Introduction This lab deals with the concept of frequency response. During steady state, sinusoidal inputs to a linear system will generate sinusoidal responses of the same frequency, but of different amplitude and phase angles. This is due to a time lag between input and output generated. These differences are functions of frequency. In this experiment, the frequency response test is used to identify the (linear) dynamics of a plant. It is performed by inputting a sinusoidal signal and comparing it with the sinusoidal output.

Objective of Experiment
1. To perform a frequency response test on an aircraft electro-hydraulic servo-actuator, hence determining the phase and gain margins of the servo-actuator.
2. To verify that increasing the gain by the gain margin causes instability.
Equipment and Apparatus 1. Electro-hydraulic servo-actuator 2. Amplifier unit 3. Oscilloscope (CRO) 4. Function generator (FG) 5. Hydraulic power supply (outside laboratory)

Background of Experiment This experiment utilizes a servo mechanism which is an automatic device that takes advantage of error-sensing feedback to correct and amplify the signal. The servo system comprises of a servo-actuator and an amplifier unit. In aircrafts, a similar system is used to convert electrical autopilot voltage outputs into a proportional mechanical movement. The mechanical movement is linked to the control rods, which controls the equipment the pilot uses to move the plane parts. When the output position is changed, the time delay should be minimal, but the response should not oscillate. These specifications usually determine the gain (or attenuation) and phase angle at some frequency. The motor is driven in a push-pull manner and has a continuous current flowing through it with a linear variable differential transformer (LVDT) with it. The LVDT is an electrical transformer used to measure linear displacement. When an error signal is detected, the displaced LVDT core causes current to increase in one coil and to decrease in the other: The net differential current causes the motor to move. The servo-actuator consists of a servo motor, a valve spool, and actuator and a linear potentiometer.

As the motor reacts based on the input signal, it tends to block one nozzle and unblock the other. This causes an increase in oil pressure behind the former, and a decrease behind the other. This differential is applied at the ends of the main valve spool that consequently moves a distance proportional to the electrical/mechanical input. As the spool moves, pressurized oil is supplied to one chamber of the main actuator, while oil in the other flows back to the return. The piston rod of the actuator moves and this movement is converted to a voltage by the potentiometer.

The summing amplifier receives the error between input and output and the error voltage is applied to the power amplifier.

[pic]
Summary of Experimental Procedure 1. We set sine wave frequency at 0.5 Hz, and fixed the amplitude of the function generator output at 5V peak-to-peak for the whole experiment.

2. We opened the hydraulic supply valve to allow the actuator to move in and out.

3. We measured and recorded the frequency (, voltage input Vi, voltage output Vo and (time in Table 1. The data was used to derive the gain and phase angle. We then repeated measurements for increasing frequencies of 1Hz.

4. We used the data to plot on the Nichols chart at each frequency, and read off the gain and phase margins.

5. We reset the function generator for 5V peak-to-peak square wave at about 0.5 Hz. We gradually increased the loop gain K by increasing the summing amplifier feedback resistor, until the system reached the point of instability. We converted K to (dB) using Gain = 20 log10 K, and compared this value of gain margin with that obtained from the Nichols chart.

6. From the Nichols chart, we read off the values of the open loop transfer function in Table 2. We used the data to draw the Bode gain plot and Bode phase plot on the same graph. From the resultant graph we estimated the order of the system and its time constants and discussed our results.

Experimental Data
[pic]
[pic]

Calculation:
For Table 1:
Sample : Frequency, f = 5.0 Hz Input voltage, Vi = 5.2 V Output voltage, Vo = 1.44 V Time, Δt = 0.062s

• ω (rad/s) = 2πf = 2π x 5.000 = 31.42 rad/s

• Closed-loop Gain (dB) = 20 log (Vo / Vi) = 20 log (1.44/5.2) = -11.15 dB

• Phase angle (°) = - Δt x f x 360° = - (0.062) x 5 x 360° = -111.6°

From Nichols Chart For Table 2:

[pic]
Sample : Frequency, f = 5.0 Hz Input voltage, Vi = 5.2 V Output voltage, Vo = 1.44 V Time, Δt = 0.062s • Corresponding values of gain and phase is obtained from Nichols chart: Open loop gain = -11.8 dB Open loop phase = -123°

• Theoretical Gain margin (open loop) =Unity – Frequency at which loop gain phase is -180° =0 – (23) = 23 dB

• Open loop phase margin = (Phase angle at which open loop gain is precisely at unity value of 1.0 dB) - 180° = (-95o) - 180° = 85o

• Experimental Gain margin: Gain, K at which the system becomes unstable = 56 Experimental Gain margin = 20 log(56) dB = 34.963 dB

-From Bode Plot: • Gain (open loop) = 20 dB 20log K = 20 K = 101 = 10 • Corner frequency, ( = 42 rad/s • Time Constant, ( =1/ω= 1/42=0.024 s

Based on the shape of the graph, we incorporated the graph 1/s with 1/(τs+1) so that the resultant graph would be of the shape we observed. Open loop transfer function,

⇒[pic]

Graphs Attached
-Nichols Chart
Close loop Gain(dB) against Close loop Phase(o)

-Bode Plot
Open loop Gain(dB) against ((rad/s)
Open loop Phase(o) against ((rad/s)

Discussions and Questions 1. Frequency response of system as input frequency increases

From the experiment data, we can see that as the input frequency increased from 0.5 Hz to 15 Hz with the input voltage (Vi) being constant, V0 and Gain decreased. We have obtained the transfer function[pic].

The s in the function is related to the input frequency ω. To yield the frequency response behavior of the system, we just replace the s with ωj to simulate a harmonic input. We get[pic]. We can see clearly that as input frequency increases, H(s) will decrease. It means the gain will decrease. Because Vi is constant, Vo will also decrease.

We also can think about it in another way. For a sinusoidal input voltage, the system needs time to respond. When Vi arrives its peak value, the Vo hasn’t reached its peak value. And Vi decreases immediately before Vo can reach its peak value. And the higher the input frequency, the more the response is delayed. Thus as ω increases, Vo decreases, and the same applies to the Gain, for Gain = 20log(Vo / Vi ). Meanwhile, because the delay gets larger and larger, the phase decreases and approaches -180o.

2. System response as gain increases

Increasing the gain causes instability. A higher gain constant K causes the output to overshoot the desired value. As we increased the gain constant, the output sinusoid on the oscilloscope showed to be more and more jagged and not smooth until the critical point when it became completely jagged. In other words, when the gain constant is increased, the output overshoots, sometimes even to a greater extent, and then it oscillates about its desired value for a longer time, causing the system to destabilize. There is a critical value for the gain constant, above which the system output will fluctuate continuously about the desired value without settling down, e.g. any gain above this critical value makes the system unstable. We can estimate the critical value from the equipment directly or from the Nichols Chart. The critical value is also called gain margin. Gain margin is an indicator of whether a feedback system is stable or not. It indicates absolute stability and the degree to which the system will oscillate without limit given any disturbance. Thus we use gain margin to better gauge system response (stable/unstable) as gain increases. On a side note, phase margin indicates relative stability (the system’s tendency to oscillate during its damped response to an input change). The phase margin of 85° obtained shows the system is quite stable in response to inputs like step inputs since it’s close to the ideal phase margin of 60°.

3. Comparison of experimental and theoretical values of gain margin

The gain margin we get directly from the experiment is 34.96dB, which is more than the theoretical value of 23dB which we obtained from the Nichols Chart, by 52%. This great difference could be due to sources of errors.

The system is so sensitive that even a little change will result a huge difference. It is difficult to find the critical point acutely by manual operation. The division of reading on the control knob is so large and poorly calibrated that we can only roughly estimate the value off the markings, leading to us overestimating the gain margin. From the Nichols Chart, values obtained were based on estimations and approximations to plot the points. The Bode diagram graphs are sketched by hand, hence cannot match the accuracy of a computer-generated graph.

4. Transfer function and order of system

The system is second order if we take into account the experimental errors in this experimental context. But if we ignore the errors, we find it seems to be first order. In this experiment, the open-loop transfer function of the aircraft electrohydraulic servo-actuator is determined through the experimental approach. The corner frequency, ω, is the boundary in the system’s frequency response where the energy flowing through the system begins to be reduced/ attenuated. Our graph has one obvious gradient change (one turning point) and when we read off the Bode plot, we obtained one value for the corner frequency ω. Since ω =1/τ, and we notice the shape of the Bode plot seems to be a combination of the shapes of the graph 1/s as well 1/(τs+1), we guess the open loop transfer function to be: [pic] at face value. This equation seems to indicate that the system is a first order system since there is only 1 ω and 1 τ. Furthermore, for a first order system, at τ , the system will reach around 63% of is final value. In our experiment, τ occurs at -14.5dB which is [pic] of the final gain (as gain increases). This value is very close to 63%, which seems to indicate our system exhibits characteristics of a first order system. Additionally, we can obtain the inverse time function from G(s) using the Laplace transform tables. [pic] The system exhibits an exponential response typical that of a first order system. g(t) approaches a constant stable value of 10 as frequency increases. As frequency ω increases, the gain decreases (becomes more negative). Since we have learnt at the earlier discussion of this experiment that greater gain causes greater instability, as the gain decreases, the exponential term decays to zero thus the system stabilizes at 10. However, we realize there are a lot of errors encountered when plotting the Bode graph (as further mentioned below, please refer to below). This raises ambiguity as to whether there could possibly be more than one corner frequency obtained ideally. If there is more than one ω, there will be more than one τ, and the system will be second order or beyond. Hence we need to find some other ways to determine the system order. When we read off the bode plot, we find that the gradient of the Bode gain plot before and after the corner frequency lies between -20 and -40 dB/decade. This is characteristic of a second order transfer function! It hints to us that our G(s) is actually a second order system. For a first order system, the phase angle will be between 0° and 90°. However, the phase angle for our system drops below -90°, hence the system is second order or more. The relative degree of the system will be more than or equal to the number of multiples of -90 degrees achieved asymptotically at the lowest point on the system’s phase plot. From our Bode phase plot, we see that the lowest point plotted has a phase of -132°. However the phase plot gradient seems to be sloping downwards and is nowhere near a horizontal asymptote yet. Since the limitations of the Nichols Chart prevented us from plotting more points for the phase angle, we can only deduce that the horizontal asymptote must be beyond -132°. Thus, since the asymptote has to be at least at -180° or beyond, –(132+n)° is a multiple of -90° by at least 2, so the system is predicted to be second order. Also, because this system oscillates when it reaches a maximum gain value, it is second order. Hence, the actually G(s) function could be second order and of the form [pic] where τ2 is not known due to the inaccuracy of the Bode plot. Thus utilizing the Laplace transform tables, the resultant g(t) could be [pic] where ω1=1/0.024. The response is non-oscillatory, starting with g(0)=0, and ending with g(infinity)=10, the maximum gain.

Error Analysis 1. On the Nichols Chart, the divisions between the adjacent lines were not that clearly defined, thus we had to estimate the values of the open gain and phase. It then resulted in imprecise and inaccurate readings. As the Bode values were obtained from the Nichols Chart, the graphs we plotted for Gain, phase, and ω are therefore being affected as well. That led to errors in K and the transfer function at last. 2. On the Nichols Chart, the maximum closed-loop gain that could be plotted was -18dB, beyond that was out of range of the chart. This meant we could not plot the gain values for ω>50 rad/s, hence we could not plot the gain and phase values on the Bode graph fully, i.e. the gain and phase plot had to be terminated when ω hit around 50 rad/s. This incomplete information means we might miss out additional corner frequencies for the gain plot, or that we cannot find the second horizontal asymptote located at the lowest phase angle for the phase plot. This will limit the information the Bode plot provides to gauge whether the system is first or second order; hence we have to rely on other methods like theory or extrapolation instead. 3. The scale we used for the Bode gain plot was not a good choice. We used a scale ranging from -30 to 50 dB, even though eventually the Bode gain plot only spanned half the paper from -15.5 to 6.9 dB. Understandably this is so there is space made for the Bode phase plot above. Yet the Bode gain plot scale is compromised, it is too small so the gain vs ω graph could not be plotted with more precise values as would have liked, affecting the graph shape and results accuracy (such as the ωc value obtained). Solution would be to plot the Bode gain plot and Bode phase plot on 2 different graph papers. This way each graph could utilize a larger scale, and comfortably fit the whole graph sheet. So the readings off the graph like the corner frequency would be more accurate. 4. The frequency of ouput voltage shown on the oscilloscope was always fluctuating unlike the input voltage which was more or less fixed at 5.2V at varying frequencies. Since the values of Vi, Vo, △time was measured by hands and eyes, they might not be very accurate, especially Vo and△time, given human reaction time of 0.02s. 5. The control knob for gain is so sensitive but the division of reading on it was too large, i.e. poorly calibrated. We had to then estimate the values. 6. Different interpretation on instability leads to different readings of K value while turning the knob. As it was based on human judgment which is subjective, it may differ among different users and not reflect the true value. 7. Operating the actuator system for a long time tends to increases the heat generation by friction; the viscosity of the fluid in the hydraulic actuator system operating under pressure will affect the actual readings. 8. Wear and tear of equipment: The potentiometer might wear out due to friction since it works by a sliding contact, hence does not work optimally, affecting the way it converts the actuator movement into voltage output. The voltage outputs obtained by not be accurate as a result. 9. Vibrations in the external environment could possibly affect the measured electrical signals since the apparatus like the oscilloscope and its channels are quite sensitive. Being driven in a push-pull manner, the motor generated a lot of vibrational noise and this could affect our readings as a result. 10. Due to the limitation of the equipment, there were also some system errors that could not be avoided. 11. There is actually other time constant, which is the time delay to change the output position by adjusting on the machinery knobs. But it can be neglected because it is very minimal as stated in the lab manual. It is very small relative to the time constant of 0.023 we obtained hence can be neglected without affecting the accuracy of our result.

Conclusion At the end of the experiment, the phase and gain margins of the servo were calculated. Through the experiment, we experienced instability in the servo system upon increasing the gain. From the gain margin, the value of gain was obtained. Therefore, the objectives of the experiment are achieved.

However, large errors are expected when performing an open-loop frequency response test on the system. Therefore, the closed loop test, which can minimize the effects of external influences on the system’s output, is used to obtain an accurate closed loop frequency response for the system. It is only so that the open-loop frequency response can be obtained using the Nichols chart. The Bode gain plot and Bode phase plot for the open-loop system can then be constructed.

The open loop transfer function of the system is [pic]which is a second order function. We further gained awareness of the realistic constraints of the experimental techniques, such as using the Nichols Chart and Bode plot, because we only managed to find one time constant even though theoretically there should be another time constant to fit the second order function.

This experiment has not only given us the chance to have hands-on application from what we learnt in class, it has also deepened our understanding of the content. We had the chance to operate on the equipment and apparatus, at the same time learning how to obtain readings and data from them. We also learnt to read and plot complex level charts and graphs with multiple axes, like the Nichols Chart and Bode Graph respectively.

This lab was conducted in groups and it created the chance for each individual to work hand in hand towards a common goal, simulating a real working experience where engineers get to work together and rely on teamwork and cooperation to succeed. This experiment has truly enhanced our understanding on the subject and given us a good foundation on the topic when we work on this subject in the future as practicing engineers.