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In this work we consider a class of non-linear 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 non-linear 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 non-linear 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 in 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 non-linear eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite dimensional linear algebra operations. ispartof: vol:44 issue:1 pages:331-358 status: Published online

International audience; A methodology is proposed for the design of sampled-data fixed-order decentralised controllers for Multiple Input Multiple Output (MIMO) Linear Time-Invariant (LTI) time-delay systems. Imperfections in the communications links between continuous-time plants ans controllers arising due to transmission time-delay, aperiodic sampling, and asynchronous sensors ans actuators are considered. We model the errors induced due to the control imperfections using an operator approach leading to a simple $L2$, stability criterion. A frequency domain-based direct optimisation approach towards controller desing is proposed in thir paper. This approach relies on the minimisation of cost functions, for stability and robustness against control imperfections, as a function of the controller or design parameters. First, the proposed method towards controller design is applied to generic MIMO LTI systems with time-delay. Second, when the delay system to be controlled has the structure of a network of coupled quasi-identical subsystems, we use a scalable algorithm to design identical decentralised controllers through network structure exploitation. Quasi-identical subsystems are identical subsystems that have non-identical uncertainties or control imperfections. By exploiting the structure, we improve the computational efficiency and scalability with the number of subsystems. The methodology has been implemented in a publicly available software, which supports system models in terms of delay differential algebraic equations. Finally, the effectiveness of the methodology is illustrated using a numerical example.

ispartof: International Journal Of Robust And Nonlinear Control vol:30 issue:10 pages:3906-3922 status: Published online

ispartof: Automatica vol:113 status: Published online

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.

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., Comment: arXiv admin note: substantial text overlap with arXiv:2003.09252, arXiv:2003.04695, arXiv:2003.05728

The paper presents a pseudospectral approach to assess the stability robustness of linear time-periodic delay systems, where periodic functions potentially present discontinuities. The considered systems are subject to linear real-valued time-periodic uncertainties affecting the coefficient matrices, and the presented method is able to fully exploit structure and potential interdependencies among the uncertainties. The assessment of robustness relies on the computation of the pseudospectral radius of the monodromy operator, namely the largest Floquet multiplier that the system can attain within a given range of perturbations. Instrumental in the adopted approach, a novel solver for the computation of Floquet multipliers is introduced, and it results into the solution of a generalized eigenvalue problem which is linear w.r.t. (samples of) the original system matrices. We provide numerical simulations for popular applications modeled by time-periodic delay systems, such as the inverted pendulum subject to an act-and-wait controller, a single-degree-of-freedom milling model and a turning operation with spindle speed variation. ispartof: TW reports status: published

This paper examines the robust (strong) H-infinity norm of a linear time-invariant system with discrete delays. The considered system is subject to real-valued, structured, Frobenius norm bounded uncertainties on the coefficient matrices. The robust H-infinity norm is the worst case value of the H-infinity norm over the realisations of the system and hence an important measure of robust performance in control engineering. However this robust H-infinity norm is a fragile measure, as for a particular realization of the uncertainties the H-infinity norm might be sensitive to arbitrarily small perturbations on the delays. Therefore, we introduce the robust strong H-infinity norm, inspired by the notion of strong stability of delay differential equations of neutral type, which takes into account both the perturbations on the system matrices and infinitesimal small delay perturbations. This quantity is a continuous function of the nominal system parameters and delays. The main contribution of this work is the introduction of a relation between this robust strong H-infinity norm and the the pseudo-spectrum of an associated singular delay eigenvalue problem. This relation is subsequently employed in a novel algorithm for computing the robust strong H-infinity norm of uncertain time-delay systems. Both the theoretical results and the algorithm are also generalized to systems with uncertainties on the delays, and systems described by a class of delay differential algebraic equations. ispartof: vol:abs/1909.07778 status: Published online

This paper examines the robust (strong) H-infinity norm of a linear time-invariant system with discrete delays. The considered system is subject to real-valued, structured, Frobenius norm bounded uncertainties on the coefficient matrices. The robust H-infinity norm is the worst case value of the H-infinity norm over the realisations of the system and hence an important measure of robust performance in control engineering. However this robust H-infinity norm is a fragile measure, as for a particular realization of the uncertainties the H-infinity norm might be sensitive to arbitrarily small perturbations on the delays. Therefore, we introduce the robust strong H-infinity norm, inspired by the notion of strong stability of delay differential equations of neutral type, which takes into account both the perturbations on the system matrices and infinitesimal small delay perturbations. This quantity is a continuous function of the nominal system parameters and delays. The main contribution of this work is the introduction of a relation between this robust strong H-infinity norm and the the pseudo-spectrum of an associated singular delay eigenvalue problem. This relation is subsequently employed in a novel algorithm for computing the robust strong H-infinity norm of uncertain time-delay systems. Both the theoretical results and the algorithm are also generalized to systems with uncertainties on the delays, and systems described by a class of delay differential algebraic equations.

We consider the nonlinear eigenvalue problem M(λ)x = 0, where M(λ) is a large parameter-dependent matrix. In several applications, M(λ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi-based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector-valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer.Math., 122 (2012), pp.169–195]. In this paper, the low-rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large-scale problems. Copyright © 2016 John Wiley & Sons, Ltd.

We consider the nonlinear eigenvalue problem: M (λ)x = 0, where M (λ) is a large parameter-dependent matrix. In several applications, M (λ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi’s method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector-valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169–195]. In this paper the low-rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large-scale problems. ispartof: Numerical Linear Algebra with Applications vol:23 issue:4 pages:607-628 status: published

We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi et al. (2014, SIAM J. Matrix Anal. Appl., 35, 699-724). This work addresses the setting when the matrix-valued function involved is very large. We describe subspace procedures that convert the original problem into a small-scale one by means of orthogonal projections and restrictions to certain subspaces, and that gradually expand these subspaces based on the optimal solutions of small-scale problems. Global convergence and superlinear rate-of-convergence results with respect to the dimensions of the subspaces are presented in the infinite dimensional setting, where the matrix-valued function is replaced by a compact operator depending on parameters. In practice, it suffices to solve eigenvalue optimization problems involving matrices with sizes on the scale of tens, instead of the original problem involving matrices with sizes on the scale of thousands., Comment: 34 pages, 2 figures

We consider a problem in eigenvalue optimization, in particular finding a local minimizer of the spectral abscissa - the value of a parameter that results in the smallest value of the largest real part of the spectrum of a matrix system. This is an important problem for the stabilization of control systems. Many systems require the spectra to lie in the left half plane in order for them to be stable. The optimization problem, however, is difficult to solve because the underlying objective function is nonconvex, nonsmooth, and non-Lipschitz. In addition, local minima tend to correspond to points of non-differentiability and locally non-Lipschitz behavior. We present a sequential linear and quadratic programming algorithm that solves a series of linear or quadratic subproblems formed by linearizing the surfaces corresponding to the largest eigenvalues. We present numerical results comparing the algorithms to the state of the art.

We present an algorithm to compute the pseudospectral abscissa for a nonlinear eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the eigenvalue and singular value functions involved. These global models follow from eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of nonlinear eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available., The Optimization in Engineering Center of the KU Leuven; Research Foundation-Flanders (FWO); Research Executive Agency of the European Union; Scientific and Technological Research Council of Turkey (TÜBİTAK); BAGEP Program of Science Academy