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By Henry M. Paynter

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1 Uncertainty and its Bounds Usually, with regard to physical and engineering insight or by executing identification algorithms, the dynamical behaviour of the process is determined, that is, the structure, order, linearity and multivariability. Considering unperturbed behaviour, the system is denoted nominal. In spite of these activities, several properties remain unknown and one has to analyze the following subjects: (i) Uncertainty of structure and parameters, including some information about the bounds of the uncertainty.

Integral of squared error norm versus frequency I = f~oo lIe(jw )1I~dw . • Minimum weighted sensitivity IIWSlloo where Wand S are the weighting matrix and sensitivity matrix of the system, respectively. 1. 9) where C is the multivariable controller and G the multivariable plant. The procedure is executed by an off-line method. 10) where G p must not exceed a certain area and the execution is on-line. Design criterion II and identification criterion 12 need not be the same. 11 ) where the implementation of the on-line identification minM 12 is preassumed.

G()] O'mm s:s IIx(s)IIF . 46) Combination with O'max[G(s)] yields O'max[G(s)] = IIG(s)lI. , the gain of a multivariable system IIG(s)x(s)IIF/llx(s)IIF is bounded between the smallest and largest singular values. e. IIG(jw)lloo, is an upper bound of the system gain. 10 Output Norm Estimates of a Multivariable System Consider a multivariable system in the frequency domain with input 1 6 = 6(jw), transfer matrix G(jw) and output y = Y(jw). 50) IThe hat symbol is used to make a distinction, only, between the signal in the time domain u(t) and in the frequency domain ii(jw) or ii( s) .

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