On Linear-Quadratic Optimal Control and Robustness of Differential-Algebraic Systems
Matthias Voigt
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Matthias Voigt, On Linear-Quadratic Optimal Control and Robustness of Differential-Algebraic Systems (2015), Logos Verlag, Berlin, ISBN: 9783832591373
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Beschreibung / Abstract
This thesis considers the linear-quadratic optimal control problem for differential-algebraic systems. In this first part, a complete theoretical analysis of this problem is presented. The basis is a new differential-algebraic version of the Kalman-Yakubovich-Popov (KYP) lemma. One focus is the analysis of the solution structure of the associated descriptor KYP inequality. In particular, rank-minimizing, stabilizing, and extremal solutions are characterized which gives a deep insight into the structure of the problem. Further contributions include new relations of the descriptor KYP inequality to structured matrix pencils, conditions for the existence of nonpositive solutions, and the application of the new theory to the characterization of dissipative systems and the factorization of rational matrix-valued functions.
The second part of this thesis focuses on robustness questions, i.e., the influence of perturbations on system properties like dissipativity and stability is discussed. Characterizations for the distance of a dissipative systems to the set of non-dissipative systems are given which lead to a numerical method for computing this distance. Furthermore, the problem of computing the H-infinity-norm of a large-scale differential-algebraic system is considered. Two approaches for this computation are introduced and compared to each other.
The second part of this thesis focuses on robustness questions, i.e., the influence of perturbations on system properties like dissipativity and stability is discussed. Characterizations for the distance of a dissipative systems to the set of non-dissipative systems are given which lead to a numerical method for computing this distance. Furthermore, the problem of computing the H-infinity-norm of a large-scale differential-algebraic system is considered. Two approaches for this computation are introduced and compared to each other.
Inhaltsverzeichnis
- BEGINN
- 1 Introduction
- 1.1 Motivation
- 1.2 Structure of this Thesis
- 1.3 System Setup
- 2 Mathematical Preliminaries
- 2.1 Matrix Theoretic Preliminaries
- 2.2 System Theoretic Basics
- 3 Linear-Quadratic Control Theory for Differential-Algebraic Equations
- 3.1 Introduction
- 3.2 State and Feedback Transformations
- 3.3 Kalman-Yakubovich-Popov Lemma
- 3.4 Even Matrix Pencils and Descriptor Lur'e Equations
- 3.5 Stabilizing, Anti-Stabilizing, and Extremal Solutions
- 3.6 Spectral Factorization
- 3.7 Nonpositive Solutions
- 3.8 Linear-Quadratic Optimal Control
- 3.9 Dissipative and Cyclo-Dissipative Systems
- 3.10 Normalized Coprime Factorizations
- 3.11 Inner-Outer Factorizations
- 3.12 Summary and Outlook
- 4 Systems with Counterclockwise Input/Output Dynamics
- 4.1 Introduction
- 4.2 Systems with Counterclockwise Input/Output Dynamics and Negative Imaginary Transfer Functions
- 4.3 Spectral Characterizations for Negative Imaginariness
- 4.4 Enforcement of Negative Imaginariness
- 4.5 Conclusions and Outlook
- 5 Computation of the Complex Cyclo-Dissipativity Radius
- 5.1 Introduction
- 5.2 The Cyclo-Dissipativity Radius
- 5.3 Perturbations of the Singular Part and the Defective In nite Eigenvalues
- 5.4 Computation of the Complex Cyclo-Dissipativity Radius
- 5.5 Numerical Results
- 5.6 Summary and Open Problems
- 6 Computation of the H1-Norm for Large-Scale Descriptor Systems
- 6.1 Introduction
- 6.2 The Pseudopole Set Approach
- 6.3 The Even Pencil Approach
- 6.4 Conclusions and Future Research Perspectives
- 7 Summary and Outlook
- Bibliography
- Index