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IMA Journal of Mathematical Control and Information 1984 1(1):95-106; doi:10.1093/imamci/1.1.95
© 1984 by Institute of Mathematics and its Applications
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On the Maximal Accuracy of Linear Optimal Systems

M. A. JOHNSON and M. J. GRIMBLE

Industrial Control Unit, Department of Electronic and Electrical Engineering, University of Strathclyde George Street, Glasgow G1 1XW

A Hilbert-space development of maximal accuracy and input squaring-down operators, both topics of interest in optimal control and linear multivariable design, is presented.

The special theory of compact self-adjoint Hilbert-space operators enables a canonical form to be developed for the optimal value of the quadratic cost function; further its asymptotic behaviour can be investigated by introducing a scalar parameter into the control weighting term. If the spectral theory is then linked to the usual orthogonal-range space/null-space Hilbert-space decomposition, the theorems for the asymptotic value of the optimal cost and the definition for maximal accuracy have a particularly simple form.

The relevance of these results to the assessment of the effectiveness of subsets of control inputs and the use of squaring-down operators is discussed. The link with the usual maximal-accuracy problem for time-invariant state-space triples is also examined.


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