Stability and control of discontinuous systems and estimation for chaotic systems

This thesis has been written as several independent papers concerning nonlinear dynamic systems.;Numerical computation of Lyapunov exponents for systems described by ordinary differential equations with discontinuous right-hand sides is discussed. Known methods for computing Lyapunov exponents are extended to such systems by using jump conditions for the variational equation and treating sliding behavior explicitly. A computer program that computes Lyapunov exponents for linear plants controlled by bang-bang control according to a linear switching law is described. An example involves the chaotic stabilization of the longitudinal dynamics of a helicopter.;On-off sampled-data regulation of a stable first-order linear plant is discussed. Two control laws are considered--a linear switching function and an optimal control which minimizes mean squared error from the command at sampling instants. Analysis shows the two controls lead to the same class of limit cycles but different mode-locking (fractal) Devil's staircases.;Application of the extended Kalman filter to plants with chaotic dynamics is discussed. The behavior of the filter is explored by an approximate covariance analysis and numerical experiment. The existence of approximate asymptotic gains suggests the construction of an approximate extended Kalman filter whose gains are precomputed functions of the state estimate. Numerical experiments with the Henon quadratic map are used as illustration for the approximate analysis and to demonstrate an actual extended Kalman filter operating on a chaotic system.;The estimation of parameters in physically motivated models of chaotic systems from measured time series is discussed. The applicability of the output error, equation error, and the more general output prediction error methods is considered. It is suggested that a simple ad hoc feedback version of the last-named method is particularly appropriate. Estimation of parameters in a numerical simulation of a damped, sinusoidally driven pendulum is used as an example.;Finally, speculation is made about the existence of chaotic solutions to continuous-time optimal control problems.