Optimal Control and Shape Design: Theory and Applications

Abstract

This work focuses on the spectrum of problems connected with the analysis and the development of computational tools and models for engineering and scientific applications. This includes: (i) reduced order modeling techniques; (ii) linear and nonlinear feedback control design methodologies and real-time implementation; and (iii) shape optimization techniques. Excluding shape optimization techniques, most of the research herein can be seen as extensions of linear quadratic regulation (LQR) techniques. First, we consider the synthesis of control methodologies for the attenuation of beam vibrations caused by a narrow-band exogenous force. By a narrow-band exogenous force we mean periodic force over a narrow frequency band or an exact harmonic. The control methods under consideration are based on the minimization of two specific quadratic cost functionals. One of these cost functionals is a typical time domain cost functional constrained by an affine plant. The other is a cost functional that is frequency dependent. These control methods have been used successfully in various applications but this investigation differs in that it emphasizes the development of real-time control methodologies based on reduced order models derived from physical first principles. Specifically, an integral component of this research is the proper orthogonal decomposition (POD) reduction technique and its application to real-time control of beam vibrations.

The second LQR extension involves a particular nonlinear control methodology that mimics standard LQR formulation. State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and state estimators for a broad class of nonlinear regulator problems. The technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Then LQR techniques are used on the state-dependent coefficients to formulate a suboptimal control law. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This work addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. A previous numerical method, which is based on the Taylor series, works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method, introduced here, can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and feasibility of the SDRE technique for the design of nonlinear compensator based feedback controllers.

Finally, this work also includes an optimization technique in which the objective is attained via alterations to the physical geometry of the system. This optimization framework, to be considered in the context of electron guns, is known as optimal shape design. Optimal shape design has been used in a number of applications including wing design, magnetic tape design, and nozzle design, among others. In this investigation, we use the methods of shape optimization to design the shape of the cathode of an electron gun. The dynamical equations modeling the electron particle path as well as the generalized shape optimization problem will be presented. Illustrative examples of the technique on gun designs that were previously limited to spherical cathodes will be given.