Control Systems 10ES43 - ATME College of Engineering, Mysuru

DEPT. of ECE, ATMECE, MYSURU 1

Control Systems

(15EC43)

Department of Electronics and Communication

ATME College of Engineering, Mysuru.


INDEX

1.1 Objectives ......................................................................................................................................... 6

1.2 Introduction ....................................................................................................................................... 6

1.3 Types of control systems: ................................................................................................................. 8

1.4 Block Diagrammatic Representation .............................................................................................. 12

1.4.2 Advantages of Block Diagram Representation: ......................................................................................14

1.5 Block diagram reduction technique ................................................................................................ 14

1.5.1 Block diagram rules .................................................................................................................................15

1.6 Applications of the control systems ................................................................................................ 16

1.7 Feed back Control System .............................................................................................................. 17

1.9 Properties of Systems ...................................................................................................................... 19

1.10 Transfer Function .......................................................................................................................... 19

1.11 Signal Flow Graphs (SFG)............................................................................................................ 21

1.12 Masons gain formula..................................................................................................................... 23

1.13 Outcomes ...................................................................................................................................... 26

1.14 Recommended Questions .............................................................................................................. 26

2.1 Objectives ....................................................................................................................................... 28

2.2 Introduction ..................................................................................................................................... 28

2.3 Response to a Unit Step Input –First Order .................................................................................... 29

2.4 Response to a Unit Ramp Input ...................................................................................................... 32

2.5 Response to a Unit Step Input - Second Order System .................................................................. 33

2.6 Proportional, Integral and Derivative Controller (PID Control) : ................................................... 36

2.7 Outcomes ........................................................................................................................................ 38

2.8 Recommended questions ................................................................................................................ 38

3.1 Objectives ........................................................................................ Error! Bookmark not defined.

3.2 Introduction ...................................................................................... Error! Bookmark not defined.

3.3 Concept of stability: ......................................................................... Error! Bookmark not defined.




MODULE -1

Introduction to Control Systems, Types of Control Systems, Effect ofFeedback Systems, Different

ial equation of Physical Systems Mechanical systems,Electrical systems, Analogous systems. Block

diagram and signal flow graphs: transfer functions, block diagram algebra and signal flow graph.

Structure

1.1 Objectives 6

1.2 Introduction 6

1.3 Types of control systems: 8

1.4 Block Diagrammatic Representation 12

1.4.2 Advantages of Block Diagram Representation: 14

1.5 Block diagram reduction technique 14

1.5.1 Block diagram rules 15

1.6 Applications of the control systems 16

1.7 Feed back Control System 17

1.9 Properties of Systems 19

1.10 Transfer Function 19

1.11 Signal Flow Graphs 21

1.12 Masons gain formula 23

1.13 Recommended Questions 26


1.1 Objectives

Students should be able to make measurements of a system and determine a transfer function.

Open loop and closed loop systems and Electrical analogy of physical systems.

Students will know the Block diagram representation Open loop and closed loop systems and

Electrical analogy of physical systems.

Block diagram reduction technique along with Signal Flow graph using the Mason’s formula.

1.2 Introduction

A control system is one which can control any quantity of interest in a machine, mechanism or

other equipment in order to achieve the desired performance or output.(or) A control system is an

interconnection of components connected or related in such a manner as to command, direct, or

regulate itself or another system. For example consider, the driving system of an automobile. Speed

of the automobile is a function of the position of its accelerator. The desired speed can be maintained

(or a desired change in speed can be achieved) by controlling pressure on the accelerator pedal. This

automobile driving system (accelerator, carburetor and engine-vehicle) constitutes a control system.

Control systems find numerous and widespread applications from everyday to extraordinary in

science, industry, and home. Here are a few examples:

(a) Home heating and air-conditioning systems controlled by a thermostat

(b) The cruise (speed) control of an automobile

(c) Manual control

(i) Opening or closing of a window for regulating air temperature or air quality

(ii) Activation of a light switch to regulate the illumination in a room

(iii) Human controlling the speed of an automobile by regulating the gas supply to the engine

(d) Automatic traffic control (signal) system at roadway intersections

(e) Control system which automatically turns on a room lamp at night, and turns it off in Day light

The general block diagram of a control system is shown below..


Fig.1.1. Block diagram of a control system.

The above method of representation of a control system is known as block diagram representation

where in each block represents an element, a plant, mechanism, device etc., whose inner details are

not indicated. Each block has an input and output signal which are linked by a relationship

characterizing the block. It may be noted that the signal flow through the block is unidirectional.

Basic Control System components : The basic control system components are objectives i.e inputs or

actuating signals to the system and the Output signals or controlled variables etc.The control system

will control the outputs in accordance with the input signals. The relation between these components

is shown in the block diagram.

Fig.1.2 Block control system Components

The components of the control system changes as we move from open loop control system to closed

loop control systems. In a closed loop control system, the feedback control network play an important

role in getting the correct output.

The general block diagram of a control system with feedback is shown below. The error detector

compares a signal obtained through feedback elements, which is a function of the output response,

with the reference input. Any difference between these two signals gives an error or actuating signal,

which actuates the control elements. The control elements in turn alter the conditions in the plant in

such a manner as to reduce the original error.

Control System

Input Signals Output Signals


Fig.1.3 .General block diagram of an automatic control system.

1.3 Types of control systems:

There are basically two types of control systems (i) the open loop system and the (ii) closed loop

System. They can both be represented by block diagrams. A block diagram uses blocks to represent

processes, while arrows are used to connect different input, process and output parts.

(a) Open loop Control System: A system which do not possess any feedback network, and

contains only the input and output relationship is known as a open loop control system.

.

Fig.4. Open loop control system

Examples of the open loop control systems are washing machines, light switches, gas ovens, burglar

alarm system etc.The drawback of an open loop control system is that it is incapable of making

automatic adjustments. Even when the magnitude of the output is too big or too small, the system

can’t make the necessary adjustments. For this reason, an open loop control system is not suitable for

use as a complex control system.

(b) Closed loop control system: A closed loop system is one which uses a feed back control

between input and output. A closed loop control system compares the output with the expected

Process Input Output


result or command status, and then it takes appropriate control actions to adjust the input

signal. Therefore, a closed loop system is always equipped with a sensor, which is used to

monitor the output and compare it with the expected result.

Fig.5.Closed Loop Control system

The output signal is fed back to the input to produce a new output. A well-designed feedback system

can often increase the accuracy of the output.

Examples for closed loop systems are air conditioners, refrigerators, automatic rice cookers,

automatic ticketing machines, etc. For example An air conditioner, uses a thermostat to detect the

temperature and control the operation of its electrical parts to keep the room temperature at a preset

constant.

One advantage of using the closed loop control system is that it is able to adjust its output

automatically by feeding the output signal back to the input. When the load changes, the error signals

generated by the system will adjust the output suitably. The limitation of a closed loop control

systems is they are generally more complicated and thus also more expensive to design.

(c) Linear versus Nonlinear Control Systems : Linear feedback control systems are idealized models

fabricated by the analyst purely for the simplicity of analysis and design

When the magnitudes of signals in a control system are limited to ranges in which system components

exhibit linear characteristics (i.e., the principle of superposition applies), the system is essentially

linear.

But when the magnitudes of signals are extended beyond the range of the linear operation, depending

on the severity of the nonlinearity, the system should no longer be considered linear. For instance,

amplifiers used in control systems often exhibit a saturation effect when their input signals become

large; the magnetic field of a motor usually has saturation properties. Other common nonlinear effects


found in control systems are the backlash or dead play between coupled gear members, nonlinear

spring characteristics, nonlinear friction force or torque between moving members, and so on. Quite

often, nonlinear characteristics are intentionally introduced in a control system to improve its

performance or provide more effective control.

Also the analysis of linear systems is easy and lot of mathematical solutions is available for their

simplification.

Nonlinear systems, on the other hand, are usually difficult to treat mathematically, and there are no

general methods available for solving a wide class of nonlinear systems. In practice , first a linear-

system is modeled by neglecting the nonlinearities of the system and the designed controller is then

applied to the nonlinear system model for evaluation or redesign by computer simulation.

Distinguish between Open loop and closed loop control systems

Open Loop System Closed Loop System

1. An open loop system has the ability to

perform accurately, if its calibration is

good. If the calibration is not perfect its

performance will go down.

2. It is easier to build.

3. In general it is more stable as the feed

back is absent .

4. If-non-linearity’s are present , the

system operation is not good.

5. Feedback is absent

Example :

(i) Traffic control system

(ii) Control of furnace for coal

heating

(iii) An Electric washing machine

1. A closed loop system has got the ability

to perform accurately because of the

feedback.

2. It is difficult to build.

3. Less stable Comparatively

4. Even under the presence of non-

linearity’s the system operates better

than open loop system.

5. Feedback is present

Example:

(i) Pressure control system

(ii) Speed control system

(iii) Temperature control system

(d) Time-Invariant control Systems:


When the parameters of a control system do not change with respect to time during the operation of

the system, the system is called a time-invariant system. In practice, most physical systems contain

elements that drift or vary with time. For example, the winding resistance of an electric motor will

vary when the motor is first being excited and its temperature is rising. Another example of a time-

varying system is a guided-missile control system in which the mass of the missile decreases as the

fuel on board is being consumed during flight. Although a time-varying system without nonlinearity

is still a linear system, the analysis and design of this class of systems are usually much more complex

than that of the linear time-invariant systems.

(e) Continuous and Discrete Data Control Systems: A continuous-data system is one in which the

signals at various parts of the system are all functions of the continuous time variable t. The signals in

continuous-data systems may be further classified as ac or dc. In control systems the ac control

system, means that the signals in the system are modulated by some form of modulation scheme. A

dc control system, on the other hand, simply implies that the signals are un-modulated, but they are

still ac signals according to the conventional definition. The schematic diagram of a closed loop dc

control system is shown below. Typical waveforms of the signals in response to a step-function input

are shown in the figure. Typical components of a dc control system are potentiometers, dc amplifiers,

dc motors, dc tachometers, and so on.

Fig 1.6 Block diagram of Continuous and Discrete Data Control Systems

In ac control systems, the signals are modulated i.e the information is transmitted by an ac carrier

signal. Here the output controlled variable behaves similarly to that of the dc system. In this case, the


modulated signals are demodulated by the low-pass characteristics of the ac motor. Ac control

systems are used extensively in aircraft and missile control systems in which noise and disturbance

often create problems. By using modulated ac control systems with carrier frequencies of 400 Hz or

higher, the system will be less susceptible to low-frequency noise. Typical components of an ac

control system are synchros, ac amplifiers, ac motors, gyroscopes, accelerometers etc.

(f) Discrete Data Control System

If the signal is not continuously varying with time but it is in the form of pulses then the control

system is called Discrete Data Control System. If the signal is in the form of pulse data, then the

system is called Sampled Data Control System. Here the information supplied intermittently at

specific instants of time. This has the advantage of Time sharing system. On the other hand, if the

signal is in the form of digital code, the system is called Digital Coded System. Here use of Digital

computers, micro processors or microcontrollers are made use of such systems and are analyzed by

the Z- transform theory.

1.4 Block Diagrammatic Representation

It is a representation of the control system giving the inter-relation between the transfer function of

various components. The block diagram is obtained after obtaining the differential equation &

Transfer function of all components of a control system. The arrow head pointing towards the block

indicates the i/p & pointing away from the block indicates the o/p.

Suppose G(S) is the Transfer function then G(S) = C(S) / R(S)

Fig.1.7. Block Diagram Representation of a Transfer Function

After obtaining the block diagram for each & every component, all blocks are combined to get a

complete representation. It is then reduced to a simple form with the help of block diagram algebra.

1.4.1 Basic elements of a block diagram

Blocks

G (S)

R(S) C(S)


Transfer functions of elements inside the blocks

Summing points

Take off points

Arrow

Fig 1.8. Basic elements of a Block Diagram

A control system may consist of a number of components. A block diagram of a system is a pictorial

representation of the functions performed by each component and of the flow of signals. The elements

of a block diagram are block, branch point and summing point.

(a) Block :

In a block diagram all system variables are linked to each other through functional blocks. The

functional block or simply block is a symbol for the mathematical operation on the input signal to the

block that produces the output.

Fig.1.9 Block

(b) Summing point:

The blocks are used to identify many types of mathematical operations, like addition and subtraction

and represented by a circle, called a summing point. As shown below diagram a summing point may

have one or several inputs. Each input has its own appropriate plus or minus sign. A summing point

has only one output and is equal to the algebraic sum of the inputs

BLOCK INPUT

OUTPUT


Fig.1.10 Summing Point

Individual & Overall performance can be studied A takeoff point is used to allow a signal to be used

by more than one block or summing point

(c)Arrow – associated with each branch to indicate the direction of flow of signal

1.4.2 Advantages of Block Diagram Representation:

It is always easy to construct the block diagram even for a complicated system

Function of individual element can be visualized

Over all transfer function can be calculated easily

1.4.3 Limitations of a Block Diagram Representation :

No information can be obtained about the physical construction

Source of energy is not shown

1.5 Block diagram reduction technique

Because of the simplicity and versatility, the block diagrams are often used by control engineers to

describe all types of systems. A block diagram can be used simply to represent the composition and

interconnection of a system. Also, it can be used, together with transfer functions, to represent the

cause-and-effect relationships throughout the system. Transfer Function is defined as the relationship

between an input signal and an output signal to a device.

Procedure to solve Block Diagram Reductions:

Step 1: Reduce the blocks connected in series


Step 2: Reduce the blocks connected in parallel

Step 3: Reduce the minor feedback loops

Step 4: Try to shift take off points towards right and Summing point towards left

Step 5: Repeat steps 1 to 4 till simple form is obtained

Step 6: Obtain the Transfer Function of Overall System

1.5.1 Block diagram rules

(1) Blocks in Cascade [Series]: When two blocks are connected in series , their resultant transfer

function is the product of two individual transfer functions.

(2) Combining blocks in Parallel: When two blocks are connected parallel as shown below ,the

resultant transfer function is equal to the algebraic sum (or difference) of the two transfer

functions.This is shown in the diagram below.

(3) Eliminating a feed back loop: The following diagram shows how to eliminate the feed back loop

in the resultant control system


(4) Moving a take-off point beyond a block: The effect of moving the takeoff point beyond a block is

shown below.

(5) Moving a Take-off point ahead of a block: The effect of moving the takeoff point ahead of a

block is shown below.

1.6 Applications of the control systems

There are various applications of control systems which include biological propulsion; locomotion;

robotics; material handling; biomedical, surgical, and endoscopic ; aeronautics; marine and the

defense and space industries. There are also many household and industrial application examples of

the control systems, such as washing machine, air conditioner, security alarm system and automatic

ticket selling machine, etc.

(i) Washing machine:

The most commonly used house hold application is the washing machine. It comes under automatic

control system ,where the machine automatically starts to pour water, add washing powder, spin and

wash clothes, discharge wastewater, etc. After the completion of all the procedures, the washing

machine will stop the operation.

However, this kind of machine only operates according to the preset time to complete the whole

washing process. It ignores the cleanness of the clothes and does not generate feedback. Therefore,

this kind of washing machine is of open loop control system.


(ii). Air conditioner

The air conditioner is used to automatically control the temperature of the room.In the air conditioner

the coolant circulated in the machine will absorb heat indoor, then it will be transported from the

vaporization device to cooling device. The hot air is then blown to outdoor by a fan. There is an

adjustable temperature device equipped in the air conditioner for the users to adjust the extent of

cooling. When the temperature of the cool air is lower than the preset one, the controller of the air

conditioner will stop the operation of the compressor to cease the circulation of the coolant. The

temperature sensor installed near the vaporization device will continuously measure the indoor

temperature, and send the results to the controller for further processing.This operation will come

under closed loop control system. The simple block diagram of air conditioner system is shown

below.

1.7 Feed back Control System

The feedback control system is represented by the following block diagram .In the diagram feedback

signal is denoted by B(S) and the output is C(S).The input function is denoted by R(S).

The open loop gain of the system is G(S) and the feedback loop gain H(S). Then the feedback

signal b(s) is given by

B(S) = H(S). C(S)


1.8 Transfer function:

The input- output relationship in a linear time invariant system is given by the transfer function. For a

time invariant system it is defined as the ratio of Laplace transform of the out to the Lapalce

transform of the input

The important features of the transfer functions are,

The transfer function of a system is the mathematical model expressing the differential

equation that relates the output to input of the system.

The transfer function is the property of a system independent of magnitude and the nature of

the input .

The transfer function includes the transfer functions of the individual elements. But at the

same time, it does not provide any information regarding physical structure of the system

If the transfer function of the system is known, the output response can be studied for various

types of inputs to understand the nature of the system

It is applicable to Linear Time Invariant system.

It is assumed that initial conditions are zero.

It is independent of i/p excitation.

It is used to obtain systems o/p response.

If the transfer function is unknown, it may be found out experimentally by applying known

inputs to the device and studying the output of the system

From the above block diagram G(S) = C(S) / E(S) &

E(S) = R(S) – B(S)

So, C(S) = G(S) .E(S)

= G(S)[ R(S)- B(S)

= G(S) [ R(S) – H(S).C(S)

Therefore,

C(S)/R(S) = G(S) / [1+ G(S).H(S)]


This is the transfer function of the closed loop control system

1.9 Properties of Systems

For any control system to understand its performance the following properties are very important.

(i).Linearity: A system is said to be linear if it follows both the law of addtivity and law of

homogeneity. The system which does not follow the law of homogeneity and additively is called a

non-linear system.

If input x1(t) produces response y1(t) and input x2(t) produces response y2(t) then the scaled and

summed input a1x1(t) +b1x2(t) produces the scaled and summed response a1y1(t) +b1y2(t) where a1 and

a2 are real scalars. It follows that this can be extended to an arbitrary number of terms, and so for real

numbers.

(ii) Time Invariance : A system with input x(t) and output y(t) is time-invariant if x(t- t0) is creates

output y ( t – t0) for all inputs x and shifts t0.

(iii) Causality : A system is causal, if the output y(t) at time t is not a function of future inputs and it

depends only on the present and past inputs . All analog systems are causal and all memeoryless

systems are causal.

If the system is causal, then this implies h (t) = 0, t < 0. Alternatively, h[n] =0, n < 0.

(iv) Stability : A system is said to be a stable if for every bounded-input there exists a bounded output

. 1.10 Transfer Function

For a open loop control system shown below the transfer function is the ratio of Laplace transform of

the out-put to the Laplace transform of the input.

G (t) C (t) R (t)


The Laplace transform of the Input is R(S) and the Laplace transform of the output is

C(S) .

So,the Transfer function of the system is G(S) = C(S) / R(S)

Example : Find the transfer function of the following RC circuit

The Laplace transformed Network is shown above. From the circuit we can write that

Vo(S) = 1/ Cs * I(S) & Vi (S ) [ R + 1/Cs ]

Cs * Vo (S) = Vi (S) / [ R + 1/Cs ]

Vo(S )/ Vi (S ) = 1/ { [ R + 1/Cs ]} Cs

Vo(S )/ Vi (S ) = 1/ { [ 1 + RCs ]} = 1/ {

1+ τ s ]} Where τ = RC

Or the Transfer function G (S) = 1/ [1+

τ s]


1.11 Signal Flow Graphs (SFG)

The block diagram method is a useful tool for simplifying the representation of a control system. But

when there are more than two feed back loops and if there exists inter-coupling between feedback

loops, and when a system has more than one input and one output, the block diagram approach is very

complex. Hence an alternate method is proposed by S.J. Mason. This method is called signal flow

graphs. In these graphs each node represents a system variable & each branch connected between two

nodes acts as Signal Multiplier. The direction of signal flow is indicated by an arrow.

A signal flow graph is a diagram that represents a set of simultaneous equations. It consists of a graph

in which nodes are connected by directed branches. The nodes represent each of the system variables.

A branch connected between two nodes acts as a one-way signal multiplier: the direction of signal

flow is indicated by an arrow placed on the branch, and the multiplication factor (transmittance or

transfer function) is indicated by a letter placed near the arrow.

So,in the figure above , the branch transmits the signal x1 from left to right and multiplies it by the

quantity a in the process. The quantity a is the transmittance, or transfer function.

Flow-Graph Definitions : A node performs two functions:

Addition of the signals on all incoming branches and Transmission of the total node signal

(the sum of all incoming signals) to all outgoing

branches

There are three types of nodes .They are Source nodes , Sink nodes and Mixed nodes

Source nodes (independent nodes) : These represent independent variables and have only outgoing

branches. In Fig. 5.21, nodes u and v are source nodes.


Sink nodes (dependent nodes): These represent dependent variables and have only incoming

branches. In Fig (a), nodes x and y are sink nodes.

Mixed nodes (general nodes): These have both incoming and outgoing branches. In Fig. (a), node w

is a mixed node. A mixed node may be treated as a sink node by adding an out going branch of unity

transmittance, as shown in Fig (b), for the equation x = au +bv

and w = cx = cau + cbv

Fig(a) Fig (b)

A path is any connected sequence of branches whose arrows are in the same direction and A

forward path between two nodes is one that follows the arrows of successive branches and in which a

node appears only once. In Fig.(a) the path uwx is a forward path between the nodes u and x.

Flow-Graph Algebra : The following rules are useful for simplifying a signal flow graph:

Series paths (cascade nodes). Series paths can be combined into a single path by multiplying the

transmittances as shown in Fig ( A ).

Path gain. The product of the transmittances in a series path.

Parallel paths. Parallel paths can be combined by adding the transmittances as shown in Fig(B).

Node absorption. A node representing a variable other than a source or sink can be eliminated as

shown in Fig (C).

Feedback loop. Aclosed path that starts at a node and ends at the same node.

Loop gain. The product of the transmittances of a feedback loop.


These results are shown diagrammatically in the following figures (A) ,(B) and C) where the original

diagram and equivalent diagrams are shown.

1.12 Masons gain formula

The relationship between an input variable and an output variable of a signal flow graphis given by

the net gain between input and output nodes and is known as overall gain ofthe system. Masons gain

formula is used to obtain the over all gain (transfer function) of signal flow graphs. According to

Mason’s gain formula Gain is given by

1 11

k

PP

Where, Pk is gain of k th

forward path and Δ is determinant of graph. Here the Δ is given by

Δ = 1-(sum of all individual loop gains)+(sum of gain products of all possible combinations of two

non touching loops –sum of gain products of all possible combination of three non touching loops)

Δk is cofactor of kth

forward path determinant of graph with loops touching k th

forward path. It is

obtained from Δ by removing the loops touching the path Pk.


Finding transfer function from the system flow graphs is explained below by example.

Example1 : Obtain the transfer function of the system whose signal flow graph is shown below.

There are two forward paths: One is Gain of path 1: P1=G1 and the other is Gain of path 2: P2=G2

There are four loops with loop gains:

L1=-G1G3, L2=G1G4, L3= -G2G3, L4= G2G4

There are no non-touching loops.

Δ = 1+G1G3-G1G4+G2G3-G2G4

Forward paths 1 and 2 touch all the loops. Therefore, Δ1= 1, Δ2= 1

So,the transfer function T is given by


Example 2 : Obtain the transfer function of C(s) /R(s) of the system whose signal flow graph shown

below.

From the system flow graph it is clear that

There is one forward path, whose gain is: P1=G1G2G3

There are three loops with loop gains:

L1=-G1G2H1, L2=G2G3H2, L3= -G1G2G3

There are no non-touching loops: Δ = 1-G1G2H1+G2G3H2+G1G2G3

Forward path 1 touches all the loops. Therefore, Δ1= 1.

The transfer function T is given by

( ) 1 1 1 2 3

( ) 1 1 2 1 1 3 2 1 2 3

C S P G G G

R S G G H G G H G G G

System Stability:

The study of stability of a control system is very important to understand the performance . This

means that the system must be stable at all times during operation. Stability may be used to define the

usefulness of the system. Stability studies include absolute & relative stability. Absolute stability is

the quality of stable or unstable performance. Relative Stability is the quantitative study of stability.


The stability study is based on the properties of the Transfer Function. In the analysis, the

characteristic equation is very important ,which describe the transient response of the system. From

the roots of the characteristic equation, following conclusions about the stability can be drawn.

(1) When all the roots of the characteristic equation lie in the left half of the S-plane, the system

response due to initial condition will decrease to zero at time Thus the system will be termed as a

stable system.

(2) When one or more roots lie on the imaginary axis & there are no roots on the RHS of

S-plane, the response will be oscillatory without damping. Such a system will be termed as critically

stable.

(3) When one or more roots lie on the RHS of S-plane, the response will exponentially increase

in magnitude and there by the system will be Unstable.

1.13 Outcomes

Able to design basic structure of control system.

Students will analyze the application of Open loop and closed loop systems.

Students should be able to design basic structure of control system using Block

diagram reduction technique.

Analyze the importance of Signal Flow graph using the Mason’s formula.

1.14 Recommended Questions

1. Name three applications of control systems.

2. Name three reasons for using feedback control systems and at least one reason for not

using them.

3. Give three examples of open- loop systems.

4. Functionally, how do closed – loop systems differ from open loop systems.

5. State one condition under which the error signal of a feedback control system would not

be the difference between the input and output.


6. Name two advantages of having a computer in the loop.

7. Name the three major design criteria for control systems.

8. Name the two parts of a system‘s response.

9. Physically, what happens to a system that is unstable?

10. Instability is attributable to what part of the total response.

11. What mathematical model permits easy interconnection of physical systems?

12. To what classification of systems can the transfer function be best applied?

13. What transformation turns the solution of differential equations into algebraic

manipulations ?

14. Define the transfer function.

15. What assumption is made concerning initial conditions when dealing with transfer

functions?

16. What do we call the mechanical equations written in order to evaluate the transfer

function ?

17. Why do transfer functions for mechanical networks look identical to transfer functions

for electrical network.

Resources:

lpsa.swarthmore.edu/Analogs/ElectricalMechanicalAnalogs.htm

www.newagepublishers.com/samplechapter/001712.pdf

www.youtube.com/watch?v=h_CDoUIYrC0

https://www.scribd.com/doc/7291532/Class-5-6-Analogous-Systems

https://www.google.co.in/?gfe_rd=cr&ei=1r0KVfuYGqHV8gfK-YH4Cg&gws_rd=ssl#

www.dartmouth.edu/~sullivan/22files/System_analogy_all.pdf

MODULE-2

TIME RESPONSE OF FEEDBACK CONTROL SYSTEMS

Standard test signals, Unit step response of the First and Second order Systems. Time

response specifications, steady state errors and error constants. Introduction to PI, PD and PID

controllers.

Structure:

Control Systems 10ES43


2.1 Objectives 28

2.2 Introduction 28

2.3 Response to a Unit Step Input –First Order 29

2.4 Response to a Unit Ramp Input 32

2.5 Response to a Unit Step Input - Second Order System 33

2.6 Proportional, Integral and Derivative Controller (PID Control) : 36

2.7 Outcomes 38

2.8 Recommended questions 38

2.1 Objectives

Get an understanding of the standard test signals

Able to analyze the response of First and second order systems

Understand the different types of steady state errors

2.2 Introduction

(a)Time response analysis

It is an equation or a plot that describes the behavior of a system and gives information about it

with respect to time response specification as overshooting, settling time, peak time, rise time

and steady state error. Time response is formed by the transient response and the steady state

response.

Time response = Transient response + Steady state response.

Transient time response or Natural response describes the behavior of the system in its first short

time until arrives the steady state value. If the input is step function then the output or the

response is called step time response and if the input is ramp, the response is called ramp time

response .. etc.

(b)Transient Response



The transient response is defined as the part of the time response that goes to zero as time

becomes very large. Thus yt(t) has the property

Lim y(t) = 0

t -->∞

The time required to achieve the final value is called transient period. The transient response may

be exponential or oscillatory in nature. Output response consists of the sum of forced response

(form the input) and natural response (from the nature of the system).The transient response is

the change in output response from the beginning of the response to the final state of the

response and the steady state response is the output response as time is approaching infinity (or

no more changes at the output). The behavior of a system in transient state is shown below.

(c) Steady State Response: The steady state response is the part of the total response that

remains after the transient has died out. For a position control system, the steady state

response when compared to with the desired reference position gives an indication of the

final accuracy of the system. If the steady state response of the output does not agree with

the desired reference exactly, the system is said to have steady state error.

2.3 Response to a Unit Step Input –First Order

Consider a feedback system with G)s) = 1/τs as shown below



The closed loop transfer function of the system is given by

For a unit step input R (s) = 1 / s and the output is given by

Inverse Laplace transformation yields

The plot of c(t) Vs t is shown below

The response is an exponentially increasing function and it approaches a value of unity as



t --- > ∞

At t = τ the response reaches a value,

which is 63.2 percent of the steady value. This time, τ is known as the time constant of the

system. One of the important characteristics about the system is its speed of response or how

fast the response is approaching the final value. The time constant τ is indicative of this measure

and the speed of response is inversely proportional to the time constant of the system.Another

important characteristic of the system is the error between the desired value and the actual value

under steady state conditions. This quantity is known as the steady state error of the - system and

is denoted by ess.

The error E(s) for a unity feedback system is given by

For the system under consideration G(s) = 1 / τs and R(s) = 1 /s Therefore

As t ~ ∞ e (t) ~ 0 . Thus the output of the first order system approaches the reference input,

which is the desired output, without any error. In other words, we say a first order system tracks

the step input without any steady state error.



2.4 Response to a Unit Ramp Input

To study the response of a unit ramp let us consider a feedback system with G)s) = 1/τs as shown below

The response of the system for which , R(s) = 1 / s2

is given by

The time response is obtained by taking inverse Laplace transform of above equation

Differentiating the above equation we get

This is the response of the system to a step input. Thus no additional information about the

speed of response is obtained by considering a ramp input.So, no additional information about

the speed of response is obtained by considering a ramp input.

But the effect on the steady state error is given by



Thus the steady state error is equal to the time constant of the system. The first order system,

therefore, can not track the ramp input without a finite steady state error. If the time constant is

reduced not only the speed of response increases but also the steady state error for ramp input

decreases. Hence the ramp input is important to the extent that it produces a finite steady state

error.

The response of a first order system for unit ramp input is shown below.

2.5 Response to a Unit Step Input - Second Order System

Let us consider a type 1, second order system as shown in Fig. below. Since G(s) has one pole at

the origin, it is a type one system.



The closed loop transfer function is give by,

The transient response of any system depends on the poles of the transfer function T(s). The

roots of the denominator polynomial in s of T(s) are the poles of the transfer function. Thus the

denominator polynomial of T(s), given by

D(s) = τs2 + S + K

is known as the characteristic polynomial of the system and D(s) = 0 is known as the

characteristic equation of the system. The above Eqn. is normally put in standard from, given

by,



The poles of T( s), or, the roots of the characteristic equation

S2 + 2δ ωn s + ωn

2 = 0

are given by,

Where is known as the damped natural frequency of the system. If δ > 1,

the two roots s1, s2 are real and we have an over damped system. If δ = 1, the system is known as

a critically damped system. The more common case of δ < 1 is known as the under damped

system.

Steady State Errors : One of the important design specifications for a control system is the

steady state error. The steady state output of any system should be as close to desired output as

possible. If it deviates from this desired output, the performance of the system is not satisfactory

under steady state conditions. The steady state error reflects the accuracy of the system. Among

many reasons for these errors, the most important ones are the type of input, the type of the

system and the nonlinearities present in the system. Since the actual input in a physical system is

often a random signal, the steady state errors are obtained for the standard test signals, namely,

step, ramp and parabolic signals.

Error Constants : Let us consider a feedback control system as shown below.



The error signal E (s) is given by E (s) = R (s) - H (s) C (s)

But C (s) = G (s) E (s)

From the above equations we have

Applying final value theorem, we can get the steady state error ess as,

The above equation shows that the steady state error is a function of the input R(s) and the open

loop transfer function G(s). Let us consider various standard test signals and obtain the steady

state error for these inputs.

2.6 Proportional, Integral and Derivative Controller (PID Control) :

A Proportional–Integral–Derivative (PID) controller is a three-term controller which is

considered as a standard controller in industrial settings. . It can be found in virtually all kinds of

control equipments, either as a stand-alone (single-station)controller or as a functional block in

Programmable Logic Controllers (PLCs)and Distributed Control Systems (DCSs).

An integral control eliminates steady state error due to a velocity input, but its effect on dynamic

response is 'difficult to predict as the system order increases to three. It is known that a

derivative term in the forward path improves the damping in the system. One of the best-known

controllers used in practice is the PID controller, where the letters stand for proportional,



integral,and derivative. The integral and derivative components of the PID controller have

individual performance implications, and their applications require an understanding ofthe basics

of these elements. Hence a suitable combination of integral and derivative controls results in a

proportional, integral and derivate control, usually called PID control. The transfer function of

the PID controller is given by,

The diagram below gives the ideal PID controller.

The overall forward path transfer function is given by,

and the overall transfer function is given by,



Proper choice of Kp, KD and Kr results in satisfactory transient and steadystate responses. The

process of choosing proper Kp, KD, at Kr for a given system is known as tuning of a P ID

controller.

2.7 Outcomes

Student will be able to analyze the different test signals and apply it to design the first order and

second order systems.

Understand the effect of different steady state errors as the type and order of the system changes

Able to understand the working of P, PI and PID controllers

2.8 Recommended questions



MODULE – 3

Stability analysis: Concepts of stability, Necessary conditions for Stability, Routh stability

criterion, Relative stability analysis: more on the Routh stability criterion, Introduction to Root -

Locus Techniques, The root locus concepts, Construction of root loci.

Structure

3.1 Objective

3.2 Introduction

3.3 Concepts of stability

3.4 Necessary conditions for Stability

3.5 Routh stability criterion

3.6 Relative stability analysis

3.7 Introduction to Root - Locus Techniques

3.8 The root locus concepts

3.9 Construction of root loci



3.1 Objectives

Students should be able to make measurements of a system and determine the stability of

the system.

Students will know the Necessary conditions for Stability.

Students will know the Relative stability analysis.

Students will know the concepts of root locus and should be able to Construct root loci.

3.2 Introduction

Every System, for small amount of time has to pass through a transient period. Whether system

will reach its steady state after passing through transients or not. The answer to this question is

whether the system is stable or unstable. This is stability analysis.

For example, we want to go from one station to other. The station we want to reach is our

final steady state. The traveling period is the transient period. Now anything may happen during

the traveling period due to bad weather, road accident etc., there is a chance that we may not

reach the next station in time. The analysis of whether the given system can reach steady state

after passing through the transients successfully is called the stability analysis of the system.

3.3 Concept of Stability

Consider a system i.e a deep container with an object placed inside it as shown in fig 3.1.

Fig 3.1: Concept of absolute Stability

Now, if we apply a force to take out the object, as the depth of container is more, it will oscillate

& settle down again at original position.

Assume that force required to take out the object tends to infinity i.e always object will

oscillate when force is applied & will settle down but will not come out such a system is called

absolutely stable system. No change in parameters, disturbances, changes the output.



Now consider a container which is pointed one, on which we try to keep a circular object. In this

object will fall down without any external application of force. Such system is called Unstable

system which is shown in Fig 3.2.

Fig 3.2: Concept of Unstable System

While in certain cases the container is shallow then there exists a critical value of

force for which the object will come out of the container.

Fig 3.3: Concept of Conditionally Stable System

As long as F < F critical object regains its original position but if F > F critical object will come out.

Stability depends on certain conditions of the system; hence system is called conditionally stable

system.

Pendulum where system keeps on oscillating when certain force is applied. Such systems

are neither stable nor unstable & hence called critically stable or marginally stable systems

Stability of control systems:

The stability of a linear closed loop system can be determined from the locations of closed loop

poles in the S-plane.

If the system has closed loop T.F



Output response for unit step input R(S) = 1/S

Find out partial fractions

If the closed loop poles are located in left half of s-plane, Output response contains exponential

terms with negative indices will approach zero & output will be the steady state output.

i.e

Transient output = 0

Such systems are called absolutely stable systems.

Now let us have a system with one closed loop pole located in right half of s- plane

C (t) = - 1.25 + 0.833e 2t

+ 0.416e -4 t

Here there is one exponential term with positive in transient output

Therefore Css = - 1.25



From the above table, it is clear that output response instead of approaching to steady

state value as t -> ∞ due to exponential term with positive index, transients go on increasing in

amplitude.

So such system is said to be unstable. In such system output is uncontrollable &

unbounded one. Output response of such system is as shown in fig 3.4.

Fig 3.4: Output Response of unstable system

For such unstable systems, if input is removed output may not return to zero. And if the input

power is turned on, output tends to ∞. If no saturation takes place in system & no mechanical

stop is provided then system may get damaged.

If all the closed loop poles or roots of the characteristic equation lie in left of s-plane,

then in the output response contains steady state terms & transient terms. Such transient terms

approach to zero as time advances eventually output reaches to equilibrium & attains steady state

value. Transient terms in such system may give oscillation but the amplitude of such oscillation

will be decreasing with time & finally will vanish. So output response of such system is shown in

fig 3.5 (a) & (b).



Fig 3.5: Steady state response of a system

BIBO Stability: This is bounded input bounded output stability.

Definition of stable system:

A linear time invariant system is said to be stable if following conditions are satisfied.

1. When system is excited by a bounded input, output is also bounded & controllable.

2. In the absence of input, output must tend to zero irrespective of the initial conditions.

Unstable system:

A linear time invariant system is said to be unstable if,

1. For a bounded input it produces unbounded output.

2. In the absence of input, output may not be returning to zero. It shows certain output without

input.

Besides these two cases, if one or more pairs simple non repeated roots are located on the

imaginary axis of the s-plane, but there are no roots in the right half of s-plane, the output

response will be undamped sinusoidal oscillations of constant frequency & amplitude. Such

systems are said to be critically or marginally stable systems.

Critically or Marginally stable systems:

A linear time invariant system is said to be critically or marginally stable if for a bounded input

its output oscillates with constant frequency & Amplitude. Such oscillation of output is called

Undamped or Sustained oscillations.

For such system one or more pairs of non-repeated roots are located on the imaginary axis as

shown in fig 3.6(a). Output response of such systems is as shown in fig 3.6(b).



Fig 3.6(a) & (b): Location of one or more non repeated roots on imaginary axis and the output

response

If there are repeated poles located purely on imaginary axis system is said to be unstable which is

as shown in the figure 3.7 (a) & (b).

Fig 3.7 (a) & (b): Location of repeated roots purely on imaginary axis and the output response

Conditionally Stable:

A linear time invariant system is said to be conditionally stable, if for a certain condition if a

particular parameter of the system, its output is bounded one. Otherwise if that condition is

violated output becomes unbounded system becomes unstable. i.e. Stability of the system

depends the on condition of the parameter of the system. Such system is called conditionally

stable system.

S-plane can be divided into three zones from stability point of view which is as shown in the

below fig 3.8.



Fig 3.8: S-plane divided into three zones from stability point of view

Relative Stability:

The system is said to be relatively more stable or unstable on the basis of settling time. System is

said to be more stable if settling time for that system is less than that of other system. The

settling time of the root or pair of complex conjugate roots is inversely proportional to the real

part of the roots. So far the roots located near the J axis, settling time will be large. As the roots

move away from J axis i.e towards left half of the s-plane settling time becomes lesser or smaller

& system becomes more & more stable. So the relative stability improves.



Fig 3.9: Location of roots on left side of RH Plane and the output response





3.4 Necessary conditions for Stability and Hurwitz Criterion:

This represents a method of determining the location of poles of a characteristics equation with

the respect to the left half & right half of the s-plane without actually solving the equation.

The T.F. of any linear closed loop system can be represented as,

Where ‘a’ and ‘b’ are constants

To find the closed loop poles we equate F(s) =0. This equation is called as Characteristic

Equation of the system.

F(s) = a0 sn + a1 s

n-1 + a2 s

n-2 +. . . .+ an= 0



Thus the roots of the characteristic equation are the closed loop poles of the system which decide

the stability of the system.

3.4.1 Necessary Conditions:

In order that the above characteristic equation has no root in right of s-plane, it is necessary but

not sufficient that,

1. All the coefficients off the polynomial have the same sign.

2. None of the coefficients vanishes i.e. all the powers of ‘S’ must be present in descending order

from ‘n’ to zero

These conditions are not sufficient.

3.4.2 Hurwitz’s Criterion:

The sufficient condition for having all roots of characteristics equation in left half of s-plane is

given by Hurwitz. It is referred as Hurwitz criterion. It states that:

The necessary & sufficient condition to have all roots of characteristic equation in left

half of s-plane is that the sub-determinants Dk, k = 1,2,3,. . .n obtained from Hurwitz’s

determinant ‘H’ must all be positive.

Method of forming Hurwitz’s determinant:

The order is n*n where n = order of characteristic equation. In Hurwitz determinant, all

coefficients with suffices greater than ‘n’ or negative suffices must all be replaced by zeros.

From Hurwitz determinant subdeterminants, Dk, k = 1,2,3,. . .n must be formed as follows:



For the system to be stable, all above determinants must be positive.

Example 3.1: Determine the stability of the given characteristics equation by Hurwitz,s method.

F(s) = s3

+ s2

+ s1 + 4 = 0 is characteristic equation.

Solution: a0 = 1, a1 = 1, a2 = 1, a3 = 4, n = 3

As D2 & D3 are negative, given system is unstable.

3.4.3 Disadvantages of Hurwitz’s Method:

1. For higher order system, to solve the determinants of higher order is very complicated & time

consuming.

2. Number of roots located in right half of s-plane for unstable system cannot be judged by this

method.

3. Difficult to predict marginal stability of the system.

Due to these limitations, a new method is suggested by the scientist Routh called Routh’s

Method. It is also called Routh-Hurwitz method.

3.5 Routh’s Stability Criterion

It is also called Routh’s array method or Routh’s Hurwitz method.



Routh suggested a method of tabulating the coefficients of characteristic equation in a particular

way. Tabulation of coefficients gives an array called Routh’s Array.

Consider the general characteristic equation as,

F(s) = a0 sn + a1 s

n-1 + a2 s

n-2 +. . . .+ an

Method of forming an array:

Coefficients of first two rows are written directly from characteristics equation.

From these two rows next rows can be obtained as follows.

From 2nd

& 3rd

row, 4th

row can be obtained as

This process is to be continued till the coefficient for s0 is obtained which will be an. From this

array stability of system can be predicted.



3.5.1 Routh’s Criterion

The necessary and sufficient condition for system to be stable is “All the terms in the first

column of Routh’s array must have the same sign. There should not be any sign change in the

first column of Routh’s array.”

If there are sign changes existing then,

1. System is unstable.

2. The number of sign changes equals the number of roots lying in the right half of the s-plane.

Example 3.2: Examine stability of F(S) = S3+6S

2+11S+6=0

Solution: a0 = 1, a1 = 6, a2 =11, a3 = 6, n = 3

As there is no sign change in the first column, system is stable.

Example 3.3: Examine stability of F(S) = s3 + 4s2 + s + 16 = 0

Solution: a0 =1, a1 = 4, a2 = 1, a3 = 16

As there are two sign changes, system is unstable.

Number of roots located in the right half of s-plane = number of sign changes = 2.



3.6 Special Cases of Routh’s Criterion

3.6.1 Special case 1:

First element of any of the rows of Routh’s array is zero and the same remaining row contains at

least one non-zero element.

Effect: The terms in the new row become infinite and Routh’s test fails.

Eg: s5 + 2s

4 + 3s

3 + 6s

2 + 2s + 1 = 0

Following two methods are used to remove above said difficulty.

First method: Substitute a small positive number ‘€’ in place of a zero occurred as a first

element in a row. Complete the array with this number ‘€’. Then examine the sign change by

taking Consider above Example.

To examine sign change,



Routh’s array is,


Second method: To solve the above difficulty one more method can be used. In this, replace

‘S’ by ‘1/Z’ in original equation. Taking L.C.M rearrange characteristic equation in descending

powers of ‘Z’. Then complete the routh’s array with this new equation in ‘Z’ & examine the

stability with this array.

Consider F(s) = s5

+ 2s4 + 3s

3 + 6s

2 + 2s + 1 = 0

Put s = 1 / Z

Z5 + 2Z

4+ 6Z

3+3Z

2+2Z+ 1 = 0




Special case 2:

All the elements of a row in Routh’s array are zero

Effect: The terms of the next row cannot be determined & Routh’s test fails

This indicates no availability of coefficient in that row.

Procedure to eliminate this difficulty:

1. Form an equation by using the coefficients of row which is just above the row of zeros.

Such an equation is called an Auxillary equation denoted as A(s). For above case such

an equation is,

A(s) = ds4 + es

2 + f

Note that the coefficients of any row are corresponding to alternate powers of ‘s’ starting from

the power indicated against it.

So ‘d’ is coefficient corresponding to s4 so first term is ds

4 of A(s).

Next coefficient ‘e’ is corresponding to alternate power of ‘s’ from 4 i.e. S2 .Hence the term es

2

& so on.

2. Taking derivative of auxillary equation with respect to ‘s’ i.e.



3. Replace row of zeros by the coefficients of

4. Complete the array of zeros by the coefficients.

Importance of auxillary equation:

Auxillary equation is always the part of original characteristic equation. This means the roots of

the auxillary equation are some of the roots of original characteristics equation. Not only this but

roots of auxillary equation are the most dominant roots of the original characteristic equation,

from the stability point of view. The stability can be predicted from the roots of A(s)=0 rather

than the roots of characteristic equation as the roots of A(s) = 0 are the most dominant from the

stability point of view. The remaining roots of the characteristic equation are always in the left

half & they do not play any significant role in the stability analysis.

e.g. Let F(s) = 0 is the original characteristic equation of say order n = 5.

Let A(s) = 0 be the auxillary equation for the system due to occurrence of special case 2 of the

order m = 2.

Then out of 5 roots of F(s) = 0, the 2 roots which are most dominant (dominant means

very close to imaginary axis or on the imaginary axis or in the right half of s-plane) from the

stability point of view are the 2roots of A(s) = 0. The remaining 5 2 = 3 roots are not significant

from stability point of view as they will be far away from the imaginary axis in the left half of s-

plane.

The roots of auxillary equation may be,

1. A pair of real roots of opposite sign i.e.as shown in the fig. 3.10 (a).



Fig 3.10 (a) Fig 3.10 (b)

2. A pair of roots located on the imaginary axis as shown in the fig. 3.10(b).

3. The non-repeated pairs of roots located on the imaginary axis as shown in the fig. 3.10 (c).

Fig 3.10 (C) Fig 3.10 (d)

4. The repeated pairs of roots located on the imaginary axis as shown in the Fig.3.10 (d).

Hence total stability can be determined from the roots of A(s) = 0, which can be out of

four types shown above.

Change in criterion of stability in special case 2:

After replacing a row of zeros by the coeffeicient of dA(s)/ds, Complete the Routh’s array

But now, the criterion that, no sign in 1st column of array for stability, no longer remains

sufficient but becomes a necessary. This is because though A(s) is a part of original characteristic

equation, dA(s)/ds is not, which is in fact used to complete the array.

So if sign change occurs in first column, system is unstable with number of sign changes

equal to number of roots of characteristics equation located in right half of s-plane.

But there is no sign changes, system cannot be predicted as stable. And in such case

stability is to be determined by actually solving A(s) = 0 for its roots. And from the location of

roots of A(s) = 0 in the s-plane the system stability must be determined. Because roots A(s) = 0

are always dominant roots of characteristic equation.



Application of Routh’s Criterion:

Relative stability analysis :

If it is required to find relative stability of system about a line s = - i.e. how many roots are

located in right half of this line s = , the Routh’s method can be used effectively.

To determine this form Routh’s aray, shift the axis of s-plane & then apply Routh’s array

i.e substitute s=s1- (= constant) in characteristic equation. Write polynomial in terms of s1.

Complete array from this new equation. The number of sign changes in first column is equal to

number of roots those are located to right of the vertical line s = - .

Determining range of values of K:

In practical system, an amplifier of variable gain K is introduced.

The closed loop transfer function is

Hence the characteristic equation is

F(s) = 1+ KG(s) H(s) = 0

Advantages of Routh’s array are as follows:

1. Stability of the system can be judged without actually solving the characteristic equation.

2. No evaluation of determinants, which saves calculation time.

3. For unstable system it gives number of roots of characteristic equation having positive real

part.

4. Relative stability of the system can be easily judged.



5. By using the criterion, critical value of system gain can be determined hence frequency of

sustained oscillations can be determined.

6. It helps in finding out range of values of K for system stability.

7. It helps in finding out intersection points of roots locus with imaginary axis.

Limitations of Routh’s criterion:

1. It is valid only for real coefficients of the characteristic equation.

2. It does not provide exact locations of the closed loop poles in left or right half of s-plane.

3. It does not suggest methods of stabilizing an unstable system.

4. Applicable only to linear system.

Ex. S6 + 4S5 +3S4- 16S2- 64S -48 = 0 Find the number of roots of this equation with positive

real part, zero real part & negative real part



Therefore One sign change & system is unstable. Thus there is one root in R.H.S of the s plane

i.e. with positive real part. Now solve A(s) = 0 for the dominant roots

A(s) = 3s4 -48 =0

Put S2 = Y

3Y2 = 48

Y2 =16,

Y = 16 = 4

So S = 2j are the two parts on imaginary axis i.e. with zero real part. Root in R.H.S. indicated

by a sign change is S = 2 as obtained by solving A(s) = 0. Total there are 6 roots as n = 6.

Roots with Positive real part = 1

Roots with zero real part = 2

Roots with negative real part = 6 2 1 = 3

Ex.: For unity feed back system,

Sol: Characteristic equation, 1 + G (s) H (s) = 0 & H(s) = 1



makes row of s0 as row of zeros but K = 0 cannot be marginal value because for K = 0, constant

term in characteristic equation becomes zeros ie one coefficient for s0 vanishes which makes

system unstable instead of marginally stable.



Ex : Using Routh Criterion, investigate the stability of a unity feedback system

whose open loop transfer function is

Sol: The characteristic equation is



Now e -sT

can be Expressed in the series form as

Trancating the series & considering only first two terms we get

esT

= 1 sT

s2 + s + 1 sT = 0

s2 + s (1- T) + 1 = 0

So Routh’s array is

1-T > 0 for stability

T < 1

This is the required condition for stability of the system.

Ex : 5 Determine the location of roots with respect to s = -2 given that

F(s) = s4 + 10 s

3 + 36s

2 + 70s + 75

Sol: shift the origin with respect to s = -2

s = s1 - 2

(s - 2 ) 4 + 10 (s’- 2)

3 + 36(s’- 2 )

2 + 70 ( s’- 2) + 75 = 0



Two sign change, there are two roots to the right of s = -2 & remaining ‘2’ are to the left

of the line s = -2. Hence the system is unstable.

Recommended Questions:

1. Explain briefly how system depends on poles and zeros.

2. Mention the necessary condition to have all closed loop poles in LHS of S-Plane

3. Explain briefly the hurwitz’s Criterion.

4. Explain briefly the Routh’s Stability Criterion.

5. Examine the stability of given equation using Routh’s method

s3+6s

2 + 11s + 6 =0

6. Examine the stability of given equation using Routh’s method

s5 + 2s

4 + 3s

3 + 6s

2 + 2s + 1 = 0

7. Using Routh Criterion, investigate the stability of a unity feedback system whose open loop

transfer function is



3.7 Introduction to Root Locus Techniques

The characteristics of the transient response of a closed loop control system are related to

location of the closed loop poles. If the system has a variable loop gain, then the location of the

closed loop poles depends on the value of the loop gain chosen. It is important, that the designer

knows how the closed loop poles move in the s-plane as the loop gain is varied. W. R. Evans

introduced a graphical method for finding the roots of the characteristic equation known as root

locus method. The root locus is used to study the location of the poles of the closed loop transfer

function of a given linear system as a function of its parameters, usually a loop gain, given its

open loop transfer function. The roots corresponding to a particular value of the system

parameter can then be located on the locus or the value of the parameter for a desired root

location can be determined from the locus. It is a powerful technique, as an approximate root

locus sketch can be made quickly and the designer can visualize the effects of varying system

parameters on root locations or vice versa. It is applicable for single loop as well as multiple loop

system.

3.8 Root Locus Concept

To understand the concepts underlying the root locus technique, consider the second order

system shown in Fig.3.11.

Fig 3.11.Second order control system

The open loop transfer function of this system is

Where, K and a are constants. The open loop transfer function has two poles one at origin s = 0

and the other at s = -a. The closed loop transfer function of the system shown in Fig.3.11 is

The characteristic equation for the closed loop system is obtained by setting the denominator of

the right hand side of Eqn.(2) equal to zero. That is,



The second order system under consideration is always stable for positive values of a and K but

its dynamic behaviour is controlled by the roots of Eqn.(3) and hence, in turn by the magnitudes

of a and K, since the roots are given by

From Eqn.(4), it is seen that as the system parameters a or K varies, the roots change. Consider a

to be constant and gain K to be variable. As K is varied from zero to infinity, the two roots s1 and

s2 describe loci in the s-plane. Root locations for various ranges of K are:

1) K= 0, the two roots are real and coincide with open loop poles of the system s1 =

0, s2 = -a.

2) 0 K < a2/4, the roots are real and distinct.

3) K= a2/4, roots are real and equal.

4) a2/4 < K < , the rots are complex conjugates

The root locus plot is shown in Fig.3.12

Fig. 3.12 Root loci of s2+as+K as a function of K

Figure 3.12 has been drawn by the direct solution of the characteristic equation. This procedure

becomes tedious. Evans graphical procedure helps in sketching the root locus quickly.

The characteristic equation of any system is given by

Where, (s) is the determinant of the signal flow graph of the system given by Eqn.(5).

Where, Pmr is gain product of mth possible combination of r non touching loops of the graph.

The characteristic equation can be written in the form.



For single loop system shown in Fig.3.13

Where, G(s) H(s) is open loop transfer function in block diagram terminology or transmittance in

signal flow graph terminology.

Fig 3.13 Single loop feedback system

From Eqn.(7) it can be seen that the roots of the characteristic equation (closed loop poles)occur

only for those values of s where

Since, s is a complex variable, Eqn.(9) can be converted into the two Evans conditions given

below.

Roots of 1+P(s) = 0 are those values of s at which the magnitude and angle condition given by

Eqn.(10) and Eqn.(11). A plot of points in the complex plane satisfying the angle criterion is the

root locus. The value of gain corresponding to a root can be determined from the magnitude

criterion.

To make the root locus sketching certain rules have been developed which helps in visualizing

the effects of variation of system gain K ( K > 0 corresponds to the negative feedback and K < 0

corresponds to positive feedback control system) and the effects of shifting pole-zero locations

and adding in a new set of poles and zeros.

GENERAL RULES FOR CONSTRUCTING ROOT LOCUS

1) The root locus is symmetrical about real axis. The roots of the characteristic equation are

either real or complex conjugate or combination of both. Therefore their locus must be

symmetrical about the real axis.



2) As K increases from zero to infinity, each branch of the root locus originates from an open

loop pole (n nos.) with K= 0 and terminates either on an open loop zero (m nos.) with K =

along the asymptotes or on infinity (zero at ). The number of branches

terminating on infinity is equal to (n- m).

3) Determine the root locus on the real axis. Root loci on the real axis are determined by open

loop poles and zeros lying on it. In constructing the root loci on the real axis choose a test point

on it. If the total number of real poles and real zeros to the right of this point is odd, then the

point lies on root locus. The complex conjugate poles and zeros of the open loop transfer

function have no effect on the location of the root loci on the real axis.

4) Determine the asymptotes of root loci. The root loci for very large values of s must be

asymptotic to straight lines whose angles are given by

5) All the asymptotes intersect on the real axis. It is denoted by a , given by

6) Find breakaway and breakin points. The breakaway and breakin points either lie on the real

axis or occur in complex conjugate pairs. On real axis, breakaway points exist between two

adjacent poles and breakin points exist between two adjacent zeros. To calculate these



polynomial ds/dK =0 must be solved. The resulting roots are the breakaway / breakin points. The

characteristic equation given by Eqn.(7), can be rearranged as

The breakaway and breakin points are given by

Note that the breakaway points and breakin points must be the roots of Eqn.(15), but not all roots

of Eqn.(15) are breakaway or breakin points. If the root is not on the root locus portion of the

real axis, then this root neither corresponds to breakaway or breakin point. If the roots of

Eqn.(15) are complex conjugate pair, to ascertain that they lie on root loci, check the

corresponding K value. If K is positive, then root is a breakaway or breakin point.

7) Determine the angle of departure of the root locus from a complex pole

Angle of departure from a complex p =180o

- (sum of angles of vectors to a complex pole in question from other poles)

+ (sum of angles of vectors to a complex pole in question from other zeros) (16)

8) Determine the angle of arrival of the root locus at a complex zero

Angle of arrival at complex zero =180o

- (sum of angles of vectors to a complex Zeros in question from other Zeros)

+ (sum of angles of vectors to a complex Zeros in question from other pole) (17)

9) Find the points where the root loci may cross the imaginary axis. The points where the root

loci intersect the j axis can be found by

a) Use Routh’s Hurwitz cretion

b) letting s = in the characteristic equation , equating both the real part and imaginary part to

zero, and solving for and K. The values of thus found give the frequencies at which root loci

cross the imaginary axis. The corresponding K value is the gain at each crossing frequency.



10) The value of K corresponding to any point s on a root locus can be obtained using the

magnitude condition, or

Phase Margin And Gain Margin Of Root Locus

Gain Margin

It is a factor by which the design value of the gain can be multiplied before the closed loop

system becomes unstable.

The Phase Margin

Problem No 1

Sketch the root locus of a unity negative feedback system whose forward path transfer function

Is G(s) = k/s

Solution:

1) Root locus is symmetrical about real axis.

2) There are no open loop zeros (m = 0). Open loop pole is at s = 0 (n = 1). One branch of root

locus starts from the open loop pole when K = 0 and goes to asymptotically when K .

3) Root locus lies on the entire negative real axis as there is one pole towards right of any point

on the negative real axis.

4) The asymptote angle is



Angle of asymptote is

5) Centroid of the asymptote is

6) The root locus does not branch. Hence, there is no need to calculate the break points.

7) The root locus departs at an angle of -180 from the open loop pole at s = 0.

8) The root locus does not cross the imaginary axis. Hence there is no imaginary axis cross over.

The root locus plot is shown in Fig.3.14

Fig 3.14: Root locus plot of K/s

Comments on stability:

The system is stable for all the values of K > 0. Th system is over damped.



Problem No 2

The open loop transfer function is given below. Sketch the root locus plot

Solution:


2) There is one open loop zero at s=-2.0(m=1). There are two open loop poles at

s=-1, -1(n=2). Two branches of root loci start from the open loop pole when

K= 0. One branch goes to open loop zero at s =-2.0 when K -> and other goes to (open loop

zero ) asymptotically when K .

3) Root locus lies on negative real axis for s -2.0 as the number of open loop poles plus

number of open loop zeros to the right of s=-0.2 are odd in number.


Angle of asymptote is


6) The root locus has break points.



The root loci brakesout at the open loop poles at s=-1, when K =0 and breaks in onto the real axis

at s=-3, when K=4. One branch goes to open loop zero at s=-2 and other goes to along the

asymptotically.

7) The branches of the root locus at s=-1, -1 break at K=0 and are tangential to a line s= - 1+j0

hence depart at 90.

8) The locus arrives at open loop zero at 180.

9) The root locus does not cross the imaginary axis, hence there is no need to find the imaginary

axis cross over. The root locus plot is shown in Fig.3.15.

Fig 3.15: Root locus plot for K(S+2)/(S+1)2


System is stable for all values of K > 0. The system is over damped for K > 4. It is critically

damped at K = 0, 4.

Problem No 3

The open loop transfer function is given below. Sketch the root locus.

Solution:




2) There are one open loop zero at s=-4(m=1). There are two open loop poles at s=0, -2(n=2).

Two branches of root loci start from the open loop poles when K= 0. One branch goes to open

loop zero when K and other goes to infinity asymptotically when

and other goes to infinity asymptotically when

3) Entire negative real axis except the segment between s=-4 to s=-2 lies on the root locus.


Angle of asymptote are


6) The brake points are given by dK/ds =0.

7) Angle of departure from open loop pole at s =0 is 180o. Angle of departure from pole at s=-2.0

is 0 .

8) The angle of arrival at open loop zero at s=-4 is 180o.

9) The root locus does not cross the imaginary axis. Hence there is no imaginary cross over.

The root locus plot is shown in fig.3.16



Fig 3.16: Root locus plot of K(s+4)/s(s+2)


System is stable for all values of K.

Problem No 4

The open loop transfer function is given below. Sketch the root locus.

Solution:


2) There is one open loop zero at s = -0.2(m=1). There are three open loop poles at

s = 0, 0, -3.6(n=3). Three branches of root loci start from the three open loop poles when K= 0

and one branch goes to open loop zero at s = -0.2 when and other two go to

asymptotically when .

3) Root locus lies on negative real axis between -3.6 to -0.2 as the number of open loop poles plus open zeros to the right of any point on the real axis in this range is odd.




Angle of asymptote are


6) The root locus does branch out, which are given by dK/ds =0.

The root loci brakeout at the open loop poles at s = 0, when K =0 and breakin onto the real axis

at s=-0.432, when K=2.55 One branch goes to open loop zero at s=-0.2 and other goes breaksout

with the another locus starting from open loop ploe at s= -3.6. The break point is at s=-1.67 with

K=3.66. The loci go to infinity in the complex plane with constant real part s= -1.67.

7) The branches of the root locus at s=0, 0 break at K=0 and are tangential to imaginary axis or

depart at 90. The locus departs from open loop pole at s=-3.6 at 0.

8) The locus arrives at open loop zero at s=-0.2 at 180 .

9) The root locus does not cross the imaginary axis, hence there is no imaginary axis cross over.

The root locus plot is shown in Fig.3.17.



Fig 3.17 Root locus plot of K(s+0.2)/s2(s+3.6)


System is stable for all values of K. System is critically damped at K= 2.55, 3.66. It is under

damped for 2.55 > K > 0 and K >3.66. It is over damped for 3.66 > K >2.55.

Recommended Questions:

1. Give the general rules for constructing root locus.

2. Define Phase margin and Gain margin of root locus.

3. Sketch the root locus of a unity negative feedback system whose forward path transfer

function is G(s) =K/s.

4. The open loop transfer function is given below. Sketch the root locus plot.

5. The open loop transfer function is given below. Sketch the root locus.

6. The open loop transfer function is given below. Sketch the root locus.



7. The open loop transfer function is given below. Sketch the root Locus



MODULE 4

FREQUENCY DOMAIN ANALYSIS AND STABILITY

4.1 Bode plots

Bode plots are graphs of the steady state response of stable continuous time Linear time

invariant systems for sinusoidal inputs, plotted as change in magnitude and phase versus

frequency on logarithmic scale. Bode plots are a visual description of the system. So, these

Bode plots are used for the representation of sinusoidal transfer function . In this

representation the magnitude of G(jω) in db, i.e, 20 log G(jω) is plotted against 'log ω'.

Similarly phase angle of G(jω) is plotted against 'log ω' .Hence the abscissa is logarithm of the

frequency and hence the plots are known as logarithmic plots. The plots are named after the

famous mathematician H. W. Bode.

The transfer function G(jω) can be written as G(jω) = G(jω)∟Ф( ω) Where Ф( ω) is the angle

G(jω) .

Since G(jω) consists of many multiplicative factors in the numerator and denominator it is

convenient to take logarithm of G(jω) to convert these factors into additions and subtractions,

which can be carried out easily.

4.1.1 Magnitude Plot

To plot the magnitude plot , magnitude is plotted against input frequency on a logarithmic

scale.It can be approximated by two lines and it forms the asymptotic (approximate) magnitude

Bode plot of the transfer function:

(i) for angular frequencies below ωc it is a horizontal line at 0 dB since at low frequencies the

term ω / ωc is small and can be neglected, making the decibel gain equation above equal to

zero,



(ii) for angular frequencies above ωc it is a line with a slope of −20 dB per decade since at

high frequencies the term ω / ωc dominates and the decibel gain expression above simplifies to

-20 log (ω / ωc ) which is a straight line with a slope of −20 dB per decade.

These two lines meet at the corner frequency. From the Bode plot, it can be seen that for

frequencies well below the corner frequency, the circuit has an attenuation of 0 dB,

corresponding to a unity pass band gain, i.e. the amplitude of the filter output equals the

amplitude of the input. Frequencies above the corner frequency are attenuated – the higher the

frequency, the higher the attenuation.

4.1.2 Phase plot

The phase Bode plot is obtained by plotting the phase angle of the transfer function given by

Φ = - tan –

(ω/ωc ) Versus ω ,

Where, ω and ωc are the input and cutoff angular frequencies respectively.

For input frequencies much lower than corner, the ratio ω/ωc is small and therefore the phase

angle is close to zero. As the ratio increases the absolute value of the phase increases and

becomes –45 degrees when ω= ωc . As the ratio increases for input frequencies much greater

than the corner frequency, the phase angle asymptotically approaches −90 degrees. The

frequency scale for the phase plot is logarithmic.

The main advantage of using the Bode plots is that multiplication of magnitudes can be

converted into addition. Also a simple method of plotting an approximate log-magnitude curve is

obtained. It is based on asymptotic approximations. Such approximation by straight line

asymptotes is sufficient if only rough information on the frequency response characteristics is

needed. The phase angle curves can be drawn easily if a template for the phase angle curve of

1+jω is available. Expanding the low frequency range ,by use of a logarithmic scale for the

frequency is very advantageous ,since characteristics at low frequencies are most important in

practical systems. The only limitation is ,it is not possible to draw the curve ,right down to zero

frequency because of the logarithmic frequency (log 0 = ∞) .But this do not give any serious

problem.

4.2 Polar plot

This plot is drawn to study the frequency response in polar co-ordinates. The polar plot of a

sinusoidal transfer function G(jω) is a plot of the magnitude of G(jω) Vs the phase of G(jω) on

polar co-ordinates as ω is varied from 0 to ∞. Polar graph sheet has concentric circles and radial

lines. Concentric circles represent the magnitude. Radial lines represents the phase angles. In



polar sheet +ve phase angle is measured in ACW from 00

and -ve phase angle is measured in

CW from 00

.

To sketch the polar plot of G(jω) for the entire range of frequency ω, i.e., from 0 to infinity, there

are four key points that usually need to be known:

The start of plot where ω = 0,

The end of plot where ω = ∞,

where the plot crosses the real axis, i.e., Im(G(jω)) = 0, and

where the plot crosses the imaginary axis, i.e., Re(G(jω)) = 0.

Procedure: The following steps are involved in the plotting of Polar graphs

Express the given expression of OLTF in (1+sT) form.

Get the expressions for | G(jω)H(jω)| & G(jω)H(jω).

Tabulate various values of magnitude and phase angles for different values of ω ranging

from 0 to ∞.

Usually the choice of frequencies will be the corner frequency and around corner

frequencies. Choose proper scale for the magnitude circles.

Fix all the points in the polar graph sheet and join the points by a smooth curve.

Write the frequency corresponding to each of the point of the plot.

4.2.1 Gain Margin

Gain Margin is defined as “the factor by which the system gain can be increased to drive the

system to the verge of instability”.

For stable systems, ωgc < ωpc

│G(j)H(j)│ at ω= ωpc < 1

GM = in positive dB .

More positive the Gain Margin , more stable is the system

For marginally stable systems, ωgc = ωpc │G(j)H(j)│ at ω=ωpc = 1

GM = 0 dB

For Unstable systems, ωgc > ωpc

│G(j)H(j)│ at ω=ωpc > 1

GM = in negative dB



Gain is to be reduced to make the system stable

The important conclusions are :

If the gain is high, the GM is low and the system’s step response shows high overshoots

and long settling time.

On the contrary, very low gains give high GM and PM, but also causes higher ess(Error),

higher values of rise time and settling time and in general give sluggish response.

Thus we should keep the gain as high as possible to reduce ess and obtain acceptable

response speed and yet maintain adequate GM and PM.

As a thumb rule an adequate GM of 2 ( ie 6 dB ) and a PM of about 30 is generally

considered sufficient enough.

If the gain of the system is increased by a factor 1/β, then the G(j)H(j) at ω=ωpc becomes

β(1/β) = 1 and hence the │G(j)H(j)│ locus pass through -1+j0 point driving the system to

the verge of instability

4.2.2 PHASE MARGIN

Phase Margin is defined as “the additional phase lag that can be introduced before the system

becomes unstable”.

Let ‘A’ be the point of intersection of │G(j)H(j)│ plot and a unit circle centered at the

origin.

Draw a line connecting the points ‘O’ & ‘A’ and measure the phase angle between the

line OA and +ve real axis. This angle is the phase angle of the system at the gain cross

over frequency.ے G(jωgc)H(jgc) = Фgc

If an additional phase lag of PM is introduced at this frequency, then the phase angle

G(jωgc)H(jωgc) will become 180 and the point ‘A‘ coincides with (-1+j0) driving the

system to the verge of instability.

This additional phase lag is known as the Phase Margin.



For a stable system, the phase margin is positive.

A Phase margin close to zero corresponds to highly oscillatory system

A polar plot may be constructed from experimental data or from a system transfer

function

If values of ω are marked along the contour, a polar plot gives the same information as a

Bode plot

Usually, the shape of a polar plot is very important as it gives much information about the

system.

4.3 Relative and absolute stability A stable system is a dynamic system with a bounded response to a bounded input. A necessary

and sufficient condition for a feedback system to be stable is that all the poles of the system

transfer function have negative real parts

A system is considered marginally stable if only certain bounded inputs will result in a bounded

output. In practical systems, it is not sufficient to know that the system is stable but a stable

system must meet the specifications of relative stability which is a quantitative measure of how

fast the transients die out in the system.

Relative stability of a system is usually defined in terms of two design parameters-phase margin

and gain margin.The relative stability of a system can be defined as the property that is measured

by the relative real part of each root or pair of roots. The relative stability of a system can also be

defined in terms of the relative damping coefficients of each complex root pair and, therefore, in

terms of the speed of response and overshoot instead of settling time.



4.4 Time Domain Analysis Vs Frequency Domain Analysis

Variable frequency, sinusoidal signal generators are readily available and precision

measuring instruments are available for measurement of magnitude and phase angle. The

time response for a step input is more difficult to measure with accuracy.

It is easier to obtain the transfer function of a system by a simple frequency domain

test.Obtaining transfer function from the step response is more tedious.

If the system has large time constants, it makes more time to reach steady state at each

frequency of the sinusoidal input. Hence time domain method is preferred over frequency

domain method in such systems.

In order to do a frequency response test on a system, the system has to be isolated and the

sinusoidal signal has to be applied to the system. This may not be possible in systems

which can not be interrupted. In such cases, a step signal or an impulse signal may be

given to the system to find its transfer function. Hence for systems which cannot be

interrupted, time domain method is more suitable.

The design of a controller is easily done in the frequency domain method than in time

domain method. For a given set of performance measures in frequency domain , the

parameters of the open loop transfer function can be adjusted easily.

The effect of noise signals can be assessed easily in frequency domain rather than time

domain.

The most important advantage of frequency domain analysis is the ability to obtain the

relative stability of feedback control systems. The Routh Hurwitz criterion is essentially

a time do main method which determines the absolute stability of a system.

Since the time response and frequency response of a system are related through Fourier

transform , the time response can be easily obtained from the frequency response. The

correlation between time and frequency response can be easily established so that the

time domain performance measures can be obtained from the frequency domain

specifications and vice versa.



4.4 Frequency Response of a Control System :

To study the frequency response of a control system let us consider second order system with the

transfer function,

The steady state sinusoidal response is obtained by substituting s = j ω in the above equation

Normalizing the frequency ω, with respect to the natural frequency ωn by defining a variable

We have,

From the above equation the magnitude and angle of the frequency response is obtained as,

and

The time response for a unit sinusoidal input with frequency OJ is given by,

The magnitude and phase of steadystate sinusoidal response for variable frequency can be

plotted and are shown in the Fig. (a) and (b).



An important performance measure, in frequency domain, is the bandwidth of the system.

From Fig. , we observe that for δ < 0.707 and U > Ur the magnitude decreases monotonically.

The frequency Ub where the magnitude becomes 0.707 is known as the cut off frequency. At

this frequency, the magnitude will be



4.5 Nyquist Criteria

A stability test for time invariant linear systems can also be derived in the frequency domain. It is

known as Nyquist stability criterion. It is based on the complex analysis result known as

Cauchy’s principle of argument.

The Nyquist stability criterion relates the location of the roots of the characteristic equation to

the open-loop frequency response of the system. In this, the computation of closed-loop poles is

not necessary to determine the stability of the system and the stability study can be carried out

graphically from the open-loop frequency response. Therefore experimentally determined open-

loop frequency response can be used directly for the study of stability. When the feedback path is

closed. The Nyquist criterion provides the information on absolute and relative stability of the

system.



Let us suppose that the system transfer function is a complex function. By applying Cauchy’s

principle of argument to the open-loop system transfer function, we will get information about

stability of the closed-loop system transfer function and arrive at the Nyquist stability criterion

The importance of Nyquist stability lies in the fact that it can also be used to determine the

relative degree of system stability by producing the so-called phase and gain stability margins.

These stability margins are needed for frequency domain controller design techniques

The Nyquist plot is a polar plot of the function D(s) = 1 = G(s).H(s)

The Nyquist criterion states that the number of unstable closed-loop poles is equal to the

number of unstable open-loop poles plus the number of encirclements of the origin of the

Nyquist plot of the complex function D(s).

4.6 Control system Compensators

A control system is usually required to meet three time response specifications, namely, steady

state accuracy, damping factor and settling time. To get the desired design with minimum errors

and to adjust the parameters of the overall system to satisfy the design criterion an additional

subsystem called 'Compensator ‘ must be used.

This compensator may be used in series with the plant in the forward path or in the feedback

path shown in Fig. below. The compensation in the forward path is known as series or cascade

compensation and the later is known as feedback compensation. The compensator may be a

passive network or an active network.

Series compensation Feedback compensation

In general the series compensation is much simpler than the feed back compensation.But the

series compensation frequently require additional amplifiers to increase the gain and to provide

isolation.i.e to avoid power dissipation the series compensator is inserted in the lowest energy

point in the feed forward path.In general the number of components required in feed back

compensation will be less than the number of components in series compensation.



There are three important types of compensators.(i) Lead Compensator (ii) Lag compensator and

(iii) Lag-Lead compensator.

In a network ,when a sinusoidal input signal ei is applied at its input and if the steady state

output eo has a phase lead then the network is called a lead network. Similarly In the network for

a sinusoidal input ei ,if the steady state output has a phase lag ,then the network is called a lag

network.

Similarly in a network for a sinusoidal input .if the steady state out put has both phase lag and

lead ,but different frequency regions ,then the network is called lag-lead network. Generally the

phase lag occurs at low frequency regions and the phase lead occurs at higher frequency region.

A compensator having the characteristics of a lead network , lag network or lead-lag network is

called a lead compensator ,lag compensator or lag-lead compensator.

The lead compensators, lag compensators and lag-lead compensators are be designed either

using electronic components like operational amplifiers or using Root Locus methods. The first

type are called electronic lag compensators .The second type are Root locus compensators. The

Root locus compensators are have many advantages over electronic type.

Lead compensation basically speeds up the response of the control system and increase the

stability of the system. Lag compensation improves the steady –state accuracy of the system but

reduces the speed of the response. If improvements in both transient response and steady state

response is required ,both lead compensator and lag compensator are used simultaneously .In

general using a single lag-lead compensator is always economical.

Lag-lead compensation combines the advantage of lag and lead compensation. Since, the lag –

lead compensator possesses two poles and two zeros ,such a compensation increases the order of

the system by 2 ,unless the cancellation of the poles and zeros occurs in the compensated system.

4.6.1 Lead Compensator

It has a zero and a pole with zero closer to the origin. The general form of the transfer function of

the load compensator is,



Subsisting



Transfer function s

4.6.2 Lag Compensator

It has a zero and a pole with the zero situated on the left of the pole on the negative real axis. The

general form of the transfer function of the lag compensator is

Therefore, the frequency response of the above transfer function will be



Now comparing with

Therefore, the transfer function is



4.6.3 Lag-Lead Compensator

The lag-lead compensator is the combination of a lag compensator and a lead compensator. The

lag-section is provided with one real pole and one real zero, the pole being to the right of zero,

whereas the lead section has one real pole and one real zero with the zero being to the right of

the pole.

The transfer function of the lag-lead compensator will be

The figure below shows the lag lead compensator



The above transfer functions are comparing with

Then



Therefore, the transfer function is given by



MODULE 5

STATE VARIABLE ANALYSIS

Analysis and the design of feedback control systems by the classical design methods (root locus

and frequency domain methods) based on the transfer function approach are inadequate and not

convenient. So, the development of the state-variable approach, took place .This methos has

the following advantages over the classical approach.

1. It is a direct time-domain approach. Thus, this approach is suitable for digital computer

computations

2. It is a very powerful technique for the design and analysis of linear or nonlinear, time-variant

or time-invariant, and SISO or MIMO systems.

3. In this technique, the nth

order differential equations can be expressed as ‘n’ equations of first

order. Thus, making the solutions easier.

4. Using this approach, the system can be designed for optimal conditions with respect to given

performance indices.

5.1 Definitions:

State : The state of a dynamical system is a minimal set of variables x1(t), x2(t)x3(t) …xn(t)

such that the knowledge of these variables at t = t0 (initial condition), together with the

Knowledge of inputs u1(t), u2(t), u3(t)… um (t) for t ≥0, completely determines the behavior of

the system for t < t0.

State-Variables :

The variables x1(t),x2(t),x3(t)… xm(t) such that the knowledge of these variables at t = t0 (initial

condition), together with the knowledge of inputs u1(t), u2(t), u3(t)… um(t) for t ≥ t0, completely

determines the behavior of the system for t < t0 ; are called state-variables. In other words, the

variables that determine the state of a dynamical system, are called state-variables.

State-Vector :

If n state variables x1(t), x2(t), x3(t)… xn(t) are necessary to determine the behavior of a

dynamical system, then these n state-variables can be considered as n components of a vector

x(t), called state-vector.

State-Space : The n dimensional space, whose elements are the n state-variables, is called state-

space. Any state can be represented by a point in the state-space.



5.2 State equation of a linear time-invariant system

For a general system of the Figure shown below the state representation can be arranged in the

form of n first-order differential equations as

……(5.1)

Integrating equation 5.1 , we get

Where I = 1,2,3……..n

Thus, the n state-variables and, the state of the system can uniquely be determined at any t < tn,

provided each state-variable is known at t = tn and all the m control forces are known throughout

the interval t 0 to t.

The n differential equations of 1 may be written in vector form as

………… (5.2)



Equation 5. 2 is the state equation for time-invariant systems. However, for time-varying

systems, the function vector f(.) is dependent on time as well, and the vector equation may be

given as ……………….(5.3)

Equation (5.3) is the state equation for time-varying systems.

The output y(t) can, in general, be expressed in terms of the state vector x(t) and input vector u(t)

as ,

For time-invariant systems :

------------------------ -----(5.4)

For time-varying systems: ----------------(5.5)

5.3 STATE MODEL OF A LINEAR SINGLE-INPUT-SINGLE-OUTPUT

SYSTEM

The state model of a linear single-input-single-output system can be written as

(5.6)

(5.7)

Where



d = Transmission Constant , u(t) = Input or Control Variable (scalar)

and, y(t) = Output Variable (scalar).

A very important conclusion is that the derivatives of all the state-variables are zero at the

equilibrium point. Thus, the system continues to lie at the equilibrium point unless otherwise

disturbed.

5.6 SOLUTION OF STATE EQUATIONS FOR LINEAR TIME-

INVARIANT SYSTEMS

The state equation of a linear time-invariant system is given by

For a homogeneous (unforced) system

So , we have ………..(5.8)

Take the Laplace Transform on both sides

…………… (5.9)

The above may also be written as

Taking the inverse Laplace transform --------------(5.10)



This equation (3) gives the solution of the LTI homogeneous state Equation. It can be observed

that the initial state x(0) at t = 0, is driven to a state x(t) at time t. This transition in state is carried

out by the matrix exponential eAt. Because of this property, eA

t is termed as the State Transition

Matrix and is denoted by ø(t).

5.7 Determination of transfer function using SVA

Let us consider the following state space equation ,

Now, take the Laplace Transform (with zero initial conditions)

We want to solve for the ratio of Y(s) to U(s), which gives the transfer function.So, we need to

remove Q(s) from the output equation. We solve the state equation for Q(s)

The matrix Φ(s) is called the state transition matrix. Now , put this into the output equation

Now solving for transfer function ,we get

This is the method of determining the transfer function from state variable analysis (SVA).

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Control Systems 10ES43 - ATME College of Engineering, Mysuru