Change search
Refine search result
12 1 - 50 of 52
CiteExportLink to result list
Permanent link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Rows per page
  • 5
  • 10
  • 20
  • 50
  • 100
  • 250
Sort
  • Standard (Relevance)
  • Author A-Ö
  • Author Ö-A
  • Title A-Ö
  • Title Ö-A
  • Publication type A-Ö
  • Publication type Ö-A
  • Issued (Oldest first)
  • Issued (Newest first)
  • Created (Oldest first)
  • Created (Newest first)
  • Last updated (Oldest first)
  • Last updated (Newest first)
  • Disputation date (earliest first)
  • Disputation date (latest first)
  • Standard (Relevance)
  • Author A-Ö
  • Author Ö-A
  • Title A-Ö
  • Title Ö-A
  • Publication type A-Ö
  • Publication type Ö-A
  • Issued (Oldest first)
  • Issued (Newest first)
  • Created (Oldest first)
  • Created (Newest first)
  • Last updated (Oldest first)
  • Last updated (Newest first)
  • Disputation date (earliest first)
  • Disputation date (latest first)
Select
The maximal number of hits you can export is 250. When you want to export more records please use the Create feeds function.
  • 1.
    Avventi, Enrico
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Wahlberg, Bo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    ARMA Identification of Graphical Models2013In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 58, no 5, p. 1167-1178Article in journal (Refereed)
    Abstract [en]

    Consider a Gaussian stationary stochastic vector process with the property that designated pairs of components are conditionally independent given the rest of the components. Such processes can be represented on a graph where the components are nodes and the lack of a connecting link between two nodes signifies conditional independence. This leads to a sparsity pattern in the inverse of the matrix-valued spectral density. Such graphical models find applications in speech, bioinformatics, image processing, econometrics and many other fields, where the problem to fit an autoregressive (AR) model to such a process has been considered. In this paper we take this problem one step further, namely to fit an autoregressive moving-average (ARMA) model to the same data. We develop a theoretical framework and an optimization procedure which also spreads further light on previous approaches and results. This procedure is then applied to the identification problem of estimating the ARMA parameters as well as the topology of the graph from statistical data.

  • 2.
    Avventi, Enrico
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Wahlberg, Bo
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Graphical Models of Autoregressive Moving-Average Processes2010In: The 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010), 2010Conference paper (Refereed)
    Abstract [en]

    Consider a Gaussian stationary stochastic vector process with the property that designated pairs of components are conditionally independent given the rest of the components. Such processes can be represented on a graph where the components are nodes and the lack of a connecting link between two nodes signifies conditional independence. This leads to a sparsity pattern in the inverse of the matrix-valued spectral density. Such graphical models find applications in speech, bioinformatics, image processing, econometrics and many other fields, where the problem to fit an autoregressive (AR) model to such a process has been considered. In this paper we take this problem one step further, namely to fit an autoregressive moving-average (ARMA) model to the same data. We develop a theoretical framework which also spreads further light on previous approaches and results.

  • 3. Byrnes, C. I.
    et al.
    Fanizza, Giovanna
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    The covariance extension equation revisited2005In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05, 2005, p. 7924-7930Conference paper (Refereed)
    Abstract [en]

    In this paper we study the steady state form of a discrete-time matrix Riccati-type equation, connected to the rational covariance extension problem and to the partial stochastic realization problem. This equation, however, is non-standard in that it lacks the usual kind of definiteness properties which underlie the solvability of the standard Riccati equation. Nonetheless, we prove the existence and uniqueness of a positive semidefinite solution. We also show that this equation has the proper geometric attributes to be solvable by homotopy continuation methods, which we illustrate in several examples.

  • 4.
    Byrnes, Christopher
    et al.
    KTH, Superseded Departments, Mathematics.
    Enqvist, Per
    KTH, Superseded Departments, Mathematics.
    Lindquist, Anders
    KTH, Superseded Departments, Mathematics.
    Cepstral coefficients, covariance lags, and pole-zero models for finite data strings2001In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 49, no 4, p. 677-693Article in journal (Refereed)
    Abstract [en]

    One of the most widely used methods of spectral estimation in signal and speech processing is linear predictive coding (LPC). LPC has some attractive features, which account for its popularity, including the properties that the resulting modeling filter i) matches a finite window of n + 1 covariance lags, ii) is rational of degree at most n, and iii) has stable zeros and poles. The only limiting factor of this methodology is that the modeling filter is "all-pole," i.e., an autoregressive (AR) model. In this paper, we present a systematic description of all autoregressive moving-average (ARMA) models of processes that have properties i)-iii) in the context of cepstral analysis and homomorphic filtering. Indeed, we show that each such ARMA model determines and is completely determined by its finite windows of cepstral coefficients and covariance lags. This characterization has an intuitively appealing interpretation of a characterization by using measures of the transient and the steady-state behaviors of the signal, respectively. More precisely, we show that these nth-order windows form local coordinates for all ARMA models of degree n and that the pole-zero model can be determined from the windows as the unique minimum of a convex objective function. We refine this optimization method by first noting that the maximum entropy design of an LPC filter is obtained by maximizing the zeroth cepstral coefficient, subject to the constraint i). More generally, we modify this scheme to a more well-posed optimization problem where the covariance data enters as a constraint and the linear weights of the cepstral coefficients are "positive"-in a sense that a certain pseudo-polynomial is positive-rather succinctly generalizing the maximum entropy method. This new problem is a homomorphic filter generalization of the maximum entropy method, providing a procedure for the design of any stable, minimum-phase modeling filter of degree less or equal to n that interpolates the given covariance window We conclude the paper by presenting an algorithm for realizing these biters in a lattice-ladder form, given the covariance window and the moving average part of the model. While we also show how to determine the moving average part using cepstral smoothing, one can make use of any good a priori estimate for the system zeros to initialize the algorithm. Indeed, we conclude the paper with an example of this method, incorporating an example from the literature on ARMA modeling.

  • 5.
    Byrnes, Christopher
    et al.
    KTH, Superseded Departments, Mathematics.
    Enqvist, Per
    KTH, Superseded Departments, Mathematics.
    Lindquist, Anders
    KTH, Superseded Departments, Mathematics.
    Identifiability and well-posedness of shaping-filter parameterizations: A global analysis approach2002In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 41, no 1, p. 23-59Article in journal (Refereed)
    Abstract [en]

    In this paper, we study the well-posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients, covariance lags, or Markov parameters. For example, we determine whether there exists a shaping filter with a prescribed window of Markov parameters and a prescribed window of covariance lags. We show that several such problems are well-posed in the sense of Hadamard; that is, one can prove existence, uniqueness (identifiability), and continuous dependence of the model on the measurements. Our starting point is the global analysis of linear systems, where one studies an entire class of systems or models as a whole, and where one views measurements, such as covariance lags and cepstral coefficients or Markov parameters, from data as functions on the entire class. This enables one to pose such problems in a way that tools from calculus, optimization, geometry, and modern nonlinear analysis can be used to give a rigorous answer to such problems in an algorithm-independent fashion. In this language, we prove that a window of cepstral coefficients and a window of covariance coefficients yield a bona de coordinate system on the space of shaping filters, thereby establishing existence, uniqueness, and smooth dependence of the model parameters on the measurements from data.

  • 6.
    Byrnes, Christopher
    et al.
    KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Georgiou, T. T.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Megretski, A.
    Generalized interpolation in H-infinity with a complexity constraint2006In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 358, no 3, p. 965-987Article in journal (Refereed)
    Abstract [en]

    In a seminal paper, Sarason generalized some classical interpolation problems for H-infinity functions on the unit disc to problems concerning lifting onto H-2 of an operator T that is defined on K=H-2 circle minus phi H-2 (phi is an inner function) and commutes with the (compressed) shift S. In particular, he showed that interpolants (i.e., f is an element of H-infinity such that f(S)=T) having norm equal to parallel to T parallel to exist, and that in certain cases such an f is unique and can be expressed as a fraction f=b/a with a, b is an element of K. In this paper, we study interpolants that are such fractions of K functions and are bounded in norm by 1 (assuming that parallel to T parallel to<1, in which case they always exist). We parameterize the collection of all such pairs (a, b)is an element of K x K and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where phi is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

  • 7.
    Byrnes, Christopher I.
    et al.
    Washington Univ, St Louis.
    Fanizza, Giovanna
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    A homotopy continuation solution of the covariance extension equation2005In: Lecture notes in control and information sciences, ISSN 0170-8643, E-ISSN 1610-7411, Vol. 321, p. 27-42Article in journal (Refereed)
  • 8. Byrnes, Christopher I.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    A note on the Jacobian conjecture2008In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 136, no 9, p. 3007-3011Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the Jacobian conjecture for a map f of complex a. ne spaces of dimension n. It is well known that if f is proper, then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if f is proper onto its image.

  • 9. Byrnes, Christopher I.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    IMPORTANT MOMENTS IN SYSTEMS AND CONTROL2008In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 47, no 5, p. 2458-2469Article in journal (Refereed)
    Abstract [en]

    The moment problem matured from its various special forms in the late 19th and early 20th centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. In particular, the theory of systems and control is no exception, where the applications have historically been to circuit theory, optimal control, robust control, signal processing, spectral estimation, stochastic realization theory, and the use of the moments of a probability density. Many of these applications are also still works in progress. In this paper, we consider the generalized moment problem, expressed in terms of a basis of a finite-dimensional subspace P of the Banach space C[a, b] and a "positive" sequence c, but with a new wrinkle inspired by the applications to systems and control. We seek to parameterize solutions which are positive "rational" measures in a suitably generalized sense. Our parameterization is given in terms of smooth objects. In particular, the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a Euclidean space and which is the domain of some canonically defined functions. The analysis of these functions, and related maps, yields interesting corollaries for the moment problem and its applications, which we compare to those in the recent literature and which play a crucial role in part of our proof.

  • 10. Byrnes, Christopher I.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Interior point solutions of variational problems and global inverse function theorems2007In: International Journal of Robust and Nonlinear Control, ISSN 1049-8923, E-ISSN 1099-1239, Vol. 17, no 5-6, p. 463-481Article in journal (Refereed)
    Abstract [en]

    Variational problems and the solvability of certain nonlinear equations have a long and rich history beginning with calculus and extending through the calculus of variations. In this paper, we are interested in 'well-connected' pairs of such problems which are not necessarily related by critical point considerations. We also study constrained problems of the kind which arise in mathematical programming. We are also interested in interior minimizing points for the variational problem and in the well-posedness (in the sense of Hadamard) of solvability of the related systems of equations. We first prove a general result which implies the existence of interior points and which also leads to the development of certain generalization of the Hadamard-type global inverse function theorem, along the theme that uniqueness quite often implies existence. This result is illustrated by proving the non-existence of shock waves for certain initial data for the vector Burgers' equation. The global inverse function theorem is also illustrated by a derivation of the existence of positive definite solutions of matrix Riccati equations without first analysing the nonlinear matrix Riccati differential equation. The main results on the existence of solutions to geometrically constrained well-connected pairs are then presented and illustrated by a geometric analysis of the existence of interior points for linear programming problems.

  • 11. Byrnes, Christopher I.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    The generalized moment problem with complexity constraint2006In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 56, no 2, p. 163-180Article in journal (Refereed)
    Abstract [en]

    In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which begins with a reformulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly convex optimization problem. Engineering applications typically demand a solution that is the ratio of functions in certain finite dimensional vector space of functions, usually the same vector space that is prescribed in the generalized moment problem. Solutions of this type are hinted at in the classical text by Krein and Nudelman and stated in the vast generalization of interpolation problems by Sarason. In this paper, formulated as generalized moment problems with complexity constraint, we give a complete parameterization of such solutions, in harmony with the above mentioned results and the engineering applications. While our previously announced results required some differentiability hypotheses, this paper uses a weak form involving integrability and measurability hypotheses that are more in the spirit of the classical treatment of the generalized moment problem. Because of this generality, we can extend the existence and well-posedness of solutions to this problem to nonnegative, rather than positive, initial data in the complexity constraint. This has nontrivial implications in the engineering applications of this theory. We also extend this more general result to the case where the numerator can be an arbitrary positive absolutely integrable function that determines a unique denominator in this finite-dimensional vector space. Finally, we conclude with four examples illustrating our results.

  • 12. Byrnes, Christopher I.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    The moment problem for rational measures: convexity in the spirit of Krein2009In: MODERN ANALYSIS AND APPLICATIONS: MARK KREIN CENTENARY CONFERENCE / [ed] Adamyan, V; Berezansky, Y; Gohberg, I; Gorbachuk, M; Gorbachuk, V; Kochubei, A; Langer, H; Popov, G, Birkhäuser Verlag, 2009, Vol. 190, p. 157-169Conference paper (Refereed)
    Abstract [en]

    The moment problem as formulated by Krein and Nudel'man is a beautiful generalization of several important classical moment problems, including the power moment problem, the trigonometric moment problem and the moment problem arising in Nevanlinna-Pick interpolation. Motivated by classical applications and examples, in both finite and infinite dimensions, we recently formulated a new version of this problem that we call the moment problem for positive rational measures. The formulation reflects the importance of rational functions in signals, systems and control. While this version of the problem is decidedly nonlinear, the basic tools still rely on convexity. In particular, we present a solution to this problem in terms of a nonlinear convex optimization problem that generalizes the maximum entropy approach used in several classical special cases.

  • 13.
    Byrnes, Christopher
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Important moments in systems, control and optimizations2009In: Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on, IEEE , 2009, p. 91-96Conference paper (Refereed)
    Abstract [en]

    The moment problem matured from its various special forms in the late 19th and early 20th Centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. In particular, the theory of systems and control is no exception, where the applications have historically been to circuit theory, optimal control, robust control, signal processing, spectral estimation, stochastic realization theory and the use of the moments of a probability density. Many of these applications are also still works in progress. In this paper, we consider the generalized moment problem, expressed in terms of a basis of a finite-dimensional subspace β of the Banach space C[a, b] and a "positive" sequences c, but with a new wrinkle inspired by the applications to systems and control. We seek to parameterize solutions which are positive "rational" measures, in a suitably generalized sense. Our parameterization is given in terms of smooth objects. In particular, the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a Euclidean space and which is the domain of some canonically defined functions. Moreover, on these spaces one can derive natural convex optimization criteria which characterize solutions to this new class of moment problems.

  • 14.
    Byrnes, Christopher
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    On the Stability and Instability of Padé Approximants2010In: Lecture notes in control and information sciences, ISSN 0170-8643, E-ISSN 1610-7411, Vol. 398, p. 165-175Article in journal (Refereed)
    Abstract [en]

    Over the past three decades there has been interest in using Pade approximants K with n = deg(K) < deg(G) = N as "reduced-order models" for the transfer function G of a linear system The attractive feature of this approach is that by matching the moments of G we can reproduce the steady-state behavior of G by the steady-state behavior of K. for certain classes of Inputs Indeed, we illustrate this by finding a first-order model matching a fixed set of moments for G. the causal inverse of a heat equation A key feature of this example is that the heat equation is a minimum phase system, so that its inverse system has a stable transfer function G and that K can also be chosen to be stable On the other hand, elementary examples show that both stability and instability can occur in reduced order models of a stable system obtained by matching moments using Pade approximants and, in the absence of stability, it does not make much sense to talk about steady-state responses nor does it make sense to match moments In this paper, we review Pack approximains. and their intimate relationship to continued fractions and Riccati equations, in a historical context that underscores why Pade approximation, as useful as it is, is not an approximation in any sense that reflects stability. Our main results on stability and instability states that if N >= 2 and l, r >= 0 with 0<l+r=n<N there is a non-empty open set U-lr of stable transfer functions G, having infinite Lebesque measure, such that each degree n proper rational function K matching the moments of G has e poles lying in C- and r poles lying in C+ The proof is constructive.

  • 15.
    Fanizza, Giovanna
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Nagamune, Ryozo
    A global analysis approach to passivity preserving model reduction2006In: Proceedings Of The 45th IEEE Conference On Decision And Control, Vols 1-14, 2006, p. 3399-3404Conference paper (Refereed)
    Abstract [en]

    Passivity-preserving model reduction for linear time-invariant systems amounts to approximating a positive-real rational transfer function with one of lower degree. Recently Antoulas and Sorensen have proposed such a model-reduction method based on Krylov projections. The method is based on an observation by Antoulas (in the single-input/single-output case) that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros, then the approximant is also positive real. However, this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by Byrnes, Georgiou and Lindquist, namely the maximum-entropy (central) solution. By tuning the interpolation points and the spectral zeros, as prescribed by this theory, one is able to obtain considerably better reduced-order models.

  • 16.
    Fanizza, Giovanna
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Nagamune, Ryozo
    Univ British Columbia, Dept Mech Engn, Vancouver.
    Passivity-preserving model reduction by analytic interpolation2007In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 425, no 2-3, p. 608-633Article in journal (Refereed)
    Abstract [en]

    Antoulas and Sorensen have recently proposed a passivity-preserving model-reduction method of linear systems based on Krylov projections. The idea is to approximate a positive-real rational transfer function with one of lower degree. The method is based on an observation by Antoulas (in the single-input/single-output case) that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros, then the approximant is also positive real. However, this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by Byrnes, Georgiou and Lindquist, namely the maximum-entropy (central) solution. By tuning the interpolation points and the spectral zeros, as prescribed by this theory, one is able to obtain considerably better reduced-order models. We also show that, in the multi-input/multi-output case, Sorensen's algorithm actually amounts to tangential Nevanlinna-Pick interpolation.

  • 17. Georgiou, T. T.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    On time-reversibility of linear stochastic models2014In: IFAC Proceedings Volumes (IFAC-PapersOnline), IFAC Papers Online, 2014, p. 10403-10408Conference paper (Refereed)
    Abstract [en]

    Reversal of the time direction in stochastic systems driven by white noise has been of central importance throughout the development of stochastic realization theory, filtering and smoothing. Similarly, in connection with certain problems in the theory of moments, the duality induced by time reversal was the key to deriving a compact parametrization and an interpretation of solutions. By combining ideas from these two lines of development we present herein a general view and a simplified account of time-reversal in stochastic models.

  • 18.
    Georgiou, Tryphon
    et al.
    University of Minnesota, United States.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    The separation principle in stochastic control, redux2013In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 58, no 10, p. 2481-2494Article in journal (Refereed)
    Abstract [en]

    Over the last 50 years, a steady stream of accounts have been written on the separation principle of stochastic control. Even in the context of the linear-quadratic regulator in continuous time with Gaussian white noise, subtle difficulties arise, unexpected by many, that are often overlooked. In this paper we propose a new framework for establishing the separation principle. This approach takes the viewpoint that stochastic systems are well-defined maps between sample paths rather than stochastic processes per se and allows us to extend the separation principle to systems driven by martingales with possible jumps. While the approach is more in line with "real-life" engineering thinking where signals travel around the feedback loop, it is unconventional from a probabilistic point of view in that control laws for which the feedback equations are satisfied almost surely, and not deterministically for every sample path, are excluded.

  • 19.
    Georgiou, Tryphon
    et al.
    University of Minnesota.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Two alternative views on control design with degree constraint2005In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05, IEEE , 2005, p. 3645-3650Conference paper (Refereed)
    Abstract [en]

    The purpose of this note is to highlight similarities and differences between two alternative methodologies for feedback control design under constraints on the McMillan degree of the feedback system. Both sets of techniques focus on uniformly optimal designs. The first is based on the work of Gahinet and Apkarian, and Skelton, Iwasaki, Grigoriades and their co-workers, while the other is based on earlier joint work of the authors with C. I. Byrnes.

  • 20. Georgiou, Tryphon T.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    A convex optimization approach to ARMA modeling2008In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 53, no 5, p. 1108-1119Article in journal (Refereed)
    Abstract [en]

    We formulate a convex optimization problem for approximating any given spectral density with a rational one having a prescribed number of poles and zeros (n poles and m zeros inside the unit disc and their conjugates). The approximation utilizes the Kullback-Leibler divergence as a distance measure. The stationarity condition for optimality requires that the approximant matches n + 1 covariance moments of the given power spectrum and m cepstral moments of the corresponding logarithm, although the latter with possible slack. The solution coincides with one derived by Byrnes, Enqvist, and Lindquist who addressed directly the question of covariance and cepstral matching. Thus, the present paper provides an approximation theoretic justification of such a problem. Since the approximation requires only moments of spectral densities and of their logarithms, it can also be used for system identification.

  • 21. Georgiou, Tryphon T.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Likelihood Analysis of Power Spectra and Generalized Moment Problems2017In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 62, no 9, p. 4580-4592Article in journal (Refereed)
    Abstract [en]

    We develop an approach to the spectral estimation that has been advocated by [ A. Ferrante et al., "Time and spectral domain relative entropy: A new approach to multivariate spectral estimation,"IEEE Trans. Autom. Control, vol. 57, no. 10, pp. 2561-2575, Oct. 2012.] and, in the context of the scalar-valued covariance extension problem, by [P. Enqvist and J. Karlsson, "Minimal itakurasaito distance and covariance interpolation," in Proc. 47th IEEE Conf. Decision Control, 2008, pp. 137-142]. The aim is to determine the power spectrum that is consistent with given moments and minimizes the relative entropy between the probability law of the underlying Gaussian stochastic process to that of a prior. The approach is analogous to the framework of earlier work by Byrnes, Georgiou, and Lindquist and can also be viewed as a generalization of the classical work by Burg and Jaynes on the maximum entropy method. In this paper, we present a new fast algorithm in the general case (i.e., for general Gaussian priors) and show that for priors with a specific structure the solution can be given in closed form.

  • 22. Georgiou, Tryphon T.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre. Shanghai Jiao Tong Univ, Peoples R China.
    Optimal Estimation With Missing Observations via Balanced Time-Symmetric Stochastic Models2017In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 62, no 11, p. 5590-5603Article in journal (Refereed)
    Abstract [en]

    We consider data fusion for the purpose of smoothing and interpolation based on observation records with missing data. Stochastic processes are generated by linear stochastic models. The paper begins by drawing a connection between time reversal in stochastic systems and all-pass extensions. A particular normalization (choice of basis) between the two time-directions allows the two to share the same orthonormalized state process and simplifies the mathematics of data fusion. In this framework, we derive symmetric and balanced Mayne-Fraser-like formulas that apply simultaneously to continuous-time smoothing and interpolation, providing a definitive unification of these concepts. The absence of data over subintervals requires in general a hybrid filtering approach involving both continuous-time and discrete-time filtering steps.

  • 23. Georgiou, Tryphon T.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Remarks on control design with degree constraint2006In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 51, no 7, p. 1150-1156Article in journal (Refereed)
    Abstract [en]

    The purpose of this note is to highlight similarities and differences between two alternative methodologies for feedback control design under constraints on the McMillan degree of the feedback system. Both sets of techniques focus on uniformly optimal designs. The first is based on the work of Gahinet and Apkarian and that of Skelton et al., while the other is based on earlier joint work of the authors with C. I. Byrnes.

  • 24. Hanebeck, U. D.
    et al.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Moment-based dirac mixture approximation of circular densities2014In: IFAC Proceedings Volumes (IFAC-PapersOnline), 2014, p. 5040-5048Conference paper (Refereed)
    Abstract [en]

    Given a circular probability density function, called the true probability density function, the goal is to find a Dirac mixture approximation based on some circular moments of the true density. When keeping the locations of the Dirac points fixed, but almost arbitrarily located, we are applying recent results on the circulant rational covariance extension problem to the problem of calculating the weights. For the case of simultaneously calculating optimal locations, additional constraints have to be deduced from the given density. For that purpose, a distance measure for the deviation of the Dirac mixture approximation from the true density is derived, which then is minimized while considering the moment conditions as constraints. The method is based on progressive numerical minimization, converges quickly and gives well-distributed Dirac mixtures that fulfill the constraints, i.e., have the desired circular moments.

  • 25.
    Karlsson, Johan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM.
    Georgiou, Tryphon
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    The inverse problem of analytic interpolation with degree constraint2006In: Proceedings Of The 45th IEEE Conference On Decision And Control, Vols 1-14, 2006, p. 559-564Conference paper (Refereed)
    Abstract [en]

    In [7], (6] a theory for degree-constrained analytic interpolation was developed in terms of the minimizers of certain convex entropy functionals. In the present paper, we introduce and study relevant inverse problems. More specifically, we answer the following two questions. First, given a function f which satisfies specified interpolation conditions, when is it that f can be obtained as the minimizer of a suitably chosen entropy functional? Second, given a function g, when does there exist a suitably entropy functional so that the unique minitnizer f which is subject to interpolation constraints also satisfies vertical bar f vertical bar = vertical bar g vertical bar on the unit circle. The theory and answers to these questions suggest an approach to identifying interpolants of a given degree and of a given approximate shape.

  • 26.
    Karlsson, Johan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Georgiou, Tryphon T.
    Department of Electrical Engineering, University of Minnesota.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    The Inverse Problem of Analytic Interpolation With Degree Constraint and Weight Selection for Control Synthesis2010In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 55, no 2, p. 405-418Article in journal (Refereed)
    Abstract [en]

    The minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint, and all solutions to this interpolation problem arise in this way by a suitable choice of weights. Selecting appropriate weights is pertinent to feedback control synthesis, where interpolants represent closed-loop transfer functions. In this paper we consider the correspondence between weights and interpolants in order to systematize feedback control synthesis with a constraint on the degree. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights. This leads to a new procedure for feedback control synthesis.

  • 27.
    Karlsson, Johan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    On Degree-Constrained Analytic Interpolation With Interpolation Points Close to the Boundary2009In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 54, no 6, p. 1412-1418Article in journal (Refereed)
    Abstract [en]

    In the recent article [4], a theory for complexity-constrained interpolation of contractive functions is developed. In particular, it is shown that any such interpolant may be obtained as the unique minimizer of a (convex) weighted entropy gain. In this technical note we study this optimization problem in detail and describe how the minimizer depends on weight selection and on interpolation conditions. We first show that, if, for a sequence of interpolants, the values of the entropy gain of the interpolants converge to the optimum, then the interpolants converge in H-2, but not in H-infinity This result is then used to describe the asymptotic behavior of the interpolant as an interpolation point approaches the boundary of the domain of analyticity. For loop shaping to specifications in control design, it might at first seem natural to place strategically additional interpolation points close to the boundary. However, our results indicate that such a strategy will have little effect on the shape. Another consequence of our results relates to model reduction based on minimum-entropy principles, where one should avoid placing interpolation points too close to the boundary.

  • 28.
    Karlsson, Johan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Stability-preserving rational approximation subject to interpolation constraints2008In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 53, no 7, p. 1724-1730Article in journal (Refereed)
    Abstract [en]

    A quite comprehensive theory of analytic interpolation with degree constraint, dealing with rational analytic interpolants with an a priori bound, has been developed in recent years. In this paper, we consider the limit case when this bound is removed, and only stable interpolants with a prescribed maximum degree are sought. This leads to weighted H-2 minimization, where the interpolants are parameterized by the weights. The inverse problem of determining the weight given a desired interpolant profile is considered, and a rational approximation procedure based on the theory is proposed. This provides a tool for tuning the solution to specifications. The basic idea could also be applied to the case with bounded analytic interpolants.

  • 29.
    Karlsson, Johan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Stable rational approximation in the context of interpolation and convex optimization2007In: Proceedings Of The 46th IEEE Conference On Decision And Control, Vols 1-14, 2007, p. 2214-2221Conference paper (Refereed)
    Abstract [en]

    A quite comprehensive theory of analytic interpolation with degree constraint, dealing with rational interpolants with an a priori bound, has been developed in recent years. In this paper we consider the limit case when this bound is removed, and only stable interpolants with a prescribed maximum degree are sought. This leads to weighted H-2 minimization, where the interpolants are parameterized by the weights. The inverse problem of determining the weight and the interpolation points given a desired interpolant profile is considered, and a rational approximation procedure based on the theory is proposed. This provides a tool for tuning the solution to specifications. The basic idea could also be applied to the case with bounded interpolants.

  • 30.
    Karlsson, Johan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong University, China.
    Ringh, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    The Multidimensional Moment Problem with Complexity Constraint2016In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 84, no 3, p. 395-418Article in journal (Refereed)
    Abstract [en]

    A long series of previous papers have been devoted to the (one-dimensional) moment problem with nonnegative rational measure. The rationality assumption is a complexity constraint motivated by applications where a parameterization of the solution set in terms of a bounded finite number of parameters is required. In this paper we provide a complete solution of the multidimensional moment problem with a complexity constraint also allowing for solutions that require a singular measure added to the rational, absolutely continuous one. Such solutions occur on the boundary of a certain convex cone of solutions. In this paper we provide complete parameterizations of all such solutions. We also provide errata for a previous paper in this journal coauthored by one of the authors of the present paper.

  • 31.
    Kuroiwa, Yohei
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    A benchmark problem in robust control with degree constraint: two-sided Nudelman interpolation with complexity constraint approachManuscript (preprint) (Other academic)
  • 32.
    Kuroiwa, Yohei
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Bi-tangential Nevanlinna-Pick interpolation with a complexity constraint2006In: Proceedings of the 17th International Symposium on the Mathematical Theory of Networks and Systems: Kyoto, 2006, 2006, p. 2541-2563Conference paper (Other academic)
  • 33.
    Lindquist, Anders
    KTH, Superseded Departments, Mathematics.
    A covariance extension approach to identification of time series2000In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 36, no 3, p. 379-398Article in journal (Refereed)
    Abstract [en]

    In this paper we consider a three-step procedure far identification of time series, based on covariance extension and model reduction, and we present a complete analysis of its statistical convergence properties. A partial covariance sequence is estimated from statistical data. Then a high-order maximum-entropy model is determined, which is finally approximated by a lower-order model by stochastically balanced model reduction. Such procedures have been studied before, in various combinations, but an overall convergence analysis comprising all three steps has been lacking. Supposing the data is generated from a true finite-dimensional system which is minimum phase, it is shown that the transfer function of the estimated system tends in H-infinity to the true transfer function as the data length tends to infinity, if the covariance extension and the model reduction is done properly. The proposed identification procedure, and some variations of it, are evaluated by simulations.

  • 34.
    Lindquist, Anders
    KTH, Superseded Departments, Mathematics.
    A new approach to spectral estimation: A tunable high-resolution spectral estimator2000In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 48, no 11, p. 3189-3205Article in journal (Refereed)
    Abstract [en]

    Traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates. Here, we present a new approach to spectral estimation, which is based on the use of filter banks as a means of obtaining spectral interpolation data, Such data replaces standard covariance estimates, A computational procedure for obtaining suitable pole-zero (ARMA) models from such data is presented. The choice of the zeros (MA-part) of the model is completely arbitrary. By suitably choices of filterbank poles and spectral zeros, the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum.

  • 35.
    Lindquist, Anders
    KTH, Superseded Departments, Mathematics.
    Generalized entropy criterion for Nevanlinna-Pick interpolation with degree constraint2001In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 46, no 6, p. 822-839Article in journal (Refereed)
    Abstract [en]

    In this paper, we present a generalized entropy criterion for solving the rational Nevanlinna-Pick problem for n + 1 interpolating conditions and the degree of interpolants bounded by n, The primal problem of maximizing this entropy gain has a very well-behaved dual problem. This dual is a convex optimization problem in a finite-dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n, The criterion requires a selection of a monic Schur polynomial of degree n, It follows that this class of monic polynomials completely parameterizes all such rational interpolants, and it therefore provides a set of design parameters for specifying such interpolants. The algorithm is implemented in state-space form and applied to several illustrative problems in systems and control, namely sensitivity minimization, maximal power transfer and spectral estimation.

  • 36.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Kalman's Influence on My Scientific Work: Some Recollections and Reflections2017In: IEEE CONTROL SYSTEMS MAGAZINE, ISSN 1066-033X, Vol. 37, no 2, p. 156-157Article in journal (Refereed)
    Abstract [en]

    I first met Rudolf Kalman in Vienna, Austria, in the spring of 1972. I had recently finished my Ph.D. at the Royal Institute of Technology, Stockholm, Sweden, and I was invited to give a talk on my recent results in stochastic control theory at a small workshop that Kalman also attended. Apparently, Kalman was favorably impressed with my talk because he took me out for dinner the same evening and immediately invited me to come to Florida for the coming academic year. Kalman had just moved from Stanford to the University of Florida, and this is how I became his first postdoctoral associate at his new Center for Mathematical Systems Theory in the fall of 1972.

  • 37.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Kullback-Leibler approximation of spectral density functions2003In: IEEE Transactions on Information Theory, ISSN 0018-9448, E-ISSN 1557-9654, Vol. 49, no 11, p. 2910-2917Article in journal (Refereed)
    Abstract [en]

    We introduce a Kullback-Leibler-type distance between spectral density functions of stationary stochastic processes and solve the problem of optimal approximation of a given spectral density P by one that is consistent with prescribed second-order statistics. In general, such statistics are expressed as the state covariance of a linear filter driven by a stochastic process whose spectral density is sought. In this context, we show i) that there is a unique spectral density P which minimizes this Kullback-Leibler distance, ii) that this optimal approximate is of the form psi/Q where the "correction term" Q is a rational spectral density function, and iii) that the coefficients of Q can be obtained numerically by solving a suitable convex optimization problem. In the special case where psi = 1, the convex functional becomes quadratic and the solution is then specified by linear equations.

  • 38.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Matrix-valued Nevanlinna-pick interpolation with complexity constraint: An optimization approach2003In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 48, no 12, p. 2172-2190Article in journal (Refereed)
    Abstract [en]

    Over the last several years, a new theory of Nevanlinna-Pick interpolation with complexity constraint has been developed for scalar interpolants. In this paper we generalize this theory to the matrix-valued, case, also allowing for multiple interpolation points. We parameterize a class of interpolants consisting of "most interpotants" of no higher degree than the central solution in terms of spectral zeros. This is a complete parameterization, and for each choice of interpolant we provide a convex optimization problem for determining it. This is derived in the context of duality theory of mathematical programming. To solve the convex optimization problem, we employ a homotopy continuation technique previously developed for the scalar case. These results can be applied to many classes of engineering problems,and, to illustrate this, we provide some,examples. In particular, we apply our method to a benchmark problem in multivariate robust control. By constructing a controller satisfying all design specifications but having only half the McMillan degree of conventional H-infinity controllers, we demonstrate the advantage of the proposed method.

  • 39.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    New Directions and Applications in Control Theory2005Collection (editor) (Refereed)
  • 40.
    Lindquist, Anders
    KTH, Superseded Departments, Mathematics.
    On the duality between filtering and Nevanlinna-Pick interpolation2000In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 39, no 3, p. 757-775Article in journal (Refereed)
    Abstract [en]

    Positive real rational functions play a central role in both deterministic and stochastic linear systems theory, as well as in circuit synthesis, spectral analysis, and speech processing. For this reason, results about positive real transfer functions and their realizations typically have many applications and manifestations. In this paper, we study certain manifolds and submanifolds of positive real transfer functions, describing a fundamental geometric duality between filtering and Nevanlinna Pick interpolation. Not surprisingly, then, this duality, while interesting in its own right, has several corollaries which provide solutions and insight into some very interesting and intensely researched problems. One of these is the problem of parameterizing all rational solutions of bounded degree of the Nevanlinna-Pick interpolation problem, which plays a central role in robust control, and for which the duality theorem yields a complete solution. In this paper, we shall describe the duality theorem, which we motivate in terms of both the interpolation problem and a fast algorithm for Kalman filtering, viewed as a nonlinear dynamical system on the space of positive real transfer functions. We also outline a new proof of the recent solution to the rational Nevanlinna Pick interpolation problem, using an algebraic topological generalization of Hadamard's global inverse function theorem.

  • 41.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Shanghai Jiao Tong Univ, Dept Automat, Shanghai, Peoples R China.
    Partial Realization Theory and System Identification Redux2017In: 2017 11TH ASIAN CONTROL CONFERENCE (ASCC), IEEE , 2017, p. 1946-1950Conference paper (Refereed)
    Abstract [en]

    Some twenty years ago we introduced a nonstandard matrix Riccati equation to solve the partial stochastic realization problem. In this paper we provide a new derivation of this equation in the context of system identification. This allows us to show that the nonstandard matrix Riccati equation is universal in the sense that it can be used to solve more general analytic interpolation problems by only changing certain parameters. Such interpolation problems are ubiquitous in systems and control. In this context we also discuss a question posed by R.E. Kalman in beginning of the 1970s.

  • 42.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Prediction-error approximation by convex optimization2007In: Modeling, Estimation and Control:Festschrift in honor of Giorgio Picci on the occation of his sixty-fifth birthday, BERLIN: SPRINGER-VERLAG BERLIN , 2007, Vol. 364, p. 239-249Chapter in book (Refereed)
    Abstract [en]

    This paper is dedicated to Giorgio Picci on the occasion of his 65th birthday. I have come to appreciate Giorgio not only as a great friend but also as a great scholar. When we first met at Brown University in 1973, he introduced me to his seminal paper [29] on splitting subspaces, which became the impetus for our joint work on the geometric theory of linear stochastic systems [23,24,25,26]. This led to a life-long friendship and a book project that never seemed to converge, but now is close to being finished [27]. I have learned a lot from Giorgio. The present paper grew out of a discussion in our book project, when Giorgio taught me about the connections between prediction-error identification and the Kullback-Leibler criterion. These concepts led directly into the recent theory of analytic interpolation with complexity constraint, with which I have been deeply involved in recent times. I shall try to explain these connections in the following paper.

  • 43.
    Lindquist, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Avventi, Enrico
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM.
    Wahlberg, Bo
    KTH, School of Electrical Engineering (EES), Automatic Control.
    Graphical Models of Autoregressive Moving-Average ProcessesIn: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523Article in journal (Other academic)
  • 44.
    Lindquist, Anders G.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Picci, Giorgio
    The Circulant Rational Covariance Extension Problem: The Complete Solution2013In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 58, no 11, p. 2848-2861Article in journal (Refereed)
    Abstract [en]

    The rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite-dimensional positive-definite Toeplitz matrix the northwest corner of which is given. The circulant rational covariance extension problem considered in this paper is a modification of this problem to partial stochastic realization of periodic stationary process, which are better represented on the discrete unit circle Z(2N) rather than on the discrete real line Z. The corresponding matrix completion problem then amounts to completing a finite-dimensional Toeplitz matrix that is circulant. Another important motivation for this problem is that it provides a natural approximation, involving only computations based on the fast Fourier transform, for the ordinary rational covariance extension problem, potentially leading to an efficient numerical procedure for the latter. The circulant rational covariance extension problem is an inverse problem with infinitely many solutions in general, each corresponding to a bilateral ARMA representation of the underlying periodic process. In this paper, we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them. A procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented.

  • 45.
    Lindquist, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Georgiou, Tryphon
    Weight selection for gap robustness with degree-constrained controllers2008In: Proc. 47th IEEE Conference on Decision and Control, 2008, p. 4127-4134Conference paper (Refereed)
    Abstract [en]

    In modern robust control, control synthesis may be cast as an interpolation problem where the interpolant relates to robustness and performance criteria. In particular,robustness in the gap fits into this framework and the magnitude of the corresponding interpolant dictate the robustness to perturbations of the plant as a function of frequency. In this paper we consider the correspondence between weighted entropy functionals and minimizing interpolants in order to find appropriate interpolants for e.g. control synthesis. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights. These results are used in order to systematize feedback control synthesis to obtain frequency dependentrobustness bounds with a constraint on the controller degree. 

  • 46.
    Lindquist, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre. Shanghai Jiao Tong University, Shanghai, China .
    Masiero, C.
    Picci, G.
    On the multivariate circulant rational covariance extension problem2013In: 2013 IEEE 52nd Annual Conference on Decision and Control (CDC), IEEE conference proceedings, 2013, p. 7155-7161Conference paper (Refereed)
    Abstract [en]

    Partial stochastic realization of periodic processes from finite covariance data leads to the circulant rational covariance extension problem and bilateral ARMA models. In this paper we present a convex optimization-based theory for this problem that extends and modifies previous results by Carli, Ferrante, Pavon and Picci on the AR solution, which have been successfully applied to image processing of textures. We expect that our present results will provide an enhancement of these procedures.

  • 47.
    Lindquist, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Picci, Giorgio
    Modeling of Periodic Time Series by Bilateral ARMA Representations2014In: INTERNATIONAL WORK-CONFERENCE ON TIME SERIES (ITISE 2014), 2014, p. 861-865Conference paper (Refereed)
    Abstract [en]

    In this extended abstract for an oral presentation we describe a moment-based approach to modeling of stationary, periodic time series from a finite sequence of covariance lags. We present a complete parameterization of a family of solutions and a convex optimization procedure to determine each solution, which is seen to be represented as a bilateral ARMA model.

  • 48.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong Univ, Dept Automat, Shanghai, Peoples R China; Shanghai Jiao Tong Univ, Sch Math, Shanghai, Peoples R China.
    Further results on multidimensional rational covariance extension with application to texture generation2017In: 2017 IEEE 56TH ANNUAL CONFERENCE ON DECISION AND CONTROL (CDC), IEEE , 2017Conference paper (Refereed)
    Abstract [en]

    The rational covariance extension problem is a moment problem with several important applications in systems and control as, for example, in identification, estimation, and signal analysis. Here we consider the multidimensional counterpart and present new results for the well-posedness of the problem. We apply the theory to texture generation by modeling the texture as the output of a Wiener system. The static nonlinearity in the Wiener system is assumed to be a thresholding function and we identify both the linear dynamical system and the thresholding parameter.

  • 49.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong University, China.
    Multidimensional rational covariance extension with applications to spectral estimation and image compression2016In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 54, no 4, p. 1950-1982Article in journal (Refereed)
    Abstract [en]

    The rational covariance extension problem (RCEP) is an important problem in systems and control occurring in such diverse fields as control, estimation, system identification, and signal and image processing, leading to many fundamental theoretical questions. In fact, this inverse problem is a key component in many identification and signal processing techniques and plays a fundamental role in prediction, analysis, and modeling of systems and signals. It is well known that the RCEP can be reformulated as a (truncated) trigonometric moment problem subject to a rationality condition. In this paper we consider the more general multidimensional trigonometric moment problem with a similar rationality constraint. This generalization creates many interesting new mathematical questions and also provides new insights into the original one-dimensional problem. A key concept in this approach is the complete smooth parameterization of all solutions, allowing solutions to be tuned to satisfy additional design specifications without violating the complexity constraints. As an illustration of the potential of this approach we apply our results to multidimensional spectral estimation and image compression. This is just a first step in this direction, and we expect that more elaborate tuning strategies will enhance our procedures in the future.

  • 50.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong University, China.
    Multidimensional rational covariance extension with approximate covariance matching2018In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 56, no 2, p. 913-944Article in journal (Refereed)
    Abstract [en]

    In our companion paper [A. Ringh, J. Karlsson, and A. Lindquist, SIAM T. Control Opton., 54 (2016), pp. 1950-1982] we discussed the multidimensional rational covariance extension problem (RCEP), which has important applications in image processing and spectral estimation in radar, sonar, and medical imaging. This is an inverse problem where a power spectrum with a rational absolutely continuous part is reconstructed from a finite set of moments. However, in most applications these moments are determined from observed data and are therefore only approximate, and the RCEP may not have a solution. In this paper we extend the results of our companion paper to handle approximate covariance matching. We consider two problems, one with a soft constraint and the other one with a hard constraint, and show that they are connected via a homeomorphism. We also demonstrate that the problems are well-posed and illustrate the theory by examples in spectral estimation and texture generation.

12 1 - 50 of 52
CiteExportLink to result list
Permanent link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf