H _∞ tracking with commands generated by unstable autonomous systems having initial conditions in finite hyper-rectangles

This paper presents a novel approach to H _∞ analysis of tracking problems, where the command signal is generated by a possibly unstable autonomous generating system (AGS), whose dynamics corresponds to the practically expected commands and whose output results only from its initial conditions (ICs). An unstable AGS (e.g. double integrator) allows generation of realistic, common command-signals that cannot be created by standard shaping filters acting on white noise or on finite-energy signals. The analysis addresses linear time-invariant possibly unstable systems in a finite-horizon setting. The unknown ICs of both the system and the AGS are assumed to lie in finite hyper-rectangles, analogously to the standard treatment of parameter-uncertainty. It is shown that the dependence of the system's performance on such ICs is convex; this fact leads to theorems assuring a H _∞ performance bound over the two ICs uncertainty regions (jointly). A solution to an 'inverse problem', i.e. finding the maximum-volume ICs hyper-rectangles for a given performance level, is also presented. The example addresses an aircraft responding to a variety of mixed step and ramp vertical acceleration commands.