Control System Manual

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.

Department of Electrical and Electronics Engineering, DCE


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=3.3f using MATLAB M – File.



MATLAB PROGRAM FOR UNIT IMPULSE PRSPONSE:MATLAB PROGRAM FOR UNIT IMPULSE PRSPONSE:

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

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 are

1. Bode Plot2. Polar plot (Nyquist plot)3. Nichols plot4. M and N circles5. 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 a sinusoidal 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.

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 also called 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

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 or fast 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 gain cross 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).




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)gridtitle (‘BODE DIAGRAM’)

Problem 2Problem 2Draw the Nyquist plot for G(s) =




num = [0 0 0]den = [1 1 0]nyquist (num,den)v = [-2,2,-5,5]axis (v)gridtitle (‘Nyquist Plot’)

Problem 2Problem 2Obtain the root focus plot of the given open loop T.F is

G(s) H (s) =


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 ::

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 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 voltage Vi = input voltage Ti = integral time constant Td = derivative time constant Kp = gain constant


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 diagram2. 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.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 ::

APPARATUS REQUIRED:APPARATUS REQUIRED:

System employed with MATLAB 6.5

THEORY:THEORY:

The control systems are designed to perform specific taskes. When

performance specification are given for single input. Single output linear time

invariant systems. Then the system can be designed by using root locus or

frequency response plots.

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

specifications. In practical system. Adjustment of gain alone will not be sufficient

to meet the given specifications. In many cases, increasing the gain may result poor

stability or instability. In such case, it is necessary to introduce additional devices

or component in the system to alter the behavior and to meet the desired

specifications. Such a redesign or addition of a suitable device is called

compensations. A device inserted into the system for the purpose or satisfying the

specifications is called compensator. The compensator behavior introduces pole &

zero in open loop transfer function to modify the performance of the system.

The different types of electrical or electronic compensators used are lead

compensator and lag compensator.

In control systems compensation required in the following situations.

1. When the system is absolutely unstable then compensation is required

to stabilize the system and to meet the desired performance.



2. When the system is stable. Compensation is provided to obtain the

desired performance.

LAG COMPENSATOR:

A compensator having the characteristics of a lag network is called a lag

compensator. If a sinusoidal signal is applied to a lag network, then in steady state

the output will have a phase lag with respect input.

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:

PROCEDURE:

With out compensator:

1. Make the connection as per the circuit diagram.

2. Apply the 2V p-p sin wave input and observe the waveform.

3. Very the frequency of the sin wave input and tabulate the values of xo and yo

4. Calculated gain and phase angle.

5. Draw the bode plot.

With lag compensator:

1. From the bode plot find the new gain crossover frequency.

2. Find out values and writ the frequency function. G(s).

3. From the transfer function calculated R1, R2 and C.

4. Set the amplifier gain at unity.

5. Insert the lag compensator with the help of passive components and

determine the phase margin of the plant.

6. Observe the step response of the compensated system.

PROCEDURE:

1. Enter the command window of MATLAB.

2. Create a New M-File by selecting file New M-File.

3. Type and save the program.

4. Execute the program by pressing F5 or Debug Run.



5. View the results.

6. Analyze the Results.

With lead compensator:

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. 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;figurestep(de*sys_cl, t)sys_cl=feedback (pitch,10);de=0.2;t=0:0.01:10;bode(sys_cl, t)grid ontitle ( '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;figurestep (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:

To determine the transfer function of the DC shunt motor

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 Ish

2 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 Control D.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.

Km

Transfer Function = S (1 + TmS)

Km = 1 / Avg Kb

Tm = JRa / Kb Kt

Kt = T / Ia

Eb = V-Ia Ra

Constant ValuesJ = 0.039 Kg2mB = 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.

KTransfer Function = Js2 (1 + s)

K = Kt / Rf

= Lf / Rf = V Zf2 – Rf

2 / 2f / Rf

= 2N / 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 Kb

Sl.No If Ia S1 S2 N V T Eb Kb = Eb /

Avg Kb =

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 CONTROL DC SHUNT MOTOR:DC SHUNT MOTOR:

Finding KFinding Kbb


Sl.No Volt Va Current Ia Ra = Va / Ia


1. 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.

Finding RFinding Raa

1. 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 Rff

1. 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 Z Finding Zff

1. 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.

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 terminal

and 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 Kt

l

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) Rf (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 Ktt

l

Sl.No If Ia S1 S2 T( N – m) N (rpm)

MODEL GRAPH:MODEL GRAPH:


If

T Ktl = T / IfT

Field Current

T

Ia

T Kt = T / Ia

Armature Current


MODEL CALCULATION:MODEL CALCULATION:

Result:


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Control System Manual