Time Response

- Transient and Steady-State Response Analysis

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Time Response
- First Order System
- Second Order System

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Time Response-Transient and Steady-
State Response Analysis

Typical Test Signals:

Step functions,
ramp functions,
acceleration functions,
impulse functions,
sinusoidal functions
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Time Response: First Order System
Unit-Step Response of First-Order Systems:

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Time Response: First Order System
Unit-Step Response of First-Order Systems:

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Time Response: First Order System
Unit-Step Response of First-Order Systems:

System showing input and output

pole-zero plot of the system

Note:
• A pole of the input function generates the form of the
forced response
• A pole of the transfer function generates the form of the
natural response

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Time Response: First Order System
Unit-Ramp Response of First-Order Systems:

error signal e(t)

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Time Response: First Order System
Unit-Ramp Response of First-Order Systems:

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Time Response: First Order System
Unit-Impulse Response of First-Order Systems:

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Illustration
A system has a transfer function

Find the time constant, Tc , settling time, Ts , and rise time, Tr .

Time Response: Second Order System

Unit-Step Response of First-Order Systems:

Overdamped
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Time Response: Second Order System
Unit-Step Response of First-Order Systems:

Underdamped

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Time Response: Second Order System
Unit-Step Response of First-Order Systems:

Undamped

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Time Response: Second Order System
Unit-Step Response of First-Order Systems:

Critically damped

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Time Response: Second Order System
Unit-Step Response of First-Order Systems:

Damped frequency of oscillation, ωd .

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General Second Order System
Unit-Step Response of Second-Order Systems:

response would be an undamped sinusoid when

Natural Frequency, ωn
Frequency of oscillation of the system
without damping

Damping Ratio, ζ

General second-order transfer

Pole location:
function finally looks like

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General Second Order System
Unit-Step Response of Second-Order Systems:

Second-order response as a function of damping ratio

Case 1: When Damping Ratio, ζ=0

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General Second Order System
Unit-Step Response of Second-Order Systems:

Second-order response as a function of damping ratio

Case 2: When Damping Ratio, 0<ζ<1: Underdamped Second Order System

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General Second Order System
Case 2: When Damping Ratio, 0<ζ<1: Underdamped Second Order System

Unit-Step Response of Second-Order Systems:

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General Second Order System
Case 2: When Damping Ratio, 0<ζ<1: Underdamped Second Order System

Unit-Step Response of Second-Order Systems:

Rise time tr:

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General Second Order System
Unit-Step Response of Second-Order Systems:

Case 3: When Damping Ratio, ζ=1

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General Second Order System
Unit-Step Response of Second-Order Systems:

Case 2: When Damping Ratio, ζ>1

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Time Response: Second Order System
Unit-Step Response of Second-Order Systems:

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Definitions of Transient-Response Specifications
Unit-Step Response of Second-Order Systems:

1. Delay time, td
2. Rise time, tr
3. Peak time, tp
4. Maximum overshoot, Mp
5. Settling time, ts

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Definitions of Transient-Response Specifications
Unit-Step Response of Second-Order Systems:
1. Delay time, td
2. Rise time, tr
3. Peak time, tp
4. Maximum overshoot, Mp
5. Settling time, ts

Peak time, tp

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Definitions of Transient-Response Specifications
Unit-Step Response of Second-Order Systems:
1. Delay time, td
2. Rise time, tr
3. Peak time, tp
4. Maximum overshoot, Mp
5. Settling time, ts

Maximum overshoot, Mp

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Definitions of Transient-Response Specifications
Unit-Step Response of Second-Order Systems:
1. Delay time, td
2. Rise time, tr
3. Peak time, tp
4. Maximum overshoot, Mp
5. Settling time, ts

Settling time, ts

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Illustration
Consider the system shown in Figure, where ζ=0.6 and wn=5 rad/sec. Obtain the rise time tr,
peak time tp, maximum overshoot Mp , and settling time ts when the system is subjected to
a unit-step input.
 Impulse Response of Second Order System
Second-Order System with Impulse Input:

Case 1: Second-Order Underdamped System with Impulse Input:

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Impulse Response of Second Order System
Second-Order System with Impulse Input:

Case 2: Second-Order Critically-damped System with Impulse Input:

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Impulse Response of Second Order System
Second-Order System with Impulse Input:

Case 3: Second-Order Overdamped System with Impulse Input:

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Impulse Response of Second Order System
Second-Order Underdamped System with Impulse Input:

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Second-Order System
Second-Order Underdamped System

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Second-Order System
Second-Order Underdamped System

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Higher-Order System
Cascading of Second Order Underdamped System with First Order System

Case 1 Case 3
Case 2

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System Response with Zeros
Cascading of Second Order Underdamped System: Impact of adding a zero to the second-order system.

Case 1: When Zero is added in first in the left half-plane

If the zero is far from the poles, then a is large compared to b and c,

When we add a zero to the transfer function, i.e. (s+a)T(s) the Laplace transform of the
response will be
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System Response with Zeros
Cascading of Second Order Underdamped System: Impact of adding a zero to the second-order system.

Case 1: When Zero is added in first in the left half-plane

When we add a zero to the transfer function, i.e. (s+a)T(s) the Laplace
transform of the response will be

derivative scaled
of original version of
response the original
response

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System Response with Zeros
Cascading of Second Order Underdamped System: Impact of adding a zero to the second-order system.

Case 2: When Zero is added in the right half-plane (if a is negative):

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Step response of a nonminimum-phase system
System Response with Zeros
Cascading of Second Order Underdamped System: Impact of adding a zero to the second-order system.

Case 3: Pole- Zero Cancellation: When Zero is added at the same location of Pole

Residue of pole at s=-3.5 is not

negligible compared to the other
The partial-fraction expansion of C1(s) is residues. Thus, a second-order step
response approximation cannot be
made for C1(s)

Residue of the pole at s=-4.01 is negligible compared to to the other residues.

Thus, a second-order step response approximation is applicable by neglecting
the response generated by the pole at -4.01.

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Error analysis
Steady state error analysis

Case 1: When input is unit-step: steady-state error of the system for a unit-step input is
Positional Error Coefficient Kp
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Error analysis
Steady state error analysis

Case 2: When input is Ramp-input: steady-state error of the system for a ramp input is

Velocity Error Coefficient Kv

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Error analysis
Steady state error analysis

Case 2: When input is Parabolic-input: steady-state error of the system for a parabolic input is

Acceleration Error Coefficient Ka

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Error analysis
Steady state error analysis

For a type 0 system,

For a type 1 or higher system,

for type 0 systems

Positional Error Coefficient Kp

Velocity Error Coefficient Kv for type 1 or higher systems

Acceleration Error Coefficient Ka 43

Error analysis
Static Velocity Error Constant Kv

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Error analysis
Static Acceleration Error Constant Ka

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Error analysis
Static Acceleration Error Constant Ka

For a type 0 system

For a type 1 system,

For a type 2 system,

For a type 3 or higher system, 46

Error analysis
Static Acceleration Error Constant Ka

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Error analysis
Static Acceleration Error Constant Ka

Steady-State Error in Terms of Gain K

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Illustration 1
Consider the system defined by G(s) with the negative unity feedback, Determine wn, wd, ζ,
First Overshoot time, First Undershoot time, First maximum overshoot Mp , First %
Undershoot, settling time ts, number of cycles completed before reaching steady state
condition.

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20 𝑠+1 𝑠+5
𝑆𝑦𝑠𝑡𝑒𝑚: 𝐺 𝑠 = , 𝑤𝑖𝑡ℎ 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑢𝑛𝑖𝑡𝑦 𝑓𝑒𝑒𝑑𝑏𝑎𝑐𝑘
𝑠+1 𝑠+5
Illustration 2
𝐾
The maximum overshoot of a unity feedback system with forward path gain 𝐺 𝑠 =
𝑠 𝑠𝑇+1

% Mp is to be reduced from 60% to 20%, the system input is a unit step input.
Determine the factor by which K should be reduced.
Illustration 3
50 2
A Feedback Control System is described by 𝐺 𝑠 = , H 𝑠 =
𝑠 𝑠+2 (𝑠+5) 𝑠

Determine it’s steady state error for ramp input.

Illustration 4
𝐾
Forward gain of a system is Control System 𝐺 𝑠 = , H 𝑠 =1
𝑠 2 𝑠+20 (𝑠+30)

Determine the value of K to limit steady state error to 10, when the input r(t)=1+10t+20t2 is
applied to the system.
Illustration 5
The measurement conducted by servomechanism show error response to step input e 𝑡 =
1.66𝑒 −8𝑡 sin(6𝑡 + 37°), Calculate it’s ζ ad undamped natural frequency, settling time ts for
5% band.
Illustration 6
The pole location of second order system is s=-4+j2 and -4-j2. Determine its output response
c(t) if input r(t)=3u(t). Also calculate its settling time for 5% band.
Illustration 7
A second order underdamped system exhibit 15% maximum overshoot when applied with a
step input of 2u(t) and then the system attain steady state value at t=5 sec. Again at t=10 sec,
the input changes to 1u(t). Plot its output waveform and write equation for the output
waveform.
Illustration 8
𝐾(𝑠+2)
Open loop transfer function of a unity feedback system is given by 𝐺 𝑠 = .
𝑠 3 +𝛽𝑠 2 +4𝑠+1
Determine the value of K and β when ζ=0.2 and wn= 3 rad/sec.
Illustration 9
(𝑠+𝛽)
Closed loop transfer function of a unity feedback system is given by T 𝑠 = .
𝑠 2 +α𝑠+𝛽
Determine it’s steady state error for (i) Step input (ii) Ramp input.
Controllers
- Proportional Controller (P-Controller)
- Integral Controller (I-Controller)
- Derivative Controller (D-Controller)
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