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Selected Topics in Controls:Focus on Controller Design using

Classical and ModernControl Techniques

Lecture # 1

Dr. Muhammad Bilal Kadri

Transient behavior of first and second order systems

Specifications of the control performance are given in terms of transient and steady state behavior.

We will look at a simple first and second order systems to understand the transient characteristics.

A brief analysis of impact of pole locations on transient performance will also be discussed.

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Pole Locations for second order systems

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Time Responses for second order system

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Second Order Under-damped Responses

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Transient Specifications

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Relationship between damping ratio and %OS

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Relationship between Rise time and Damping ratio

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Pole plot for second order underdamped systems

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Second order system: Time domain specifications

2

2

2 2

1

2

( )2

4) :

)% x100

) :1

n

n n

s

n

n

G ss s

a Settling Time T

b OS e

c Peak Time Tp

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Lines of constant %OS, Tp (peak time) and Ts (settling time)

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Step Responses of Second-order under-damped system

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Steady State Analysis

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Types of test inputs

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Steady State Error Equation

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Steady State Error (Step and Ramp input)

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Types of systems (0,I and II)

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Relationships between input, system type, static error constants, and steady-

state errors

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Sensitivity

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Mathematical Expression

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Why Root Locus?

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Impact of feedback gain on stability

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Root Locus Basics

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Pole locations as a function of gain (K)

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Root Locus for a feedback system

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Example 1

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Example 1 (contd)

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Matlab Code

Num=poly([-3 -4])

Den=poly([-1 -2])

Ts1=tf(Num,Den)

rlocus(Num,Den)

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Comparison Table

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Approximate the third order with a second order system without zeros. 1.52% overshoot corresponds to damping

ratio of 0.8. Draw a line on the root locus. Three points will intersect the 0.8 line.

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Root Sensitivity

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Example

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Figure 8.4 (for reference)

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Class Assignment (Calculate it yourself)

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