This paper presents an approach to robustness analysis for nonlinear feedback systems. We pursue a notion of model uncertainty based on the closeness of input-output trajectories which is not tied to a particular uncertainty representation, such as additive, parametric, structured, etc. The basic viewpoint is to regard systems as operators on signal spaces. We present two versions of a global theory where stability is captured by induced norms or by gain functions. We also develop local approaches (over bounded signal sets) and give a treatment for systems with potential for finite-time escape. We compute the relevant stability margin for several examples and demonstrate robustness of stability for some specific perturbations, e.g., small-time delays. We also present examples of nonlinear control systems which have zero robustness margin and are destabilized by arbitrarily small gap perturbations. The paper considers the case where uncertainty is present in the controller as well as the plant and the generalization of the approach to the case where uncertainty occurs in several subsystems in an arbitrary interconnection.
Bibliographical noteFunding Information:
Manuscript received April 12, 1996; revised March 10, 1997. Recommended by Associate Editor, J. Shamma. This work was supported in part by the NSF, AFOSR, EPSRC, and NATO. T. T. Georgiou is with the Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455 USA. M. C. Smith is with the Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, U.K. Publisher Item Identifier S 0018-9286(97)06594-X.
- Gap metric
- Nonlinear systems
- Robust control