Keywords:Infinite Dimensional Systems Theory Abstract: In classical control theory, we usually have a state equation or system and just one control, with the mission of achieving a predetermined goal. Sometimes, the goal is to minimize a cost function in a prescribed family of admissible controls; this is the optimal control viewpoint. A more interesting situation arises when several (in general, conflictive or contradictory) objectives are considered. This may happen, for example, if the cost function is the sum of several terms and it is not clear how to average. It can also be expectable to have more than one control acting on the equation. In this talk, we present an overview of the known results on this subject for the heat equation. We will recall the results of Araruna and collaborators where hierarchic exact controllability results were established for linear and semilinear heat equations. In this research, and in the seminal papers by J.-L. Lions, the main idea is to work with one primary control (the leader) and one or several secondary controls (the followers). For each possible leader, the associated followers try to minimize a functional (or reach equilibrium if there is more than one cost objective function). Then, the leader is chosen such that the associated state satisfies a final time constraint. We will present the recent result with E. Fernández-Cara et al., where we accomplish optimal control and controllability tasks with a hierarchy of controls. This time, however, the controllability goal will be commended to the follower, while the choice of the leader will be subject to an optimal control problem. It will be seen that this makes the problem more difficult to handle (essentially because we must work all the time in a very restrictive class of leader controls).

Keywords:Optimization : Theory and Algorithms, Operator Theoretic Methods in Systems Theory Abstract: A trace polynomial is a polynomial in noncommuting variables and traces of their products. It is positive if its evaluations on all symmetric matrices, or more generally, self-adjoint operators from tracial von Neumann algebras, attain only positive semidefinite values. A Positivstellensatz for positive univariate trace polynomials is presented, and a characterization of trace-positive multivariate noncommutative polynomials is discussed.

Institute of Mathematics for Industry, Kyushu University

Keywords:Optimization : Theory and Algorithms Abstract: We focus on computing certified upper bounds for the positive maximal singular value (PMSV) of a given matrix.

The PMSV problem boils down to maximizing a quadratic polynomial on the intersection of the unit sphere and the nonnegative orthant. We provide a hierarchy of tractable semidefinite relaxations to approximate the value of the latter polynomial optimization problem as closely as desired. This hierarchy is based on an extension of Polya's representation theorem. Doing so, positive polynomials can be decomposed as weighted sums of squares of s-nomials, where s can be a priori fixed (s=1 corresponds to monomials, s=2 corresponds to binomials, etc.). This in turn allows us to control the size of the resulting semidefinite relaxations.

Keywords:Operator Theoretic Methods in Systems Theory, Infinite Dimensional Systems Theory, Optimization : Theory and Algorithms Abstract: Let A be a unital commutative R-algebra, K a closed subset of the character space of A, and B a linear subspace of A. For a linear functional L:B --> R, we investigate conditions under which L admits an integral representation with respect to a positive Radon measure supported in K. When A is equipped with a submultiplicative seminorm, we employ techniques from the theory of positive extensions of linear functionals to prove a criterion for the existence of such an integral representation for L. When no topology is prescribed on A, we identify suitable assumptions on A, K, B and L which allow us to construct a seminormed structure on A, so as to exploit our previous result to get an integral representation for L. Our main theorems allow us to extend some well-known results on the Classical Truncated Moment Problem, the Truncated Moment Problem for point processes, and the Subnormal Completion Problem for 2-variable weighted shifts. We also analyze the relationship between the Full and the Truncated Moment Problem in our general setting; we obtain a suitable generalization of Stochel's Theorem which readily applies to Full Moment Problems for localized algebras.

Keywords:Operator Theoretic Methods in Systems Theory, Optimization : Theory and Algorithms Abstract: We will consider the multidimensional truncated p times p Hermitian matrix-valued moment problem. We will prove a characterisation of truncated p times p Hermitian matrix-valued multisequence with a minimal positive semidefinite matrix-valued representing measure via the existence of a flat extension, i.e., a rank preserving extension of a multivariate Hankel matrix (built from the given truncated matrix-valued multisequence). Moreover, the support of the representing measure can be computed via the intersecting zeros of the determinants of matrix-valued polynomials which describe the flat extension. We will also use a matricial generalisation of Tchakaloff's theorem due to the first author together with the above result to prove a characterisation of truncated matrix-valued multisequences which have a representing measure. When p = 1, our result recovers the celebrated flat extension theorem of Curto and Fialkow. The bivariate quadratic matrix-valued problem will be explored in detail.

Keywords:Port-Hamiltonian Systems, Infinite Dimensional Systems Theory Abstract: Mini Course on Infinite-dimensional port-Hamiltonian systems organised by Birgit Jacob and Timo Reis. The theory of port-Hamiltonian systems provides a geometric modelling framework for systems of various physical domains, such as mechanics, electrodynamics and thermodynamics. This approach has its roots in analytical mechanics and starts from the principle of least action, and proceeds, via the Euler-Lagrange equations and the Legendre transform, towards the Hamiltonian equations of motion. This class is further closed under network interconnection. That is, coupling of port-Hamiltonian systems again leads to a port-Hamiltonian system, whence it further allows to describe multi-physical systems, i.e., systems obtained by interaction of several physical domains. The port-Hamiltonian approach further allows the qualitative solution behavior, since it provides an energy balance. Modelling of port-Hamiltonian dynamics may result in various different types of equations, such as ordinary differential equations, differential-algebraic equations, partial differential-algebraic equations and partial differential equations. The latter two types can be reformulated as infinite-dimensional systems, which results in a need for a wide theory of infinite-dimensional port-Hamiltonian systems.

The aim of this mini-course is to give a tutorial on the theory and practice of infinite-dimensional port-Hamiltonian systems. We will provide basics of modelling, analysis and numerics for this class. In particular, we will treat the following questions in the course:

What are practical examples of infinite-dimensional port-Hamiltonian systems? How is modelling of physical systems by infinite-dimensional port-Hamiltonian systems been done? What is known about analysis of infinite-dimensional port-Hamiltonian systems? What are appropriate numerical tools for infinite-dimensional port-Hamiltonian systems? What are open problems for infinite-dimensional port-Hamiltonian systems?

Swiss Federal Institute of Technology (ETH) Zurich

Keywords:System Identification, Linear Systems, Signal Processing Abstract: The existing proofs of the fundamental lemma (J.C. Willems, et al. Control Lett., 54(4), 325-329, 2005) use arguments by contradiction and do not give insight into the assumptions of controllability and persistency of excitation of the input. We present an alternative constructive proof that reduces the required persistency of excitation and characterizes the nongeneric cases in which the extra persistency of excitation beyond the time horizon is needed.

Keywords:Linear Systems, Stochastic Control and Estimation Abstract: Jan Willems and co-authors introduced the characterization of finite-time behaviors of linear systems via the image of Hankel matrices already in 2005. The increasing popularity and research interest of data-driven control techniques has catalyzed the use of this result – which is commonly known as Willems' fundamental lemma – for predictive control and beyond. In this note, we recap recent results on a stochastic extension of the fundamental lemma from~ Pan et al., 2021. Specifically, we leverage the framework of polynomial chaos expansions to derive a computationally tractable stochastic extension of the fundamental lemma.

Keywords:System Identification, Linear Systems, Algebraic Systems Theory Abstract: We state necessary and sufficient conditions for one finite length input-output trajectory to determine uniquely (modulo state-space isomorphisms) a minimal linear, deterministic input-state-output system, given an upper bound on the state dimension.

Keywords:System Identification, Robust and H-Infinity Control, Linear Systems Abstract: Modeling and control of dynamical systems rely on measured data, which contains information about the system. Finite data measurements typically lead to a set of system models that are unfalsified, i.e., that explain the data. The problem of data-informativity for stabilization or control with quadratic performance is concerned with the existence of a controller that stabilizes all unfalsified systems or achieves a desired quadratic performance. Recent results in the literature provide informativity conditions for control based on input-state data and ellipsoidal noise bounds, such as energy or magnitude bounds. In this paper, we consider informativity of input-state data for control where noise bounds are defined through the cross-covariance of the noise with respect to an instrumental variable; bounds that were introduced originally as a noise characterization in parameter bounding identification. The considered cross-covariance bounds are defined by a finite number of hyperplanes, which induce a (possibly unbounded) polyhedral set of unfalsified systems. We provide informativity conditions for input-state data with polyhedral cross-covariance bounds for stabilization and H-2/H-infinity control through vertex/half-space representations of the polyhedral set of unfalsified systems.

Keywords:Optimal Control, Stochastic Control and Estimation, Linear Systems Abstract: Entropy regularization, or a maximum entropy method for optimal control has attracted much attention especially in reinforcement learning due to its many advantages such as a natural exploration strategy and robustness against disturbances. Nevertheless, for safety-critical applications, it is crucial to suppress state uncertainty due to the stochasticity of high-entropy control policies and dynamics to an acceptable level. To achieve this, we consider the problem of steering a state distribution of a deterministic discrete-time linear system to a specified one at final time with entropy-regularized minimum energy control. We show that this problem boils down to solving coupled Lyapunov equations. Based on this, we derive the existence, uniqueness, and explicit form of the optimal policy.

Keywords:Systems on Graphs, Optimization : Theory and Algorithms, Stochastic Control and Estimation Abstract: Recently, there has been a large interest in the theory of optimal transport and its connections to the Schrödinger bridge problem. In this work we generalize some of these results to multi-marginal optimal transport problems when the cost function decouples according to a tree structure. In particular, the entropy regularized multi-marginal optimal transport problem can be seen as a Schrödinger bridge problem on the same tree. Moreover, based on this, we extend efficient algorithms for the bi-marginal problem to the multi-marginal setting where the cost function decouples according to a tree structure. Such problems appear in several applications of interest such as barycenter and tracking problems. A common approach for solving these problems is by utilizing pairwise regularization. However, we show that the multi-marginal regularization introduces less diffusion which is favorable in many applications.

Keywords:Nonlinear Filtering and Estimation, Stochastic Control and Estimation, Stability Abstract: This paper is concerned with optimality and stability analysis of a family of ensemble Kalman filter (EnKF) algorithms. EnKF is commonly used as an alternative to the Kalman filter for high-dimensional problems, where storing the covariance matrix is computationally expensive. The algorithm consists of an ensemble of interacting particles driven by a feedback control law. The control law is designed such that, in the linear Gaussian setting and asymptotic limit of infinitely many particles, the mean and covariance of the particles follow the exact mean and covariance of the Kalman filter. The problem of finding a control law that is exact does not have a unique solution, reminiscent of the problem of finding a transport map between two distributions. A unique control law can be identified by introducing control cost functions, that are motivated by the optimal transportation problem or Schr"odinger bridge problem. The objective of this paper is to study the relationship between optimality and long-term stability of a family of exact control laws. Remarkably, the control law that is optimal in the optimal transportation sense leads to an EnKF algorithm that is not stable.

Keywords:Stochastic Control and Estimation, Stochastic Modeling and Stochastic Systems Theory, System Identification Abstract: The problem to reconcile observed marginal distributions with a given prior was posed by E. Schro ̈dinger in 1932/32, and is now known as the Schroo ̈dinger Bridge Problem. It represents a stochastic counterpart of the Optimal Mass Transport (OMT). In either setting, the problem to interpolate between “unbalanced” marginals has been approached by introducing source/sink terms into the transport equations, in an adhoc manner, chiefly driven by applications in image registration. In the present work we developed a formalism to interpolate between “unbalanced” marginals in the original spirit of E. Schro ̈dinger, seeking the most likely transport of particles that may vanish along their path between given end points in time. In this, we develop a Schro ̈dinger system of equations that accounts for losses, by allowing particles to “jump” into a coffin state according to a suitable probabilistic law. The solution of the Schro ̈dinger system allows constructing a stochastic evolution that reconciles the given unbalanced marginals.

Keywords:Optimization : Theory and Algorithms, Stochastic Modeling and Stochastic Systems Theory Abstract: We propose a distributed nonparametric algorithm for solving measure-valued optimization problems with additive objectives. Such problems arise in several contexts in stochastic learning and control including Langevin sampling from an unnormalized prior, mean field neural network learning and Wasserstein gradient flows. The proposed algorithm comprises a two-layer alternating direction method of multipliers (ADMM). The outer-layer ADMM generalizes the Euclidean consensus ADMM to the Wasserstein consensus ADMM, and to its entropy-regularized version Sinkhorn consensus ADMM. The inner-layer ADMM turns out to be a specific instance of the standard Euclidean ADMM. The overall algorithm realizes operator splitting for gradient flows in the manifold of probability measures.

Keywords:Stochastic Control and Estimation, Optimization : Theory and Algorithms, System Identification Abstract: This manuscript introduces a regression-type formulation for approximating the Perron-Frobenius Operator by relying on distributional snapshots of data. These snapshots may represent densities of particles. The Wasserstein metric is leveraged to define a suitable functional optimization in the space of distributions. The formulation allows seeking suitable dynamics so as to interpolate the distributional flow in function space. A first-order necessary condition for optimality is derived and utilized to construct a gradient flow approximating algorithm. The framework is exemplified with numerical simulations.

Keywords:Port-Hamiltonian Systems Abstract: In this talk, we will develop extended balancing and its structure preservation possibilities for linear systems, as well as extended balancing theory for nonlinear systems in the contraction framework. For the latter, we introduce the concept of the extended differential observability Gramian and inverse of the extended differential controllability Gramian for nonlinear dynamical systems and show their correspondence with generalized differential Gramians. We also provide how extended (differential) balancing can be utilized for model reduction to get a smaller apriori error bound in comparison with generalized (differential balancing). We will focus on preserving the structure of a port-Hamiltonian system with help of extended balancing in both the linear and nonlinear systems setting.

Keywords:Coding Theory Abstract: The densities of codes with certain properties have always been of interest in classical coding theory, in particular to understand how many of such codes exist and how likely a random code will have the prescribed properties. Further applications of density results of codes appear in code-based cryptography, where it is important that the set of codes with a certain property is large enough to outgo brute force attacks. In this talk we will present various density results for optimal or close-to-optimal codes in different metric spaces with different types of linearity. In particular, we will show when optimal codes in the Hamming, rank and sum-rank metric are dense and when they are sparse.

Keywords:Machine Learning and Control Abstract: Recent radical evolution in distributed sensing, computation, communication, and actuation has fostered the emergence of cyber-physical network systems. Regardless of the specific application, one central goal is to shape the network collective behavior through the design of admissible local decision-making algorithms. This is nontrivial due to various challenges such as the local connectivity, system complexity and uncertainty, limited information structure, and the complex intertwined physics and human interactions.

In this talk, I will present our recent progress in formally advancing the systematic design of distributed coordination in network systems via harnessing special properties of the underlying problems and systems. In particular, we will present three examples and discuss three type of properties, i) how to use network structure to ensure the performance of the local controllers; ii) how to use the information and communication structure to develop distributed learning rules; iii) how to use domain-specific properties to further improve the efficiency of the distributed control and learning algorithms.

Keywords:Stochastic Modeling and Stochastic Systems Theory, Multidimensional Systems, Nonlinear Filtering and Estimation Abstract: The paper reports results on the modeling of related stochastic variables, based on a finite fully ordered sequence of higher order moments (or correlations), and using mutually independent parameters that characterize all solutions that interpolate the given (or measured) data. The results are obtained by determining properties of the hierarchical generalized Hankel matrix of the moments. A system theoretic approach is used to derive the results. It appears that an extension of the related Hamburger-Jacobi orthogonal polynomials to the multivariate case does not suffice to yield a parametrization, but a further reduction of the recursive Cholesky factorization of the moment matrix does.

Keywords:Operator Theoretic Methods in Systems Theory, Signal Processing, Computations in Systems Theory Abstract: The classical sampling formula associated with the names of Shannon, Whittaker, Nyquist and Kotelnikov, will be exhibited as a special case of a general sampling formula in the setting of reproducing kernel Hilbert spaces of entire functions due to Louis de Branges. Other applications of these spaces and generalizations to spaces of vector valued entire functions will also be discussed briefly.

Keywords:Linear Systems, Mathematical Theory of Networks and Circuits, Hybrid Systems Abstract: Switches in electrical circuits may lead to Dirac impulses in the solution; a real world example utilizing this effect is the spark plug. Treating these Dirac impulses in a mathematically rigorous way is surprisingly challenging. This is in particular true for arguments made in the frequency domain in connection with the Laplace transform. A survey will be given on how inconsistent initials values have been treated in the past and how these approaches can be justified in view of the now available solution theory based on piecewise-smooth distributions.

Keywords:Mathematical Theory of Networks and Circuits, Feedback Control Systems Abstract: Positive real Odd rational functions, PO, (a.k.a. Foster or Lossless) correspond to the driving point immittance of reactive, i.e L-C, circuits. We here show that through PO functions one can model a feedback-loop connection (irrespective of the nature of each block). This suggests a research problem: Extend one of the classical scalar circuits synthesis schemes like Brune, Darlington, Bott-Duffin or Foster to the realm of possibly elaborate network of feedback loops. A challenge here is that the respective PO functions are of several non-commuting variables.

Keywords:Optimal Control, Mechanical Systems, Linear Systems Abstract: An explicit algorithm will be presented for computing the H2 norm of a single input single output system from the coefficients in its transfer function. The algorithm follows directly from Cauchy's residue theorem, and the most computationally intensive step involves solving a polynomial Diophantine equation. This can be efficiently solved using subresultant sequences in a fraction-free variant of the extended Euclidean algorithm. The coefficients in these subresultant sequences correspond to the Hurwitz determinants, whereby a stability test can be obtained alongside computing the H2 norm with little additional computational effort. Implementations of the algorithm symbolically, in exact arithmetic, and in floating-point arithmetic will be presented. The accompanying talk will demonstrate an example application on the design of passive train suspension systems that optimise passenger comfort. The example will demonstrate the algorithm's greater robustness and computational efficiency relative to H2 norm algorithms requiring the computation of the controllability or observability Gramians. The more general application of the techniques to the realisation of optimal lumped-parameter networks will also be discussed.

Keywords:Mechanical Systems, Nonlinear Systems and Control, Optimal Control Abstract: Motivated by the advantages of structure-preserving integration for applications ranging from molecular dynamics to astrodynamics, geometric integration has been brought into optimal control in the past two decades. Advantages over conventional methods have been shown in biomechanics, robotics, automotive applications, and space mission design. The implicit midpoint method, that is a member of the class of symplectic (partitioned) Runge-Kutta methods but also possess a variational derivation and thus is symmetry-preserving, is widely used due to its many favorable properties. In particular, efficient computations can be achieved by coarse discretizations of state and control signals, since structure preservation does not have to be ensured by small step sizes, as it is the case in conventional methods. Then, specific input parametrizations become an issue when implementing optimized signals in control architectures. We show numerical studies for piecewise linear control signals used inenergy optimal control problems.

Keywords:Optimal Control, Mechanical Systems Abstract: The optimal control of mechanical systems satisfies an optimal control version of Noether's theorem. Accordingly, there exist generalized momentum maps on the level of the optimal control problem which are conserved if the system has symmetry. For constrained mechanical systems different approaches to define the necessary optimality conditions are known. These approaches will be compared with respect to their capability to preserve the generalized momentum maps. In addition to that, a discretization approach will be proposed which is capable to preserve the generalized momentum maps.

Keywords:Mechanical Systems, Nonlinear Systems and Control, Numerical and Symbolic Computations Abstract: We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (General Equation for the NonEquilibrium Reversible-Irreversible Coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g., Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.

Keywords:Port-Hamiltonian Systems, Mathematical Theory of Networks and Circuits, Physical Systems Theory Abstract: A prevalent theme throughout science and engineering is the ongoing paradigm shift from isolated systems to open and interconnected systems. Port-Hamiltonian theory developed as a synthesis of geometric mechanics and network theory. The possibility to model complex multiphysical systems via interconnection of simpler components is often advertised as one of its most attractive features. The development of a port-Hamiltonian modelling language however remains a topic which has not been sufficiently addressed. We report on recent progress towards the formalization and implementation of a modelling language for exergetic port-Hamiltonian systems. Its diagrammatic syntax is the operad of undirected wiring diagrams with an interpretation akin to bond graphs. Together with a port-Hamiltonian semantics defined as an operad functor, this enables a modular and hierarchical approach to model specification.

Keywords:Port-Hamiltonian Systems, Infinite Dimensional Systems Theory, Numerical and Symbolic Computations Abstract: In this article, we present the port-Hamiltonian representation, the structure preserving discretization and the resulting finite-dimensional state space model of one-dimensional filaments based on a mixed finite element formulation. Due to the fact that the equations of motion of a filamentous body are based on the theory of geometrically nonlinear mechanical systems, the port-Hamiltonian formulation is expressed by means of its co-energy (effort) variables. The resulting port-Hamiltonian state space model features a quadratic Hamiltonian and the nonlinearity is reflected in the state dependence of its interconnection matrix. Numerical experiments generated with FEniCS illustrate the properties of the resulting finite element models.

Keywords:Machine Learning and Control, Neural Networks Abstract: This talk discusses the role that systems theory plays in unveiling fundamental limitations of learning algorithms and architectures when used to control a dynamical system, and in suggesting strategies for overcoming these limitations. As an example, a feedforward neural network cannot stabilize a double integrator using output feedback. Similarly, a recurrent NN with differentiable activation functions that stabilizes a non-strongly stabilizable system must be itself open loop unstable, a fact that has profound implications for training with noisy, finite data. A potential solution to this problem, motivated by results on stabilization with periodic control, is the use of neural nets with periodic resets, showing that indeed systems theoretic analysis is instrumental in developing architectures capable of controlling certain classes of unstable systems. The talk will finish by arguing that when the goal is to learn control oriented models, the loss function should reflect closed loop, rather than open loop model performance, a fact that can be accomplished by using gap-metric motivated loss functions.

Keywords:Linear Systems, Robust and H-Infinity Control, Feedback Control Systems Abstract: In this extended abstract we consider input-output systems described by higher order difference equations, also called autoregressive systems. We assume that we have input-output data obtained from an underlying true, but unknown, system. The problems we then consider is to determine on the basis of these data whether this unknown system is stable. We also deal with the problem of determining whether a stabilizing controller exists, and, if so, to determine one using only the data. In order to tackle these problems we heavily rely on methods from the behavioral approach to systems and control, in particular the notion of quadratic difference form.

Keywords:Model Predictive Control, Optimal Control, Linear Systems Abstract: Recently, data-driven predictive control of linear systems has received wide-spread research attention. It hinges on the fundamental lemma by Willems et al. In a previous paper, we have shown how this framework can be applied to predictive control of linear time-invariant descriptor systems. In the present paper, we present a case study wherein we apply data-driven predictive control to a discrete-time descriptor model obtained by discretization of the power-swing equations for a nine-bus system. Our results show the efficacy of the proposed control scheme and they underpin the prospect of the data-driven framework for control of descriptor systems.

Keywords:Model Predictive Control Abstract: In this paper, we present a data-driven distributed model predictive control (MPC) scheme to stabilise the origin of dynamically coupled discrete-time linear systems subject to decoupled input constraints. The local optimisation problems solved by the subsystems rely on a distributed adaptation of the Fundamental Lemma by Willems et al., allowing to parametrise system trajectories using only measured input-output data without explicit model knowledge. For the local predictions, the subsystems rely on communicated assumed trajectories of neighbours. Each subsystem guarantees a small deviation from these trajectories via a consistency constraint. We provide a theoretical analysis of the resulting non-iterative distributed MPC scheme, including proofs of recursive feasibility and (practical) stability. Finally, the approach is successfully applied to a numerical example.

Keywords:Hybrid Systems, Stability, Systems on Graphs Abstract: We consider stability analysis of constrained switching linear systems in which the dynamics is unknown and whose switching signal is constrained by an automaton. We propose a data-driven Lyapunov framework for providing probabilistic stability guarantees based on data harvested from observations of the system. By generalizing previous results on arbitrary switching linear systems, we show that, by sampling a finite number of observations, we are able to construct an approximate Lyapunov function for the underlying system. Moreover, we show that the entropy of the language accepted by the automaton allows to bound the number of samples needed in order to reach some pre-specified accuracy.

Max Planck Institute for Dynamics of Complex Technical Systems

Keywords:Large Scale Systems, System Identification Abstract: This work aims at tackling the problem of learning surrogate models from noisy time-domain data by means of matrix pencil-based techniques, namely the Hankel and Loewner frameworks. A data-driven approach to obtain reduced order state-space models from time-domain input-output measurements for linear time-invariant (LTI) systems is proposed. This is accomplished by combining the aforementioned model order reduction (MOR) techniques with the signal matrix model (SMM) approach. The proposed method is illustrated by a numerical example consisting of a high-order building model.

Keywords:Linear Systems, Large Scale Systems, Computations in Systems Theory Abstract: We present an extension of the Loewner framework, an established data-driven reduction, and identification method. This will be referred to as the one-sided Loewner framework since only one set of interpolation conditions are explicitly and exactly matched. For the other set of conditions, approximated interpolation is imposed. We describe how to explicitly characterize new interpolation conditions, derived from the latter set. We also show connections to the iterative AAA algorithm. Typical applications include constructing reduced models from frequency response data measured from systems in electronics or mechanical engineering. We illustrate the application of the main method on a large-scale benchmark example.

Keywords:Large Scale Systems, System Identification, Linear Systems Abstract: In this extended abstract, we present a time-domain data-driven technique for model reduction by moment matching of linear systems. We propose an algorithm, based on the so-called swapped interconnection, that (asymptotically) approximates an arbitrary number of moments of the system from a single time-domain sample. A family of reduced-order models that match the estimated moments is derived. Finally, the use of the proposed algorithm is demonstrated on the problem of model reduction of an atmospheric storm track model.

Keywords:Nonlinear Systems and Control Abstract: The model reduction problem by least squares moment matching is studied. A recent time-domain characterization of least squares moment matching for linear systems is used to define a notion of least squares moment matching for nonlinear systems. Models achieving least squares moment matching are shown to minimize an a priori error bound on the worst case r.m.s. gain of an error system with respect to a given family of signals, thus providing new insights on the linear theory.

Keywords:Large Scale Systems, Nonlinear Systems and Control Abstract: In this work, we introduced extended differential balancing, which is a model reduction approach for nonlinear dynamical systems in the contraction framework. We utilized the solutions of two time-varying LMIs to arrive at a balanced realization of the associated variational system to perform the truncation of less important states of the original system. One of the main contributions of the work is to show a computationally tractable way of performing model reduction providing tighter aprioiri error bounds in comparison with generalized differential balancing. On the other hand, we introduce generalized controllability function and generalized observability function for continuous-time stable nonlinear systems. We also propose a balanced realization for nonlinear port-Hamiltonian systems in which the generalized energy functions are balanced as well as the Hamiltonian of the corresponding system is in diagonal form. Moreover, the reduced order model obtained by truncation preserves the port-Hamiltonian structure.

Keywords:Optimization : Theory and Algorithms Abstract: We consider polynomials of a few linear forms and show how exploit this type of sparsity for optimization on some particular domains like the Euclidean sphere or a polytope. Moreover, a simple procedure allows to detect this form of sparsity and also allows to provide an approximation of any polynomial by such sparse polynomials.

Keywords:Algebraic Systems Theory Abstract: Given a real algebraic curve in projective space, we study the computational problem of deciding whether there exists a hyperplane meeting the curve in real points only. More generally, given any divisor on such a curve, we may ask whether the corresponding linear series contains an effective divisor with totally real support. This translates into a particular type of parametrized real root counting problem that we wish to solve exactly.

We will focus on examples and some general results and conjectures, based on recent work with Huu Phuoc Le and Dimitri Manevich.

Keywords:Algebraic Systems Theory, Applications of Algebraic and Differential Geometry in Systems Theory Abstract: We present recent results joint with Mario Kummer (TU Dresden) on convex hulls of curves. We see a large family of examples where these convex hulls turn out to be hyperbolicity cones. For convex hulls of elliptic curves, we are able to show that these hyperbolicity cones are spectrahedra, generalizing previous results by Henrion and Scheiderer.

Keywords:Optimization : Theory and Algorithms, Applications of Algebraic and Differential Geometry in Systems Theory Abstract: For a fixed number of n+1 (n≥1) variables and even degree 2d (d≥1), the SOS cone of all real forms representable as finite sums of squares (SOS) of half degree d real forms is included in the PSD cone of all positive semidefinite (PSD) real forms. Hilbert (1888) states that both cones coincide if and only if n+1=2, d=1 or (n+1,2d)=(3,4). In this talk, we discuss necessary or sufficient conditions to extend local positive semidefiniteness of real quadratic forms along projective varieties generated by s (s≥0) real quadratic forms. Those conditions allow us to construct an explicit filtration of intermediate cones between the SOS and PSD cone along the Veronese variety. Indeed, the latter is known to be a projective variety finitely induced by real quadratic forms. We analyze this filtration for proper inclusions. In fact, after applying an inductive argument, it suffices to investigate the situation for a truncated subfiltration of the former. A result of Blekherman et al. (2016) on projective varieties of minimal degree permits us to handle the first inclusion in the constructed filtration. Generalizing this observation, we are able to show that the first n+1 cones in the peculiar filtration coincide. Finally, we lay out the situation in the basic non Hilbert case of quaternary quartics by identify exactly two strictly separating intermediate cones in the constructed filtration of the SOS and PSD cone via considerations of real forms based on techniques due to Robinson (1969) and Choi and Lam (1977). This is a work in progress with Salma Kuhlmann und Charu Goel.

Keywords:Applications of Algebraic and Differential Geometry in Systems Theory Abstract: The relationship between the cone of positive semidefinite (psd) real forms and its subcone of sums of squares (sos) of forms is of fundamental importance in real algebraic geometry and optimization, and has been studied extensively (see for instance Marshall (2008)). The study of this relationship goes back to the 1888 seminal paper of Hilbert, where he gave a complete characterisation of the pairs (n,2d) for which a psd n-ary 2d-ic form can be written as sos. In this talk we discuss how this relationship changes under the additional assumptions of invariance on the given forms, i.e. when we consider the induced action of a real finite reflection group on the ring of polynomials. We will see that in equivariant situations Hilbert’s classification does not remain true in general and depends on the group action, the degree and the number of variables.

Max Planck Institute for Mathematics in the Sciences

Keywords:Optimization : Theory and Algorithms Abstract: Convex Algebraic Geometry lives at the intersection of Convex Geometry, Optimization, Algebraic Geometry and Real Algebra. Classically, convex geometry has been studied from an analytical point of view. Here, we approach it using tools from real and complex algebraic geometry, with a focus on semialgebraic convex bodies, beyond polytopes.

Keywords:Robust and H-Infinity Control, Nonlinear Systems and Control, Large Scale Systems Abstract: In this paper, we present a global solution to a nonlinear H-infinity optimal control problem for control-affine systems with an associated potential function. We also give a closed form expression for an optimal controller and demonstrate its potential for sparsity. This paper thus advances a recent result which considers the same problem restricted to systems with symmetric state matrix and nonlinear input matrix. We further apply the main result to obtain a simpler and more intuitive statement for a class of systems capable of modeling nonlinear buffer networks.

Keywords:Nonlinear Systems and Control, Robust and H-Infinity Control, Operator Theoretic Methods in Systems Theory Abstract: The Scaled Relative Graph (SRG) is a generalization of the Nyquist diagram that may be plotted for nonlinear operators, and allows nonlinear robustness margins to be defined graphically. This abstract explores techniques for shaping the SRG of an operator in order to maximize these robustness margins.

Keywords:Nonlinear Systems and Control, Algebraic Systems Theory Abstract: We show that any asymptotic observer for a non-linear kinematic system on a differentiable manifold contains a full internal model of the plant.

Keywords:Feedback Control Systems, Linear Systems Abstract: A recent result in (Incremona et al., 2022) put forward an architecture of internal model based controllers, in which the stabilizer can be fully separated from the internal model. In this paper we propose a parametrized implementation of this controller, which isolates a parameter shaping properties of the exosystem. We show that with this implementation the closed-loop dynamics have an affine dependence on the parameter. As such, the closed-loop system remains stable even under arbitrary variations of the parameter, as long as it remains bounded. We demonstrate that this property is beneficial for adding an adaptation mechanism to adjust parameters of the internal model.

Keywords:Applications of Algebraic and Differential Geometry in Systems Theory, Nonlinear Systems and Control, Feedback Control Systems Abstract: In this paper, we give normal forms for flat two-input control-affine systems in dimension five that admit a flat output depending on the state only (we call systems with that property x-flat systems). We discuss relations of x-flatness in dimension five with static and dynamic feedback linearization and show that if a system is x-flat it becomes linearizable via at most three prolongations of a suitably chosen control. Therefore x-flat systems in dimension five can be, in general, brought into normal forms generalizing the Brunovsky canonical form. If a system becomes linear via at most two-fold prolongation, the normal forms are structurally similar to the Brunovsky form: they have a special triangular structure consisting of a linear chain and a nonlinear one with at most two nonlinearities. If a system becomes linear via a three-fold prolongation, we obtain not only triangular structures but also a nontriangular one, and face new interesting phenomena.

Keywords:Delay Systems, Linear Systems Abstract: In this note we study discrete-time dead-time compensation from the viewpoint of the observer-based design procedure. We show that the discrete equivalent of the observer-predictor architecture can be derived ab initio via classical state-feedback and observer arguments under mild assumptions. The resulting observer is reduced order and we show that this choice is justifiable even if corresponding state measurement channels are noisy.

Keywords:Port-Hamiltonian Systems, Infinite Dimensional Systems Theory, Systems on Graphs Abstract: In this talk we consider power networks consisting of loads and generators which are interconnected via transmission lines. Here we use a distributed model for the transmission lines and provide a port-Hamiltonian formulation as a boundary control system and show the exponential stability and a power-balance equation for classical solutions.

Keywords:Port-Hamiltonian Systems, Infinite Dimensional Systems Theory, Operator Theoretic Methods in Systems Theory Abstract: Recently, port-Hamiltonian (pH) representations have been developed for multi-phase flow models, such as the Two-Fluid Model and the zero-slip Drift Flux Model (DFM), with non-quadratic Hamiltonian functionals, by eliminating constraints and writing a partial differential-algebraic system as a system with (only) partial differential equations. However, the existing multi-phase modelling framework is not modular enough since mathematical computations have to be performed again for even a small change, say a different governing equation of state, in the model description. Furthermore, a pH representation of the general DFM still does not exist, and the complicated, non-linear models may not always be amenable to the pH model formulation as per the current state-of-the-art. To this end, we make efforts towards developing a general pH descriptor formalism for non-linear multi-phase flow dynamics.

Keywords:Port-Hamiltonian Systems, Infinite Dimensional Systems Theory Abstract: A port-Hamiltonian formulation of a general class of interacting particle systems and its corresponding mean-field partial-differential equation is discussed. To establish the port-Hamiltonian structure of the interacting particle systems a specific variable transformation is employed. It turns out that an appropriate retransformation of the characteristics corresponding to the mean-field partial differential equation yields again a port-Hamiltonian structure.

Keywords:Port-Hamiltonian Systems, Infinite Dimensional Systems Theory Abstract: We regard port-Hamiltonian systems on multidimensional spatial domains. We show that there are multiple (slightly different) Dirac structures assigned to such systems. Moreover, we point out that not every Dirac structure admits well-posedness of the corresponding system.

Keywords:Port-Hamiltonian Systems, Infinite Dimensional Systems Theory, Dissipativity Abstract: We present a framework to formulate infinite dimensional port-Hamiltonian systems by means of system nodes, which provide a very general and powerful setting for unbounded input and output operators that appear, e.g., in the context of boundary control or observation. One novelty of our approach is that we allow for unbounded and not necessarily coercive Hamiltonian energies. To this end, we construct finite energy spaces to define the port-Hamiltonian dynamics and give an application in case of multiplication operator Hamiltonians where the Hamiltonian density does not need to be positive or bounded. In order to model systems involving differential operators on these finite energy spaces, we show that if the total mass w.r.t. the Hamiltonian density (and its inverse) is finite, one can define a unique weak derivative.

Keywords:Port-Hamiltonian Systems, Infinite Dimensional Systems Theory, Stability Abstract: The multiplier approach is applied to a class of port-Hamiltonian systems with boundary dissipation to establish exponential decay. The exponential stability of port-Hamiltonian systems has been studied and sufficient conditions obtained. Here the decay rate Me^{-alpha t} is established with M and alpha are in terms of system parameters. This approach is illustrated by several examples, in particular, boundary stabilization of a piezoelectric beam with magnetic effects.

Keywords:Optimal Control, Feedback Control Systems, Numerical and Symbolic Computations Abstract: The policy iteration method is a classical algorithm for solving optimal control problems. We introduce a policy iteration method for Mean Field Games systems and we prove, under a classical monotonicity assumption on the coupling cost, the convergence of this procedure to the solution of the problem.

Keywords:Nonlinear Systems and Control, Stochastic Modeling and Stochastic Systems Theory Abstract: The goal of this talk is to present a rigorous derivation of a macroscopic traffic flow model with a bifurcation or a local perturbation from a microscopic one. The microscopic model is a simple follow-the-leader with random parameters. The random parameters are used as a statistical description of the road taken by a vehicle and its law of motion. The limit model is a deterministic and scalar Hamilton-Jacobi equation on a network with a flux limiter, the flux-limiter describing how much the bifurcation or the local perturbation slows down the vehicles. The proof of the existence of this flux limiter relies on a concentration inequality and on a delicate derivation of a super-additive inequality.

Keywords:Optimal Control, Optimization : Theory and Algorithms, Nonlinear Systems and Control Abstract: We study some models of evolutive deterministic mean field games with finite time horizon where the Hamiltonian is not coercive in the gradient term because the dynamic of the generic player has some forbidden directions. We study the existence of weak solutions and their representation by means of relaxed equilibria in the Lagrangian setting which are described by a probability measure on optimal trajectories.

Keywords:Optimal Control Abstract: In this talk we study the numerical approximation of deterministic Mean Field Games where the dynamics of a typical agent is non-linear with respect to the state variable and affine with respect to the control variable. Particular instances of the problem considered here are MFGs with control on the acceleration. Our main result is the convergence of solutions of this approximation towards MFG equilibria.