# Control System Response in Time Domain

**Topics:**Step response, Control theory, Transfer function

**Pages:**16 (3931 words)

**Published:**October 12, 2012

System Response in Time Domain|

Name| Student No| Time Slot| Signature|

1 Johan Jarvi| | Monday| |

Tuesday| |

Wednesday| |

Thursday| 13:00 |

Friday| |

| |

2 Lachlan Hutch| | | |

| | |

| We, by signing this page, declare that the work presented in this report is all work done by us, unless appropriate reference has been made to the work of others. We acknowledge that should this not be the case the report will receive zero marks and due action may be taken.| | Lab Number: 1| | |

Demonstrator| |

Submitted on| |

Mark Received| |

Experiment 1

1.1. Requirements of Experiment 1

Experiment 1 consisted of finding the damping ratio ζ, natural frequency ωn, settling time Ts, peak time Tp and percent overshoot %OS, as well as the dominant pole pair were to be found for three different second order systems, given by the transfer functions:

These variables should then be found through two different methods, one which consisted of writing a program in MATLAB to analytically evaluate each variable, and another method where the response characteristcs were then supposed to be found directly from the response curve by using MATLAB built in functions such as cursors. The final part of experiment one would be to use the Electromechanical Servomechanism Virtual Laboratory (ESVL) to generate the same three response curves only this time using a virtual reality type view. The same response characteristics were then to be obtained through reading off the oscilloscope output of the ESVL.

1.2. Introduction (Background)

In order to fully understand the first experiment of this laboratory, it is important to have some understanding of what a transfer function is, and how changing different parameters of the first order or second order systems should change the behavior of the response curve.

By changing parameters of the first order system, the speed of the response curve will change, however varying the parameters of a second order system may give large changes in the way the response curve behaves.

It is important to note that the ideal second order system has the following mathematical equation:

eq. 1.2.1

And that the pole locations are given by:

eq. 1.2.2

The locations of the poles govern the appearance of the transient response. Changing the responses to be either Overdamped, Underdamped, Undamped or Critically Damped.

In this experiment various response characteristics are to be obtained, which were explained in section 1.1. Therefore it is also crucial to have an understanding of what these different response characteristics actually mean, and that will be shown in the image on the following page, along with mathematical equations to calculate each value.

This response curve nicely demonstrates the meaning of each variable calculated later in the experiment. Some basic knowledge of what these different things mean may also be necessary; hence an explanation will be provided below.

Peak time, Tp: is the time required to reach the first, or maximum peak. In the above figre that is shown with Tmax, same location where the maximum overshoot occurs.

Percent Overshoot, %OS: the amount by which that particular waveform overshoots its steady state value. Shown on the figure by “maximum overshoot”

Settling time, Ts: this is the time required for the transient’s damped oscillations to reach and stay within +/-2% of the steady state value.

And Rise time, Tr: this is the time required for the waveform to go from 10% of the final value to 90% of the final value.

The last thing that may be important in order to understand what the various variables mean when it comes to the ESVL section of the experiment would be to have an understanding of what the different coefficients are in terms of the DC motor that the ESVL...

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