There are two competing generalisations of the H_\infty norm available for nonlinear systems, being the incremental gain, and the induced L_2 norm. The incremental gain has many advantages over the induced norm in a theoretical sense, but it appears to be a quite conservative measure. For the class of systems studied, open loop unstable plants cannot be stabilised in an incremental gain sense, which appears to limit the incremental gain's usefulness as an analytical tool.
The methods used to give bounds for the operator norms were developed elsewhere, although some have only appeared in the dissertation [Rom95]. For continuous time systems, the Small Gain Theorem (Circle Criterion) estimate, the bound given in the paper by Liu, Chitour, Sontag [LCS93], the bounds for the induced norm using dissipation functions as calculated by the method outlined in [Rom94], and the bounds for the incremental gain using linear matrix inequalities developed in [Rom95] are compared herein for the general first order plant.
The behaviour of the induced norm as the plant pole crosses from the closed left half plane to the open right half plane is pathological if one takes the input set to be L_2. The introduction of bounded input sets causes the induced norm to behave in a much more sensible fashion. It is believed that the use of such sets is necessary to give non-conservative analysis. This has been noted by other researchers, as in the paper by Lin, Saberi and Teel [LST].
Analysis techniques developed in [Rom95] are used to compute the induced l_2 norm of a first order discrete time example. Many of the results on continuous time induced norm computation in [Rom94] have been extended to discrete time, but it is not possible to give the results in as clean a form.
Finally, a preliminary examination of the state feedback synthesis problem has been made. Two differing approaches to state feedback synthesis are illustrated for a first order example. The first method is an extension of that which appears in the paper [vdS92] by A.J. van der Schaft, and which relies on the use of differential inequalities. Even the first order example shows some difficulties with a naive extension of the methods in [vdS92] to this class of systems. The second method relies on a parameterised piecewise linear controller, and the analysis undertaken to fix suitable parameter values. The advantage of this approach is that it allows one to cleanly force the linearisation of the controller to be the linear controller derived by the much better understood linear design methodology, and the goal of the nonlinear synthesis is to best preserve the linear operating characteristics. In other words, this is viewed to be a preliminary attempt of using nonlinear H_\infty methods to synthesize anti-windup schemes.
[LST] Z. Lin, A. Saberi, and A.R. Teel. Simultaneous L_p-stabilization and Internal stabilization of linear systems subject to input saturation - state feedback case. Submitted for journal publication.
[LCS93] W.Liu, Y.Chitour, and E.D. Sontag. Remarks on finite gain stabilizability of linear systems subject to input saturation. In Proc. IEEE Conf. Dec. and Cont., pages 1808--1813, 1993.
[Rom94] B.G. Romanchuk. On the computation of the induced L_2 norm of single input linear systems with saturation. In Proc. IEEE Conf. Dec. and Cont., pages 1427--1432, 1994.
[Rom95] B.G. Romanchuk. Input-Output Analysis of Feedback Loops with Saturation Nonlinearities. Ph. D. Dissertation, University of Cambridge, 1995.
[vdS92] A.J. van~der Schaft. L_2-gain analysis of nonlinear systems and nonlinear state feedback H_\infty control. IEEE Trans. Automat. Contr., 37(6):770--784, 1992.