Universal formula for robust stabilization of affine nonlinear multistable systems

In this paper, the problem of robust stabilization of affine nonlinear multistable systems with respect to disturbance inputs is studied. The results are obtained using the framework of input-to-state stability (ISS) and integral input-to-state stability (iISS) for systems with multiple invariant sets. The notions of ISS and iISS control Lyapunov functions as well as the small control property are extended within the multistability framework. It is verified that the universal control formula can be applied to yield the ISS (iISS) property to the closed- loop system. The efficiency of the proposed control Lyapunov function in the multistable sense is illustrated in two academic examples.

[1]  G. Leonov,et al.  Hidden attractors in dynamical systems , 2016 .

[2]  Rodolphe Sepulchre,et al.  Differential analysis of nonlinear systems: Revisiting the pendulum example , 2014, 53rd IEEE Conference on Decision and Control.

[3]  Romeo Ortega,et al.  A relaxed characterization of ISS for periodic systems with multiple invariant sets , 2017, Eur. J. Control.

[4]  V. I. Vorotnikov.  Partial stability and control , 1998 .

[5]  U. Feudel,et al.  Control of multistability , 2014 .

[6]  Rodolphe Sepulchre,et al.  Analysis of Interconnected Oscillators by Dissipativity Theory , 2007, IEEE Transactions on Automatic Control.

[7]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[8]  Alexander L. Fradkov,et al.  Control of oscillatory behavior of multispecies populations , 2012 .

[9]  Frank Allgöwer,et al.  Bistable Biological Systems: A Characterization Through Local Compact Input-to-State Stability , 2008, IEEE Transactions on Automatic Control.

[10]  Eduardo Sontag,et al.  Untangling the wires: A strategy to trace functional interactions in signaling and gene networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Denis Efimov,et al.  A condition of CLF existence for affine systems , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[12]  Eduardo D. Sontag,et al.  Input to state stability and allied system properties , 2011 .

[13]  David Angeli,et al.  An almost global notion of input-to-State stability , 2004, IEEE Transactions on Automatic Control.

[14]  V. Vorotnikov.  On stability and stabilization of motion with respect to a part of the variables , 1982 .

[15]  Denis V. Efimov.  Global Lyapunov Analysis of Multistable Nonlinear Systems , 2012, SIAM J. Control. Optim..

[16]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .

[17]  Eduardo Sontag,et al.  New characterizations of input-to-state stability , 1996, IEEE Trans. Autom. Control..

[18]  Eduardo D. Sontag,et al.  Monotone systems under positive feedback: multistability and a reduction theorem , 2005, Syst. Control. Lett..

[19]  David Angeli,et al.  The ISS approach to the stability and robustness properties of nonautonomous systems with decomposable invariant sets: An overview , 2016, Eur. J. Control.

[20]  David Angeli,et al.  Characterizations of Input-to-State Stability for Systems With Multiple Invariant Sets , 2015, IEEE Transactions on Automatic Control.

[21]  Eduardo Sontag.  Comments on integral variants of ISS , 1998 .

[22]  M Laurent,et al.  Multistability: a major means of differentiation and evolution in biological systems. , 1999, Trends in biochemical sciences.

[23]  Eduardo Sontag,et al.  On characterizations of the input-to-state stability property , 1995 .

[24]  Denis Efimov,et al.  UNIVERSAL FORMULA FOR OUTPUT ASYMPTOTIC STABILIZATION , 2002 .

[25]  Denis Efimov,et al.  Robust Global Synchronization of Brockett Oscillators , 2019, IEEE Transactions on Control of Network Systems.

[26]  N. Rouche,et al.  Stability Theory by Liapunov's Direct Method , 1977 .

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

[28]  David Angeli,et al.  A characterization of integral input-to-state stability , 2000, IEEE Trans. Autom. Control..

[29]  Z. Artstein.  Stabilization with relaxed controls , 1983 .

[30]  Daniel Liberzon,et al.  Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation , 2002, Syst. Control. Lett..

[31]  Eduardo Sontag.  A universal construction of Artstein's theorem on nonlinear stabilization , 1989 .