Polytopic control Lyapunov functions for robust stabilization of a class of nonlinear systems

Abstract We develop a method for computing a region in state space over which a nonlinear system is guaranteed by a given polytopic control Lyapunov function to be stable in closed loop under some appropriate control law. For systems which are nonlinear in only a few state variables, the procedure is computationally tractable; the computation time required to evaluate stability over each cone comprising a level set of the Lyapunov function is exponential in the number of “nonlinear states” but otherwise polynomial in the dimension of the full state space. Control constraints and robustness to bounded disturbances are easily incorporated.

[1]  Hiromasa Haneda,et al.  Computer generated Lyapunov functions for a class of nonlinear systems , 1993 .

[2]  George Bitsoris,et al.  Comparison principle, positive invariance and constrained regulation of nonlinear systems , 1995, Autom..

[3]  Eduardo Sontag.  A Lyapunov-Like Characterization of Asymptotic Controllability , 1983, SIAM Journal on Control and Optimization.

[4]  S. Battilotti,et al.  Stabilization via dynamic output feedback for systems with output nonlinearities , 1994 .

[5]  Marc Wayne McConley.  A computationally efficient Lyapunov-based procedure for control of nonlinear systems with stability and performance guarantees , 1997 .

[6]  Eugênio B. Castelan,et al.  Eigenstructure assignment for state constrained linear continuous time systems , 1992, Autom..

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

[8]  Franco Blanchini,et al.  Nonquadratic Lyapunov functions for robust control , 1995, Autom..

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

[10]  Jean B. Lasserre,et al.  Reachable, controllable sets and stabilizing control of constrained linear systems , 1993, Autom..

[11]  Robert K. Brayton,et al.  Constructive stability and asymptotic stability of dynamical systems , 1980 .

[12]  V. Vittal,et al.  Computer generated Lyapunov functions for interconnected systems: Improved results with applications to power systems , 1983, The 22nd IEEE Conference on Decision and Control.

[13]  Per-olof Gutman,et al.  Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states , 1984, The 23rd IEEE Conference on Decision and Control.

[14]  J. Tsinias.  Versions of Sontag's “Input to state stability condition” and the global stabilizability problem , 1993 .

[15]  Yuandan Lin,et al.  A universal formula for stabilization with bounded controls , 1991 .

[16]  Yuandan Lin,et al.  Recent results on Lyapunov-theoretic techniques for nonlinear stability , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[17]  F. Blanchini,et al.  Constrained stabilization of continuous-time linear systems , 1996 .

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

[19]  G. Bitsoris,et al.  Constrained regulation of linear continuous-time dynamical systems , 1989 .

[20]  P. Kokotovic,et al.  Inverse Optimality in Robust Stabilization , 1996 .

[21]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[22]  E. Feron,et al.  A control Lyapunov function approach to robust stabilization of nonlinear systems , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).