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Optimization of Sparse Control Structures in Multivariable Systems

Babazadeh, Maryam | 2016

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 48636 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Nobakhti, Amin
  7. Abstract:
  8. In this thesis, the optimal control structure selection and design of sparse multi-variable control systems is addressed. A fundamental challenge which frequently emerges in engineering, social, and economic sciences, is the optimal selection of a subset of elements, in order to maximally fulfil a design objective. In practice, it is required to solve this underlying selection problem in conjunction with a non-linear or non-convex optimization which is designed to ensure desired performance. The requirement to solve these two problems simultaneously is what makes it inherently difficult; one which has thus far eluded efforts to develop a systematic means of determining its solution. In spite of the lack of such methodologies, similar variations of this problem appear in a wide range of applications. These include the simple control structure selection problem, constrained sensors and actuators placement, distributed control via band limited communication link, and input/output selection.This thesis begins by demonstrating that the existing methodologies devised to solve the problem, especially for control structure selection, have significant deficiencies and may in some cases rely on incorrect assumptions. The review of sparse optimization methodologies,mostly developed in signal processing and machine learning, utilizes the l1 norm heuristic in order to induce sparsity in the optimization solution. However, application of the l1 norm heuristic in control structure selection, comes with significant reservations which are detailed in this thesis via illustrating examples.Accordingly, a systematic approach for the design of sparse dynamic output feedback control structures for Linear Time Invariant (LTI) systems is presented in this thesis. The combinatorial nature of the problem, coupled with the malign interaction between the degree of sparsity and optimality, are the main challenges for which effective solutions are formulated in this thesis. The proposed solutions in this, and subsequent sections, are centered around a framework of Semi-definite Programs (SDP). The methodology is further extended to optimal control structure selection in uncertain systems with structured, norm-bounded and polytopic uncertainties.Subsequently, the sparse optimal control design with guaranteed stability of the closedloop system is considered. A new methodology for the state feedback control problem with H2 norm minimization is proposed based on a new globallyconvergent algorithm which iteratively solves a set of smaller SDPs. The algorithm guarantees the stability of the closedloop system as well as the convergence of the solution to a stationary point of the original problem. The proposed method has several applications; for instance to provide optimal structure selection for distributed systems, and interconnected networks.The remainder of the thesis is dedicated to improvements of the l1-norm approximation when the feasible set belongs to a semi-definite cone. It is shown that the gap between the l1 norm and l0 norm may lead to inappropriate selection of non-zero decision variables.Thus a new framework to induce sparse solutions in convex problems with linear matrix inequality (LMI) constraints is proposed. The presented method is applicable to both openloop and closed-loop designs obtained in earlier sections of the thesis. The proposed functionoutperforms the l1 norm when referenced against optimal choice of non-zero elements. As a case study, the previously developed pre-compensator design is formulated based on the proposed sparsity promotion scheme and the results confirm the superiority of the presented method in optimal control structure selection
  9. Keywords:
  10. Convex Optimization ; Semidefinite Optimization ; Sparsity ; Control Structure ; Distributed System ; Multivariable System

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