1. Introduction
Control theory provides a systetmatic approach to design feedback loops that are stable and settle quickly to their steady state values.
In many computing systems, control theory is used widely in problem such as adjusting scheduling priorites, memory allocations, and network bandwith allocation and flow control and TCP/IP
2. Control Theoty Fundamentals
The magic of feedback: create a system that perform well from components that perform poorly.
— Karl Astrom
2.1. Notions
Because we are talking about computer system, so the time is discrete. Here are some symbols
- time: $k$
- reference input: $r(k)$, the desired value of measured output
- control error: $e(k)$, the difference between reference input and the measured output
- control output $u(k)$, the output of controller. This is a parameter in the target system
- disturbance input $d(k)$, any changes that affects the way in which the $u(k)$ influences the measured output
- measured output $y(k)$
- noise input $n(k)$ changes the measured output produced by the target system
2.2. Purpose
- regulatory control: ensure the measured output is close to the reference input
- disturbance rejection
- optimization: obtain the best value of the measured output
2.3. Good Properties
SASO
- stable
- accurate
- short settling time
- no overshoot
3. Example 1 IBM Lotus Domino Server
Goal: Control RIS in the server
- Model how MaxUsers (output of the controller) affects RIS
Construct the model by applying least squares regression to the data obtained from off-line experiments data. And the resulting model is
$$
y(k)=0.43 y(k-1)+0.47 u(k-1)
$$
Then use Z-transform to get the transfer function
$$
G(z)=\frac{0.47}{z-0.43}
$$
- Construct a controller, we want to the whole system’s transfer function has following properties:
- the pole of transfer function close to 0
- the stead state gain to be 1