Stabilization of Nonlinear PDE's and Applications to Control of Flows

Abstract

We have contributed in three general areas that are relevant to high performance control of aerospace vehicles. We have developed systematic techniques for real time optimization using the method of extremum seeking. This method is now highly general applicable to multivariable problems, to problems with time-varying parameters, to problems involving slow dynamics where fast convergence is demanded, to problems where the objective is convergence to any value of the gradient (not just zero). to problems that involve limit cycles (unsteady extrema). and to problems that evolve in discrete time. Examples of applications that we have pursued include formation flight, compressor stall and surge, arid thermoacoustic combustion instabilities. In 2003 we published a research monograph on extremum seeking. In the area of flow control we have pioneered the use of stabilization for drag reduction and mixing. Employing Lyapunov techniques, we have solved flow control problems in channels, pipe geometries, flows around bluff bodies, and jet flows. In 2002 we published the first book dedicated to control algorithm design for fluid flows. For a broad class of linear parabolic distributed parameter systems we pioneered a back-stepping method for solving the boundary control stabilization problems. This is the first method that yields explicit solutions for both the control laws and for the closed-loop solutions. The implications of this are impossible to overstate-the design is direct and free of numerical issues, and the well posedness analysis is trivial (it is a bonus to the explicit design process). For several nonlinear PDEs we solved global stabilization problems in the presence of parametric uncertainties and input dynamics, paving the way for a future general theoretical development for nonlinear PDEs.

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Document Details

Document Type
Technical Report
Publication Date
May 01, 2003
Accession Number
ADA417657

Entities

People

  • Miroslav Krstić

Organizations

  • University of California, San Diego

Tags

Communities of Interest

  • Space
  • Weapons Technologies

DTIC Thesaurus Topics

  • Algorithms
  • Applied Mathematics
  • Boundaries
  • Combustion
  • Compressors
  • Control Systems
  • Difference Equations
  • Differential Equations
  • Dynamics
  • Equations
  • Fluid Flow
  • Formation Flight
  • Hypervelocity Flow
  • Instability
  • Jet Flow
  • Mathematics
  • Optimization

Readers

  • Control Systems Engineering.
  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Fluid Mechanics and Fluid Dynamics.

Technology Areas

  • Space
  • Space - Spacecraft Maneuvers