Off-axis absolute stability criteria and μ-bounds involving non-positive-real plant-dependent multipliers for robust stability and performance with locally slope-restricted monotonic nonlinearities

One of the most important aspects of the control design and evaluation process is the analysis of feedback systems for robust stability and performance. Over the past several year significant attention has been devoted to the use of small gain tests and other norm-based methods. Unfortunately, due to their dependence on norms, these tests do not allow the inclusion of phase information regarding the system uncertainty. Phase information, here, refers to the characterization of the phase of the modeling uncertainty in the frequency domain. As discussed in 11,121, in the time domain phase information is captured by restricting the allowable time variation of the uncertain system parameters. Recent research [10-131 has led to the development of new robustness analysis and synthesis tools that are significantly less conservative than previous methods due to their inclusion of phase information. Specifically, these robustness analysis and synthesis tools are based upon quadratic Lyapunov functions that are a function of the parametric uncertainty, i.e., parameter-dependent Lyapunov functions. In contrast to analysis and synthesis methods based on fixed quadratic Lyapunov functions such as H_∞ theory, these tests guarantee robust stability by means of a family of Lyapunov functions and do not apply to arbitrarily time-varying uncertainties and hence possess the potential for exploiting phase information [11,13-151.