As the analysis often relies on state space methods, an initial examination is made of the controllability/observability properties of this class of systems. A key difference to the case of purely linear systems is that one cannot globally stabilise an asymptotically unstable plant with a purely bounded feedback, which justifies the need for semi-global stability results.
Attention is then turned to operator based analysis, which is an extension of the sort used in linear H_\infty robust control theory. One generalisation of the $\Hf$ norm of a linear operator is the induced 2-norm of the nonlinear operator. The computation of this norm by the use of dissipation functions is examined in this dissertation.
By restricting the nonlinearity present to be the piecewise linear standard saturation function, the test conditions on a candidate dissipation function for a single input system are found to be a set of differential inequalities. Dissipation theory is also applied to discrete time systems. The test condition which results for discrete time systems is in a more implicit form, however. A suboptimal existence theorem is developed for a subclass of systems which shows the equivalence of finite gain stability and the convergence of a specific dissipation function.
A second generalisation of the H_\infty norm of a linear system is the incremental gain of the nonlinear system. A test condition to bound the incremental gain is developed using the solution of linear matrix inequalities. The computational advantage of using the easily computed linear matrix inequality solution is offset by the fact that it appears to be a quite conservative measure. A general construction is performed on discrete time systems which proves that open loop unstable plants cannot be stabilised in an incremental gain sense. Although the analysis has been made on a restricted class of systems, the basic principle of this lack of stabilisability seems readily generalisable.
Finally, analytic examples of the use of the techniques developed in this dissertation are examined. The inter-relationship between stability, the incremental gain and the induced norm is illustrated. Additionally, the importance of the use of bounded input sets in the induced norm computation is made evident.