Broadly defined, controls are used to modify the behavior of a system. For example, an unstable system such as an inverted pendulum can be actuated upon to keep the pendulum upright. Another example is changing the behavior of already stable systems, such as damping a resonant mode in a pendulum system to keep the pendulum bob still.
A basic control system consists of the thing that will be controlled, often referred to as the
“plant”, a sensor that reports the current behavior of the plant, an actuator that can act on the plant, and a filter that will determine how to use the sensor signal to create an actuator signal. In the case of either type of pendulum, the pendulum itself is the plant, and one can sense the current angle of the pendulum relative to vertical, and actuate by moving the location of the support point in a horizontal direction. The control filters are used to determine how much to move the support point, and in which direction. Here I will briefly describe benefits and drawbacks of different kinds of controls, but for a more thorough introduction see, for example, [166].
Controls are used widely throughout the LIGO detectors. One of the most fundamental uses is to control the length of the laser cavities, to keep them within the linear range of the sensors, so that we can detect gravitational waves. They are also used to control the frequency and intensity of the main pre-stabilized laser, provide seismic isolation, control the angular motion of cavities, and in many other applications.
A.1 Feedback controls
Feedback is arguably the most common form of control system. A sensor is used to determine the current position (or velocity, or other variable) of the system, and is compared to a desired setpoint.
The difference between these is the error signal of the system. A control filter (in this thesis we only consider linear control filters, although in general they can be non-linear) is used to transform the error signal into a control signal. The control signal is fed to some kind of actuator that can affect the system. The system responds to this actuation, and the new position is determined by the sensor.
All of these components together make a “feedback loop”. Where the loop has high gain, the error signal will be forced to zero. Note that this nulling of the error signal implies that the loop will inject any noise from the sensor (e.g. electronics noise) into the loop.
Often, the sensor, actuator and plant are pre-determined and fixed, while the control filter must be chosen by the user. The total dynamics of the system must be “stable”. If the plant, sensor or actuator require a high-order set of equations to describe them, it can be challenging to create a control filter that stabilizes the system. On the other hand, feedback systems do not require exquisite knowledge of the fixed components of the system. A corollary to this is that feedback systems can be designed such that they are robust against small variations in the plant. Obviously this does not remain true if the plant changes drastically (such as a sign flip in the optical plant of the interferometers, as described in Section4.4).
For many systems, feedback control is appropriate, but since it requires that the disturbance pass through the system before it is sensed and actuated upon, it may not be the best solution for all systems. For example, if noise is non-linearly coupled into the system (like angular fluctuations of a mirror causing scattered light, which will couple back into other areas of the interferometer in potentially unknown ways), we prefer to eliminate the noise before the coupling can take place. In cases such as this, feedforward (as described in SectionA.2) may be a better choice.
A.1.1 Calculating effect of feedback
FigureA.1shows a very simple feedback system. The plant, or system dynamics (eg. pendulum) is labeledPin the green block. The sensor (eg. measurement of pendulum angle) is labeledSin the blue block. The actuator (eg. to move the pendulum) is labeledAin the red block. The control filter, Gis represented by the yellow block, and is what will be used to decide how to create an actuation signal from the sensor.
We will assume that we are only dealing with linear time-invariant systems, and that our analysis will be entirely in the frequency domain, where we can find that the effect of two systems in series is the same as the product of the two frequency responses.
If location ain Figure A.1 represents an input disturbance δx, location b will represent the residual noise after stabilizationδxs. Locationbis referred to as the error point of the feedback loop, and locationcis the control point of the loop.
To determine the effect of the loop, we want to calculateδδxxs. The traditional way of calculating this quantity involves writing out the equation for each summing node, and solving. Here,
δxs=δx+δxs(GAPS). (A.1)
Control Filter G
Actuator A Sensor
S
System Dynamics
"Plant"
P
a b c
d e
f
Figure A.1: Cartoon of how feedback control works. Sense disturbance after the plant, use a controller (designed by the user) to decide how to actuate on the plant.
We isolateδxs,
δxs(1−GAPS)=δx (A.2)
and then divide to find
δxs
δx = 1
1−GAPS, (A.3)
which is the canonical form for the closed loop effect of a feedback loop. While it is possible to solve more complicated systems (such as those shown in Figure4.12and Figure4.15) with this technique, it can become very burdensome.
Instead, we will recognize that we can equivalently write out a system of equations, where each equation relates one numbered location in the loop diagram to each adjacent location.
b=a+ f c=Gb d=Ac e=Pd f =Se
(A.4)
We can add a source term and write this system of equations in matrix form,
~
vsteady state=MT~vsteady state+~vin (A.5)
where
~ v=
a b c d e f
(A.6)
and
MT =
0 0 0 0 0 0
1 0 0 0 0 1
0 G 0 0 0 0
0 0 A 0 0 0
0 0 0 P 0 0
0 0 0 0 S 0
. (A.7)
Note that other derivations may writeMTas justM, but for consistency with Section3.1.2, we will call itMT(the transpose ofM). Solving EquationA.5for~vsteady state
~
vin , we find
~vsteady state
~ vin
=
1−MT−1
. (A.8)
This “transfer matrix” tells us the transfer function from any point to any other point in the loop (even if we do not have the ability to actually measure that particular transfer function). For example, to compare to EquationA.3, we can extract the [2,1] element from EquationA.8which will give us the transfer element from locationato locationb. This is
~
vsteady state
~ vin
[2,1]= 1
1−GAPS, (A.9)
which is the same result as the original method of calculating the closed loop transfer function.
While the system shown in FigureA.1 is simple to solve, this nodal matrix technique is easily extended to much more complex systems, such as those shown in Figure4.12or Figure4.15.
A.2 Feedforward controls
As discussed above, some systems may require a different type of control system to prevent dis- turbances from propagating through the system. Figure A.2 shows a basic block diagram of a feedforward system that can accomplish this. An external disturbance affect the plant through a coupling transfer function, but if we are able to witness that disturbance independently from the
plant, we can construct a filter to send through an actuator and cancel the disturbance’s effects before it affects the plant. Note that for some noises (such as quantum shot noise or radiation pressure noise) we do not have the ability to independently measure the noise, and so cannot apply feedforward to suppress those noise contributions.
Feedforward Filter W
Actuator A Sensor
S
System Dynamics
"Plant"
P Disturbance
Coupling Transfer Function
T
Figure A.2: Cartoon of how feedforward control works. The disturbance directly affects the plant, but it is also sensed by an external “witness” sensor. The output of the sensor is fed through a filter to the actuator, and cancels the effect of the disturbance before it arrives at the plant. Dynamics between the disturbance and the plant are the primary features that the feedforward filter must account for.
A challenge with the feedforward control topology is that it requires very precise knowledge of the system. In particular, one must very carefully measure the transfer function from the input of the actuator to the input of the plant, so that can be removed from the calculation of the feedforward filter. The ability to measure this transfer function can be a limiting factor in the ability of the feedforward system to subtract the noise. The error goes as
σ=
r1−C
2NC, (A.10)
whereCis the coherence of the measurement, andNis the number of averages in that measurement.
So, if we only measure the actuator to 10 %, we will only be able to subtract approximately a factor of 10 of the noise. If we want to subtract a factor of 100, we must measure to better than 1 % (assuming no other limitations, such as the noise of the witness sensor, become limiters).