Telescope Axis Servo Control
D. Rowe 11/11/07
Linear Model
Figure 1: The control system model
Figure 1 shows a first-order linear model of a telescope axis servo control system. A motor having torque constant cτ and voltage constant cv drives an ideal speed reducer having reduction ratio N. The reducer drives the telescope axis without backlash. The motor plus reducer produce a torque on the axis given by
where i is the current into the motor,
.
Rm is the resistance of the motor winding, vm is the back emf of the motor due to its rotational velocity, and v is the output voltage of the PID controller. Since
where ωa is the rotational velocity of the axis, we have
(1)
Let Ia be the rotational inertia of the axis and τe be an externally-applied disturbing torque. Then the acceleration of the axis position angle, θa, is given by
(2)
The proportional-integral-derivative (PID) controller produces an output voltage given by
(3)
where θ0 is the desired axis position angle and θa is the actual position angle. The three PID coefficients, to be determined by the desired characteristics of the servo system, are a0, the proportional coefficient, a1, the derivative coefficient and a2, the integral coefficient.
Equations (1) through (3) form a third-order linear differential equation in the axis position angle, θa.
Frequency Domain Behavior
We analyze the system in the frequency domain by taking the Laplace transforms of (1), (2) and (3) and combining. After some algebra,
, (4)
where ω0 is the natural frequency of the servo loop and is given by
. (5)
δ is the damping coefficient, given by
, (6)
s is the normalized complex frequency,
, (7)
and
. (8)
Response Due to Externally Applied Torque
Equation (4) can be written
(9)
The squared modulus of H(s) is
(10)
where Ω is the normalized frequency, i.e.,
. (11)
Using equations (9) and (10), in Figures 2, 3, and 4 we plot the frequency response of the telescope’s axis to an applied external torque. In these graphs the x-axis is the excitation frequency normalized to the natural frequency of the loop. The y-axis is the transfer function in dB, relative to the last term in equation (9). Note that when γ is greater than zero the transfer function is greater than one is some part of the spectrum. From these graphs, a good compromise between peaking and low frequency suppression (DC offset) is found when δ = 1, and γ =0.2.
Example
The telescope has a rotational inertia of 10 kg-m^2 and the altitude axis has a loop natural frequency of 10 Hz. The loop is set up with δ = 1, and γ = 0.2. An external sinusoidal torque is applied at 1 Hz with peak-to-peak amplitude of 1 N-m. From Figure 3 we see that the normalized transfer function is -6 dB (one-half in amplitude). Combining this with equation (9), the peak-to-peak position error at the axis is 2.6 arcseconds, or approximately 1 arcsecond RMS.
Figure 2: Normalized transfer function for externally applied torque to axis position error. γ varies from 0.0 to 0.6 and δ is fixed at 0.7.
Figure 3: Normalized transfer function for externally applied torque to axis position error. γ varies from 0.0 to 0.6 and δ is fixed at 1.0.
Figure 4: Normalized transfer function for externally applied torque to axis position error. γ varies from 0.0 to 0.6 and δ is fixed at 1.4.
Servo Error Transfer Function
The servo error transfer function is given by
. (12)
The squared modulus of F(s) is
(13)
This transfer function is plotted in Figure 5.
Figure 5: Servo error transfer function. γ varies from 0.0 to 0.6 and δ is fixed at 1.0.
Current Quantization Jitter
Assume that the motor in Figure 1 is driven with finite current resolution, such as by a PWM output stage. The driver output current is quantized into Nq quantization levels from –full scale to +full scale. Let the motor torque be Tmax when the current is at full scale. Thus, the torque is discrete and quantized into = 2Tmax /Nq levels. Further assume that the output driver is sampled at rate fs.
Torque quantization causes noise to be injected into the servo system, which appears as position error at the axis. The quantization noise power spectral density (PSD) will appear uniformly distributed (white) over the loop bandwidth, as long as the sampling rate is much larger than the loop bandwidth. From sampling theory, the torque PSD is given by
(14)
In equation (14), the units for torque PSD are N-m/rt(Hz).
The torque noise is transferred to position error like any other externally applied torque. Equation (9) gives the transfer function. To find the RMS jitter due to torque noise, we multiply the PSD in equation (14) with the transfer function in equation (9) and integrate the position noise PSD over all frequencies up to the Nyquist frequency of the sampler. For the cases with low peaking (γ < 0.2) this integral is roughly equal to:
(15)
where is the RMS jitter in position at the axis, in radians.
Example
A 10-bit PWM current driver is updated 500 times per second. It drives a motor that has full scale torque of 30 N-m. The servo loop has a loop bandwidth of 10 Hz and the axis has rotational inertia of 10 kg-m^2. From (15) the jitter is ~ 0.012 arcseconds RMS due to the quantization error inherent in the PWM D/A converter. This is an approximation. The exact answer will depend on the details of the loop transfer function.
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