Keywords: Control of Distributed Parameter Systems, Aerospace and Avionic Systems, Large Scale Systems
Abstract: Wind farms comprise a network of dynamical systems coupled through interactions with the turbulent atmospheric boundary layer (ABL). The ABL dynamics govern the velocity field that enters the farm as well as the turbulent mixing that regenerates energy for extraction at downstream rows, thereby playing a fundamental role in wind farm power production. Understanding the dynamic interactions between turbines, wind farms, and the ABL is therefore critical for improving wind farm performance. This talk introduces a suite of models that leverage ABL information to provide improved models of both static and dynamic conditions in the wind farm. An example of such a dynamic model in a control setting provides a case study wherein turbines are viewed as actuators that adjust the flow field to collectively produce more efficient power tracking performance.

Keywords: Nonlinear Systems and Control, Dissipativity, Port-Hamiltonian Systems
Abstract: Based on van der Schaft(2024), this paper aims at extending to the nonlinear case the theory of reciprocal linear systems as developed in Willems(1972).

Keywords: Nonlinear Systems and Control, Dissipativity, Stability
Abstract: A new definition of discrete-time negative imaginary (NI) systems is provided. This definition characterizes the dissipative property of a zero-order hold sampled continuous-time NI system. Under some assumptions, asymptotic stability can be guaranteed for the closed-loop interconnection of an NI system and an output strictly negative imaginary system, with one of them having a one step advance. In the case of linear systems, we also provide necessary and sufficient frequency-domain and LMI conditions under which the definition is satisfied. Also provided is a simple DC gain condition for the stability results in the linear case.

Keywords: Nonlinear Systems and Control, Feedback Control Systems, Stability
Abstract: In this paper, we propose employing battery-based feedback control and nonlinear negative imaginary (NI) systems theory to reduce the need for such expansion. By formulating a novel Lur'e-Postnikov-like Lyapunov function, stability results are presented for the feedback interconnection of two single nonlinear NI systems, while output feedback consensus results are established for the feedback interconnection of two networked nonlinear NI systems based on a network topology. This theoretical framework underpins our design of battery-based control in power transmission systems. We demonstrate that the power grid can be gradually transitioned into the proposed NI systems, one transmission line at a time.

Keywords: Nonlinear Systems and Control, Neural Networks
Abstract: Motivated by neuromorphic computing applications, this work considers electrical circuits comprising memristors and capacitors, connected to external sources. We derive a model describing the dynamic behaviour of such a circuit and show that, for given initial conditions and a fixed input signal, the voltages across the memristors converge towards zero, implying that the conductance values of the memristors converge to a fixed value.

Keywords: Nonlinear Systems and Control, Robust and H-Infinity Control, Feedback Control Systems
Abstract: We present incremental stability analysis of nonlinear feedback systems based on the recent notion of scaled relative graph (SRG). The essence of our proposed analysis is that the separation of SRGs of two open-loop systems on the complex plane implies closed-loop incremental stability. The main result generalizes the existing SRG separation theorem for stable open-loop systems which was based on a critical assumption of a chordal property. By comparison, our analysis allows possibly unstable open-loop systems and does not require the chordal assumption. Moreover, an L2e-counterpart to an SRG, referred to as a hard SRG, is proposed in parallel, based upon which we recover the existing incremental passivity theorem.

Keywords: Nonlinear Systems and Control, Stability
Abstract: Dynamic multipliers can be used to establish the input/output stability of a Lurye system but there exist examples in the literature where Lipschitz continuity is lost. In this extended abstract we discuss what properties dynamic multipliers still guarantee and the implications for practical applications. Several questions remain open: not least whether the existing examples also imply loss of continuity.

Keywords: Computations in Systems Theory, System Identification, Linear Systems
Abstract: We present a novel data-driven reformulation of iterative SVD-rational Krylov algorithm (ISRK), in its original formulation a Petrov-Galerkin (two-sided) projection-based iterative method for model reduction combining rational Krylov subspace (on one side) with Gramian/SVD based subspaces (on the other side). We show that at each step of ISRK, we do not necessarily require access to the original system matrices, but only to input/output data. In this context, data represent samples of the system’s transfer function, evaluated at particular values (frequencies). Numerical examples illustrate the efficiency of the new data-driven formulation.

Keywords: Machine Learning and Control, Linear Systems, Neural Networks
Abstract: A state-space model (SSM) uses linear, time-invariant (LTI) systems to model complicated time series data, such as weather and language. Given a training dataset, one applies a gradient-based algorithm to train an SSM in the frequency domain. To make training efficient, one leverages a simplified (e.g., diagonal) structure of the state matrices in the LTI systems. The latent states need to be meaningful; hence, the LTI systems are often initialized by a particular class of matrices called the HiPPO matrices, which store the Legendre coefficients of the input into the state vector. Unfortunately, such matrices cannot be stably diagonalized. While earlier solutions propose to modify the initial HiPPO matrices, the modification leads to a non-smooth transfer function that causes instability in the frequency domain. In this work, we propose an alternative solution by approximately diagonalizing the original HiPPO matrices in a backward stable way. This method results in a robust model that performs better than the earlier ones on many large test datasets. Additionally, we monitor the evolution of the LTI systems throughout training. We make observations about the change in the transfer functions in different parts of the frequency space, which demonstrate how an SSM is trained and suggest possible future improvements in the model. We conclude by providing a new research direction of SSMs through the lens of the Hankel singular values, which leads to a more robust parameterization scheme of an LTI system using its Hankel operator.

Keywords: Numerical and Symbolic Computations
Abstract: Classical linear-subspace model reduction (MOR) techniques are widely used whenever speed-ups of parametric problems or real-time evaluations are needed. However, when the underlying problem at hand admits slowly decaying Kolmogorov n-widths, resulting reduced-order models of low-dimension can yield inaccurate results. To overcome this difficulty, MOR on manifolds has been introduced which uses nonlinear projections. In this talk, we provide a unifying geometric framework for MOR on manifolds, which allows us to geometrically understand and unify existing MOR techniques. Moreover, this includes structure-preserving MOR techniques for, e.g., Hamiltonian and Lagrangian systems.

Keywords: System Identification, Applications of Algebraic and Differential Geometry in Systems Theory, Linear Systems
Abstract: Frequency-domain system identification of SISO linear systems using least squares can be cast as sum-of-rational optimization problem. Prediction error methods based on thissum-of-rational least-squares objective suffer from nonconvexities, and may become stuck in local minima. Our work uses the sum-of-squares hierarchies of semidefinite programs based on the sum-of-rationals framework of (Bugarin, Henrion, Lasserre 2016) to find a convergent sequences of lower-bounds to the optimal mean-squared error. If the semidefinite program’s solution satisfies a rank condition, we can recover plants that are global optima for the mean squared error. The method can be extended towards identification in continuous-time and in closed-loop.

Keywords: System Identification, Linear Systems, Nonlinear Systems and Control
Abstract: We study the problem of identifying linear and nonlinear systems from data. In particular, we discuss the inference of models with certified stability properties. This includes asymptotic stability for linear systems as well as Lyapunov stability for nonlinear systems with quadratic nonlinearity. Moreover, we also study the identification of free dynamics for systems with trapping regions.

Keywords: System Identification, Linear Systems, Nonlinear Systems and Control
Abstract: This work introduces an approach for generating guaranteed stable reduced-order linear time-invariant (LTI) models in a completely automated, data-driven, and efficient manner. The presented method is based on the AAA algorithm for rational fitting of scalar transfer functions. We propose a computationally efficient multi-input multi-output extension of the AAA algorithm, and we combine the resulting algorithm with a post-processing stability enforcement step that is formulated in terms of a small convex problem. A numerical example from electronics engineering attests to the method’s effectiveness.

Keywords: Nonlinear Systems and Control, Port-Hamiltonian Systems
Abstract: This extended abstract suggests control approaches for the trajectory-tracking problem in two classes of weakly coupled electromechanical systems. To this end, we formulate these systems within the port-Hamiltonian framework. Then, the notion of a contractive port-Hamiltonian system is used to develop tracking control approaches. These methods, derived from the Interconnection and Damping Assignment Passivity Based Control approach, eliminate the need to solve partial differential equations and use coordinate transformations. We also investigate the effect of coupled damping on the transient performance of the closed-loop system. The applicability of the proposed approaches is shown through simulations in two electromechanical applications.

Keywords: Quantum Control, Stability, Nonlinear Systems and Control
Abstract: In closed quantum systems, if controlled systems are not strongly regular or there exists at least one eigenstate that is directly uncoupled to the target state, then such systems control will become degenerate cases. This paper proposes a quantum control method designed by an implicit Lyapunov function to transfer states of the quantum systems in these two degenerate cases, especially the method proposed is suitable for multi-control Hamiltonian systems. The state distance is selected as the Lyapunov function in this paper. Numerical simulation experiments on a 4-level system are done, and the experiment results indicate the effectiveness of the implicit Lyapunov control method proposed for degenerate cases and multi-control Hamiltonians.

Keywords: Port-Hamiltonian Systems, Infinite Dimensional Systems Theory, Dissipativity
Abstract: Port-Hamiltonian systems were recently extended to include implicitly defined energy and energy ports thanks to a (Stokes-)Lagrange subspace. Here, we study the equivalent port-Hamiltonian representations of two systems with damping, written using either a classical Hamiltonian or a Stokes-Lagrange subspace. Then, we study the Timoshenko beam and Euler-Bernoulli models, the latter being the flow-constrained version of the former, and show how they can be written using either a Stokes-Dirac or Stokes-Lagrange subspace related by a transformation operator. Finally, it is proven that these transformations commute with the flow-constraint projection operator.

Keywords: Port-Hamiltonian Systems, Infinite Dimensional Systems Theory, Operator Theoretic Methods in Systems Theory
Abstract: Very recently, Jacob and Laasri obtained strong results on the solvability and well-posedness of infinite-dimensional time-varying linear port-Hamiltonian systems with boundary control and boundary observation. In this paper, we complement their results by giving a relatively non-technical discussion on the solvability of linear, infinite-dimensional time-varying linear systems not necessarily of boundary control type. The presentation is an overview of two recent papers by the author. The focus is on port-Hamiltonian systems and the theory is illustrated on a system with a delay component in the state dynamics.

Keywords: Port-Hamiltonian Systems, Infinite Dimensional Systems Theory, Stability
Abstract: Port-Hamiltonian systems (pHS) provide a useful tool for modelling physical systems such as e.g. flexible beams within mechanical systems. We study the question of when a distributed port-Hamiltonian system is bounded-input bounded-output (BIBO) stable, continuing recent research on subtleties of this classical notion for infinite-dimensional systems. Analysing the transfer function of this system class, we provide sufficient conditions for BIBO stability for a sub-class of pHS.

Keywords: Nonlinear Systems and Control, Mechanical Systems, Applications of Algebraic and Differential Geometry in Systems Theory
Abstract: In this paper, we examine the exact linearization of configuration flat Lagrangian control systems in generalized state representation with p degrees of freedom and p−1 control inputs by quasi-static feedback of its generalized state. We formally introduce generalized Lagrangian control systems, which are obtained when configuration variables are considered as inputs instead of forces. This work presents all possible lengths of integrator chains achieved by an exact linearization with a quasi-static feedback law of the generalized state that allows for rest-to-rest transitions. We show that such feedback laws can be systematically derived without using Brunovský states.

Keywords: Nonlinear Systems and Control, Numerical and Symbolic Computations, Optimal Control
Abstract: A finite-horizon nonlinear optimal control problem is considered. Stat-quad duality is used to generate an equivalent problem with linear dynamics and running cost that is quadratic in state with an additional term that is nonlinear in newly introduced control state variables. The new problem form is used to obtain a representation of the value function in terms of staticization over a set of quadratic functions, where the coefficients of the quadratic functions consist of the solutions to certain ODEs. A novel numerical method is indicated for solution of the resulting staticization problem; the method leverages the low dimensionality of nonlinearity. An example is included.

Keywords: Optimal Control, Machine Learning and Control, Infinite Dimensional Systems Theory
Abstract: We establish a novel theoretical connection between optimization problems arising in scientific machine learning (SciML) and generalized Hopf formulas, which represent the solution to certain Hamilton-Jacobi partial differential equations (HJ PDEs). Namely, we show that solving certain regularized learning problems is equivalent to solving an optimal control problem and its associated HJ PDE. We leverage these connections to design new efficient training approaches for SciML based on existing HJ PDE solvers. As a first exploration of these connections, we consider linear regression problems and develop a new Riccati-based methodology, which demonstrates potential computational and memory advantages over conventional learning approaches.

Keywords: Optimal Control, Nonlinear Systems and Control
Abstract: We consider approximations of differential games in fractional order systems governed by Caputo differential equations. It is known that value functionals of past state trajectories are characterized as viscosity solutions of the Isaacs partial differential equations (PDEs) defined on that trajectory space. Motivated by the Euler approximation method of Caputo differential equations, we propose discrete-time approximations of the differential games. Using stability-type arguments of viscosity solutions with the generators of the path-dependent dynamic programming operators, we show that the discrete-time approximations converge to a viscosity solution of the Isaacs PDEs on the trajectory space as the size of the time-discretization goes to 0.

Keywords: Optimal Control, Nonlinear Systems and Control
Abstract: We consider a new class of repeated zero-sum games in which the payoff of one player is the escape rate of a dynamical system which evolves according to a nonexpansive nonlinear operator depending on the actions of both players. Considering order preserving finite dimensional linear operators over the positive cone endowed with Hilbert's projective (hemi-)metric, we recover the matrix multiplication games, introduced by Asarin et al., which generalize the joint spectral radius of sets of nonnegative matrices and arise in some population dynamics problems (growth maximization and minimization). We establish a two-player version of Mañé's lemma characterizing the value of the game in terms of a nonlinear eigenproblem. This generalizes to the two-player case the characterization of joint spectral radii in terms of extremals norms. This also allows us to show the existence of optimal strategies of both players.

Keywords: Optimal Control, Nonlinear Systems and Control
Abstract: Although numerical schemes exist for approximating the value of an optimal control problem, the curse-of-dimensionality limits their application in practice. In this paper, super- and subsolutions of a Hamilton-Jacobi equation are used to characterise upper and lower bounds of a corresponding value function that hold locally over sublevel, superlevel, or ‘thick’ level sets of the bounding functions. In reachability analysis, these bounds may facilitate the tractable computation of inner and outer approximations of reachable sets. The value of Mayer problems for time-varying, continuous-time nonlinear systems with input constraints are considered.

Keywords: Linear Systems, Systems on Graphs
Abstract: We implement and analyze the linear systems solver created by Koutis, Miller and Peng which allows to solve Lx = b, where L is the Laplacian of a weighted, undirected graph with n nodes and m edges, in almost-linear time O(mlogn). We obtain that the solver is close to theoretical performances, running in almost-linear time. However, it has a large hidden constant, which makes it noncompetitive against methods with higher asymptotic complexity in most test cases. When applied to solving discretized partial differential equations, we observe that The KMP solver is better than Conjugate Gradient.

Keywords: Mathematical Theory of Networks and Circuits
Abstract: This extended abstract characterizes the graphical properties of an optimal topology with minimal Laplacian energy under the constraint of fixed numbers of vertices and edges, and devises an algorithm to construct such connected optimal graphs.

Keywords: Mathematical Theory of Networks and Circuits, Discrete Event Systems, Computations in Systems Theory
Abstract: In this paper we aim to verify in Coq basic properties of compositional behavioural frameworks for dynamic systems. We analyse systems by their external behaviour rather than a state space model, and use the parallel composition operator to build complex systems from simpler subsystems. It is important that the composition is well-defined and avoids undesirable behaviour such as deadlocks. To ensure the our framework is correct, we formulate and prove the results in the proof assistant Coq. We consider discrete-time deterministic systems and timed-event systems, with a view to eventually proving results on a framework for hybrid systems.

Keywords: Mathematical Theory of Networks and Circuits, Neural Networks, Dissipativity
Abstract: It is shown that the port behavior of a resistor–diode network corresponds to the solution of a monotone deep equilibrium network (a neural network in the limit of infinite depth), giving a parsimonious construction of a neural network in analog hardware.

Keywords: Mathematical Theory of Networks and Circuits, Operator Theoretic Methods in Systems Theory, Systems on Graphs
Abstract: We consider transport processes on metric graphs with time-dependent velocities and show that, under continuity assumption of the velocity coefficients, the corresponding non-autonomous abstract Cauchy problem is well-posed by means of evolution families and evolution semigroups.

Keywords: Infinite Dimensional Systems Theory, Nonlinear Systems and Control, Stochastic Control and Estimation
Abstract: Koopman operators linearize nonlinear dynamics, making their spectral analysis crucial. Dynamic Mode Decomposition (DMD) is a popular method for this, but challenges arise due to the operator's infinite-dimensional action. These include spurious modes and verifying decompositions, especially in stochastic systems where Koopman operators assess observable expectations. Our approach incorporates variance to tackle these issues, using a DMD-like matrix to approximate residual and variance sums. This enables accurate spectral computation for stochastic Koopman operators and introduces variance-pseudospectra for statistical coherence. We validate with convergence results and practical examples, including neural recordings from mice, showcasing new insights beyond standard expectation-based dynamical models.

Keywords: Machine Learning and Control, Nonlinear Systems and Control, Stochastic Modeling and Stochastic Systems Theory
Abstract: We present an efficient approach to modeling dynamical systems with complex long-time behaviour by approximating the Koopman operator on reproducing kernel Hilbert spaces. To maintain computational efficiency, we employ low-rank approximation techniques based on random Fourier features (RFF). We present effective model validation techniques and illustrate the method's success using molecular dynamics simulation data.

Keywords: Nonlinear Systems and Control, Operator Theoretic Methods in Systems Theory, Stochastic Modeling and Stochastic Systems Theory
Abstract: We provide novel probabilistic error bounds on traditional and kernel-based Extended Dynamic Mode Decomposition based on both i.i.d. and ergodic sampling. In addition, we illuminate the RKHS invariance under the Koopman operator of stochastic systems.

Keywords: Operator Theoretic Methods in Systems Theory, Systems on Graphs, Machine Learning and Control
Abstract: Data-driven methods for the identification of the governing equations of dynamical systems or the computation of reduced surrogate models play an increasingly important role in many application areas such as physics, chemistry, biology, and engineering. Given only measurement or observation data, data-driven modeling techniques allow us to gain important insights into the characteristic properties of a system, without requiring detailed mechanistic models. However, most methods assume that we have access to the full state of the system, which might be too restrictive. We show that it is possible to learn certain global dynamical features from local observations using delay embedding techniques, provided that the system satisfies a localizability condition -- a property that is closely related to the observability and controllability of linear time-invariant systems.

Keywords: System Identification, Linear Systems, Large Scale Systems
Abstract: The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems, the main reason being the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. Until now, the situation where for large-scale systems, we (i) only have access to partial observations (i.e., measurements, as is very common for experimental data) or (ii) deliberately perform coarse graining (for efficiency reasons) has not been treated to its full extent. In this paper, we address the pitfall associated with this situation, that the classical EDMD algorithm does not automatically provide a Koopman operator approximation for the underlying system if we do not carefully select the number of observables. Moreover, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to massively increase the model efficiency. We also briefly draw a connection to domain decomposition techniques for partial differential equations and present numerical evidence using the Kuramoto--Sivashinsky equation.