linear feedback controller analysis - 18.1

18. CONTROL SYSTEM ANALYSIS

Topics:

Objectives:

18.1INTRODUCTION

18.2CONTROL SYSTEMS

•Control systems use some output state of a system and a desired state to make control decisions.

•In general we use negative feedback systems because,

-they typically become more stable

-they become less sensitive to variation in component values

-it makes systems more immune to noise

• Consider the system below, and how it is enhanced by the addition of a control system.

linear feedback controller analysis - 18.2

		Control variable
INPUT					OUTPUT
				SYSTEM
(e.g. θ gas)					(e.g. velocity)
				(e.g. a car)

vdesired			verror		Driver or	θ gas		car	vactual
	+	_			cruise control

The control system is in the box and could be a driver or a cruise control (this type is known as a feedback control system)

Figure 18.1 An example of a feedback controller

Human rules to control car (also like expert system/fuzzy logic):

1.If verror is not zero, and has been positive/negative for a while, increase/decrease θ gas

2.If verror is very big/small increase/decrease θ gas

3.If verror is near zero, keep θ gas the same

4.If verror suddenly becomes bigger/smaller, then increase/decrease θ gas.

5.etc.

Figure 18.2 Rules for a feedback controller

linear feedback controller analysis - 18.3

•Some of the things we do naturally (like the rules above) can be done with math-

ematics

18.2.1PID Control Systems

•The basic equation for a PID controller is shown below. This function will try to compensate for error in a controlled system (the difference between desired and actual output values).

u = K	e + K	i∫	edt + K	de
				-----
c				d dt

Figure 18.3 The PID control equation

• The figure below shows a basic PID controller in block diagram form.

	proportional			PID Controller
V		Kp( e)			V
				+V
	integral		+
+	e	Ki( ∫e)		u
			+	amp	motor
-	derivative		+	-V
	K	d
		----e
		d dt
Figure 18.4 A block diagram of a feedback controller

linear feedback controller analysis - 18.4

e.g.

		dverror
θ		----------------
	gas = Kcverror + Ki∫verrordt + Kd	dt

Rules 2 & 3

Rule 4

(general difference)

(Immediate error)

Rule 1

(Long term error)

Kc

Ki Relative weights of components

Kd

This is a PID Controller

Proportional

Integral

Derivative

For a PI Controller

θ gas = Kcverror + Ki∫verrordt

For a P Controller

θ gas = Kc verror

For a PD Controller
dverror
θ gas = Kcverror + Kd	----------------
	dt

• The PID controller is the most common controller on the market.