` `
An open-loop transfer function for pointing control of a two-inertia
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 a DC servo motor which produces the torque needed
to rotate the satellite. The closed-loop system is stable with closed-loop
poles at s = -0.0199+/-j0.0688, -0.0840+/-j0.9876, -10.0004.

Uncompensated System Root Locus

` `
Specifications for the system are: (1) steady-state error for a ramp input
= 0.2; (2) settling time for a step input of approximately 40 seconds.

` `
The given system is Type 1. The velocity error constant for the given system
is 0.1276, so the steady-state error for a ramp input is 7.84. Therefore,
the compensator must reduce the steady-state by a factor of about 40 to
satisfy the error specification. The step response is too slow, so the
dominant closed-loop poles must be moved to the left.

Uncompensated System Step Response

` `
To achieve the settling time and steady-state error specifications,
a two-step procedure will be used. First, a phase lead compensator will be
designed to move the dominant branches of the root locus to satisfy the
settling time specification. Second, a phase lag compensator will be
designed to reduce the steady-state error to the required value.

` `
To achieve a settling time of 40 seconds for the step response, the real
part of the dominant closed-loop poles should be at approximately s = -0.1.
The ratio of the imaginary part of the dominant pole to the real part will
affect the percent overshoot. Although no specification is given on
overshoot in this problem, the point s = s_{1} = -0.1 + j0.1 is chosen. The
root locus must be made to go through this point, and then this point must
be made a closed-loop pole.

Exploded View of Uncompensated Root Locus

` `
The phase angle of the given satellite model at the selected point s_{1} is
104.9 degrees. Therefore, for s_{1} to be on the root locus, the compensator
must have an angle of +75.1 degrees at the point s_{1}. The compensator zero
must add more than this amount since the compensator pole will subtract its
phase angle from that of the zero. The zero is placed directly below the
chosen point, z_{c_lead} = -0.1. The choice of location for the compensator
zero is somewhat arbitrary -- the normal rule of thumb would place it at or to the left of the plant pole
at s = -0.04 (so that the chosen point s_{1} will represent a dominant
closed-loop pole) and yet provide an angle greater than 75.1 degrees at s_{1}.
With the chosen zero location, the angle from the zero to s_{1} is 90
degrees, so the compensator pole must provide an angle
of 14.9 degrees at s_{1}. The distance to the left of s_{1} for the pole to
give that angle is 0.376; therefore, the compensator pole is located at
s = -0.476. The lead compensator's gain must make the magnitude of the
given system in series with the lead compensator equal to 1 at the point s_{1}.
The magnitude at that point is 0.0799, so the lead compensator gain is 12.5101. The closed-loop poles are at s = -0.1+/-j0.1,
-0.3402, -0.0721+/-j0.9599, -10.0003. This system is stable and satisfies the settling
time specification.

Lead Compensated System Root Locus

Lead Compensated System Root Locus -- Exploded
View

Lead Compensated Step Response

` `
The purpose of the lag compensator is to increase the effective gain at
the point s = s_{1} without changing the root locus at that point or
changing the dominance of that point. The ratio of compensator zero to
pole is the amount by which the error must be reduced. The lead-compensated
system has a velocity error constant K_{v} = 0.335, so the steady-state error
is 2.985. The lag compensator must reduce that by a factor of 14.9248 in
order to meet the specification. The value of * alfa* for the lag
compensator is equal to 14.9248. The compensator zero is chosen to
the right of the point s_{1} by a factor of 100, and the pole is to the right
of that by the factor of *alfa*. Their locations are s = -0.001 and
s = -6.7003e-05, respectively. The closed-loop poles are located at
s = -0.10028e-03, -0.0992+/-j0.1004, -0.3408, -0.0721+/-j0.9599, -10.0000. The
velocity error constant of the system is now 5, so the steady-state error
specification is satisfied, and the step response is virtually unchanged
from the lead-compensated response. The effect of the closed-loop pole at
s=-0.10028e-03 is almost cancelled by the compensator zero at s = -0.001, and
the other closed-loop poles are very close to their previous locations.
The point s_{1} is no longer a closed-loop pole, but the movement of that pole
is negligible.

Lag-Lead Compensated Root Locus -- Exploded
Once

Lag-Lead Compensated Root Locus -- Exploded
Twice

Lag-Lead Compensated Step Response

` `
The steady-state ramp error for the given system is approximately 8 and for
the lead-compensated system it is approximately 3. Both of those systems
are in steady-state long before the final lag-lead-compensated. For the lag-lead compensated system,
the settling time in the ramp response is long due to the
very low frequency pole. It takes several thousand seconds for the
lag-lead compensated system to reach steady state.

Original, Intermediate, and Final Ramp Responses

The model for this example was adapted from an example
in *Feedback Control of Dynamic Systems*, 2nd Edition, by G.F.
Franklin, J.D. Powell, and A. Emami-Naeini, Addison-Wesley, 1991 using
certain numerical values from the Hubble Space Telescope.

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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