Bode's integral: Extensions in linear time-varying and nonlinear systems

Bode's integral formula describes a fundamental limitation in feedback control system design. Although Bode's original result dealt with single-input, single-output, continuous-time, linear, time-invariant systems that are open-loop stable, it has now been extended in numerous ways, including unstable and multi-input multi-output systems. Because the limitations singled out in Bode's integral are so fundamental, it is natural to expect that similar constraints must exist for systems that are not linear or time-invariant. However, extending this result to classes of systems that do not admit transfer functions, for example time-varying or nonlinear systems, presents several difficulties.; In this dissertation, we present extensions for linear time-varying systems and a class of nonlinear systems. For linear time-varying systems, we first present analogues of the logarithmic integral found in Bode's result, based We use the dichotomy spectrum as an extension of system pole/zero dynamics for these systems. Our study shows that a constraint exists on the closed-loop input-output description of these systems, similar to Bode's integral formula for linear time-invariant systems. Possible extensions of Bode's complementary sensitivity integral for linear time-varying systems also are discussed.; For nonlinear systems, we use the difference of conditional entropy rate between input and output of the system as an analogue of the logarithmic integral in Bode's result. It is shown that this difference is zero for the sensitivity operator of a stable nonlinear system that possesses fading memory from both the input to output and output to input. For an unstable system, we factor its sensitivity operator into an all-pass factor and a minimum-phase factor, and show that the resulting difference of conditional entropy of the signal passing through the minimum-phase factor is non-negative.; Because the fading memory condition is essential in the analysis of nonlinear systems, we present a necessary condition and a sufficient condition under which the nonlinear dynamical systems considered possess fading memory. The necessary condition is based on the relationship between continuity of the system and the fading memory condition. The sufficient condition comes from a Lyapunov stability theorem.