Optimal Pole Placement in Time-Dependent Linear Systems

In this paper a problem of optimal modal control for general linear, time-dependent systems is posed and solved. We review the recent modal decomposition of the general linear system. We then pose the problem of controlling one unstable mode, with a specified new Lyapunov exponent, but minimizing the mean-square gain function. The method is applied to the shallow-angle re-entry of a Delta Clipper-like vehicle, and the modal decomposition leads to both stability information and characterization of how the trajectory is unstable. The unstable Lyapunov exponent is moved so that the trajectory is stable, as verified by closed-loop Lyapunov exponent analysis. When implemented within the original nonlinear system, the controller can handle initial errors at the kilometer level. Only 4% modulation of the original drag factor is required for control. I. Introduction T HE problem of control of constant coefficient linear systems is well understood, forming the basis of most of control theory. Control of time-dependent systems is not yet at the same level of understanding. Fairly comprehensive discussion of the time-periodic case have appeared: see Breakwell et al. 1 for the linear quadratic regulator for periodic systems and Calico and Wiesel2 for pole placement in periodic systems. Recently, the current author has successfully produced the modal decomposition for a general timedependent linear system3 and introduced the first pole placement algorithm for such systems. In this paper we extend the method to optimal pole placement and will successfully control a shallowangle re-entry trajectory by this new technique.