[Journal] IEEE Transactions on Automatic Control. Vol. 48.
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829–833, Apr. 1999. R. E. Skelton, T. Iwasaki, and K. M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design. New York: Taylor Francis, 1997. M. Vidyasagar, Control System Synthesis: A Factorization Approach. Cambridge, MA: MIT Press, 1985. G. Zames and B. A. Francis, “Feedback, minimax sensitivity, and optimal robustness,” IEEE Trans. Automat. , vol. AC-28, pp. 585–600, May 1985. H H Jie Chen (S’87–M’89–SM’98) was born in The People’s Republic of China in 1963. S. E. A. D. degree in electrical engineering, all from The University of Michigan, Ann Arbor, in 1985, 1987, and 1990, respectively.
48, NO. 8, AUGUST 2003 From Lemma 4, there exists a matrix Let the spectral decomposition of such that be given by where is unitary due to skew symmetry of the complex conjugate of . We then have for all where the last condition is equivalent to , and denotes . From Lemma 4, It can be shown that when approaches infinity, the pamust approach zero and we recover the standard rameter Kalman–Yakubovich–Popov lemma [16]–[18]. The finite frequency condition in Theorem 1 can be generalized to the case where the frequency range is any interval of the form by utilizing the fact that can be a complex matrix.
Lemma 4: Consider a conjugate symmetric function is analytic and has no zero in . 2) Authorized licensed use limited to: Peking University. Downloaded on November 16, 2009 at 22:51 from IEEE Xplore. Restrictions apply. 1334 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 8, AUGUST 2003 Proof of Theorem 3: The proof is technically similar to that for Theorem 1, but requires attention to different aspects of the problem. 4), we obtain . That is and Since , and is min. This imum phase, it follows that fact then enables us to write the equation shown at the bottom such that of the page.