Gap metrics, representations, and nonlinear robust stability

Various alternative definitions for the nonlinear L/sub 2/- and /spl nu/-gap metrics are studied. The concept of /spl beta/-conjugacy and multiplicative homogeneity are introduced to relate the metrics to each other and to compare the stability margins of nonlinear feedback loops expressed in terms of the norms of complementary parallel projections. Left and right representations for the graph of a nonlinear system are studied. A new definition of "normalized" is introduced for left representations. Formulas for the gap metrics as the norm of the product of left and right representations are derived.

[1]  Mathukumalli Vidyasagar,et al.  Robust controllers for uncertain linear multivariable systems , 1984, Autom..

[2]  K. Glover,et al.  Robust stabilization of normalized coprime factor plant descriptions with H/sub infinity /-bounded uncertainty , 1989 .

[3]  Arjan van der Schaft,et al.  The class of stabilizing nonlinear plant controller pairs , 1996, IEEE Trans. Autom. Control..

[4]  J. Helton,et al.  H∞ control for nonlinear systems with output feedback , 1993, IEEE Trans. Autom. Control..

[5]  Glenn Vinnicombe,et al.  A /spl nu/-gap distance for uncertain and nonlinear systems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[6]  Raimund J. Ober,et al.  On the gap metric and coprime factor perturbations , 1993, Autom..

[7]  Tryphon T. Georgiou,et al.  The parallel projection operators of a nonlinear feedback system , 1993 .

[8]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[9]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[10]  G. Vinnicombe.  Frequency domain uncertainty and the graph topology , 1993, IEEE Trans. Autom. Control..

[11]  M. James,et al.  Robust H∞ output feedback control for nonlinear systems , 1995, IEEE Trans. Autom. Control..

[12]  Keith Glover,et al.  Robust control design using normal-ized coprime factor plant descriptions , 1989 .

[13]  A. Isidori,et al.  Disturbance attenuation and H/sub infinity /-control via measurement feedback in nonlinear systems , 1992 .

[14]  Pierpaolo Soravia.  ${\cal H}_\infty$ Control of Nonlinear Systems: Differential Games and Viscosity Solutions , 1996 .

[15]  Tryphon T. Georgiou,et al.  On the computation of the gap metric , 1988 .

[16]  Arjan van der Schaft,et al.  Robust stabilization of nonlinear systems via stable kernel representations with L 2 -gain bounded uncertainty , 1995 .

[17]  A. Isidori.  H∞ control via measurement feedback for affine nonlinear systems , 1994 .

[18]  Tryphon T. Georgiou,et al.  Robust stability of feedback systems: a geometric approach using the gap metric , 1993 .

[19]  T. Georgiou,et al.  Optimal robustness in the gap metric , 1990 .

[20]  T. Georgiou,et al.  Robustness analysis of nonlinear feedback systems: an input-output approach , 1997, IEEE Trans. Autom. Control..

[21]  van der Arjan Schaft,et al.  Normalized coprime factorizations and balancing for unstable nonlinear systems , 1994 .

[22]  M. R. James,et al.  Partially Observed Differential Games, Infinite-Dimensional Hamilton--Jacobi--Isaacs Equations, and Nonlinear $H_\infty$ Control , 1996 .

[23]  T. Basar,et al.  H∞-0ptimal Control and Related Minimax Design Problems: A Dynamic Game Approach , 1996, IEEE Trans. Autom. Control..

[24]  David J. N. Limebeer,et al.  Linear Robust Control , 1994 .

[25]  M. James,et al.  Extending H-infinity Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives , 1987 .

[26]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[27]  Brian D. O. Anderson,et al.  Robust Stabilization of Nonlinear Systems via Normalized Coprime Factor Representations , 1998, Autom..

[28]  A. Schaft.  L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences 218 , 1996 .

[29]  Louis R. Hunt,et al.  Right coprime factorizations and stabilization for nonlinear systems , 1993, IEEE Trans. Autom. Control..