This proposal outlines a comprehensive program in an emerging area of control of nonlinear distributed parameter systems. The motivating practical problems come from stabilization of unstable fluid flows, ubiquitous in today's advanced consumer products and automotive, aerospace, and marine vehicles. These applications abound with nonlinearities of superlinear growth and instabilities in the interior of a spatial domain where controls can act only from the boundary. These issues have successfully been dealt with in a finite-dimensional setting, and are uniquely positioned to address them for distributed parameter systems.
This proposal advocates the development of two types of methods. First, methods based on backstepping, which employ either the measurement of the state in the interior of the spatial domain, or a nonlinear observer, for stabilization of highly unstable systems with boundary controls. The control action is typically propageted from the boundary into the interior of the domain by viscosity/diffusion mechanisms present in many disstributed parameter systems.
The second method is based on simple feedbacks involving only boundary measurements. Such feedbacks are in the form of Jurdjevic-Quinn feedback laws that add damping for asymptotic stabilization of systems with known control Lyapunov functions. An example is shown of a fluid flow problem in which a J-Q type controller is developed for low Reynolds number and demonstrate its effectiveness for a high, realistic value of Reynolds number, many orders of magnitude higher than the design value. In the same setting, this proposal proposes to go beyond equilibrium stablization, to enforcement of non-equilibrium quasi-steady motion. An example of such a control objective is mixing in fluids where one wants to achieve random motion of particles by deterministic means. This proposal plans to dramatically depart from conventional approaches to this problem which are all of open-loop/motion-planning type. A feedback law will be shown that raises both the turbulent kinetic energy and the vorticity of the fluid (uniformly in space).
Several theoretical advances have been produced in stabilization of uncertain nonlinear systems (resulting in several best paper awards and a new book on this topic), as well as transition (industrial experiments with our control algorithms and a patent offered for licensing consideration to a leading aeroengine manufacturer.