Tuning Based on Penalty Functions

The tables use three different tuning criteria-integral of absolute error (lAE), integral of square of error (ISE) and integral over time of the absolute error (ITAE). Not used by Smith and Murrill, another function can be defined - integral over time of the square of the error (ITSE). Each of these is a form of penalty function representing the size and duration of error. The tuning methods aim to minimise the penalty. 

We remove the sign of the error when integrating. Otherwise positive errors would be cancelled by negative errors, so that even a sustained oscillation would incur a penalty close to zero. The penalty functions are defined as follows. 

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