arXiv subjects

Wim Michiels

Publications and source records attributed to Wim Michiels.

At least 19 recordsLinked to original sources

arXiv2026

Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be evaluated with little computational effort for a given eigenpair, assuming the matrix perturbations are measured by the spectral or Frobenius norm. We also show how symmetric perturbations can be exploited in the analysis. By means of two numerical experiments we demonstrate that problems incorporating eigenvector nonlinearities potentially need to be treated with additional care, when compared to the linear or eigenvalue-nonlinear theory.

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arXiv2026

A numerical approach to the co-design of PID controllers and low-pass filters for time-delay systems

This paper addresses the numerical optimization of proportional-integral-derivative (PID) controllers for linear time-invariant systems with delays, where the derivative action is implemented using a low-pass filter. While performance assessment is often based on the spectral abscissa of the ideal PID-controlled system, the inclusion of a derivative filter fundamentally alters the closed-loop spectral properties and cannot be treated as a post-processing step. In particular, the spectral abscissa of the filtered closed-loop system may differ significantly from that of its unfiltered counterpart, potentially affecting both stability and performance. We propose a systematic numerical design framework in which the PID gains and the filter constant are optimized simultaneously by directly minimizing the spectral abscissa of the filtered closed-loop system. Treating the filter as an integral part of the control design allows us to reconcile robustness at high frequencies, in the sense of mitigating fragility issues due to approximate identities, with performance at low frequencies, in addition to counter measurement noise amplification. At the end of the presentation, numerical examples illustrate the proposed approach and highlight the benefits of controller-filter co-design. The results apply to general linear systems with input and/or state delays and are valid for both single-input single-output (SISO) and multi-input multi-output (MIMO) configurations.

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arXiv2026

On hyperbolic PDEs, filtered feedback control laws, and fractal-like stability crossing curves

The paper addresses the boundary control of a class of hyperbolic PDEs, based on an equivalent representation in terms of an integral-difference equation. The situation is considered where direct compensation of reflection terms induces a fragile closed-loop system, in the sense of lack of strong stability. This is theoretically resolved by adding a low-pass filter to the control law, but the choice of its cut-off frequency is crucial in balancing robustness at high frequencies and performance at low frequencies. First, the maximum stability interval in parameter $T$ is determined, with $T$ the inverse of the filter's cutoff frequency. Next, model mismatch on the PDE parameters is considered and a sufficient stability condition is derived in terms of allowable mismatch and cut-off frequency, satisfied in a region in the combined parameter space with a conic shape around $T=0$. Finally, this qualitative behavior is confirmed by exact stability charts for a special case where all model mismatch is contained into one parameter. It is highlighted that the set of stability crossing curves exhibits a fractal-like structure, which is explained using a limit system with discrete delays.

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arXiv2026

Computing and Optimizing the $H^2$-norm of Delay Differential Algebraic Systems

We present a Lanczos tau method for the approximation and optimization of the $H^2$-norm of time-delay systems described by semi-explicit delay differential algebraic equations. The soundness of this approach is proven under the assumption of a finite strong $H^2$-norm. Furthermore, we prove convergence if the rational approximation of the exponential underlying the discretization is well-behaved and the discretization is stability preserving. Numerical results suggest that, for multiple delays, the method converges at cubic rate in the discretization degree for systems of retarded type and linearly for those of neutral type. In the single delay case, we note geometric convergence of the $H^2$-norm for systems of both retarded and neutral type when a symmetric basis is chosen. Explicit formulas are derived for the gradient of the approximation with respect to system parameters and delays. These allow us to compute the entire gradient using only about double the computational time of approximating the $H^2$-norm alone. We illustrate how these can be used to synthesize robust feedback controllers and stable approximate models. The article is concluded by a discussion of how the presented results extend and improve for approximations based on splines. We note acceleration of the convergence rate by about two orders for such a choice. Finally, we prove that a Lanczos tau method using a spline based on Legendre orthogonal polynomials preserves stability and guarantees convergence of the $H^2$-norm.

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arXiv2025

Linearizing a nonlinear eigenvalue problem with quadratic rational eigenvector nonlinearities

Nonlinear eigenvalue problems with eigenvector nonlinearities (NEPv) are algebraic eigenvalue problems whose matrix depends on the eigenvector. Applications range from computational quantum mechanics to machine learning. Due to its nonlinear behavior, existing methods almost exclusively rely on fixed-point iterations, the global convergence properties of which are only understood in specific cases. Recently, a certain class of NEPv with linear rational eigenvector nonlinearities has been linearized, i.e., the spectrum of the linear eigenvalue problem contains the eigenvalues of the NEPv. This linear problem is solved using structure exploiting algorithms to improve both convergence and reliability. We propose a linearization for a different class of NEPv with quadratic rational nonlinearities, inspired by the discretized Gross-Pitaevskii equation. The eigenvalues of this NEPv form a subset of the spectrum of a linear multiparameter eigenvalue problem which is equivalent to a system of generalized eigenvalue problems expressed in terms of operator determinants. A structure exploiting Arnoldi algorithm is used to filter a large portion of spurious solutions and to accelerate convergence.

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arXiv2025

Integrated design of system structure and delayed resonator towards efficient non-collocated vibration absorption

The problem of non-collocated vibration absorption by a delayed resonator is addressed with emphasis on system fatigue resistance and energy efficiency of control actions. The analysis is performed for a system consisting of an arbitrary large series of flexibly linked single-degree-of-freedom masses. For the stage where the vibration of the target mass is fully absorbed by the non-collocated resonator, key forces, motion amplitudes and potential energies across the system structure are assessed. Next, a complete parameter set of the resonator gain and delay is derived, and the actuation force and power needed by the resonator for the full vibration absorption is determined. The derived quantities are utilized in forming an optimization problem to balance minimal risk of fatigue across the system structure and power needed by the resonator, under the closed loop stability and parameter constraints. Next to the gain and delay of the resonator, selected structural parameters of the system are used as variables in the constrained nonlinear optimization problem. Experimental and numerical case studies are included to demonstrate benefits of the proposed integrated structural and control design.

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arXiv2025

Delayed dynamic-feedback controller design for multi-frequency vibration suppression

We present a methodology for designing a dynamic controller with delayed output feedback for achieving non-collocated vibration suppression with a focus on the multi-frequency case. To synthesize the delay-based controller, we first remodel the system of equations as a delay-differential algebraic equation (DDAE) in such a way that existing tools for design of a static output feedback controller can be easily adapted. The problem of achieving non-collocated vibration suppression with sufficient damping is formulated as a constrained optimization problem of minimizing the spectral abscissa in the presence of zero-location constraints, with the constraints exhibiting polynomial dependence on its parameters. We transform the problem into an unconstrained one using elimination, following which we solve the resulting non-convex, non-smooth optimization problem.

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arXiv2024

The Lanczos Tau Framework for Time-Delay Systems: Pad\'e Approximation and Collocation Revisited

We reformulate the Lanczos tau method for the discretization of time-delay systems in terms of a pencil of operators, allowing for new insights into this approach. As a first main result, we show that, for the choice of a shifted Legendre basis, this method is equivalent to Pad\'e approximation in the frequency domain. We illustrate that Lanczos tau methods straightforwardly give rise to sparse, self nesting discretizations. Equivalence is also demonstrated with pseudospectral collocation, where the non-zero collocation points are chosen as the zeroes of orthogonal polynomials. The importance of such a choice manifests itself in the approximation of the $H^2$-norm, where, under mild conditions, super-geometric convergence is observed and, for a special case, super convergence is proved; both significantly faster than the algebraic convergence reported in previous work.

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arXiv2023

Analysis and controller-design of time-delay systems using TDS-CONTROL. A tutorial and manual

TDS-CONTROL is an integrated MATLAB package for the analysis and controller-design of linear time-invariant (LTI) dynamical systems with (multiple) discrete delays, supporting both systems of retarded and neutral type. TDS-CONTROL is based on a state-space representations for these TDSs, although functionality is provided to obtain such a formulation from a frequency-domain description of the system. Firstly, the package offers various functionality for analyzing such systems, like methods for computing the spectral abscissa, the H-infinity norm, the pseudospectral abscissa, and the distance to instability. Furthermore, as TDS-CONTROL is designed with neutral time-delay systems in mind, it has the appealing feature that the sensitivity of certain quantities (such as the spectral abscissa) with respect to infinitesimal delay perturbations can explicitly be taken into account. Secondly, TDS-CONTROL also allows to design fixed-order dynamic output feedback controllers. The corresponding controller-design algorithms are based on minimizing the spectral abscissa, the H-infinity norm, or a combination of both with respect to the free controller parameters by solving a non-smooth, non-convex optimization problem. As a strictly negative spectral abscissa is a necessary and sufficient condition for stability, the presented design methods are thus not conservative. It is also possible to impose structure on the controller, enabling the design of decentralized and proportional-integral-derivative (PID) controllers. Furthermore, by allowing the plant to be described in delay descriptor form (i.e., the system's dynamics are given in terms of delay differential algebraic equations), acceleration feedback and Pyragas-type and delay-based controllers can also be considered.

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arXiv2022

A structure preserving shift-invert infinite Arnoldi algorithm for a class of delay eigenvalue problems with Hamiltonian symmetry

In this work we consider a class of delay eigenvalue problems that admit a spectrum similar to that of a Hamiltonian matrix, in the sense that the spectrum is symmetric with respect to both the real and imaginary axis. More precisely, we present a method to iteratively approximate the eigenvalues of such delay eigenvalue problems closest to a given purely real or imaginary shift, while preserving the symmetries of the spectrum. To this end the presented method exploits the equivalence between the considered delay eigenvalue problem and the eigenvalue problem associated with a linear but infinite-dimensional operator. To compute the eigenvalues closest to the given shift, we apply a specifically chosen shift-invert transformation to this linear operator and compute the eigenvalues with the largest modulus of the new shifted and inverted operator using an (infinite) Arnoldi procedure. The advantage of the chosen shift-invert transformation is that the spectrum of the transformed operator has a "real skew-Hamiltonian"-like structure. Furthermore, it is proven that the Krylov space constructed by applying this operator, satisfies an orthogonality property in terms of a specifically chosen bilinear form. By taking this property into account during the orthogonalization process, it is ensured that even in the presence of rounding errors, the obtained approximation for, e.g., a simple, purely imaginary eigenvalue is simple and purely imaginary. The presented work can thus be seen as an extension of [V. Mehrmann and D. Watkins, "Structure-Preserving Methods for Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils", SIAM J. Sci. Comput. (22.6), 2001], to the considered class of delay eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite dimensional linear algebra operations.

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arXiv2020

Tensor-Krylov method for computing eigenvalues of parameter-dependent matrices

In this paper we extend the Residual Arnoldi method for calculating an extreme eigenvalue (e.g. largest real part, dominant,...) to the case where the matrices depend on parameters. The difference between this Arnoldi method and the classical Arnoldi algorithm is that in the former the residual is added to the subspace. We develop a Tensor-Krylov method that applies the Residual Arnoldi method (RA) for a grid of parameter points at the same time. The subspace contains an approximate Krylov space for all these points. Instead of adding the residuals for all parameter values to the subspace we create a low-rank approximation of the matrix consisting of these residuals and add only the column space to the subspace. In order to keep the computations efficient, it is needed to limit the dimension of the subspace and to restart once the subspace has reached the prescribed maximal dimension. The novelty of this approach is twofold. Firstly, we observed that a large error in the low-rank approximations is allowed without slowing down the convergence, which implies that we can do more iterations before restarting. Secondly, we pay particular attention to the way the subspace is restarted, since classical restarting techniques give a too large subspace in our case. We motivate why it is good enough to just keep the approximation of the searched eigenvector. At the end of the paper we extend this algorithm to shift-and-invert Residual Arnoldi method to calculate the eigenvalue close to a shift $σ$ for a specific parameter dependency. We provide theoretical results and report numerical experiments. The Matlab code is publicly available.

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arXiv2020

Subspace method for multiparameter-eigenvalue problems based on tensor-train representations

In this paper we solve $m$-parameter eigenvalue problems ($m$EPs), with $m$ any natural number by representing the problem using Tensor-Trains (TT) and designing a method based on this format. $m$EPs typically arise when separation of variables is applied to separable boundary value problems. Often, methods for solving $m$EP are restricted to $m = 3$, due to the fact that, to the best of our knowledge, no available solvers exist for $m>3$ and reasonable size of the involved matrices. In this paper, we prove that computing the eigenvalues of a $m$EP can be recast into computing the eigenvalues of TT-operators. We adapted the algorithm in \cite{Dolgov2014a} for symmetric eigenvalue problems in TT-format to an algorithm for solving generic $m$EPs. This leads to a subspace method whose subspace dimension does not depend on $m$, in contrast to other subspace methods for $m$EPS. This allows us to tackle $m$EPs with $m > 3$ and reasonable size of the matrices. We provide theoretical results and report numerical experiments. The MATLAB code is publicly available.

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arXiv2020

A scalable controller synthesis method for the robust control of networked systems

This manuscript discusses a scalable controller synthesis method for networked systems with a large number of identical subsystems based on the H-infinity control framework. The dynamics of the individual subsystems are described by identical linear time-invariant delay differential equations and the effect of transport and communication delay is explicitly taken into account. The presented method is based on the result that, under a particular assumption on the graph describing the interconnections between the subsystems, the H-infinity norm of the overall system is upper bounded by the robust H-infinity norm of a single subsystem with an additional uncertainty. This work will therefore briefly discuss a recently developed method to compute this last quantity. The resulting controller is then obtained by directly minimizing this upper bound in the controller parameters.

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arXiv2020

Analysis and Design of Strongly Stabilizing PID Controllers for Time-Delay Systems

This paper presents the analysis of the stability properties of PID controllers for dynamical systems with multiple state delays, focusing on the mathematical characterization of the potential sensitivity of stability with respect to infinitesimal parametric perturbations. These perturbations originate for instance from neglecting feedback delay, a finite difference approximation of the derivative action, or neglecting fast dynamics. The analysis of these potential sensitivity problems leads us to the introduction of a `robustified' notion of stability called \emph{strong stability}, inspired by the corresponding notion for neutral functional differential equations. We prove that strong stability can be achieved by adding a low-pass filter with a sufficiently large cut-off frequency to the control loop, on the condition that the filter itself does not destabilize the nominal closed-loop system. Throughout the paper, the theoretical results are illustrated by examples that can be analyzed analytically, including, among others, a third-order unstable system where both proportional and derivative control action are necessary for achieving stability, while the regions in the gain parameter-space for stability and strong stability are not identical. Besides the analysis of strong stability, a computational procedure is provided for designing strongly stabilizing PID controllers. Computational case-studies illustrating this design procedure complete the presentation.

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arXiv2020

Tuning an H-Infinity Controller with a Given Order and a Structure for Interconnected Systems with Delays

An eigenvalue based framework is developed for the H-infinity norm analysis and its norm minimization of coupled systems with time-delays, which are naturally described by delay differential algebraic equations (DDAEs). For these equations H-infinity norms are analyzed and their sensitivity with respect to small delay perturbations is studied. Subsequently, numerical methods for the H-infinity norm computation and for designing controllers minimizing the H-infinity norm with a prescribed structure or order, based on a direct optimization approach, are briefly addressed. The effectiveness of the approach is illustrated with a software demo. The chapter concludes by pointing out the similarities with the computation and optimization of characteristic roots of DDAEs.

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arXiv2020

Eigenvalue based algorithms and software for the design of fixed-order stabilizing controllers for interconnected systems with time-delays

An eigenvalue based framework is developed for the stability analysis and stabilization of coupled systems with time-delays, which are naturally described by delay differential algebraic equations. The spectral properties of these equations are analyzed and their stability properties are studied, taking into account the effect of small delay perturbations. Subsequently, numerical methods for stability assessment and for designing stabilizing controllers with a prescribed structure or order, based on a direct optimization approach, are briefly addressed. The effectiveness of the approach is illustrated with a software demo. The paper concludes by pointing out the similarities with the computation and optimization of H-infinity norms.

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arXiv2020

Fixed-Order H-infinity Optimization of Time-Delay Systems

H-infinity controllers are frequently used in control theory due to their robust performance and stabilization. Classical H-infinity controller synthesis methods for finite dimensional LTI MIMO plants result in high-order controllers for high-order plants whereas low-order controllers are desired in practice. We design fixed-order H-infinity controllers for a class of time-delay systems based on a non-smooth, non-convex optimization method and a recently developed numerical method for H-infinity norm computations.

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arXiv2020

Computation of Extremum Singular Values and the Strong H-infinity Norm of SISO Time-Delay Systems

We consider the computation of H-infinity norms for Single-Input-Single-Output (SISO) time-delay systems, which are described by delay differential algebraic equations. Unlike the iterative level set methods in the literature, we present a novel numerical method to compute the H-infinity norm. This method requires solving one eigenvalue problem of at most twice the size of the eigenvalue problem in every iteration of a level set method, but in practice often considerably lower. We first show that the computation of extrema of the transfer function can be turned into the computation of the imaginary axis zeros of a transcendental function. We compute these zeros by a predictor-corrector type algorithm. It is known that the H-infinity norm of delay differential algebraic systems, which can model both retarded and neutral type systems, might be sensitive with respect to arbitrarily small delay perturbations. This recently led to the concept of strong H-infinity norms, which explicitly take into account such small delay perturbations. We present a direct numerical method to compute the strong H-infinity norm of SISO time-delay systems. Our algorithm is applicable to the closed-loop system of interconnections (series, parallel, feedback, junctions) of time-delay systems and/or controllers.

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