Other Time Domain Specifications

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Command Reponse

NameSymbolDefinition
Decay ratio The ratio between two consecutive maxima of the error for a step change in setpoint or load. The value , which is called quarter amplitude damping, has been used traditionally
Steady state error The percentage error between the reference input and actual output of a linear system.

Integrated Error

The integrated absolute error (IAE) is given by the integral

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A related criterion is the the integrated error (IE)

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The criteria IE and IAE are the same if the error does not change sign. Notice that IE can be very small even if the error is not. For IE to be relevant it is necessary to add conditions that ensure that the error is not too oscillatory. The criterion IE is a natural choice for control of quality variables for a process where the product is sent to a mixing tank. The criterion may be strongly misleading, however, in other situations. It will be zero for an oscillatory system with no damping.

The integrated squared error (ISE) is defined as

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There are many other criteria, for example the time multiplied absolute error defined by

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Input/Output Measures

There are other criteria that take account of both input and output signals for example the quadratic criterion

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where is a weighting factor.

The criteria IE and QE can easily be computed analytically, simulations are however required to determine IAE.