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7. CONTINUOUS CONTROL SYSTEMS

7.1 CONTROL 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.

		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)

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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.

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

7.1.1 PID 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).

	de
u = Kce + Ki∫ edt + Kd	-----
	dt

• 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
		d	----e
			dt

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e.g.

θ

= K

v

+ K

v

dt + K

dverror

gas

c

error

i∫

error

----------------

d

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		= K		v		+ K
	gas		c		error			----------------
							d	dt

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

7.1.2 Analysis of PID Controlled Systems With Laplace Transforms

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1. We can rewrite the control equation as a ratio of output to input.

θ

= K

v

+ K

v

dt + K

dverror

gas

c

error

i∫

error

----------------

d

dt

θ

gas

K

+ K

i∫

dt + K

d

------------ =

c

d

----

verror

dt

Then do a Laplace transform

d

→

s

dx

sx

----

----- →

dt

dt

∫ dt

=

1

∫ xdt

=

x

--

-

s

s

L

θ gas

=

Kc +

Ki

+ Kds

The transfer function

------------

----

verror

s

2. We can also develop a transfer function for the car.

	F = Aθ gas					= 10θ			gas
		F			= 10		Transfer function for engine and transmission. (Laplace
	---------
	θ	gas					transform would be the same as initial value.)
							d2x		dv
	F =				Ma =	M
							-------		= M-----
							dt	2	dt
	F	=			d
	--				M----
	v				dt
			v		1
	L	---			------			Transfer function for acceleration of car mass
			F		= Ms

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3. We want to draw the system model for the car.

θ gas		10	F		1	vactual
					------
					Ms

•The ‘system model’ is shown above.

•If θ gas is specified directly, this is called ‘open loop control’. This is not desirable, but much simpler.

•The two blocks above can be replaced with a single one.

θ gas		10	vactual
		------
		Ms

4.If we have an objective speed, and an actual speed, the difference is the ‘system error’

verror = vdesired – vactual

‘set-point’ - desired system operating point

5. Finally, knowing the error is verror, and we can control θ gas (the control variable), we can select a control system.

verror		Controller	θ gas

L		θ gas		= Kc	Ki	+ Kds
	------------				+ ----
		verror			s

*The coefficients can be calculated using classical techniques, but they are more commonly approximated by trial and error.