On Kalman-Yakubovich-Popov lemma for stabilizable systems

Abstract

1089the case, problem (15) is equivalent to problem (17). We further note that the inequality in (29) is equivalent to Both in the case of trace and log-determinant, the function f (X) is concave on the cone of positive-definite matrices. This implies that the optimal value of X; Z are X = Xopt , Z = Zopt , as claimed. ACKNOWLEDGMENTThis note has benefited from interesting discussions and valuable input from several people, including R. Balakrishnan, S. Boyd, E. Féron, A. Kurzhanski, and R. Tempo. The authors particularly thank A. Nemirovski for his help regarding Section III-C. Useful comments from the reviewers and the Associate Editor are also gratefully acknowledged. Control, vol. 8, no. 4-5, pp. 377-400, 1998. [9] L. El Ghaoui and G. Calafiore, "Confidence ellipsoids for uncertain linear equations with structure," in 38th Conf. Decision Control, vol. 2, Phoenix, AZ, Dec. 1999, pp. 1922 , "Worst-case simulation of uncertain systems," Control, vol. 65, no. 5, pp. 847-866, 1996. REFERENCES[19] K. M. Nagpal and P. P. Khargonekar, "Filtering and smoothing in an H setting," IEEE Trans. Automat. Contr., vol. 36, pp. 152-166, Feb. 1991. [20] A. Nemirovski, "Proof of self-concordance of a barrier function," unpublished, Jan. On Kalman-Yakubovich-Popov Lemma for Stabilizable SystemsJoaquín Collado, Rogelio Lozano, and Rolf Johansson Abstract-The Kalman-Yakubovich-Popov (KYP) Lemma has been a cornerstone in System Theory and Network Analysis and Synthesis. It relates an analytic property of a square transfer matrix in the frequency domain to a set of algebraic equations involving parameters of a minimal realization in time domain. This note proves that the KYP lemma is also valid for realizations which are stabilizable and observable.

The Yakubovich–Kalman–Popov lemma and stability analysis of dynamic output feedback systems

“…[6]. [23,24]}there may exist Lyapunov weighting matrices P ¼ P T 50 without any controllability assumption satisfied. Whereas such insights previously remained unexploited, our new results provide observers with constructive procedures to find such a P ¼ P T 50 in the context of the Popov criterion and the Circle criterion.…”

“…Except for the non-minimality statement, the Popov Criterion and its proof remain unchanged with an SPR notion without minimality [23,27]. The transfer function HðsÞ has a state-space realization SfA; B; ZkCA þ kC; ZkCB þ I m g}cf.…”

“…A key question is whether a non-SPR transfer function may be brought to SPR form at the expense of minimality. Whereas Meyer [2] demonstrated early insights that controllability is not necessary for the YKP lemma and constructive proofs for stabilizable systems recently appeared [23,24], few constructive methods for dynamic output feedback relevant to the Lur'e problem appear available [25].…”

Strictly positive real systems based on reduced-order observers

“…Implicitly, Rantzer [26], in a novel proof based on convexity properties and linear algebra, did not require minimality of the state space representation of Z(s). It was not until recently [8] that the minimality relaxation was explicitly proved in an algebraic fashion so that a state space representation can have noncontrollable modes and still satisfy the SPR conditions provided that the uncontrollable modes are stable. Other interesting properties of SPR systems and comparisons are presented in [29], [18].…”

Vibration control for active seat suspension systems via dynamic output feedback with limited frequency characteristic

“…The generalized KYP lemma is useful for the analysis and synthesis of practical application problems [3,11,15,24].…”