Stability of uncertain systems using Lyapunov functions with non-monotonic terms

Abstract This paper is concerned with the problem of robust stability of uncertain linear time-invariant systems in polytopic domains. The main contribution is to present a systematic procedure to check the stability of the uncertain systems by using an arbitrary number of quadratic functions within higher order derivatives of the vector field in the continuous-time case and higher order differences of the vector field in the discrete-time case. The matrices of the Lyapunov function appear decoupled from the dynamic matrix of the system in the conditions. This fact leads to sufficient conditions that are given in terms of Linear Matrix Inequalities defined at the vertices of the polytope. The proposed method does not impose sign condition constraints in the quadratic functions that compose the Lyapunov function individually. Moreover, some of the quadratic functions do not decrease monotonically along trajectories. However, if the sufficient conditions are satisfied, then a monotonic standard Lyapunov function that depends on the dynamics of the uncertain system can be constructed a posteriori. Numerical examples from the literature are provided to illustrate the proposed approach.

[1]  Graziano Chesi,et al.  Exact robust stability analysis of uncertain systems with a scalar parameter via LMIs , 2013, Autom..

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

[3]  Jin Bae Park,et al.  Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems , 2011, Syst. Control. Lett..

[4]  Alessandro Astolfi,et al.  Dynamic Lyapunov functions , 2013, Autom..

[5]  Dimitri Peaucelle,et al.  S-Variable Approach to LMI-Based Robust Control , 2014 .

[6]  J. Lasserre,et al.  On parameter-dependent Lyapunov functions for robust stability of linear systems , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[7]  Amir Ali Ahmadi,et al.  Non-monotonic Lyapunov functions for stability of discrete time nonlinear and switched systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[8]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[9]  Tetsuya Iwasaki,et al.  Lyapunov-based exact stability analysis and synthesis for linear single-parameter dependent systems , 2010, Int. J. Control.

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

[11]  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.

[12]  Carsten W. Scherer,et al.  LMI Relaxations in Robust Control , 2006, Eur. J. Control.

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

[14]  Graziano Chesi.  Instability Analysis of Uncertain Systems via Determinants and LMIs , 2015, IEEE Transactions on Automatic Control.

[15]  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.

[16]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[17]  Valter J. S. Leite,et al.  An improved LMI condition for robust D-stability of uncertain polytopic systems , 2003, Proceedings of the 2003 American Control Conference, 2003..

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

[19]  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..

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

[21]  B. Barmish.  Necessary and sufficient conditions for quadratic stabilizability of an uncertain system , 1985 .

[22]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

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

[24]  J. Geromel,et al.  A new discrete-time robust stability condition , 1999 .

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

[26]  Robert E. Skelton,et al.  Stability tests for constrained linear systems , 2001 .

[27]  Graziano Chesi.  On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems , 2008, Autom..

[28]  D. Henrion,et al.  QUADRATIC SEPARATION FOR FEEDBACK CONNECTION OF AN UNCERTAIN MATRIX AND AN IMPLICIT LINEAR TRANSFORMATION , 2005 .

[29]  Amir Ali Ahmadi,et al.  On higher order derivatives of Lyapunov functions , 2011, Proceedings of the 2011 American Control Conference.