A broad class of natural and man-made systems--weather phenomena, turbulence, vortices on sphere, Internet traffic, etc.--exhibit "complicated" phenomena that could easily be misinterpreted as random if it weren't for a more refined modeling analysis revealing that the apparent randomness is in fact produced by nonlinearity in the dynamics. Over the past ten years, this observation has led to a flare up of interest in "chaotic" phenomena, although the basic premise that nonlinear dynamics could lead to random-looking phenomena is at the heart of ergodic theory and traces back to Poincare. In fact, several of the popular attributes of chaos can be dispensed of for generating "complicated" phenomena, and we adopted the point of view of A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge, 1997, that "complicated asymptotic dynamics" starts with nontrivial recurrence.
There are many possible interpretations of "controlling complicated asymptotic dynamics," depending on the application in mind, and this has created a host of new control paradigms. In some applications, like high power lasers, there is a need to enforce the periodicity of the behavior. In some other applications, like mixing, quite on the contrary, there is the need to destroy periodic behavior and enforce transitivity. Finally, an entirely new control paradigm, inspired from physiology, is to navigate within the attractor so as to avoid fixed or periodic points.
These various control objectives can somehow be unified under the trajectory tracking paradigm, where the trajectory to be tracked could be either a natural trajectory of the system (attractors are very rich in trajectories of all kinds) or a composite trajectory made up of pieces of natural trajectories assembled together. To be more specific, the trajectory to be followed is given by
x(k+1)=f(x(k),0)
whereas the controlled trajectory is
z(k+1)=f(z(k),u(k))
where u(k) is a small control action. Probably the best illustrative example is coming from astrodynamics, where x(k) would be the motion of such a light celestian body as an asteroid or a Trojan body of Jupiter and z(k) would be the trajectory of a spacecraft in a rendez-vous mission; if the gravitational attraction between the spacecraft and the celestian body is negligible, the motion of the spacecraft is the same as the motion of the celestian body, except for the propulsion force u(k) which is steering the spacecraft to the celestian body, that is,
e(k)=z(k)-x(k) -> 0
Given a trajectory x(k) to be tracked, the approach is to linearize the tracking error e(k) along every point of the reference trajectory, leading to a tracking error that takes the form of a linear system with its parameters varying according to the dynamics x(k) of the reference trajectory. This kind of linear systems have been called Linear Dynamically Varying (LDV). The tracking objective is attained by forcing, via either Linear-Quadratic or H-Infinity techniques, the error to go asymptotically zero. In both the LQ and H-Infinity cases, the compensator is of the spatially varying type, that is,
u(k)=K(x(k))e(k)
and is provided via the solution P(x), defined over the whole attractor, of a Functional Algebraic Riccati Equation (FARE). The functionality of this equation stems from the fact that it links the solutions P(x), P(f(x)) at two successive points of the attractor. Numerical techniques for solving the FARE have been developed.
It turns out that the linear controller, constructed to stabilize the LDV system, also stabilizes the nonlinear system in some neighborhood of the nominal trajectory. Closer inspection reveals that the linearization error is a bounded feedback around the LDV plant, so that the H-Infinity approach has the definite advantage of allowing for the attenuation of the linearization error, and by the same token the amplification of the domain of attraction.
Extension of this technique to continuous nonlinear dynamics over Riemannian manifods has been achieved. In this case, the compensator is provided by the solution to a Partial Differential Riccati Equation (PDRE) defined over the manifold.
As benchmark examples to try out and evaluate control schemes for "complicated asymptotic dynamics," we have considered the followings;
This research was supported by National Science Foundation grant ECS-98-02594.