control system

control system

The sole objective of

control system

is to generate feasible inputs
to the plant (e.g.

dynamic systems

) such that it will operate as it
intended to under a wide range of operating conditions.

Examples of control systems: Cruise control, auto pilot, rice cooker.

For a general finite dimensional dynamic system in its ODE form,

x

˙

=

f

⁢

(

x

⁢

(

t

)

,

t

)

+

g

⁢

(

x

⁢

(

t

)

,

u

⁢

(

t

)

,

t

)

,

(1)

y

=

h

⁢

(

x

⁢

(

t

)

,

t

)

,

where

x

∈

ℝ

n

is the state,

y

∈

ℝ

l

is the
output and

u

∈

ℝ

m

is the control input of the system.
In the control literature, equation

1

is general referred
as the plant, where the function

f

:

ℝ

n

×

ℝ

→

ℝ

n

governs the

system dynamics

,

g

:

ℝ

m

×

ℝ

n

×

ℝ

→

ℝ

n

determines how the input (control signals)
influence the state

x

via

actuators

(e.g. gas turbine) and

h

:

ℝ

n

×

ℝ

→

ℝ

l

determines how the state generates the output signal. If

m

is equal to

n

, the plant is

fully actuated

. If

m

<

n

then the plant is

under actuated

and otherwise the plant is

over actuated

. For a plant that is not
explicitly dependent in time

t

, such system is called a

Autonomous

system. The main
differece between a control system and a general dynamic system is
the additional signal

u

⁢

(

t

)

.

For example, to control an airplane, the control system has to control
the

thrust

,

flaps

,

aileron

and

rudder

, which they
are the control signals of the system

u

. Those control input
influence the system state

x

such as

speed

(with thrust),

attitude

and

orientation

(with flaps, aileron and rudder).
To physically alter the state of the airplane, actuators such as
gas turbines are needed, which are controlled by the control signals

u

.

The control signal

u

can be generated in a

closed-loop

fashion
or

open-loop

fashion. An open-loop control system generates

u

with the user, or operator supplied reference state

x

d

or output

y

d

only; meanwhile closed-loop control system uses both reference
and

feedback

signals that are usually measured from

sensors

. In the airplane attitude control example, the desired
attitude is usually represented in roll-pitch-yaw angle
representation, and these signals are measurable by attaching
sensors to the flaps, aileron and rudder. In engineering
practice, only closed-loop control systems should be used, since
open-loop systems are not

robust

against uncertainties, modeling
errors and measurement errors.

If a closed-loop control system is based on state feedback, such
contol system is called a

state-feedback

control system. By the
same token, a

output-feedback

control system is based on output
feedback only. Notice that output signals are available for feedback
by

definition

, however in reality not all the states are mesurable. If
a state-feedback control system with all the states available for
feedback, it is called a

full-state feedback

system and otherwise
is call

partial-state feedback

system, which usually requires a

state
observer

(e.g.

Kalman filter

) to estimate the unavailable states.

To illustrate the simple concept of control systems, we will use a
simple example. A truck driver is required to travel 1000 Km in 10
hours. To relive the stress on the driver’s heel, he has placed a stick
to the gas paddle so the car travels at

100

⁢

Km

/

h

. Under perfect
conditions, the driver will reach the destination in the allocated
time. However, a certain section of the road is up-hill, so the truck
slowed down by a considerable amount and will not arrive it’s
destination in time. To remedy this problem, the driver ’implemented’
a simple solution using the speed-o-meter such that the gas paddle position

p

s

⁢

e

⁢

t

of the truck is now depends on the current speed

v

c

⁢

u

⁢

r

⁢

r

⁢

e

⁢

n

⁢

t

of the truck,

p

s

⁢

e

⁢

t

=

-

K

⁢

(

v

c

⁢

u

⁢

r

⁢

r

⁢

e

⁢

n

⁢

t

-

100

⁢

Km

/

h

)

,
where

K

is just an adjustable

parameter

. So if the truck is

running

too slow (e.g. up-hill),

p

s

⁢

e

⁢

t

will be

positive

(more gas to the
engine) hence speed will increase to maintain the desired speed, so
vice-versa for the down-hill case.

In this example, we have outlined all the major components of a
typical control system:

•

Actuator: engine,

•

Sensor: speed-o-meter,

•

Plant: truck,

•

Control input: gas paddle,

•

Control objective: 1000 Km in 10 hrs,

•

Control law:

p

s

⁢

e

⁢

t

=

-

K

⁢

(

v

c

⁢

u

⁢

r

⁢

r

⁢

e

⁢

n

⁢

t

-

100

⁢

K

⁢

m

/

h

)

,

The science aspect of control systems is the study of
design, synthesis and analysis of control systems using mathematical
concepts, and the engineering aspect of controls systems is to
implement, construct and adjust the control system according to
real-life situation and limitations.

Title

control system

Canonical

name

ControlSystem

Date of creation

2013-03-22 14:22:59

Last modified on

2013-03-22 14:22:59

Owner

ppirrip (5555)

Last modified by

ppirrip (5555)

Numerical id

9

Author

ppirrip (5555)

Entry type

Definition

Classification

msc 93A10

Related topic

SystemDefinitions

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