Wim Michiels

Quoc Tran Dinh, Suat Gumussoy, Wim Michiels et al.

A novel optimization method is proposed to minimize a convex function subject to bilinear matrix inequality (BMI) constraints. The key idea is to decompose the bilinear mapping as a difference between two positive semidefinite convex mappings. At each iteration of the algorithm the concave part is linearized, leading to a convex subproblem.Applications to various output feedback controller synthesis problems are presented. In these applications the subproblem in each iteration step can be turned into a convex optimization problem with linear matrix inequality (LMI) constraints. The performance of the algorithm has been benchmarked on the data from COMPleib library.

Elias Jarlebring, Simen Kvaal, Wim Michiels

Consider a symmetric matrix $A(v)\in\RR^{n\times n}$ depending on a vector $v\in\RR^n$ and satisfying the property $A(αv)=A(v)$ for any $α\in\RR\backslash{0}$. We will here study the problem of finding $(λ,v)\in\RR\times \RR^n\backslash\{0\}$ such that $(λ,v)$ is an eigenpair of the matrix $A(v)$ and we propose a generalization of inverse iteration for eigenvalue problems with this type of eigenvector nonlinearity. The convergence of the proposed method is studied and several convergence properties are shown to be analogous to inverse iteration for standard eigenvalue problems, including local convergence properties. The algorithm is also shown to be equivalent to a particular discretization of an associated ordinary differential equation, if the shift is chosen in a particular way. The algorithm is adapted to a variant of the Schrödinger equation known as the Gross-Pitaevskii equation. We use numerical simulations toillustrate the convergence properties, as well as the efficiency of the algorithm and the adaption.

Elias Jarlebring, Karl Meerbergen, Wim Michiels

The partial Schur factorization can be used to represent several eigenpairs of a matrix in a numerically robust way. Different adaptions of the Arnoldi method are often used to compute partial Schur factorizations. We propose here a technique to compute a partial Schur factorization of a nonlinear eigenvalue problem (NEP). The technique is inspired by the algorithm in [8], now called the infinite Arnoldi method. The infinite Arnoldi method is a method designed for NEPs, and can be interpreted as Arnoldi's method applied to a linear infinite-dimensional operator, whose reciprocal eigenvalues are the solutions to the NEP. As a first result we show that the invariant pairs of the operator are equivalent to invariant pairs of the NEP. We characterize the structure of the invariant pairs of the operator and show how one can carry out a modification of the infinite Arnoldi method by respecting the structure. This also allows us to naturally add the feature known as locking. We nest this algorithm with an outer iteration, where the infinite Arnoldi method for a particular type of structured functions is appropriately restarted. The restarting exploits the structure and is inspired by the well-known implicitly restarted Arnoldi method for standard eigenvalue problems. The final algorithm is applied to examples from a benchmark collection, showing that both processing time and memory consumption can be considerably reduced with the restarting technique.

Loïc Michel, Wim Michiels, Xavier Boucher

A new "model-free" control methodology is applied for the first time to power systems included in microgrids networks. We evaluate its performances regarding output load and supply variations in different working configuration of the microgrid. Our approach, which utilizes "intelligent" PI controllers, does not require any converter or microgrid model identification while ensuring the stability and the robustness of the controlled system. Simulations results show that with a simple control structure, the proposed control method is almost insensitive to fluctuations and large load variations.

Vilhelm Peterson Lithell, Victor Janssens, Elias Jarlebring et al.

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.

Wim Michiels, Bin Zhou

A delay Lyapunov matrix corresponding to an exponentially stable system of linear time-invariant delay differential equations can be characterized as the solution of a boundary value problem involving a matrix valued delay differential equation. This boundary value problem can be seen as a natural generalization of the classical Lyapunov matrix equation. We present a general approach for computing delay Lyapunov matrices and H2 norms for systems with multiple discrete delays, whose applicability extends towards problems where the matrices are large and sparse, and the associated positive semidefinite matrix (the ``right-hand side' for the standard Lyapunov equation), has a low rank. In contract to existing methods that are based on solving the boundary value problem directly, our method is grounded in solving standard Lyapunov equations of increased dimensions. It combines several ingredients: i) a spectral discretization of the system of delay equations, ii) a targeted similarity transformation which induces a desired structure and sparsity pattern and, at the same time, favors accurate low rank solutions of the corresponding Lyapunov equation, and iii) a Krylov method for large-scale matrix Lyapunov equations. The structure of the problem is exploited in such a way that the final algorithm does not involve a preliminary discretization step, and provides a fully dynamic construction of approximations of increasing rank. Interpretations in terms of a projection method directly applied to a standard linear infinite-dimensional system equivalent to the original time-delay system are also given. Throughout the paper two didactic examples are used to illustrate the properties of the problem, the challenges and methodological choices, while numerical experiments are presented at the end to illustrate the effectiveness of the algorithm.

Diego Torres-García, Wim Michiels

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.

Koen Ruymbeek, Karl Meerbergen, Wim Michiels

In applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters $ω$ and we are interested in the minimal eigenvalue of a matrix pencil $(A,B)$ with $A,B$ symmetric and $B$ positive definite. If $ω$ can be interpreted as the realisation of random variables, one may be interested in statistical moments of the minimal eigenvalue. In order to obtain statistical moments, we need a fast evaluation of the eigenvalue as a function of $ω$. Since this is costly for large matrices,we are looking for a small parametrized eigenvalue problem whose minimal eigenvalue makes a small error with the minimal eigenvalue of the large eigenvalue problem. The advantage, in comparison with a global polynomial approximation (on which, e.g., the polynomial chaos approximation relies), is that we do not suffer from the possible non-smoothness of the minimal eigenvalue. The small scale eigenvalue problem is obtained by projection of the large scale problem. Our main contribution is that for constructing the subspace we use multiple eigenvectors as well as derivatives of eigenvectors.We provide theoretical results and document numerical experiments regarding the beneficial effect of adding multiple eigenvectors and derivatives.

Fatih Kangal, Karl Meerbergen, Emre Mengi et al.

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.