chapter 6 part a slides

7/31/2019 Chapter 6 Part a Slides

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CHAPTER 6:

COMPENSATION

TECHNIQUES

(Part A)


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Introduction

Control system design involves the following three

steps:

Determine what the system should do and how

to do it (design specifications).

Determine the controller or compensator

configuration relative to how it is connected to

the controlled process.

Determine the parameter values of the

controller to achieve the design goal.


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Design Specifications

Used to describe what the system should do and

how it is done.

Examples of specifications: relative stability,steady-state accuracy (error), transientresponse, and frequency-responsecharacteristics.


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Frequency Domain Design Graphical tools such as Bode plot, Nyquist plot, Nichols

chart.

Advantages:

High order systems do not pose any particularproblem.

Pure time delay (e-sT) only affects the phase response.

Pure time delay does not need to be an integermultiple of the sampling interval.

Disadvantage:

Final measure of system performance more commonlyspecified as a time domain requirement.


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Time Domain Design

Includes transient response, limits on control signal,

integrated absolute error (area between the curve of the

desired response and that of the actual response).

Advantages:

Final measure of system performance more commonly specified as

a time domain requirement.

Commonly applied in auto-tuning using relays.

Disadvantages:

Feasible analytically only for second order systems.

Pure time delay has to be an integer multiple of the sampling

interval.


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Controller Configurations

Most conventional design methods rely on the fixed-

configuration design.

Control efforts involve the modification or compensation

of the systems performance characteristics. The general design using fixed configuration is also called

compensation.

A compensator is an additional component or circuit thatis inserted into a control system to compensate for a

deficient performance.

Examples: PID-type controllers, lag, lead and lag-leadcompensators


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Controller Configurations in

Control System Compensation

Series or Cascade compensation

Feedback compensation

State Feedback ControlCompensation

Series-feedback compensation


8/44

Controller Configurations in

Control System Compensation

Forward Compensation with series compensation

Feedforward compensation


9/44

Fundamental Pr inciples of

DesignController configuration> Controller type>

Controller parameter values

Controller parameter values are typically thecoefficients of one or more transfer functions

making up the controller.

Can be selected only if the process transfer

function is known. Determine how individual parameter values

influence the design specifications and system

performance.


10/44


DesignGeneral guidelines:

Complex-conjugate polesof the closed-loop transferfunction lead to a step response that is underdamped. If

all system poles are real, the step response isoverdamped.

The response of a system is dominated by the polesclosest to the origin in the s-plane. Transient due tothose poles farther to the left decay faster.

The farther to the left in the s-plane the systemsdominant poles are, the faster the system will respondand the greater its bandwidth will be.

The farther to the left in the s-plane the systems

dominant poles are, the larger its internal signals will be.


11/44


DesignGeneral guidelines (cont.):

When a pole and zero of a system transfer functionnearly cancel each other, the portion of the system

response associated with the zero-pole pair will have asmall magnitude.

Time and frequency domain specifications are loosely

associated with each other.

Rise Time:

Phase Margin:

Resonance Peak:

10

16.260.0

n

rt

degrees60marginphase0100

marginphase

0.707

12

1

2

rM


12/44

Cascade Compensation

Networks Compensator GC(s) is cascaded with the unalterable

process GP(s).

GC(s) can be chosen to alter the shape of the root locus.

In general,

Problem reduces to the selection of the zeros and poles

ofGC(s).

n

j

j

m

i

i

C

ps

zsK

sG

1

1

)(

)(

)(


13/44

Process: Second order prototype with T.F.

Controller: PD type with the T.F.

Control signal applied to the process

Design with PD (Propor tional-

Der ivative) Controller

)2()(

2

n

nP

sssG

)()( CDPc zsKsKKsG

dt

tdeKteKtu DP

)()()(


14/44

Electronic-circuit realization

of the PD controller

For the two-op-amp circuit , the input impedance of stage 1,

Output voltage of stage 1,

11

1

1

1

11

1

1

1 1

11

RsC

R

R

RsCsC

R

inin ERsCR

RE

RsC

R

RE 11

1

2

11

1

21 1

1


15/44

Output voltage of stage 2,

Transfer function of op-amp circuit:

Transfer function of PD controller:

Comparing with PD equation

Advantage: Only two op-amps are needed.

Disadvantage: Does not allow independent selection ofKPandKD, as they are commonly dependent on R2.

inERsC

R

REE 11

1

210 1

sCRR

R

sE

sEsG

in

c 12

1

20

)(

)()(

sKKsG DPc )(

1212 / CRKRRK DP


16/44

For the three-op-amp circuit ,

Stage 1:

Stage 2:

inER

RE

1

21 indd ECsRE 2

R

E

R

E

R

E 021 000

0

1

2 EECsRE

R

Rinddin


17/44

Transfer function of op-amp circuit:

Transfer function of PD controller:

Forward-path transfer function of the compensated system:

PD control is Equivalent to adding a simple zero at s = -KP/KDto

the forward-path transfer function.

sCR

R

R

sE

sEsG dd

in

C 1

20

)(

)()(

sKKsG DPc )(

n

DPnPC

ss

sKKsGsG

sE

sYsG

2)()(

)(

)()(

2


18/44

Summary of Effects of PD

Control Improves dampingand reduces maximum overshoot

Reduces rise time and settling time

Increases BW

Improves gain margin, phase margin and resonancepeak

May accentuate noise at higher frequencies

Not effective for lightly damped or initially unstable

system May require a relatively large capacitor in circuit

implementation


19/44

Example 6.1

Consider a second order model of an aircraft attitude

control system as follows:

Performance specifications:

Steady-state error due to unit ramp input 0.000443

Maximum overshoot 5%

Rise time tr 0.005s

Settling time ts 0.005s

)2.361(

4500

)( ss

K

sG


20/44

Uncompensated system:

The ramp error constant =

Steady-state error due to a unit ramp input,

ess 0.000443

ess= 1/Kv361.2/(4500K) 0.000443 => K 181.17.

Characteristic equation

Natural Frequency: rad/s

Damping ratio: (quite low)

2.361

4500)(lim

0

KssGK

sv

0)17.181(45002.3612 ss08152652.3612 ss

92.902815265 n

2.02

2.361

n

%7.52)-1/100exp(-overshootMaximum 2


21/44

System compensated with PD controller

- Inserting PD controller in forward-path of the system

- The damping and maximum overshoot are improved- Maintain the essdue to the unit ramp input at 0.000443

With the PD controller and K= 181.17, the forward-path transferfunction is

The closed-loop transfer function is

Effects of the PD controller:

Add a zero at s= -KP/KDto the closed-loop transfer function.

Increase the damping term.

)2.361(

)(815265

)(

)(

)(

ss

sKK

s

s

sGDP

e

y

pD

DP

r

y

KsKs

sKK

s

s

815265)8152652.361(

)(815265

)(

)(2


22/44

The ramp error constant is

The steady state error due to a unit ramp input is

ess= 1/Kv= 0.000443/KP.

Characteristic equation

Arbitrarily set KP= 1. (Which acceptable from the essrequirement)

=> Increased damping!

If we wish to have critical damping = 1, KD= 0.001772.

P

P

sv K

K

ssGK 1.22572.361

815265

)(lim0

0815265)8152652.361(2 PD KsKs

rad/s92.902815265 n

DD K

K46.4512.0

84.1805

8152652.361


23/44

Unit step responses of the attitude control

system with and without PD control


24/44

Table below gives the results for KP= 1, KD= 0, 0.0005, 0.00177 and0.0025.

Performance requirements are all satisfied with KD 0.00177.

Constraints on KD:

Large KDcorresponds to large BW, which may cause high-

frequency noise problem. The capacitor value in the op-amp circuit implementation should

not be too large.

PD controller decreases the maximum overshoot and settling time

KD t r (s) ts (s) Max. overshoot (%)

0 0.00125 0.0151 52.2

0.0005 0.00144 0.0076 25.7

0.00177 0.00119 0.0015 4.2

0.0025 0.00103 0.0013 0.7


25/44

A certain industrial plant synthesizes a chemical product from raw

materials at high temperature in a reactor. It is required to design a

controller for the system in order to control the temperature of the

reactor, which is considered an important parameter affecting the

quality of the chemical product. The transfer function between its

input (desired temperature) and the output (actual temperature) is

estimated to be

a) The system is known to be lightly damped. Find the damping ratio

of the system in closed-loop without any controller.

b) Design a Proportional-Derivative (PD) controller in order toimprove the damping ratio by five times of the original value

found in part (a). It is also required that the steady-state error in

response to a unit ramp input be equal to 0.01.

c) State, in general, two positive effects of improving the damping

ratio of a lightly damped system.

)2.0(

1)(

ss

sGP


26/44

Process: Second order prototype with T.F.:

Controller: PI type with the T.F.:

Control signal applied to the process:

)2()(

2

n

nP

sssG

Design with PI Controller

s

zsK

s

KKsG CIPC

)(

dtteKteKtu IP )()()(


27/44

Electronic-circuit realization

of the PI controller

For the two-op-amp circuit , the transfer function,

Comparing with PI equation:

sCRR

R

sE

sEsG

in

c

211

20 1

)(

)()(

211

2 1CR

KR

RK IP


28/44

For the three-op-amp circuit ,

Transfer function:

Comparing with PI equation:

sCRR

R

sE

sEsG

iiin

C

1

)(

)()(

1

20

ii

IPCR

KR

RK

1

1

2


29/44

The forward-path transfer function of the compensated

system is

Immediate effects of PI controller:

Adds a zero at s= -KI/KPto the forward-path transfer function. Adds a pole at s= 0 to the forward-path transfer function.

System type is increased from type-1 to type-2.

System order increased from second order to third order.

)2(

)()()()(2

2

n

IPnPC

ss

KsKsGsGsG


30/44

Advantages and disadvantages of PI control:

Reduces rise time

Increases settling time

Decreases BW

Filters out high frequency noise

May need a large capacitor value

Feasible method of designing the PI controller: Select the zero at s = -KI/KPso that it is relatively close to the

origin and away from the most significant poles of the process.

KPand KIshould both be relatively small.

A zero close to the origin provides the effects of pole-zero

cancellation. Improves stability by reducing phase lag.

s

sKKK

s

KKsG

I

PI

IPC

1

)(


31/44

The smaller KI/ KPis, the faster approaches +90degrees.

KPshould be small to avoid a large gain at phase crossoverfrequency.

Increase stability.

I

P

K

K1tan

I

PC

K

KjG 1tan90)(

222

2

1

|)(|PII

PI

C

KKK

KK

jG


32/44

Example 6.2

Consider the second order attitude control systemdiscussed in Example 6.1. Applying the PI controller, the

forward-path transfer function of the system becomes

Time domain performance requirements:

Steady-state error due to parabolic input t2us(t)/2 0.2

Maximum overshoot 5%

Rise time tr 0.01s

Settling time ts 0.02s

)2.361(

)/(4500)()()(

2

ss

KKsKKsGsGsG PIPPC


33/44

System compensated with PI controller

The parabolic error constant is

The steady state error constant is

Set K= 181.17 (The value used in Example 6.1)

=> KI 0.002215 (Minimum value ofKI= 0.002215 )

I

I

PIP

ssa

KK

KK

ss

KKsKKssGsK

46.122.361

4500

)2.361(

)/(4500lim)(lim

2

2

0

2

0

0.2)(08026.01

Ia

ss

KKK

e


34/44

Characteristic equation of the closed-loop system:

Rouths test:

Stable for 0 < KI

/KP

< 361.2.

Arbitrarily select

08152658152652.361 23 IP

KsKss

I

IP

I

P

Ks

KKs

Ks

Ks

815265

2.361/815265815265

8152652.361

8152651

0

1

2

3

2.361P

I

K

K

10P

I

K

K


35/44

Forward-path:

Starting points (poles): 0, 0, -361.2

Ending points (zeros): -10, ,

Intersection of asymptotes:

)2.361()10(815265

)2.361()10()17.181(4500)(

22

sssK

sssKsG PP

13

zerosfiniteofsum-polesfiniteofsum

175

2

(-10)-361.2-


36/44

Breakaway points:

s= 0, -21, -175

0ds

dKP

)10(815265

)2.361(2

s

ssKP

010

4.7223

)10(

2.361

815265

12

2

23

s

ss

s

ss

ds

dKP

072242.3912 23 sss


37/44

Root loci with KI/KP= 10, KPvaries


38/44

Assume we wish to have a relative damping ratio of 0.707

From the root loci, KP= 0.08 and KI= 0.8

At the design point, the three characteristic equation roots are

at

s = -10.605 -175.3 + j175.4 and -175.3 j175.4

Relationship between complex conjugate poles, and n

Consider a pair of complex conjugate poles given by

Roots =

0222 nnss

2

222

12

442

nn

nnnj


39/44

Table below gives the attributes of the unit step responses of

the system with PI control for various values ofKI/KP, with KP=

0.08.

KI/ KP KI KP Maximum overshoot

(%)

tr (s) ts (s)

0 0 1.00 52.7 0.00135 0.015

20 1.60 0.08 15.16 0.0074 0.049

10 0.80 0.08 9.93 0.0078 0.0294

5 0.40 0.08 7.17 0.0080 0.023

2 0.16 0.08 5.47 0.0083 0.0194

1 0.08 0.08 4.89 0.0084 0.0114

0.5 0.04 0.08 4.61 0.0084 0.0114

0.1 0.008 0.08 4.38 0.0084 0.0115


40/44

Unit step responses of the system in Example 6.2 with PI control


41/44

A process has a transfer function of

The process is to be controlled in closed loop using

a PI controller. Design the controller such that the

steady-state error in response to a ramp input is

10% of the magnitude of the ramp. The closed-

loop zero should be placed at 10.

5

1)(

s

sGP


42/44

Design with PID Controller

We can regard design of controllers as a filter design problem PD controller => High-pass filter

PI controller => Low-pass filter

To improve both the ess(which is achieved using PI controller)and transient response (which is achieved using PD controller)

independently, a PID controller can be used PID controller => band-attenuate filter

Transfer function of the PID controller:

s

zszsK

s

KsKsKsK

s

KKsG

leadlagIPD

D

I

pC

))(()(

2

dtde

KdteKeKu DIp


43/44

Bode plot for PID controller with KP=KI=KD= 1


44/44

When designing a PID controller for a given system, followthe steps shown below to obtain a desired response:

Obtain an open-loop response and determine what

needs to be improved

Add a proportional control to improve the rise time

Add a derivative control to improve the overshoot

Add an integral control to eliminate the steady-state

error

Adjust each ofKP, KIand KDuntil you obtain a desired

overall response

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