control system analysis, Exercises of Control Systems

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CONTROL SYSTEM ENGINEERING-I

Department of Electrical Engineering

VEER SURENDRA SAI UNIVERSITY OF TECHNOLOGY,

ODISHA, BURLA

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This document does not claim any originality and cannot be used as a substitute for

prescribed textbooks. The information presented here is merely a collection by the committee

members for their respective teaching assignments. Various sources as mentioned at the end

of the document as well as freely available material from internet were consulted for

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disclaim any implied warranties of merchantability or fitness for a particular purpose. The

committee members shall be liable for any loss of profit or any other commercial damages,

including but not limited to special, incidental, consequential, or other damages.

6.0 Root Locus Technique

6.1 Angle and Magnitude Criterion

6.2 Properties of Root Loci

6.3 Step by Step Procedure to Draw Root Locus Diagram

6.4 Closed Loop Transfer Function and Time Domain response

6.5 Determination of Damping ratio, Gain Margin and Phase Margin from Root Locus

6.6 Root Locus for System with transportation Lag.

6.7 Sensitivity of Roots of the Characteristic Equation.

7.0 Frequency Domain Analysis.

7.1 Correlation between Time and frequency response

7.2 Frequency Domain Specifications

7.3 Polar Plots and inverse Polar plots

7.4 Bode Diagrams

7.4.1 Principal factors of Transfer function

7.4.2 Procedure for manual plotting of Bode Diagram

7.4.3 Relative stability Analysis

7.4.4 Minimum Phase, Non-minimum phase and All pass systems

7.5 Log Magnitude vs Phase plots.

7.6 Nyquist Criterion

7.6.1 Mapping Contour and Principle of Argument

7.6.2 Nyquist path and Nyquist Plot

7.6.3 Nyquist stability criterion

7.6.4 Relative Stability: Gain Margin, and Phase Margin

7.7 Closed Loop Frequency Response

7.7.1 Gain Phase Plot

7.7.1.1 Constant Gain(M)-circles

7.7.1.2 Constant Phase (N) Circles

7.7.1.3 Nichols Chart

7.8 Sensitivity Analysis in Frequency Domain

• MODULE#

Design : The process of conceiving or inventing the forms, parts, and details of system to

achieve a specified purpose.

Simulation: A model of a system that is used to investigate the behavior of a system by

utilizing actual input signals.

Optimization: The adjustment of the parameters to achieve the most favorable or

advantageous design.

Feedback Signal : A measure of the output of the system used for feedback to control the

system.

Negative feedback : The output signal is feedback so that it subtracts from the input signal.

Block diagrams : Unidirectional, operational blocks that represent the transfer functions of

the elements of the system.

Signal Flow Graph (SFG): A diagram that consists of nodes connected by several directed

branches and that is a graphical representation of a set of linear relations.

Specifications: Statements that explicitly state what the device or product is to be and to do.

It is also defined as a set of prescribed performance criteria.

Open-loop control system: A system that utilizes a device to control the process without

using feedback. Thus the output has no effect upon the signal to the process.

Closed-loop feedback control system: A system that uses a measurement of the output and

compares it with the desired output.

Regulator : The control system where the desired values of the controlled outputs are more or

less fixed and the main problem is to reject disturbance effects.

Servo system: The control system where the outputs are mechanical quantities like

acceleration, velocity or position.

Stability: It is a notion that describes whether the system will be able to follow the input

command. In a non-rigorous sense, a system is said to be unstable if its output is out of

control or increases without bound.

Multivariable Control System : A system with more than one input variable or more than

one output variable.

Trade-off: The result of making a judgment about how much compromise must be made

between conflicting criteria.

1.2. Classification

1.2.1. Natural control system and Man-made control system:

Natural control system: It is a control system that is created by nature, i.e. solar

system, digestive system of any animal, etc.

Man-made control system: It is a control system that is created by humans, i.e.

automobile, power plants etc.

1.2.2. Automatic control system and Combinational control system:

Automatic control system: It is a control system that is made by using basic theories

from mathematics and engineering. This system mainly has sensors, actuators and

responders.

Combinational control system: It is a control system that is a combination of natural

and man-made control systems, i.e. driving a car etc.

1.2.3. Time-variant control system and Time-invariant control system:

Time-variant control system: It is a control system where any one or more

parameters of the control system vary with time i.e. driving a vehicle.

Time-invariant control system: It is a control system where none of its parameters

vary with time i.e. control system made up of inductors, capacitors and resistors only.

1.2.4. Linear control system and Non-linear control system:

Linear control system: It is a control system that satisfies properties of homogeneity

and additive.

 Homogeneous property: f^  x^ ^ y ^ ^ f^  x^   f^  y 

 Additive property: f   x   f  x 

Non-linear control system: It is a control system that does not satisfy properties of

homogeneity and additive, i.e.  

3

f x  x

1.2.5. Continuous-Time control system and Discrete-Time control system:

Continuous-Time control system: It is a control system where performances of all

of its parameters are function of time, i.e. armature type speed control of motor.

Discrete - Time control system: It is a control system where performances of all of

its parameters are function of discrete time i.e. microprocessor type speed control of

motor.

1.2.6. Deterministic control system and Stochastic control system:

Deterministic control system: It is a control system where its output is predictable

or repetitive for certain input signal or disturbance signal.

Stochastic control system: It is a control system where its output is unpredictable or

non-repetitive for certain input signal or disturbance signal.

1.2.7. Lumped-parameter control system and Distributed-parameter control system:

Lumped-parameter control system: It is a control system where its mathematical

model is represented by ordinary differential equations.

Distributed-parameter control system: It is a control system where its mathematical

model is represented by an electrical network that is a combination of resistors,

inductors and capacitors.

1.2.8. Single-input-single-output (SISO) control system and Multi-input-multi-output

(MIMO) control system:

SISO control system: It is a control system that has only one input and one output.

MIMO control system: It is a control system that has only more than one input and

more than one output.

1.2.9. Open-loop control system and Closed-loop control system:

Open-loop control system: It is a control system where its control action only

depends on input signal and does not depend on its output response.

 Expensive than that of open-loop control system

 Complicate for maintenance

 Less stable operation than that of open-loop control system

1.3.3. Comparison between Open-loop and Closed-loop control systems:

It is a control system where its control action depends on both of its input signal and

output response.

Sl.

No.

Open-loop control systems Closed-loop control systems

1 No feedback is given to the control system A feedback is given to the control system

2 Cannot be intelligent Intelligent controlling action

3

There is no possibility of undesirable

system oscillation(hunting)

Closed loop control introduces the

possibility of undesirable system

oscillation(hunting)

4

The output will not very for a constant

input, provided the system parameters

remain unaltered

In the system the output may vary for a

constant input, depending upon the

feedback

5

System output variation due to variation in

parameters of the system is greater and the

output very in an uncontrolled way

System output variation due to variation in

parameters of the system is less.

6 Error detection is not present Error detection is present

7 Small bandwidth Large bandwidth

8 More stable Less stable or prone to instability

9 Affected by non-linearities Not affected by non-linearities

10 Very sensitive in nature Less sensitive to disturbances

11 Simple design Complex design

12 Cheap Costly

1.4. Servomechanism

It is the feedback unit used in a control system. In this system, the control variable is

a mechanical signal such as position, velocity or acceleration. Here, the output signal

is directly fed to the comparator as the feedback signal, b(t) of the closed-loop control

system. This type of system is used where both the command and output signals are

mechanical in nature. A position control system as shown in Fig.1.3 is a simple

example of this type mechanism. The block diagram of the servomechanism of an

automatic steering system is shown in Fig.1.4.

Fig.1.3. Schematic diagram of a servomechanism

Fig.1.4. Block diagram of a servomechanism

Examples:

 Missile launcher

 Machine tool position control

 Power steering for an automobile

 Roll stabilization in ships, etc.

1.5. Regulators

It is also a feedback unit used in a control system like servomechanism. But, the

output is kept constant at its desired value. The schematic diagram of a regulating

CHAPTER# 2

2. Control System Dynamics

2.1. Definition: It is the study of characteristics behaviour of dynamic system, i.e.

(a) Differential equation

i. First-order systems

ii. Second-order systems

(b) System transfer function: Laplace transform

2.2. Laplace Transform: Laplace transforms convert differential equations into algebraic

equations. They are related to frequency response.

 ^  ^ ^

0

st

x t X s x t e dt





  

L (2. 1 )

    

0

st

x t X s x t e dt





  

L (2. 2 )

No. Function

Time-domain

x(t)=

ℒ

• 1 {X(s)}

Laplace domain

X(s)= ℒ{x(t)}

1 Delay δ(t-τ) e

• τs

2 Unit impulse δ(t) 1

3 Unit step u(t)

s

1

4 Ramp t 2

1

s

5

Exponential

decay

e

• αt

s  

6

Exponential

approach

 

t

e

 

1 

(  )

ss 

7 Sine sin ωt

2 2

s 

8 Cosine cos ωt

2 2

s  

s

9 Hyperbolic

sine

sinh αt

2 2

s 

10 Hyperbolic

cosine

cosh αt

2 2

s  

s

11 Exponentiall

y decaying

sine wave

e t

t





sin



2 2 (  ) 



s  

12 Exponentiall

y decaying

cosine wave

e t

t 

 cos



2 2

( )

s

s

2.3. Solution of system dynamics in Laplace form: Laplace transforms can be solved using

partial fraction method.

A system is usually represented by following dynamic equation.

A s

N s

B s

The factor of denominator, B(s) is represented by following forms,

i. Unrepeated factors

ii. Repeated factors

iii. Unrepeated complex factors

(i) Unrepeated factors

N s A B

s a s b s a s b

A s b B s a

s a s b

By equating both sides, determine A and B.

Example 2.1:

Expand the following equation of Laplacetransform in terms of its partial fractionsand obtain

its time-domain response.

2

( )

( 1)( 2)

s

Y s

s s



 

Solution:

The following equation in Laplacetransform is expandedwith its partial fractions as follows.

s A B

s s s s

s A s B s

s s s s

By equating both sides, A and B are determined as A  2, B  4. Therefore,

2 4

( )

( 1) ( 2)

Y s

s s

  

 

Taking Laplace inverse of above equation,

2

( ) 2 4

t t

y t e e

 

  

(ii) Unrepeated factors

2 2 2

N s A B A B s a

s a s a s a s a

By equating both sides, determine A and B.

Example 2.2:

Expand the following equation of Laplacetransform in terms of its partial fractionsand obtain

its time-domain response.

2

s

Y s

s s

Solution:

The following equation in Laplacetransform is expandedwith its partial fractions as follows.

2 2

2

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

s A B C

s s s s s

  

    

By equating both sides, A and B are determined as A  2, B  4. Therefore,

2

2 4 4

( )

( 1) ( 1) ( 2)

Y s

s s s

   

  

Taking Laplace inverse of above equation,

2

( ) 2 4 4

t t t

y t te e e

  

   

2

( ) 2 4 4

t t t

y t te e e

  

   

Applying final value theorem,

lim

s

s s

s j s j



CHAPTER# 3

3. Transfer Function

3.1. Definition: It is the ratio of Laplace transform of output signal to Laplace transform of input

signal assuming all the initial conditions to be zero, i.e.

Let, there is a given system with input r(t) and output c(t) as shown in Fig.3.1 (a), then its

Laplace domain is shown in Fig.3.1 (b). Here, input and output are R(s) and C(s) respectively.

(a) (b)

(c)

Fig.3.1. (a) A system in time domain, (b) a system in frequency domainand (c) transfer function with differential

operator

G(s) is the transfer function of the system. It can be mathematically represented as follows.

 

 

  zero initial condition

C s

G s

R s

 Equation Section (Next) (3. 1 )

Example 3.1: Determine the transfer function of the system shown inFig.3.2.

Fig.3.2. a system in time domain

Solution:

Fig.3.1 is redrawn in frequency domain as shown in Fig.3.2.

Fig.3.2. a system in frequency domain

s=- 1 - j

s=-1+j

Fig.3.3. pole-zero map

3.3. Properties of Transfer function:

 Zero initial condition

 It is same as Laplace transform of its impulse response

 Replacing ‘s’ by

d

dt

in the transfer function, the differential equation can be obtained

 Poles and zeros can be obtained from the transfer function

 Stability can be known

 Can be applicable to linear system only

3.4. Advantages of Transfer function:

 It is a mathematical model and gain of the system

 Replacing ‘s’ by

d

dt

in the transfer function, the differential equation can be obtained

 Poles and zeros can be obtained from the transfer function

 Stability can be known

 Impulse response can be found

3.5. Disadvantages of Transfer function:

 Applicable only to linear system

 Not applicable if initial condition cannot be neglected

 It gives no information about the actual structure of a physical system

CHAPTER# 4

4. Description of physical system

4.1. Components of a mechanical system: Mechanical systems are of two types, i.e. (i)

translational mechanical system and (ii) rotational mechanical system.

4.1.1. Translational mechanical system

There are three basic elements in a translational mechanical system, i.e. (a) mass, (b)

spring and (c) damper.

(a) Mass: A mass is denoted by M. If a force f is applied on it and it displays

distance x , then

2

2

d x

f M

dt

 as shown in Fig. 4. 1.

Fig. 4. 1. Force applied on a mass with displacement in one direction

If a force f is applied on a massM and it displays distance x 1 in the direction of f and

distance x 2

in the opposite direction, then

2 2

1 2

2 2

d x d x

f M

dt dt

as shown in Fig.4.2.

M

X 1

f

X 2

Fig.4. 2. Force applied on a mass with displacement two directions

(b) Spring: A spring is denoted by K. If a force f is applied on it and it displays

distance x , then f  Kx as shown in Fig.4.3.

Fig.4.3. Force applied on a spring with displacement in one direction

If a force f is applied on a springK and it displays distance x 1 in the direction of f and

distance x 2

in the opposite direction, then (^)   1 2

f  K x  x as shown in Fig.4.4.