Multipliers for memoryless incrementally positive MIMO nonlinearities

This thesis concerns stability analysis of systems featuring incrementally positive multi-input-multi-output (MIMO) nonlinearities. Classes of multipliers, i.e. convolution operators, that preserve positivity of such nonlinearities are derived. Stability of the system on hand can then be established using the standard integral quadratic constraint (IQC) theorem.; The single-input-single-output (SISO) case of such problems has received a considerable attention over the last four decades, the celebrated works of Zames, Falb, Popov and Willems being the cornerstone results. However, a rigorous generic treatment of the MIMO case has been conspicuous by its absence. This thesis is devoted to a study of the MIMO case. Some special instances of the MIMO case are considered as well. The idea is to characterize such a nonlinearity based on the energy content of its input signal and the energy content of its output signal, i.e. based on the IQC's satisfied by the nonlinearity. Multipliers that preserve positivity of a nonlinearity prescribe IQC's satisfied by the nonlinearity of interest. The larger the class of such multipliers, the better it is for the lesser is the conservatism in IQC based stability analysis. The key thesis contribution is the characterization of the largest possible classes of multipliers that preserve positivity of the nonlinearities of interest. As a consequence, it reduces conservatism in IQC based stability analysis of such systems to a minimum possible.; The thesis has three main technical results. The first result demonstrates the MIMO case failure of the celebrated 1968 Zames-Falb framework and extends the framework to cover the MIMO case. The second result extends the 1971 Willems work on positivity preservation of SISO monotone nonlinearities to the case of repeated SISO monotone nonlinearities. This marks an incremental improvement in the admirable result derived recently by D'Amato et al. The third result demonstrates that although they preserve positivity of monotone nonlinearities, stability multipliers such as Popov multipliers and Zames-Falb multipliers may fail to preserve incremental positivity thereof. This result implies that although stability of the system on hand may be determined using these multipliers, it is doubtful whether the same can be said about continuity thereof.; Stability conditions that use our results are expressible as linear matrix inequalities. These can be tested using software packages such the LMI Control Toolbox, available for use with MATLAB. Potential applications of the thesis work include the least conservative estimate of robustness margin in anti-reset windup schemes, and the design of service allocation algorithms at a server (or servers in a multi user environment.