Lab Report 3_EELE3314

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3314L

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Apr 3, 2024

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Running head: EELE3314L LAB FOR CONTROL SYSTEMS 1 EELE3314L Lab for Control Systems LAB #3 Frequency Response Characteristics
2 EELE3314L LAB FOR CONTROL SYSTEMS Results Procedure 4.1 Measured Magnitude M at w = 1rad/s (dB) Calculated Kv =10^M/20 Calculated Kv = sim sG(s) -20.1 0.0989 0.0999 Estimated Ess = 1/kv Measured Ess from response plot 10 9.9 (approximately 10) Procedure 4.2 Required Ess Calculated Kv =1/Ess Calculated K 0.05 20 200.2 Measured Magnitude at w = 1rad/s (dB) Calculated Kv =10^(M/20) Measured Ess from response plot 25.9 19.7 0.05 Procedure 4.3 Bode plot Measured Mr (dB) Measured Magnitude at wr (rad/s) Measured Mr linear 5.28 12.8 1.8365 Calculated ζ Calculated wn (rad/s) 0.2839 13.9759 Calculated Mp (dB) Calculated Ts (s) 39.4494% 1.0081 Procedure 4.3 Response plot Measured Mp (dB) Measured Ts (s) 39.4% 0.994 Procedure 4.3 Root Locus Measured Mp (dB) Calculated Ts (s) (using measured values) 39.8% 1.017 Procedure 4.4 Measured Phase Margin (degrees) Measured Gain Margin (Linear) From Graph 89.4 From Matlab Command 89.3711 1101.1 Procedure and Analysis
3 EELE3314L LAB FOR CONTROL SYSTEMS 1. Unity Feedback System 1.1 Code for all of Part 1 1.2 Kv of Unity Feedback System
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4 EELE3314L LAB FOR CONTROL SYSTEMS From the Bode Diagram, the value for magnitude at 1rad/s was approximately -20.1dB. Shown in the calculations below, the Kv that responds to this value of -20.1dB magnitude is 0.0989 (approximately 0.1). This value was verified using the calculation as well and found to be nearly identical. From the Bode Plot Verify using Calculation
5 EELE3314L LAB FOR CONTROL SYSTEMS 1.3 Steady State Error of Unity Feedback System
6 EELE3314L LAB FOR CONTROL SYSTEMS In this part of the lab, the steady state error was found mathematically using the value of Kv and known equation for unit ramp response for a type 1 system, which is 1/Kv. Mathematically, the steady state error was found to be 10. In order to confirm the correctness of this estimate, the unit ramp response of the closed-loop system was plotted on the same graph as the input unit ramp waveform using MatLab. At steady state, the difference in values between the input unit ramp waveform and the unit ramp response of the closed-loop system will be equal to the steady state error. From the MatLab plot, this error was found to be 9.9, i.e. approximately 10, as mathematically expected. Mathematically obtaining steady state error Graphically obtaining steady state error in MatLab
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7 EELE3314L LAB FOR CONTROL SYSTEMS 2. Finding Control Gain K to Achieve Steady State Error of 0.05
8 EELE3314L LAB FOR CONTROL SYSTEMS 2.1 Code for all of Part 1 2.2 Mathematically Determining K and Confirming it Using the Magnitude Plot
9 EELE3314L LAB FOR CONTROL SYSTEMS Using the knowledge that 1/Kv is the unit ramp relation for steady state error, meaning that 1/steady state error is equal to Kv, the formula for Kv using the limit of the transfer function multiplies by s as s approaches zero was used to find K. This procedure is shown below and the value of K that met the required 0.05 steady state error was found to be 200.2. The corresponding value of Kv that was found mathematically was 20dB, which was verified using the value of gain at the frequency (omega) of 1rad/s on the magnitude plot. As it was difficult to read this value at exactly 1rad/s, the value at 1.01rad/s was used and found to be 25.9. The calculations were done to show that the Kv corresponding to this gain value was 19.7, which was very close to the value of 20 determined mathematically. Mathematically obtaining K for steady state error of 0.05 o First find Kv value o Then find K using known Kv and transfer function
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10 EELE3314L LAB FOR CONTROL SYSTEMS Bode plot for both K*Gp(s) and Gp(s)
11 EELE3314L LAB FOR CONTROL SYSTEMS o Calculating the Kv from the value of gain at 1rad/s on K*Gp(s) magnitude plot Using the value of 25.9dB read on the magnitude plot, when K = 200.2, the value of Kv was determined to be 19.7.
12 EELE3314L LAB FOR CONTROL SYSTEMS 2.3 Comparing the Bode Plots of K*Gp(s) and Gp(s) It is known from the theory that when the gain K of transfer function is increased, only the magnitude plot is impacted. An increase in K will result in a shift of the magnitude plot upward on the bode plot, so that it may begin its response at the value of K on the magnitude axis. For Gp(s) alone (i.e. a gain of 1), the starting point of the bode plot is 0dB, which corresponds to 20log(1). When the gain is increased to K =200.2, the new starting point of the magnitude plot is 46.03dB, which corresponds to 20log(200.2). While the starting magnitude in dB changes with an increase in K, the shape of the magnitude plot determined by the zeroes and poles of the transfer function are still unchanged (only the dB values at which these changes occur change). This phenomena can be observed in the bode plot figure above. As mentioned, the gain K only impacts the magnitude plot starting position, meaning the phase plot is identical for both K*Gp(s) and Gp(s). This is because K is a 0 th order term in the numerator and its associated phase on the phase plot is always 0 degrees, regardless of the value of K. As such, one would expect that the phase plot for K*Gp(s) and Gp(s) to be identical, which is precisely what is observed in the above figure.
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13 EELE3314L LAB FOR CONTROL SYSTEMS 2.4 Confirming the Steady State Error of 0.05 Using MatLab In order to confirm that the steady state error was indeed 0.05. The unit ramp response of the closed-loop system was plotted on the same graph as the input unit ramp waveform using MatLab. At steady state, the difference in values between the input unit ramp waveform and the unit ramp response of the closed-loop system will be equal to the steady state error. From the MatLab plot, this error was found to be 0.05, as mathematically expected. Below are the plots for the overall graph and a zoomed in version to show the difference between the two waveforms, which is indeed 0.05. Graphically obtaining input ramp waveform and ramp response in MatLab
14 EELE3314L LAB FOR CONTROL SYSTEMS Graphically representing steady state error in MatLab
15 EELE3314L LAB FOR CONTROL SYSTEMS 3. Percent Overshoot and Settling Time of the Unit Step Response of the Closed Loop System 3.1 Matlab Code for all Three Methods
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16 EELE3314L LAB FOR CONTROL SYSTEMS 3.2 Method One: Using Bode Plot and Formulas to Find Percent Overshoot and Setting Time Using MatLab, the bode plot of the closed-loop system with K = 200.2 was obtained. From the magnitude plot, the dB value at the peak of the magnitude plot corresponded to Mr and the frequency at this point corresponded to r; these values were found to be 5.28 dB and 12.8rad/s, respectively. The value of Mr was then converted to linear scale so that it could be used in the equations for damping ratio and natural frequency (Mr_linear = 1.8365). Using the formulas provided the values were found to be = 0.2839, n = 13.9759 rad/s, Ts = 1.0081 s, and %OS = 39.45% in MatLab, which were almost identical to the values determined when the calculations were done by hand. Graphically obtaining the value of Mr and the resonant frequency r
17 EELE3314L LAB FOR CONTROL SYSTEMS Using MatLab to Calculate Values of , n, Ts, and %OS Mathematically Finding Values of , n, Ts, and %OS to Confirm MatLab Results
18 EELE3314L LAB FOR CONTROL SYSTEMS
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19 EELE3314L LAB FOR CONTROL SYSTEMS
20 EELE3314L LAB FOR CONTROL SYSTEMS 3.3 Method Two: Using the Step Response to Find Percent Overshoot and Setting Time Using MatLab, the step response of the closed-loop system with K = 200.2 was obtained. From the step response plot, the different between the steady state value and the peak is the overshoot Mp; to convert this value to %OS (or %Mp), it is simply multiplied by 100%. The settling time is the time at which the output reaches within 2% (or 5% is another option, but Matlab uses 2% for settling time) of its final value. MatLab plots allow these values to be displayed on the step response as shown in the figure below. From this figure, percent overshoot was found to be 39.4% and settling time found to be 0.994s. These very slight discrepancies in values are likely due to the lack of precision in the values provided from the step response plot. Graphically obtaining the value of %Overshoot and Settling time from Step Response
21 EELE3314L LAB FOR CONTROL SYSTEMS 3.4 Method Three: Using the Root Locus to Find Percent Overshoot and Setting Time Using MatLab, the root locus of the open-loop system without K was obtained and from the root locus, the values for percent overshoot and settling time were found. It was impossible to get the cursor to point to exactly K=200.2 (shown in the first two figures below), so the value of K = 200 was used. Using this data at this gain value, the percent overshoot was found to be 39.8% and settling time (determined from the values of zeta = 0.281 and natural frequency = 14rad/s) was found to be 1.017s. These very slight discrepancies in values are due to the lack of precision in the values provided from the root locus plot. Graphically obtaining the value of %Overshoot and Settling time from Root Locus
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22 EELE3314L LAB FOR CONTROL SYSTEMS
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23 EELE3314L LAB FOR CONTROL SYSTEMS Calculating Settling Time As expected, all three methods obtained almost identical results for setting time and percent overshoot, with some small discrepancies due to precision and rounding error.
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24 EELE3314L LAB FOR CONTROL SYSTEMS 4. Step Response of the Closed-Loop System Using Gain Margin 4.1 Determining Gain and Phase Margin Using the initial system of Gp(s), with no gain Gc(s) included, the open-loop Bode plot was obtained. From the magnitude plot, the gain margin was found to be 60.8dB and from the phase plot, the phase margin was found to be 89.4 degrees. Using the MatLab command for margins, the gain margin and phase margin values were more accurately found to be 60.8365dB and 89.3711 degrees, respectively. This more accurate value for gain margin was used in the subsequent section. MatLab Code
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25 EELE3314L LAB FOR CONTROL SYSTEMS Graphically obtaining GM from Bode Plot for Open-Loop System without gain K o Converting dB to Linear
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26 EELE3314L LAB FOR CONTROL SYSTEMS Using MatLab Command to Get Gain Margin and Phase Margin Open-Loop System without gain K
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27 EELE3314L LAB FOR CONTROL SYSTEMS 4.2 Step Response of Closed-Loop System for Different Values of Gc(s) 4.2.1 Code for All Three Cases of Gc(s)
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28 EELE3314L LAB FOR CONTROL SYSTEMS 4.2.2 Step Response of Closed-Loop System for Gc(s) = GM The first case explored was when the gain Gc(s) was set to the value of the gain margin. In this case, the oscillations remained constant at the same amplitude as the step response oscillated to infinity. This response indicated that the system was on the exact point of stability and the poles were located exactly on the jw axis. The GM used to plot the step response below was the most precise value determined by MatLab. When the less precise value of GM was used, the system experienced a slight decrease in oscillations, which indicates that this GM value does indeed correspond to jw axis and when it varies slightly it moves off the axis. Step Response of Closed-Loop System for Gc(s) = GM
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29 EELE3314L LAB FOR CONTROL SYSTEMS 4.2.3 Step Response of Closed-Loop System for Gc(s) = GM - 1 The second case explored was when the gain Gc(s) was set to the value of the gain margin minus one. In this case, the oscillations were very large at the beginning but quickly decreased in amplitude as the system reached its steady state. This response indicated that the system was stable and the poles were located far in the left half plane. Step Response of Closed-Loop System for Gc(s) = GM – 1
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30 EELE3314L LAB FOR CONTROL SYSTEMS 4.2.4 Step Response of Closed-Loop System for Gc(s) = GM + 1 The third case explored was when the gain Gc(s) was set to the value of the gain margin plus one. In this case, the oscillations began to grow at t = 0 and continue to grow in amplitude as the step response oscillated to infinity. This indicated that the system was unstable and had poles located far in the right half plane. Step Response of Closed-Loop System for Gc(s) = GM + 1
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31 EELE3314L LAB FOR CONTROL SYSTEMS Conclusion The purpose of this lab was to study the relation between the frequency and time-domain characteristics of LTI systems, including the factors that impact the stability of a system. An additional objective was to learn more about bode plots and their related MatLab functions. During the lab, students explored various components of a system and how they impacted the frequency response of the systems. The system being modeled in the lab was a DC motor Gp(s) with a related controller Gc(s) that were connected in a unity feedback configuration. Three types of unity feedback systems exist: type 0, type 1, and type 2. These types correspond to the number of pure integrators in the forward path of the feedback system. Each type of unity feedback system has its own steady state errors and error constants associated with it. In this experiment, only type 1 systems were explored. In the first part of this lab, the relation between the bode plot of a type 1 open loop system and steady state error was explored. Given an intended steady state error value, designers can design a system that may achieve this intended error and test this using the bode plot of the open loop system. This is achieved using the velocity constant Kv, which is equal to the inverse of the steady state error of the system for its ramp response. By adjusting this Kv value in accordance with the intended error of the ramp function, the system can be designed to have specified error values. This Kv value is directly related to the bode plot, as the linear value for gain at the frequency of 1rad/s. This relation allowed design specifications for ramp error (an intended steady state ramp error of 0.05) to be confirmed during the experiment. Similar procedures can be performed to design a system with specific steady state errors for the unit step and unit parabolic inputs and their corresponding K values, however, in this lab only the unit ramp error was explored. Following the exploration of the bode plot, the various ways to determine the
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32 EELE3314L LAB FOR CONTROL SYSTEMS characteristics of a system (overshoot, damping coefficient, setting time etc,) using MatLab were explored. In this part of the lab, the relationship between the bode plot of the closed loop system, step response of the closed loop system, and root locus of the open loop system were explored. Interestingly, it was learned that the root locus is especially useful in controller design, as it enables designers to determine how the natural frequency, damping coefficient, and overshoot of a system is impacted when its value of K is adjusted. These characteristics determine the stability of the system and how it responds to different inputs, and as such are highly important in the controller design process. The final portion of the lab involved determining the relation between the gain margin and the stability of the system. It was found that the value of gain margin of the open loop system Gp(s) determined the regions of stability of the system for the closed loop system. When the gain K of the closed loop system is equal to the gain margin, the poles of the closed loop transfer function lie on the jw axis. When it is less than the gain margin, the poles of the closed loop system reside in the left half plane (system is stable) and when it is greater than the gain margin, the poles of the closed loop system reside in the right half plane (system is unstable). Practically speaking, the stability margins (gain and phase margin) are required in a system to leave room for potential deviations/uncertainties in the system. When there are lower margins, the system is less stable and the poles are closer to the right-half plane, which was indeed observed in the experiment. Overall, the performance of this lab was successful. The lab helped students to understand the factors that impact the stability of a system and how this impacts their frequency response. Furthermore, the lab introduced students to important functions of MatLab that will assist students in designing control systems for this class and general circuits for future courses.
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33 EELE3314L LAB FOR CONTROL SYSTEMS
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