We construct optimally robust port-Hamiltonian realizations of a given rational transfer function that represents a passive system. We show that the realization with a maximal passivity radius is a normalized port-Hamiltonian one. Its computation is linked to a particular solution of a linear matrix inequality that defines passivity of the transfer function, and we provide an algorithm to...
Port-Hamiltonian (pH) systems are a natural way to model many physical processes. Numerous specialized numerical techniques have been created to take advantage of and maintain the structure of pH systems, such as model order reduction (MOR) and space- and time-discretization. In this work, we focus on the structure-preserving MOR of linear time-varying (LTV) pH systems. LTV systems appear...
We consider cables that interact with the electromagnetic field through radiation in a bidirectional way. We show that the port-Hamiltonian framework is well-suited to model this interaction. The cable is described by the telegraph equations, while the electromagnetic field is (unsurprisingly) described by Maxwell’s equations. The coupling goes via boundary conditions for the electric and...
Port-Hamiltonian systems have emerged as a fundamental framework for modeling a wide range of physical phenomena due to their energy-based structure and inherent stability properties. To address the complexity of such models, various structure-preserving model reduction techniques have been developed for both linear and nonlinear port-Hamiltonian dynamics. In this work, we focus on bilinear...
The efficient solution of large-scale multiterm linear matrix equations is a challenging task in numerical linear algebra, and it is a largely open problem. The topic has attracted great interest within the numerical community in the past decade, thanks to the applicability of linear matrix equations in a growing number of applications. In this talk we will discuss recent developments in...
The Discrete Empirical Interpolation Method (DEIM) is an important tool in model reduction, significantly reducing the cost of evaluating a high-dimensional function if the function value is known to be nearly contained in a low-dimensional subspace. On a matrix level, DEIM requires one to select a well-conditioned square submatrix $U(I,:)$ of an orthonormal basis $U \in \mathbb R^{n\times...
This contribution will address relationship between spectra of preconditioned self-adjoint PDE operators associated with boundary value problems and their approximations using generalized matrix eigenvalue problems arising from discretization. We will recall several results from operator theory literature, compare them with results on the generalised eigenvalue problem in the numerical PDE...
We provide a systematic theory of singular pencils $\lambda E-A$, with (possibly unbounded) operator coefficients in a Hilbert space. Apparently, the situation is more complicated than in the finite dimensional case. Several equivalent statements connected to the Kronecker canonical form become essentially different when the dimension is infinite.
We show the relation of these concepts to...
Any (real or complex) square matrix $M$ can be split into $M=H+S$, where $H$ is the Hermitian and $S$ is the skew-Hermitian part of $M$. Interestingly, this splitting occurs naturally and with a physical meaning in applications that involve energy-based modeling using DAE systems in dissipative Hamiltonian form (dHDAEs). The applicability of this modeling approach has been demonstrated in...
Robust controllers that stabilize dynamical systems even under disturbances and noise are often formulated as solutions of nonsmooth, nonconvex optimization problems. While methods such as gradient sampling can handle the nonconvexity and nonsmoothness, the costs of evaluating the objective function may be substantial, making robust control challenging for dynamical systems with...
This talk concerns the challenge of identifying states of dynamic systems that operate across multiple time scales, ranging from youth to old age, drawing on personal experiences of the speaker in collaboration with V. Mehrmann.
The central state identification challenge arises from the fact that data is only accessible at the finest time scales, and may be rapidly forgotten (e.g.,...
Consider the linear time-invariant time-delay system (TDS)
$$ \begin{align*} \frac{\mathrm{d}\mathbf{x}(t)}{\mathrm{d}t} &= A_0 \mathbf{x}(t) + \sum_{j=1}^{N_1} A_j \mathbf{x}(t - \tau_j) + \sum_{j=1}^{N_2} B_j \mathbf{u}(t- t_j), \\\\ \mathbf{y}(t) &= \sum_{j=1}^{N_3} C_j \mathbf{x}(t-s_j) + \sum_{j=1}^{N_4} D_j \mathbf{u}(t-h_j), \end{align*} $$ where $\mathbf{x}(t) \in...
A classical topic in analog design is the study of input-output systems with extra structure. Recently, this interest is renewed by motivations stemming from e.g. neuro-computing, dynamical networks, and scalable control. In this talk we will discuss various, often related, symmetry structures for standard linear systems, including gradient systems, relaxation systems and input-output...
We consider a matrix polynomial dependent on several parameters. The minimization of its spectral abscissa, the real part of its rightmost eigenvalue, over the parameters is motivated by stability considerations on the associated higher-order linear control system, yet comes with computational challenges especially due to the non-Lipschitz nature of the spectral abscissa. We instead propose...
We consider linear time-invariant continuous control systems that are bounded real, also known as scattering passive. Our main theoretical contribution is to show the equivalence between such systems and port-Hamiltonian (PH) systems whose factors satisfy certain linear matrix inequalities. Based on this result, we propose a formulation for the problem of finding the nearest bounded real...
In this presentation, we explore the challenge of computing the stability region for a class of singular nonlinear continuous- and discrete-time systems. Under specific conditions, we derive topological properties of the stability region. After providing key characterizations of its boundary, we introduce a practical, direct method for computing this region. The proposed method involves...
We show how invariant subspaces will change when a defective matrix is perturbed. We focus on the case when an invariant subspace corresponding to the same size Jordan blocks of a single eigenvalue is perturbed. The perturbations are characterized in terms of fractional orders. As an application, we study a class of Hamiltonian matrices that are related to Riccati inequalities. We show how the...
It is well known that eigenvalues of a Hermitian matrix pencil $L(\lambda) = \lambda A - B,$ occur in pairs $(\lambda, \bar{\lambda})$ with eigenvalues, if any, on the extended real line being the ones where the symmetry breaks down. We refer to such eigenvalues as critical eigenvalues and study the problem of finding optimal Hermitian perturbations of $L(\lambda)$ when it is of even size...
Block diagonalization algorithms have many applications, including computation of reciprocal condition numbers for eigenvalues and invariant or deflating subspaces of matrices or matrix pencils, respectively, fast computation of matrix functions, or of the response of linear time invariant standard or descriptor systems. For matrix pencils, this algorithm is applied to a matrix pair in a...
Port-Hamiltonian systems extend Hamiltonian systems to incorporate network structure and energy exchanges through ports, enabling the modeling of open and interconnected systems from various physical domains. The interconnection of network components often leads to differential-algebraic equations (DAEs), which also include algebraic constraints, for example Kirchhoff's laws. To ensure that...
Abstract. Infinite-dimensional state-space systems with boundary control and observation pose significant analytical challenges due to the typically unbounded nature of input and output maps. A common approach to establishing a solution theory, such as via a variation of constants formula, relies on the admissibility of input and output operators—an assumption that is often difficult to verify...
At the beginning of the era of electronic computing there was a big effort to produce software to make the newly constructed hardware useful. In the area of scientific computing, one need that was recognized early on was for efficient and reliable methods to compute the eigenvalues of a matrix. This need was met around 1960 by the so-called QR algorithm, especially the implicitly-shifted...
The "index of hypocoercivity" is defined via a coercivity-type estimate for the self-adjoint/skew-adjoint parts of the generator, and it quantifies `how degenerate' a hypocoercive evolution equation is, both for ODEs and for evolutions equations in a Hilbert space. We show that this index characterizes the polynomial decay of the propagator norm for short time and illustrate these concepts for...
The complete eigenstructure, or structural data, of a rational matrix R(s) is comprised by its invariant rational functions, both finite and at infinity, which determine its finite and infinite pole and zero structures, and by the minimal indices of its left and right null spaces. These quantities arise in many applications in control theory and have been thoroughly studied in numerous...
