Exponential Stability of Gradient Systems with Applications to Nonlinear-in-Control Design Methods

Exponential stability analysis for gradient systems is the primary focus of this paper. Sufficient conditions are derived that guarantee exponential stability for both autonomous and parameter-dependent gradient systems. These conditions require boundedness of singular values of a Jacobian matrix, uniformly in the system state space. The reported theoretical results are subsequently applied to design tracking controllers for a class of nonlinear-in-control dynamical systems. The design is carried out using time-scale separation techniques.

[1]  Mehrdad Saif,et al.  An iterative learning observer for fault detection and accommodation in nonlinear time‐delay systems , 2006 .

[2]  Terry L. Friesz,et al.  Day-To-Day Dynamic Network Disequilibria and Idealized Traveler Information Systems , 1994, Oper. Res..

[3]  I. Sandberg.  Global inverse function theorems , 1980 .

[4]  Robert M. Sanner,et al.  Gaussian Networks for Direct Adaptive Control , 1991, 1991 American Control Conference.

[5]  L. Praly,et al.  Adaptive nonlinear regulation: estimation from the Lyapunov equation , 1992 .

[6]  J. Nazuno.  Haykin, Simon. Neural networks: A comprehensive foundation, Prentice Hall, Inc. Segunda Edición, 1999 , 2000 .

[7]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[8]  C. Desoer,et al.  Global inverse function theorem , 1972 .

[9]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[10]  Liqun Qi,et al.  A novel neural network for variational inequalities with linear and nonlinear constraints , 2005, IEEE Transactions on Neural Networks.

[11]  Simon Haykin,et al.  Neural Networks: A Comprehensive Foundation , 1998 .

[12]  Anders Lindquist,et al.  Interior point solutions of variational problems and global inverse function theorems , 2007 .

[13]  Xing-Bao Gao,et al.  Exponential stability of globally projected dynamic systems , 2003, IEEE Trans. Neural Networks.

[14]  E. Lavretsky,et al.  Dynamic inversion of multi-input nonaffine systems via time-scale separation , 2006, 2006 American Control Conference.