Riccati Equation Approaches for Robust Stability and Performance Analysis Using the Small Gain, Positivity, and Popov Theorems

In recent years, small gain (or H_∞) analysis has been used to analyze feedback systems for robust stability and performance. However, since small gain analysis allows uncertainty with arbitrary phase in the frequency domain and arbitrary time variations in the time domain, it can be overly conservative for constant real parametric uncertainty. More recent results have led to the development of robustness analysis tools, such as extensions of Popov analysis, that are less conservative. These tests are based on parameter-dependent Lyapunov functions, in contrast to the small gain test which is based on a fixed quadratic Lyapunov function. This paper uses a benchmark problem to compare Popov analysis to small gain analysis and positivity analysis (a special case of Popov analysis that corresponds to a fixed quadratic Lyapunov function). The state space version of these tests, based on Riccati equations, are implemented using continuation algorithms. The results show that the Popov test is significantly less conservative than the other two tests and for this example is completely nonconservative in terms of its prediction of robust stability.