Introduction to Second Order Systems

April 2nd, 2014 | Posted in Panel Building
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Next article of Nasir from his tutorial on Control Systems.

Introduction

As we discussed earlier we have two methods of analyzing the working and functioning of a control system named as:

  • Time domain analysis
  • Frequency domain analysis

The time domain analyzes the functioning of the system on basis of time. This analysis can only be applied when nature of input plus mathematical model of the control system is known. Expressing the main input signals is not an easy task and cannot be determined by simple equations. There are two components of any system’s time response, which are:

  • Transient response
  • Steady state response

In order to judge the functioning and behavior of a system, typical and standard test signals are used. The characteristics of an input signal are constant acceleration, constant velocity, a sudden change or a sudden shock. We have already discussed four types of test signals i.e.

  • Impulse
  • Step
  • Ramp
  • Parabolic

We already discussed first in detail in the previous article. It said that the system whose input-output equation is a differential equation is called a system. In this article we will be focusing on second .


Systems

Introduction to Second Order Systems 1

The order of a differential equation is the highest degree of derivative present in that equation. A system whose input-output equation is a second order differential equation is called Second Order System.

There are a number of factors that make second order systems important. They are simple and exhibit oscillations and overshoot. Higher order systems are based on second order systems. In case of mechanical second order systems, energy is stored in the form of inertia whereas in case of electrical systems, energy can be stored in a capacitor or inductor.

Standard form of second order system is given by:

Introduction to Second Order Systems 2

Where:

  • ωn Is the natural frequency
  • Introduction to Second Order Systems 3is the damping ratio
  • If 0< Introduction to Second Order Systems 3 <1, system is named as Damped System
  • If < Introduction to Second Order Systems 3 =1, system is named as Critically Damped System
  • If < Introduction to Second Order Systems 3 >1, system is named as Over Damped System


Response of a Second order system

We analyze the responses in second order systems in undamped, under damped, critically damped and over damped cases. Let us have a look on these:

  • 1. Step response of Second-order systems:

Critically Damped Case: (Introduction to Second Order Systems 3 =1)

Introduction to Second Order Systems 4


Two poles are equal. That means:

Introduction to Second Order Systems 5

In a unit step input, we have:

Introduction to Second Order Systems 6

And output is:

Introduction to Second Order Systems 7

Steady-state error: e (∞) = 0


Over Damped case: (Introduction to Second Order Systems 3 > 1)

Introduction to Second Order Systems 8


We can write the transfer function of a second-order system by factoring the denominator as:

Introduction to Second Order Systems 9

Taking the inverse Laplace transform yields the time response:

Introduction to Second Order Systems 10

The unit-step time response is:

Introduction to Second Order Systems 11


Damped case: (Introduction to Second Order Systems 3 < 1)

For a damped case in which 0 < Introduction to Second Order Systems 3 < 1 time response is given by:

Introduction to Second Order Systems 12

Introduction to Second Order Systems 13


  • 1. Ramp response of a second-order system:

Again we have 3 cases here that are:

Introduction to Second Order Systems 3 = 1, critically damped case

Introduction to Second Order Systems 3 > 1, over damped case

0 < Introduction to Second Order Systems 3 < 1, under damped case

The Laplace transform of a unit-ramp input is R(s) = 1/s^2

The output is given by:

Introduction to Second Order Systems 14


  • 2. Impulse response of a second-order system:

The Laplace transform of a unit-impulse input is R(s) = 1.

The unit impulse response is give by:

Introduction to Second Order Systems 15

Conclusion

In this tutorial we have discussed second order systems and their responses. As we are now done with the order of systems we will move on to Transient Response Analysis of Control Systems. Stay tuned for more insight on control systems.

Nasir.

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Comment(s) to “Introduction to Second Order Systems”

  1. Ancy Albert says:

    Can you explain about undamped second order systems in detail?

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