Department of Electronics Telecommunications and Informatics University of Aveiro Subject: Sistemas de Controlo II (Control Systems II) 2014/2015 Lecture 1 (12/Feb 2015) Petia Georgieva ([email protected]

) SC2 2014/2015 1

•

SYLLABUS

0. Introduction and historical perspectives

Continuous (analog) control systems - Petia Georgieva

Chapter 1. Review of the main topics of control system theory (SC1) (chapters I-VII of the book of Prof. Melo) Chapter 2. Basic compensators (chapter VIII) Chapter 3. PID compensator (chapter VIII) Chapter 4. State-feedback control (chapters X, XI) • • • • •

Digital control systems - Telmo Cunha

Chapter 5. Design of discrete-time controllers through emulation Chapter 6. Controller design in the discrete time Chapter 7. Direct digital control Chapter 8. Identification of discrete-time models through the least squares technique Chapter 9. RST controllers – polynomial approach

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• •

0. Introduction

System – An interconnection of elements and devices for a desired purpose.

Control System – An interconnection of elements and devices that will provide a desired response.

• The interaction is defined in terms of i.

System input ii. System output iii. Disturbances

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Historical Perspectives of Control Systems

1. Ancient Greece (1 to 300 BC) - water float regulation, water clock, automatic oil lamp 2. Cornellis Drebbel (17 th - Temperature control century) 3. Industrial revolution (18 th century) – development of steam engine. How to control the speed of rotation of the engine without human intervention ?

4. 18th Century (1769 ?) James Watt’s flyball governor for the speed control of a steam engine (the first system for automatic control of a machine).

Watt’s Flyball Governor (speed limiter)

Historical Perspectives of Control Systems

Laplace (1749-1827) and Fourier (1758-1830) – developed essential mathematical framework for theoretical analysis. Maxwell (1868) in his paper “On governors” developed the differential eq. model of the governor , linearized around an equilibrium point and proved that the stability of the system depends on the roots of a characteristics equation having negative real parts. Hurwitz (1875) and Routh (1905) - developed stability criteria for linear systems.

Lyapunov (1983), russian mathematician, developed stability criteria for non-linear systems.

Bell Telephone Labs (1930) - work on feedback amplifier design based on the concept of frequency response and backed by the maths of complex variables. Nyquist (1932) developed stability criteria using frequency domain methods. Bode (1945) and Nickols extended it to the most comonly used control system design in the frequncy domain. Evans (1948) –based on the work of Maxwell and Routh, Evans developed the Root Locus method to display grafically the roots of the characteristic eq.

SC1 (Routh- Hurwitz table & Root Locus method) Routh- Hurwitz table Root Locus method ( Matlab rlocus)

Open-Loop Control System Single Input-Single Output (SISO) Example of Open Loop Control System: Missile Launcher

Example of Open Loop Control System : control of the speed of a turntable

Closed-Loop Control System Single Input-Single Output (SISO) Has a negative feedback to compare the actual output to the desired output response.

Disturbance Set-point or Reference input + + Error

Controller

Controlled Signal Manipulated Variable

Actuator

+ + +

Process

Actual Output

Sensor

Feedback Signal

Example of a SISO Closed Loop Control System (negative feedback) Missile Launcher System

Example of a Closed Loop Control System model of the national income

Example of a Closed Loop Control System of the speed of a turntable

Multi Input Multi Output (MIMO) Closed Loop Control System

Desired Output Response Controller Process Output Variables Measurement

Example of MIMO Closed Loop Control System Boiler Generator

Example of a Modern MIMO Control System

The Future of Control Systems

CAMBADA IEETA ROBOTICS – part of the future control systems 2008: World champion of middle size robocup 2009-2015: Leading position in all national&intern. competitions 18

Chapter 1:Review of the main topics of control system theory (SC1)

1. Linear system modeling 2. Time (transient) response of 1 st and 2 nd order systems 3. Stability (Routh-Hurwitz) 4. Control architectures (open-loop; closed-loop) 5. Negative feedback systems 6. Root-locus 7. Steady-state response (steady-state error)

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What is a Mathematical Model?

A set of mathematical equations (e.g., differential equations) that describes the input-output behavior of a system.

What is a model used for?

•System Simulation •Prediction/Forecasting •Control System Design •Design/Performance Evaluation

Dynamic Systems

A system is said to be dynamic if its current output may depend on the past history as well as the present values of the input variables. Mathematically,

y

(

t

)

 

[

u

(



), 0

  

t

]

u

: Input,

t

: Time

Example: A moving mass

y u

Model: Force=Mass x Acceleration

M

  

u

M

Example of a Dynamic System

Velocity-Force:

v

(

t

) 

y

 (

t

)  ( 0 )  1

M

0 

t u

(  )

d



Position-Force:

y

(

t

) 

y

( 0 ) 

t

( 0 )  1

M

0

t t

  1

u

(  0 )

d



dt

1

Therefore, this is a dynamic system. If the torque force (bdy/dt) is included, then

M



b

 

u

Homework –find the models (SC1 examples) 24

Mathematical Modeling Basics

• Mathematical model of a real world system is derived using a combination of physical laws (1 st principles) and/or experimental tools. • Physical laws are used to determine the model structure (linear or nonlinear) and order.

• The parameters of the model are often estimated and/or validated experimentally.

• Mathematical model of a dynamic system is often expressed as a system of differential (difference in the case of discrete-time systems) equations.

Mathematical Modeling Basics

• A nonlinear model is often linearized about a certain operating point. • Model reduction (or approximation) may be needed.

• Numerical values of the model parameters are often approximated from experimental data by curve fitting.

Example: Accelerometer

Consider the mass-spring-damper (may be used as accelerometer or seismograph) system:

x u

M f s

(y): the spring reaction force; nonlinear function of y=x-u

f d

(y): the torque reaction force; nonlinear function of y=x-u

Newton’s 2nd law

Mx

  

u

  

f d



f s

Linearized model:

My



by



ky



Mu

Transfer Function (TF)

Transfer Function is the algebraic input-output relationship of a linear time-invariant system in s domain.

U G(s) Y

Example: Accelerometer System

My



by



ky

 

Ms

2

Ms

2 

bs



k

,

s



d dt

Comments on TF

• Transfer Function is a property of the system independent from the input-output signals.

• It is an algebraic representation of differential equations applying the Laplace Transformation.

• Systems from different disciplines (e.g., mechanical, chemical, electrical) may have the same transfer function

Mixed Systems

• Most real world systems (processes, plants) are of the mixed type, e.g., electromechanical, hydromechanical, etc • Each subsystem within a mixed system can be modeled as single discipline system first.

• The subsystems are integrated into the entire system applying for example the block diagram rools. • Overall mathematical model may be assembled into a system of equations, or a transfer function.

Electro-Mechanical Example

Input: voltage u Output: Angular velocity 

R

a

u

Elecrical Subsystem (loop method):

i

a

L

a

B dc



J u



R i a a



L a di a dt



e , b

Mechanical Subsystem

e - emf voltage b T motor



J

 

B ω

Electro-Mechanical Example

Power Transformation:

Torque-Current:

T motor



K t i a

Voltage-Speed:

e b



K b ω u R

a

i

a

L

a

dc

where K t : torque constant, K b : velocity constant For an ideal motor

K t



K b

Combing previous equations results in the following mathematical model:

B

  

L J a di a



dt



B ω R a i K t a



i a

K

b

  0 

u

Transfer Function of Electromechanical Example

Taking Laplace transform of the system’s differential equations with zero initial conditions gives:

R

a

L

a

    

L a Js

s

 

B R a



W



I

a

(

(s)

s

-

) 

K t K

b W I a

(

s

) (

s

)  0 

U

(

s

)

u i

a

K t B

 Eliminating I

a

yields the input-output transfer function of 2nd order system.

W(s)



U(s) L a Js 2

 

JR a K t



BL a

 

BR a



K t

K b

Reduced Order Model

Assuming small inductance, L

a

 0 we get a transfer function of 1st order system.

Ω(s) U(s)

K

 

Js

  

K R t a t



K b



K R t a

 

R a

 

Ts K

system gain 

t

K b

R a

1 ,

T



J

t

K b

R a

 time constant

What is the Control System Engineer trying to achieve?

• First, to understand well the application in order to apply a suitable control system.

• A good control system has to – generate a response quickly and without strong oscillations (good transient response), – – have low error once settled (good steady-state response), and will not oscillate wildly or damage that system (stability).

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Summary

The central problem in control is to find a technically feasible way to act on a given process so that the process behaves, as closely as possible, to some desired behavior. Furthermore, this approximate behavior should be achieved in the face of uncertainty of the process and in the presence of uncontrollable external disturbances acting on the process.

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BIBLIOGRAPHY (in Portugues)

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BIBLIOGRAPHY

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