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Advanced Control Theory

Review of Control System

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Advanced Control TheoryChibum Lee -Seoultech

Introduction to control


Control System

Control system: An interconnection of components

forming a system configuration that will provide a

desired system response

Relay/SCR

TemperatureSensor

Target Temperature

Heater

ControllerRood


Example of Control

Automobile driving


Types of controller

Typical controllers

• Embedded Controller :

Specially designed controller for specified systems

MPU, MCU, DSP, FPGA Cost for development useful for mass production

• PC Controller

Easy development, costly, easy access

• PLC (Programmable Logic Controller)

Widely used in general automation systems, sequence control

Use Ladder diagram, Modular composition


Control System Terminology

Systems

• A physical set of components that takes a signal, and

produces a signal

Signals

• A function representing some variable that contains some

information about the behavior of a system.

Systemsignal A signal B



Systems

• Plant: The physical object to be controlled.

Terminology comes from chemical process plants

like oil refineries or power plants.

• Sensor: The device that allows you to measure variables for

plant monitoring and for a variety of other purpose

In mechanical systems, it measures pressure, force, speed,

position, etc.

• Cf) Actuator: The device that causes the process to provide

the output. The device that provides the motive power to

the process

7



Systems

• Controller: The device or operation that gives commands to

the plant.

These commands are usually based on the current performance

of the plant compared to the desired performance of the plant.

The controller may or may not include the actuator, the device

that enforces the controllers commands on the plant.

Plant

Sensor

Controller

8



Signals

• Reference: What you’d like the measured output of your

controlled plant to be.

e.g. speed setting for your cruise control.

• Feedback (Measured output): Variables that are coming out

of your plant that are being compared to your reference.

e.g. speed measurement of your car from wheel speed sensors



Signals

• (Feedback) Error: Difference between your reference and

your feedback signals.

e.g. 10 km/h difference between reference and feedback.

• (Plant) Input: The signal that is passed from the controller

to the plant to affect some action.

e.g. Throttle variation.

• (Plant) Output: all the signals coming out of the plant.

Note: not all outputs are used in feedback



Systems and Signals

• Plant, Controller, and Sensor all systems.

usually have some dynamic or differential equation description.

transfer function representation for these systems.

• Signals carry information between these elements.

Plant

Sensor

Controllerreference error input output

feedback



Disturbance

• Disturbance: An unwanted signal that is not accessible to

the controller and tends to adversely affect the systems

output

System or signal ???

Plant

Sensors

Controller

disturbance


Control system diagram with disturbance and noise

Control system diagram with inner and outer loop



Closed vs. Open loop control

Closed loop control

• The difference of reference and

feedback are used as a means

of control.

Open loop control

• The output has no effect on control action

• usually simpler with fewer components

• Examples would be timers; e.g. toaster.

Gc Gp

Gc Gp

Closed-loop (with feedback)

Open-loop (without feedback)


Open-loop example



Closed-loop example




Comparison

Open-loop system Closed-loop system

Cost no sensors simple and

less expensive

Stabilitya stable plant cannot go unstable

a stable plant can go unstable, but an unstable plant can go stable

Performance

Disturbances or modeling mismatch can cause errors recalibration are

needed

more robust to disturbances and uncertainty They usually have better

performance



•Open loop •Closed loop

+

-

+

+

+

+

10% error 0.01% error

+

-

+

+

+

+

with model uncertainty


1. Study the plant and the control objectives.

2. Model the plant (if necessary, simplify the model.)

3. Decide the variables to be controlled (controlled outputs), the

measurements(sensor) and manipulated variables (actuators)

4. Select the control configuration and the type of controller.

5. Decide on performance specifications

6. Design a controller.

7. Analyze the resulting controlled system

8. Simulate the resulting controlled system

9. Choose hardware and software and implement the controller.

10. Test and validate the control system Hardware

labs

Control design procedure

Software

labs


Laplace transform


Why Laplace transform?

• Powerful for solving linear ODEs with initial values

• Easy to deal step and Dirac delta functions

• Comfortable for discontinued or complex periodic functions

Laplace Transforms

Easy

Difficult

AE Solver

Laplace Transform

InverseLaplace

Transform

DE SolverOriginal Prob.(differential eqn.)

Solution to original Prob.

S-domain Prob.(algebraic eqn.)

Solution to S-domain Prob.

T-domain

S-domain


Definition

Laplace transform of a function f(t)

• Integral transform

Inverse Laplace transform of F(s)

j

j

stdsF(s)eπj2

1

0

stF s f e f t dt

1 F f t


Laplace transform

The Laplace transform is a linear operation.

• Obeys the principle of superposition.

)]([)]([ )()(

))()(()()(

2211

0

22

0

11

0

22112211

txLatxLadttxaedttxae

dttxatxaetxatxaL

stst

st


• Shifting (s-domain)

• Shifting (t-domain)

• Differentiation

• Integration

• Final value theorem

• Initial values theorem )(lim)0(

)(lim)(

)(])([

)0(][][

)0()(][

)()]([

)()]([

0

0

2

2

ssFf

ssFf

s

sFdfL

fdt

dfsL

dt

fdL

fssFdt

dfL

sFeTtfL

asFtfeL

s

s

t

sT

at

Properties of Laplace transform


Laplace Transform usage in control system

2 primary uses for Laplace Transforms.

Compact representation of:

• (a) Signals

• (b) Systems

Plant

Sensors

Controller

disturbance


Signals

Step

Ramp

0

1

t au t a

t a

ase

u t as

2000

000

11

)()(

sdte

sdt

s

e

s

et

dts

edt

s

etdttesF

ststst

ststst


Signals

Sinusoid

Cf)

sin(t)ejt e jt

2 j

Euler’s formula

22

0

0

11

2

1

2

sin][sin)(

sjsjsj

dtej

ee

dtettLsF

sttjtj

st

)sin()( ttf

)cos()( ttf 22

11

2

1][cos)(

s

s

jsjstLsF


Signals

Impulse function

0,

0,0)()(

t

tttf

1)(

dtt

)}(1)(1{

1lim)(

0

ttt

1lim1

lim011

lim

11lim)(1)(1{

1lim

)}(1)(1{1

lim

)(lim)]([)(

000

000

00

00

|

s

se

s

e

s

e

s

s

e

sdtetdtet

dtett

dtettLsF

sss

ststst

st

st


Laplace transform

Signals


Systems

Systems are abstract representations of

dynamical phenomena.

1st order

Consider a free response would be considering

just the system.

systemaffectingsignalsystem

Faxx

1)0( ;0 txaxx


Signals and Systems

How signals affect systems and how systems

generate signals Convolution

Convolution

• Commutative Law:

• Distributive Law:

• Associative Law:

•

• Unusual Properties of Convolution:

• Convolution Theorem

0

t

f g t f g t d

f g g f

1 2 1 2f g g f g f g

f g v f g v

000 ff

1f f

f g f g


Signals and Systems

How does signal f affect system g?

signalsystemsignal

nconvolutio

system

)( )()()( sFsGtftg

output)(sG

)(tfinput


Convolution

1st order example again

In the time domain

systemaffectingsignalsystem

Faxx

systemaffectingsignal

systemofdynamics

responsesignalsystem

sFas

sX )( 1

)(

t

ta dfetx0

)()( Convolution


Ex. Solve

• LHS:

• RHS:

• Output:

'' , 0 1, ' 0 1y y t y y

Signals and Systems

output1

1)(

2

ssG

ttf )(Ramp input

,1)0'(,1)0( yy

System with IC

ttet sinh

1

1)( :TF)0'()0(

2

2

ssGYysyYs ,1)0'(,1)0( yy

2

1)()(

ssFttf

response IC))()(()( 1 sFsGLty


Signals and Systems

Visual explanation

• Example 1

• Example 2

system)( signal,)( sFsG


Modeling of Control systems


Mathematical Models

Models are key elements in the design and analysis of

control systems qualitative mathematical model

We must make a compromise b/w the simplicity of the

model vs. the accuracy of the results of analysis

???)t(u )t(y


Linear vs. nonlinear system

Linear system: the principle of superposition holds

• Linearity in mathematics

Let V and W be vector spaces over the same field K.

A function f: V → W is said to be a linear map if for any 2 vectors x and y

in V and any scalar α in K, the following conditions are satisfied:

• Linearity in system

A general system can be described by operator H, that maps an input

x(t) as a function of t to an output y(t) a type of black box description.

Linear systems satisfy the properties of superposition and homogeneity.

additivity

homogeneity


Linear Time Invariant System

Linear Time Invariant = Linear & time invariant

A time-invariant (TIV) system is one whose output does not depend explicitly on time.• If the input signal x(t) produces an output y(t), then

any time shifted input, x(t+), results in a time-shifted output y(t+ )

• Time invariant means that the coefficients in the differential equations are constant and don’t change with respect to time.

We can apply impulse response & Laplace transform in LTI system


Time-Varying Models

A time-varying system is a system that is not

time invariant its output depend explicitly

upon time

• Eg. a spacecraft control system.

The mass of fuel consumption changes due to fuel

consumption

Dynamic system

Linear Nonlinear

LinearTime

Invariant

LinearTime

Varying

Our focus


Transfer Functions

• Assuming zero initial conditions, take the Laplace

Transform of both sides

)()(

)()()()(

2

2

sFsYkbsms

sFskYsbsYsYms

)(1

)(

)(2

sGkbsmssF

sY

output

input

G(s)input output

TransferFunction


Transfer Functions

Transfer Function: the ratio of the Laplace

transform of the input and output of a linear

time-invariant system with zero initial conditions

and zero-point equilibrium.

Rational function in the complex variables

• Let x(t) : input , y(t) : output

xbxbxbxb

yayayaya

m

m

m

m

n

n

n

n

01

)1(

1

)(

01

)1(

1

)(

(n > m)


Transfer Functions

n-th order system since the highest power in the

denominator is n.

Note:

• limited to time-invariant, differential equation

• independent of the input magnitude. (homogeneity)

• no information on physically structure. (MKS and RLC)

01

1

1

01

1

1

ICs zero

)(

)(

][

][)( :TF

asasasa

bsbsbsb

sX

sY

inputL

outputLsG

n

n

n

n

m

m

m

m


System Poles and Zeros

• Roots of N(s)=0 : the system zeros z1, z2, …, zm

• Roots of D(s)=0 : the system poles p1, p2, …, pn

Note

• (System) Poles and zeros: real or either complex conjugate pairs

))(())((

))(())((

)(

)(

)(

121

121

01

1

1

01

1

1

nn

mm

n

n

n

n

m

m

m

m

pspspsps

zszszszsK

sD

sN

asasasa

bsbsbsbsG

numerator

denominator


Feedback Systems


Feedback Systems

Compared to open loop system, feedback control has

the following advantages:

• Decreased sensitivity of the system to variations in the parameters

of the process

• Improved rejection of the disturbance

• Improved measurement noise attenuation

• Improved reduction of the state-state error of the system

• Easy control and adjustment of the transient response of the

system


Feedback Systems

Closed-loop system subject to a disturbance and

a measurement noise

• Assume LTI system

(Mostly H(s) 1)

U(s) V(s)

Ym(s)


System Transfer Function

Define Tracking error:

If we consider input signals separately and use

principle of superposition, the output is given by

)()()( sYsRsE

)()()()(1

)()()()(

)()()(1

)(

)()()()(1

)()()(

sNsHsGsG

sHsGsGsD

sHsGsG

sG

sRsHsGsG

sGsGsY

pc

pc

pc

p

pc

pc



Assuming H(s)=1 , internal signals are given by

• Output

• Measured output

• Control input

)()()(1

)()()(

)()(1

)()(

)()(1

)()()( sN

sGsG

sGsGsD

sGsG

sGsR

sGsG

sGsGsY

pc

pc

pc

p

pc

pc

)()()(1

1)(

)()(1

)()(

)()(1

)()()( sN

sGsGsD

sGsG

sGsR

sGsG

sGsGsY

pcpc

p

pc

pc

m

)()()(1

)()(

)()(1

)()()(

)()(1

)()( sN

sGsG

sGsD

sGsG

sGsGsR

sGsG

sGsU

pc

c

pc

pc

pc

c



Gang of 4

Why “sensitivity” transfer function?

)()(/)(

)(/)(sS

sGsG

sTsT

pp

: Complimentary sensitivity transferfunction

: Sensitivity transfer function

)()(1

1

)()(1

)(

)()(1

)(

)()(1

)()(

sGsGsGsG

sG

sGsG

sG

sGsG

sGsG

pcpc

c

pc

p

pc

pc

SsGsG

TsGsG

sGsG

pc

pc

pc

)()(1

1

)()(1

)()(

the ratio of the change in the system TFto the change of a plant TF


Performance Specification

Good control tracking error small

• Good reference tracking

• Good disturbance rejection

• Good noise attenuation

)()()(1

)()()(

)()(1

)()(

)()(1

1)(

)()()()(

sNsGsG

sGsGsD

sGsG

sGsR

sGsGsE

sT

pc

pc

sSsG

pc

p

sS

pc

p

1)()()(1

1

)(

)(

sS

sGsGsR

sE

pc

1)()()()(1

)(

)(

)(

sSsG

sGsG

sG

sD

sEp

pc

p

1)()()(1

)()(

)(

)(

sT

sGsG

sGsG

sN

sE

pc

pc


Performance Specifications

Achieving good reference tracking, disturbance rejection,

noise attenuation simultaneously is not possible

• Algebraic limitation

Choice between Good reference tracking and noise attenuation

Solution?

• Separate the frequency components in reference, disturbance,

and noise

1)()(1

)()(

)()(1

1)()(

sGsG

sGsG

sGsGsTsS

cp

cp

cp


Time Domain Analysis



After the mathematical model of the system is obtained,

analysis of system performance is needed.

• Input signal to a control system is unknown

but what if to use known test input signal

• A correlation b/w the response characteristics of a system to a

typical test input signal and the capability of the system to cope

with actual input signals

SystemInput r(t) Output y(t)



Time response

• Transient response: goes from IC to final state

• Steady state response: the system output behaves as t

approaches infinity

)()()( tytyty sstr

SystemInput(t) Output(t)


1st Order Systems

Standard form ryyT

1

1

)(

)(

)()()1(

TssR

sY

sRsYTs

TF of 1st order system

Y(s) Y(s)


1st Order Systems –Step response

Unit step response w/ zero initial conditions

Ts

s

Ts

T

ssTssY

1

11

1

1

1

1

1)(

ssR

tUtr s

1)(

)()(

0t,1)( / Ttety

1


1st Order Systems –Step response

Characteristic feature of a 1st order system:

• At t=T,

ie. the response y(t) is 63.2% of its total change

• At t=0, the slope of the response

632.0 368.00.1 1)()( 1 eTyty

Te

Tdt

dy

t

Tt

t

11

0

/

0

T : time constant

y(t)


1st Order Systems –Ramp response

Unit ramp response w/ zero initial conditions

1

1

1

1

1)(

2

2

2

Ts

T

s

T

s

sTssY

0 t,)( / TtTeTtty

2

1)(

)()(

ssR

ttUtr s


1st Order Systems –Ramp response

The error between the reference & the output.

• Smaller T smaller ess

ess= T

Te(t)e)e(

)eT(

TeTtt

y(t)r(t)e(t)

tss

t/T

t/T

lim

1

y(t)

y(t)


1st Order Systems –Impulse response

Unit impulse response

1

1)(

TssY

0t,1

)( / TteT

ty

1)()()( sRttr

y(t)

y(t)


1st Order Systems –Impulse response

Compare

• Time response to an impulse reference signal is identical to

an initial condition response with zero reference.

TyyyT

1)0( w/ 0

1

1)(

1)()1(

0)()0()(

1

sTsY

sYsT

sYYTssTY

T

0)0( w/ (0) yyyT

1

1)(

1)()1(

1)()(

sTsY

sYsT

sYssTY


2nd Order System

Closed-loop

22

2

22

2

)(

)(

nn

n

ss

J

Ks

J

Bs

J

K

KBsJs

K

sR

sY

Y(s) (=(s))

Standard form of 2nd order system

JK

B

J

Kn

2,


2nd Order System

Solve the following differential equation

• Case 1.

• Case 2.

0)0(,0)0(,)(2 2 xxtxxx nn

1)0(,0)0(,02 2 xxxxx nn


2nd Order System

Dynamic behavior of 2nd order system can be described

in terms of damping ratio and natural frequency n

• Characteristic equation (denominator of TF=0)

• Poles (root of characteristic equation)

22

2

2)(

)(

nn

n

sssR

sY

02 22 nnss

2

21 1, nn jss

10


2nd Order System

Pole location

1, ,1 If

, ,1 If

1, ,10 If

, ,0 If

2

21

21

2

21

21

nn

n

nn

n

ss

ss

jss

jss


2nd Order System-Step response

For underdamped case,

22222

22

2

)(1)(

1

1

2)(

dn

d

dn

n

nn

n

ss

s

s

ssssY

10

)1

(tan re whe)sin(1

1

sin1

cos1)(

21

2

2

te

tetety

d

t

d

t

d

t

n

nn

21 nd

1)(lim

tyt



For no damped case,

2222

2 11)(

nn

n

s

s

ssssY

0

tty ncos1)(

1)(lim

tyt

y(t)



For critically damped case,

ssssssY

n

n

nn

n 1

)(

1

2)(

2

2

22

2

1

)1(1)( tety n

tn

1)(lim

tyt



For overdamped case,

)1()1(

1

1

)1)(1()(

22

22

2

nn

nnnn

n

sss

ssssY

212

)1(

22

)1(

22

21

22

121

)1(12

1

)1(12

11)(

s

e

s

e

eety

tsts

n

tt nn

1)(lim

tyt

,1

1, 2

21 nnss


2nd Order System-Impulse response

Impulse response or Initial condition response

For

For

22

2

2)(

nn

n

sssY

tety d

tn n

sin

1)(

2

0for 0)(lim

tyt

10

1

t

nntety

2)(


Higher Order Systems

Unit step response

01

1

1

01

1

1

)(

)(

asasasa

bsbsbsb

sR

sYn

n

n

n

m

m

m

m

)2(2

1)(

1)(

1

response ord 2nd

22

2

1

response ord1st

01

1

1

01

1

1

nrqss

csb

ps

a

s

a

sasasasa

bsbsbsbsY

r

k kkk

kkkkjkq

j j

j

n

n

n

n

m

m

m

m

r

k

kk

t

kkk

t

k

q

j

tp

j tectebeaaty kjkjj

1

22

1

1sin1cos)(


Higher Order Systems


Example – Dominant pole

Ex.

)25.6)(256(

)5.2(5.62

)(

)(2

sss

s

sR

sY

)256(

)5.2(10

)(

)(2

ss

s

sR

sY


Transient-Response specification (2nd ord. sys.)

Transient Response Characteristics to a unit step input

• Delay time, td : time required for reach 50%

• Rise time, tr : time required for rise from 10% to 90% or

from 0% to 100%

• Peak time, tp : time required for reach peak value

• Maximum overshoot, Mp :

• Settling time, ts: time required for reach

and stay 2% or 5% of

final value

y(t)

)()( yty p



Rise time:

tr

y(t)

2

2

2

1tansin

1cos

)sin1

(cos11)(

rdrdrd

rdrd

t

r

ttt

ttety rn

tetety d

t

d

t nn

sin1

cos1)(2



01

tan1

01

tan01

10For

1tan

1

21

21

2

21

d

r

d

r

t

t

???



• From system pole

22

2

2)(

)(

nn

n

sssR

sY

2

21 1, nn jss

)(11

tan1

2

1

dd

rt

n

nsangle2

1

1

1tan)(


y(t)


Peak Time

tp

,3,2,,00)sin(0)sin(

0)sin()cos()cos()sin(

0

)sin()cos()cos()sin()(

pdpdd

d

nnpd

t

pddpdnpdpd

d

nn

t

pddpd

d

ndt

pdpd

d

nt

np

ttte

tttte

ttettety

pn

pn

pnpn

tetety d

t

d

t nn

sin1

cos1)(2


y(t)

tetety d

t

d

t nn

sin1

cos1)(2


Maximum

Overshoot

Mp

%100)(

)()(PO)Overshoot(%

)sin(1

)cos(

1)(1)(

2211

2

y

yty

eee

ytyM

p

d

pp

n

n

dn


y(t)


Settling

Time

ts

• Approximation (comes from envelop function )

criterion 5%for 5

criterion 2%for 4

n

s

n

s

t

t

tetety d

t

d

t nn

sin1

cos1)(2

211

tne



y(t)

)1

(tan re whe)sin(1

1

sin1

cos1)(

21

2

2

te

tetety

d

t

d

t

d

t

n

nn


Parameter Selection –example1

Y(s)

Y(s)

J

K

KJ

KKb

ssKsKKbJs

K

sR

sY

nh

nn

n

h

,2

2)(

)(22

2

2


Parameter Selection –example1

criterion) (5% sec 86.13

, criterion) (2% sec 48.24

sec 65.01

1tan

sec 178.02

m,N 5.12,2

53.31

11

456.061.11

2.0

2

21

2

22

2

1 2

n

s

n

s

nd

r

hnnh

n

nd

p

p

tt

t

K

BKJKJK

J

K

KJ

KKb

t

eM


Steady-State Errors in Feedback Systems

Unity feedback system

)()()(1

lim)(lim)(lim

)()()(1

1)()()(

)()()(1

)()()(

00sR

sGsG

sssEtee

sRsGsG

sYsRsE

sRsGsG

sGsGsY

pcsst

ss

pc

pc

pc

number) type:())(())((

))(())(()()(

121

121 Npspspspss

zszszszsKsGsG

ll

N

mmpc


Unit step input


0or ,1For

1

1

)()(

)()(lim)()(lim ,0For

constant)error position :(1

1

)()(lim1

1

1

)()(1lim)(lim)(lim

1

0

1

00

0

00

ssp

p

ss

l

m

spc

sp

p

ppcs

pcsst

ss

eKN

Ke

pspss

zszsKsGsGKN

KKsGsG

ssGsG

sssEtee


Unit ramp input


0or)()(

)()(lim ,2For

1

)()(

)()(lim ,1For

or

0)()(


constant)error velocity :(1

)()(lim

1

)()(

1lim

1

)()(1lim)(lim)(lim

1

1

0

1

1

1

0

1

0

1

00

0

0

200

ss

l

N

m

sv

v

ss

l

m

sv

ss

l

m

spc

sv

v

vpcs

pcs

pcsst

ss

epspss

zszsKsKN

Ke

pspss

zszsKsKN

e

pspss

zszsKssGssGKN

KKsGssGsGssGs

ssGsG

sssEtee



Type 1 system response to a ramp input

y(t)

y(t)


Unit parabolic(acceleration) input


0or)()(

)()(lim ,3For

1

)()(

)()(lim ,2For

or

0)()(


constant)error on accelerati:(1

)()(lim

1

)()(

1lim

1

)()(1lim)(lim)(lim

1

12

0

1

2

12

0

1

12

0

2

0

2

0

220

300

ss

l

N

m

sa

a

ss

l

m

sa

ss

l

N

m

spc

sa

a

apcs

pcs

pcsst

ss

epspss

zszsKsKN

Ke

pspss

zszsKsKN

e

pspss

zszsKssGsGsKN

KKsGsGssGsGss

ssGsG

sssEtee


Type 2 system response to a parabolic input


y(t)

y(t)

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