control systems Transfer function

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Sri Lanka Institute of Information Technology

Faculty of Engineering

Department of Electrical and Computer

Engineering

EC3501: Control Systems Laboratory

Lab 01: Transfer Functions

Name/ Index Number : De Silva O.R.M. (EN14536458)

Name/ Index Number : Kaushalya S. A. D. T. P. (EN14535468)

Group : Group 03

Submission Date : 07th March 2016


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Table of ContentsList of Figures ................................................................................................................................. 2

List of Tables .................................................................................................................................. 3

List of Equations ............................................................................................................................. 3

Introduction ..................................................................................................................................... 4

Task 1 .............................................................................................................................................. 4

Exercise 1 ........................................................................................................................................ 5

1. Transfer Function of G1....................................................................................................... 5

2. Transfer Function of G2.......................................................................................................... 6

3. Transfer Function of G3.......................................................................................................... 7

Exercise 2 ........................................................................................................................................ 8

1. Step Response of Function of G1 ........................................................................................... 8

2. Step Response of Function of G2 ......................................................................................... 10

3. Step Response of Function of G3 ......................................................................................... 12

Exercise 3 ...................................................................................................................................... 14

Exercise 4 ...................................................................................................................................... 18


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List of FiguresFigure 1 - Pole-Zero Map for Transfer Function G1 ...................................................................... 5

Figure 2 - Pole-Zero Map for Transfer Function G2 ...................................................................... 6

Figure 3 - Pole-Zero Map for Transfer Function G3 ...................................................................... 7

Figure 4 - Step Response for Transfer Function G1 ....................................................................... 8

Figure 5- Step Response for Transfer Function G2 ...................................................................... 10

Figure 6- Step Response for Transfer Function G3 ...................................................................... 12

Figure 7 - Simulink Model to Obtain Step Response ................................................................... 14

Figure 8 - Under-Damped Response ............................................................................................. 14

Figure 9 - Simulink Model to Obtain Step Response ................................................................... 15

Figure 10 - Critically-Damped Response ..................................................................................... 15

Figure 11 - Simulink Model to Obtain Step Response ................................................................. 16

Figure 12 - Undamped Response .................................................................................................. 16

Figure 13 - Varying Damping Ratio for a Second Order System ................................................. 17

Figure 14 - Step Response for G1(S) ............................................................................................ 19



Figure 17 - Impulse Response for G3(S) ...................................................................................... 23

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List of TablesTable 1 - Result Comparison Table for G1 Step Response .......................................................... 10

Table 2 - Result Comparison Table for G2 Step Response .......................................................... 12

Table 3 -Result Comparison Table for G3 Step Response ........................................................... 13

List of EquationsEquation 1 - General Second Order System Equation .................................................................... 8

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IntroductionControl systems play an important role in most of electro mechanical systems. To control the

system precisely, it’s important to design the control system accurately. This lab introduces the

analysis of control systems and determine their stability for step inputs. The Laplace transform is

frequently used in analyzing the systems since it’s convenient to simplify the equations. The

system equations are modelled and simulated using MATLAB to analyze.

Task 1We can construct a transfer function in Matlab window by assigning the numerator or

denominator coefficient vector s or directly by the following ‘tf ’ function as below


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Exercise 1

1. Transfer Function of G1

Figure 1 - Pole-Zero Map for Transfer Function G1

Since the real part of poles of the function is positive, the system is unstable. The imaginary partof the poles accounts for the oscillation in the time domain function. According to Pole-Zero

Map the poles are situated in the right side of the plane making the system unstable.


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The poles are situated in the imaginary axis. Therefore, the system is marginally stable andsystem will oscillate with a constant amplitude determined by the initial conditions


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.


The poles have positive real part corresponds to an exponentially increasing component C −σt

in the homogeneous response; thus defining the system to be unstable.


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Exercise 2General form of a second order system:

1. Step Response of Function of G1

Figure 4 - Step Response for Transfer Function G1

=

+ + =

.

+ 2 +

Equation 1 - General Second Order System Equation


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Calculation

= 21

2 + 16 + 21

=

21

2 + 8 + 212

= 21/2

= 21/2

= 2

= 8

2 21/2

= 8

√ 42 = 1.2344

= 1 0.4167 + 2.917

= 1 0.41671.2344 + 2.9171.2344

21/2 = 1.5216

= 4

= 4

1.2344 ∗ √ 10.5 = 1

ℎ = −

− × 100

ℎ = − ×.

. × 100 → :

= lim→∞

= lim→

= lim→

21

2 + 16 + 21 = 1

Below table compares the calculated result vs the simulated result.


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Calculation

= 2

3 + 4 + 6

=

2

3 + 43 + 2

= 0.333 0.378−. sin1.247 + 1.080

= 2

= √ 2

= 2

=43

2√ 2

= 23√ 2

= 0.4714

= 1 0.4167 + 2.917

= 1 0.41670.4714 + 2.9170.4714

√ 2

= 1.0265

= 4

= 4

0.4714 ∗ √ 2 = 6.00005

ℎ = −

− × 100

ℎ = − ×.

. × 100 = 18.651%

=

1

=

2 × 1 0.4714 = 2.5188

= () = 0.333 0.378−.×. sin1.247 × 2.5188 + 1.080 = 0.3278


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Table 2 - Result Comparison Table for G2 Step Response

Parameter Simulated Result Calculated Result

Rise Time 1. 1222s 1.0265s

Settling Time 5.8461s 6.00005sOvershoot 18.6513% 18.651%

Peak 0.3955 0.3278

Peak Time 2.5180s 2.5188s

3. Step Response of Function of G3

Figure 6- Step Response for Transfer Function G3


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Calculation

= √ 6

+ 8 + √ 6

= 2.4495

+ 8 + 2.4495

= 0.9994 1.0337−. + 0.0441−.

= √ 6

= √ 6

= 2

=

8

2 √ 6

= 4

1.565 = 2.5557

ℎ = −

− × 100

ℎ = − ×.

. × 100 → :

= lim→∞

= lim→

= lim→

√ 6

+ 0.319 × + 7.681 = 0.9996


Table 3 -Result Comparison Table for G3 Step Response

Parameter Simulated Result Calculated Result

Rise Time 6.8946s -

Settling Time 12.4010s -

Overshoot 0% -

Peak 0.9988 0.9996

Peak Time 21.2892s -


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Exercise 3

To obtain the step responses of the following transfer functions, following models were

created using Simulink.

From the scope block step response can be obtained as follows.

1.

Figure 8 - Under-Damped Response

Figure 7 - Simulink Model to Obtain Step Response


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2.

Figure 10 - Critically-Damped Response

Figure 9 - Simulink Model to Obtain Step Response


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Effect of Varying Damping Ratio on a Second Order System

Figure 13 - Varying Damping Ratio for a Second Order System


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Exercise 4

Transient response of a system is response of a system to a change from equilibrium. A system issaid to be in a transient state when a process variable has been changed and the system has not

yet reached steady-state. Steady state response is the response that can be achieved in the

equilibrium state of the control system.

Transient means changing. Transient state is an interval of time in which our system is either

"warming up" or taking its time to respond to a disturbance. Steady is the opposite of transient.

Steady state is a condition where our system continues with an easily predictable behavior and

few values of it are changing if any. Transient state response is a description of how the system

functions during transient state. Steady state response is a description of how the system

functions during steady state.

2.

T1(s)-The poles in the transfer function is in left side of the pole zero plot.Therefore the system is

stable .

Poles are as follows:

-2+0.5i , -2-0.5i

Transfer function for poles

=

1

+ 4 + 4.25


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Figure 14 - Step Response for G1(S)

This system is much more stable .There is no overshoot in the system.


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T2(s): The poles are in the left side of the pole zero plot. Therefore, the system is stable.

Poles are:

(-1+1i), (-1-1i)

The Transfer function :

= 1

+ 2 + 2



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In this system the step response slightly overshoot therefore the system is much more stable.

T3(s): The poles are in the imaginary axis of the pole zero plot. Therefore, the system is

marginally stable.

Poles:(2i, -2i)

The transfer function:

= 1

+ 4


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This marginally stable system is oscillating until 250s.



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T4(s):One pole has a positive value in the real axis .Therefore the system is unstable.

Poles:(0.5,-0.5)

Figure 17 - Impulse Response for G3(S)


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The system is oscillating and the system is unstable.

control systems Transfer function

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