingTravel on closed path containing in thein the domain in the domain in theclockwise directionclockwise direction

inin domain,domain, travel on closed path encircling the origin in thetravel on closed path encircling the origin in thecounter clockwisecounter clockwise directiondirection

Every encirclement of Every encirclement of in s domain leads to one encirclement orin s domain leads to one encirclement orthe origin inthe origin in domaindomain

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For a general transfer functionFor a general transfer function

For every zero inside the closed path in theFor every zero inside the closed path in the domain, every domain, everyclockwise encirclement around the path gives one clockwiseclockwise encirclement around the path gives one clockwiseencirclement of the origin in theencirclement of the origin in the domaindomain

For every pole inside the closed path in theFor every pole inside the closed path in the domain, everydomain, everyclockwise encirclement around the path gives one counterclockwise encirclement around the path gives one counterclockwise encirclement of the origin in theclockwise encirclement of the origin in the domaindomain

The number of clockwise encirclements of the origin inThe number of clockwise encirclements of the origin indomaindomain is equal to number of zerosis equal to number of zeros less the number ofless the number ofpolespoles inside the closed path in the inside the closed path in the domain. domain.

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To assess closed-loop stability:To assess closed-loop stability: Need to identify unstable poles of the closed-loop systemNeed to identify unstable poles of the closed-loop system

That is, the number of poles in the RHPThat is, the number of poles in the RHP

Consider the transfer functionConsider the transfer function

Poles of the closed-loop system are the zeros ofPoles of the closed-loop system are the zeros of

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PathPath of interest in the domain must encircle the entireof interest in the domain must encircle the entireRHPRHP

• Travel around a half semi-circle or radius that encircles the entire RHP• For a proper transfer function

• Every point on the arc of radius are such that

collapse to a single point in domain

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Path of interest is the Path of interest is the Nyquist Nyquist pathpath

Consider the Consider the Nyquist Nyquist plot ofplot of

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Compute the poles ofCompute the poles of

AssumeAssume thatthat it has it has unstable polesunstable poles

Count the number of clockwise encirclements of the origin of theCount the number of clockwise encirclements of the origin of theNyquist Nyquist plot ofplot of Assume that itAssume that it makes makes clockwise encirclementsclockwise encirclements

The number of unstable zeros of The number of unstable zeros of (the number of unstable(the number of unstablepoles of poles of the closed-loop system)the closed-loop system)

Therefore the closed-loop system is unstableTherefore the closed-loop system is unstable ifif

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ConsiderConsider the open-loop transfer functionthe open-loop transfer function

has the same poleshas the same poles asas

The origin inThe origin in domaindomain corresponds to the point (-1,0) in thecorresponds to the point (-1,0) in thedomaindomain

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Nyquist Nyquist StabilityStability

In terms of theIn terms of the open-loop transfer functionopen-loop transfer function

The number of polesThe number of poles ofof givesgives

The number of clockwise encirclements ofThe number of clockwise encirclements of (-1,0)(-1,0) ofofgives the number of clockwise encirclements of the origingives the number of clockwise encirclements of the origin ofof

The number of unstable polesThe number of unstable poles of the closed-loop system is givenof the closed-loop system is givenbyby

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Nyquist Stability Criterion

“If N is the number of times that the Nyquist plot encircles the point (-1,0) in the complex plane in the clockwise direction, and P is thenumber of open-loop poles of GOL that lie in the right-half plane, thenZ=N+P is the number of unstable poles of the closed-loopcharacteristic equation.”

Strategy

1. Compute the unstables poles of2. Substitute s=jω in GOL(s)3. Plot GOL(jω) in the complex plane4. Count encirclements of (-1,0) in the clockwise direction

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Nyquist Criterion

Consider the transfer function

and the PI controller

The open-loop transfer function is

No unstable polesOne pole on the Nyquist path (must go around small circle around

the origin in the right half plane)

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There are 2 clockwise encirclements of (-1,0)There are 2 clockwise encirclements of (-1,0)

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Counting encirclements must account for removal of originCounting encirclements must account for removal of origin

the closed-loop system is unstablethe closed-loop system is unstable

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Nyquist Criterion






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Nyquist Criterion






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Bode StabilityBode Stability

Stability margins for linear systemsStability margins for linear systems

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Stability Considerations

Control is about stability

Considered exponential stability of controlled processes using: Routh criterion Direct Substitution Root Locus Bode Criterion (Restriction on phase angle) Nyquist Criterion

Nyquist is most general but sometimes difficult to interpret

Roots, Bode and Nyquist all in MATLAB

Polynomial (no dead-time)

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Stability MarginsStability Margins

Stability margins for linear systemsStability margins for linear systems

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Stability MarginsStability Margins

Gain margin:Gain margin: LetLet be the amplitude ratio ofbe the amplitude ratio of at the criticalat the critical

frequencyfrequency

Phase margin:Phase margin: LetLet be the phase shift ofbe the phase shift of at the frequency whereat the frequency where

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Stability MarginStability Margin

Documents

CHEE434 notes nyquiststability 2011 · Nyquist Stability Criterion “If N is the number of times that the Nyquist plot encircles the point (-1,0) in the complex plane in the clockwise