Skew Toeplitz Theory and Pseudorational Transfer Functions

A state space version of the skew Toeplitz theory for (distributed) H" optimization theory is presented. The approach combines this theory with the realization theory for pseudorational transfer functions. 1 Time-Domain Interpretation of Sarason Space In [15], we established a strong link between the skew Toeplitz H" theory of 12, 4, 5, 6, 9, 121 and its realization in the time domain. The fundamental motivation is that while the former is primarily a frequency domain theory it is nontrivial, but certainly desirable, to establish its time-domain counterpart for distributed parameter systems. Roughly speaking, the relationship is as follows: Consider the problem of weighted sensitivity H"-optimization: where W E H" and B ( s ) is an inner function. Let X" := (BH2)' = H 2 e B H Z . We know from Sarason's theorem [lo] that p is precisely the norm of the compression of W ( s ) to SE. See (121 and the references therein. The crucial step is to give a suitable (time-domain) representation for X " . In the L domain, Ahern and Clerk [l] gave a representation for H ( B ) := H Z @ BH', and the associated compressed shift S ( B ) := nSlH(B) where S denotes the unilateral right shift in H 2 and n : H 2 4 H ( B ) orthogonal projection. While it is possible to obtain results in the z domain and then transform them into the s domain, we here present a different route. We give a direct interpretation in the s domain using realization theory developed in 113, 141. The key observation is that if B ( s ) does not have any singular part, and its zeros satisfy certain growth conditions, then it arises from the Laplace transform of a distribution with compact support. In other words, it falls into the category of pseudorational transfer functions, and we have a concrete realization procedure based on such a distribution [13]. In [13], using the eigenfunction completeness, we have shown that the above problem is reducible to the limiting case of Sevanlinna-Pick interpolation. A drawback is that it involves infinitely many interpolation conditions while in the skew Toeplitz theory, it can be reduced to essentially finitely many conditions. In this note we show that for delay-differential 5ystems a more compact treatment is possible. 'This author was supported in part by the Tateishi Science Foundation. 'This author was supported in part by grants from the National Sciencr Foundation DhlS-8811084 and ECS-9122106, the Air Force Offirr of Scientific Research F49620-94-1-00S8DEF, and by the Army Renearrh Ofice DAAL03-91-G-0019 and DAAH04-934-0332, 0-7803-1 968-0/94$4.0001994 IEEE Allen Tannenbaum Department of Electrical Engineering University of Minnesota Minneapolis, Minnesota 55455 USA