Local stability analysis of nonlinear systems

This paper considers local stability properties of systems comprising stable linear time-invariant operators in combination with a scalar nonlinearity. We consider those nonlinearities whose gain can be related to the peak value of their input signal. It is assumed that the nonlinearity has some nominal gain for small signals (ie with peak value less than some number), and that the gain then increases for larger inputs. It is shown that there is a class of exogenous inputs, characterised by their energy, such that all signals in the system are bounded, and the effective gain of the nonlinearity is no greater than the nominal value. It is further shown that, providing a stated condition is satisfied, there is a larger class of exogenous inputs, again characterised by their energy, such that all signals in the system are bounded. This condition is shown to be an inequality between known parameters of the nonlinearity and the H2and H∞-norms of the linear parts of the system.