RICCATI EQUATION APPROACHES FOR SMALL GAIN, POSITIVITY, AND POPOV ROBUSTNESS ANALYSIS

In recent years, small gain (or H(infinity)) analysis has been used to analyze feedback systems for robust stability and performance. Howeve r, since small gain analysis allows uncertainty with arbitrary phase i n the frequency domain and arbitrary time variations in the time domai n, it can be overly conservative for constant real parametric uncertai nty. More recent results have led to the development of robustness ana lysis tools, such as extensions of Popov analysis, that are less conse rvative. These tests are based on parameter-dependent Lyapunov functio ns, in contrast to the small gain test, which is based on a fixed quad ratic Lyapunov function. This paper uses a benchmark problem to compar e Popov analysis with small gain analysis and positivity analysis (a s pecial case of Popov analysis that corresponds to a fixed quadratic Ly apunov function). The state-space versions of these tests, based on Ri ccati equations, are implemented using continuation algorithms. The re sults show that the Popov test is significantly less conservative than the other two tests and for this example is completely nonconservativ e in terms of its prediction of robust stability.