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In this paper we consider structure-preserving model reduction of second-order systems using a balanced truncation approach. Several sets of singular values are introduced for such systems, which lead to different concepts of balancing and different second-order balanced truncation methods. A comparison of these methods with other second-order balanced truncation techniques is presented. We also show that, in general, none of the existing structure-preserving balanced truncation methods for second-order systems preserves stability in the reduced models. Numerical examples are given that demonstrate the properties of the new methods.

Different concepts related to controllability of differential-algebraic equations are described. The class of systems considered consists of linear differential-algebraic equations with constant coefficients. Regularity, which is, loosely speaking, a concept related to existence and uniqueness of solutions for any inhomogeneity, is not required in this article. The concepts of impulse controllability, controllability at infinity, behavioral controllability, strong and complete controllability are described and defined in time-domain. Equivalent criteria that generalize the Hautus test are presented and proved. Special emphasis is placed on normal forms under state space transformation and, further, under state space, input and feedback transformations. Special forms generalizing the Kalman decomposition and Brunovsky form are presented. Consequences for state feedback design and geometric interpretation of the space of reachable states in terms of invariant subspaces are proved.

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We present an extension of the positive real and bounded real balanced truncation model reduction methods to large-scale descriptor systems. These methods are based on balancing the solutions of the projected Lur'e matrix equations. Important properties of these methods are that, respectively, passivity and contractivity are preserved in the reduced-order models and that there exist approximation error bounds. We also discuss the numerical solution of the projected Lur'e equations. Numerical examples are given.

SUMMARYIn this work we consider the problem of multiport passive reciprocal network synthesis by descriptor systems. A numerical method is presented that leads to circuit equations in modified nodal analysis.