Selected publications of A.C. Antoulas

1. New results on the algebraic theory of linear systems: the solution of the cover problems Linear Algebra and Its Applications, Special issue on Linear Systems and Control, edited by P.A. Fuhrmann, 50, pp. 1-43 (1983).

Abstract. A wide class of control problems can be formulated in terms of subspace inclusions called the cover problems. This paper presents a general solution of the these problems both in state-space and polynomial terms. The unifying element turns out to be the partial realization problem. The contribution of this paper is that of parametrizing all solutions keeping track of their complexity at the same time.

2. A new approach to synthesis problems in linear systems IEEE Transactions on Automatic Control, AC-30, pp. 465-474 (1985).

Abstract. A very general synthesis problem is formulated in terms of a nonlinear equation with rational entries and solved. The novelty consists in the fact that all admissible compensators are parametrized and classified according to their complexity. The key for doing this is on the one hand, the use of polynomial matrix factorizations, and on the other the conversion of the problem to an appropriate partial realization problem.

3. On recursiveness and related topics in linear systems IEEE Transactions on Automatic Control, AC-31, pp. 1121-1135 (1986).

Abstract. This paper is concerned with a cascade decomposition of linear systems. It explores the close relationships between a number of topics like: matrix linear fractional representations, nested feedback interconnections formal power series, recursive realization of matrix impulse response sequences matrix continued fractions a matrix Euclidean algorithm, etc. As a corollary the celebrated Berlekamp-Massey algorithm is extended to the matrix case. The natural tool for achieving this is a certain polynomial unimodular matrix of size p+m which can be associated to every system with m inputs and p outputs.

4. On the scalar rational interpolation problem with B.D.O. Anderson, IMA J. of Mathematical Control and Information, Special Issue on Parametrization problems, edited by D. Hinrichsen and J.C. Willems, 3, pp. 61-88 (1986).

Abstract. Rational interpolation is treated as a generalization of (partial) realization. It is shown that the Loewner matrix an old tool for studying the interpolation problem generalizes the Hankel matrix in this respect. Moreover, the parametrization of all solutions to the rational interpolation problem with the complexity (McMillan degree) as parameter is achieved for the first time.

5. Rational interpolation and the Euclidean Algorithm Linear Algebra and Its Applications, 108, pp. 157-171 (1988).

Abstract. The (scalar) rational interpolation problem with a different degree of complexity (namely the sum of the numerator and the denominator degrees as opposed to the maximum between the numerator and the denominator degrees) is considered. It is shown that the Euclidean algorithm and the degrees of the successive quotients, provide the key to parametrizing all solutions where the complexity is the parameter defined above. This clarifies many aspects of the Pade and the Cauchy approximation problems not well understood in the literature.

6. On the stable rational interpolation problem with B.D.O. Anderson, Linear Algebra and Its Applications, Special Issue on Linear Control Theory, 122/123/124, pp. 301-329 (1989).

Abstract. Various classical results in rational interpolation theory are revisited and reinterpreted using the Loewner matrix introduced above. An important result in this regard is the fact that the celebrated Nevanlinna-Pick algorithm consists of nothing more than unconstrained minimal interpolation of the original set of data together with a mirror-image set of data.

7. The cascade structure in system theory in Three decades of mathematical system theory, edited by H. Nijmeijer and J.M. Schumacher, Springer Lecture Notes in Control and Information Science, 135, pp. 1-18 (1989).

Abstract. It is pointed out that many seemingly diverse results in system theory can be explained in a unified way using the cascade interconnection of two-port systems as a tool. In particular this refers to passive network synthesis results like Darlington synthesis and Inverse Scattering on the one hand, and recursive realization results on the other.

8. State space and polynomial approaches to rational interpolation with B.D.O. Anderson, in Progress in Systems and Control Theory III: Realization and Modeling in System Theory, M.A. Kaashoek, J.H. van Schuppen and A.C.M. Ran Editors, Birkhäuser (1990), pp. 73-82.

Abstract. Further insight is provided on the use of the Loewner matrix in the study of the scalar rational interpolation problem. In particular all solutions are constructed both in a state-space and a polynomial setting.

9. Rational interpolation and state-variable realizations B.D.O. Anderson and A.C. Antoulas, Linear Algebra and Its Applications, Special Issue on Matrix problems, 137/138, pp. 479-509 (1990).

Abstract. The construction of solutions of the matrix rational interpolation problem in state-space form, is derived using the (block) Loewner matrix. This theory generalizes the construction of realizations by means of the (block) Hankel matrix.

10. On minimal realizations Systems and Control Letters, 14: 319-324 (1990).

Abstract. The machinery introduced in the recursiveness paper 25 is shown to lead to a new test for minimality of (both scalar and matrix) realizations based on the degrees of the entries of an appropriate Bezout identity.

11. On the solution of the minimal rational interpolation problem with J.A. Ball, J. Kang, and J.C. Willems, Linear Algebra and Its Applications, Special Issue on Matrix Problems, 137/138: 511-573 (1990).

Abstract. This paper provides for the first time, the solution of a general matrix rational interpolation problem with the (McMillan) degree as measure of complexity; the computation of minimal interpolants follows as a corollary. This is done by defining directly from the data a pair of matrices. All admissible interpolant degrees are then obtained from the reachability indices of this pair. Moreover the corresponding linear dependencies of the columns of the reachability matrix of this pair provide a parametrization of all interpolants of appropriate complexity. These results are extended to a very general bitangential interpolation problem by the appropriate definition of a second pair of matrices.

12. Rational interpolation and Prony's method with J.C. Willems, Analysis and Optimization of Systems, J.L. Lions and A. Bensoussan, eds, Springer-Verlag, Lecture Notes in Control and Info. Science, 144: 297-306 (1990).

Abstract. Using the tools introduced in the above paper the widely used Prony's method of data fitting is re-examined. It is actually shown that the concept of reachability allows a complete characterization of all solutions to that problem. It also illustrates how to avoid mis-using this method (often encountered in the literature).

13. Mathematical System Theory: The influence of R.E. Kalman edited book, Springer-Verlag Berlin Heidelberg, 605 pages (1992).

Abstract. This book is a Festschrift containing a collection of papers written by leaders in the field, surveying the influence of R. E. Kalman's ideas in mathematical system theory. It was published on the occasion of Kalman's 60th birthday.

14. The realization problem: Linear deterministic systems with T. Matsuo and Y. Yamamoto, in Mathematical System Theory, A.C. Antoulas Editor, pp. 191-212, Springer-Verlag (1992).

Abstract. The realization problem for linear deterministic systems is reviewed both for finite- and infinite-dimensional systems. This is the problem of constructing the state from impulse response data.

15. A behavioral approach to linear exact modeling with J.C. Willems, IEEE Transactions on Automatic Control, AC-38: 1776-1802 (1993).

Abstract. Using the modeling framework introduced by the second author results obtained earlier by the first author on the recursiveness of the realization problem, are generalized to exact modeling of linear systems based on arbitrary time series. The main tool for achieving this sweeping generalization is again a polynomial matrix of size p+m which is attached to any system with m inputs and p outputs.

16. Noise identification in approximate modeling with Y. Yamamoto (in japanese), Journal of the Society of Instrument and Control Engineers (SICE), 32: 718-722 (1993).

Abstract. The problem of identifying all possible noise terms in a given set of data is addressed. The answer termed generalized least squares (GLS) turns out to be a generalization of the well known least squares (LS) and of the total least squares (TLS) schemes.

17. A nonlinear approach to the aircraft tracking problem with R.H. Bishop, Journal of Guidance Control, and Dynamics, 17: 1124-1130 (1994).

Abstract. This paper grew out of work done for Oerlikon-Contraves AG, Zürich. It describes the application of recent advances in non-linear filtering to the problem of tracking a maneuvering aircraft. In comparison to the extended Kalman Filter currently in use the advantages of the new so-called geometric filter, are guaranteed convergence and greatly reduced computational time requirements.

18. Recursive modeling of discrete-time time series in IMA volume on Linear Algebra for Control, P. van Dooren and B.W. Wyman Editors, Springer Verlag, vol. 62: 1-20 (1993).

Abstract. The approach to modeling inspired by the behavioral framework consists in treating all measurements on an equal footing not distinguishing between inputs and outputs. Consequently the initial search is for autonomous models. In the linear time invariant case, the main result guarantees the existence of a minimal complexity autonomous generating model; this means that all other models can be explicitly constructed from this generating model. Among them in most cases the so-called input-output controllable models are of interest. The main purpose of this paper is to show how these models can be constructed in a an easy-to-implement recursive way.

19. A unifying framework for linear exact modeling Proc. 12th IFAC Triennial World Congress, Sydney, Pergamon Press (1993).

Abstract. Exact modeling problems satisfying constraints of complexity stability and norm boundedness, are cast and solved into a common framework. The data are exponential trajectories. Rational interpolation turns out to be a special case of this modeling problem. The main tool is the most powerful model a central concept in the behavioral framework and its representations, called the generating systems.

20. A new approach to modeling for control Linear Algebra and Its Applications, Third Special Issue on Systems and Control, 203-204: 45-65 (1994).

Abstract. The behavioral approach to exact modeling leads to the concept of generating systems. It turns out that all exact models are parametrized by means of a feedback interconnection of a generating system with an arbitrary system. This approach is extended in the present paper to the approximate modeling case. The main tool for achieving this is the Hankel norm approximation applied to the generating system. This set-up has the additional feature that it allows us to address at the same time the control problem with little additional effort.

21. The cascade structure for lossless systems with M. Mansour, Comptes Rendus de l'Académie des Sciences Paris, 319, Série II, p. 1001-1008 (1994).

Abstract. It is shown that a certain decomposition of a stable system into a cascade of stable two-port subsystems has the property that the elimination of any number of these subsystems preserves stability. This is equivalent to the elimination of rows and columns from the associated Mansour matrix. The decomposition also gives a natural way for separating fast and slow modes of the original system.

22. On the approximation of Hankel matrices in Operators Systems and Linear Algebra, edited by U. Helmke, D. Prätzel-Wolters and E. Zerz, pages 17-23, Teubner Verlag (1997).

Abstract. {\rm In this paper we examine the problem of optimal approximation in the 2-norm of a finite square Hankel matrix by a Hankel matrix of rank one and provide a necessary and sufficient condition for its solvability.

23. On behaviors rational interpolation, and the Lanczos algorithm with E.J. Grimme and D.C. Sorensen, Proc. 13th IFAC Triennial World Congress, San Francisco, Pergamon Press (1996).

Abstract. This paper explores the interconnections between two methods which can be used to obtain rational interpolants. The first method: the behavioral approach, constructs a generating system in the frequency domain which explains a given data set. The second method: the rational Lanczos algorithm, can be used to construct a rational interpolant for a system defined by (potentially very high-order) state-space equations. This paper works to merge the theoretical attributes of the behavioral approach with the theoretical and computational properties of rational Lanczos. As a result it lays the foundation for the computation of reduced-order stabilizing controllers through rational interpolation.

24. Approximation of linear operators in the 2-norm Linear Algebra and Its Applications, Special Issue on Challenges in Matrix Theory, 278: 309-316 (1998).

Abstract. The problems of approximating in the $2$-induced norm linear oparators which are (1) finite-dimensional unstructured, and (2) infinite-dimensional structured (Hankel) have been solved. The solutions of these two problems exhibit striking similarities. These similarities suggest the search of a unifying framework for the approximation of linear operators in the 2-induced norm.

25. Optimizing the EV of the data matrix: a case study in non-smooth analysis with A. Astolfi, Report EE Dept., Imperial College London, 1998.

Abstract. In this work we study some properties of the extrema of the EV's associated to the data covariance matrix arising in the robust identification problem for discrete time finite dimensional linear systems. It is well known that the EV's are continuous but in general, non-differentiable functions of the parameters of the matrix. As a consequence the maximization of a pre-specified EV cannot be performed using tools from smooth analysis and is typically performed numerically using LMI's methods. Nevertheless we show that non-differentiable extrema have a simple interpretation enabling their detection. Finally the convexity properties of the EV's of the data covariance matrix are studied.

26. Controllability and Observability with E.D. Sontag and Y. Yamamoto, in Wiley Encyclopedia of Electrical and Electronics Engineering, edited by J.G. Webster, volume 4: 264-281 (1999).

27. Approximation of linear dynamical systems in Wiley Encyclopedia of Electrical and Electronics Engineering, edited by J.G. Webster, volume 11: 403-422 (1999).

Abstract. The last two articles are Invited papers prepared for the Wiley Encyclopedia of Electrical and Electronics Engineering.

28. On the choice of inputs in identification for robust control with B.D.O. Anderson, Automatica, 35: 1009-1031 (1999).

Abstract. The thesis that noisy identification has close ties to the study of the singular value decomposition of perturbed matrices is investigated. In particular by assuming an upper bound on the norm of the perturbation one can obtain a convex parametrization of an uncertain family of systems which contains the system generating the data. In this approach the second-smallest singular value of the data matrix becomes a quantity of great importance as it provides an upper bound for the size of the uncertain family. This yields a new tool leading to the design of input functions which are optimal or persistently exciting from the point of view of identification for robust control.

29. Approximation of large-scale dynamical systems A.C. Antoulas, Draft, to appear, SIAM Press (2003).

Abstract. This textbook presents two classes of approximation methods for linear dynamical systems which lead to explicit and easy-to-implement algorithms and have great potential for applications. The important feature of the first approach is that the approximation error can be quantified while for the second its numerical stability.

30. A survey of model reduction methods for large-scale systems A.C. Antoulas, D.C. Sorensen, and S. Gugercin, Structured Matrices in Operator Theory, Numerical Analysis, Control, Signal and Image Processing,'' Contemporary Mathematics, AMS publications, 2001.

Abstract. An overview of model reduction methods and a comparison of the resulting algorithms are presented. These approaches are divided into two broad categories, namely SVD based and moment matching based methods. It turns out that the approximation error in the former case behaves better globally in frequency while in the latter case the local behavior is better.

31. Lyapunov, Lanczos, and Inertia A.C. Antoulas and D.C. Sorensen, Linear Algebra and Its Applications, 326: 137-150 (2001).

Abstract. We present a new proof of the inertia result associated with Lyapunov equations. Furthermore we present a connection between the Lyapunov equation and the Lanczos process which is closely related to the Schwarz form of a matrix. We provide a method for reducing a general matrix to Schwarz form in a finite number of steps (O(n3)). Hence, we provide a finite method for computing inertia without computing eigenvalues. This scheme is unstable numerically and hence is primarily of theoretical interest.

32. Approximation of large-scale dynamical systems: An overview A.C. Antoulas and D.C. Sorensen, Technical Report, February 2001.

Abstract. In this paper we review the state of affairs in the area of approximation of large-scale systems. We distinguish among three basic categories, namely the SVD-based, the Krylov-based and the SVD-Krylov-based approximation methods. The first two were developed independently of each other and have distinct sets of attributes and drawbacks. The third approach seeks to combine the best attributes of the first two.

33. The Sylvester equation and approximate balanced reduction D.C. Sorensen and A.C. Antoulas, Fourth Special Issue on Linear Systems and Control, Edited by V. Blondel, D. Hinrichsen, J. Rosenthal, and P.M. van Dooren, Linear Algebra and Its Applications, {\bf 351-352}: 671-700 (2002).

Abstract. The purpose of this paper is to investigate methods for the iterative computation of approximately balanced reduced order systems. The resulting approach is both completely automatic once an error tolerance is specified and yields an error bound. This is to be contrasted with existing projection methods, namely PVL (Pad\'e via Lanczos) and rational Krylov, which do not satisfy these properties. Our approach is based on the computation and approximation of the {\it cross gramian} of the system. The cross gramian is the solution of a Sylvester equation and therefore some effort is dedicated to the study of this equation with some new insights. Our method produces a rank $k$ approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the orignal system. It is well suited to large scale problems because there are no matrix factorizations of the large (sparse) system matrix. Only matrix-vector products are required.

34. On the decay rate of the Hankel singular values and related issues A.C. Antoulas, D.C. Sorensen, and Y. Zhou, Systems and Control Letters, vol 46 (5), pp. 323-342 (2002).

Abstract. This paper investigates the decay rate of the Hankel singular values of linear dynamical systems. This issue is of considerable interest in model reduction by means of balanced truncation, for instance, since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error. The decay rate involves a new set of invariants associated with a linear system, which are obtained by evaluating a modified transfer function at the poles of the system. These considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two Lyapunov equations. The related problem of determining the decay rate of the eigenvalues of the solution to one Lyapunov equation will also be addressed. Very often these eigenvalues like the Hankel singular values, are decaying rapidly. This fact has motivated the development of several algorithms for computing low rank approximate solutions to Lyapunov equations. However, until now, conditions assuring rapid decay have not been well understood. Such conditions are derived here by relating the solution to a numerically low rank Cauchy matrix determined by the poles of the system.

35. Frequency domain representation and singular value decomposition A.C. Antoulas, UNESCO EOLSS (Encyclopedia for the Life Sciences), Contribution 6.43.13.4, June 2001.

Abstract. This contribution reviews the external and the internal representations of linear time-invariant systems. This is done both in the time and the frequency domains. The realization problem is then discussed. Given the importance of norms in robust control and model reduction, the final part of this contribution is dedicated to the definition and computation of various norms. Again, the interplay between time and frequency domain norms is emphasized.

36. The issue of consistency in identification for robust control S. Gugercin, A.C. Antoulas, and H.P. Zhang, Technical Report, July 2001.

Abstract. Given measured data generated by a discrete-time linear system we propose a model consisting of a linear, time-invariant system affected by norm-bounded perturbation. Under mild assumptions, the plants belonging to the resulting uncertain family form a convex set. The approach depends on two key parameters: an a priori given bound of the perturbation, and the input used to generate the data. It turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs. Two model validation problems in this identification scheme are explored, namely the worst and the best invalidation problems. It turns out that while the former is a max-min optimization problem subject to a spherical constraint, the latter is a quadratic optimization problem with a quadratic and a convex constraint.

37. A modified low-rank Smith method for large-scale Lyapunov equations S. Gugercin, D.C. Sorensen, and A.C. Antoulas, Numerical Algorithms, Vol. 32, Issue 1, pp. 27-55, January 2003.

Abstract. In this note we present a modified cyclic low-rank Smith method to compute low-rank approximations to solutions of Lyapunov equations arising from large-scale dynamical systems. Unlike the original cyclic low-rank Smith method introduced by Penzl, the number of columns required by the modified method in the approximate solution does not necessarily increase at each step and is usually much lower than in the original cyclic low-rank Smith method. The modified method never requires more columns than the original one. Upper bounds are established for the errors of the low-rank approximate solutions and also for the errors in the resulting approximate Hankel singular values. Numerical results are given to verify the efficiency and accuracy of the new algorithm.