Structured dissipativeness and absolute stability of nonlinear systems

The paper considers a problem of absolute stability for a nonlinear uncertain system with structured uncertainty. An S-procedure for nonlinear systems is established and, using this, the paper shows that the absolute stability problem is equivalent to an absolute stability problem for a nonlinear uncertain system with unstructured uncertainty. The paper also introduces a new notion of structured dissipativeness for nonlinear systems. This notion extends the standard notion of dissipativeness in that it allows for multiple supply functions. The paper shows that a nonlinear uncertain system containing structured uncertainties will be absolutely stable if and only if a corresponding structured dissipativeness condition is satisfied. It is also shown that structured dissipativeness is equivalent to the standard notion of dissipativeness with a parameter dependent supply function.