On symmetric rational transfer functions
TL;DR: In this article, a detailed study of symmetric transfer functions is presented, and a detailed analysis of transfer functions can be found in Section 5.1.1]...
About: This article is published in Linear Algebra and its Applications.The article was published on 1983-04-01 and is currently open access. It has received 87 citations till now.
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TL;DR: In this paper, a generic class of flexible systems, characterized by finitely many lightly damped harmonic oscillators, is analyzed by means of the "open-loop principal component analysis", that is, singular value analysis and Gramian balancing.
Abstract: A generic class of flexible systems, characterized by finitely many lightly damped harmonic oscillators, is analyzed by means of the "open-loop principal component analysis," that is, singular value analysis and Gramian balancing. As the main result, it is shown that, as the damping ratio goes to zero, the balanced state coordinates are decoupled and coincide with the modal coordinates. Further, simple formulas expressing the "asymptotic singular values" as functions of the modal parameters are derived.
81 citations
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TL;DR: In this article, a technique for balancing a system in a closed-loop fashion is developed, which is referred to as "LQG balancing", for it assumes that the system to be balanced is closed up with a standard LQG feedback loop.
Abstract: In this paper, a technique for balancing a system in a closed-loop fashion is developed. This technique is referred to as ‘LQG balancing’, for it assumes that the system to be balanced is closed up with a standard LQG feedback loop. This paper focuses on LQG balancing of symmetric passive systems; such systems are used to model large vibrating structures with collocated rate sensors and actuators. A balancing characterization of reciprocity, passivity and losslessness is provided. A new method for reducing both the plant and the LQG compensator is developed. It is shown, using a hyperstability argument, that the loop made up with the full plant and any reduced LQG compensator is stable and has acceptable feedback properties.
77 citations
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TL;DR: In this paper, the model reduction problem for state-space symmetric systems is investigated, and it is shown that several model reduction methods, such as balanced truncations, balanced truncation which preserves the DC gain, optimal and sub-optimal Hankel norm approximations, inherit the state space symmetric property.
69 citations
01 Jan 1979
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01 Jun 1984
TL;DR: In this article, the Routh-Hurwitz problem of singular pencils of matrices has been studied in the context of systems of linear differential equations with variable coefficients, and its applications to the analysis of complex matrices have been discussed.
Abstract: Volume 2: XI. Complex symmetric, skew-symmetric, and orthogonal matrices: 1. Some formulas for complex orthogonal and unitary matrices 2. Polar decomposition of a complex matrix 3. The normal form of a complex symmetric matrix 4. The normal form of a complex skew-symmetric matrix 5. The normal form of a complex orthogonal matrix XII. Singular pencils of matrices: 1. Introduction 2. Regular pencils of matrices 3. Singular pencils. The reduction theorem 4. The canonical form of a singular pencil of matrices 5. The minimal indices of a pencil. Criterion for strong equivalence of pencils 6. Singular pencils of quadratic forms 7. Application to differential equations XIII. Matrices with non-negative elements: 1. General properties 2. Spectral properties of irreducible non-negative matrices 3. Reducible matrices 4. The normal form of a reducible matrix 5. Primitive and imprimitive matrices 6. Stochastic matrices 7. Limiting probabilities for a homogeneous Markov chain with a finite number of states 8. Totally non-negative matrices 9. Oscillatory matrices XIV. Applications of the theory of matrices to the investigation of systems of linear differential equations: 1. Systems of linear differential equations with variable coefficients. General concepts 2. Lyapunov transformations 3. Reducible systems 4. The canonical form of a reducible system. Erugin's theorem 5. The matricant 6. The multiplicative integral. The infinitesimal calculus of Volterra 7. Differential systems in a complex domain. General properties 8. The multiplicative integral in a complex domain 9. Isolated singular points 10. Regular singularities 11. Reducible analytic systems 12. Analytic functions of several matrices and their application to the investigation of differential systems. The papers of Lappo-Danilevskii XV. The problem of Routh-Hurwitz and related questions: 1. Introduction 2. Cauchy indices 3. Routh's algorithm 4. The singular case. Examples 5. Lyapunov's theorem 6. The theorem of Routh-Hurwitz 7. Orlando's formula 8. Singular cases in the Routh-Hurwitz theorem 9. The method of quadratic forms. Determination of the number of distinct real roots of a polynomial 10. Infinite Hankel matrices of finite rank 11. Determination of the index of an arbitrary rational fraction by the coefficients of numerator and denominator 12. Another proof of the Routh-Hurwitz theorem 13. Some supplements to the Routh-Hurwitz theorem. Stability criterion of Lienard and Chipart 14. Some properties of Hurwitz polynomials. Stieltjes' theorem. Representation of Hurwitz polynomials by continued fractions 15. Domain of stability. Markov parameters 16. Connection with the problem of moments 17. Theorems of Markov and Chebyshev 18. The generalized Routh-Hurwitz problem Bibliography Index.
9,334 citations