B_lecture6 Time-domain Specifications Automatic control System

Time-Domain Analysis of Control Systems

Introduction

Because time is used as an independent variable in most control systems, it is usually of interest to evaluate the output response with respect to time, or simply, the time response.

In the analysis problem, a reference input signal is applied to a system, and the performance of the system is evaluated by studying the system response in the time domain.

The inputs to many practical control systems are not

exactly known ahead of time.

In many cases, the actual inputs of a control system

may vary in random fashion with respect to time.

(For instance, in a radar-tracking system for anti-aircraft

missiles, the position and speed of the target to be tracked

may vary in an unpredictable manner).

It is difficult to design a control system so that it will

perform satisfactorily to all possible forms of input

signals.

Introduction

For the purpose of analysis and design, it is necessary to

assume some basic types of test inputs so that the

performance of a system can be evaluated.

By selecting these basic test signals properly, not only is

the mathematical treatment of the problem systematized,

but the response due to these inputs allows the prediction

of the systems performance to other more complex inputs.

In the design problem, performance criteria may be

specified with respect to those test signals so that the

system may be designed to meet the criteria.

Introduction

Typical Test Signals For The Time Response of Control Systems

Step-Function Input:

0 0

0 )(

t

tRtr

)(tr

t

R

0)()( tuRtror s

The step function is very useful as a signal since its initial

instantaneous jump in amplitude reveals a great deal

about a systems quickness in responding to inputs with abrupt changes.

)(tus is the unit-step function

Ramp-Function Input:

0 0

0 )(1)(

t

ttRttRtr

0 t

)(tr

Slope=R

The ramp function has ability to test how the system would

respond to a signal that changes linearly with time.


Parabolic-Function Input:

0 0

0 2)(1

2)(

2

2

t

ttR

ttR

tr

0 t

)(tr

The parabolic function represents a signal that is one order

faster than the ramp function.


1). Unit step response

s

ssRs1

ssLth

11

Typical Responses to Typical Test Signals of

Control Systems

2). Unit ramp response

2

1

sssRssCt

2

1 1

ssLtct

3). Impulse response

sssRssK 1 sLtk 1

Relationship between these responses

t

0

t

t20

1 1( ) ( ) ( ) , h(t) ( )

1 1( ) ( ) ( ) , c (t) ( )t

H s S K s k ds s

C s S H s h dss

dt

tdcssCsH

dt

tdhssHsK

tt

)(h(t) , )()(

)(k(t) , )()(

Typical Responses to Typical Test Signals of

Control Systems

{ ( ) , ( ) , ( ) }k t impulse response h t step response c t ramp responset

The Unit-Step Response

and Time-Domain Specification

Basic and macroscopically requirements to

design a control system:

The system must be stable (stability)First requirement.

The control should be accurate (accuracy).

The response should be quick-acting (rapidity).

For linear control systems,the characterization of the

transient response is often done by use of the response of

a linear control system when the input is a unit-step

function.



Many control systems are dominated by a second order

pair of poles. So look at time response (to a unit-step

input) of

2

2 22n

n n

C ss

R s s s

The response of a system could be : )()()( tCtCsC st

Transient portion and steady-state portion )(tCt )(tCs



The typical uint-step response of a second-order system

Percent overshot %100% max

ss

ss

y

yy

)(ty

t

maxy

ssy

0

overshoot

pt Peak time

Peak overshoot is important, both because it is a measure

(to a degree) of stability, and for practical reasons, overshoot

should be minimized (think of an elevator!).

For under-damped systems

Settling Time: The settling time is defined as the time

required for the step response to decrease and stay

within a specified percentage of its final value.

A frequently used figure is 5 percent.

)(ty

t0

00.1

95.0

st

05.1




Setting time is a measure of rapidity (quick-acting ) of the system.

)(ty

t0

00.1

50.0

dt

Delay Time: The delay time is defined as the time

required for the step response to reach 50 percent

of its final value.




Rise Time: For under-damped systems with an overshoot, the

rise time is defined as the time required for the step response

to rise from 0 to 100% of its final value. Rise time is a measure of rapidity (quick-acting ) of the system.

)(ty

t0

00.1

rt




Rise Time: If the system is over-damped, then the peak time

is not defined, and the 10-90% rise time is normally used. Rise time is a measure of rapidity (quick-acting ) of the system.

)(ty

t0

00.1

rt

10.0

90.0




)(ty

t0

sse

Steady-State Error: The steady-state error of a

system response is defined as the discrepancy

between the output and the reference input when

the steady state(t) is reached.



1sse c


Steady-State Error is a measure of accuracy of the system.



=2% or 5%


Each of the above parameters may be important in the

design of the control system.

B_lecture6 Time-domain Specifications Automatic control System

Documents

Transcript of B_lecture6 Time-domain Specifications Automatic control System