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Module 2Analysis of continuous time systems

1. Time domain solution of first and second ordersystem.

2. Steady state error

3. Concept of stability - Routh- Hurwitz techniques.4. Bode diagrams. Concept of phase margin and gain

margin

5 Root locus, polar plots and theory of Nyquist

criterion.

6. Theory of lag, lead and lag-lead compensators.

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Typical Test Signals.

The commonly used test input signals are step functions rampfunctions, acceleration functions, impulse functions, sinusoidalfunctions, and white noise.

With these test signals, mathematical and experimental analysesof control systems can be carried out easily, since the signals arevery simple functions of time.

In analyzing and designing control systems, we must have a basisof comparison of performance of various control systems.

This basis may be set up by specifying particular test input signalsand by comparing the responses of various systems to these inputsignals.

Many design criteria are based on the response to such test signalsor on the response of systems to changes in initial conditions

(without any test signals). The use of test signals can be justified because of a correlation

existing between the response characteristics of a system to atypical test input signal and the capability of the system to copewith actual input signals.

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Which of these typical input signals to use for analyzingsystem characteristics may be determined by the formof the input that the system will be subjected to mostfrequently under normal operation.

If the inputs to a control system are gradually changingfunctions of time, then a ramp function of time may bea good test signal.

Similarly, if a system is subjected to suddendisturbances, a step function of time may be a goodtest signal;

for a system subjected to shock inputs, an impulsefunction may be best.

Once a control system is designed on the basis of testsignals, the performance of the system in response toactual inputs is generally satisfactory.

The use of such test signals enables one to compare the

performance of many systems on the same basis.

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Transient Response and Steady-State Response.

The time response of a control system consists of twoparts:

(a) transient response(b) steady-state response.

By transient response, we mean that which goes fromthe initial state to the final state.

By steady-state response, we mean the manner inwhich the system output behaves as t approachesinfinity.

The system response may be written as:

the first term on the right-hand side of the equation isthe transient response.

the second term is the steady-state response.

)()()( tytyty SStr

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FIRST-ORDER SYSTEMS

Consider the first-order system shown in Figure as

shown by following Block diagram:

The input-output relationship is given by:

The initial conditions are assumed to be zero.

)1(1

1

)(

)()( TssR

sYsCL

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Unit-Step Response of First-Order Systems

Since the Laplace transform ofthe unit-step function is 1/s, substituting R(s)=1/sinto (1), we get:

Expanding Y(s) into partial fractions

gives:

sTssY 1

1

1)(

)2(1

11

11)(

T

ss

TsT

ssY

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Taking the inverse Laplace transform of (2), we

obtain:

Equation (3) states that initially the output y(t) is

zero and finally it becomes unity.

One important characteristic of such an exponentialresponse curve y(t) is that at t = T, the value of y(t)

is 0.632, or the response y(t) has reached 63.2% of

its final value. This may be easily seen by substituting t = T in y(t)

in (3).

Time t = T is called as Time Constant of the system.

)3(0,1)(

tforety T

t

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the smaller the time constant T, the faster the

system response.

Another important characteristic of the exponential

response curve is that the slope of the tangent line

at t = 0 is 1/T, since:

The output would reach the final value at t = T if it

maintained its initial speed of response.

From (4), we can see that the slope of the response

curve y(t) decreases monotonically from 1/T at t = 0

to zero at .

)4(11

)(00

TeTtydt

d

t

T

t

t

t

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The exponential response curve y(t) given by (3) isshown in Figure:

In one time constant, the exponential response curvehas gone from 0 to 63.2% of the final value.

In two time constants, the response reaches 86.5% ofthe final value.

At t=3T, 4T, and 5T, the response reaches 95%, 98.2%,

and 99.3%, respectively, of the final value.

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for t = 4T, the response remains within 2% of the

final value.

As seen from (3), the steady state is reached

mathematically only after an infinite time.

In practice, however, a reasonable estimate of the

response time is the length of time the response

curve needs to reach and stay within the 2% line of

the final value, or four time constants.

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Unit-Ramp Response of First-Order Systems.

Since the Laplace transform of the unit-ramp

function is:

we obtain the output of the system as:

Expanding Y(s) into partial fractions gives

Taking the inverse Laplace transform of (5), we

obtain:

2

1

)( ssR

1

11)(

2

Tss

sY

)5(1

1)(

2

2

Ts

T

s

T

ssY

)6(0)(

tforTeTtty T

t

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The error signal e(t) is:

As t approaches infinity, approaches zero, and

thus the error signal e(t) approaches T.

The unit-ramp input and the system output areshown in Figure

)1(

)()()(

T

t

T

t

eTTeT

tytrte

T

t

e

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The error in following the unit-ramp input is equal

to T for sufficiently large value of t.

The smaller the time constant T, the smaller the

steady-state error in following the ramp input.

Unit-Impulse Response of First-Order Systems.

For the unit-impulse input, R(s)=1 and the output of

the system of can be obtained as:

The inverse Laplace transform of (7) gives

The response curve given by (8) is shown in Figure:

)7(1

1)(

TssY

)8(01

)(

tforeT

ty Tt

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An Important Property of Linear Time-InvariantSystems.

For the unit-ramp input, the output y(t) is:

For the unit-step input, which is the derivative of

unit-ramp input, the output y(t) is:

0)(

tforTeTtty Tt

0,1)(

tforety Tt

Fi ll f th it i l i t hi h i th

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Finally, for the unit-impulse input, which is thederivative of unit-step input, the output y(t) is:

Comparing the system responses to these three inputsclearly indicates that the response to the derivative of

an input signal can be obtained by differentiating theresponse of the system to the original signal.

the response to the integral of the original signal canbe obtained by integrating the response of the system

to the original signal and by determining theintegration constant from the zero-output initialcondition.

This is a property of linear time-invariant systems.Linear time-varying systems and nonlinear systems donot possess this property.

0

1

)(

tforeTty T

t