February 6, 1996 The main goal of the azimuth and elevation servos on the E.O. Smith Telescope is to reduce the amount of position error during tracking. Since the position command has relatively low acceleration and very low velocity during tracking, the response to these inputs becomes unimportant. Although, the response to disturbance inputs is very important, and is the driving factor in the design of the servos. The Telescope has been designed to operate with wind speeds up to 35 MPH, and it is desired to have an overall tracking error of no more than 0.05 arcseconds with winds approaching this speed. We can assume that the error due to torque disturbances from bearings and friction is negligible compared to that of winds that we may expect. Since the elevation axis has significantly less inertia than the azimuth, it will require a higher servo bandwidth for disturbance rejection.
Where J is the moment of inertia of the elevation axis, wn is the cutoff frequency of the low-pass filter, Kp is the proportional gain, Ki is the integral gain, and Kd is the derivative gain. The moment of inertia of the elevation axis is 1700 lbf-ft-sec^2. The value for Ki and Kd are computed for a given value of compensator break frequency, wpid, by calculating the real and imaginary components of the complex zero.
We will set the low-pass filter cutoff frequency to twice the desired servo bandwidth. The PID compensator break frequency will be set to 0.4 times the desired bandwidth. Then the proportional gain is adjusted until the magnitude response crosses zero db at the desired bandwidth. This results in an open-loop response with low phase margin. This is characteristic of a servo designed for steady state accuracy. If we use this method, the results will be consistent as we change the bandwidth.
The open-loop response of the system with a bandwidth of 20 Hz. is shown below:
The forward transfer function is simply the mechanical inertia.
The feedback path relative to the disturbance input includes the compensators and filters, power amplifier, motor torque factor, and drive ratio.
The closed-loop disturbance response becomes:
Multiplying the disturbance transfer function by the torque disturbance spectrum results in the servo error spectrum shown below.
Using the preceding model, the RMS error was calculated for a variety of wind speeds and servo bandwidths. The frequency range of the integral was 0-100 Hz for each case. The following table and graphs summarize the results:
RMS error (arcseconds)
BW(Hz) 15 MPH 20 MPH 25 MPH 30 MPH 35MPH
10 0.110 0.195 0.305 0.439 0.506
15 0.037 0.066 0.103 0.149 0.203
20 0.017 0.030 0.048 0.069 0.092
30 0.005 0.010 0.015 0.024 0.032
40 0.003 0.005 0.008 0.012 0.016
The main reason why the EOST requires a higher bandwidth than most other existing telescopes is the fact that it is small in comparison. The WIYN elevation axis has 33.4 times more inertia than the EOST, and the WIYN is considered to be fairly small and light. A telescope axis with a lower moment of inertia will inherently have less disturbance rejection capability and will require higher servo loop gains to compensate for this. In order to achieve these high servo loop gains (specifically at low frequencies), the bandwidth must be increased. In fact, the servo bandwidth should be much higher than the frequency band of the disturbance that is being rejected. Another reason why the EOST requires such a high bandwidth is the fact that the telescope will be completely exposed to the wind. Most other telescopes are at least partially protected from the wind by enclosures.
There are several sources of uncertainty in this analysis. First, the wind torque model may be different at our location. It is possible that the air flow may be more laminar on top of the 30+ ft. tower than it is for the MMT which the wind spectrum is based on. Also, the servo model is an approximation and does not include notch filters or the effects of sampling. And, the PID break frequency is fixed at 0.4 times the bandwidth. The actual system will have more flexibility in the design parameters. But, the model does give a good estimate of bandwidth.
We have only been able to achieve an elevation servo bandwidth of 7 Hz with the telescope mounted on the temporary support stand in Bethel. The reason for this limited bandwidth is that there are mechanical resonances in the 30 to 100 Hz range. Adding filters to attenuate these resonances causes phase lag at the crossover frequency which limits the bandwidth. It is estimated that the lowest resonance should be at least four times the desired bandwidth.
We suspect that many of the resonances that we have seen are caused by the temporary stand. We hope that when the telescope is mounted on the concrete tower, most of the worst resonances will disappear. Unfortunately, we know of only two ways to determine the telescope's response on the tower. One is to perform a detailed finite element analysis on the whole telescope. (Only parts of the telescope currently have FEA models.) This would be costly and time consuming. The other is to wait until the telescope is mounted on the tower, and measure the response. If we then determine that there are resonances below 100 Hz, we can locate the causes and fix them with the telescope mounted on the tower. This is risky since there may be fundamental problems that would be difficult and expensive to fix. Also, it is often difficult to determine the cause of a resonance by measurement.