Section 8 Frequency Response Designweb.engr.oregonstate.edu/~webbky/MAE4421_files... · Plot the open‐loop Bode plot of the uncompensated system for -L1 Locate frequency where phase

MAE 4421 – Control of Aerospace & Mechanical Systems

SECTION 8: FREQUENCY‐RESPONSE DESIGN

K. Webb MAE 4421

Introduction2

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Introduction

In a previous section of notes, we saw how we can use root‐locus techniques to design compensators

Two primary objectives of compensation Improve steady‐state error Proportional‐integral (PI) compensation Lag compensation

Improve dynamic response Proportional‐derivative (PD) compensation Lead compensation

In this section of notes, we’ll learn to design compensators using a system’s open‐loop frequency response We’ll focus on lag and lead compensation

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Improving Steady‐State Error4

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Improving Steady‐State Error

Consider the system above with a desired phase margin of 50°

According to the Bode plot: 130° at

3.46 /

Gain is 12.1at

Set 12.1 4for desired phase margin

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Can read the position constant directly from the Bode plot:

Note that , as

desired

Gain margin is

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Steady‐state error to a constant reference is1

1 0.154 → 15.4%

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Let’s say we want to reduce steady‐state error to

Required position constant

10.05 1 19

Increase gain by 4x Bode plot shows desired position constant

But, phase margin has been degraded significantly

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Step response shows that error goal has been met But, reduced phase margin results in significant overshoot and ringing

Error improvement came at the cost of degraded phase margin

Would like to be able to improve steady‐state error without affecting phase margin Integral compensation Lag compensation

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Integral Compensation10

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PI Compensation

Proportional‐integral (PI) compensator:

1 1

Low‐frequency gain increase Infinite at DC System type increase

For ≫ 1/ Gain unaffected Phase affected little PM unaffected

Susceptible to integrator overflow Lag compensation is often

preferable

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Lag Compensation12

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Lag Compensation

Lag compensator11 , 1

Objective: add a gain of at low frequencies without affecting phase margin

Lower‐frequency pole: 1/ Higher‐frequency zero: 1/ Pole/zero spacing determined by For ≪ 1/

Gain: ~20 log Phase: ~0°

For ≫ 1/ Gain: ~0 Phase: ~0°

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Lag Compensation vs.

Gain increased at low frequency only Dependent on DC gain: 20log

Phase lag added between compensator pole and zero 0° 90° Dependent on

Lag pole/zero well below crossover frequency Phase margin unaffected

11

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Lag Compensator Design Procedure

Lag compensator adds gain at low frequencies without affecting phase margin

Basic design procedure: Adjust gain to achieve the desired phase margin Add compensation, increasing low‐frequency gain to achieve desired error performance

Same as adjusting gain to place poles at the desired damping on the root locus, then adding compensation Root locus is not changed Here, the frequency response near the crossover frequency is not changed

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Lag Compensator Design Procedure

1. Adjust gain, , of the uncompensated system to provide the desired phase margin plus 5°…10° (to account for small phase lag added by compensator)

2. Use the open‐loop Bode plot for the uncompensated system with the value of gain set in the previous step to determine the static error constant

3. Calculate as the low‐frequency gain increase required to provide the desired error performance

4. Set the upper corner frequency (the zero) to be one decade below the crossover frequency: 1/ /10 Minimizes the added phase lag at the crossover frequency

5. Calculate the lag pole: 1/6. Simulate and iterate, if necessary

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Lag Example – Step 1

Design a lag compensator for the above system to satisfy the following requirements 2% for a step input % 12%

First, determine the required phase margin to satisfy the overshoot requirement

lnln

0.559

100 55.9° Add ~10° to account for compensator phase at

65.9°

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Plot the open‐loop Bode plot of the uncompensated system for 1

Locate frequency where phase is180° 114.1°

This is , the desired crossover frequency

2.5 /

Gain at is 8.4 → 0.38

Increase the gain by 1/ 8.4 → 2.63

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Gain has now been set to yield the desired phase margin of 65.9°

Use the new open‐loop bode plot to determine the static error constant

Position constant of the uncompensated system given by the DC gain:

11.14 → 3.6

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Calculate to yield desired steady‐state error improvement Steady‐state error:

11 0.02

The required position constant:

11 49 → 50

Calculate as the required position constant improvement

13.9 → 14

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Lag Example – Steps 4 & 5

Place the compensator zero one decade below the crossover frequency, 2.5 /

1/ 0.25 /4

The compensator pole:

1/ .

1/ 0.018 /

Lag compensator transfer function

11

144 156 1

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Bode plot of compensated system shows:

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Lag compensator adds gain at low frequencies only

Phase near the crossover frequency is nearly unchanged

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Steady‐state error requirement has been satisfied

Overshoot spec has been met Though slow tail makes overshoot assessment unclear

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Lag Compensator – Summary

11

Higher‐frequency zero: / Place one decade below crossover frequency,

Lower‐frequency pole: / sets pole/zero spacing

DC gain: 20 log

Compensator adds low‐frequency gain Static error constant improvement Phase margin unchanged

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Improving Dynamic Response26

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Improving Dynamic Response

We’ve already seen two types of compensators to improve dynamic response Proportional derivative (PD) compensation Lead compensation

Unlike with the lag compensator we just looked at, here, the objective is to alter the open‐loop phase

We’ll look briefly at PD compensation, but will focus on lead compensation

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Derivative Compensation28

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PD Compensation

Proportional‐Derivative (PD) compensator:1

Phase added near (and above) the crossover frequency Increased phase margin Stabilizing effect

Gain continues to rise at high frequencies Sensor noise is amplified Lead compensation is usually preferable

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Lead Compensation30

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Lead Compensation

With lead compensation, we have three design parameters: Crossover frequency, Determines closed‐loop bandwidth, ; risetime, ; peak time,

; and settling time,

Phase margin, PM Determines damping, , and overshoot

Low‐frequency gain Determines steady‐state error performance

We’ll look at the design of lead compensators for two common scenarios, either Designing for steady‐state error and phase margin, or Designing for bandwidth and phase margin

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Lead Compensation

Lead compensator11 , 1

Objectives: add phase lead near the crossover frequency and/or alter the crossover frequency

Lower‐frequency zero: 1/ Higher‐frequency pole: 1/ Zero/pole spacing determined by For ≪ 1/

Gain: ~0 Phase: ~0°

For ≫ 1/ Gain: ~20 log 1/ Phase: ~0°

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Lead Compensation vs.

11 , 1

determines: Zero/pole spacing

Maximum compensator phase lead,

High‐frequency compensator gain

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Lead Compensation –

, zero/pole spacing, determines maximum phase lead

sin11

Can use a desired to determine

1 sin1 sin

occurs at 1

1

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Lead Compensation – Design Procedure

1. Determine loop gain, , to satisfy either steady‐state error requirements or bandwidth requirements:a) Set to provide the required static error constant, orb) Set to place the crossover frequency an octave below the desired

closed‐loop bandwidth

2. Evaluate the phase margin of the uncompensated system, using the value of just determined

3. If necessary, determine the required PM from or overshoot specifications. Evaluate the PM of the uncompensated system and determine the required phase lead at the crossover frequency to achieve this PM. Add ~10° additional phase – this is

4. Calculate from 5. Set . Calculate from and

6. Simulate and iterate, if necessary

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Double‐Lead Compensation

A lead compensator can add, at most, of phase lead

If more phase is required, use a double‐lead compensator

11

For phase lead over , / must be very large, so typically use double‐lead compensation

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Lead Compensation – Example 1

Consider the following system

Design a compensator to satisfy the following for a ramp input

Here, we’ll design a lead compensator to simultaneously adjust low‐frequency gain and phase margin

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Lead Example 1 – Steps 1 & 2

The velocity constant for the uncompensated system islim→

lim→ 1

Steady‐state error is1

0.1

10 Adding a bit of margin

12 Bode plot shows the resulting

phase margin is 16.4°

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Lead Example 1 – Step 3

Approximate required phase margin for Design for 13%

First calculate the required damping ratioln

ln0.545

Approximate corresponding PM, and add correction factor

100 10° 64.5°

Calculate the required phase lead

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Calculate from 1 sin1 sin 0.147

Set , as determined from Bode plot, and calculate

3.4 /1 1

3.4 0.1690.7687

The resulting lead compensator transfer function is

11 12

0.7687 10.1130 1

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120.7687 10.1130 1

The lead compensator Bode plot

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Lead‐compensated system: 48.5° 7.2 /

High‐frequency compensator gain increased the crossover frequency Phase was added at the

previous crossover frequency PM is below target

Move lead zero/pole to higher frequencies Reduce the crossover

frequency increase Improve phase margin

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As predicted by the insufficient phase margin, overshoot exceeds the target % 20.9% 15%

Redesign compensator for higher Improve phase margin Reduce overshoot

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The steady‐state error requirement has been satisfied

Will not change with compensator redesign Low‐frequency gain will not be changed

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Iteration yields acceptable value for 5.5rad/sec Maintain same zero/pole spacing, , and, therefore, same

Recalculate zero/pole time constants:1 1

5.5 0.1470.4742

0.147 ⋅ 0.4742 0.0697

The updated lead compensator transfer function:

120.4742 10.0697 1

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Crossover frequency has been reduced 5.58 /

Phase margin is close to the target 58.2°

Dip in phase is apparent, because is now placed at point of lower open‐loop phase

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Overshoot requirement now satisfied

% 14.7% 15%

Low‐frequency gain has not been changed, so error requirement is still satisfied

Design is complete

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Again, consider the same system

Design a compensator to satisfy the following ( )

Now, we’ll design a lead compensator to simultaneously adjust closed‐loop bandwidth and phase margin

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The required damping ratio for 10% overshoot is

0.5912

Given the required damping ratio, calculate the required closed‐loop bandwidth to yield the desired settling time

. 1 2 4 4 2

7.52 /

We’ll initially set the gain, , to place the crossover frequency, , one octave below the desired closed‐loop bandwidth

/2 3.8 /

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Plot the Bode plot for Determine the loop gain at the desired crossover frequency

23.3

Adjust so that the loop gain at the desired crossover frequency is

123.3 14.7

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Phase margin for the uncompensated system:

14.9°

Required phase margin to satisfy overshoot requirement:

100 59.1°

Add 10° to account for crossover frequency increase

69.1°

Required phase lead from the compensator

54.2°

Generate a Bode plot using the gain value just determined

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Calculate zero/pole spacing, , from required phase lead, 1 sin1 sin 0.1040

Calculate zero and pole time constants

0.8228

0.0855 The resulting lead compensator transfer

function:11

14.70.8228 10.0855 1

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Bode plot of the compensated system

Substantially below target

Crossover frequency is well above the desired value /

Iteration will likely be required

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Overshoot exceeds the specified limit % 19.1° 10%

Settling time is faster than required 0.98 1.2

Iteration is required Start by reducing the target

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Must redesign the compensator to meet specifications Must increase PM to reduce overshoot Can afford to reduce crossover, , to improve PM

Try various combinations of the following Reduce crossover frequency, Increase compensator zero/pole frequencies, Increase added phase lead, , by reducing

Iteration shows acceptable results for: 2.4 / 3.4 / 52°

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Redesigned lead compensator:

6.270.8542 10.1013 1

Phase margin:62°

Crossover frequency:4.84 /

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Dynamic response requirements are now satisfied

Overshoot:% 8%

Settling time:1.09

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Lead compensator adds gain at higher frequencies Increased crossover frequency

Faster response time

Phase added near the crossover frequency Improved phase margin

Reduced overshoot

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Step response improvements: Faster settling time Faster risetime Significantly less overshoot and ringing

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Lead‐Lag Compensation

If performance specifications require adjustment of: Bandwidth Phase margin Steady‐state error

Lead‐lag compensation may be used11

11

Many possible design procedures – one possibility:1. Design lag compensation to satisfy steady‐state error and

phase margin2. Add lead compensation to increase bandwidth, while

maintaining phase margin

Documents

Section 8 Frequency Response Designweb.engr.oregonstate.edu/~webbky/MAE4421_files... · Plot the open‐loop Bode plot of the uncompensated system for -L1 Locate frequency where phase