Servomechanism and System Identification School of Electrical and Electronic Engineering Coursework (EEE8074)
Name: Sohail Ahmed Student no 140645978 Submitted to Prof. Kabita Adhikari Date of Submission: 10/June/2015
Table of contents Abstract................................................................................................................................................... ................................................................................................................................................... 4 Introduction .................................................................... .......................................................................................................................................... ........................................................................ .. 5 Chapter 1 ............................................................. ................................................................................................................................... ................................................................................... ............. 6 Servomechanisms ................................................................... ............................................................................................................................... ............................................................ 6 Aims and objectives: ....................................................................................................................... ....................................................................................................................... 6 1.1 Background ............................................................................................................................... ............................................................................................................................... 6 1.2 Milestone 1- Servo Gain Constant K m Determination ........................................................... 6 Discussion and analysis ......................................................................................................... ........................................................................................................... .. 7 1.3 Milestone 2: Basic closed speed control ............................................................... ............................................................................... ................ 8 Discussion and Analysis......................................................... ........................................................................................................... .................................................. 8 1.4 Milestone 3: Basic closed loop position control ................................................................... 9 Discussion and Analysis......................................................... ........................................................................................................... .................................................. 9 1.5 Milestone 4: Closed loop with position and velocity ffeedback eedback .......................................... 10 Discussion and analysis ......................................................................................................... ......................................................................................................... 10 Chapter 2 ............................................................. ................................................................................................................................... ................................................................................. ........... 11 System Identification ........................................................................................................................ ........................................................................................................................ 11 Aims and Objectives:...................................................................... ..................................................................................................................... ............................................... 11 2.1 Background ............................................................................................................................. ............................................................................................................................. 11 2.2 Milestone 1: Position control system analysis .................................................................... .................................................................... 12 Discussion and analysis ......................................................................................................... ......................................................................................................... 12 2.3 Milestone2: Close loop speed control S ystem identification ............................................. 12 Milestone (2.1): Close loop speed control system identification at attenuator (0.3) .......... 12 Milestone (2.2) Close loop speed control system identification at attenuator (1) .............. 13 Discussion and analysis ......................................................................................................... ......................................................................................................... 14 2.4 Milestone3: System identification for Close loop position control .................................... 14 Millstone (3.1) Close loop position control system identification at attenuator (0.3) ......... 14 Millstone (3.2) Close loop position tr ansfer function attenuator at (1) ............................... ............................... 15 Discussion and analysis ......................................................................................................... ......................................................................................................... 16 2.5 Milestone 4 Simulation ....................................................................................................... ....................................................................................................... 17 Discussion and Analysis......................................................... ......................................................................................................... ................................................ 17 Discussion and Results .......................................................................................................... .......................................................................................................... 17 Conclusion............................................................................................................................................. .............................................................................................................................................18 18
2
References ............................................................................................................................................ 19 Appendix ............................................................................................................................................... 20 Experiment 1: Milestone 1 ................................................................................................................ 20 Milestone 3 ....................................................................................................................................... 21 Milestone 4 ....................................................................................................................................... 22 Experiment 2: .................................................................................................................................... 23 Milestone 2(1) ................................................................................................................................... 23
3
Abstract This report deals with basics of servomechanism and System identifications application to servo system. System identification is a process carried out to identify the parameters and transfer functions of different unknown systems. The methods used are divided into two main types, offline and online. Two methods of offline system identification to identify the parameter and transfer function of servo system are applied. The step response and bode plot method, step response method have some limitation in case of systems is first order ore those without overshoot, so this method applied for just sake of understanding. To identify the parameter and transfer function of such system bode plots are used and results verified through Matlab.
4
Introduction DC servo motors are simple Dc motors which are developed with servomechanism for different applications, these motors are very common in electrical vehicles and robotics. These are mostly used in high precision applications. Servomechanism is basically a control mechanism which consist of at least one mechanical component, in which feedback is provided to vary the input. In the practical world sometimes system parameters and transfer function of system are not available. In order to analyse the characteristics of such systems we need to identify the parameters and find the transfer function of system. System identification is a field which deals with identification of unknown parameters and transfer function of such systems. System identification is broadly categorised into two main parts - online system identification which involve more sophisticated algorithms requiring computers and offline system identification which basically rely on conventional techniques of step response and bode plots. First of all the parameter of servomotor are identified such as time and electro-mechanical time constant. Actually we are going to use bode plot to identify the system transfer function due to some limitation of step response, before that in order to understand why step response cannot be used in some exceptional cases experimental results are obtain to prove it. In second part we are going to use bode that overcome the disadvantage of step response to identify the system transfer function and prove these with simulation results from Matlab[1, 2].
5
Chapter 1 Servomechanisms Aims and objectives:
Identification of servomotor parameters Importance of tachometer and Basic speed loop control system setup Construction of basic position control system Construction of position control system with velocity feedback[1]
1.1 Background Servomechanism is control mechanism which is used to control the device on the basis of an error signal obtained by the taking difference of feedback signal and reference in put signal. DC motor controlled by servo is called DC servo motor. Two types of servomechanism are we going to study speed and position control The transfer function of DC servomotor (DC motor and servo amplifier) is given below equation 1.2. Figure 1.SA150+DCM150F Block Diagram[1]
= VVmt = K+mS×τKmt ………1.2
In equation (1) ωm represent the angular velocity of DC motor, V m is the servomotor input voltage Km is gain of servo amplifier, τm is the electro- mechanical time constant. Where K t=Vt/ ωm tachometer sensitivity and Vt is tachometer final voltage. The servo gain constant can be obtained by using equation 1.2 and 1.4[1].
Km = Vm VVmt = K1m ×SτKmt
Equation(1.2)
Equation(1.3)
Equation(1.4) t ∞;s=0
Km = VmV×t Vt
This experiment have four milestones 1 2 3 4
Servo Gain Constant Km Determination Basic closed loop speed control Basic closed loop position control system set up Closed loop position control with velocity feedback
1.2 Milestone 1- Servo Gain Constant K m Determination By setting up servo equipment according to according to fig (1.1) and measuring input voltage(Vm) at different voltage levels ranging from 0 to +3 and measuring the tachometer output voltage (Vt) by the help of oscilloscope and angular speed (ω m) and then changing the polarity of input voltage and again measuring the both input, output voltages and angular speed. By the help of measured values and using the equation (1.2) and (1.4) we can determine the K m by including the effect of tachometer and without tachometer different ways. The voltage level after which motor start running at constant speed called dead band voltage of motor and for this motor it is measured around
6
1.289V[1]. All the results are given in table 1. SA150D
Vm (Ch1)
(n) RPM
(ω) Rad/Sec
Vt (Ch2)
T#1 T#1 T#1 T#1 T#2 T#2 T#2 T#2
1.2 1.5 2 2.5 -1.2 -1.5 -2 -2.5
2890 2900 2900 2900 2960 2960 2950 2940
302.68 303.73 303.73 303.73 -310.01 -310.01 -308.96 -307.92
8 8.1 8.1 8.1 -8.1 -8.1 -8.1 -8.1
Kt =Vt/ω =/ Km=vt/Vm∗Kt 0.0264 0.0267 0.0267 0.0267 0.0261 0.0261 0.0262 0.0263
252.23 202.48 151.86 121.49 258.34 206.67 154.48 123.17
252.23 202.48 151.86 121.49 258.34 206.67 154.48 123.17
Table 1. Servomotor Parameter The graphical relation between input voltage and speed are shown in fig (2) and (3) Servomotor Time Constant τ m Determination
Time constant of servo motor can be calculated by changing the input to square wave of 0.25Hz and by the help of oscilloscope we calculated time constant at point where angular speed(ω) is 63.2% of final value. We repeat this process for two positive values of square wave amplitude and vice versa. All the results of reading and calculation are given in table 1.2 and scoops of all the results are given in appendix Experiment 1 milestone 1[1].
Forward
Reverse
Square Wave Amplit ude (Vm)
Square Wave Frequenc y (F) Hz
Tachomet er (Vt)
Time Constant (Tm) mSec
ω
1.5
0.25
7.85
126
303.73
202.49
2
0.25
7.8
126
303.73
151.87
-1.5
0.25
-7.75
126
-310.01
206.67
-2
0.25
-7.75
128
-308.96
154.48
Km= (ω/V m)
Figure 2.speed vs voltage (+iv)
Table 2.Time constant determination
The average value of time constant calculated from above readings is 127msec. Figure 3.speed vs voltage (-iv) Discussion and analysis This main purpose of this experiment is to check the effect of tachometer on system parameter. Tachometer is basically output sensor which measure the speed and provide the feedback to control for feedback control system. The parameter of tachometer that can affect the system parameter is sensitivity of tachometer and it has small value which remain constant and is ignorable.in this experiment we calculated system parameter without taking in consideration the effect of tachometer and with tachometer .we find out that our parameter remained same in both cases. This proves that i n feedback control tachometer act like unity feedback and does not affect system parameter. 7
1.3 Milestone 2: Basic closed speed control To construct closed loop speed control the feedback signal proportional to speed is provided as feedback from tachometer (GT150X) to the comparator (OA150A) and the feedback selector is set to 100Kohm resistor comparator. Comparator compares the reference V ref to the feedback and
Speed Rev/min 200 400 600 800 1000 1200 1400 1600 1800 2000 Figure 4.Basic closed loop speed control
Resistor (Feedback) 100K 100K 100K 100K 100K 100K 100K 100K 100K 100K
Vref (Ch1) 0.48 1.05 1.58 2.09 2.58 3.01 3.90 4.17 5.10 5.49
Verror (Ch2) 0.094 0.112 0.131 0.148 0.167 0.171 0.189 0.198 0.228 0.24
Table 3.Speed, Resistance, V (ref),V (error)
Provide Verror as input to servomotor (SA150D+DCM150). The whole setup is completed as according to fig 4.The positive side of tachometer is connected to OA150A as input and other side is grounded. Those are determined by increasing the reference voltage without feedback and by the help of voltmeter measuring the voltage across tachometer. Motor speed range is determine by tachometer which is 200rev/min to 2000rev/min. Several reading of motor speed at 200 to 2000rev/min are taken from tachometer and reference voltage(Vref ) and Error voltage (Verror ) is determined by the help of oscilloscope ch1 and ch2 at these specific speed values. All the measured values are given in table 1.3.The graphical relation speed against reference voltage and speed against error voltage are shown in fig 5 and 6[1].
Verror Vs Speed (ωm)
Vref Vs Speed (ωm)
3000
3000
n i m / 2000 v e 1000 R )
n i m / 2000 v e 1000 R )
ω ( d e e p S
ω ( d e e p S
m
m
0
0.09
0.14
0.19
0.24
Verror (V)
Figure 5.Verror (v) vs speed (rev/min)[1]
0.29
0
0.48
2.48
4.48
6.48
Vref (V)
Figure 6.Vref (v) vs speed (rev/min)[1]
Discussion and Analysis In this experiment basic speed feedback control is i mplemented by providing the simply feedback from tachometer which measure the output speed. The difference between feedback and reference value is error value which is used as in put for servomotor to control the speed. It is clear from the graph given in fig 1.6 and 1.7 that speed increased linearly with both reference and error inputs. In order to control 8
the output speed according to reference value this error signal should be zero ideally. But practically it’s not possible. So by using different controllers we try to reduce the error value to match the reference value to the desired output. 1.4 Milestone 3: Basic closed loop position control Closed loop position control is implemented as shown in fig 1.8.The input (150H) and output potentiometer (150X) at their midpoint indicates zero voltage. The input potentiometer is offset by 30 degree and as soon it happen output potentiometer start is following the input this proves our connection are correct. Closed loop transfer function is
⁄×(n) = s+nsn+n
Figure 7.Position control with velocity feedback[1]
Square wave of amplitude 2V (p-p) and frequency of 0.5Hz is provided as input from function generator. The ch1 and ch2 of oscilloscope are connected the input and output terminal to observe the response by changing the value of forward path gain (K1) and velocity feedback gain (K2).By setting the K2=0 and changing the value of K1=1,2,4,8 and 10 the value of peak time(Tp) ,time period(T) and percentage overshoot(P.O) are measured by using oscilloscope at given values of K1.the value of damping ratio (ζ) is calculated by using equation (1.5)[1] .
P.O=
− √ −
Peak Time (tP)
Time period (T)
100
Forward Path Gain (K1)
Tachometer Feedback Gain (K2)
………………………. (1.5)
Percentage Overshoot P.O (%)
1 0 0 0.00 0 2 0 214 0.00 7.5 4 0 160 0.00 27.2 8 0 160 0.00 20.6 10 0 152 206.00 28 * Damping Ratio has been calculated using equation 1.5 Table 4.values of K1, K2, ( ), T ( ), P.O (%)
Settling time(tS) s
Natural frequency(ωn) rad/s
0 0.33 0.49 0.405 0.478
0 19.01 21.24 21.97 22.29
*Damping Ratio ξ
0 0.636108 0.382807 0.449232 0.375703
Scopes are given in appendix experiment1, milestone3 and measurements in are given in table 4.
Discussion and Analysis in order to implement basic closed loop potion control in this part of experiment we remove speed feedback which was represented by K2 and we investigated the effect of forward path gainK1 on the system stability by determine the value of damping ratio at different values of K1.As it is clearly seen in the table 4 that increasing the value of K1 causes the decrees in damping ratio which in term causes 9
more overshoot in output and lead system to instability. This happen because of increase in natural frequency of system which make system faster but cause more overshoot and oscillation and make system to unstable if damping ratio decreased to very low value. That is undesirable in position control to have oscillation and overshoot because when you are going to align our system to certain position these oscillation make it difficult achieve the desired results. Normally damping ratio should be between 0.4 and 0.8 to achieve good results. The overshoot 2 to 6% is acceptable because practically it’s very hard to get zero overshoot.in this arrangement we only have one parameter K1 which effect both damping ration and natural frequency simultaneously so it is very difficult achieve desire results.in order to overcome this problem in next experiment we are going to use speed feedback to control natural frequency and damping ratio independently. System transfer function can be identified by this method but due to limitation in case of no overshoot and first order system bode plot are preferred. 1.5 Milestone 4: Closed loop with position and velocity feedback The closed loop with position and velocity feedback is implemented as shown in fig 1.8 in previous milestone. This time we set the forward gain K1=10 and change the value of velocity feedback gain K2 in order to analyze the effect of K2 on system output response. For different values of K2=1, 2, 4, 8 and 10 output is analyzed on oscilloscope and peak time (Tp), time period (T) and percentage overshoot(P.O) are obtained from oscilloscope all scopes are given in appendix experiment 1, milestone4. All the measurements and calculation are given in table 5[1].
Forward Path Gain (K1)
Tachometer Feedback Gain (K2)
Peak Time (Tp) mSec
Period Time (T) mSec
Percentage Overshoot P.O (%)
10
1
0
0
0
10
2
0
0
0
10
4
0
0
0
10
8
0
0
0
0
0
0
10 10 Table 5.values of K2, K1, Tp, T, P.O and ζ
Discussion and analysis In closed loop position control with velocity feedback the results shown in table 1.5 clearly indicates that in this case there is no overshoot recorded because of velocity feedback gain which provide independent control of damping ratio without affecting natural frequency. To apply step response method for system transfer function identification we require the some sort of overshoot to get value of damping ratio these results shows the limitation of step response in system identification.in order to identify the such type of systems we use bode plots.in next experiment system identification done with help of bode plots.
10
Chapter 2 System Identification Aims and Objectives:
Identification of unknown transfer function by bode plot
Transfer function parameterising by using bode plot[1]
2.1 Background System identification is the process of determining the equations which govern a systems dynamic response. Depending upon our analysis, we may wish to find the differential equations which characterize the system, the state space model, or t he transfer function model of the system. Broadly speaking, there are two categories of system identification; on line and off line. In this experiment off line system identification is used for parameter identification of servomotor in which a series of measurements are taken and through calculation and approximated mathematical model is developed. This typically leads us to identifying an approximate system transfer function. Frequency domain (bode plot) approach is used for identification. Two types of models are common in the field of system identification. Grey box model: The model of system about which we already have some information about the underlying dynamics or some physical parameters is called grey box model. Black box model: The model of system about which we don’t have any information about the dynamics of system is called black box model. There two types of servomechanism one to produce the controlled speed as output and other controlled position. We are going to identify the parameter of servomotor which falls under the category of black box model. The general open loop transfer function of first order system in frequency domain is (2.1) (2.2) G(s) =KLF/1+Sτm τm=1/ωc In this experiment we are using the same servomotor that we used in previous experiment the open loop transfer function of servomotor will be same as given in equation 2.1. Generally the values of K LF and τm can be calculated From the bode plot of magnitude plot. K LF is the value of magnitude at lowest frequency also known as low Figure 8.bode plot first order frequency gain. The value of ωc is determined from graph as shown in fig 2.1.τm can be calculated by using the equation 2.2. The open loop transfer function of position control is given by the product of speed controller transfer function G(s) and the speed/position conversion P(s). P(s) =66.5/s……… (2.2) This experiment have four millstones Position control system analysis. Analysis of closed loop speed control transfer function at different attenuator values by bode plot. Analysis of position control transfer function at different attenuator values by bode pot. Matlab simulation of obtained transfer function[1].
‐
‐
‐
11
2.2 Milestone 1: Position control system analysis Position control system is constructed as show in figure 2.2 by closing the feedback switch on the control unit. The low frequency square wave (+/-5volts) is provided as input from function generator. The value of velocity feedback attenuator (0.2, 0.4, 0.6) t he steady state error and maximum overshoots are measured by using oscilloscope. All the measured values are given in table 6[1].
Tachometer Feedback Gain 0.2 0.3 0.4 0.6 Figure 9.Closed loop position control[1]
Frequency (F) Hz 0.5 0.5 0.5 0.5
Steady State Error 0 0 -0.2 -0.3
Over shoo t (%) 5 1 1 0
Table 6.Vlaues of K2, SSE and overshoot (%)[1]
Discussion and analysis
We already discussed in experiment 1 that the velocity feedback gain in position control increase the damping without effecting the natural frequency. Similarly in this case it clearly seen the result obtained that increasing the gain the overshoot decreases. Steady state error remain zero, because it doesn’t affect steady state value. The small negative values mention in table are due to oscilloscope limitation which are small enough to be ignored.so it is proved again that velocity feedback for position control improve the response by adding additional damping. The application where high precision is required we use position control with velocity feedback in order to avoid oscillation. 2.3 Milestone2: Close loop speed control System identification In this milestone we are going to analyses the closed loop speed transfer function for identification of system by the help of bode plot at two different values of velocity feedback attenuator. In order determine the effect of attenuator values on system transfer function. Milestone (2.1): Close loop speed control system identification at attenuator (0.3) In this milestone we are going to use same system model that was constructed in previous milestone. Closed loop speed control is established by removing position Tachometer Function Sine Vin Vout Gain Phase feedback from comparator. The Feedback Generator Wave Shift value of attenuator used for Gain (K2) Amplitude Frequency velocity back is fixed as 0.3 and (F) Hz sine wave of frequency 0.1Hz and 0.3 1.0V 0.1 1.09 3.5 3.2 0 amplitude of 1.0V is provided as 0.3 1.0V 0.3 1.09 3.5 3.2 -0.5 reference input from function 0.3 1.0V 1 1.09 3.5 3.2 -2 generator. The output response is 0.3 1.0V 3 1.13 3.5 3.1 -10.8 observed on oscilloscope. 0.3 1.0V 10 1.13 1.9 1.7 -42 Magnitude and phase angle are 0.3 1.0V 30 1.13 1.9 1.7 -112 measured by oscilloscope. Table7.Value of K2, Amplitude, F, , ,Gain and Phase shift , , Same process is repeated for frequency values of 0.3, 1.0, 3.0, 10.0, 30.0Hz. All the results obtained are given shown in table 6 and bode plot of frequency against magnitude and phase angle are given in fig 10 and 11.
12
Bode Plot 4.0
e d u t i 2.0 n g a M
0.0 0
10
20
30
40
Frequency
Figure 10.Magnitude vs Frequency
Figure 11.Phase vs Frequency
We can determine the system transfer function of by using the general equation (2.1) of first order system. From bode plot of magnitude vs frequency the value of K LF=3.2 and ωc=6.284 rad/s and from equation (2.2) τm=1/ωc = 0.16.Putting these value in (2.1) system transfer function[1].
G(s)=19.38/ 6.25+S…………………………….(2.3) Milestone (2.2) Close loop speed control system identification at attenuator (1)
In this experiment we are going to use same procedure that we used in previous milestone for identification of system transfer function by using speed feedback but this time attenuator value is set to 0.1.All the measurement and the calculation regarding natural frequency and damping ratio are given
Tachometer Feedback Gain (K2)
Function Generator Amplitude
Sine Wave Frequency (F) Hz
Vin
Vout
Gain
Phase Shift
Oversho ot
1 1 1 1 1 1 1
1.0V 1.0V 1.0V 1.0V 1.0V 1.0V 1.0V
0.1 0.3 1 3 10 30 40
1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.99 0.99 0.98 1 1 0.59 0.42
0.99 0.99 0.98 1.00 1.00 0.59 0.42
0 0 -4 -6 -8 -74.9 -110
0 0 0.3 0 0 0 0
, , ,Gain and Phase shift, P.O
Table 8.Value of K2, Amplitude, F
in table 8 and the bode plot of the magnitude and phase against frequency are given in fig 2.6 and 2.7
We can determine the systems transfer function by using the general equation (2.1) of first order system. From bode plot of magnitude vs frequency the value of K LF=1 and ωc=75.4 rad/s and from equation (2.2) τm=1/ωc = 0.01333.
And putting these value in (2.1) system transfer function[1] G(s)1 =75.2/ 75.2+S……………………………. ( 2.4)
13
Figure 12.Magnitude vs Frequency
Figure 13.Phase vs Frequency
Discussion and analysis By looking at the results obtained from the experiment for closed loop speed control at different frequency values and attenuator value at (0.3) and (1) it is obvious that increase in the value of attenuator is directly affecting the magnitude and natural frequency of system. At low value of 0.3 the value of magnitude is high and natural frequency is small as compared to 1 because the attenuator is connected with feedback which is acting as derivative which tend to damp the amplitude of output, in other word output magnitude is decreased as compared to value at 0.3.due to which system reaches its steady state faster, which mean high natural frequency. Because the time constant of system is inversely proportional to natural frequency. System become faster rise time is decreased due to high value of natural frequency. In Matlab section we are going to prove these results by plotting step response. 2.4 Milestone3: System identification for Close loop position control In this millstone we are going to identify the close loop position control system with velocity feedback at two different at two different values of attenuator.in order to determine the impact of attenuator on system. Millstone (3.1) Close loop position control system identification at attenuator (0.3) We use the same system that was used in millstone (2.1.)In order to obtain the closed loop position transfer function we turn the feedback switch on to establish the closed loop position control.by keeping the value of attenuator at 0.3 parameter for closed loop transfer are calculated by using oscilloscope and given in table 9.
Tachometer Feedback Gain (K2)
Function Generator Amplitude
Square Vin Vout Gain Phase Wave Shift Frequency (F) Hz 0.3 3.0V 0.1 1.05 1.06 1.0 0 0.3 3.0V 0.3 1.05 1.06 1.0 -0.5 0.3 3.0V 1 1.05 1.05 1.0 -2.7 0.3 3.0V 3 1.05 1.03 1.0 -24 0.3 3.0V 8 1.05 0.88 0.8 -162 0.3 3.0V 10 1.05 0.55 0.5 -206 Table 9.Value of K2 (0.3), Amplitude, F, , Gain and Phase shift, P.O
,
Overshoot
0 0 0.1 3.3 2.4 0
14
The bode plot of magnitude against frequency and phase against frequency are given in fig 14 and 15[1].
Figure 14.Magnitude vs Frequency
Figure 15.Phase vs Frequency
Now with position feedback our System become 2nd order and we know the general equation for 2nd order system is G(s)=Kωn2/S2+2ζωnS+ωn2………(2.6) In order to obtained closed loop position transfer function we find the value of K DC gain damping ratio (ζ) and natural frequency (ω n) using equation 2.5, 2.6 and 2.7[2].
... (2.5)
DC gain=
= ∗− …… (2.6) ωn= ω-90 ……… (2.7)
M (0) =magnitude at
=0
M (-90) =magnitude when phase -90
ω-90=frequency at which phase plot is at -90
From bode plot the values of above parameters are M (0) =1.0
M (-90) =0.95
DC gain=1.122
ω-90=25rad/sec
=0.50
The closed transfer function by using equation 2.6 and putting values is
G(s) =701.3/S2+25S+625 Millstone (3.2) Close loop position transfer function attenuator at (1) In this experiment we are going to use the same procedure as we did in milestone (2.2) for closed loop position control but this time we are going to change the attenuator gain from 0.3 to 0.1 in order
Tachometer Feedback Gain (K2) 1 1 1 1 1 1
Function Generator Amplitude 1.0V 1.0V 1.0V 1.0V 1.0V 1.0V
Table 10 Value of K2 (1), Amplitude, F,
Square Wave Frequency (F) Hz 0.1 0.3 0.5 1 1.5 2
Gain
Phase Shift
1.01 1.01 1.01 1.01 1.01 1.01
0.95 0.92 0.64 0.62 0.6 0.4
0.94 0.91 0.63 0.61 0.59 0.40
0 -8 -21 -70 -85 -102
, , Gain and Phase shift, P.O
15
check the effect on transfer function all the measurement and calculation are given in table 2.5 and bode plot magnitude and phase against frequency are given in fig 16 and 17[1].
Figure17.Phase vs Magnitude
Figure 16.Magnitude vs Frequency
In order to obtained closed loop position transfer function we find the value of K DC gain damping ratio (ζ) and natural frequency ( ) using equation 2.11, 2.12 and 2.13[2].
DC gain=
... (2.11) M (0) =magnitude at jω=0
= ∗− …… (2.12) ωn= ω-90 ……… (2.13)
M (-90) =magnitude when phase -90
ω-90=frequency at which phase plot is at -90
From bode plot the values of above parameters are M (0) =0.99 DC gain=1.120
M (-90) =0.64
ω-90=9.43rad/sec
=0.520
The closed transfer function by using equation 2.6 and putting values is
G(s)1 =100/S2+10S+89 Discussion and analysis In these two experiments closed position control with velocity feedback is implemented at two different value of attenuator in order to account for effect of attenuator on system performance.
At low value of attenuator at 0.3 the value of damping ratio is smaller than at attenuator value of 1.Because this attenuator is used in speed feedback so it work as derivative gain. This value directly influence the performance of system increase in attenuator value decrease natural frequency due to increase in damping ratio mean system become slower due to increase in rise time of system. Increase in damping ratio cause decrease in percentage overshoot of system but settling time of system will be increased. These results will be validated in Matlab section.
16
2.5 Milestone 4 Simulation
Figure 17.Close loop speed Control Discussion and Analysis These results basically validate the results obtained theoretically. That increase in feedback gain for speed causes decrease in amplitude due to decrease in amplitude make system faster because the required value is smaller. Thus system natural f requency also increased.
Figure 18.Closed loops position step response Discussion and Results These results prove our theoretical results that increasing the value of feedback gain increases the value of damping ratio, make system slower and increase the settling time. Both the system r each the steady state value of 1.12.
17
Conclusion It is concluded that, first half deal with understanding of servomechanism, servomotor parameter identification, construction of basic close loop speed and position feedback systems and application of step response technique of system identification to get the transfer function of servo system. But due to limitation of step response in case of first order and system with no overshoot (position control with velocity feedback), this technique is not used for system transfer function identification, all the procedure implemented for understanding. In second half bode plots are used for system identification for both close loop speed and position. Same procedure is repeated for construction of close loop speed and position loops. By applying different frequency signal and at different values of velocity feedback attenuator transfer functions are calculated. These values shows the effect of change in attenuator on system transfer function. These changes in transfer function also effect system performance which are measured in term of settling time, rise time, percentage overshoot and steady state value. These transfer function plotted in Matlab and theoretical results are verified.
18
References [1] [2]
K. Adhakari, Servomechnisms and System Identification(SSI)( course notes). Newcastle: Newcaslte University, 2015. M. Armstrong, Digital Control System(course notes). Newcastle: Newcastle University, 2015.
19
Appendix Experiment 1: Milestone 1
Figure19.Deadband Voltage
Figure21.Time Constant at 2 Volts
Figure 20.Time Constant at1.5 Volts
Figure22.Time constant at -1.5 Volts
Figure22.Time constant at -2.0 Volts
20
Milestone 3
Figure23.System Response at K=1
Figure24.System Response at K=2
Figure 24.System Response at K=4
Figure25.System Response at K=8
Figure26.System Response at K=10
21
Milestone 4
Figure 27.System Response at K1=10 and K=1
Figure 28.System Response at K1=10 and K2=2
Fi ure 29.S stem Res onse at K1=10 and
Figure 30.System Response at K1=10 and K=8
Figure 31.System Response at K1=10 and K=10
22
Experiment 2: Milestone 2(1)
Figure 32.System Response Gain=0.3, F=0.1 Hz
Figure 32.System Response Gain=0.3, F=3 Hz
Fi ure 32.S stem Res onse Gain=0.3, F=1 Hz
Figure 32.System Response Gain=0.3, F=10 Hz
Figure 32.System Response Gain=0.3, F=30 Hz
23
