chee 319 process dynamics and control - …€¦ · 1 chee 319 process dynamics and control...

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CHEE 319Process Dynamics and Control

Frequency Response ofFrequency Response ofLinear Control SystemsLinear Control Systems

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Frequency ResponseFrequency Response

Response of a linear control systemResponse of a linear control system

to a sinusoidal input signal as a function of frequencyto a sinusoidal input signal as a function of frequency

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FirstFirst Order ProcessOrder Process

Sinusoidal responseSinusoidal response

Solving forSolving for ,, and and taking the inverse and and taking the inverse Laplace Laplace transformtransform

Sinusoidal response is the ultimate response of theSinusoidal response is the ultimate response of the 1st1storder systemorder system

rearrangingrearranging

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First order ProcessFirst order Process

Response to a sinusoidal input signal

Recall: Sinusoidal input yields sinusoidal outputcharacterized by and

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

2

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First Order ProcessFirst Order Process

Sinusoidal response of a 1st order process to a sinusoidalSinusoidal response of a 1st order process to a sinusoidalinput signal is:input signal is: Sinusoidal with amplitudeSinusoidal with amplitude

and phase lagand phase lag

Important characteristics of the sinusoidal response of the processImportant characteristics of the sinusoidal response of the processare the amplitude ratioare the amplitude ratio

and the phase lag, both functions of frequency and the phase lag, both functions of frequency . .

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Second Order ProcessSecond Order Process

Sinusoidal ResponseSinusoidal Response

wherewhere

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Frequency Response

Q: Do we “have to” take the Laplace inverse to compute theAR and phase shift of a 1st or 2nd order process?

No

Q: Does this generalize to all transfer function models?

Yes

Study of transfer function model response to sinusoidal inputsis called “Frequency Domain Response” of linearprocesses.

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Frequency Response

Some facts for complex number theory:

i) For a complex number:

It follows that where

such that

Re

Im

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Frequency Response

Some facts:ii) Let and then

iii) For a first order process

Let

Then

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Frequency Response

Main Result:

The response of any linear process to a sinusoidalinput is a sinusoidal.

The amplitude ratio of the resulting signal is given by theModulus of the transfer function model expressed in thefrequency domain, .

The Phase Shift is given by the argument of the transferfunction model in the frequency domain.

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For a general transfer function

Frequency Response summarized by

where is the modulus of G(jω) and ϕ is the argument of G(jω)

Note: Substitute for s=jω in the transfer function.

G s r sq s

e s z s zs p s p

sm

n( ) ( )

( )( ) ( )

( ) ( )= =

! !! !

!"1

1

L

L

G j G j e j( ) ( )! ! "=

G j( )!

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Frequency Response

The facts:

For any linear process we can calculate the amplitude ratioand phase shift by:

i) Letting s=jω in the transfer function G(s)

ii) G(jω) is a complex number. Its modulus is the amplitude ratio ofthe process and its argument is the phase shift.

iii) As ω, the frequency, is varied that G(jω) gives a trace (or a curve)in the complex plane.

iv) The effect of the frequency, ω, on the process is the frequencyresponse of the process.

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Frequency Response

Examples:

1. Pure Capacitive Process

2. Dead Time

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Examples:Examples:

3. Lead unit3. Lead unit

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Frequency Response

Examples:

4. n process in series

Frequency response of

therefore

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Frequency Response

Examples:

5. n first order processes in series

6. First order plus delay

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Frequency Response

To study frequency response, we use two types ofgraphical representations

1. The Bode Plot:Plot of (in dB)vs. on loglog scalePlot of (in degrees) vs. on semilog scale

2. The Nyquist Plot:Plot of the trace of in the complex plane

Plots lead to effective stability criteria and frequency-based design methods

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Bode PlotsBode Plots

First order processFirst order process

High frequency asymptote

High frequencyPhase shift

Bandwidth

Low frequency asymptote

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Bode plotsBode plots

Characteristic of the Bode plotCharacteristic of the Bode plot low frequency asymptotelow frequency asymptote

High frequency asymptote (ifHigh frequency asymptote (if ))

Yields a line onYields a line on loglog loglog scale,scale,

The phase shift at high frequenciesThe phase shift at high frequencies

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Bode plotBode plot

Characteristics of the Bode plot:Characteristics of the Bode plot: Cut-off frequencyCut-off frequency - this is the frequency - this is the frequency at whichat which

For first order systemsFor first order systems

The cut-off frequency is also the The cut-off frequency is also the bandwidthbandwidth of the first order of the first ordersystemsystem

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Bode PlotBode Plot

First order unstable systemFirst order unstable system

Frequency responseFrequency response

Amplitude ratio and phase shift:Amplitude ratio and phase shift:

SameSame amplitude ratio but positive phase shiftamplitude ratio but positive phase shift

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Bode plotBode plot

First order unstableFirst order unstable systemsystem

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Bode plotBode plot

Lead unit:Lead unit:

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Bode plotBode plot

Lead unit (right half plane zero)Lead unit (right half plane zero)

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FiltersFilters Pass bandPass band is the range of frequencies where is the range of frequencies where the signals passthe signals pass

through the system at the same degree of amplificationthrough the system at the same degree of amplification LowLow pass filterpass filter is a is a dynamical system with a pass band in the lowdynamical system with a pass band in the low

frequeny frequeny rangerange High pass filterHigh pass filter is a dynamical system with a pass band in the high is a dynamical system with a pass band in the high

frequency rangefrequency range Band pass filterBand pass filter is a dynamical system with ais a dynamical system with a passpass band over aband over a

certain range of frequenciescertain range of frequencies Bandwidth Bandwidth is the width of the frequency interval over the passis the width of the frequency interval over the pass

band of the filterband of the filter

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(Ideal) Low pass filter:(Ideal) Low pass filter: With pass band gain With pass band gain and bandwidthand bandwidth

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High-pass filter:High-pass filter: Pass band gainPass band gain andand cut-off frequencycut-off frequency

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Pass band filter:Pass band filter: Pass band gain Pass band gain and cut-off frequenciesand cut-off frequencies

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For second order processesFor second order processes

Frequency response is given byFrequency response is given by

OverdampedOverdamped

Critically dampedCritically damped

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Second order processSecond order process

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Bode plotBode plot

Characteristics:Characteristics: High frequency limitHigh frequency limit

High frequency limitHigh frequency limit hashas slope -2 in Bode plotslope -2 in Bode plot

Underdamped Underdamped process displays aprocess displays a maximummaximum amplification greateramplification greaterthan the pass band gainthan the pass band gain

at the resonant frequency,at the resonant frequency,

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Bode plotBode plot

Overdamped Overdamped system:system:

Slope=-2

Slope=-1

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Bode plotBode plot

Characteristics:Characteristics:

Amplitude and phase shift have two inflection points. One for eachAmplitude and phase shift have two inflection points. One for eachpole of the systempole of the system

OverallOverall asymptotic behavior is second order.asymptotic behavior is second order.

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Bode PlotBode Plot

Overdamped Overdamped processprocess withwith stable zerosstable zeros

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Bode plotBode plot

Overdamped Overdamped processprocess withwith stable zerosstable zeros

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Bode plotBode plot

ZerosZeros reduce the relative orderreduce the relative order High frequency asymptote is a straight line (on High frequency asymptote is a straight line (on loglogloglog) with slope) with slope

equal to the equal to the realtive realtive orderorder

LHP zeros increase the phase of the systemLHP zeros increase the phase of the system

RHP zeros decrease the phase of the systemRHP zeros decrease the phase of the system Also called Also called nonminimum nonminimum phasephase systemssystems because they exhibitbecause they exhibit

more phase lag than any other system with the same ARmore phase lag than any other system with the same AR

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Bode plotBode plot

Delayed system:Delayed system:

e.g. First order plus delaye.g. First order plus delay

AR is unchanged but the phase shift has no highAR is unchanged but the phase shift has no high frequencyfrequencyasymptoteasymptote

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Bode plotBode plot

First order plus delay:First order plus delay:

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Bode plotBode plot

System with pole at zero (integrator)System with pole at zero (integrator)

No finite dc gain

Slope = -2

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Bode plotBode plot

Overall characteristics of Bode plotsOverall characteristics of Bode plots dc Gain is thedc Gain is the value ofvalue of AR AR at at

Slope of high frequency asymptote isSlope of high frequency asymptote is the negative of the relativethe negative of the relativeorder of the systemorder of the system

Inflection points inInflection points in AR AR and the phase shiftand the phase shift correspond to poles andcorrespond to poles andzeros of the transfer functionzeros of the transfer function

Each unstable poles and stable zeros lead to positiveEach unstable poles and stable zeros lead to positive phase shift ofphase shift of90 degrees as90 degrees as

Each stable pole and Unstable zerosEach stable pole and Unstable zeros reducereduce phase shift phase shift by 90by 90degrees asdegrees as

DelayDelay leads to a phase shift ofleads to a phase shift of asas

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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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CHEE 319Process Dynamics and Control

Frequency DomainAnalysis of Control Systems

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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 of GOL(s)2. 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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There areThere are no clockwise encirclements of (-1,0)no clockwise encirclements of (-1,0)

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Nyquist Criterion






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There areThere are many clockwise encirclements of (-1,0)many clockwise encirclements of (-1,0)

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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

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Surprise Quiz (20 minutes)Surprise Quiz (20 minutes)

Consider the nonlinear systemConsider the nonlinear system

Using pole-placement, synthesize a controlUsing pole-placement, synthesize a control such thatsuch thatthe characteristic polynomial of the closed-loop system has thethe characteristic polynomial of the closed-loop system has theformform

(choose such that a unique solution exists)(choose such that a unique solution exists) 7/107/10

Compute the complementary sensitivity function and determine ifCompute the complementary sensitivity function and determine ifthe closed-loop system has zero steady-state errorthe closed-loop system has zero steady-state error 3/103/10

chee 319 process dynamics and control - …€¦ · 1 chee 319 process dynamics and control...

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