Kalman-Popov-Yakubovich lemma and the S-procedure: A historical essay

A history of two fundamental results of the mathematical system theory—the Kalman-Popov-Yakubovich lemma and the theorem of losslessness of the S-procedure—was presented. The studies directly concerned with these statements were reviewed. The recent publications using the theorem of losslessness of the S-procedure to derive the Kalman-Popov-Yakubovich lemma and its generalizations were analyzed.

[1]  F. Hausdorff.  Der Wertvorrat einer Bilinearform , 1919 .

[2]  Paul Pinsler.  Über das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen , 1936 .

[3]  L. L. Dines.  On the mapping of quadratic forms , 1941 .

[4]  Fehmi Cirak,et al.  ZEITSCHRIFT FÜR ANGEWANDTE MATHEMATIK UND MECHANIK , 1952, Über Stammfaktoren bei ternären quadratischen Formen.

[5]  R. E. Kalman,et al.  Contributions to the Theory of Optimal Control , 1960 .

[6]  L. Brickman.  ON THE FIELD OF VALUES OF A MATRIX , 1961 .

[7]  V. A. Yakubovich.  ON ONE CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS , 1961 .

[8]  Hans Schneider,et al.  Some theorems on the inertia of general matrices , 1962 .

[9]  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.

[10]  K. Meyer,et al.  LIAPUNOV FUNCTIONS FOR THE PROBLEM OF LUR'E. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[11]  B. Zh. Maygarin.  The absolute stability of nonlinear automatic control systems , 1965 .

[12]  S. Lefschetz.  Stability of nonlinear control systems , 1966 .

[13]  P. Parks,et al.  Liapunov redesign of model reference adaptive control systems , 1966 .

[14]  B. Anderson.  A SYSTEM THEORY CRITERION FOR POSITIVE REAL MATRICES , 1967 .

[15]  P. K. C. Wang,et al.  Theory of Stability and Control for Distributed Parameter Systems (a Bibliography) , 1968 .

[16]  Brian D. O. Anderson,et al.  Algebraic Structure of Generalized Positive Real Matrices , 1968 .

[17]  J. Lions,et al.  Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles , 1968 .

[18]  P. Faurre,et al.  Identification par Minimisation d'une Representation Markovienne de Processus Aleatoire , 1970 .

[19]  P. K. C. Wang,et al.  A control-theoretic approach to the plasma confinement problem , 1970 .

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

[21]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[22]  L. Silverman,et al.  On the discrete time algebraic Riccati equation , 1973 .

[23]  R. Monopoli.  The Kalman-Yacubovich lemma in adaptive control system design , 1973 .

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

[25]  B. Molinari.  The Stabilizing Solution of the Algebraic Riccati Equation , 1973 .

[26]  V. Andreev,et al.  Synthesis of optimal controls for amplitudeimpulse systems in the problem of minimizing the mean value of a functional of quadratic type , 1973 .

[27]  Alexander L. Fradkov,et al.  Duality theorems for certain nonconvex extremal problems , 1973 .

[28]  Jan C. Willems,et al.  On the existence of a nonpositive solution to the Riccati equation , 1974 .

[29]  B. Molinari.  Conditions for nonpositive solutions of the linear matrix inequality , 1975 .

[30]  J. William Helton.  A Spectral Factorization Approach to the Distributed Stable Regular Problem; the Algebraic Riccati Equation , 1976 .

[31]  P. Moylan,et al.  The stability of nonlinear dissipative systems , 1976 .

[32]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[33]  V. V. Chalam,et al.  Lyapunov Redesign of Model Reference Adaptive Control Systems (Part I) , 1979 .

[34]  Frequency theorem and the Lur'e equation , 1979 .

[35]  P. Moylan,et al.  Dissipative Dynamical Systems: Basic Input-Output and State Properties , 1980 .

[36]  P. Lancaster,et al.  Existence and uniqueness theorems for the algebraic Riccati equation , 1980 .

[37]  Alain Bensoussan,et al.  Impulse Control and Quasi-Variational Inequalities , 1984 .

[38]  Franco Flandoli,et al.  Riccati Equation Arising in a Boundary Control Problem with Distributed Parameters , 1984 .

[39]  R. Triggiani,et al.  Algebraic Riccati equations arising in boundary/point control: A review of theoretical and numerical results Part II: Approximation theory , 1990 .

[40]  Irena Lasiecka,et al.  Algebraic Riccati equations arising in boundary/point control: A review of theoretical and numerical results Part I: Continuous case , 1990 .

[41]  J. Willems.  Paradigms and puzzles in the theory of dynamical systems , 1991 .

[42]  S. Bittanti,et al.  The Riccati equation , 1991 .

[43]  A. Isidori,et al.  Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems , 1991 .

[44]  D. Hill,et al.  Dissipative nonlinear systems: basic properties and stability analysis , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[45]  Lihua Xie,et al.  On the Discrete-time Bounded Real Lemma with application in the characterization of static state feedback H ∞ controllers , 1992 .

[46]  V. Yakubovich.  Nonconvex optimization problem: the infinite-horizon linear-quadratic control problem with quadratic constraints , 1992 .

[47]  Michael Heiss,et al.  Inverse Passive Learning of an Input-Output-Map Through Update-Spline-Smoothing , 1992, 1992 American Control Conference.

[48]  Ian R. Petersen,et al.  Nonlinear versus linear control in the absolute stabilizability of uncertain systems with structured uncertainty , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[49]  Lagrange multipliers method in robust control: the l/sup 1/-setting , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[50]  Pascal Gahinet,et al.  H/sub /spl infin// design with pole placement constraints: an LMI approach , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[51]  P. Gahinet.  Explicit controller formulas for LMI-based H/sub /spl infin// synthesis , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[52]  P. Gahinet,et al.  A linear matrix inequality approach to H∞ control , 1994 .

[53]  Alexander I. Barvinok,et al.  Problems of distance geometry and convex properties of quadratic maps , 1995, Discret. Comput. Geom..

[54]  Henry Wolkowicz,et al.  Indefinite Trust Region Subproblems and Nonsymmetric Eigenvalue Perturbations , 1995, SIAM J. Optim..

[55]  David J. Hill,et al.  Nonlinear adaptive control of feedback passive systems , 1995, Autom..

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

[57]  Luciano Pandolfi,et al.  Input dynamics and nonstandard riccati equations with applications to boundary control of damped wave and plate equations , 1995 .

[58]  Equivalent Conditions for the Solvability of the Nonstandard LQ -Problem for Pritchard--Salamon Systems , 1995 .

[59]  Ruth F. Curtain.  The Kalman-Yakubovich-Popov lemma for Pritchard-Salamon systems , 1996 .

[60]  Vlad Ionescu,et al.  GENERALIZED CONTINUOUS-TIME RICCATI THEORY , 1996 .

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

[62]  Damir Z. Arov,et al.  Passive linear stationary dynamical scattering systems with continuous time , 1996 .

[63]  Pascal Gahinet,et al.  Explicit controller formulas for LMI-based H∞ synthesis , 1996, Autom..

[64]  P. Gahinet,et al.  H∞ design with pole placement constraints: an LMI approach , 1996, IEEE Trans. Autom. Control..

[65]  Marc Teboulle,et al.  Hidden convexity in some nonconvex quadratically constrained quadratic programming , 1996, Math. Program..

[66]  N. E. Barabanov.  On quadratic stabilizability of linear dynamical systems , 1996 .

[67]  O. E. Flippo,et al.  Duality and sensitivity in nonconvex quadratic optimization over an ellipsoid , 1996 .

[68]  Francesca Bucciy.  Frequency Domain Stability of Nonlinear Feedback Systems with Unbounded Input Operator , 1997 .

[69]  L. Pandolfi.  The Kalman-Popov-Yakubovich Theorem: an overview and new results for hyperbolic control systems , 1997 .

[70]  Andrey E. Barabanov,et al.  Canonical Matrix Factorisation and Polynomial Riccati Equations , 1997, Eur. J. Control.

[71]  Quasidissipative systems with one or several supply rates , 1997, 1997 European Control Conference (ECC).

[72]  An optimal control problem with unbounded control operator and unbounded observation operator where the Algebraic Riccati Equation is satisfied as a Lyapunov equation , 1997 .

[73]  C. Scherer,et al.  Multiobjective output-feedback control via LMI optimization , 1997, IEEE Trans. Autom. Control..

[74]  A. Rantzer,et al.  System analysis via integral quadratic constraints , 1997, IEEE Trans. Autom. Control..

[75]  Boris Polyak.  Convexity of Quadratic Transformations and Its Use in Control and Optimization , 1998 .

[76]  Luciano Pandolfi,et al.  Dissipativity and the Lur'e Problem for Parabolic Boundary Control Systems , 1998 .

[77]  A. Matveev.  Spectral Approach to Duality in Nonconvex Global Optimization , 1998 .

[78]  A. Megretski,et al.  Integral quadratic constraints for monotonic and slope restricted diagonal operators , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[79]  Luciano Pandolfi,et al.  The Kalman-Yakubovich-Popov theorem for stabilizable hyperbolic boundary control systems , 1999 .

[80]  Rf Curtain.  Old and new perspectives on the positive-real-lemma in systems and control theory , 1999 .

[81]  D. J. Clements,et al.  A State-Space Approach to Indefinite Spectral Factorization , 2000, SIAM J. Matrix Anal. Appl..

[82]  L. Pandolfi.  Recent results on the Kalman-Popov-Yakubovich problem , 2000 .

[83]  Irena Lasiecka,et al.  Extended algebraic Riccati equations in the abstract hyperbolic case , 2000 .

[84]  Petre Stoica,et al.  MA estimation in polynomial time , 2000, IEEE Trans. Signal Process..

[85]  T. Iwasaki,et al.  Generalized S-procedure and finite frequency KYP lemma , 2000 .

[86]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[87]  R. Curtain,et al.  The Popov criterion for strongly stable distributed parameter systems , 2001 .

[88]  L. Pandolfi.  Factorization of the Popov function of a multivariable linear distributed parameter systems in the non-coercive case: a penalization approach , 2001 .

[89]  J.-B. Hiriart-Urruty,et al.  Permanently Going Back and Forth between the ``Quadratic World'' and the ``Convexity World'' in Optimization , 2002 .

[90]  Volker Mehrmann,et al.  Existence, Uniqueness, and Parametrization of Lagrangian Invariant Subspaces , 2001, SIAM J. Matrix Anal. Appl..

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

[92]  F. Bucci.  Bucci THE NON-STANDARD LQR PROBLEM FOR BOUNDARY CONTROL SYSTEMS , 2003 .

[93]  H. Abou-Kandil,et al.  Matrix Riccati Equations in Control and Systems Theory , 2003, IEEE Transactions on Automatic Control.

[94]  Romeo Ortega,et al.  On the solvability of extended Riccati equations , 2004, IEEE Transactions on Automatic Control.

[95]  Shinji Hara,et al.  Generalized KYP lemma: unified frequency domain inequalities with design applications , 2005, IEEE Transactions on Automatic Control.

[96]  Alexander L. Fradkov.  Conic S-Procedure And Constrained Dissipativity , 2005 .

[97]  D. Arov,et al.  The Infinite-Dimensional Continuous Time Kalman - Yakubovich - Popov Inequality , 2005, CDC/ECC.

[98]  Shinji Hara,et al.  Time domain interpretations of frequency domain inequalities on (semi)finite ranges , 2005, Syst. Control. Lett..

[99]  S. Gusev,et al.  Extended Kalman-Yakubovich-Popov Lemma in a Hilbert Space and Fenchel Duality. , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[100]  D. Arov,et al.  The Infinite-Dimensional Continuous Time Kalman–Yakubovich–Popov Inequality , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[101]  L. Rodman,et al.  Matrix Riccati Equations in Control and Systems Theory [Book Report] , 2005, IEEE Control Systems.

[102]  V. Yakubovich,et al.  Stability of Stationary Sets in Control Systems With Discontinuous Nonlinearities , 2004, IEEE Transactions on Automatic Control.

[103]  S. V. Gusev.  The Fenchel duality, S-procedure, and the Yakubovich-Kalman lemma , 2006 .

[104]  A. Matveev.  Theory of optimal control in the works of V.A. Yakubovich , 2006 .

[105]  A. Churilov,et al.  Frequency methods in the theory of pulse-modulated control systems , 2006 .

[106]  Alexander L. Fradkov,et al.  Method of passification in adaptive control, estimation, and synchronization , 2006 .

[107]  M. R. Liberzon.  Essays on the absolute stability theory , 2006 .

[108]  Damir Z. Arov,et al.  The Kalman-Yakubovich-Popov inequality for discrete time systems of infinite dimension , 2006 .

[109]  V. Ugrinovskii,et al.  Stochastic problems of absolute stability , 2006 .

[110]  Tamás Terlaky,et al.  A Survey of the S-Lemma , 2007, SIAM Rev..