Chapter 2: Introduction to the Control of SISO Systems · Chapter 2: Introduction to the Control of...

Chapter 2:

Introduction to the

Control of SISO Systems

Control Automático

3º Curso. Ing. IndustrialEscuela Técnica Superior de Ingenieros

Universidad de Sevilla

(Some of the illustrations are borrowed from : Modern Control Systems (Dorf and Bishop)

2

Outline of the presentation

1. Dynamical Systems

2. Single Input-Single Output Systems (SISO Systems)

3. Identification of Dynamic Systems

4. Equilibrium points. Steady state characteristic

5. Linearization

6. Control scheme

7. Basic control actions

3

Dynamical systems

� System: object composed by a number of interrelated parts. Theproperties of the system are determined by the relationshipsbetween its different parts.

� Dynamical: its state varies with time

� Signal or variable: every magnitude that evolves with time

4

Basic notions

� We understand the system to be part of thereal world with a boundary with the outsideenvironment.

� Types of signals:

� Input signals: they act upon the system and are responsible for its future evolution.

� Output signals: they are the signals to be measured (and controlled). They represent the effect of the system on its environment.

� Internal variables: all the remaining

variables

� Examples:

xx

x

x xxx

xx

xx x

x

x

states

5

Basic notions

� Types of inputs: (from a technological point of view )� Manipulated variables: their evolution can be manipulated and fixed to a

desired value � Disturbances: are often regarded as uncontrolled being determined by the

environment in which the system resides (weather variations, process feed quality variations, …)

� Parameters of the system: magnitudes that characterize the system. They

allow one to distinguish between systems with similar structural and functional characteristics.

� Example: distinguish between parameters and signals of the systems

corresponding to the illustrations above.

6

Basic notions

� Models:

� Representation of the system that enables

its study.

� Physical representation (scaled-models)

� Mathematical representation (dynamic equations)

� Purposes of a model:

� Prediction of the evolution of the system

� Analysis of the behavior of the system

� Analysis of the effect of the variation of a parameter

� Analysis of the effect of the inputs on the evolution of the system

Modeling error

7

Modelling of Dynamical Systems

� Trade off between the accuracy of the model and its simplicity

� The type of model should be chosen according to the desired functionalities and purposes � Analysis

� Objective: cualitative analysis of the system’s behaviour.� This analysis can be a difficult task. � The model should be as simple as possible, but reflecting the main characteristics

and properties of the dynamics.

� Simulation� Objective: prediction of the evolution of the system.� This is normally a simpler task than the analysis (it can be solved by means of

numerical integration).� The model should have a degree of detail capable of yielding small prediction

errors.

ErrorComplexity

8

Simulation of systems

� Numerical Integration of the differential equations

� Discretization of time {t0, t1, t2,…}

� Integration step

� Computation of the outputs {y0, y1, y2,…}

� Example: Euler Method

−= )()(

1)( ty

A

Ktq

Aty p

&

−+= −−− 1k

p

1k1kk yA

Kq

A

1hyy

� Initialization : y0=y(0)

� For k=1 to N

� tk=k h

�

� End

Model

Input Output

Initial conditions

SIMULATOR

9

System Representation

• Inputs• Manipulated inputs:

• Cold water valve xf• Hot water valve xc

• Disturbances• Ambient temperatureTa• Temperatures Tc y Tf• Pressure at the pipes

of cold and hot water• Outputs

• Temperature of tank T• Water level in tank h

• Measurements:• Metal resistance termometer • Pressure sensor

Tc

xc qf

Tf

qc

qsT

h T

Tm

hm

xf

Ta

10

System Representation

xc

xf

Tah

T Tm

hm

System SensorsActuator

qcqf

∆Pvr

Tm

xc qf

Tf

qc

qsT

h T

hm

xf

Ta

11

Single Input-Single Output

Systems





5. Linearization

6. Control scheme


12

Linear systems representation

• Differential equation: it models the dynamics of a lumped parameter linear system in continuous time.

• Laplace transform:

mnmodelsCausal

nequationtheofOrder

tubdt

tdub

dt

tudb

dt

tudbya

dt

tdya

dt

tyda

dt

tydmmm

m

m

m

nnn

n

n

n

≥

++++=++++ −−

−

−−

−

:

:

)()(

...)()()(

...)()(

11

1

1011

1

1

systemu(t) y(t)

G(s)U(s) Y(s)G(s)U(s)

13

Frequency response

� Steady-state output for sinusoidal input

� G(jw) characterizes the frequency response of the system

� Fourier Series expansion ⇒ G(jw) characterizes the system

systemu(t) y(t)

14

Graphic plots

� Objective: Graphic plot of

� Bode Diagram:

2 semi-logarithmic scalar plots

� Magnitude

� Phase

-120

-100

-80

-60

-40

-20

0

Mag

nitu

de (

dB)

10-2

10-1

100

101

102

103

-180

-135

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequenc y (rad/s ec )

-120

-100

-80

-60

-40

-20

0

Mag

nitu

de (

dB)

10-2

10-1

100

101

102

103

-180

-135

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequenc y (rad/s ec )

15

Example

qc

Tm

Ta

T

Caldera

xc

-

-

16

Identification of Dynamic Systems





5. Linearization

6. Control scheme


17

Identification

� Obtaining a model from the temporal response of the system � Model parameters (for a given structure of the model)� Parametric model

� Structure and parameters (unknown model)� Black box identification

� Analysis of the system’s output corresponding to

different test input signals

� Impulse response

� Step response

� Sinousoidal response

18

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

tiempo

y

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300

1

2

3

4

5

6

7

8

9

10

tiempo

u

Step input signal Output of the system

G(s)?

Identification based on the step response

19

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

tiempo

y

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300

1

2

3

4

5

6

7

8

9

10

tiempo

u

Step input signal Output of the system

Characteristic response of a first order system:Exponential evolution with non zero slope at the instant corresponding to the step jump


20

Candidate Transfer Function

sK

sGττττ+

=1

)(

Two parameters:K?

?ττττ


21

K: it is obtained from the steady state :

32

6

13

28 ==−−=

∆∆=

uy

K

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

tiempo

y

2=∆u

6=∆y

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300

1

2

3

4

5

6

7

8

9

10

tiempo

u


22

τ : it is obtained from the transitory response

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300

0.51

1.52

2.53

3.54

4.55

5.56

6.57

7.58

8.59

9.510

tiempo

y

6=∆y

ττττ

78.363.0 =∆⋅ y


23

Frequency based identification

� G(s) can be determined from the experimental Bode Diagram

� Determination of the frequency range:� Step response: Characteristic time constant of the system

� Other factors:� Frequency range of noise

� Sampling time

systemu(t) y(t)

24

Frequency identification of a tank

Operating point:

Qs

H

k

k

h(t)

Válvulah

To Workspace

Sine Wave

Scopesqrt

MathFunction

1s

Integrator

h0 Constant1

q0 Constant

1/A

1/A

Qs

H

k

k

h(t)

Válvulah

To Workspace

Sine Wave

Scopesqrt

MathFunction

1s

Integrator

h0 Constant1

q0 Constant

1/A

1/A

25

10-4

10-3

10-2

10-1

100

10

15

20

25

30

35

10-4

10-3

10-2

10-1

100

-100

-80

-60

-40

-20

0


26

10-4

10-3

10-2

10-1

100

10

15

20

25

30

35

10-4

10-3

10-2

10-1

100

-100

-80

-60

-40

-20

0

Bode ExperimentalBode sistema aprox.

1/τ

Ke (dB)


27

Equilibrium points. Steady state

characteristic





5. Linearization

6. Control scheme


28

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

Transitory and steady state response

Steady state Transitory responseresponse

Steady state responseTransitory response

29

Equilibrium point

The equilibrium point is reached when the derivative of vs is zero. That is, when ve = vs

30

Equilibrium point

Uniqueness of the equilibrium point for linear systems:

• Given an input, for example ve= 1 volt, the system will evolve till it

reaches a unique equilibrium point that corresponds to theoutput v

s=1 volt.

•If the input is ve= 2 volts, then the system evolves till it reaches an

equilibrium point that corresponds in this case to an output vs=2

volts.

• For a given input, there is only one equilibrium point.

31

Steady state characteristic

Relationship between the input and the output in the steady state regimen.

Example:

ve

vs

In steady state:

32

R+ _

V


The steady state characteristic can be often obtained in an experimental way:

For example: DC Motor

Input: Applied voltage V (volts)

Output: Angular velocity (r.p.s.) revolutions per second

33


V(v) R(r.p.s.)

0 0

1 0

2 0.2

3 1.3

4 3.2

5 5.1

6 6.5

7 7.2

8 7.4

9 7.4

Applying different voltages at the input and measuring the revolutions per second in steady state:

R+ _

V

34


Graphic representation of the steady state characteristic

1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9R

V

35


Some considerations for the analysis of the steady state characterisitic

1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9R

V

Zone of linear behaviour

Zone of non linear behaviour

36

Static gain

u

yK static ∆

∆=

The static gain allows one to determine which is the final increment at the output of the system due to a given increment in the input.

systemu(t) y(t)

37

Static gain

0 5 10012

3456

78

Consider the following data, obtained from the step response of the system. Which is the static gain ?

0 5 10012

3456

78

?staticKsystemu(t) y(t)

38

Static Gain

1=∆u

0 5 10012

3456

78

0 5 10012345678u y

3=∆y

1=∆u

1

5

2

53

1

3

12

25 ≠≠==−−=

∆∆= staticstaticstatic KK

u

yK

39

Static Gain

• The steady state characteristic of a system allows one to determine which is its static gain at each operating point (equilibrium point): It is given by the slope of the curve.

1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9y

u

u

yK static ∆

∆=

40

Static gain• In the zone corresponding to a linear behaviour, the static gain characteristic has a constant slope. Therefore, in this zone the static gain is constant regardless of the operating point

1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9y

u

Linear zone: same static gain Kstatic for every operating point

Zones of non linear behaviour: Kstatic depends on the operating point

41

Linearization





5. Linearization

6. Control scheme


42

Linear dynamic systems:

Superposition principle

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

u1

y1

u2 y2

u1+u2

y1+y2

Linear system

Linear system

Linear system

43

Superposition principle (it is not applicable for non linear systems)

0 5 10 15 20 25 300

2

4

6

8

10

12

0 5 10 15 20 25 300

2

4

6

8

10

12

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

u1

u2 y2

ut=u1+u2

0 5 10 15 20 25 300

2

4

6

8

10

12

y1

yt=y1+y2/

Non Linear system

Non Linear system

Non Linear system

44

System Linearization

� Objective:

� Obtaining approximated linear models from non linear ones.

� Operating poing:

� Equilibrium point at which the linearization is done.

� Properties:

� It represents in a correct way the system in a neigborhod of the

equilibrium point.

� Outside of the region of applicability of the linearized model, the error

might be too large.

45

Linealización de sistemas

Las variables incrementales dependendel punto de funcionamiento elegido

46

Linealización de sistemas

47

Example

Operating point:

Defining incremental variables

Modeling error

48

Illustrative example

� Good approximation around the

equilibrium point

� For larger deviations, the linear

model might incurr in large errors

� All the signals evolve around their

value at the equilibrium point

49

Control scheme





5. Linearization

6. Control scheme


50

Feedback control

Controller

Manipulatedvariable

Controlled outputActuator System

Sensor

Measured signal

-y(t)

error

e u

Reference

Negative feedback:

↑↑↑↑e � ↑↑↑↑y � ↓↓↓↓e Compensation for the error(if not, unstable)

51

Controller gain

� The controller should guarantee a positive gain, that is, ↑↑↑↑e � ↑↑↑↑y � Positive gain:

� If ↑↑↑↑u � ↑↑↑↑y, then ↑↑↑↑e � ↑↑↑↑u

� Negative gain:

� If ↑↑↑↑u � ↓↓↓↓y, then ↑↑↑↑e � ↓↓↓↓ u

h

h

52

Linearization and control

Linearizedmodel

u(t) y(t)

Plant

+u0

U(t)

-

Y(t)

y0

y(t)u(t)

u(t)u0

U(t)y(t)

y0

Y(t)

53

Control of linearized systems

e(t) = (R(t)-y0)-(Y(t)-y0)= R(t)-Y(t)

Plant

+u0

U(t)

-

Y(t)u(t)Controller

R(t) e(t)

Equivalent (linear) control system

Controller

G(s)C(s)

Gs(s)

Ga(s)

Plant

Sensor

-

+R E U Y

Ym

VG(s)C(s)

Gs(s)

Ga(s)

Sensor

-

+G(s)G(s)C(s)C(s)

Gs(s)Gs(s)

Ga(s)Ga(s)

Actuator

Sensor

-

+R E U Y

Ym

V

54

Basic control actions





5. Linearization

6. Control scheme


55

Basic control terms

� Relay based control

� Proportional term

� Integral term

� Derivative term

56

Relay based control

� On-Off control

� Control law

� If e(t)>0, u(t)=umax

� If e(t)<0, u(t)=umin

� Oscillatory behavior

� Drives the system to the reference point

� Relay control with hysteresis

� Reduces oscillatory behavior

� Increasing the band of the

hysteresis reduces the frequency and

increases the amplitude

SystemU(t)

-

Y(t)R(t) e(t)

Relay

e

uumax

umin

57

Level Control of a vessel

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Histéresis de anchura 0.04

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Histéresis de anchura 0.08

Qs

H

10

k

Válvulah

To Workspace

Step1

Step

Scope

Rele

r

Referencia

sqrt

MathFunction

1s

Integrator

1/5

1/A

58

Proportional term

� Control law

Proportionalband

umax

umin

e

u

u0

System

+u0

U(t)

-

Y(t)Kp

R(t) e(t)

59

Proportional term

� Properties:

� Reduces oscillatory behavior

� BP=0% � Relay control

� It eliminates the tracking error of the step-response for the equilibrium reference u0

� In general it does not eliminate the tracking error of the step response for arbitrary references

60

Proportional level control of a Vessel

Qs

H

10

k

Válvulah

T o Workspace

Step1

Step

Scope

r

Referencia

sqrt

M athFunction

1s

Integrator

10

Gain

7.0711 Constant

1/5

1/A

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Kp=10

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Kp=10

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Kp=100

61

Integral term

� PI control law

SystemU(t)

-

Y(t)PI

R(t) e(t) • Eliminates the tracking error of the step-response for arbitrary references

• Increases oscillatory behavior (may lead to instability)

62

Integral term

� Adapts the value of u0

� If the closed-loop system is stable then

u(t) bounded � bounded � e(t) → 0

System

+ u0

U(t)

-

Y(t)

KpR(t)

e(t)

1er order(K=1, t=Ti)

63

PI level control of a vessel

Qs

H

10

k

Válvula

1

s+1

Transfer Fcn

h

To Workspace

Step1

Step

Scope

r

Referencia

sqrt

MathFunction

1s

Integrator

100

Gain

1/5

1/A

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Kp=100 T

i=1

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Kp=100, Ti=0.1

64

Derivative term

� PD control law

� Predicts future evolution of the error

� May improve transient

� Amplifies high-frequency noise

System

+u0

U(t)

-

Y(t)PD

R(t) e(t)

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Chapter 2: Introduction to the Control of SISO Systems · Chapter 2: Introduction to the Control of...

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