CHAPTER 2.0 INTRODUCTION TO CONTROL SYSTEMS · 2010-09-11 · CHAPTER 2.0 INTRODUCTION TO CONTROL SYSTEMS ... The history of feedback control system begun as early as in 1769 when

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KJM597 Control Systems Faculty of Mechanical Engineering

UiTM Shah Alam

CHAPTER 2.0 INTRODUCTION TO CONTROL SYSTEMS

Control systems can be placed into three broad functional groups:

Monitoring systems, such as Supervisory Control and Data Acquisition (SCADA) systems, which

provide information about the process state to the operator;

Sequencing systems, used where some process must follow a pre-defined sequence of discrete

events;

Closed-loop systems, which is widely taught in engineering course, are typically implemented to

give some process a set of desired performance characteristics

The history of feedback control system begun as early as in 1769 when James Watt’s steam engine and

governor are developed. The Watt stem engine often used to mark the beginning of the Industrial

Revolution in England. The revolution of automatic control system continues in which the first ever

autonomous rover vehicle, known as Sojourner was invented in 1997.

In summary below is the history of feedback control system

1769 - James Watt’s flyball governer

Figure 2.0: James Watt’s flyball governer

1868 - J. C. Maxwell’s model of governer

1927 - H. W. Bode’s feedback amplifiers

1932 - H. Nyquist’s stability theory

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1954 - George Devol’s robot design

1970 - State-variable models and optimal control theory

1980 - Robust control system design

1997 - First ever autonomous rover vehicle “Sojourner”

Figure 2.1: Sojourner

But before we go into further details, we have to know control systems’ terms and concepts. The

frequently used terms and concepts are as follow:

Automation - The control of a process by automatic means Control system - An interconnection of components forming a system

configuration that will provide a desired response Controlled variable

- Quantity or condition that is measured and controller. Normally it is the output of the system

Manipulated variable

- Quantity or condition that is varied by the controller so as to affect the value of the controlled variable

Plant - A plant is a piece of equipment, perhaps just a set of machine parts functioning together, the purpose of which to perform a particular operation. Any physical object to be controller (such as heating furnace, a chemical reactor etc) is called a plant

Processes - A process can be defined as a natural, progressively continuing operation or development marked by a

Info: The mobile Sojourner had a mass

of 10.5kg and 0.25 square meter solar

array

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series of gradual changes that succeed one another in a relatively fixed way and lead towards a particular result or end

Disturbances - A disturbance is a signal which tends to adversely affect the value of the output of the system. If a disturbance is generated within the system, it is called internal; which an external disturbance is generated outside the system.

Feedback control

- Feedback control is an operation which in the presence of disturbances, tends to reduce the difference between the output of a system and the reference input and which does so on the basis of the difference.

Feedforward - Feedforward has a reference signal which is act as an additional input.

Source: AAMI, Fac of Mech Eng., UiTM

Figure 2.2: Input-output configuration of control system (souce: AAMI)

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Figure 2.3: Input-output configuration of a closed-loop control system (source: AAMI)

2.1 OPEN LOOP AND CLOSED-LOOP SYSTEMS

2.1.1 Open Loop Control System

A system is said to be an open loop system when the system’s output has no effect on the control

action. In open loop system, the output is neither measured nor fed back for comparison with the input.

Figure 2.4: Open loop control system

An open loop control system utilizes an actuating device (or controller) to control the process directly

without using feedback as shown in Figure 2.4.

The advantages and the disadvantages of an open-loop control system is tabulated in table 2.1 below

ADVANTAGES DISADVANTAGES

Simple and ease of maintenance Disturbances and changes in calibration cause errors Less expensive

Stability is not a problem Output may be different from what is desired Convenient when output is hard to

measure

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2.1.2 Closed-loop control system

A system that maintains a prescribed relationship between the output and the reference input is called a

closed-loop system or a feedback control system. The system uses a measurement of the output and

feedback of the signal to compare it with the desired output.

Figure 2.5: Closed loop control system

In a closed-loop control system, the actuating error signal, which is the difference between the input

signal and the feedback signal, is fed to the controller so as to reduce the error and bring the output of

the system to a desired value.

2.1.3 Comparison between open loop and closed-loop control system.

The table below shows the comparison between the two systems:

OPEN LOOP CLOSED LOOP

System stability is not a major problem, therefore easier to build

The use of feedback makes the system response relatively insensitive to external disturbances and internal variations in system parameters

Use open loop only when the inputs are known ahead of time and there is no disturbances

System stability is a major problem because the system tends to overcorrect errors that can cause oscillations or changing amplitude.

2.2 TRANSFER FUNCTION

The transfer function of a linear system is defined as the ratio of the Laplace transform of the output

variable to the Laplace transform of the input variable, with all initial conditions assumed to be zero. The

Transfer function of a system (or element) represents the relationship describing the dynamics of the

system under consideration. A transfer function may be defined only for a linear, stationary (constant

parameter) system. A non-stationary system often called a time-varying system, has one or more time-

varying parameters, and the Laplace transformation may not be utilized. Furthermore, a transfer

function is an input-output description of the behavior of a system. Thus the transfer function

description does not include any information concerning the internal structure of the system and its

behavior.

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2.2.1 The Transfer function of linear systems

The transfer function of a LTI system is defined as the Laplace transform of the impulse response, with

all the initial conditions set to zero.

)]([)( tgLsG

The transfer function is related to the Laplace transform of the input and the output through the

following relation:

)(

)()(

sR

sYsG

where all the initial conditions set to zero, and )(sY and )(sR are the Laplace transform of )(ty and

)(tr respectively.

Although the transfer function of a linear system is defined in terms of the impulse response, in practice,

the input-output relation of a linear time-invariant system with continuous–data input is often described

by the differential equation, so it is more convenient to derive the transfer function directly from the

differential equation.

Let us consider that the input-output relation of a linear time-invariant system is described by the

following nth-order differential equation with constant real coefficients:

)()(

.....)()(

)()(

......)()(

011

1

1011

1

1 trbdt

tdrb

dt

trdb

dt

trdbtya

dt

tdya

dt

tyda

dt

tydm

m

mm

m

mn

n

nn

n

To obtain the transfer function of the linear system that is represented by Eq. (2.3), we simply take the

Laplace transform on both sides of the equation and assume zero initial conditions. The result is

R(s)bsbsbsbY(s)asasas m

m

m

m

n

n

n

01

1

101

1

1

The transfer function between )(tr and )(ty is given by:

01

1

1

01

......

..............

)(

)()(

asasas

bsbsb

sR

sYsG

n

n

n

m

m

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The transfer function is said to be strictly proper if nm . If nm then the transfer function is proper.

It is improper if nm .

Characteristic Equation: The characteristic equation of a LTI system is defined as the equation

obtained by setting the denominator polynomial of the transfer function to zero. Thus, the

characteristic equation of the system described by the Eq. (2.4) is

0a 01

1

1

assas n

n

n

Later, we shall show that the stability of a linear single-input single-output system is governed

completely by the roots of the characteristic equation.

2.2.2 Transfer function of multivariable system

The definition of a transfer function is easily extended to a system with multiple inputs and outputs. A

system of this type is often referred to as a multivariable system. Figure 2.6 shows a control system with

two inputs and two outputs.

Figure 2.6: General block representation of a two-input, two-output system

Since the principle of superposition is valid for linear systems, the total effect on any output due to all

the inputs acting simultaneously is obtained by adding up the outputs due to each input acting alone.

Thus, using transfer function relations we can write the simultaneous equations for the output variables

as

)()()()()(

)()()()()(

2221212

2121111

sRsGsRsGsY

sRsGsRsGsY

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where )(sG ij is the transfer function relating the ith output to the jth input variable. Thus

)(

)(

sR

sYG

j

iij

In general, for j inputs and i outputs, we can write the simultaneous equations for the output variables

as

)(

)(

)(

)()()(

)()()(

)()()(

)(

)(

)(

2

1

21

22221

11211

2

1

sR

sR

sR

sGsGsG

sGsGsG

sGsGsG

sY

sY

sY

jijii

j

j

i

It is convenient to express Eq. (2.7) in a matrix-vector form

G(s)R(s)Y(s)

where

)(

)(

)(

)(2

1

sY

sY

sY

sY

i

is the i 1 transformed output vector; whereas

)(

)(

)(

)(2

1

sR

sR

sR

sR

j

is the j 1 transformed input vector; and

)()()(

)()()(

)()()(

)(

21

22221

11211

sGsGsG

sGsGsG

sGsGsG

sG

ijii

j

j

is the i j transfer-function matrix.

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2.3 DEFINITION OF STABILTY

A stable system is defined as a system which gives a bounded output in response to a bounded input.

The concept of stability can be illustrated by considering a circular cone placed on a horizontal surface,

as shown in Fig. 2.7 and Fig. 2.8.

Figure 2.7: The stability of a cone.

----------------------------------------------------------------------------------------------------

Figure 2.8: Stability in the s-plane.

The stability of a dynamic system is defined in a similar manner. Let u(t), y(t), and g(t) be the input,

output, and impulse response of a linear time-invariant system, respectively. The output of the system is

given by the convolution between the input and the system's impulse response. Then

0

)()()( dgtuty

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This response is bounded (stable system) if and only if the absolute value of the impulse response, g(t),

integrated over an infinite range, is finite. That is

0

)( dg

Mathematically, Eq. (4.24) is satisfied when the roots of the characteristic equation, or the poles of G(s),

are all located in the left-half of the s-plane.

A system is said to be unstable if any of the characteristic equation roots is located in the right-half of

the s-plane. When the characteristic equation has simple roots on the j-axis and none in the right-half

plane, we refer to the system as marginally stable.

The following table illustrates the stability conditions of a linear continuous system with reference to the

locations of the roots of the characteristic equation.

STABILITY CONDITION LOCATION OF THE ROOTS

Stable All the roots are in the left-half s-plane

Marginally stable of marginally unstable At least one simple root and no multiple

roots on the j-axis; and no roots in the right-half s-plane.

Unstable At least one simple root in the right-half s-plane or at least one multiple-order root

on the j-axis.

The following examples illustrate the stability conditions of systems with reference to the poles of the

closed-loop transfer function M(s).

321

20)(

ssssM

Stable

)22)(1(

)1(20)(

2

sss

ssM

Unstable due to the pole at s = 1

)4)(2(

)1(20)(

2

ss

ssM

Marginally stable or marginally unstable due to s =

j2.

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)10()4(

10)(

22

sssM

Unstable due to the multiple-order pole at s = j2.

2.3.1 Open loop and Closed loop stability

A system is open-loop stable if the poles of the loop transfer function G(s)H(s) are all in the left hand side of s-plane.

Figure 2.9: A typical closed-loop system

A system is closed0loop stable (or simply stable) if the poles of the closed-loop transfer function (or zeros of 1+G(s)H(s) are all in the left hand side of s-plane

2.4 BASIC CONTROL ACTIONS

The following six basic control actions are very common among industrial automatic controllers:

1. Two-position or on-off controller 2. Proportional controller 3. Integral controller 4. Proportional-plus-integral controller 5. Proportional-plus-derivative controller 6. Proportional-plus-derivative-plus-integral controller

2.4.1 Two-position of on-off control action

In a two-position control system, the actuating element has only two fixed positions which are, in many cases, simply on and off. Two-position or on-off control is relatively simple and inexpensive and, for this reason, is very widely used in both industrial and domestic control systems.

Let the output signal from the controller be m(t) and the actuating error signal be e(t). In two position control, the signal m(t) remains at either a maximum or minimum value, depending on whether the actuating error signal is positive or negative, so that

𝑚 𝑡 = 𝑀1 𝑓𝑜𝑟 𝑒(𝑡) > 0

= 𝑀2 𝑓𝑜𝑟 𝑒(𝑡) < 0

+- e(s)

yysp H(s)

PlantController

G(s)

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Where 𝑀1and 𝑀2, are constants. The minimum value 𝑀2, is usually either zero or −𝑀1. Two-position controllers are generally electrical devices, and an electric, solenoid-operated valve is widely used in such controller. Pneumatic proportional controller with very high gain act as two-position controller and are sometimes called pneumatic two-position controller.

Figure 2.10 show the block diagrams for two-position controller. The range through which the actuating error signal must move before the switching occurs is called the differential gap.

Figure 2.10: Two-position controller

2.4.2 Proportional controller

For a controller with proportional control action, the relationship between the output of the controller m(t) and the actuating error signal e(t) is

𝑚 𝑡 = 𝐾𝑝𝑒(𝑡)

or, in Laplace Transform

𝑀(𝑠)

𝐸(𝑠)= 𝐾𝑝

Where 𝐾𝑝 , is termed the proportional sensitivity or the gain.

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Whatever the actual mechanism may be and whatever the form of the operating power, the proportional controller is essentially an amplifier with and adjustable gain.

The proportional action has the following two properties:

1. Reduce rise time 2. Does not eliminate steady state error

Example 2.1:

Given a system consist of mass-spring and damper

a) The second order PDE is: b) Taking the LT c) The TF is therefore: d) Let M=1kg, b=10N.s/m, k=20 N/m & F(s)=1, therefore X(s) / F(s): e) From the Transfer Function, the DC gain is: f) Corresponding to the steady state error of: g) The settling time is:

b

M

x

F

k

Open Loop Response

Time (sec)

Dis

pla

ce

me

nt

(m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

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P control (K) reduces the rise time, increases the overshoot and reduces the steady state error.

h) The closed-loop transfer function of the system with P controller is X(s)/F(s)=G/(1+G): i) Let the P gain (K) equal 300

Rise time and ss error reduced, slightly reduced settling time but increased overshoot.

2.4.3 Integral controller

In a controller with integral control action, the value of the controller output m(t) is changed at a rate proportional to, the actuating error signal e(t). That is

𝑑𝑚(𝑡)

𝑑𝑡= 𝐾𝑖𝑒(𝑡)

Therefore; 𝑚 𝑡 = 𝐾𝑖 𝑒 𝑡 𝑑𝑡𝑡

0

Where 𝐾𝑖 is an adjustable constant. The transfer function of the integral controller is

𝑀(𝑠)

𝐸(𝑠)=

𝐾𝑖

𝑠

If the value of e(t) is doubled, then the value of m(t) varies twice as fast. For zero actuating error, the value of m(t) remains stationary.

Closed Loop Step : K = 300

Time (sec)

Dis

pla

ce

me

nt

(m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

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The integral controller has the following properties:

1. Proportional controllers often give a steady-state error. Integral controller arose from trying to add a “reset” term to the control signal to eliminate steady state error. In other words, the integral controller “resets” the bias error from the P controller.

2. Gives large gain at low frequencies resulting in “beating down” load disturbances. 3. May make the transient response worse. 4. Controller phase starts out at -90° and increases to 0° at the break frequency. This phase lag can

be compensated by derivative action.

The integral controller act as “automatic reset” as shown in figure 2.11

Figure 2.11: Automatic reset action

Almost always used in conjunction with P control.

Figure 2.12: PI control

The integral term may be expressed in (i) 𝑇𝑖 and (ii) 𝑘𝑖

The integral term 𝑇𝑖 is known as the integral time constant. 𝑇𝑖 = ∞ corresponds to pure (proportional) gain.

The integral term 𝑘𝑖 is known as integral gain (e.g: in MATLAB)

The relationship between 𝑇𝑖 and 𝑘𝑖 is as follows:

𝑘𝑖

𝑠=

𝐾

𝑇𝑖𝑠

+-

ysp yplantK

load disturbance

1sTi

u

e

+-

ysp yplant

Kload disturbance

1sTi

u

e K

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Example 2.2:

a) I control reduces the rise time, increases both settling time and overshoot, and eliminates the steady-state error

b) The closed-loop transfer function of the system with a PI controller is: X(s)/F(s) = ______________ .

c) Let k = 30 and ki = 70. P gain (k) was reduced because the I controller also reduces the rise time and increases the overshoot as does the P controller (double effect).

2.4.4 Derivative controller

Introducing a derivative controller will add damping and in doing so:

1. increases system stability (add phase lead)

2. reduces overshoot

3. generally improves transient response

A derivative controller may able to provide anticipative action but derivative action can make the system become noisy.

Almost always used in conjunction with P control.

Figure 2.12: PD control

The integral term may be expressed in (i) 𝑇𝑑 and (ii) 𝑘𝑑

The integral term 𝑇𝑑 is known as the derivative time constant.

Closed Loop Step : K = 30, Ki = 70

Time (sec)

Dis

pla

ce

me

nt

(m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

+-

ysp yplantc

load disturbance

KTd s

sTd /N1+

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The integral term 𝑘𝑑 is known as derivative gain (e.g: in MATLAB)

The relationship between 𝑇𝑑 and 𝑘𝑑 is as follows:

𝑘𝑑𝑠 = 𝐾𝑇𝑑𝑠

Example 2.3:

a) D control reduces both settling time and overshoot. b) The closed-loop transfer function of the system with a PD controller is:

X(s)/F(s)=______________ c) Let k = 300 and kd = 10.

d) Reduced overshoot and settling time, small effect on rise time and ss error

Closed Loop Step : K = 300, Kd = 10

Time (sec)

Dis

pla

ce

me

nt

(m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

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2.4.5 PID controller

In some system the commonly implemented controller consist of the P, I and D control action. We call

this type of controller as PID controller.

Figure 2.13: PID control

The standard form of PID controller according to ISA (Instrument Society of America) is as follows:

𝐺𝑐 𝑠 = 𝐾(1 +1

𝑠𝑇𝑖+ 𝑇𝑑𝑠)

Or 𝐺𝑐 𝑠 = 𝐾 +𝑘𝑖

𝑠+ 𝑘𝑑𝑠

Closed Loop Step : K = 350, Ki = 300, Kd = 50

Time (sec)

Dis

pla

ce

me

nt

(m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

K

1/( )Tsi

ysp

T sd

G s( )u

-

+ e y

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Example 2.4:

a) The closed-loop transfer function of the system with a PID controller is: X(s)/F(s) = (kd s

2 +ks+ki )/(s3 + (10+kd)s2 + (20+k)s + ki ) b) Let k = 350, ki = 300 and kd = 50.

c) No overshoot, fast rise and settling time and no steady-state error

2.4.6 PID tuning

Introducing the P, I and D controller has certainly proven to contribute some effect to our system’s response. These effects are summarized as in table below.

CLOSED LOOP RESPONSE

RISE TIME OVERSHOOT SETTLING TIME

SS ERROR

K Decrease Increase Small change Decrease

𝑘𝑖 =𝐾

𝑇𝑖

Decrease Increase Increase Eliminate

𝑘𝑑 = 𝐾𝑇𝑑 Small change Decrease Decrease Small change

When you are designing a PID controller for a given system, follow the steps shown below to obtain a desired response.

1. Obtain an open-loop response and determine what needs to be improved 2. Add a proportional control to improve the rise time 3. Add a derivative control to improve the overshoot 4. Add an integral control to eliminate the steady-state error 5. Adjust each of K, Ki, and Kd until you obtain a desired overall response referring to the table

shown previously to find out which controller controls what characteristics.

Closed Loop Step : K = 350, Ki = 300, Kd = 50

Time (sec)

Dis

pla

ce

me

nt

(m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

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6. It is not necessary to implement all three controllers (P, I & D) into a single system. For example, if a PI controller gives a good enough response, then you don't need to add D control to the system. Simple is better.

Documents

CHAPTER 2.0 INTRODUCTION TO CONTROL SYSTEMS · 2010-09-11 · CHAPTER 2.0 INTRODUCTION TO CONTROL SYSTEMS ... The history of feedback control system begun as early as in 1769 when