Chapter 3 Time-Domain Analysis of the linear systems

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Chapter 3 Time-Domain Analysis of the linear
systems

• Introduction
• Stability and Algebraic criteria
• Analysis of stable error
• First-order system analysis
• Second-order system analysis
• High-order system analysis

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3.1 Introduction
3.1.1 Basic and macroscopically requirements to
design a control system.

• 1) The system must be stable (stability)
• First
  requirement.
• 2) The control should be accurate (accuracy).
• 3) The response should be quick-acting
  (rapidity).

3.1.2 Basic assumption conditions for analyzing
a linear control system
The response of a system could be
zero-state zero-input response response
transient steady-state portion portion
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3.1 Introduction
3.1.3 Typical test input signal

• 2. Which types of the
• test input signal ?
• Step input signal
• Ramp input signal
• Parabolic input
• signal
• Pulse input signal
• Sinusoidal input
• signal

• 1. why to research the test input signal?
• The actual input signal of the system is
  multifarious, normally a standard test input
  signal should be chosen?to analyze the system
  performance.
• Allow the designer to compare several designs
  project.
• Many input signals of the control systems are
  similar to the test signals.

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3. Typical test input signal
3.1 Introduction

• 1) Step input signal

2) Ramp input signal
R(s)A/s
A1 unity step input
A1 unity ramp function
Application test signal of the constant-input
systems
Application some input-tracking systems
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3.1 Introduction

• 3) Parabolic input signal

4) Pulse input signal
A 1 unity parabolic signal
Application same as the Ramp signal
Such as Impactive disturbance
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3.1 Introduction

• 5) Sinusoidal signal

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3.1 Introduction
3.1.4 Relationship between impulse response and
other responses

• 1. For the typical input signals

Theorem
proof
as above
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3.1 Introduction

If g(t) ?impulse response of a linear system.
g(t) L-1 G(s)
h(t) ? step response of a linear system.
ct(t) ? ramp response of a linear system.
ctt (t) ? parabolic response of a linear system
2. For any signal r(t) we have the
Convolution theorem
then
example
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3.1.5 The transient performance specifications
of a control system
3.1 Introduction

• For different system, the research aim is
  different.
• For example, the tracking servo control systems
• are different from the constant-regulating
  systems.

• performance measure in terms of the step
  responses
• of the systems.

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Performance specification definition
A
Setting time ts
B
Peak time tp
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3.1 Introduction

• 1) Rise time tr

2) Peak time tp
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3.1 Introduction

• 3) Percent overshoot

4) Setting time ts
5) Delay time td
s ? smoothness of the response.
tr?tp?td ? rapidity of the response.
ts ? rapidity of the transient process
of the response.
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3.2 Stability analysis of the linear systems
(Stability the most important performance
for a control system)
3.2.1 What is the Stability of a system ?
3.2.2 The sufficient and necessary conditions of
the sta- bility for a linear system.
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3.2 Stability analysis of the linear systems
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The sufficient and necessary conditions of the
stability for a linear system

Graphic representation
Unstable region
Stable region
The relationship between the systems
stability and the position of poles in S-plane.
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3.2.3 Routh Criterion
3.2 Stability analysis of the linear systems

• for a control system

Then
Characteristic equation of the system
1G(s)H (s)0
Assume
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3.2 Stability analysis of the linear systems

• We have

we need all roots of the characteristic
equation lie in the left-half of s-plane for a
stable system.
For the equation
The question is how could we know the roots
all lie in the left-half of s-plane?
Routh do it like this
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3.2 Stability analysis of the linear systems

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3.2 Stability analysis of the linear systems

• conclusion(Routh Criterion)

1) All elements of the first column of the
Routh-table(array) are positive . The system
is and must is stable.
Necessary,
sufficient
2) The number of roots with positive real
parts is equal to the number of changes in sign
of the first column of Routh-table .
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3.2 Stability analysis of the linear systems

Example 3.2.2
Example 3.2.1
Unstable . 2 roots With
positive Real parts
Stable
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3.2 Stability analysis of the linear systems
A unity feedback system, Open-loop
Example 3.2.3
solution
0ltklt8, the system is stable
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3.2 Stability analysis of the linear systems
some or other element is equal to zero in the
Routh-table.
Example 3.2.4
s4 1 1 1 s3
2 2 s2 0 1 s1

egt0
s0 1
Unstable. 2 roots with positive real parts.
Note
Can use a infinitesimal egt0 substituting the
zero element in the first column.
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There are all zero elements in some row of
Routh-table.
Example 3.2.5
Make auxiliary polynomial
0
0
multiplied by 1/2
no effect result
Unstable, no root in the right-half of
s-plane. but there are two pair of roots in the
imaginary axis of s-plane
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3.2 Stability analysis of the linear systems

• Solving the auxiliary equation, we have

The order of the auxiliary polynomial is
always even indicating the number of symmetrical
root pairs .
Note
Two inferences about Routh-criterion
1) The characteristic equation is short of one or
more than one items The system must be unstable .
Example
2) The coefficient of the characteristic equation
are different in sign. The system must be
unstable.
Example
unstable, The coefficient are different in
sign.
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3.3 steady-stable error Analysis of the linear
system

• Connection to next part