Alternating projection techniques for multiobjective control

Karolos M Grigoriadis, Purdue University

Abstract

Modern control systems need to satisfy stringent performance requirements on multiple system outputs. In this dissertation, new formulations and computational tools are proposed to solve multiobjective control design problems. By utilizing the simple geometric structure of these problems, simple and effective computational algorithms based on alternating projections are proposed for a solution. At first, the multiobjective covariance control design problem is formulated as a feasibility problem to find a matrix in the intersection of the assignability and output performance constraint sets. Analytical expressions for the orthogonal projections onto the constraint sets are derived and alternating projection techniques are proposed for a numerical solution. The low-order covariance control design problem is also examined. Analytical expressions are obtained for the covariance controllers which minimize the required control effort. A geometric formulation is also provided for control problems described by Linear Matrix Inequalities (LMIs), and the corresponding expressions for the orthogonal projections are derived. Alternating projections are applied to solve fixed-order LMI control problems, such as stabilization and $\rm H\sb\infty$ control. Mixed $\rm H\sb2-H\sb\infty$ control problems are treated in frequency domain using alternating projections. The required projections are computed using fast Fourier transform techniques. Alternating projections are also proposed to improve the Q-Markov COVER identification method. The expressions for the orthogonal projections onto the realizability constraint sets are derived. Finally, a redesign procedure is proposed where both the plant and the controller are redesigned to minimize the required control effort, and preserve either the closed-loop system matrix, or the closed-loop covariance matrix. A convex quadratic optimization solves this problem. An iterative plant and controller redesign is proposed to integrate the plant and the controller design steps. This technique has guaranteed convergence. The significance of this research is twofold. First, it introduces the use of alternating projection techniques to treat multiobjective control and identification problems. These techniques appear to be very effective and easy to implement. Second, the last part of the research emphasizes the importance of combining the plant design and the controller design to obtain a better overall design. This integrated design can be achieved with efficient algorithms.

Degree

Ph.D.

Advisors

Skelton, Purdue University.

Subject Area

Aerospace materials

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