Signals Maximising the Integral-Square-Error in the Process of Models Optimisation

Abstract

The previous chapter was dedicated to the determination of maximum mapping errors by means of signals with one constraint or two simultaneous constraints imposed upon them. In this chapter we will consider the problem of minimisation of such errors. It is easy to see that for a given order and type of simplified model minimisation of the maximum mapping error is reduced to the solutions known in the domain of parametric optimisation. However, commonly applied methods of such optimisation refer mainly to the minimisation of a chosen objective function to a standard input signal, which in the majority of cases, is the unit step input. A good illustration of this can be seen through the example in [71] which presents six different optimum second order models of a seventh order real system of a supersonic transport aircraft pitch rate, obtained for six different objective functions and for a common unit step input signal. It should be noted however, that the parameters of the optimum models as well as the values of maximum errors depend equally on the chosen objective function and the input signals applied in the optimisation process. For this reason, the parameters of the optimum models obtained for one of the objective functions using the chosen standard signal can still significantly differ from the model parameters for that objective function with a different input signal. In order to make the optimisation results independent of input signal shape or the minimised mapping errors based on them, the optimisation of simplified models will be accomplished by replacing the standard signals with signals which maximise the chosen error functional.