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MAYURBHANJ SCHOOL OF ENGINEERING�ELECTRICAL ENGINEERING DEPARTMENT

AY: 2021 – 22

6TH Semester

Control System Engineering

Unit – 5

Time Response Analysis

Dr. Mrutyunjay Das

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INTRODUCTION

  • After deriving mathematical model of the system, analysis of its performance is needed.
  • The responses to various types of inputs are analysed to study different specifications e.g. Accuracy, Stability, Oscillations, etc.
  • Time response is defined as to how a system behaves in accordance with time when a specified test signal is applied.
  • It has two parts.
  • Transient response
  • Steady state response

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INTRODUCTION

  • Time response

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INTRODUCTION

  • Time response analysis is also called time domain analysis. Here, we study the response, i.e. the output as a function of time.
  • Total time response c(t) of a control system consists of transient response (dynamic response) ctr(t) and steady state response css(t).

c(t) = ctr(t) + css(t)

  • where    c(t) = total time response
  •           ctr(t) = transient response
  •         css(t) = steady-state response.

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TRANSIENT RESPONSE�

  • After applying input to the control system, output takes certain time to reach steady state. So, the output will be in transient state till it goes to a steady state. Therefore, the response of the control system during the transient state is known as transient response.
  • The transient response will be zero for large values of ‘t’. Ideally, this value of ‘t’ is infinity and practically, it is five times constant.
  • Mathematically, we can write it as

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STEADY STATE RESPONSE�

  • The part of the time response that remains even after the transient response has zero value for large values of ‘t’ is known as steady state response. This means, the transient response will be zero even during the steady state.

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STANDARD TEST SIGNALS�

  • Unit Impulse Signal
  • Unit Step Signal
  • Unit Ramp Signal
  • Unit Parabolic Signal

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UNIT IMPULSE SIGNAL

  • A unit impulse signal, δ(t) is defined as
  • δ(t)=0 for t≠0

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  • And 

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  • The following figure shows unit impulse signal.

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UNIT STEP SIGNAL

  • A unit step signal, u(t) is defined as
  • u(t)=1; t≥0

=0; t<0

  • Following figure shows unit step signal.

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UNIT RAMP SIGNAL

  • A unit ramp signal, r(t) is defined as
  • r(t)=t; t≥0

=0; t<0

  • We can write unit ramp signal, r(t) in terms of unit step signal, u(t)as
  • r(t) = tu(t)

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UNIT PARABOLIC SIGNAL

  • A unit parabolic signal, p(t) is defined as,

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  • We can write unit parabolic signal, p(t) in terms of the unit step signal, u(t) as,

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RESPONSE OF THE FIRST ORDER SYSTEM

  • We know that the transfer function of the closed loop control system has unity negative feedback as,
  • Substitute,   in the above equation.

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IMPULSE RESPONSE OF FIRST ORDER SYSTEM

  • Consider the unit impulse signal as an input to the first order system.
  • So, r(t)=δ(t), R(s)=1

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  • Rearrange the above equation in one of the standard forms of Laplace transforms.

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  • Apply inverse Laplace transform on both sides.

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STEP RESPONSE OF FIRST ORDER SYSTEM

  • Consider the unit step signal as an input to first order system.
  • So, r(t)=u(t),
  • Consider the equation, 
  • Substitute,     in the above equation.

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  • Do partial fractions of C(s).

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  • Comparing both sides of the equation,

A = 1 and B = - T

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STEP RESPONSE OF FIRST ORDER SYSTEM

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Applying inverse Laplace transform on both the sides.

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  • The unit step response, c(t) has both the transient and the steady state terms.
  • The transient term in the unit step response is

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  • The steady state term in the unit step response is 

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RAMP RESPONSE OF FIRST ORDER SYSTEM

  • Consider the unit ramp signal as an input to the first order system

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  • By partial fraction method,

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  • Comparing both sides of the equation,

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RAMP RESPONSE OF FIRST ORDER SYSTEM

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  • The transient term in the unit ramp response is

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  • The steady state term in the unit ramp response is

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PARABOLIC RESPONSE OF FIRST ORDER SYSTEM

  • Consider the unit parabolic signal as an input to the first order system

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  • By partial fraction,

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  • On simplification,

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PARABOLIC RESPONSE OF FIRST ORDER SYSTEM

  • The transient term in the unit parabolic response is

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  • The steady state term in the unit parabolic response is

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Thank You