Regional Analysis of Slope-Restricted Lurie Systems

This paper considers the stability analysis of nonlinear Lurie type systems where the nonlinearity is both (locally) sector and slope restricted. Convex conditions for verifying stability, computing outer estimates of reachable sets, and upper bounds on the induced <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_2$</tex-math></inline-formula> gain in a local or global domain are proposed. The conditions use a Lyapunov function that is quadratic on both the states and the nonlinearity and has an integral term on the nonlinearity. Numerical examples outline the benefits of the proposed approach.

[1]  P. Falb,et al.  Stability Conditions for Systems with Monotone and Slope-Restricted Nonlinearities , 1968 .

[2]  S. Weissenberger,et al.  Application of results from the absolute stability problem to the computation of finite stability domains , 1968 .

[3]  On the stability of feedback systems with one diffrentiable nonlinear element , 1966 .

[4]  P. Park.  A revisited Popov criterion for nonlinear Lur'e systems with sector-restrictions , 1997 .

[5]  J. Suykens,et al.  An absolute stability criterion for the Lur'e problem with sector and slope restricted nonlinearities , 1998 .

[6]  Matthew C. Turner,et al.  Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches , 2016, Eur. J. Control.

[7]  Luca Zaccarian,et al.  Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations , 2009, Syst. Control. Lett..

[8]  Luca Zaccarian,et al.  A common framework for anti-windup, bumpless transfer and reliable designs , 2002, Autom..

[9]  J. A. Walker,et al.  Finite regions of attraction for the problem of Lur'e. , 1967 .

[10]  Aristide Halanay,et al.  Absolute stability of feedback systems with several differentiable non-linearities , 1991 .

[11]  Michael G. Safonov,et al.  Computer-aided stability analysis renders Papov criterion obsolete , 1987 .

[12]  Stephen P. Boyd,et al.  Analysis of linear systems with saturation using convex optimization , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[13]  Guang Li,et al.  Lyapunov functions for the multivariable Popov criterion with indefinite multipliers , 2009, Autom..

[14]  Fan Zhang,et al.  Dynamic Feedback Synchronization of Lur'e Networks via Incremental Sector Boundedness , 2016, IEEE Transactions on Automatic Control.

[15]  R. O'Shea.  An improved frequency time domain stability criterion for autonomous continuous systems , 1966, IEEE Transactions on Automatic Control.

[16]  M. Safonov,et al.  Zames-Falb multipliers for MIMO nonlinearities , 2000 .

[17]  P. Olver.  Nonlinear Systems , 2013 .

[18]  John Hauser,et al.  Dynamics of a driven Stirling engine , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[19]  G. V. S. Raju,et al.  Absolute stability of control systems with many sector and slope-restricted non-linearities , 1974 .

[20]  Ross Drummond,et al.  Positivity conditions of Lyapunov functions for systems with slope restricted nonlinearities , 2016, 2016 American Control Conference (ACC).

[21]  Tingshu Hu,et al.  Absolute stability with a generalized sector condition , 2004, IEEE Transactions on Automatic Control.

[22]  Carsten W. Scherer,et al.  Zames–Falb Multipliers for Invariance , 2017, IEEE Control Systems Letters.

[23]  Alexandre Megretski,et al.  New results for analysis of systems with repeated nonlinearities , 2001, Autom..

[24]  U. Jönsson.  Stability analysis with Popov multipliers and integral quadratic constraints , 1997 .

[25]  J. P. Lasalle,et al.  Absolute Stability of Regulator Systems , 1964 .

[26]  Alexander Lanzon,et al.  LMI searches for anticausal and noncausal rational Zames-Falb multipliers , 2014, Syst. Control. Lett..

[27]  Luca Zaccarian,et al.  Global Stability and Finite L2m-Gain of Saturated Uncertain Systems via Piecewise Polynomial Lyapunov Functions , 2013, IEEE Transactions on Automatic Control.

[28]  Matthew C. Turner,et al.  ℒ2 gain bounds for systems with slope-restricted nonlinearities , 2010, Proceedings of the 2010 American Control Conference.

[29]  PooGyeon Park,et al.  Stability criteria of sector- and slope-restricted Lur'e systems , 2002, IEEE Trans. Autom. Control..

[30]  V. R. Sastry,et al.  Finite regions of attraction for the problem of Lur'e† , 1971 .