Motor Modeling and Position Control Lab 3 MAE 334
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1 Motor ing and Position Control Lab 3 MAE 334 Evan Coleman April, 23 Spring 23 Section L9
2 Executive Summary The purpose of this experiment was to observe and analyze the open loop response of a DC servo motor. Before actually measuring the response of a motor, several simulations were performed using Simulink. Simulations of both open and closed loop systems were completed. It was found that the response of the open loop system was unpredictable and not always reminiscent of the input signal. The closed loop response was found to be a much better representation of the input signal. To model the open loop response of the motor, several different input signals were used. First, the motor was subjected to several step inputs each of a different amplitude. It was found that the motor responded very similarly to the Simulink open loop model. The velocity had a tendency to overshoot its target value. Next, the motor was subjected to several input signals of varying velocity. Such signals included a sine wave, square wave and sawtooth wave. Different amplitudes and frequencies were also tested. It was found that the motor s position could not be accurately recorded using an open loop system. The velocity overshoot was not a constant value; this caused the position to change at a rate that was different than the input signal. Because of this, there was no way to accurately determine the position of the motor. Methodology The first step of this experiment was to model the DC servo motor using Simulink. The transfer function of the system was input into Simulink and modeled as an open loop system to determine both the velocity and position responses. Next, the motor was modeled using a closed loop proportional derivative (PD) system. Unlike the open loop system, the closed loop system takes into account the feedback from the motor, in this case, velocity and position information. The motor measures this information using a tachometer. Next, the actual response of a DC servo motor was measured by subjecting it to an input from Simulink. A power supply, servo motor plant, and data acquisition board were all connected to a computer running Simulink. First, the motor was subjected to several step inputs of varying amplitudes. Both the position and velocity responses of the motor were recorded. Then the motor was subjected to several, more complex, signals. A total of six trials were performed. Two each of sine, square and sawtooth waves each; once with a low amplitude and frequency, and then with a slightly higher amplitude and frequency. Discussion of Results ing of An Open Loop System In an effort to model the DC servo motor using Simulink, the transfer function much first be determined. It was found to be Ω l (s) V m (s) = K τs + where K is the steady-state gain of the system which is the product of various properties of the motor such as the gearbox efficiency, motor efficiency, gear ratio and motor-torque constant and τ is the time constant of the system. The gain was determined to be.763 Volts and the time constant was.275 seconds. This transfer function was input to Simulink to model the velocity output of the system s ideal response as shown in figure. () 2
3 2 Simulated Step Response of Open Loop System (Velocity) Input: Voltage V m (t) Output: Angular Velocity ω(t) Figure : Velocity Response of As shown, the velocity of the motor begins at rad/s and increases to about.8 rad/s. Because this model is of an open loop system, the actual response of the motor cannot be used to fix its response. This is why the velocity of the motor overshoots the target velocity. The position of the motor can also be modeled as an open loop system. For this, the transfer function becomes Ω(s) V m (s) = K s(τs + ) The response using this transfer function was modeled with Simulink and is shown in figure 2. (2) 2 Simulated Step Response of Open Loop System (Position) Input: Voltage V m (t) Output: Angular Position θ(t) Figure 2: Position Response of This shows that the position of the motor increased mainly linearly, indicating a constant velocity. There is a small portion of non-linear behavior near t =, that is from when the motor s velocity increased from to its final value of about.8 rad/s. 3
4 ing of A Closed Loop System Using a closed loop system allows the error of the output to be used in correcting the input signal. A proportional-derivative controller (PD) is one way to do this. A PD controller provides an input that is proportional to the error of the output and its derivative. This means that the input to the motor will be corrected based on the error of its output. A new Simulink model was created to use a PD controller to model the theoretical response of a DC motor and different proportional (k p ) and derivative (k d ) coefficients were tested. The first test performed, as shown in figure 3, uses only the proportional part of the PD controller. Simulated Step Response of Closed Loop System (Position) Input: Target Position θ d (t) Output: Angular Position θ(t) Figure 3: Closed Loop Response of (k p =, k d = ) As shown, using only the proportional part of the PD controller, the response is already much closer to the input than the open loop response. In this example, the motor stops moving when it reaches its desired target position of rad while the open loop system kept moving, basically ignoring the input. It can be shown that the response is greatly impacted by the value of the proportional coefficient, k p. Figure 4 shows the response with a much greater proportional coefficient. Simulated Step Response of Closed Loop System (Position) Input: Target Position θ d (t) Output: Angular Position θ(t) Figure 4: Closed Loop Response of (k p =, k d = ) While this example has a much smaller time constant than the previous one, it greatly overshoots its target 4
5 position. The higher proportional coefficient also creates some oscillations in the system response. Overall, this example stabilizes around the same time as the previous example, so it would seem that the proportional coefficient does not affect that. A test was also run using only the derivative part of the PD controller as shown in figure 5. Simulated Step Response of Closed Loop System (Position) Input: Target Position θ d (t) Output: Angular Position θ(t) Figure 5: Closed Loop Response of (k p =, k d = ) As shown, the use of a derivative coefficient drastically decreases the time constant of the response. However, the lack of a proportional coefficient causes the system to never reach its target position. At this point, it is evident that both a proportional and derivative coefficient must be used to achieve the desired response. Figure 6 shows this. Simulated Step Response of Closed Loop System (Position) Input: Target Position θ d (t) Output: Angular Position θ(t) Figure 6: Closed Loop Response of (k p =, k d = ) The above response closely resembles the response of figure 3. It seems that the addition of a derivative coefficient reduces the amount of overshoot in the response. However, this example takes slightly longer to stabilize than that of figure 3. Figure 7 shows an example using both a higher proportional coefficient and derivative coefficient. 5
6 Simulated Step Response of Closed Loop System (Position) Input: Target Position θ d (t) Output: Angular Position θ(t) Figure 7: Closed Loop Response of (k p =, k d = 5) In this example the increased proportional coefficient does cause some overshoot, but that in combination with the increased derivative coefficient causes a much smaller time constant and drastically quicker settling time. These values seems to be the optimal coefficients for this type of system. Negative values of each coefficient were also tested and all seemed to show similar responses. Figure 8 shows a positive derivative coefficient and a negative proportional coefficient. Simulated Step Response of Closed Loop System (Position) Input: Target Position θ d (t) Output: Angular Position θ(t) Figure 8: Closed Loop Response of (k p =, k d = 5) As shown, the system simply fails with negative coefficients. This is because using a negative gain is essentially flipping the signs of the system. This causes the error of the output to be used to correct the input in the wrong direction, thus making the error even worse. Step Response of A A DC servo motor connected to a computer running Simulink was subjected to several step inputs of different step sizes. Figure 9 shows the response to a step input with a step size of.8. 6
7 Fitted Response Output Input Figure 9: Step Response of (Step Size =.8) The above figure shows the response of the model, the actual response, and the fitted response of the motor. As shown, the velocity of the motor oscillates quite a bit, but it does have a relatively short time constant and settling time. One thing to note is that with the absence of a PD controller, the motor never actually achieves its target velocity. Similar to the open loop theoretical response where the target velocity was overshot. The time constant of the above response was calculated in two ways. One, by calculating the time that it took to reach 63.2% of it s final value. And two, by using the curve fitting tool in MATLAB. The time constants were found to be.27 s and.4 s respectively. This is an error of 34.%. The gain was also calculated using two methods. The first method was to divide the steady state value by the step size and the second was to again use MATLAB s curve fitting tool. The gains were found to be.8833 and.888 respectively. Various characteristics of the response for several different step sizes are shown in table. Table : Motor Response Characteristics Experiment Number Step Input Amplitude [volts] Data Sampling Rate [Hz] Final Steady State Velocity, ω [rad/s] Estimated Gain from Time Series Data, K Estimated Time Constant from Time Series Data, τ [s] Gain from Curve Fitting Tool, K Time Constant from Curve Fitting Tool, τ [s] Time Constant from week model, τ [s] Estimated/Fit Time Constant Error, % Estimated/Fit Gain Error, %
8 As shown in the above table, the time constants for the experimentally determined data and the theoretically determined data from week are very close. They are nearly identical for a step size of.6, and for the other step sizes, it is only off by about. seconds. The plots for the step sizes not shown in this section can be found in appendix A. ing the Physical System Another way to measure the response of a DC motor is to subjected it to a signal with a varying angular velocity, ω. For this experiment, a motor was subjected to signals in the form of sine, square and sawtooth waves of various amplitudes and frequencies. A sample of these responses are shown in this section and the rest can be found in appendix B. Figure shows the motor s velocity response to a sine wave Figure : for a Sine Input (Amp = 5, Freq =.2 Hz) As shown, the motor responded very similarly to the model. The only difference is that when the motor changes direction, the actual response is somewhat slower than the model. This causes the motor s velocity to overshoot the target velocity. It would make sense that for a higher frequency, the motor would take longer to react to the change in input because the value changes much more quickly. This hypothesis is confirmed in figure 8
9 Figure : for a Sine Input (Amp =, Freq =.5 Hz) As shown, there is a larger gap between the model and response at the peaks than there were in the previous figure. This gap is also present in the angular position response shown in figure 2. 2 Angular Position Response Figure 2: Angular Position Response for a Sine Input (Amp =, Freq =.5 Hz) 9
10 The position response gives a whole new meaning to the overshoot that was observed in the velocity response. Because the overshoot at each peak is of a slightly different magnitude each time, this causes inconsistencies in the position reading from the servo motor. Since the velocity is not changing consistently or uniformly, the position response can no longer be trusted. This phenomenon has an effect similar to interference in that it gives the data a drift. The position seems to move farther past the target value with each oscillation. Velocity overshoot is also prevalent when subjecting the motor to other types of input signals, such as a square wave as shown in figure Figure 3: for a Square Input (Amp = 5, Freq =.2 Hz) As can be seen in the above figure, the velocity, once again, overshoots the target value. The overshoot is a characteristic of an open loop system. Since the motor cannot readjust its input signal based on the error of its output, as it would with a PD controller, the motor overshoots its target value. The drift is also prevalent in the position response of the square wave. A figure demonstrating this can be found in appendix B. Finally, the motor was subjected to a velocity input in the form of a sawtooth wave as shown in figure 4.
11 Figure 4: for a Sawtooth Input (Amp = 5, Freq =.2 Hz) The velocity overshoot is present here as well, but there is also another interesting characteristic of the sawtooth wave response. Whenever the motor switches directions (ω = ), it stops moving completely for a short period of time. It is unclear what caused this, but a possibility is an error in the Simulink model. Another possibility is that the static friction in the motor prevented it from moving right away. However, if this were the case, there would have been a similar instance of this in the sine wave response which there was not. Conclusions This experiment found that closed loop systems are much more preferable than open loop systems. Motors modeled using an open loop system only respond based on their input. Because of this, they have a tendency to overshoot their target values which can have other negative effects such as a position drift. Closed loop systems, on the other hand, have the ability to react to their output or the error in their output. They are then able to use this error to correct their input signal. It was found that using a PD controller with certain coefficients produced a response much closer to the input signal than the open loop system did. An open loop system that uses a varying signal as its input, such as a sine wave, can be subject to drift. This occurs because the motor s velocity overshoots its target value at each peak, and because this overshoot is not a constant value, the position of the motor is no longer accurate since its velocity is not being varied uniformly. In conclusion, open loop systems are inconsistent and prone to error.
12 Appendix A Motor Step Responses Fitted Response Output Input Figure 5: Step Response of (Step Size =.6).2 Fitted Response Output Input Figure 6: Step Response of (Step Size =.) Fitted Response Output Input Figure 7: Step Response of (Step Size =.2) 2
13 Appendix B Motor Signal Responses 2 Angular Position Response Figure 8: Angular Position Response for a Sawtooth Input (Amp = 5, Freq =.2 Hz) Figure 9: for a Sawtooth Input (Amp =, Freq =.5 Hz) Angular Position Response 4 Angular Position Response Figure 2: Angular Position Response for a Sawtooth Input (Amp =, Freq =.5 Hz) Figure 2: Angular Position Response for a Sine Input (Amp = 5, Freq =.2 Hz) 3
14 5 Angular Position Response 5 Angular Position Response Figure 22: Angular Position Response for a Square Input (Amp = 5, Freq =.2 Hz) Figure 23: Angular Position Response for a Square Input (Amp =, Freq =.5 Hz) Figure 24: for a Square Input (Amp =, Freq =.5 Hz) 4
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