A Brief Introduction toA Brief Introduction to Control Theory · • Linear time-invariant...

A Brief Introduction toA Brief Introduction toControl Theory

Harald Paulitsch

Classification of Control Systems (2)

Introduction to Control Theory Harald Paulitsch 2

Classification of Control Systems

bull Open-loop control systemscontrol value only depends on input signalndash control value only depends on input signal

bull Closed-loop control systemsndash with feedback loop

l ll d ldquof db k l rdquorArr also called ldquofeedback control systemsrdquo

Introduction to Control TheoryHarald Paulitsch

3

Advantages of Feedback Control SystemsSystems

bull System output can be made to automatically follow theSystem output can be made to automatically follow the temporal value of the input function

bull The system performance is less sensitive to variations of parameter values

bull The system performance is less sensitive to unwanted disturbancesdisturbances


Disadvantages of Feedback Control SystemsSystems

bull Potential of instability (stability is a major design concern)Potential of instability (stability is a major design concern)bull Additional components for providing the feedback signal

are required (measurement)


Laplace Transformation (1)

Calculation in the Laplace domain (s)

intinfin

minussdot==

+=

stdtetxtxLsX

js ωσ

)()()(

int

intinfin+

sdot==

j

dtetxtxLsX

σ1

)()()(

1

0

intinfinminus

minus sdotsdotsdot

==j

stdsesXj

sXLtxσπ

)(2

1)()( 1


Laplace Transformation (2)Important Laplace Transformations

x(t) X(t) = Lx(t)x(t) X(t) = Lx(t)

δ(t) 1

1 or σ(t) 1 s1 or σ(t) 1 s

t 1 s2

1n tn 1 sn+11n t 1 s

eamiddot t 1 (s ndash a)


The Transfer Functionbull The transfer function G(s) is defined as the ratio of the

Laplace transformed output signal Y(s) to the input signal p p g ( ) p gX(s)

G(s) = Y(s) X(s)In the time domain the transfer function is a mapbull In the time domain the transfer function is a map

f(t) rarr f(t)

g(t) y(t)x(t)G(s) Y(s)X(s)


The Transfer Function (2)

bull Time invariancef(t-t ) rarr f(t-t )f(t-t0) rarr f (t-t0)

bull Linearityaf (t) +bf (t) rarr af (t) +bf (t)a f1(t) +b f2(t) rarr a f 1(t) +b f 2(t)

bull However almost no system is truly linear and time-invariant But it can be useful totime-invariant But it can be useful to approximate systems as such


The Transfer Function (3)Building blocks of transfer functions g(t) x(t) rarr y(t) G(s)

bull Proportional elementy(t) = KP x(t) G(s) = Y(s)X(s) = KP

I t ti l tbull Integrating elementy(t) = KI int x(t) dt G(s) = KI s

bull Differential elementbull Differential elementy(t) = KD dx(t)dt G(s) = KD s



bull first order delay (PT1)first order delay (PT1)T0yrsquo(t) + y(t) = KPx(t) G(s) = KP (1 + sT0)( ) P ( 0)

bull second order delay (PT2)T01yrsquorsquo(t) + T02yrsquo(t) + y(t) = KPx(t)01 y (t) 02 y (t) y(t) P (t)G(s) = KP (1 + sT02 + s2T01)


Testing the Transfer Function (1)

bull When applying a test function the output allows to reason about the transfer functionto reason about the transfer function

bull Possible test functionsdirac signal δ(t)ndash dirac signal δ(t) δ(t) = (infin if t=0 otherwise 0) Lδ(t) = 1

t d d t i l (t)ndash standard step signal σ(t)σ(t) = (1 if t ge 0 otherwise 0) Lσ(t) = 1s



bull Proportional elementy(t) = KP x(t)

bull Integrating elementy(t) = KI int x(t) dt



bull Differential elementy(t) = KD dx(t)dt

bull First order delayT0yrsquo(t) + y(t) = KPx(t)



bull Second order delay

(pict re taken from Wikipedia)


(picture taken from Wikipedia)

Transfer Function of Control System

bull Transfer function of parallel components

)()()( 21 sGsGsG +=


Transfer Function of Control System (2)

bull Transfer function of serial components

)()()( 21 sGsGsG sdot=



bull Transfer function of feedback loop

01 2

1 2 0

G (s)G (s) G (s)G(s)

1 G (s) G (s) 1 G (s)sdot

= =+ sdot +


Stability Nyquist Criterion

bull Plotting the transfer function of the opened control system as Nyquist Plot (separate axis forcontrol system as Nyquist Plot (separate axis for real and imaginary part)

bull Critical point of F(ω) [-1 j0]bull Critical point of F(ω) [-1j0]bull Amplitude passage ωD1 |F(ωD1)| = 1bull Phase passage ω arc(F(ω )) = πbull Phase passage ωD2 arc(F(ωD2)) = πbull Stable behavior ωD1 lt ωD2


Stability Nyquist Criterium (2)Im

ωD1

-1 Re

ω=0ω=infincritical point

ωD2 ReD2

ω gt


Stability Bode Plot

Q S

bull A

bull P


Designing a Control System in the Time Domain using the Step ResponseDomain using the Step Responsebull S

ndash

1x(t)

t

bull S

T

T

K


Designing a Control System in the Time Domain using the Step Response (2)Domain using the Step Response (2)

y(t)Ks

bull Tu hellipeffective delay times

bull Ta hellipeffectivecompensation time

K ibull Ks hellipgain

tTu Ta


Designing a Control System in the Time Domain using the Step Response (3)Domain using the Step Response (3)bull Based on the values Tu Ta and Ks the rules by

[Chien Hrones and Reswick] can be used to[Chien Hrones and Reswick] can be used to find a first setting of the controllerndash supports P PI and PID controller typesndash supports P PI and PID controller typesndash settings for 0 and 20 allowed overshooting

depending on the control path fine tuning mayndash depending on the control path fine tuning may be required


Rules from Chien Hrones and Reswick


Boundary conditions of Designing a Control System by Rules using the Step Response bull Requires time invariance and linearity of

t l thcontrol pathbull The described method is for analog controllersbull Method gives an approximate starting configurationbull Method gives an approximate starting configurationbull Possible optimizations either manually or using

sophisticated algorithms (eg genetic algorithm)p g ( g g g )


Designing a Digital Controller (PID)

bull Position algorithmn ee

bull Velocity algorithms

nnD

n

isiInPn T

eeKTeKeKu 1minusminus++= sum

y g

S li ti T t b ffi i tl lls

nnnDsnInnPn T

)ee(eKTeK)e(eKΔu 211

2 minusminusminus

minusminus++minus=

bull Sampling time Ts must be sufficiently smallTs le 01 TaTs le 025 Tu


Bang-Bang Control

bull 2 states only (onoff)M it d fbull Magnitude of hysteresis Δ

bull Extension three level controllerlevel controller


Fuzzy Logic ndash Fuzzy Control

bull Multi-valued logicI t d f t f l d f b hi tbull Instead of true or false degree of membership to a ldquoFuzzy SetrdquoM b hi f tibull Membership function

]10[)( X ]10[)( rarrXxAμ


Example Fuzzy Variable ldquotemprdquo- 5 fuzzy sets (very cold cold hellip)- micro ld(x) = 0 12microcold(x) = 012- micromoderate(x) = 033 very

cold cold moderate hot very hot

0 7 8 9654321

014033


0 7 8 9654321

Operators onF S tFuzzy Sets

(x)μ1(x)μ AA minus=

(x)]micro(x)min[micro (x)micro BABA =cap

( )μ( )μ AAnot

( )]micro( )[micro( )micro BABAcap

( )]( )[( ) (x)]micro(x)max[micro (x)micro BABA =cup


Process flow of fuzzy systems

controller

Control Path


Example Steam Turbine

1

verycold cold moderate hot very

hot

Input Variables temperature and pressure

0

1

T0 T7 T8 T9T6T5T4T3T2T1

Output Variable throttle setting


Example Fuzzyficationvery very

1


hot

0 6

1) Temperature Input xmicrocold(x) = 06microvery cold(x) = 0

0T0 T7 T8 T9T6T5T4T3T2T1

06

x

hellipmicrovery_hot(x) = 0

2) P I t2) Pressure Input ymicrolow(y) = 037microok(y) = 013

( ) 0microweak(y) = 0hellipmicrohigh(y) = 0


Example Rule Base

1) IF temperature IS cold AND pressure IS low THENthrottle IS positivethrottle IS positive

2) IF temperature IS cold AND pressure IS ok THENthrottle IS zerothrottle IS zero

3) IF temperature IS cold AND pressure IS strong THENthrottle IS negative

4) hellip


)

Example Rule Evaluation (MaxMin)1) IF temperature IS cold AND pressure IS low THEN

throttle IS positivemicrocold(x) = 06

2) IF temperature IS cold AND pressure IS ok THEN

microlow(y) = 037

) p pthrottle IS zero

microcold(x) = 06micro (y) = 0 13


microok(y) = 013


throttle IS negative

Example Aggregation (MaxMin)


Example Defuzzyficationbull Generation of crisp output value by use of ldquocenter of

gravityrdquo method

center of gravity


Summarybull Linear time-invariant systems simplify control systems

designgbull Time or frequency response can be used for designing

control systemsbull Requirements to control systems

ndash StabilityLimited overshootingndash Limited overshooting

ndash Sufficient dynamics


Summary contbull Bang-bang or three level controllers for simple taskbull Mathematical model may be too difficult or expensive to

gain Fuzzy control offer an alternativebull Fuzzy control requires a priori knowledge

Fuzzy control requires 4 stepsbull Fuzzy control requires 4 stepsndash Fuzzyficationndash Rule evaluationRule evaluationndash Aggregationndash Defuzzyfication


y

Dankeschoumlna esc ouml

A Q ti Any Questions


Neural Controllers

bull Learning algorithm operates online or offlinebull Success is not guaranteed


Fuzzy Control vs Neural ControlIncorporates (expert) knowledge Learns

Knowledge must be gained Black box behaviour

T i i l b i d tTuning is laborious and not a formal method

Tuning possibly fails Training possibly fails

Both do not require a mathematical model of the process



Cooperative neuro-fuzzy models

bull Pre- or postprocessing neural network(modifies the in or output of the fuzzy controller)(modifies the in- or output of the fuzzy controller)

N l t k dj t t thbull Neural network adjusts or even generates the membership functions or the rules


Hybrid neuro-fuzzy controller


Classification of Control Systems (2)







3












intinfin

minussdot==

+=

stdtetxtxLsX

js ωσ

)()()(

int

intinfin+

sdot==

j

dtetxtxLsX

σ1

)()()(

1

0

intinfinminus

minus sdotsdotsdot

==j

stdsesXj

sXLtxσπ

)(2

1)()( 1




δ(t) 1

1 or σ(t) 1 s1 or σ(t) 1 s

t 1 s2







f(t) rarr f(t)





































)()()( 21 sGsGsG +=








01 2

1 2 0



= =+ sdot +







ωD1

-1 Re


ωD2 ReD2

ω gt


Stability Bode Plot

Q S

bull A

bull P



ndash

1x(t)

t

bull S

T

T

K



y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers





















3












intinfin

minussdot==

+=

stdtetxtxLsX

js ωσ

)()()(

int

intinfin+

sdot==

j

dtetxtxLsX

σ1

)()()(

1

0

intinfinminus

minus sdotsdotsdot

==j

stdsesXj

sXLtxσπ

)(2

1)()( 1




δ(t) 1

1 or σ(t) 1 s1 or σ(t) 1 s

t 1 s2







f(t) rarr f(t)





































)()()( 21 sGsGsG +=








01 2

1 2 0



= =+ sdot +







ωD1

-1 Re


ωD2 ReD2

ω gt


Stability Bode Plot

Q S

bull A

bull P



ndash

1x(t)

t

bull S

T

T

K



y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers



























intinfin

minussdot==

+=

stdtetxtxLsX

js ωσ

)()()(

int

intinfin+

sdot==

j

dtetxtxLsX

σ1

)()()(

1

0

intinfinminus

minus sdotsdotsdot

==j

stdsesXj

sXLtxσπ

)(2

1)()( 1




δ(t) 1

1 or σ(t) 1 s1 or σ(t) 1 s

t 1 s2







f(t) rarr f(t)





































)()()( 21 sGsGsG +=








01 2

1 2 0



= =+ sdot +







ωD1

-1 Re


ωD2 ReD2

ω gt


Stability Bode Plot

Q S

bull A

bull P



ndash

1x(t)

t

bull S

T

T

K



y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers


















δ(t) 1

1 or σ(t) 1 s1 or σ(t) 1 s

t 1 s2







f(t) rarr f(t)





































)()()( 21 sGsGsG +=








01 2

1 2 0



= =+ sdot +







ωD1

-1 Re


ωD2 ReD2

ω gt


Stability Bode Plot

Q S

bull A

bull P



ndash

1x(t)

t

bull S

T

T

K



y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers



















f(t) rarr f(t)





































)()()( 21 sGsGsG +=








01 2

1 2 0



= =+ sdot +







ωD1

-1 Re


ωD2 ReD2

ω gt


Stability Bode Plot

Q S

bull A

bull P



ndash

1x(t)

t

bull S

T

T

K



y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers


















































)()()( 21 sGsGsG +=








01 2

1 2 0



= =+ sdot +







ωD1

-1 Re


ωD2 ReD2

ω gt


Stability Bode Plot

Q S

bull A

bull P



ndash

1x(t)

t

bull S

T

T

K



y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers






















01 2

1 2 0



= =+ sdot +







ωD1

-1 Re


ωD2 ReD2

ω gt


Stability Bode Plot

Q S

bull A

bull P



ndash

1x(t)

t

bull S

T

T

K



y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers





















ωD1

-1 Re


ωD2 ReD2

ω gt


Stability Bode Plot

Q S

bull A

bull P



ndash

1x(t)

t

bull S

T

T

K



y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers
















Stability Bode Plot

Q S

bull A

bull P



ndash

1x(t)

t

bull S

T

T

K



y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers

















ndash

1x(t)

t

bull S

T

T

K



y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers

















y(t)Ks




tTu Ta















nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers





























nnD

n

isiInPn T


y g


nnnDsnInnPn T


2 minusminusminus

minusminus++minus=



Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers
















Bang-Bang Control






]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers


















]10[)( X ]10[)( rarrXxAμ




0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers


















0 7 8 9654321

014033


0 7 8 9654321




( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers



















( )μ( )μ AAnot





controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers

















controller

Control Path



1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers

















1


hot


0

1





1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers

















1


hot

0 6



06

x





Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers
















Example Rule Base




4) hellip


)




microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers



















microlow(y) = 037




microok(y) = 013






gravityrdquo method

center of gravity












y




Neural Controllers



















gravityrdquo method

center of gravity












y




Neural Controllers


























y




Neural Controllers



















Neural Controllers
















A Brief Introduction toA Brief Introduction to Control Theory · • Linear time-invariant...

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Transcript of A Brief Introduction toA Brief Introduction to Control Theory · • Linear time-invariant...