Bode Design

7/31/2019 Bode Design

1/11

1

Amme 3500 :

System Dynamics & Control

Design via Frequency Response

Dr. Stefan B. Williams

Slide 2Dr. Stefan B. Williams Amme 3500 : Introduction

Course OutlineWeek Date Content Assignment Notes

1 1 Mar Introduction

2 8 Mar Frequency Domain Modelling

3 15 Mar Transient Performance and the s-plane4 22 Mar Block Diagrams Assign 1 Due

5 29 Mar Feedback System Characteris tics

6 5 Apr Root Locus Assign 2 Due

7 12 Apr Root Locus 2

8 19 Apr Bode Plots No Tutorials

26 Apr BREAK

9 3 May Bode Plots 2

10 10 May State Space Modeling Assign 3 Due

11 17 May State Space Design Techniques

12 24 May Advanced Control Topics

13 31 May Review Assign 4 Due

14 Spare

Slide 3Dr. Stefan B. Williams Amme 3500 : Bode Design

Frequency Response

In week 7 we looked at modifying thetransient and steady state response of asystem using root locus design techniques Gain adjustment (speed, steady state error) Lag (PI) compensation (steady-state error) Lead (PD) compensation (speed, stability)

We will now examine methods for designingfor a particular specification by examining thefrequency response of a system

We still rely on approximating CL behaviouras 2nd Order

Slide 4

Time vs. Freq. Domain Analysis

Control system performance generally judged bytime domain response to certain test signals

(step, etc.)

Simple for < 3 OL poles or ~2nd order CLsystems.

No unified methods for higher-order systems. Freq response easy for higher order systems

Qualitatively related to time domain behaviourMore natural for studying sensitivity and noise

susceptibility

Dr. Michael V. Jakuba Amme 3500 : FrequencyResponse


2/11

2

Slide 5

Frequency Response

Specifications Crossover frequency: Gain Margin: Phase Margin: Bandwidth (CL specification) Less intuitive than RL, but easier to draw for

high order systems.Dr. Michael Jakuba Amme 3500 : Bode Design

1)( =cjG

== 180)(whereat1)(s.t.(dB) jGjKGK

)180 cG(jPM =

dB3)( =BWjG cBWc 2


Transient Response via Gain

The root locus demonstrated that we canoften design controllers for a system viagain adjustment to meet a particular

transient response

We can effect a similar approach using thefrequency response by examining the

relationship between phase margin and

damping

Slide 7

Gain Adjustment and the

Frequency Response

Dr. Michael Jakuba Amme 3500 : Bode Design

2

( )( 2 )

n

n

G ss s

!

"!=

+

2

2 2( )

2

n

n n

T ss s

=

+ +

1c

2c

1BW

2BW

20 dB


Design using Phase Margin

Recall that the Phase Margin is closelyrelated to the damping ratio of the system

For a unity feedback system with open-loop function

We found that the relationship betweenPM and damping ratio is given by

2

( )

( 2 )

n

n

G s

s s

!

"!

=

+

1

2 4

2tan

2 1 4

PM!

! !

"

=

" + +


3/11

3



PhaseMargin



Given a desiredovershoot, we canconvert this to arequired damping

ratio and hence PM

Examining the Bodeplot we can find thefrequency that gives

the desired PM



The design procedure therefore consistsof

Draw the Bode Magnitude and phase plotsDetermine the required phase margin from the

percent overshoot

Find the frequency on the Bode phasediagram that yields the desired phase margin

Change the gain to force the magnitude curveto go through 0dB


Phase Margin Example

For the following position control system shownhere, find the preamplifier gain K to yield a 9.5%

overshoot in the transient response for a stepinput


4/11

4


BodeDiagram

Frequency (rad/sec)

Phase(deg)

Magnitude(dB

)

-150

-100

-50

0

50

10-2

10-1

100

101

102

103

-270

-225

-180

-135

-90

Phase Margin Example

BodeDiagram

Frequency (rad/sec)

Phase(deg)

Magnitude(dB

)

-150

-100

-50

0

50

10-2

10-1

100

101

102

103

-270

-225

-180

-135

-90

Draw the Bode plot For 9.5% overshoot, z=0.6

and PM must be 59.2o

Locate frequency with therequired phase at 14.8

rad/s

The magnitude must beraised by 55.3dB to yield

the cross over point at this

frequency

This yields a K = 583.9


Designing Compensation

As we saw previously, not allspecifications can be met via simple gainadjustment

We examined a number of compensatorsthat can bring the root locus to a desired

design point

A parallel design process exists in thefrequency domain


Designing Compensation

In particular, we will look at the frequencycharacteristics for the

PD Controller Lead Controller PI Controller Lag Controller

Understanding the frequency characteristics ofthese controllers allows us to select the

appropriate version for a given design


PD Controller

The ideal derivative compensator adds a puredifferentiator, or zero, to the forward path of the

control system

The root locus showed that this will tend tostabilize the system by drawing the rootstowards the zero location

We saw that the pole and zero locations giverise to the break points in the Bode plot

( ) ( )c

U s K s z = +


5/11

5


BodeDiagram

Frequency (rad/sec)

Phase(deg)

Magnitude(dB)

10-1

100

101

102

0

45

90

0

5

10

15

20

25

PD Controller

The Bode plot for a PDcontroller looks like this

The stabilizing effect isseen by the increase inphase at frequenciesabove the breakfrequency

However, the magnitudegrows with increasingfrequency and will tend toamplify high frequencynoise


Lead Compensation

Introducing a higher order pole yields the leadcompensator

This is often rewritten aswhere 1/ is the ratio between pole-zero breakpoints

The name Lead Compensation reflects the factthat this compensator imparts a phase lead

( )

( )( ) c

c

K s z

c cs pU s z p

+

+=


BodeDiagram

Frequency (rad/sec)

Phase(deg)

Magnitude(dB)

0

5

10

15

20

10-1

100

101

102

-60

-30

0

Lag Compensation

The Bode plot for a Lagcompensator looks like

this This compensator

effectively raises themagnitude for lowfrequencies

The effect of the phaselag can be minimized bycareful selection of thecentre frequency


Lag Compensation

In this case we aretrying to raise the

gain at lowfrequencies without

affecting the

stability of thesystem

Slide 36Dr. Michael Jakuba Amme 3500 : Bode Design

Lag Compensation

Nise suggests settingthe gain K for s.s.

error, then designing a

lag network to attain

desired PM.

Franklin suggestssetting the gain K for

PM, then designing a

lag network to raisethe low freq. gain w/o

affecting system

stability.

System without lag compensation,

Franklin et al. procedure


10/11

10

Slide 37Dr. Michael Jakuba Amme 3500 : Bode Design

Lag Compensation Design

The design procedure (Franklin et al.) consists ofthe following steps:1. Find the open-loop gain K to satisfy the phase margin

specification without compensation

2. Draw the Bode plot and evaluate low frequency gain3. Determine a to meet the low-frequency gain error

requirement

4. Choose the corner frequency =1/T to be one octaveto one decade below the new crossover frequency

5. Evaluate the second corner frequency =1/T6. Simulate to evaluate the design and iterate as

required


Lag Compensation Example

Returning again to the previous example, we willnow design a lag compensator to yield a ten fold

improvement in steady-state error over the gain-compensated system while keeping theovershoot at 9.5%


Lag Compensation Example

In the first example, we found the gain K=583.9 wouldyield our desired 9.5% overshoot with a PM of59.2o at14.8rad/s

For this system we find that

We therefore require a Kv of 162.2 to meet ourspecification

We need to raise the low frequency magnitude by afactor of 10 (or 20dB) without affecting the PM

0lim ( )

100583.9 16.22

3600

vs

K sG s

=

= =


Lag Compensator Example First we draw the

Bode plot with

K=583.9

Set the zero at onedecade, 1.48rad/s,

lower than the PM

frequency

The pole will be at 1/relative to this so

Bode Diagram

Frequency(rad/sec)

Phase(deg)

Magnitude(dB)

-100

-50

0

50

100

10-2

10-1

100

101

102

103

-270

-225

-180

-135

-90

Bode Diagram

Frequency(rad/sec)

Phase(deg)

Magnitude(dB)

-100

-50

0

50

100

10-2

10-1

100

101

102

103

-270

-225

-180

-135

-90

1.483( )

0.1483

0.674 110

6 .74 1

sU s

s

s

s

+=

+

+=

+


11/11

11


Lag Compensator Example

The resulting systemhas a low frequency

gain Kv of 162.2 asper the requirement

The overshoot isslightly higher thanthe desired

Iteration of the zeroand pole locationswill yield a lowerovershoot if required

Step Response

Time (sec)

Amplitude

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

System: untitled1

Time (sec): 0.187Amplitude: 1.17


Lag-Lead Compensation

As with the Root Locus designs we consideredpreviously, we often require both lead and lagcomponents to effect a particular design

This provides simultaneous improvement intransient and steady-state responses

In this case we are trading off three primary designparameters Crossover frequency c which determines bandwidth,

rise time and settling time

Phase margin which determines the damping coefficientand hence overshoot

Low frequency gain which determines steady state errorcharacteristics


Conclusions

We have looked at techniques fordesigning controllers using the frequencytechniques

There is once again a trade-off in therequirements of the system

By selecting appropriate pole and zerolocations we can influence the systemproperties to meet particular designrequirements


Further Reading

NiseSections 11.1-11.5

Franklin & PowellSection 6.7

Documents

Bode Design