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A Method of Extremum Seeking Control Based on a Time Varying Kalman Filter and its Application to Formation Flight

  • Author(s): Ryan, John Jens
  • Advisor(s): Speyer, Jason L.
  • et al.
Abstract

This dissertation presents a novel extremum seeking control method which combines a time-varying Kalman filter with a Newton Raphson algorithm. The Kalman filter is used to estimate the gradient and Hessian of a performance function. The resulting estimates are used in the Newton Raphson algorithm to guide the system to a local extremum of the performance function.

Convergence of the method to a local extremum is proven when the system is subject to noisy measurements. This is accomplished by showing that the output of the algorithm is a supermartingale. It is shown that the system will converge to an area around the extremum with a radius defined, in part, by the error covariance of the Kalman filter estimates.

The method is applied to two examples. The first utilizes a single independent parameter performance function. The second applies the method to the problem of formation flight for drag reduction. In the first example, two implementations of the method are examined. The first uses only gradient estimates. The second uses both gradient and Hessian estimates. Both implementations show good convergence in the presence of noisy measurements.

The second example is of formation flight for drag reduction. The problem is described in some detail and includes an aerodynamic development of the drag-reduction phenomenon. The problem is explored with two simulations. The first uses coefficient of induced drag as its performance function and estimates the gradient and Hessian of the performance function. It shows good convergence of the method. The second simulation first uses pitch angle and then aileron deflection as its performance function. It estimates the gradient but not the Hessian of the performance function. It also shows good convergence.

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