Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

a b s t r a c t It has already been recognized that looking for a positive definite Lyapunov function such that a high- order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher- order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.

[1]  Ricardo C. L. F. Oliveira,et al.  Parameter-Dependent LMIs in Robust Analysis: Characterization of Homogeneous Polynomially Parameter-Dependent Solutions Via LMI Relaxations , 2007, IEEE Transactions on Automatic Control.

[2]  Ricardo C. L. F. Oliveira,et al.  Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions , 2008, Syst. Control. Lett..

[3]  Graziano Chesi,et al.  Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions , 2007, Autom..

[4]  Seyyed Kamaleddin Yadavar Nikravesh,et al.  A new theorem on higher order derivatives of Lyapunov functions. , 2009, ISA transactions.

[5]  Kiriakos Kiriakidis,et al.  Robust stabilization of the Takagi-Sugeno fuzzy model via bilinear matrix inequalities , 2001, IEEE Trans. Fuzzy Syst..

[6]  Graziano Chesi,et al.  Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach , 2005, IEEE Transactions on Automatic Control.

[7]  J. Bernussou,et al.  A new robust D-stability condition for real convex polytopic uncertainty , 2000 .

[8]  D. Peaucelle,et al.  Robust Performance Analysis of Linear Time-Invariant Uncertain Systems by Taking Higher-Order Time-Derivatives of the State , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[9]  G. Chesi.  Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems , 2009 .

[10]  M. Fu,et al.  Piecewise Lyapunov functions for robust stability of linear time-varying systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[11]  Liu Hsu,et al.  LMI characterization of structural and robust stability , 1998 .

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

[13]  Graziano Chesi,et al.  Homogeneous Lyapunov functions for systems with structured uncertainties , 2003, Autom..

[14]  Pedro Luis Dias Peres,et al.  A less conservative LMI condition for the robust stability of discrete-time uncertain systems , 2001, Syst. Control. Lett..

[15]  Zongli Lin,et al.  A descriptor system approach to robust stability analysis and controller synthesis , 2004, IEEE Trans. Autom. Control..

[16]  A. L. Zelentsovsky.  Nonquadratic Lyapunov functions for robust stability analysis of linear uncertain systems , 1994, IEEE Trans. Autom. Control..

[17]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[18]  Pedro Luis Dias Peres,et al.  An LMI condition for the robust stability of uncertain continuous-time linear systems , 2002, IEEE Trans. Autom. Control..

[19]  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.

[20]  Pedro Luis Dias Peres,et al.  An improved LMI condition for robust D-stability of uncertain polytopic systems , 2003, IEEE Trans. Autom. Control..

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

[22]  Karolos M. Grigoriadis,et al.  A unified algebraic approach to linear control design , 1998 .

[23]  G. Papavassilopoulos,et al.  Biaffine matrix inequality properties and computational methods , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[24]  Ricardo C. L. F. Oliveira,et al.  LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions , 2006, Syst. Control. Lett..

[25]  Arthur R. Butz,et al.  Higher order derivatives of Liapunov functions , 1969 .