EMOSC 25: Energy-based modeling, simulation, and control of dynamical systems - Workshop in honor of Volker Mehrmann’s 70th birthday

26 May 2025, 12:00 → 28 May 2025, 15:10 Europe/Berlin

Faculty of Mathematics

Benjamin Unger, Christian Mehl, Matthias Voigt, Peter Benner, Tatjana Stykel, Alexandra Schulte

In honor of Volker Mehrmann's 70th birthday, this workshop will bring together researchers from the theory and numerics of energy-based system models and their applications. The workshop will focus on aspects such as modeling, model reduction, system identification, and system analysis, as well as analytical and numerical methods for structured simulation and control.

Registration

Participation fee is 70 Euros (to be paid in cash on conference site)

Conference Dinner

 

The conference dinner will take place at Alte Pumpe on Tuesday, May 27 starting at 7pm.

 

Monday 26 May

12:00 Welcome & Registration

13:00 Opening

Talks

1 Linear Systems with Symmetry Structures: Reciprocal, Relaxation, and Input-Output Hamiltonian systems

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 Hamiltonian systems.

Speaker: Arjan van der Schaft (University of Groningen)

2 Port-based modeling and discretization of thermo-visco-elasticity from various perspectives

Different avenues exist for the thermodynamically consistent modeling of open multi-physical systems with reversible-irreversible couplings. Thermodynamically consistent means that both the first (energy conservation) and the second (entropy production) law of thermodynamics are reflected in the model structure and the numerical schemes derived therefrom.

Founded on the port-Hamiltonian paradigm, which emerged from combining and generalizing network theory and Hamiltonian mechanics, and which is closely related to the bond graph language, several formulations have been proposed: quasi-Hamiltonian, port contact systems, reversible-irreversible port-Hamiltonian systems (riPHS), exergetic PHS or port-thermodynamic systems from a Liouville geometry point of view. The dynamcis in these formulations is in principle generated by the total energy or a homogeneous Hamiltonian as the single potential. All approaches directly include ports and interfaces with a (control) environment, whose role has recently been investigated from a geometric perspective.

Differently, in formulations like GENERIC or metriplectic systems, recently proposed in a four-bracket formulation, the dynamics is derived from two generators, the energy and the entropy, as potentials. The port-based perspective is not at the origin of these approaches, yet extensions to the open case have already been sketched. Finally, thermodynamic systems and their interconnections can be modeled via variational principles, similar to the ones for non-holonomic mechanical systems.

To mimic the fundamental physical properties of the models at a discrete level, structure-preserving numerical methods have to be constructed. In particular (but not only) for the GENERIC formulation, energy-momentum-entropy (EME) preserving schemes have been developed and successfully applied for the simulation of coupled phenomena as from thermo-visco-elasticity.

The main purpose of this talk is to present the different energy- and entropy-based formulations of a discrete thermo-visco-elastic model problem and to discuss their properties and relations, in particular in view of the free choice of thermodynamic variables, which is facilitated by a special form of GENERIC proposed by Mielke (2011). We highlight structure preservation under numerical integration on the finite-dimensional case and close with comments on spatial discretization and simulation examples.

Speaker: Paul Kotyczka (TUM)

3 Port-Hamiltonian Formulation of Cable Harnesses with Radiation Effects

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 magnetic fields at the lateral surfaces of the cables, and is determined by the voltages and currents along the transmission line. Additionally, we discuss some analytical properties, including the semigroup property of the autonomous dynamics.

Speaker: Timo Reis (Technische Universität Ilmenau)

15:00 Coffee break

Talks

4 Model reduction of bilinear port-Hamiltonian systems

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 port-Hamiltonian systems and explore novel model reduction methods that maintain their intrinsic structure. By leveraging techniques designed for bilinear systems with quadratic outputs, we propose both interpolatory and balancing-based reduction approaches. These methods ensure computational efficiency while preserving the key dynamical and physical properties of the original system, making them well-suited for large-scale applications.

Speaker: Serkan Gugercin

5 Recent developments in multiterm matrix equations solvers

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 different directions,
from the use of effective randomized strategies to subspace-rich algorithms
that conveniently leverage the problem structure.

This talk is partially based on collaborations with
Martina Iannacito, Davide Palitta, Marcel Schweitzer and Yihong Wang

Speaker: Valeria Simoncini (Universita di Bologna)

6 Adaptive Randomized DEIM

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 $U \in \mathbb R^{n\times k}$. Commonly used approaches to selecting the row index set $I$ proceed in a greedy manner. For example, Q-DEIM by Drmac and Gugercin first selects the row of maximum norm, then removes this selected row by orthogonal projection and updates $U$ before selecting the next row of maximum norm, etc. While such greedy strategies often work well, there are cases for which $\|U(I,:)^{-1}\|_2$ grows exponentially with $k$. Randomization provides a simple cure: Instead of choosing the row of maximum norm, one selects the row index randomly with probability proportional to the row norm. The selected row is removed by orthogonal projection and, in turn, the probability distribution for selecting the next row is adapted. The index set $I$ resulting from this adaptive randomized selection satisfies $\|U(I,:)^{-1}\|_2 \le \sqrt{1+k(n-k)}$ in expectation, which matches the bound obtained from solving the NP complete problem of finding the submatrix of maximum volume. The derandomization of our adaptive randomized DEIM strategy results in a deterministic algorithm recently proposed by Osinsky, which is slightly more expensive but satisfies the same bound deterministically. The talk will also consider various applications and extensions, including symplectic model reduction and Lagrangian graph bases.

Speaker: Daniel Kressner (EPFL)

Poster Blitz & Poster Section

7 A compositional, energy-based software framework for modeling mechanical, electromagnetic and thermodynamic systems

This talk introduces a compositional, energy-based modeling language for classical mechanics, electromagnetism, and irreversible thermodynamics. The formal, domain-specific language has a graphical syntax, whose expressions are morphisms in a symmetric monoidal category. This enables the straightforward modular and hierarchical composition of complex systems from simpler subsystems. Primitive subsystems are classified into energy storage, reversible energy exchange, and irreversible processes.

The proposed framework builds on established theories, including port-Hamiltonian systems, bond graph notation, and the metriplectic or GENERIC formalisms. Additionally, some category theory is applied to ensure a well-behaved, compositional structure. Unlike general-purpose modeling languages such as Modelica, which handle unstructured differential-algebraic equations, this framework is specifically designed for physical systems, having a structure that ensures adherence to principles such as the first and second laws of thermodynamics.

After summarizing the theoretical setup, the talk presents a prototype implementation of the modeling language, as a first step toward practical applications. It concludes with reflections on potential directions for future work.

Speaker: Markus Lohmayer (FAU Erlangen-Nürnberg)

8 Balanced Truncation for Bilinear-Quadratic Output Systems

We study so called bilinear-quadratic output (BQO) systems of the form $\dot{x}(t)=Ax(t)+\sum_{k=1}^mN_kx(t)u_k(t)+Bu(t)$ with initial condition $x(0)=0$ and output $y(t)=Cx(t)+\begin{bmatrix} x(t)^TM_1x(t)& \dots & x(t)^TM_px(t) \end{bmatrix}^T$, where $A\in\mathbb{R}^{n\times n}, B\in \mathbb{R}^{n\times m}, C\in\mathbb{R}^{p\times n},N_k\in \mathbb{R}^{n\times n}$ for $k=1,\dots,m$, $M_j\in\mathbb{R}^{n\times n}$ for $j=1,\dots,p$, $t\in [0,\infty)$. Here, $x(t)\in\mathbb{R}^n$ describes the state, $u(t)\in\mathbb{R}^m$ the input and $y(t)\in \mathbb{R}^p$ the output of the system. Moreover, we assume $M_j$ to be symmetric due to $x(t)^TM_jx(t)=x(t)^TM_j^Tx(t)=\frac{1}{2}x(t)^T(M_j+M_j^T)x(t)$.

We present algebraic Gramians in different variants for these BQO systems, compare them and their relations to Lyapunov equations arising from bilinear ($M_j=0$) or linear-quadratic output ($N_k=0$) systems. Next, we propose a balancing algorithm for truncation of certain states with the goal of structure-preserving model order reduction. This algorithm is tested in several numerical experiments.

Speaker: Till Peters

9 Energy Efficiency in Subway Systems through Port-Hamiltonian Formulation

Efficient mobility and sustainable transportation are crucial for driving economic growth and preserving the environment in today’s globalized society. However, the transportation sector, responsible for a substantial portion of energy consumption and CO2 emissions, presents a significant challenge for achieving sustainable and ecological transformations, especially in countries like Germany. Rail transport, a key mode of transportation, plays a major role in electricity consumption.

Despite Germany's progress in enhancing energy efficiency, there remains untapped potential for further reducing energy consumption and CO2 emissions. The EKSSE research project is focused on improving the energy efficiency of Nuremberg's and Hamburg's subway systems. This presentation outlines the project's objectives and introduces various approaches for modeling the subway system as a port-Hamiltonian system, offering a promising avenue for optimizing energy usage.

Speaker: Dorothea Hinsen (TU Berlin)

10 KLAP: KYP lemma based low rank approximation for passivation

We present a novel passivation method, called KLAP, for linear time-invariant systems based on the Kalman-Yakubovich-Popov (KYP) lemma in low-rank factorized form.
The passivation problem in our framework corresponds to finding a perturbation to a given non-passive system that renders the system passive while minimizing the $\mathcal{H}_2$ distance between the original non-passive and the resulting passive system.
We show that this problem can be formulated as an unconstrained optimization problem whose objective function can be differentiated efficiently even in large-scale settings.
We show that any global minimizer of the unconstrained problem yields the same passive system. Furthermore, we prove that, in the absence of a feedthrough term, every local minimizer is also a global minimizer. For cases involving a non-trivial feedthrough term, we analyze global minimizers in relation to the extremal solutions of the algebraic Riccati equations, which can serve as tools for identifying local minima.
To solve the resulting numerical optimization problem efficiently, we propose an initialization strategy based on modifying the feedthrough term and a restart strategy when it is likely that the optimization has converged to a local minimum.
Numerical examples illustrate the effectiveness of the proposed method.

Speaker: Jonas Nicodemus

11 Passive feedback control for nonlinear systems

Dynamical systems can be used to model a broad class of physical processes, and conservation laws give rise to system properties like passivity or port-Hamiltonian structure. An important problem in practical applications is to steer dynamical systems to prescribed target states, and feedback controllers combining a regulator and an observer are a powerful tool to do so. However, controllers designed using classical methods do not necessarily obey energy principles, which makes it difficult to model the controller-plant interaction in a structured manner. In this talk, we show that a particular choice of the observer gain gives rise to passivity properties of the controller that are independent of the plant structure. Furthermore, we state conditions for the controller to have a port-Hamiltonian realization and show that a model order reduction scheme can be deduced using the framework of nonlinear balanced truncation. Two dimensional numerical examples based on physical models will illustrate benefits and current limitations of the method.

Speaker: Attila Karsai (TU Berlin)

12 Power-based modeling of power grids

Renewable energy sources such as wind turbines and solar cells are connected to the power grid via grid-forming inverters (GFIs), which can contribute to grid stability and synchronization. Their rollout presents a significant modeling challenge, as GFIs are a relatively new and complex technology, of which there is a limited practical and theoretical understanding. In fact, physics-based modeling is often not feasible.

The so-called normal form of GFIs is a technology-neutral formulation of power grid dynamics that encompasses the space of all plausible GFIs, including established models for conventional generators. It has proven itself to be suited for grey-box modeling, system identification from data, and stability analysis of heterogeneous mixes of technologies. In the normal form approach, GFIs are voltage sources reacting to a collocated current, with the aim of providing a specified amount of power (voltage times current). Therefore, a connection to energy-based modeling lies at hand.

Energy-based modeling is often associated with a bottom-up approach, starting at the underlying physics, while the normal form is a top-down approach starting at the design goal. This contribution explores whether these two complementary approaches can be brought together.

Speaker: Jakob Niehues (Potsdam Institute for Climate Impact Research)

13 Realisation of constraints in stochastic Langevin and port-Hamiltonian dynamics

The realisation of constraints by strong confining forces is a classical theme in mechanics. Constraining a mechanical system typically leads to a differential algebraic equation of differential index 3. Recently, there has been a growing interest in studying constrained stochastic differential equations, due to their relevance in molecular dynamics, material science, computational statistics, or machine learning.
In this talk, I will discuss the realisation of algebraic constraints on stochastic differential equations with degenerate noise, specifically, Langevin-type systems and closely related port-Hamiltonian systems. The constraints are realised by adding stiff confinement terms that penalise deviations of the stochastic dynamics from the constraint surface. In doing so, we focus on two aspects: (1) the pathwise approximation of the constrained dynamics by an unconstrained one with a strong confining force, (2) the preservation of structural properties, such as stability or invariant measures.
This is joint work with Lara Neureither (Cottbus) and Upanshu Sharma (Sydney).

Speaker: Carsten Hartmann (BTU Cottbus-Senftenberg, Institut für Mathematik)

14 Scalable modeling of electro-thermal microgrids using a Reversible Irreversible port-Hamiltonian Systems Approach

As the demand for energy grows, energy systems are becoming increasingly complex. Current systems heavily rely on fossil fuels, leading to negative environmental impacts. To mitigate this, future systems are expected to feature a significant proportion of renewable sources, though these are often uncertain and intermittent. One solution to better allocate renewable resources involves enabling a stronger operational coupling among the principal energy carriers: electric, heat, and gas grids. This introduces major challenges from a control perspective, as the complexity and scale of integrated energy systems grow substantially.
In this context, port-Hamiltonian systems theory provides a systematic and intrinsically modular framework for the modeling and control of complex multi-energy systems. Although port-Hamiltonian formulations for thermodynamically reversible processes are well established, systematically and modularly representing systems with both reversibilities and irreversibilities remains an active research area.
This work addresses the challenging problem of modularly modeling complex electro-thermal microgrids featuring converter-interfaced electrical sources, heat exchangers, and heat pumps. The latter components enable strong coupling between electric and heat grids but introduce very complex thermodynamic phenomena, such as phase-changing flows and other irreversibilities.

Speaker: Arijit Sarkar (Brandenburg University of Technology Cottbus - Senftenberg)

15 Singular matrix pencils: Minimal indices through perturbation behavior

Computing the complete eigenstructure of matrix pencils is a challenging problem. Small perturbations can change both the eigenvalues with their multiplicities, as well as the minimal indices of a given pencil. Recently, however, perturbation theory was used to compute eigenvalues of singular matrix pencils. In this poster, we investigate how the behavior of a general matrix pencil under small perturbations can help determine its minimal indices.

Speaker: Andrii Dmytryshyn (Chalmers University of Technology)

16 Structured sparse regression in quadratic manifolds

Approximating field variables and data vectors from sparse samples is a key challenge for model order reduction of nonlinear dynamical systems. Without utilizing sparse samples -- a process often referred to as hyper-reduction -- the online costs of the reduced model continue to scale with the dimension of the full model. In this talk, we present a new methodology for empirical sparse regression that computes approximations as nonlinear projections onto precomputed quadratic manifolds. Empirical sparse regression on quadratic manifolds can reduce approximation errors by several orders of magnitude compared to widely used methods such as gappy proper orthogonal decomposition or discrete empirical interpolation, which rely on linear approximation spaces by several orders of magnitude. This advantage is particularly pronounced when the data represents transport-dominated or wave-like dynamics, as commonly observed in energy-conserving dynamical systems. After revisiting our greedy quadratic manifold construction, we detail the sparse regression algorithm and demonstrate its effectiveness through numerical examples, showcasing applications in Vlasov systems and a rotating detonation rocket engine.

Speaker: Paul Schwerdtner (NYU)

Tuesday 27 May

Talks

17 Spectral approximation and generalized eigenvalue problem in operator preconditioning

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 setting, and discuss the issue of spectral polution. Relationship between spectra of infinite dimensional operators and eigenvalues of the associated finite matrices will be illustrated using second order self-adjoint PDE operators.

Speaker: Zdeněk Strakoš (Charles University Prague)

18 Singular and regular operator port Hamiltonian pencils

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 solvability of the corresponding (infinite dimensional) differential-algebraic equations $E\dot x=Ax$.

While the general theory is rather complicated, it essentially simplifies for the operator pencils of type
$$ \lambda E-(J-R), $$ where $E,R$ are positive semidefinite, $J$ is skew-symmetric. Here the results are analogous to the finite dimensional situation.
In particular we give necessary and sufficient conditions for uniqueness of solutions of the corresponding Cauchy problem.

The talk is based on:

C. Mehl, V. Mehrmann, M. Wojtylak, Spectral theory of infinite dimensional dissipative Hamiltonian systems, arXiv preprint arXiv:2405.11634.

Speaker: Michał Wojtylak (Jagiellonian University, Kraków, Poland)

19 $M=H+S$

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 a variety of application areas including thermodynamics, electromagnetics, and fluid mechanics. The linear algebraic systems that arise from the linearization and/or discretization of such dHDAEs have matrices of the form $M=H+S$, where $H$ is positive definite or positive semidefinite. This class of matrices has intriguing properties. We will discuss several important examples in this context as well as efficient approaches for solving the resulting linear algebraic systems iteratively. The talk will be based on joint work with Candan Güdücü, Volker Mehrmann, Justus Ramme (all TU Berlin), and Daniel Szyld (Temple University).

Speaker: Jörg Liesen (TU Berlin)

10:30 Coffee break

Talks

20 Multi-Fidelity Robust Controller Design with Gradient Sampling

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 high-dimensional state spaces. In this work, we introduce multifidelity variants
of gradient sampling that leverage low-cost, low-fidelity models with low-dimensional state spaces
for speeding up the optimization process while nonetheless providing convergence guarantees for a
high-fidelity model of the system of interest, which is primarily accessed in the last phase of the
optimization process. Our first multifidelity method initiates gradient sampling on higher-fidelity
models with starting points obtained from cheaper, lower-fidelity models. Our second multifidelity
method relies on ensembles of gradients that are computed from low- and high-fidelity models. Numerical experiments with controlling the cooling of a steel rail profile and laminar flow in a cylinder
wake demonstrate that our new multifidelity gradient sampling methods achieve up to two orders
of magnitude speedup compared to the single-fidelity gradient sampling method that relies on the
high-fidelity model alone.

Speaker: Michael Overton (New York University)

21 Data driven dynamics in trees

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., breakfast to dinner), rendering longer trends nearly unobservable, and thus necessitating the introduction of coarser, hidden variables that are able to capture broader temporal dependencies. Our approach extends the usual framework of Kalman filtering to graphical models and has elements in common with methods for modeling multiresolution stochastic processes on dyadic trees, while avoiding some of the limitations of tree-structured models.

Speaker: Christopher Beattie (Virginia Tech)

22 Zeros, poles and system equivalence of time-delay systems

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\title{Zeros, poles and system equivalence of time-delay systems }
\author{  {\bf Rafikul Alam} and Jibrail Ali\\ email: rafik@iitg.ac.in, jibrail@iitg.ac.in  \\
    Department of Mathematics\\ Indian Institute of Technology
    Guwahati \\Guwahati-781039, INDIA }
\date{}
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\noindent
{\bf Extended abstract:} Consider the linear time-invariant time-delay system (TDS)
\begin{eqnarray} \frac{\mathrm{d}\mathbf{x}(t)}{\mathrm{d}t} &=& A_0 \mathbf{x}(t) + \sum^{N_1}{j=1} A_j \mathbf{x}(t - \tau_j) + \sum^{N_2}{j=1} B_j \mathbf{u}(t- t_j), \ \mathbf{y}(t) &=& \sum^{N_3}{j=1} C_j \mathbf{x}(t-s_j) + \sum^{N_4}{j=1} D_j \mathbf{u}(t-h_j), \end{eqnarray}
where $\mathbf{x}(t) \in \mathbb{C}^r$ and $ \mathbf{u}(t) \in \mathbb{C}^n$ are state and control vectors, respectively, at a time $t$, $ (A_i, B_i, C_i, D_i) \in \mathbb{C}^{r\times r} \times \mathbb{C}^{r\times n} \times \mathbb{C}^{m\times r} \times \mathbb{C}^{m\times n}$ and $ ( \tau_i, t_i, s_i, h_i)$ are time-delay parameters. A system matrix $\mathbf{S}(\lambda)$ and the transfer function $\mathbf{M}(\lambda)$ of the TDS are given by
$$ \mathbf{S}(\lambda) := \left[\begin{array}{c|c} A(\lambda) & B(\lambda) \\ \hline -C(\lambda) & D(\lambda)\end{array}\right] \text{ and } \mathbf{M}(\lambda) := [D(\lambda) + C(\lambda) A(\lambda)^{-1} B(\lambda)], $$where $ A(\lambda) := \lambda I_r - A_0 - \sum^{N_1}{j=1} A_j e^{-\lambda \tau_j},$ $ B(\lambda) := \sum^{N_2}{j=1} B_j e^{-\lambda t_j},$ $C(\lambda) := \sum^{N_3}{j=1} C_j e^{-\lambda s_j}$ and $ D(\lambda) := \sum^{N_4}{j=1} D_j e^{-\lambda h_j}$ are entire matrix-valued functions. Our main aim is two-fold. First, to study the canonical forms of $\mathbf{S}(\lambda)$ and $\mathbf{M}(\lambda)$ so as to analyze zeros and poles of the TDS. Second, to investigate system equivalence, that is, if $\mathbf{S}1(\lambda)$ and $\mathbf{S}_2(\lambda)$ are system matrices of time-delay systems, then investigate Rosenbrock system equivalence (written as $\mathbf{S}_1(\lambda)\sim{rse} \mathbf{S}2(\lambda)$) as well as Fuhrmann system equivalence (written as $\mathbf{S}_1(\lambda)\sim{fse} \mathbf{S}2(\lambda)$). We show that $\mathbf{M}(\lambda)$ admits a Smith-McMillan form $\Sigma{\mathbf{M}}(\lambda)$ given by \begin{equation} \Sigma_{\mathbf{M}}(\lambda) = \left[\begin{array}{ccc|c}\phi_1(\lambda)/\psi_1(\lambda) & & & \\ & \ddots& & \\ & & \phi_p(\lambda)/\psi_p(\lambda) & \\ \hline & & & 0_{m-p\times n-p} \end{array}\right],\end{equation} where $ \phi_1, \ldots, \phi_p$ and $ \psi_1, \ldots, \psi_p$ are entire functions, $\phi_j$ and $\psi_j$ are relatively prime. Further, $ \phi_j$ divides $\phi_{j+1}$ and $ \psi_{j+1}$ divides $\psi_j$ for $ j=1:p-1.$ Furthermore, $\phi_1,\cdots, \phi_p$ and $ \psi_1, \ldots, \psi_p$ are given by $$ \phi_j(\lambda) = \prod^\infty_{\ell=1} (\lambda- \lambda_\ell)^{\partial_j(\lambda_\ell)}u_{j\ell}(\lambda) \text{ and } \psi_j(\lambda) = \prod^\infty_{\ell=1} (\lambda- \mu_\ell)^{\delta_j(\mu_\ell)}v_{j\ell}(\lambda),$$where $ u_{j\ell}, v_{j\ell} $ are entire functions with no zeros in $\mathbb{C}$, $\lambda_\ell$ and $\mu_\ell$ are zeros and poles of $\mathbf{M}(\lambda)$, $\partial_j({\lambda_\ell})$ are $\delta_j(\mu_\ell)$ are appropriate non-negative integers for $j=1:p$ and $\ell \in \mathbb{N}.$ We also show that $\mathbf{M}(\lambda)$ admits a right coprime matrix-fraction description (MFD), that is, there exist entire matrix-valued functions $N(\lambda)$ and $D(\lambda)$ such that $N(\lambda)$ and $D(\lambda)$ are right coprime, $D(\lambda)$ is regular, and $$ \mathbf{M}(\lambda) = N(\lambda) D(\lambda)^{-1}.$$ Further, $\sigma_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(N)$ and $\wp_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(D),$ where $\sigma_{\mathbb{C}}(X)$ is the spectrum (set of zeros) and $\wp_{\mathbb{C}}(X)$ is the set of poles of a meromorphic matrix-valued function $X(\lambda)$. For system matrices, we show that $$\mathbf{S}_1(\lambda) \sim_{rse} \mathbf{S}_2(\lambda) \Longleftrightarrow \mathbf{S}_1(\lambda) \sim_{fse} \mathbf{S}_2(\lambda).$$ Further, if $\mathbf{S}1(\lambda)$ and $\mathbf{S}_2(\lambda)$ are {\em irreducible} then $\mathbf{S}_1(\lambda) \sim{rse} \mathbf{S}2(\lambda) \Longleftrightarrow \mathbf{S}_1(\lambda) \text{ and } \mathbf{S}_2(\lambda)$ have the same transfer function. Finally, if $\mathbf{S}(\lambda)$ is irreducible and $\Sigma{\mathbf{M}}(\lambda)$ is the Smith-McMillan form of $\mathbf{M}(\lambda)$ as given above, then we show that \begin{eqnarray*} S_A(\lambda) &=& I_{r-p} \oplus \mathrm{diag}(\psi_p(\lambda), \psi_{p-1}(\lambda), \psi_1(\lambda)) \\ S_{\mathbf{S}}(\lambda) &= & I_r \oplus \mathrm{diag}(\phi_1(\lambda), \ldots, \phi_p(\lambda)) \oplus 0_{m-p, n-p} \end{eqnarray*} are the Smith forms of $A(\lambda)$ and $\mathbf{S}(\lambda),$ respectively. Hence we show that $ \sigma_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(\mathbf{S}) $ and $ \wp_{\mathbb{C}}(\mathbf{M}) = \sigma_{\mathbb{C}}(A).$ % % % We characterize holomorphic system matrices having the same transfer function $\mathbb{M}(\lambda)$ and present, among other things, the Smith form of $\mathbf{S}(\lambda)$ and the Smith-McMillan form of $\mathbb{M}(\lambda).$ With a view to analyzing zeros and poles of the TDS, we establish relationship between canonical forms of the system matrix $\mathbf{S}(\lambda), $ the state matrix $A(\lambda)$ and the transfer function $ \mathbb{M}(\lambda).$ \vspace{2ex} \noindent {\bf References:} \begin{itemize} \item[1.] M. Frost and C. Storey, ``{A note on the controllability of linear constant delay differential systems}", {\em International J. Control,} Vol.~28, No.~5, pp.~673-679, 1978. \item[2.] M. Frost and C. Storey, ``{Further remarks on the controllability of linear constant delay-differential systems}", {\em International J. Control,} Vol.~30, No.~5, pp.~863-870, 1979. \item[3.] A. I. G. Vardulakis, {\em Linear Multivariable Control}, John Wiley $\&$ Sons Ltd., 1991.
%\item[4.] L. Weiss, ``{On the controllability of delay-differential systems}", {\em SIAM J. Control,} Vol.~5, No.~4, pp.~575-587, 1967.

\end{itemize}

\end{document}

Speaker: Rafikul Alam (IIT Guwahati)

12:30 Lunch

Talks

23 Optimal robustness of passive systems

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 construct this optimal solution. We also consider the problem of finding the nearest passive system to a given nonpassive one and provide a simple but suboptimal solution. These problems are formulated for both continuous-time and discrete-time systems and is linked to the problem of finding a realization of a rational transfer function such that its passivity radius is maximized.

Speaker: Paul Van Dooren (Université catholique de Louvain)

24 Minimization of the Pseudospectral Abscissa of a Matrix Polynomial with Applications to Damping Optimization

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 approaches to minimize the pseudospectral abscissa, the real part of the rightmost eigenvalue attainable over all perturbations of the matrix polynomial of prescribed norm. The efficiency and applicability of the proposed approaches are illustrated on several large matrix polynomials depending on parameters, especially those arising from damping optimization.

Speaker: Emre Mengi (Koc University)

25 Finding the nearest bounded-real port-Hamiltonian system

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 system to a given system and design an algorithm combining alternating optimization and Nesterov's fast gradient method. This formulation also allows us to check whether a given system is bounded real by solving a semidefinite program and provide a PH parametrization for it. We illustrate our proposed algorithms on real and synthetic data sets.

Speaker: Punit Sharma (Indian Institute of Technology Delhi, India)

26 Characterizing and Computing the Stability Region of Singular Nonlinear Dynamical Systems

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 solving an associated eigenvalue problem and performing forward and backward integration of the singular system in question. Numerical examples are included to illustrate the approach. This work extends concepts discussed in the monograph Stability Regions of Nonlinear Dynamical Systems: Theory, Estimation, and Applications (Cambridge University Press, 2015) by H.-D. Chiang and L. F. Alberto.

The talk is based on joint work with Pham Hong Quan and Le Huy Hoang.

Speaker: Hoang Linh Vu (Vietnam National University Hanoi)

16:00 Coffee break

Talks

27 Invariant subspace perturbations of Hamiltonian matrices with defective imaginary eigenvalues

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 purely imaginary eigenvalues
and their invariant subspaces are perturbed under certain structured perturbations.

Speaker: Hongguo Xu (University of Kansas)

28 Removing Critical Eigenvalues of Structured Matrix Pencils and Hamiltonian Matrices with Optimal Perturbations

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 such that the perturbed Hermitian matrix pencil is regular and if it has any critical eigenvalues, then those can be removed by a further arbitrarily small Hermitian perturbation. Additionally, the optimal perturbations are required to be real if $L(\lambda)$ is real. This leads to upper and lower bounds on the distance to a nearest regular Hermitian matrix pencil without critical eigenvalues. It is observed that when $L(\lambda)$ is randomly generated, the upper bound is sometimes equal to the lower bound and is often within a factor of $\sqrt{2}$ of the lower bound.

The work is also extended to the case of $*$-alternating and $*$-palindromic matrix pencils of even size with respect to their corresponding critical eigenvalues.

Under additional assumptions on the coefficient $A$ of $L(\lambda),$ we also study the same problem while perturbing only $B.$ In such cases, the distance is attained by the lower bound if $L(\lambda)$ is a definite pencil. However, in practice we find many examples where this happens even when $L(\lambda)$ is not a definite pencil. The techniques may also be applied to the Hamiltonian matrices to find upper and lower bounds on the distance to \emph{bounded realness}. In such cases it is proved that if the Hamiltonian matrix has only purely imaginary eigenvalues with the ones of positive type being separated from those of negative type, then the distance is attained by the lower bound. However, as experiments demonstrate, this can also happen when the Hamiltonian matrix does not satisfy this condition.

Speaker: Shreemayee Bora (Indian Institute of Technology Guwahati)

29 New Results on Block Diagonalization of Matrix Pencils

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 generalized (real) Schur form, obtained using unitary transformations. The off-diagonal blocks are annihilated by solving generalized Sylvester equations, but this involves non-unitary transformations. In order to ensure good accuracy, the numerical condition of these transformations should be bounded. Specifically, if any element of the solution of a Sylvester equation is larger than a specified constant $b$, the process of solving that equation is stopped, a new pair of diagonal blocks is selected, and another Sylvester equation is attempted to be solved.

There are several strategies to choose a new pair of blocks after each failure. For real matrices, the original strategy chooses a pair of diagonal blocks of order 1 or 2 whose eigenvalue(s) are closest to the mean of eigenvalues of the current pair of diagonal blocks. The chosen pair is moved by unitary equivalence transformations at the bottom of the current pair. Another attempt is made to solve the corresponding generalized Sylvester equation. If the new transformation matrices have all elements with magnitude bounded by $b$, then the added blocks are accepted and the same procedure is applied to the remaining part. Another strategy selects the pair of diagonal blocks whose eigenvalues are closest to the eigenvalues of the current pair. Two pairs of blocks are supposed to belong to the same cluster if all their eigenvalues satisfy an absolute or relative distance condition.

Such strategies are of the bottom-up type, and they are very efficient for matrix pencils with relatively small order and well separated eigenvalues, in which case the solver often succeeds to obtain diagonal blocks of order at most two. But for large order problems with clustered eigenvalues, the solution time can be high, due to the possibility to have a big number of unsuccessful attempts to split the blocks.

A preliminary analysis of the clustered structure of the spectrum, followed by an appropriate reordering of the eigenvalues, could improve the efficiency. Specifically, starting by the most separated eigenvalues, the solver could quickly decouple them, leaving to the end all possibly big clusters of eigenvalues. Few failed attempts to split such a cluster could signal that there is no reason to continue the computations and, therefore, finish the process with one or more large blocks. These strategies could be considered as being of ``top-down'' type. This paper investigates such a top-down technique.

Numerical tests have been performed to assess the possible performance improvement using a clustering technique. For random problems of order 100, the mean CPU time has been reduced by a factor of about 2 compared to the results using bottom-up strategies. For a real problem of order 999, with a clustered block of order 734 (for $b = 5000$), the computing time has been reduced from about 62.7 to 4.7 seconds. The symmetric chordal metric is used as a suitable ``distance'', which seems appropriate for problems that may have infinite eigenvalues. The properties of the chordal metric are also studied, and reliable and efficient algorithms for its computation are proposed.

Speaker: Vasile Sima (National Institute for Research & Development in Informatics (retired))

18:30 Conference Dinner at Alte Pumpe

Wednesday 28 May

Talks

30 Discrete gradient methods for semi-explicit port-Hamiltonian DAEs

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 these constraints are not violated, additional care is necessary when applying numerical methods to DAEs.

In this talk we discuss the application of discrete gradient methods to nonlinear port-Hamiltonian DAEs, with a focus on the case of semi-explicit DAEs. Discrete gradient methods are particularly suitable for the time discretization of port-Hamiltonian systems, since they are structure-preserving regardless of the form of the Hamiltonian, unlike other common methods whose structure-preserving characteristics are limited to quadratic Hamiltonians, like the implicit midpoint rule or other Gauss-Legendre collocation schemes. We also present numerical results to demonstrate the capabilities of our methods.

This is joint work with Philipp Kinon (KIT) and Philipp Schulze (TU Berlin).

Speaker: Riccardo Morandin (OvGU Magdeburg)

31 Structure-preserving model reduction of linear time-varying port-Hamiltonian systems

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 quite naturally in many applications, e.g. in the linearization of nonlinear systems around non-stationary reference solutions, or when some of the system parameters are time dependent. In the literature there are few works on the MOR of general LTV systems, but even fewer on the MOR of LTV-pH systems. In this talk, we introduce a general approach for or the structure-preserving MOR of LTV-pH systems based on (Petrov)- Galerkin projection. We present multiple variants of the usual balanced truncation method to obtain a reduced pH model. Numerical experiments are provided to demonstrate the effectiveness of the proposed methods.

Speaker: Karim Cherifi (Wuppertal university)

32 A System Node Approach to Port-Hamiltonian Partial Differential Equations

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 in practical applications. In this talk, we introduce a system node-based approach for port-Hamiltonian partial differential equations that (i) naturally extends the finite-dimensional input-state-output model proposed by Volker and coauthors in 2018 and (ii) seamlessly integrates boundary control and observation. The versatility of this approach is demonstrated through various examples, including beam equations, heat equations, wave equations, and Oseen equations.

Speaker: Manuel Schaller (TU Chemnitz)

11:00 Coffee break

Talks

33 Bulge chasing is pole swapping

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 variant due to John Francis. For the generalized eigenvalue problem, Moler and Stewart introduced a variant of Francis's algorithm called the QZ algorithm. These algorithms, with various bells and whistles added over the years, are still the dominant algorithms today. These are \emph{bulge-chasing} algorithms. They create bulges at one end of a (Hessenberg) matrix or pencil and chase them to the other end. A few years ago a new class of algorithms, \emph{pole-swapping} algorithms, was introduced by Camps, Meerbergen, Vandebril, and others. It turns out that pole swapping is a generalization of bulge chasing. It might happen that new pole-swapping codes will supplant the current QR and QZ codes in the major software packages. Whether this turns out to be true or not, the pole-swapping viewpoint is extremely valuable for a detailed understanding of what makes this class of algorithms, both bulge-chasing and pole-swapping, work.

Speaker: David Watkins (Washington State University)

34 Short- and long-time behavior in evolution equations: the role of the hypocoercivity index

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 Lorentz kinetic equation on a torus.

Discrete time analogues of the above systems (obtained via the mid-point rule) are contractive, but typically not strictly contractive. For this setting we introduce "hypocontractivity" and an "index of hypocontractivity" and discuss their close connection to the continuous time evolution equations.

References:
1) F. Achleitner, A. Arnold, E. Carlen, The Hypocoercivity Index for the short time behavior of linear time-invariant ODE systems, J. of Differential Equations (2023).
2) A. Arnold, B. Signorello, Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium, Kinetic and Related Models (2022).
3) F. Achleitner, A. Arnold, V. Mehrmann, E. Nigsch, Hypocoercivity in Hilbert spaces, J. of Functional Analysis (2025).

Speaker: Anton Arnold (TU Wien)

35 Polynomial and rational matrices with the invariant rational functions and the 4 sequences of minimal indices prescribed

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 references. However, R(s) has other two fundamental subspaces which, in contrast, have received much less attention in the literature. They are its column and row spaces, which also have their associated minimal indices. This work solves the problems of finding necessary and sufficient conditions for the existence of rational matrices in two scenarios: (a) when the invariant rational functions and the minimal indices of the column and row spaces are prescribed, and (b) when the complete eigenstructure together with the minimal indices of the column and row spaces are prescribed. The particular, but extremely important, cases of these problems for polynomial matrices are solved first and are the main tool for solving the general problems.

This is a joint work with Itziar Baragaña, Silvia Marcaida and Alicia Roca.

Speaker: Froilán Dopico (Universidad Carlos III de Madrid)

13:00 Closing