` `
An open-loop transfer function for attitude stabilization of a communications
satellite is shown below. The output of the transfer function
is the measured pointing angle of the satellite, and the input is the control
voltage for the reaction wheel which produces the torque needed to rotate
the satellite. The uncompensated gain crossover frequency is 0.031 r/s, and
the uncompensated phase margin is 21.2 degrees. The uncompensated phase
crossover frequency is 0.998 r/s, and the uncompensated gain margin is 20.1
(26.1 db).

Uncompensated System Bode Plots

` `
Specifications for the system are: (1) steady-state error for a ramp input
= 0.1; (2) gain crossover frequency approximately 0.05 r/s; and (3) phase
margin greater than 50 degrees.

` `
The given system is Type 1. The velocity error constant for the given system
is 0.1, so the steady-state error for a ramp input is 10. Therefore, the
compensator must provide a DC gain of 100 to satisfy the error specification.
With the compensator gain included, there are multiple gain crossover frequencies,
and the closed-loop system is unstable.

Uncompensated System with Compensator Gain Bode
Plots

` `
To achieve the gain crossover frequency and phase margin specifications,
a two-step procedure will be used. First, a phase lead compensator will be
designed to provide enough positive phase shift at 0.05 r/s to satisfy the
phase margin specification. Second, a phase lag compensator will be designed
to lower the resulting magnitude curve (given system + compensator gain +
lead compensator) to 0 db at 0.05 r/s.

` `
At 0.05 r/s, the given system (with and without compensator gain) has a
phase shift of -163.4 degrees. Therefore, the lead compensator must provide
43.4 degrees of phase shift at that frequency (50 + 10) - (180 - 163.4).
The value of * alfa* which produces that amount of phase shift is 0.1857.
The maximum phase shift will be added at 0.05 r/s, so the compensator zero
and pole are located at -0.0215 and -0.1160, respectively. Since the
magnitude curve has not been adjusted yet, the closed-loop system is still
unstable.

Lead Compensated System Bode Plots

` `
At 0.05 r/s, the lead compensated system has a magnitude of 39.2 db, so the
value of * alfa* for the lag compensator would be 91.2. Since this is much larger than the textbook
recommended value of 10 - 20, two stages of lag compensator will be used.
Therefore, * alfa* for the lag compensator is the square root of 91.2,
giving us an * alfa* of 9.55. The lag compensator zeros are placed lower in
frequency than 0.05 r/s by a factor of 20 (my choice when there are 2
stages, rather than a factor of 10). Therefore, each of the 2 compensator
zeros are at s = -0.0025, and each of the 2 poles is at s = -2.6177e-04.
This system is stable and satisfies all the specifications. The compensated
gain crossover frequency and phase margin are 0.05 r/s and 54.9 degrees,
respectively. The compensated phase crossover frequency and gain margin are 1
r/s and 3.43 (10.7 db), respectively. The steady-state error for a ramp input is
0.1, so all specifications have been satisfied. The total compensator is the
series combination of the lag and lead compensators that have been designed.

Final Lag-Lead Compensated System Bode Plots

` `
The step response for the lag-lead compensated system shows much smaller
overshoot, shorter settling time, and less oscillation. This makes the
response of the final system design much more desirable than that of the
uncompensated system.

Original and Final Step Responses

` `
The ramp response for the lag-lead compensated system shows much smaller
steady-state error than the given system, which was one of the
specifications. Steady state for the lag-lead compensated system has not
yet been reached in 1000 seconds. It takes approximately 3000 seconds
for the error to achieve its steady-state value.

Original and Final Ramp Responses

The open-loop model for this example was adapted from an
example in *Linear Control Systems* by C.E. Rohrs, J.L. Melsa, and
D.G. Schultz, McGraw-Hill, 1993.

Circuit models for implementing compensators such as these as electronic
circuits can be found in the text *Modern Control Engineering*, 3rd
Edition, by Ogata, Prentice Hall, 1997.

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*Lastest revision on
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