Stabilization of discrete-time nonlinear systems by smooth state feedback

Abstract For a class of nonlinear discrete-time systems of the form Σ: x(k+1)=ƒ(x(k))+g(x(k))u(k) , we investigate conditions under which a nonlinear system can be rendered globally asymptotically stable via smooth state feedback. Our main result is that any nonlinear system with Lyapunov-stable unforced dynamics can always be globally stabilized by smooth state feedback if suitable controllability-like rank conditions are satisfied. Several examples are presented to demonstrate the applications of the stability results developed in this paper.

[1]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[2]  Nicholas Kalouptsidis,et al.  Output Feedback Stabilization of Discrete-Time Control Systems , 1990 .

[3]  J. Carr.  Applications of Centre Manifold Theory , 1981 .

[4]  A. Isidori,et al.  Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems , 1991 .

[5]  Aristotle Arapostathis,et al.  Remarks on smooth feedback stabilization of nonlinear systems , 1988 .

[6]  Nicholas Kalouptsidis,et al.  Stability improvement of nonlinear systems by feedback , 1984 .

[7]  P. Kokotovic,et al.  A positive real condition for global stabilization of nonlinear systems , 1989 .

[8]  P. Kokotovic,et al.  Global stabilization of partially linear composite systems , 1990 .

[9]  R. E. Kalman,et al.  Control System Analysis and Design Via the “Second Method” of Lyapunov: II—Discrete-Time Systems , 1960 .

[10]  V. Jurdjevic,et al.  Controllability and stability , 1978 .

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

[12]  John Tsinias,et al.  Stabilizability of Discrete-Time Nonlinear Systems , 1989 .

[13]  A. Isidori,et al.  Local stabilization of minimum-phase nonlinear systems , 1988 .

[14]  D. Aeyels.  Stabilization of a class of nonlinear systems by a smooth feedback control , 1985 .

[15]  A. Isidori,et al.  Asymptotic stabilization of minimum phase nonlinear systems , 1991 .

[16]  Wei Lin,et al.  Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems , 1994, IEEE Trans. Autom. Control..

[17]  A. Isidori,et al.  New results and examples in nonlinear feedback stabilization , 1989 .

[18]  A. Isidori.  Nonlinear Control Systems , 1985 .