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7/30/2019 29015103 Control System Manual
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Control Systems Laboratory Manual
DIGITAL SIMULATION OF LINEAR SYSTEMDIGITAL SIMULATION OF LINEAR SYSTEM
EXPT.NOEXPT.NO :
DATEDATE :
AIM:AIM:
To simulate the time response characteristic of higher-order Multi-
input multi output (MIMO) liner system using state variable formulation.
APPARATUS REQUIRED:APPARATUS REQUIRED:
MATLAB 6.5
THEORY:THEORY:
Time Domain Specification
The desired performance characteristics of control systems are specified in
terms of time domain specification. System 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 the transient response to a units step input signal.
The transient response of a system to a unit step input depends on the initial
conditions. Therefore to compare the time response of various systems it is
necessary to start with standard initial conditions. The most practical standard is
to start with the system at rest and output and all time derivatives there of zero.
The transient response of a practical control system often exhibits damped
oscillation before reaching steady state.
The transient response characteristics of a control system to a unit step input
are specified in terms of the following time domain specifications.
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Control Systems Laboratory Manual
1. Delay time, td
2. Rise time, tr
3. Peak time, tp
4. Maximum overshoot, Mp
5. Setting time, ts
FORMULA:FORMULA:
Damped frequency of oscillation,
PROCEDURE:PROCEDURE:
7. Enter the command window of the MATLAB.
8. Create a new workspace by selecting new file.
9. Complete your model.
10.Run the model by either pressing F5 or start simulation.
11.View the results.
12.Analysis the stability of the system for various values of gain.
PROBLEM:PROBLEM:
Obtain the step response of series RLC circuit with R = 1.3K , L = 26mH and
C=3.3 f using MATLAB M File.
MATLAB PROGRAM FOR UNIT IMPULSE PRSPONSE:MATLAB PROGRAM FOR UNIT IMPULSE PRSPONSE:
Department of Electrical and Electronics Engineering, DCE
d
Risetime
=
21 1tan
=
where
21 =nd
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PROGRAM:PROGRAM:
num = [ 0 0 1 ]
den = [ 1 0.2 1 ]
impulse (num, den)
grid
title ( unit impulse response plot)
MATLAB PROGRAM FOR UNIT STEP PRSPONSE:MATLAB PROGRAM FOR UNIT STEP PRSPONSE:
PROGRAM:PROGRAM:
Format long e
num = [ 0 0 1.6e10 ]
den = [ 1 50000 1.6e10 ]
step (num, den)
grid on
title (step response of series RLC circuit)
Result:
STABILITY ANALYSIS OF LINEAR SYSTEMSTABILITY ANALYSIS OF LINEAR SYSTEM
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Control Systems Laboratory Manual
EXPT.NOEXPT.NO ::
DATEDATE ::
AIM:AIM:
(i) To obtain the bode plot, Nyquist plot and root locus of the given
transfer function.
(ii) To analysis the stability of given linear system using MATLAB.
APPARATUS REQUIRED:
System with MATLAB
THEORY:THEORY:
Frequency Response:Frequency Response:
The frequency response is the steady state response of a system when the
input to the system is a sinusoidal signal.
Frequency response analysis of control system can be carried either
analytically or graphically. The various graphical techniques available for
frequency response analysis are1. Bode Plot
2. Polar plot (Nyquist plot)
3. Nichols plot
4. M and N circles
5. Nichols chart
Bode plot:Bode plot:
The bode plot is a frequency response plot of the transfer function of a
system. A bode plot consists of two graphs. One is plot of the magnitude of asinusoidal transfer function versus log . The other is plot of the phase angle of a
sinusoidal transfer function versus log .
The main advantage of the bode plot is that multiplication of magnitude can
be converted into addition. Also a simple method for sketching an approximate log
magnitude curve is available.
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Polar plot:Polar plot:
The polar plot of a sinusoidal transfer function G (j ) on polar coordinates
as is varied from zero to infinity. Thus the polar plot is the locus of vectors
[G (j ) ] G (j ) as is varied from zero to infinity. The polar plot is alsocalled Nyquist plot.
Nyquist Stability Criterion:Nyquist Stability Criterion:
If G(s)H(s) contour in the G(s)H(s) plane corresponding to Nyquist contour
in s-plane encircles the point 1+j0 in the anti clockwise direction as many times
as the number of right half s-plain of G(s)H(s). Then the closed loop system is
stable.
Root Locus:Root Locus:
The root locus technique is a powerful tool for adjusting the location of
closed loop poles to achieve the desired system performance by varying one or
more system parameters.
The path taken by the roots of the characteristics equation when open loop
gain K is varied from 0 to are called root loci (or the path taken by a root of
characteristic equation when open loop gain K is varied from 0 to is called root
locus.)
Frequency Domain Specifications:Frequency Domain Specifications:
The performance and characteristics of a system in frequency domain are
measured in term of frequency domain specifications. The requirements of a
system to be designed are usually specified in terms of these specifications.
The frequency domain specifications are
1. Resonant peak, Mr
2. Resonant Frequency, r.
3. Bandwidth.
4. Cut off rate
5. Gain margin
6. Phase margin
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Resonant Peak, MResonant Peak, Mrr
The maximum value of the magnitude of closed loop transfer function is
called the resonant peak, Mr. A large resonant peak corresponds to a large over
shoot in transient response.
Resonant Frequency,Resonant Frequency, rr
The bandwidth is the range of frequency for which the system gain is more
than -3db. The frequency at which the gain is -3db is called cut off frequency.
Bandwidth is usually defined for closed loop system and it transmits the signals
whose frequencies are less than cut-off frequency. The bandwidth is a measured of
the ability of a feedback system to produce the input signal, noise rejection
characteristics and rise time. A large bandwidth corresponds to a small rise time orfast response.
Cut-Off Rate:Cut-Off Rate:
The slope of the log-magnitude curve near the cut off frequency is called
cut-off rate. The cut-off rate indicates the ability of the system to distinguish the
signal from noise.
Gain Margin, KGain Margin, Kgg
The gain margin, Kg is defined as the reciprocal of the magnitude of open
loop transfer function at phase cross over frequency. The frequency at witch the
phase of open loop transfer function is 180 is called the phase cross over
frequency, pc.
Phase Margin,Phase Margin,
The phase margin , is that amount of additional phase lag at the gain cross
over frequency required to bring the system to the verge of instability, the gaincross over frequency gc is the frequency at which the magnitude of open loop
transfer function is unity (or it is the frequency at which the db magnitude is zero).
PROCEDURE:PROCEDURE:
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1. Enter the command window of the MATLAB.
2. Create a new M file by selecting File New M File.
3. Type and save the program.
4. Execute the program by either pressing F5 or Debug Run.
5. View the results.6. Analysis the stability of the system for various values of gain.
Problem 1Problem 1
Obtain the bode diagram for the following system
MATLAB ProgramMATLAB Program
a = [0 1 ; -25 -4]
b = [1 1 ; 0 1]
c = [1 1 ; 1 1]
d = [0 0 ; 0 0]
bode (a, b, c, d)
grid
title (BODE DIAGRAM)
Problem 2Problem 2
Draw the Nyquist plot for G(s) =
MATLAB ProgramMATLAB Program
num = [0 0 0]
Department of Electrical and Electronics Engineering, DCE
=
+
=
2
1
2
1
2
1
2
1
2
1
425
10
10
11
425
10
x
x
y
y
y
y
x
x
x
x
)1( +ss
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den = [1 1 0]
nyquist (num,den)
v = [-2,2,-5,5]
axis (v)
gridtitle (Nyquist Plot)
Problem 2Problem 2
Obtain the root focus plot of the given open loop T.F is
G(s) H (s) =
MATLAB ProgramMATLAB Program
num = [0 0 0 0 1]
den = [11.1 10.3 5 0]
rlocus (num,den)
gridtitle [Root Locus Plot]
Result:
STUDY OF P, PI AND PID CONTROLLERSTUDY OF P, PI AND PID CONTROLLER
EXPT.NOEXPT.NO ::
DATEDATE ::
Department of Electrical and Electronics Engineering, DCE
)106.0)(5.0( 2 +++ ssss
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AIM:AIM:
To find the percentage peak overshoot and steady state error of the
given P, PI and PID
APPARATUS REQUIRED:APPARATUS REQUIRED:
MATLAB, SIMULINK
THEORY:THEORY:
A controller is similar to an amplifier. It is used in closed loop control
system to enhance the system output.
In proportional controller, the output is proportional to the input. Kp
represents the gain constant of proportional contoller.
Where Vo = output voltage
Vi = input voltage
Rf= feedback resistance
Ri = input resistance
Kp = gain constant
The control action of proportional plus integral controller is defined by the
equation
Vo (t) = Kp Vi (t) + Vi (t) (Kp / Ti)
The transfer function of the controller is
T.F = Vo (s) / Vi (s) = Kp (1 + 1 / sTi)
Where Vo = output voltage
Vi = input voltage
Ti = integral time constant
Kp = gain constant
Both Kp and Ti are adjustable. The integral time constant Ti adjusts integral control
action while changed in the values of Kp affects both the integral parts of the
control action. The inverse of integral time Ti is called the reset rate.
The reset rate is the number of time per minute that the proportional part of
the control action is duplicated. Reset rate is measured in terms of reset per minute.
The combinations of the proportional control action, integral control action and
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I
F
i
OPR
R
V
VK +== 1
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derivative combined action has the advantages of the three individual control
actions.
The transfer function of the controller is given by
Vo (s) / Vi (s) = Kp (1 + 1 / (Ti + Td))
where Vo = output voltageVi = input voltage
Ti = integral time constant
Td = derivative time constant
Kp = gain constant
PROCEDURE:PROCEDURE:
PROPORTIONAL CONTROLLER:PROPORTIONAL CONTROLLER:
1. Make the connections as per the circuit diagram2. Set the values by using knobs on the trainer as follows:
Input amplitude to 1 V (p-p)
Frequency at low value
4. For various values of Kc, observe the waveforms
PROPORTIONAL INTEGRAL CONTROLLER:PROPORTIONAL INTEGRAL CONTROLLER:
1. Make the connections as per the circuit diagram
2. Set the values by using knobs on trainer as follows
Input amplitude to 1V (p-p)
Frequency at low value and Ki to zero.
3. Keep Kc = 0.6 and increase Ki in small steps and observe the
waveforms.
PROPORTIONAL INTEGRAL DERIVATIVE CONTROLLER:PROPORTIONAL INTEGRAL DERIVATIVE CONTROLLER:
1. Make the connections as per the circuit diagrams.
2. Set the values by using knobs on trainer as following.
Input amplitude to 1V (p-p)
Kc = 0.6, Ki = 54.85, Kd = 0
Frequency at low values
3. The system shows fairly large overshoot.
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Control Systems Laboratory Manual
4. The above steps are repeated for a few non zero values of Kd.
5. The improvement in transient performance is observed using CRO
with increasing values of Kd, while the steady state error remains unchanged.
6. Calculate the values of peak overshoot and steady state error.
Result:
DESIGN OF LAG AND LEAD COMPENSATORSDESIGN OF LAG AND LEAD COMPENSATORS
EXPT.NOEXPT.NO ::
DATEDATE ::
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Lag compensation result in a improvement in steady state performance but
result in slower response due to reduced bandwidth. The attenuation due to the lag
compensator will shift the gain crossover frequency to a lower frequency point
where the phase margin is acceptable. Thus the lag compensator will reduce the
bandwidth of the system and will result in slower transient response.
Lag compensator is essentially a low pass filter and high frequency noise
signals are attenuated. If the pore introduce by compensator is cancelled by a zero
in the system, then lag compensator increase the order of the system by one.
LEAD COMPENSATOR:
A compensator having the characteristics of a lead network is called a lead
compensator. If sinusoidal signal is applied to a lead network, then in steady state
the output will have a phase lead with respect to input.
The lead compensator increase the bandwidth, which improves the speed of
response and also reduces the amount of overshoot. Lead compensation
appreciably improves the transient response, whereas there is a small change in
steady state accuracy. Generally lead compensation is provided to make an
unstable system as a stable system. A lead compensator is basically a high pass
filter and so it amplifies high frequency noise signals. If the pole is introduced by
the compensator is not cancelled by a zero in the system, then lead compensator
increases order of the system by one.
FORMULA:
Department of Electrical and Electronics Engineering, DCE
0
0
xy
ABGain ==
)/log(20 AB=
)/(sin 01 AxPhase =
)/(sin 01 By=
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4. Execute the program by either pressing F5 or Debug Run.
5. View the results.
6. Analyze the results.
MATLAB coding with Compensator:
PROGRAM:
num = [ 0 0 100 5 ];
den = [ 400 202 1 0 ];
sys = (sys)
margin (sys)
[ gm, ph, wpc, wgc ] = margin (sys)
title (BODE PLOT OF COMPENSATED SYSTEM)
MATLAB coding with out lag Compensator:
PROGRAM:
num = [ 0 0 5 ];
den = [ 2 1 0 ];
sys = tf (num, den)
bode (sys)
Margin (sys)
[ gm, ph, wpc, wgc ] = margin (sys).
title (BODE PLOT OF UNCOMPENSATED SYSTEM);
MATLAB coding with out Compensator for loop system
PROGRAM:
den=[ 1 0.739 0.921 0 ];
pitch=tf(num, den);
sys_cl=feedback (pitch,1);
de=0.2;
t=0:0.01:10;
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figure
step(de*sys_cl, t)
sys_cl=feedback (pitch,10);
de=0.2;
t=0:0.01:10;
bode(sys_cl, t)grid on
title ( 'BODE PLOT FOR CLOSED LOOP SYSTEM WITHOUT
COMPENSATOR')
MATLAB coding with Compensator for loop system
PROGRAM:
num=[1 151 0.1774 ];
den=[1 0.739 0.921 0 ];pitch=tf(num, den);
alead=200;
Tlead=0.0025;
K=0.1;
lead=tf(K*[alead*Tlead 1], [Tlead 1]);
bode(lead*pitch)
sys_cl=feedback(lead*pitch,10);
de=0.2;
t=0:0.01:10;
figure
step (de*sys_cl, t)
title('BODE PLOT FOR CLOSED LOOP SYSTEM WITH
COMPENSATOR')
Result:
TRANSFER FUNCTION OF DC SHUNT MOTORTRANSFER FUNCTION OF DC SHUNT MOTOR
EXPT.NOEXPT.NO ::
DATEDATE ::
AIM:AIM:
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To determine the transfer function of the DC shunt motor
APPARATUS REQUIRED:APPARATUS REQUIRED:
S.No Name of the Equipment Range Type Quantity
THEORY:THEORY:
Speed can be controlled by varying (i) flux per pole (ii) resistance of armature circuit and(iii) applied voltage.
It is known that N Eb. If applied voltage is kept, Eb = V IaRa will
Remain constant. Then, N 1
By decreasing the flux speed can be increased and vice versa. Hence this
method is called field control method. The flux of the DC shunt motor can be
changed by changing field current, Ish with the help of shunt field rheostat. Since
the Ish relatively small, the shunt filed rheostat has to carry only a small current,
which means Ish2 R loss is small. This method is very efficient. In non-interpolar
machines, speed can be increased by this methods up to the ratio 2: 1. In interpolar
machine, a ratio of maximum to minimum speed of 6:1 which is fairly common.
FORMULA:FORMULA:
Armature ControlArmature ControlD.C. Shunt motor:D.C. Shunt motor:
It is DC shunt motor designed to satisfy the requirements of the servomotor.
The field excited by a constant DC supply. If the field current is constant then
speed is directly proportional to armature voltage and torque is directly
proportional to armature current.
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KmTransfer Function =
S (1 + TmS)
Km = 1 / Avg Kb
Tm = JRa / Kb Kt
Kt = T / Ia Eb = V-Ia Ra
Constant Values
J = 0.039 Kg2m
B = 0.030 N / rpm
Field ControlField Control D.C. Shunt motor:D.C. Shunt motor:
It is DC shunt motor designed to satisfy the requirements of the servomotor.
In this motor the armature is supplied with constant current or voltage. Torque is
directly proportional to field flux controlling the field current controls the torque of
the motor.
K
Transfer Function = Js2 (1 + s)
K = Kt / Rf
= Lf / Rf = V Zf2 Rf2 / 2 f / Rf = 2 N / 60 T = r ( S1 S2 ) * 9.81 N-m and r = .075m
OBSERVATION TABLE FOR TRANSFER FUNCTION ARMATURE
CONTROL DC SERVO MOTOR:
Table No. 1 Finding the value of KTable No. 1 Finding the value of Kbb
Avg Kb =
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Sl.No If Ia S1 S2 N V T Eb Kb = Eb /
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Table No. 2 To find RTable No. 2 To find Raa
Avg Ra =
PRECAUTIONS:PRECAUTIONS:
At starting,
The field rheostat should be kept in minimum resistance position
PROCEDURE FOR TRANSFER FUNCTION OF ARMATURE CONTROLPROCEDURE FOR TRANSFER FUNCTION OF ARMATURE CONTROLDC SHUNT MOTOR:DC SHUNT MOTOR:
Finding KFinding Kbb1. Keep all switches in OFF position.
2. Initially keep voltage adjustment POT in minimum potential position.
3. Initially keep armature and field voltage adjustment POT in minimum
position.
4. Connect the module armature output A and AA to motor armature terminal
A and AA respectively, and field F and FF to motor field terminal F and FF
respectively.
5. Switch ON the power switch, S1, S2.
6. Set the field voltage 50% of the rated value.
7. Set the field current 50% of the rated value.
8. Tight the belt an take down the necessary readings for the table 1 to find
the value of Kb.
9. Plot the graph Torque as Armature current to find Kt.
Department of Electrical and Electronics Engineering, DCE
Sl.No Volt Va Current Ia Ra = Va / Ia
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Finding RFinding Raa1. Keep all switches in OFF position.
2. Initially keep voltage adjustment POT in minimum position.
3. Initially keep armature and field voltage adjustment POT in minimum
potential position.4. Connect module armature output A and AA to motor armature terminal
A to AA respectively.
5. Switch ON the power switch and S1.
6. Now armature voltage and armature current are taken by varying the
armature POT with in the rated armature current value.
7. The average resistance value in the table -2 gives the armature resistance.
PROCEDURE FOR TRANSFER FUNCTION OF FIELD CONTROL D.C.PROCEDURE FOR TRANSFER FUNCTION OF FIELD CONTROL D.C.
SHUNT MOTOR:SHUNT MOTOR:
Finding RFinding Rff1. Keep all switches in OFF position.
2. Keep armature field voltage POT in minimum potential position.
3. Initially keep armature and field voltage adjustment POT in minimum
potential position.
4. Connect module filed output F and FF to motor filed terminal F and FF
respectively.5. Switch ON the power, S1 and S2.
6. Now filed voltage and filed current are taken by varying the armature POT
with in the rated armature current value.
7. Tabulate the value in the table no 3 average resistance values give the fied
resistance.
Finding ZFinding Zff1. Keep all switches in OFF position.
2. Keep armature and field voltage POT in minimum position.
3. Initially keep armature and field voltage adjustment POT in minimum
position.
4. Connect module varaic output P and N to motor filed terminal F and FF
respectively.
5. Switch on the power note down reading for the various AC supply by
adjusting varaic for the table no 4.
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Finding KFinding Kttll
1. Keep all switches OFF position.
2. Initially keep voltage adjustment POT in minimum potential position.
3. Initially keep armature and field voltage adjustment POT in minimum
position.
4. Connect the module armature output A and AA to motor armature terminaland AA respectively, and field F and FF to motor field terminal F and FF
respectively.
5. Switch ON the power switch, S1 and S2.
6. Set the filed voltage at rated value (48V).
7. Adjust the armature voltage using POT on the armature side till it reaches
the 1100 rpm.
8. Tight the belt and take down the necessary reading for the table 5 Ktl
9. Plot the graph Torque as Field current to find Ktl
OBSERVATION TABLE FOR TRANSFER FUNCTION OF ARMATUREOBSERVATION TABLE FOR TRANSFER FUNCTION OF ARMATURE
CONTROL DC SERVO MOTOR:CONTROL DC SERVO MOTOR:
Table No:3 To find RTable No:3 To find Rff
Sl.No If(amp) Vf(Volt) R f (ohm)
Avg Rf =
Table No:4 To find ZTable No:4 To find Zff
Sl.NoIf(amp)
mAVf(Volt) Zf = Vf / If
Avg ZAvg Zff ==
Table No: 5 To find KTable No: 5 To find Kttl
Sl.No If Ia S1 S2 T( N m) N (rpm)
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MODEL GRAPH:MODEL GRAPH:
MODEL CALCULATION:MODEL CALCULATION:
Department of Electrical and Electronics Engineering, DCE
If
T
Kt
l= T /
IfT
Field Current
T
I
a
T
Kt = T /
Armature Current
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Result: