When designing a system, the goal is often to design an optimum system. But defining "optimum" is often not easy. One way to do this is to come up with a measure of "optimality", i.e. a performance measure.

One set of possible performance measures can be obtained from the step response. If we think of a "perfect" system, as following a step input exactly, then any difference between the step input and our output is an error, e(t). An obvious performance measure would be when that is somehow a representation of the total error (over all time) between the step input and our output.

An example of a performance measure would be the integral of the squared error:

where J is the performance measure.

MATLAB code (link to code) plots 6 different
performance measures (see the code for a description) for a second order prototype system
as a function of ζ, with ω_{n}=1. At the end of the program it prints out the
value of ζ which minimizes the performance measure. The code takes a minute or two to
run. Below is a printout of the graph, and the minimum ζ.

Color |
PerformanceMeasure |
Minimum z |

Yellow | 0.50 | |

Magenta | 0.71 | |

Red | 0.59 | |

Green | 0.67 | |

Blue | 0.75 | |

Black | 0.79 |

Note that for this broad range of performance criteria is used, the optimal ζ is between 0.5 and 0.8. Typically a value of 0.707 (1/sqrt(2)) is used -- but this value is not critical.

Performance measures can also be used on higher order systems. If a performance measure was found for a third order system, there would be two parameters to vary. The performance measures would then form a surface with the two parameters as the independent variables (for the second order system above, there was only one parameter and therefore only one independent variable). It is also possible to find performance measures for even higher order systems (though the surfaces generated are harder to visually since they use more than three dimensions).

If the time multiplied absolute error criterion

is used, we can find the optimal system for systems of many different orders. The coefficients for the characteristic equations for order 2 through 6 are given below with the constant coefficient=1 (From Ogata, Control Engineering, Prentice Hall).

s^{6} |
s^{5} |
s^{4} |
s^{3} |
s^{2} |
s | s^{0} |

1 | 1.414 | 1 | ||||

1 | 1.75 | 2.15 | 1 | |||

1 | 2.1 | 3.4 | 2.7 | 1 | ||

1 | 2.8 | 5.0 | 5.5 | 3.4 | 1 | |

1 | 3.25 | 6.60 | 8.60 | 7.45 | 3.95 | 1 |

The output for the 2nd, 4th, and 6th order optimal responses are given in the graph below. Link to MATLAB code.