Linear control and estimation of nonlinear chaotic convection: Harnessing the butterfly effect

This paper examines the application of linear optimal/robust control theory to a low-order nonlinear chaotic convection problem. Linear control feedback is found to be fully effective only when it is switched off while the state is far from the desired equilibrium point, relying on the attractor of the system to bring the state into a neighborhood of the equilibrium point before control is applied. Linear estimator feedback is found to be fully effective only when ~a! the Lyapunov exponent of the state estimation error is negative, indicating that the state estimate converges to the uncontrolled state, and ~b! the estimator is stable in the vicinity of the desired equilibrium point. The aim in studying the present problem is to understand better some possible pitfalls of applying linear feedback to nonlinear systems in a low-dimensional framework. Such an exercise foreshadows problems likely to be encountered when applying linear feedback to infinite-dimensional nonlinear systems such as turbulence. It is important to understand these problems and the remedies available in a low-dimensional framework before moving to more complex systems. © 1999 American Institute of Physics. @S1070-6631~99!01105-8#