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# Introduction and statement of the problem

DOI link for Introduction and statement of the problem

Introduction and statement of the problem book

# Introduction and statement of the problem

DOI link for Introduction and statement of the problem

Introduction and statement of the problem book

## ABSTRACT

The sensitivity minimization problem for delay systems is a difficult mathematical prob lem and for a single delay exp(—sr) this was solved in [10, 7]. Since then, more general problems have been considered in [8 , 2 0 , 21] (an introductory survey can be found in [18]). These references propose a frequency domain approach to the problem. In [18, 20, 21], the idea is to transform the continuous time problem into a discrete-time one which is then solved to yield the optimal infinite-dimensional controller. This solution is then transformed back into continuous time, where it is not only often infinite-dimensional, but improper as well. Consequently, approximations need to be made at this stage to yield an implementable (sub-optimal) controller. While this seems to yield good results in practice (see [17, 21] and [10]), a satisfactory theory for these approximations is still lacking. The difficulty with designing finite-dimensional controllers is that it is not sufficient to design good approximations to the infinite-dimensional optimal controller. One also needs to achieve a certain performance level at the same time. In [19] an approximation for certain scalar plants is proposed. However, one of the assumptions is that (1 — s E H i, which eliminates a lot of interesting plants, including exp(—sr) and exp(—sr ) /(a -f-s). Despite this, the method did work satisfactorily on the (example) plant exp(—0 .01s)/(s — 1), even though it did not satisfy the theoretical assumptions. A different approach is taken

in [24]; one first approximates the infinite-dimensional system and then designs an op timal controller for the approximating finite-dimensional plant. In order to obtain the approximation one needs to first obtain an inner-outer factorization. In general, this step has to be performed numerically which is a difficult problem in itself, even for scalar plants (see [6]). We note that in both of the above frequency domain approaches one usually needs explicit inner-outer factorizations of infinite-dimensional systems at a cer tain stage. While this is straightforward for simple systems (such as exp( - s r ) P ( s ) with P(s) rational and scalar), this is an unsolved problem for general irrational multivariable transfer functions (in [6] some numerical results are obtained). Summarizing, we can say that although the frequency domain approach provides existence results for Woo-control problems for general delay systems, a satisfactory methodology for real controller design for these systems is still missing. With the exception of [17], all the mixed-sensitivity control designs have been for scalar plants. The main problem is that of obtaining an explicit inner-outer factorization via a good numerical algorithm.