Positive real lemma: necessary and sufficient conditions for the existence of solutions under virtually no assumptions

In this note, the celebrated positive real lemma equations are considered with the purpose of obtaining necessary and sufficient conditions for the existence of solutions under the mildest system-theoretic assumptions. More precisely, necessary and sufficient conditions are established under the very weak assumption of sign-controllability. It is then shown that the order of the equations may be suitably reduced by restricting attention to a subspace related to a certain observable subsystem. This reduction, beside being interesting per se, permits us to weaken the assumption of sign-controllability. Finally, the assumptions are further weakened as to include the case when the state matrix has uncontrollable eigenvalues on the imaginary axis.

[1]  Luciano Pandolfi,et al.  On the solvability of the positive real lemma equations , 2002, Syst. Control. Lett..

[2]  A. Packard,et al.  Optimal LQG performance of linear uncertain systems using state-feedback , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[3]  J. Willems.  Least squares stationary optimal control and the algebraic Riccati equation , 1971 .

[4]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[5]  Venkataramanan Balakrishnan,et al.  Semidefinite programming duality and linear time-invariant systems , 2003, IEEE Trans. Autom. Control..

[6]  Shinji Hara,et al.  Structure/control design integration with finite frequency positive real property , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[7]  D. Youla,et al.  On the factorization of rational matrices , 1961, IRE Trans. Inf. Theory.

[8]  V. Yakubovich.  A frequency theorem in control theory , 1973 .

[9]  W. Rugh.  Linear System Theory , 1992 .

[10]  O. Brune.  Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency , 1931 .

[11]  Giorgio Picci,et al.  A geometric approach to modeling and estimation of linear stochastic systems , 2001 .

[12]  Uri Shaked,et al.  Robust discrete-time minimum-variance filtering , 1996, IEEE Trans. Signal Process..

[13]  A. Rantzer.  On the Kalman-Yakubovich-Popov lemma , 1996 .

[14]  P. Young.  Controller design with real parametric uncertainty , 1996 .

[15]  Gjerrit Meinsma,et al.  A Dual Formulation of Mixed and on the Losslessness of Scaling , 1998 .

[16]  T. Başar.  Absolute Stability of Nonlinear Systems of Automatic Control , 2001 .

[17]  Kunpeng Sun,et al.  Robust Linear Filter Design via LMIs and Controller Design with Actuator Saturation via SOS Programming , 2004 .

[18]  Lihua Xie,et al.  Robust Kalman filtering for uncertain discrete-time systems , 1994, IEEE Trans. Autom. Control..

[19]  Rolf Johansson,et al.  On the Kalman-Yakubovich-Popov Lemma for Stabilizable Systems , 2001 .

[20]  Harald K. Wimmer,et al.  On the algebraic Riccati equation , 1976, Bulletin of the Australian Mathematical Society.

[21]  R. Kálmán.  LYAPUNOV FUNCTIONS FOR THE PROBLEM OF LUR'E IN AUTOMATIC CONTROL. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[22]  M. Fu,et al.  A dual formulation of mixed μ and on the losslessness of (D, G) scaling , 1997, IEEE Trans. Autom. Control..

[23]  Roy M. Howard,et al.  Linear System Theory , 1992 .

[24]  Mary C. Brennan,et al.  on the , 1982 .

[25]  L. Pandolfi.  An observation on the positive real lemma , 2001 .

[26]  J. Willems,et al.  On the dissipativity of uncontrollable systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[27]  Lihua Xie,et al.  On robust filtering for linear systems with parameter uncertainty , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[28]  Brian D. O. Anderson,et al.  Network Analysis and Synthesis: A Modern Systems Theory Approach , 2006 .

[29]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.