DIGITAL SIMULATIONS of Linear Systems

Expt. No:Date :

DIGITAL SIMULATIONS Of Linear Systems

Aim

To digitally simulate the time response characteristics of a linear system without non linearities and to verify it manually.

APPARATUS REQUIRED

Mat lab Software

THEORY

The time characteristics of control systems are specified in terms of time domain specifications. Systems with energy storage elements cannot respond instantaneously and will exhibit transient responses, whenever they are subjected to inputs or disturbances.

The desired performance characteristics of a system of any order may be specified in terms of transient response to a unit step input signal. The transient response characteristics of a control system to a unit step input is specified in terms of the following time domain specifications

Delay time td

Rise time tr

Peak time tp

Maximum overshoot Mp

Settling time ts

PROCEDURE:

1. In MATLAB software open a new model in simulink library browser.

2. From the continuous block in the library drag the transfer function block.

3. From the source block in the library drag the step input.

4. From the sink block in the library drag the scope.

5. From the math operations block in the library drag the summing point.

6. Connect all to form a system and give unity feedback to the system.

7. For changing the parameters of the blocks connected double click the respective

block.

8. Start simulation and observe the results in scope.

9. From the step response obtained note down the rise time, peak time, peak overshoot and settling time.

10. For the same transfer function write a matlab program to obtain the step response and verify both the results.

PROGRAM

%This is a MATLAB program to find the step response

num=[0 0 25];

den=[1 6 25];

sys = tf (num,den);

step (sys);

PLOT

RESULT

Department of Electrical Engineering
Assessment of Practicals
Max Marks / Marks Obtained
Performance : / 25
Record : / 15
Viva-voce : / 10
Total / 50
Signature

Expt. No: Date:

a)STABILITY ANALYSIS OF LINEAR SYSTEMS (Bode Plot)

AIM

To obtain the bode plot for the given system whose transfer function is given as

G(S)=242(s+5)

s(s+1)(s2+5s+121)

and to find out whether the system is stable or not.

APPARATUS REQUIRED

Mat lab Software

THEORY

A Linear Time-Invariant Systems is stable if the following two notions of system stability are satisfied

When the system is excited by Bounded input, the output is also a Bounded output.
In the absence of the input, the output tends towards zero, irrespective of the initial conditions.

The following observations are general considerations regarding system stability and are

If all the roots of the characteristic equation have negative real parts, then the impulse response is bounded and eventually decreases to zero, then system is stable.

If any root of the characteristic equation has a positive real part, then system is unstable.
If the characteristic equation has repeated roots on the jω-axis, then system is unstable.
If one are more non-repeated roots of the characteristic equation on the jω-axis, then system is unstable.

BODE PLOT :

Consider a Single-Input Single-Output system with transfer function

C(s) b0 sm + b1 sm-1 + ……+ bm

R(s) a0 sn + a1sn-1 + ……+an

Where m < n.

Rule 1 A system is stable if the phase lag is less than 180˚ at the frequency for which the gain is unity (one).

Rule 2 A system is stable if the gain is less than one (unity) at the frequency for which the phase lag is 180˚.

The application of these rules to an actual process requires evaluation of the gain and phase shift of the system for all frequencies to see if rules 1 and 2 are satisfied. This is obtained by plotting the gain and phase versus frequency. This plot is called BODE PLOT. The gain obtained here is open loop gain.

The stability criteria given above represent Limits of Stability. It is well to design a system with a margin of safety from such limits to allow for variation in components and other unknown

PROGRAM

%BODE PLOT OF THE SYSTEM%

%Enter the numerator and denominator of the transfer function

num=[0 0 0 242 1210];

den=[1 6 126 121 0];

sys=tf(num,den)

%Specify the frequency range and enter the command

w=logspace(-2,4,1000);

bode(sys,w)

xlabel('Frequency')

ylabel( ' Phase angle in degrees Magnitude of G(s)')

title('Bode Plot of the system 242(s+5)/s(s+1)(s^2+5*s+121)')

%To determine the Gain Margin,Phase Margin, Gain crossover frequency and

%Phase cross over frequency

[ Gm, Pm, Wcp, Wcg ]= margin (sys)

factors. This consideration leads to the revised stability criteria, or more properly, a Margin of Safety provided to each condition. The exact terminology is in terms of a Gain Margin and Phase Margin from the limiting values quoted.

If the phase lag is less than 140˚ at the unity gain frequency, the system is stable. This then, is a 40˚ Phase Margin from the limiting values of 180˚.
If the gain is 5dB below unity (or a gain of about 0.56) when the phase lag is 180˚, the system is stable. This is 5dB Gain Margin.

ALGORITHM

Write a Program to (or using SIMULINK) obtain the Bode plot for the given system.
Access the stability of given system using the plots obtained.

PROCEDURE TO OBTAIN BODE PLOT

1. Rewrite the sinusoidal transfer function in the time constant form by replacing s by jω

2. Identify the corner frequencies associated with each factor of the transfer function.

3. Knowing the corner frequencies draw the asymptotic magnitude plot. This plot consists of straight line segments with line slope changing at each corner frequency by +20db/decade for a zero and -20db/decade for a pole. For a complex conjugate zero or pole the slope changes by + 40db/decade.

4. Draw a smooth curve through the corrected points such that it is asymptotic to the line segments. This gives the actual log-magnitude plot.

5. Draw phase angle curve for each factor and add them algebraically to get the phase plot.

manual calculations

i)The sinusoidal transfer function G (jω) is obtained by replacing s by jω in the given s domain transfer function

G(jω)=242(jω +5)

jω (jω +1)( jω 2+5 jω +121)

On comparing the quadratic factor of G(s) with standard form of quadratic

factor , ζ and ωn can be evaluated.

s2+5s+121 = s2+2ζωns + ωn2

On comparison

ωn2 = 1212ζωn= 5

ωn =11 rad/secζ = 0.227

G(jω)= 10(1+0.2jω)

jω (1+jω)( 1+0.4 jω -0.0083ω2)

ii)CORNER FREQUENCIES

The corner frequencies are ωc1=1rad/sec ωc2= 5 rad/sec and ωc3=11rad/sec

Choose a low frequency ωl such that ωl< ωc1 and choose a high frequency ωh> ωc3.

Let ωl=0.5 rad/sec and ωh=100 rad/sec

iii)MAGNITUDE PLOTS

Calculate A at ωl, ωc1, ωc2, ωc3, and ωh

Let A= | G(jω)| in db

At ω= ωl A= 20 log(10/0.5)=26.03db

ω= ωc1 , A=20log(10/1)=20db

ω= ωc2 A= -40log(5/1)+20=-7.96 db

ω= ωc3 A = -20log(11/5) - 7.96 = -14.80 db

ω= ωh A = -60log(100/11)-14.80 = - 72.3 db

These values are plotted in the semilog graph sheet taking frequency along the logarithmic scale and magnitude in db along the linear scale

iv)PHASE PLOT

The phase angle of G(jω) as a function of ω is given by

Φ = ‹G(jω) = tan-1 0.2ω -90 – tan-1 ω – tan-1 0.04ω/(1 – 0.0083ω2)

Ω / tan-1 0.2ω / tan-1 ω / tan-1 {0.04ω/
(1 – 0.0083ω2)} / Φ = ‹G(jω)
0.5 / 507 / 26.56 / 1.15 / -112
1 / 11.3 / 45 / 2.31 / -126.01
5 / 45 / 78.96 / 14.04 / -138
10 / 63.43 / 84.29 / 63.44 / -174.3
11 / 65.5 / 84.8 / 85.8 / -195.4
20 / 75.96 / 87.14 / 180-19.98=160 / -261.18
50 / 84.3 / 88.85 / 180-6=174 / -268.55
100 / 87014 / 89.43 / 180-2.9=177.1 / -269.3

These values are plotted in the semilog graph sheet taking the same frequency as before along the logarithmic scale and phase angle in degrees along the linear scale.

OUTPUT (from simulation)

242 s + 1210

------

s^4 + 6 s^3 + 126 s^2 + 121 s

Gm = 2.0273

Pm = 41.8270

Wcp = 10.0961

Wcg = 3.6322

OUTPUT (from graph)

ωgc= Φgc=3.1rad/sec

Phase margin γ=180+ Φgc = 180-134 = 46 degrees

Gain Margin = 12 db

ωpc = 10.1 rad/sec

BODE PLOT

VIVA QUESTIONS:

Define stability of Linear Time Invariant System.
Give the stability conditions of system using Pole-Zero plot.
Define Bode Plot.
What is the use of Bode Plot?
What are the conditions of stability in bode plot?
Define Stability criteria.
Define Limits of stability.
Define safe regions in stability criteria.
Define Phase margin and Gain margin.

REFERENCE

NAGRATH & GOPAL,” Control Systems”.
MATLAB User Manual .
Control Systems by Nagoor gani
Control system Engineering by S .P .Eugene Xavier and J.Joseph Cyril babu.
Control system Engineering by R.Ananda Natarajan and P.Ramesh babu.

RESULT:

Department of Electrical Engineering
Assessment of Practicals
Max Marks / Marks Obtained
Performance : / 25
Record : / 15
Viva-voce : / 10
Total / 50
Signature

Expt. No:Date:

b)STABILITY ANALYSIS OF LINEAR SYSTEMS

(Root Locus Plot)

AIM

To obtain the Root locus plot for the given system whose transfer function is given as

G(S)=K

s(s+3)(s2+3s+11.25)

APPARATUS REQUIRED

Mat lab Software

THEORY

ROOT LOCUS PLOT :

The characteristic of the transient response of a closed-loop system is related to the location of the closed loop poles. If the system has a variable loop gain, then the location of the closed-loop poles depend on the value of the loop gain chosen. A simple technique known as “Root Locus Technique” used for studying linear control systems in the investigation of the trajectories of the roots of the characteristic equation.

This technique provides a graphical method of plotting the locus of the roots in the s-plane as a given system parameter is varied over the complete range of values(may be from zero to infinity). 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 form the locus. The root locus is a powerful technique as it brings into focus the complete dynamic response of the system . The root locus also provides a measure of sensitivity of roots to the variation in the parameter being considered. This technique is applicable to both single as well as multiple-loop systems.

PROCEDURE:

Write a Program to (or using SIMULINK) obtain the Root locus plot for the given system.
Access the stability of given system using the plots obtained.

PROGRAM

%ROOT LOCUS OF THE SYSTEM%

num=[0 0 0 0 1]

den=[1 6 20.25 33.75 0]

sys=tf(num,den)

rlocus(sys)

v=[-10,10,-8,8];

axis(v)

xlabel('Real Axis')

ylabel('Imaginary Axis')

title('Root Locus of the sytem ')

title('Root Locus Plot of the system K/s(s+3)(s^2+3s+11.25))')

MANUAL CALCULATIONS

Number of poles =4, zeros = 0, number of root locus branches =4. Starting points s=0, -3 & 1.5+ j3.
Pole – zero plot is as follows

Section between 0 and -3 is part of root locus. One breakway point is between s=0 and s=-3.

Angle of asymptotes are 45,135,225 and 315 degrees
Centroid = -1.5
Three Breakway points are -1.5,-1.5 + j 1.8371
Intersection with imaginary axis s= + j2.37.
Angle of departure -90, +90.
Root locus is plotted.
Stability for 0< K<82.26 system is stable.

K=82.26 system is marginally stable.

K>82.26 system is unstable

OUTPUT

num =

0 0 0 0 1

den =

1.0000 6.0000 20.2500 33.7500 0

Transfer function:

------

s^4 + 6 s^3 + 20.25 s^2 + 33.75 s

GRAPH(from Simulation)

VIVA QUESTIONS

Define root locus technique.
What are the conditions of stability in root locus criteria?
What is the advantage of root locus technique?
Which method of stability analysis is more advantageous?
How the stability of unstable is improved?
What are the methods to improve the stability?
What is the use of compensators?
What do you mean by Root-Loci?
What is complementary Root Loci?
What are contours?
State the basic properties of Root Locus.
How would you find the number of branches of Root Loci?
How are the break away points of the root locus determined?
How is the point of intersection of the asymptotes with real axis found out.

REFERENCE

NAGRATH & GOPAL,” Control Systems”.
MATLAB User Manual .
Control Systems by Nagoor gani
Control system Engineering by S .P .Eugene Xavier and J.Joseph Cyril babu.
Control system Engineering by R.Ananda Natarajan and P.Ramesh babu.

RESULT:

Department of Electrical Engineering
Assessment of Practicals
Max Marks / Marks Obtained
Performance : / 25
Record : / 15
Viva-voce : / 10
Total / 50
Signature

Expt. No:Date:

c) STABILITY ANALYSIS OF LINEAR SYSTEMS

(NyquistPlot)

AIM

To obtain the Nyquist plot for the given system whose transfer function is given as

G(S)=50

(s+4)(s2+3s+3)

and to find out whether the system is stable or not.

APPARATUS REQUIRED

Mat lab Software

THEORY

POLAR PLOTS OR NYQUIST PLOTS:

The sinusoidal transfer function G(jω) is a complex function is given by

G(jω) = Re[ G(jω)] + j Im[G(jω)] or

G(jω) = │G(jω) │ ∟G(jω) = M ∟Φ ------(1)

From equation (1), it is seen that G(jω) may be represented as a phasor of magnitude M and phase angle Φ. As the input frequency varies from 0 to ∞, the magnitude M and phase angle Φ changes and hence the tip of the phasor G(jω) traces a locus in the complex plane. The locus thus obtained is known as

POLAR PLOT.

The major advantage of the polar plot lies in stability study of systems. Nyquist related the stability of a system to the form of these plots. Polar plots are referred as NYQUIST PLOTS.

NYQUIST stability criterion of determining the stability of a closed loop system by investigating the properties of the frequency domain plot of the loop transfer function G(s) H(s).

Nyquist stability criterion provides the information on the absolute stability of a control system as similar to Routh- Hurwitz criterion. Not only giving the absolute stability, but indicates “Degree of Stability” i.e “Relative Stability” of a stable system and the degree of instability of an unstable system and indicates how the system stability can be improved. The Nyquist stability citerion is based on a Cauchy’s Residue Theorem of complex variables which is referred to as the “principle of argument”.

Let Q(s) be a single –valued function that has a finite number of poles in the s-plane. Suppose that an arbitrary closed path Гq is chosen in the s-plane so that the path does not go through any one of the poles or zeros of Q(s); the corresponding Гq locus mapped in the Q(s) plane will encircle the origin as many times as the difference between the number of the zeros and the number of poles of Q(s) that are encircled by the s-plane locus Гq.

The principle of argument is given by

N= Z - P

Where N – number of encirclemnts of the origin made by the Q(s) –plane locus Гq.

Z – number of zeros of Q(s) encircled by the s-plane locus Гq in the s-plane.

P - number of poles of Q(s) encircled by the s-plane locus Гq in the s-plane.

ALGORITHM

Write a Program to (or using SIMULINK) obtain the Nyquist plot for the given system.
Access the stability of given system using the plots obtained.

PROGRAM

%NYQUIST PLOT

%Enter the numerator and denominator of the transfer function

num=[0 0 0 50]

den=[1 7 12 12]

sys=tf(num,den)

%Specify the frequency range and enter the command

nyquist(sys)

v=[-3 5 -7 7]

axis(v)

xlabel('Real Axis');

ylabel('Imaginary Axis');

title('Nyquist Plot of the sytem 50/(s+4)(s^2+3s+3)')

%To determine the Gain Margin,Phase Margin, Gain crossover frequency and

%phase cross over frequency

[Gm,Pm,Wcp,Wcg]=margin (sys)

v =

-3 5 -7 7

Gm =

1.4402

Pm =

11.1642

Wcp =

3.4643

Wcg =

2.9533

MANUAL CALCULATIONS:

NYQUIST

VIVA QUESTIONS

What is polar plot?
What is Nyquist plot?
Define the conditions of stability in polar plot.
What is the use and advantage of polar plot.

REFERENCE

1. NAGRATH & GOPAL,” Control Systems”.

2. MATLAB User Manual .

3. Control Systems by Nagoor gani

RESULT:

Department of Electrical Engineering
Assessment of Practicals
Max Marks / Marks Obtained
Performance : / 25
Record : / 15
Viva-voce : / 10
Total / 50
Signature

11. (a)DESIGN AND IMPLEMENTATION OF LAG COMPENSATOR

AIM

( I ) To determine the transfer function of Lag Compensator

( 2 ) To draw a frequency response plot of Lag Compensator

APPARATUS REQUIRED:

S.no / Name of the apparatus / Range / Type / Quantity
1 / CRO / 30 Mhz / - / 1
2 / AFO / 1 Mhz / - / 1
3 / Resistor / - / 1
4 / Capacitor / - / 1
5 / Breadboard / - / 1
6 / Connecting wires / - / Few

THEORY:

The control systems are designed to perform specific tasks. The specifications are

Generally related to accuracy, relative stability and speed of response. In time domain the

Transient state performance specifications are given in terms of rise time, maximum over

Shoot, settling time and damping ratio. In frequency domain, the transient state performance

Specifications are given in terms of phase margin; gain margin, resonant peak and bandwidth.

The first step in design is the adjustment of gain to meet the desired specification. In

Practical systems, adjustment of gain alone will not be sufficient to meet the given specification. In many cases increasing the gain may result in poor stability. In such cases, it is Necessary to introduce additional device or components in the system to alter the behavior and to meet the desired specifications. Such a redesign of a suitable device is called compensation.

A device inserted into the system for the purpose of satisfying the specification is

Called compensator. A compensator having the characteristics of the lag network is called

Lag compensator. It is used to improved the steady state response and reduce the bandwidth.

The main drawback of lag compensator is poor transient state.

TABULAR COLUMN

s.no / Frequency / Output voltage / Phase shift /

Gain in db

/ X1 / X2

MODEL GRAPH:

Magnitude

Frequency (Hz)

TRANSFER FUNCTION:

The R-C network shown in fiq can realize the lag compensator.

Let

Eo(s) = Output voltage

Ei(s) = Input voltage

By voltage division rule,

Eo(s) = Ei(s) ( R2+1/sC) / ( R1+R2+1/sC)

Eo(s) = Ei(s) (sCR2+1) /sC / [sC(R1+R2)+1] /sC

Eo(s) = Ei(s) (sCR2+1) / [sC(R1+R2)+1]

Eo(s) (sCR2+1)

 = 

Ei(s) [sC(R1+R2)+1]

Eo(s) (s+1/ CR2)

 =  ……………………1

Ei(s) (R1+R2)/ R2 [s+1/((R1+R2)/R2)R2C]

Eo(s) 1 (s+1/T)

 =   ……………………………………………2

Ei(s)  (s+1/T)

Comparing equation 1 & 2

We get T =R2C,

 = (R1+R2)/ R2

Where T = Time constant

 = Attenuation factor

PROCEDURE:

1.The connections are made as per the circuit diagram.

2.Switch on the main supply.

3.Set the input signal by using signal generator.

4.Vary the frequency and magnitude of the signal generator.

5.Note the different readings of the output value enter into the tabular column.

6.Verify the output waveform, it lag behind the applied voltage.

6.Switch off the supply.

7.Disconnect the circuit and return the components to the lab instructor.

RESULT:

Department of Electrical Engineering
Assessment of Practicals
Max Marks / Marks Obtained
Performance : / 25
Record : / 15
Viva-voce : / 10
Total / 50
Signature

11.(b)DESIGN AND IMPLEM ENTATION OF LEAD COMPENSATOR

AIM

(1) To determine the transfer function of Lead Compensator

(2) To draw a frequency response plot of Lead Compensator

APPARATUS REQUIRED :

THEORY:

The control systems are designed to perform specific tasks. The specification is

Generally related to accuracy, relative stability and speed of response. In time domain the

Transient state performance specifications are given in terms of rise time, maximum over

Shoot, settling time and damping ratio. In frequency domain, the transient state performance

Specifications are given in terms of phase margin; gain margin, resonant peak and bandwidth.

The first step in design is the adjustment of gain to meet the desired specification. In