Lecture 5 - Notes

January 19th, 2016

Types of Control Systems

Automatic Controller an automatic controller compares the actual value of the plant output with the reference input (desired value) determines the deviation and produces a control signal that will reduce the deviation to zero or to a small value $e(t) \text { or } E(S) \text{ is the threshold of the actuating signal } \\\\ u(t) \text{ or } U(s) \text{ is the output of the controller }$

Element being controlled only has two states (typically on or off)
Simple and inexpensive so they are widely used
differential gap: The range through which the actuating error signal must move before the switching occurs
The thermostat of a room set at 20C might turn on at 18C and turn off at 22 having a differential gap of 4C
The basic premise is the system turns on when a threshold has been reached and off when the threshold reaches a desired value

$u(t) = U_{1}, \text{for } e(t) > 0 \\\\ = U_{2}, \text{for} e(t) < 0 \\\\ U_{1} \text{ is the ON value } \\\\ U_{2} \text{ is the OFF value}$

A linear feedback control system where the controller output is proportional to the error signal
Used when a system is a relatively long response time
- Will control a system better because the error can be dampened before there is an output
Is essentially an amplifier with an adjustable gain

$$ u(t) = K_{p}e(t) \\ \mathscr{L} \text{{$u(t)$}} = \frac{U(s)}{E(s)} = K_{p} \\ K_{p} = \text{proportional gain} $$

The value of the controller output u(t) is changed at a rate proportional to the actuating error signal e(t)
This process starts slowly but overtime it begins to really eliminate errors

$\frac{du(t)}{dt} = K_{i} e(t) \\\\ \text{ or } \\\\ u(t) = K_{i}\int{0}^{t}e(t)dt$

The controller output is proportional to the rate of change of the measurement or error
The controller output is calculated by the rate of change of the deviation or error with time

$u(t) = - k_{D} \frac{dEr}{dt}$

The above controllers can be combined (through addition) to be the below combinations