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### Sensitivity

Professor Walter W. Olson

Department of Mechanical, Industrial and Manufacturing Engineering

University of Toledo

Outline of Today’s Lecture

- Review
- Static Error Constants (Review)
- Loop Shaping
- Loop Shaping with the Bode Plot
- Lead and Lag Compensators
- Lead design with Bode plot
- Lead design with root locus
- Lag design with Bode plot

- Important transfer functions
- Gang of Six
- Gang of Four

- Disturbance Rejection
- Noise Rejection
- Limitations

Static Error Constants

- If the system is of type 0 at low frequencies will be level.
- A type 0 system, (that is, a system without a pole at the origin,)will have a static position error, Kp, equal to

- If the system is of type 1 (a single pole at the origin) it will have a slope of -20 dB/dec at low frequencies
- A type 1 system will have a static velocity error, Kv, equal to the value of the -20 dB/dec line where it crosses 1 radian per second

- If the system is of type 2 ( a double pole at the origin) it willhave a slope of -40 dB/dec at low frequencies
- A type 2 system has a static acceleration error,Ka, equal to the value of the -40 dB/dec line where it crosses 1 radian per second

Loop Shaping

Error

signal

E(s)

- We have seen that the open loop transfer function, has profound influences on the closed loop response
- The key concept in loop shaping designs is that there is some ideal open loop transfer (B(s)) that will provide the design specifications that we require of our closed loop system
- Loop shaping is a trial and error process
- To perform loop shaping we can used either the root locus plots or the Bode plots depending on the type of response that we wish to achieve
- We have already considered an important form of loop shaping as the PID controller

Output

y(s)

Input

r(s)

Controller

C(s)

Plant

P(s)

Open Loop

Signal

B(s)

Sensor

-1

+

+

Loop Shaping with the Bode Plot

- The open loop Bode plot is the natural design tool when designing in the frequency domain.
- For the frequency domain, the common specifications are bandwidth, gain cross over frequency, gain margin, resonant frequency, resonant frequency gain, phase margin, static errors and high frequency roll off.

-3 db

Roll off Rate dB/dec

Resonant peak gain, dB

Bandwidth rps

Resonant peak frequency rps

Gain cross over

frequency rps

Loop Shaping with the Bode Plot

Increase of gain

also increases

bandwidth and

resonant gain

Break frequency

corresponds to the

component pole or zero

Poles bend the magnitude and phase down

Zeros bend the magnitude and the phase up

Lead and Lag Compensators

- The compensator with a transfer functionis called a lead compensator if a<b and a lag compensator if a>b
- The lead and the lag compensator can be used together
- Note: the compensator does add a steady state gain ofthat needs to be accounted for in the final design
- There are analytical methods for designing these compensators (See Ogata or Franklin and Powell)

Lead Compensator

- The lead compensator is used to improve stability and to improve transient characteristics.
- The lead compensator can be designed using either frequency response or root locus methods
- Usually, the transient characteristics are better addressed using the root locus methods
- Addressing excessive phase lag is better addressed using the frequency methods

- The pole of the system is usually limited by physical limitations of the components use to implement the compensator
- In the lead compensator, the zero and pole are usually separated in frequency from about .4 decades to 1.5 decades depending on the design

Lead Compensator (Frequency Design)

Mechanical

Lead

Compensator

Note:

the lead compensator opensup the high frequency regionwhich could cause noise problems

The Lead compensator addsphase

b1

xi

b2

x0

k

y

f

wm

Lag Compensator

- Lag compensators are used to improve steady state characteristics where the transient characteristics are adequate and to attenuate high frequency noise
- In order to not change the transient characteristics, the zero and pole are located near the origin on the root locus plot
- The starting point for the design on a root locus is to start with a pole location at about s = -0.001 and then locate the pole as needed for the desired effect

- In order to not give up too much phase, the zero and pole are located away from the phase margin frequency

Lag Compensator

Mechanical

Lag

Compensator

xi

b2

k

a

b

b1

Note that the lag compensator causes

a drop in the magnitude and phase

This could be useful in reducing bandwidth, and improving gain margin; however it might reduce phase margin

x0

Sensitivity

- Sensitivity is an evaluation of how the system responds to various signals compared to the design signal
- In general, we want the system to respond to the reference input
- We do not want the system to respond to noises and other signals that do not contribute to the accuracy of the desired output

Several Transfer Functions

N

D

Measurement

Noise

Disturbances

u

h

Y

U

R

E

+

+

Y

-1

Controller

Process

+

+

+

+

The Model

N

D

u

h

R

Y

U

E

+

+

Y

-1

Controller

Process

“Gang of Six”

Complementary

Sensitivity

Function

Load

Sensitivity

Function

+

+

+

“Gang of Four”

+

Noise

Sensitivity

Function

Sensitivity

Function

Disturbance Rejection

N

D

- We want our system designed such that the disturbances to the system are attenuated
- Harold S. Black gave us the answer: negative feedback

u

h

Y

U

R

E

+

+

Y

-1

Controller

Process

+

+

+

+

Disturbance Rejection

Disturbances with

a frequency less

than the cross over

frequency are

attenuated while

frequencies higher

would be passed

Fortunately, most load

disturbances are low

frequency and often

can be treated as step

inputs

Noise Rejection

- We would also like noise rejection
- Noise is most often high frequency signals caused by the sensors used to measure
- Noise is presented as a result of the feedback terms
- We do not have noise as defined here in an open system

- In the closed loop error, noise is multiplied by T, the complementary sensitivity function,
- In a system without a pre-filter, this is the transfer function
- For this reason high frequency roll-off is important

Limitations

- Systems with right hand side poles and zeros are inherently hard to control
- For a system with right hand side poles, pk, Bode showed that
- Improvements in one frequency region are met with deteriorations in another frequency region
- Sometimes called the waterbed effect

Summary

- Important transfer functions
- Gang of Six
- Gang of Four

- Disturbance Rejection
- Noise Rejection
- Limitations

Next: Robustness

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