Feedback and control theory

Many disciplines, from navigation and aeronautics to mechanical engineering, rely on control theory as a measure against deviations of the system from a desired state. Quantum optics makes no exception, and feedback loops are commonly applied to lasers and cavities to stabilize the frequency. Another application, more specific to optomechanics, involves the use of feedback to cool down a specific mode of oscillation of the resonator. This is known as feedback cooling, and it will be a central topic in Part II. In this section we focus on the basics of control theory [50] in order to encompass a broader class of systems, including for instance the cavity locking schemes discussed in the next section.

At the core of every system in control theory is the plant, which is the element we want to keep in a certain state. Internal dynamics or external elements may cause

+ + + 𝐾P 𝐾I 𝐾D 𝜀 𝐶 (c) 𝑦 𝑆 𝑃 𝐶 𝜀 − + 𝑟 (b) 𝑦 𝑃 𝐶 𝑟 (a)

Figure 2.8: Simple examples of control systems. (a) Open-loop control: the controller C

directly acts on the input to the plant P in order to obtain an output y as close as possible to the reference signal r. (b) Closed-loop control: some sensors S are used to feed back the output and compare it to the reference in order to create the error signal ". (c) The proportional-integral-derivative controller.

deviations from this state, and another module, the controller, is normally required to restore the desired conditions. In order to know how to act to bring the system closer to the target, rather than further away, it is essential to have the appropriate sensors to register the current state of the system. Control theory is represented well by block diagrams, where each block represents a part of the system (plant, control, etc.) and inputs and outputs are the measurable signals. In time domain the input is transformed into an output by a convolution operation, and in Fourier or Laplace domain this becomes equivalent to a multiplication. A good control should make the transfer function, given by the ratio of the output and the input of the total system, as close to unity as possible.

A relatively unsophisticated method of implementing control is outlined in Fig. 2.8a. The plant, P, has an output y that we would like to get as close to the reference r

as possible. The controller, C, uses the reference to change the input to the plant, and therefore the final output as well. This scheme makes no use of sensors, and relies on prior modelling of the system to implement control. The transfer function for this example is

y

r =P C, (2.94)

and it is straightforward to infer that a control acting like the inverse of the plant, i.e. C = P 1_{, would achieve the result sought. Unfortunately this simple, open-loop} method is useful only for very predictable systems, and is not robust against arbitrary swings that are not covered by the modelling.

A closed-loop feedback control, as in Fig. 2.8b, does not su↵er from the same issue. The key lies in the error signal, which is proportional to the di↵erence between the state of the system and the reference. The aim of the control system is to maintain the error signal as close to zero as possible. The outline of the setup includes some sensing devices, S, that send the information recorded back in order to create and update the error signal, which is obtained by subtracting the measured output from the reference. Feeding a non-vanishing error into the controller prompts a reaction that modifies the plant’s input. The changes applied through this negative feedback loop are expected to oppose the causes of the non-vanishing error in the first place, thus restoring the system to balance. With the output depending on the error asy=P C", and the error depending on the output as "=r Sy, the transfer function obtained by closing the loop is

y

r =

P C

1 +SP C. (2.95)

Even though this transfer function might seem harder to bring close to unity than Eq. 2.94, it represents a much better choice for most practical applications since there is no need to model the plant perfectly. Any disturbance, whether internal or external, is handled directly by the feedback.

There is a special type of controller that accounts for the vast majority of appli- cations because of its versatility: the proportional-integral-derivative (PID) controller (Fig. 2.8c). Starting from the error signal, the PID controller produces an output given by three terms proportional to the error itself and its integral and derivative over time:

(C⇤")(t) =KP"(t) +KI Z t 0 d⌧ "(⌧) +KD d"(t) dt . (2.96)

The reason for the presence of the proportional term is clear. If the error signal is di↵erent from 0, the controller has to act to restore the system with a strength proportional to the magnitude of the deviation. Responding only to what the error signal indicates at the present instant might not be enough, however. This is where the integral and derivative terms play their part. The integral of the error signal can detect patterns in the history of the feedback and is particularly sensitive to slow and periodic disturbances. Thanks to the accumulation over time, it is more sensitive to a constant o↵set than the proportional gain and it is therefore useful to dynamically compensate for possible deviations from the steady state. The derivative of the error signal, on the other hand, anticipates what the disturbances might be in the near

future. It can predict fast or sudden events, but it is rarely used because it could easily become unmanageable if the system is too erratic. The proportionality constants KP,

K_I and K_D designate the gains associated with each operation, and should be tuned independently to best account for the requirements of the system.