Keywords: Nonlinear Cooperative Control, Aerospace
Abstract: This paper addresses the distributed formation control problem of networked satellites operating in cooperative and competitive settings. We propose a distributed bipartite formation controller that guarantees connectivity maintenance among cooperative satellites while preventing collisions between interconnected satellites. We design a gradient-based control law using barrier-Lyapunov functions to ensure that the constraints on the inter-satellite distances are satisfied. Moreover, we establish asymptotic stability of the bipartite formation manifold for almost all initial conditions satisfying the required constraints. Finally, the proposed approach is illustrated through a numerical simulation.

Keywords: Bifurcation and Chaos, Control of Mechanical, Electrical and Process Systems
Abstract: In this paper, the vibration control of the Duffing system is studied. We specifically address the problem of using feedback control to explore the natural vibration behavior of the Duffing system (stable or bistable) under harmonic excitation. The controller should be designed to be simultaneously stabilizing and noninvasive in cases where its reference signal corresponds to a natural vibration response of the uncontrolled system. Available approaches in the literature rely on displacement and velocity measurements despite accelerometers being the most common sensors used for vibration testing. The focus of this paper is on proposing an approach for noninvasive vibration control of the Duffing system using only accelerometers whereas accelerations measurement is a priori insufficient for state estimation. In particular, we show that acceleration feedback is sufficient for noninvasive controllability of the Duffing system, and in this regard, we propose an acceleration-based control strategy that guarantees the convergence of the system response (including displacement, velocity, and acceleration) to a desired possible natural vibration response of the system. The results are validated by mathematical analysis and numerical simulation.

Keywords: Dynamical Systems Techniques, Control of Mechanical, Electrical and Process Systems
Abstract: This study explores the complex dynamics of two coupled systems, where the driver system influences the response system through coupling and additional periodic forcing. Resonances such as transmitted resonance, coupling-induced resonance, and coupling-forcing resonance are examined for their individual and combined effects on the response system's amplitude and frequency. The interplay of these phenomena is analyzed under various conditions, with particular focus on cases involving time-delayed oscillators. Additionally, coupling introduces an exchange of information between the systems, therefore a time-delayed should be taken into account for the information to travel from an oscillator to the other. Here, we propose to model the delay into the driver system so that it can propagate its delay through the entire coupled system, generating delay induced behavior into the non delayed response system. Finally, we think that the findings reveal how coupling strength, resonance dynamics, and information delays interact, offering insights into synchronization and resonance behavior in coupled systems.

Keywords: Mechatronic Systems, Control of Mechanical, Electrical and Process Systems, Nonlinear Modeling of Lumped And/Or Distributed Parameter Systems
Abstract: This paper presents an algorithm for the iterative control of a 2D array of memory piezoactuators which are used in a novel Hysteretic Deformable Mirror (HDM) Huisman et al. (2021). The HDM is composed by a mirror thin plate placed on top of piezoelectric wafers interlaced with electrodes and exhibited output remanence behavior (e.g., the presence of remnant memory when the input is signal is set to zero). The control objective in the HDM is to achieve and hold a desired mirror surface deflection by regulating the remnant output of the piezoactuators. Following recent iterative control method to achieve desired remnant output for a single piezoactuator, we present the generalization of this approach to control the remnant output of 2D array of piezoactuators with a time-multiplexed control input. Particularly, we present conditions for the convergence of the 2D output remnant to a desired 2D reference values. We show the efficacy of our method in a simulation of an HDM with 5 × 5 piezoactuators.

Keywords: Control of Mechanical, Electrical and Process Systems, Mechatronic Systems, Robustness
Abstract: Due to the high modeling effort, nonlinearities, and parameter uncertainties of hydraulic systems, manually tuned PID-controllers are still state of the art in industrial applications thanks to their simplicity and robustness. To allow more precise control, however, the controller parameters need to be adapted depending on the current working point. Since accurate physical models are typically not available, data-based models approximating the input-output behavior can be used for the adaptation. In this work, the adaptive PID-control approach of Cetin and Iplikci (2015) is extended such that it can be used to stably control a hydraulic differential cylinder with two inputs and two outputs. The controller parameters are adapted based on a gradient descent approach which minimizes the estimated future control error. For the output prediction, a nonlinear autoregressive model with exogenous inputs is used. A thorough simulative study shows that the proposed extended method gives satisfactory results in terms of transient behavior and disturbance compensation.

Keywords: Stability, Control of Mechanical, Electrical and Process Systems, Lyapunov Methods
Abstract: We propose a neural control method to provide guaranteed stabilization for mechanical systems using a novel negative imaginary neural ordinary differential equation (NINODE) controller. Specifically, we employ neural networks with desired properties as state-space function matrices within a Hamiltonian framework to ensure the system possesses the NI property. This NINODE system can serve as a controller that asymptotically stabilizes an NI plant under certain conditions. For mechanical plants with colocated force actuators and position sensors, we demonstrate that all the conditions required for stability can be translated into regularity constraints on the neural networks used in the controller. We illustrate the utility, effectiveness, and stability guarantees of the NINODE controller through an example involving a nonlinear mass-spring system.

Keywords: Dynamical Systems Techniques, Passivity, Disturbance Atténuation
Abstract: Passivity-based control is a widely applicable framework examined by many researchers. A vast amount of techniques exist, where often a proposed passive target system yields a control input to force the uncontrolled system to behave as the target system. It accomplishes such objective by reshaping the potential and kinetic energy. The kinetic energy is the most challenging one to reshape, because the target is not clear: kinetic energy influences the rate of convergence i.e. the rate at which a system reaches its equilibrium but a quantification remains difficult. This study tackles the problem of a benchmark system's kinetic energy being advantageously reshaped based on a target system. The foundation of this study will be physical insight, which will serve as the basis for the development of four target systems.

Keywords: Dynamical Systems Techniques, Robustness, Performance Issues
Abstract: Symbiotic control integrates fixed-gain and adaptive learning architectures to mitigate the effect of system uncertainties. Specifically, fixed-gain control alone requires knowledge of uncertainty bounds to achieve predictable closed-loop system behavior, while adaptive learning alone functions without such bounds but may introduce unpredictability in closed-loop system performance. To this end, symbiotic control harnesses the predictability of fixed-gain control and the adaptability of learning mechanisms to create a framework that does not rely on knowledge of system uncertainties. This paper studies symbiotic control of dynamical systems with nonparametric system uncertainties. First, we robustify the nonparametric symbiotic control framework by introducing a low-pass filter to its fixed-gain control architecture. Second, using grid-based optimization, we compute optimal parameters for the proposed fixed-gain architecture that maximizes the delay margin and minimizes the H∞ norm. In particular, when adaptive learning is off, the delay margin is calculated from the loop transfer function and the H∞ norm is derived from a transfer function with uncertainty as input and user-defined states as output. Then, the adaptive learning mechanism is added to the control loop with low gains to improve the closed-loop system performance gradually. Finally, two illustrative numerical examples are also given to demonstrate the efficacy of the proposed nonparametric symbiotic control framework.

Keywords: Dynamical Systems Techniques, Stabilization, Lyapunov Methods
Abstract: This paper presents a novel event-triggering framework to schedule state and control data transmissions, where our contribution is twofold. First, the norm-free event-triggering concept, which offers a distinct advantage over traditional norm-based methods by significantly reducing the number of data transmissions, is generalized from linear to nonlinear systems for the first time in the literature, specifically to schedule state data transmissions. Second, since the application of norm-free scheduling for both state and control data transmissions simultaneously is challenging as elaborated in the paper, a unique configuration is introduced in which norm-free state data scheduling is blended with norm-based control data scheduling. The proposed overall framework is first analyzed within a nonlinear setting and then the analysis for the linear setting is given as a special case when a nonlinear system is linearized to operate closely around an equilibrium point of interest. In addition to our system-theoretical analyses, an illustrative numerical example is also presented for the purpose of demonstrating the efficacy of the proposed event-triggering framework.

Keywords: Filter Design, State Estimation and Applications, Dynamical Systems Techniques
Abstract: This paper presents an invariant Kalman filter for estimating the relative trajectories between two dynamic systems. Invariant Kalman filters formulate the estimation error in terms of the group operation, ensuring that the error state does not depend on the current state estimate—a property referred to as state trajectory independence. This is particularly advantageous in extended Kalman filters, as it makes the propagation of the error covariance robust to large estimation errors. In this work, we construct invariant Kalman filters to the trajectory of one system relative to another. Specifically, we show that if the relative dynamics can be described solely by relative state variables, they automatically satisfy state trajectory independence, allowing for the development of an invariant Kalman filter. The corresponding relative invariant Kalman filter is formulated in an abstract fashion and is demonstrated numerically for the attitude dynamics of a rigid body.

Keywords: Dynamical Systems Techniques
Abstract: Classic control techniques typically rely on a model of the system's response to external inputs, which is difficult to obtain from first principles especially if the unknown dynamics are nonlinear. In this paper, we address this issue by presenting an approach based on the so-called signature transform, a tool that is still largely unexplored in data-driven control. We first show that the signature provides rigorous and practically effective features to represent and predict system trajectories. Furthermore, we propose a novel use of this tool on an output-matching problem, paving the way for signature-based, data-driven predictive control.

Keywords: Dynamical Systems Techniques, System Inversion
Abstract: The problem of the inversion of injective nonlinear maps when the measurements are affected by a periodic noise with zero mean is addressed by a combination of symbolic (Grobner basis) and numerical techniques (continuous–time Newton method). Six different continuous– time Newton algorithms are proposed pointing out advantages and drawbacks.

Keywords: Numerical Methods, Stability
Abstract: In this paper it is shown how to approximate rational first integrals, if any, of planar polynomial systems using only sampled measurements of their trajectories, and how to use them to characterize the overall behavior of the system in the plane, especially the stability properties of equilibrium points. To pursue this goal, first the properties of rational first integrals are characterized. Then, it is shown how to use first integrals to characterize the behavior of the polynomial system in the Poincare disk. Finally, it is shown how to determine rational first integrals, if any, in both a model-driven and a data-driven setting.

Keywords: Optimal Control, Stability, Discrete Events
Abstract: This paper proposes a novel data-driven control method to address the problem of adaptive optimal control for a class of discrete-time nonlinear systems with input time-delay. Q-learning and policy iteration (PI) are integrated to compute the approximate optimal control policy without any prior knowledge of the system dynamics. We have theoretically shown that the sequence generated by the Q-learning and PI is nonincreasing, and converges monotonically to the optimal control policy. Simulation results for the Van der Pol oscillator demonstrate the optimal control problem of nonlinear system with input time-delay can be addressed by the proposed approach.

Keywords: Stability, Small Gain Theorems, I/O Stability
Abstract: This paper studies a set-theoretic generalization of Lyapunov and Lagrange stability for abstract systems described by set-valued maps. Lyapunov stability is characterized as the property of inversely mapping filters to filters, Lagrange stability as that of mapping ideals to ideals. These abstract definitions unveil a deep duality between the two stability notions, enable a definition of global stability for abstract systems, and yield an agile generalization of the stability theorems for basic series, parallel, and feedback interconnections, including a small-gain theorem. Moreover, it is shown that Lagrange stability is abstractly identical to other properties of interest in control theory, such as safety and positivity, whose preservation under interconnections can be thus studied owing to the developed stability results.

Keywords: Stability, Stabilization, Lyapunov Methods
Abstract: In this paper, we study the set-point tracking problem for nonlinear systems for which tracking is possibly achievable on multiple equilibria. We first show that designing a controller enforcing 2-contraction is sufficient to solve this problem, provided that trajectories remain bounded. Then, we focus on unitary relative degree systems in normal form, and we design a controller using high-gain tools. We provide sufficient conditions to solve the problem, under the assumption that the zero dynamics is weakly non-minimum phase and 2-contractive. We validate our design on a frequency set-point tracking problem taken from power systems.

Keywords: Stability
Abstract: Incremental stability has become increasingly common in the context of nonlinear control. However, there is a lack of understanding of the difference between several notions of incremental stability. The objective of this paper is to show that the notion of incremental global asymptotic stability is very similar to that of incremental global exponential stability. In particular, nonexponential relaxation of the convergence rate appears only "locally" but not "globally." Some examples are provided to justify our theoretical arguments.

Keywords: Stabilization, Robustness, Lyapunov Methods
Abstract: This paper revisits the adaptive stabilization of linear uncertain systems and develops the analysis to multi-agent network frameworks. Firstly, by adopting a more general stabilizing direction and introducing a sigma modification based control design, a robust adaptation law is demonstrated to effectively handle uncertainties in the model. To enable the adaptation law design more flexible, the cancellation constraint associated with a key matrix therein is relaxed. Additionally, a novel Lyapunov function is constructed to establish sufficient conditions for stability, which are crucial for ensuring robust synchronization in multi-agent systems. Finally, a simulation example is provided to validate the proposed approach, highlighting its novelty and practical effectiveness.

Keywords: Optimal Control, Robustness, Parameter Estimation
Abstract: In this study, we consider a set-based adaptive reinforcement learning framework for the design of optimal control systems for a class of nonlinear systems with unknown dynamics. Assuming that the system has access to the measurement of a set of basis functions, the proposed set-based approach is shown to guarantee convergence to a neighbourhood of the optimal value function. In contrast to existing nonlinear reinforcement learning technique, the proposed approach does not require a parametrization of the unknown dynamics. The dynamics are estimated using a nonparametric learning techniques inspired by extremum seeking control techniques. In particular, a timescale transformation approach is proposed to estimate the drift and control component of each basis functions. The stability of the proposed control system is guaranteed if the trajectories of the system meet a persistency of excitation condition. A simulation study is conducted to demonstrate the effectiveness of the proposed approach.

Keywords: Parameter Estimation, Computational Efficiency, Small Gain Theorems
Abstract: Many problems arising in system identification, controller design, and model reduction can be formulated as nonlinear optimization problems, where the cost function depends on steady-state system responses. Solving such problems often requires the computation of numerous steady-state system responses and their gradients with respect to parameter changes, posing significant computational challenges. In this article, we focus on discrete-time nonlinear multiple-input, multiple-output Lur'e-type systems that exhibit unique, bounded, and globally exponentially stable steady-state solutions. Our main contribution is a computationally efficient mixed-time-frequency (MTF) algorithm for the computation of the steady-state response of Lur'e systems with guaranteed numerical convergence. Additionally, we propose a method for efficiently computing gradients of the cost function with respect to parameter changes, again leveraging the proposed MTF algorithm. The proposed tools offer substantial computational advantages as the underlying nonlinear optimization problems require extensive steady-state response calculations and gradient evaluations. We validate our approach with a benchmark system identification problem, demonstrating a 95% reduction in computation time.

Keywords: Parameter Estimation, Lyapunov Stability Methods, Input-To-State Stability
Abstract: In this paper, the adaptive output regulation problem is revisited for a class of nonlinear systems with an uncertain linear exosystem and a polynomial steady-state input. To address such a problem, a novel adaptive regulator design framework is proposed with a pre-processing internal model used as a filter for the purpose of establishing a steady-state linear regression equation (LRE). A gradient-based estimator then can be constructed to identify the uncertain parameters in the LRE. Thus, by appropriately incorporating such an estimator, an adaptive regulator can be developed and shown to solve the regulation problem asymptotically.

Keywords: Parameter Estimation, Quantized Feedback and Feedback with Communication Constraints
Abstract: This paper addresses the issue of parameter estimation in Cyber-Physical Systems where binary outputs are subject to data tampering attacks. A novel recursive identification algorithm is designed to estimate unknown parameters under tampered binary outputs, ensuring asymptotically convergent parameter estimation. Theoretical guarantees are provided for the algorithm's performance, including mean-square convergence, almost sure convergence, and convergence rate. Numerical simulations show the effectiveness of the proposed method.

Keywords: Nonlinear Model Predictive Control Theory and Applications, Control of Mechanical, Electrical and Process Systems, Model Based Control
Abstract: Active load balancing in battery cells has significant advantages in energy storage systems and electric vehicles (EVs). However, diversity in balancing network power electronics components, energy transfer paths, etc., leads to a multitude of architectures and topologies, each yielding varying performance, complexity, and cost. Therefore, this work implements and compares the load balancing performance of (i) series, (ii) module, and (iii) layer-based adjacent battery cell network topologies, each managed by a different nonlinear model predictive control (NMPC) strategy. The solution is built on the dynamics of active balancing networks and battery cells and incorporates high-fidelity balancing currents to characterize the load using state of charge (SOC). The simulation results, which are performed on twelve adjacent cells, reveal that load balancing in cells connected in layer-based topology is achieved in 1430 s. While series and module-based topologies achieved the load balancing objective in 2980 and 2770 s, respectively

Keywords: Nonlinear Model Predictive Control Theory and Applications, Model Based Control
Abstract: In this paper, we design offset-free nonlinear Model Predictive Control (MPC) for surrogate models based on Extended Dynamic Mode Decomposition (EDMD). The model used for prediction in MPC is augmented with a disturbance term, that is estimated by an observer. If the full information about the equilibrium of the real system is not available, a reference calculator is introduced in the algorithm to compute the MPC state and input references. The control algorithm guarantees offset-free tracking of the controlled output under the assumption that the modeling errors are asymptotically constant. The effectiveness of the proposed approach is showcased with numerical simulations for two popular benchmark systems: the van-der-Pol oscillator and the four-tanks process.

Keywords: Nonlinear Model Predictive Control Theory and Applications, Optimal Control, Transportation Systems
Abstract: The safety of Autonomous Vehicles (AVs) remains a key challenge for developers and stakeholders due to the dynamic environments in which they operate. From a control perspective, nonlinear model predictive control (MPC) is a powerful control strategy to operate in such dynamic environments. This paper evaluates the performance and compares the safety criteria for AV controllers designed using MPC and control barrier functions (CBFs) that ensure safety in obstacle avoidance through the principle of set invariance. With CBFs, there are two design approaches: CBFs formulated as a discrete-time constraint (MPC-CBF) or as a quadratic program (MPC-CBF-QP). In addition to CBFs, a more straightforward approach for obstacle avoidance is to use the safe-distance constraints in the MPC formulation (MPC-DC). The results of these three nonlinear MPC control strategies: (I) MPC-DC, (II) MPC-CBF-QP, and (III) MPC-CBF, implemented on an actual vehicle in autonomous mode, are discussed in detail. Their performance and safety conditions are compared in three common urban and highway driving scenarios: (i) avoiding static obstacles, (ii) sudden pedestrian interaction, and (iii) overtaking the lead vehicle (LV). We experimentally and theoretically show that the MPC-CBF-QP is computationally efficient and guarantees safety in all considered scenarios. MPC-DC often fails to meet safety requirements in critical scenarios, and MPC-CBF performance heavily depends on prediction, control, and CBF horizon.

Keywords: Nonlinear Model Predictive Control Theory and Applications, Stability, Model Based Control
Abstract: Extended dynamic mode decomposition (EDMD) is a popular data-driven method to predict the action of the Koopman operator, i.e., the evolution of an observable function along the flow of a dynamical system. In this paper, we leverage a recently-introduced kernel EDMD method for control systems for data-driven model predictive control. Building upon pointwise error bounds proportional in the state, we rigorously show practical asymptotic stability of the origin w.r.t. the MPC closed loop without stabilizing terminal conditions. The key novelty is that we avoid restrictive invariance conditions. Last, we verify our findings by numerical simulations.

Keywords: Adaptive Observers, Applications of Observer Design, Model Based Control
Abstract: This work investigates the passivity-based control of a class of underactuated mechanical systems with series-elastic actuator for which the actuator states are not measured. The main contribution is a new design of the integral interconnection-and-damping assignment passivity-based control that employs a dynamic extension to estimate the actuator states. Numerical simulations demonstrate the effectiveness of the new controller.

Keywords: Control of Sampled Data Systems, Numerical Methods, Geometric Methods
Abstract: The motivation for this paper is the implementation of nonlinear state feedback control, designed based on the continuous-time plant model, in a sampled control loop under relatively slow sampling. In previous work we have shown that using one-step predictions of the target dynamics with higher order integration schemes, together with possibly higher order input shaping, is a simple and effective way to increase the feasible sampling times until performance degradation and instability occur. In this contribution we present a unifying derivation for arbitrary orders of the previously used Lobatto IIIA collocation and Hermite interpolation schemes through the Hermite-Obreschkoff formula. We derive, moreover, an IDA-PBC controller for a magnetic levitation system, which requires a non-constant target interconnection matrix, and show experimental results.

Keywords: Lyapunov Stability Methods, Stabilization, Power Systems
Abstract: The paper proposes an analytical nonlinear control approach for enhancing the transient performance of lossy multi-machine power systems. The prominent feature of this approach lies in the inclusion of damping assignment, whereby the oscillation of the power system frequency is dampened out more effectively. In this regard, the proposed control approach is said to achieve enhanced transient stabilization, which is desirable in practical applications. A case study demonstrates the enhanced performance of the proposed control approach over existing nonlinear control approaches.

Keywords: Passivity, Nonlinear Modeling of Lumped And/Or Distributed Parameter Systems, Robotics
Abstract: This paper addresses the position and anti-drift control of large-scale curling Hydraulically Amplified Self-Healing Electrostatic (HASEL) actuators using the Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) methodology formulated within the port-Hamiltonian systems (PHS) framework. Expanding upon previous work on low-scale models, this study adapts the control strategy to large-scale systems, ensuring its effectiveness across scales. The proposed control law retains its structure from the low-scale case, with dimensionality remaining constant despite the increased system complexity. As in the low-scale setting, the closed-loop system mitigates the drift. Numerical simulations confirm the methodology’s effectiveness, demonstrating its capability to achieve precise position control and mitigate drift in large-scale systems.

Keywords: Nonlinear Modeling of Lumped And/Or Distributed Parameter Systems, Experiment Design, Computer Algebra-Based Methods and Tools
Abstract: Many control courses focus on linear systems, behaviour and control, whereas the real world is nonlinear. This paper introduces learning resources on nonlinear systems which staff can use in introductory courses to help students appreciate this fundamental point, without being overloaded by analytical and mathematical detail. Hence the focus is on interactive resources which display simple animations and plots while allowing users to make small parameter changes and observe the impact on behaviour and control. This paper reviews some of the new resources for non-linear systems in detail and summarises all of them alongside a discussion of how they might be embedded into a first, or indeed later, course in systems, modelling and control. The resources have been embedded into the MATLAB control101 toolbox..

Keywords: State Estimation and Applications, Nonlinear Modeling of Lumped And/Or Distributed Parameter Systems, Filter Design
Abstract: This paper presents a real-time capable algorithm for the learning of Gaussian Processes (GP) for submodels. It extends an existing recursive Gaussian Process (RGP) algorithm which requires a measurable output. In many applications, however, an envisaged GP output is not directly measurable. Therefore, we present the integration of an RGP into an Extended Kalman Filter (EKF) for the combined state estimation and GP learning. The algorithm is successfully tested in simulation studies and outperforms two alternative implementations -- especially if high measurement noise is present. We conclude the paper with an experimental validation within the control structure of a Vapor Compression Cycle typically used in refrigeration and heat pumps. In this application, the algorithm is used to learn a GP model for the heat-transfer values in dependency of several process parameters. The GP model significantly improves the tracking performance of a previously published model-based controller.

Keywords: Nonlinear Modeling of Lumped And/Or Distributed Parameter Systems, Transportation Systems, Lyapunov Methods
Abstract: This paper investigates the traffic congestion control problem for a two-lane traffic flow system consisting of a fast lane and a slow one, in which lane-changing behaviors are present and there exists the inflow fluctuation due to the on-ramp at the upstream boundary of the slow lane. The macroscopic traffic dynamics are described by a two-lane Aw-Rascle-Zhang (ARZ) traffic model, consisting of fourth-order partial differential equations (PDEs). A proportional-integral (PI) boundary feedback control law, which is implemented by regulating the outlet VSLs and a ramp metering at the inlet boundary, is designed to suppress the disturbances and to drive the traffic states of both lanes to desired states. Then sufficient conditions for ensuring the finite-gain L_2 stability of the two-lane traffic flow system are developed in terms of linear matrix inequalities (LMIs) by employing the Lyapunov function method. Finally, the proportional and integral gains are obtained by solving an optimization problem under LMI constraints, and the effectiveness of the proposed conditions is validated by numerical examples.

Keywords: Nonlinear Modeling of Lumped And/Or Distributed Parameter Systems
Abstract: We study port-control for metriplectc systems. Using the well-known representation of metriplectic systems as dissipative Hamiltonian systems with the exergy as Hamiltonian function, the corresponding port-controlled Hamiltonian systems are considered as exergy-controlled metriplectic systems. The applicability of the machinery of port-Hamiltonian systems theory, in particular interconnection of exergy controlled metriplectic systems, is studied.

Keywords: Dynamical Systems Techniques
Abstract: We study the two-sided moment matching problem for time-varying systems connected to an explicit signal generator and an explicit filter. We first provide the notion of moment in a time-varying setting for both the standard and the swapped interconnection. By leveraging these notions of moment, we derive reduced-order models that achieve two-sided moment matching. Finally, we illustrate the result on a benchmark example.

Keywords: Geometric Methods, Numerical Methods
Abstract: Model reduction simplifies complex dynamical systems while preserving essential properties. This paper revisits a recently proposed system-theoretic framework for least squares moment matching. It interprets least squares model reduction in terms of two steps process: constructing a surrogate model to satisfy interpolation constraints, then projecting it onto a reduced-order model. Using tools from output regulation theory and Krylov projections, this approach provides a new view on classical methods. For illustration, we reexamine the least-squares model reduction method by Lucas and Smith, offering new insights into its structure.

Keywords: Nonlinear Modeling of Lumped And/Or Distributed Parameter Systems, Passivity
Abstract: In this paper, we develop a scalable lumped-parameter model of a flexible structure. We opt for the port-Hamiltonian framework for the representation due to its innate capability of structure-preservation for power-preserving interconnections. We utilize a canonical transformation to reduce the computational burden associated with symbolic computations. Based on the physical discretization, we then validate the applicability of models of different orders on an example of cantilever beam. We also propose a port-Hamiltonian structure-preserving generalized balanced truncation approach for further reduction of the order.

Keywords: Numerical Methods, Nonlinear Modeling of Lumped And/Or Distributed Parameter Systems, Computational Efficiency
Abstract: In this paper, we consider model order reduction (MOR) methods for problems with slowly decaying Kolmogorov n-widths as, e.g., certain wave-like or transport-dominated problems. To overcome this Kolmogorov barrier within MOR, nonlinear projections are used, which are often realized numerically using autoencoders. These autoencoders generally consist of a nonlinear encoder and a nonlinear decoder and involve costly training of the hyperparameters to obtain a good approximation quality of the reduced system. To facilitate the training process, we show that extending the to-be-reduced system and its corresponding training data makes it possible to replace the nonlinear encoder with a linear encoder without sacrificing accuracy, thus roughly halving the number of hyperparameters to be trained.

Keywords: Algebraic Methods, Feasibility and Stability Issues, Lyapunov Methods
Abstract: This paper addresses a zero-sum differential game for stabilizing a nonlinear control system subject to external disturbances, described by an ordinary differential equation. The synthesis of optimal feedback controls requires solving the Bellman-Isaacs equation. An algebraic method, generalizing existing approaches, is proposed to solve this equation efficiently. Numerical simulations demonstrate the application of the method to a technical control system, such as a synchronous generator.

Keywords: Algebraic Methods, Observability and Observer Design
Abstract: This paper employs the polynomial/transfer function approach for nonlinear systems in order to develop a method for finding the right fraction description of nonlinear system directly from its state equations. Then necessary and sufficient conditions for nonlinear system to be accessible and observable are given in terms of such a description.

Keywords: Observability and Observer Design, Algebraic Methods
Abstract: The paper studies the observability property of nonlinear discrete-time control systems, which depend explicitly on an unknown disturbance variable. Two notions, disturbance and state observability, which correspond to the possibility of expressing the disturbance or state as a function of system output, input and a finite number of their future values, are introduced and studied. Necessary and sufficient conditions are found for disturbance and state observability in two cases: when no additional assumptions are made and when the disturbance is assumed to be constant. Then, two methods for state and/or disturbance estimation are proposed. Finally, the theory is illustrated by two examples.

Keywords: Stability, Linear Algebraic Methods
Abstract: Nonlinear systems with stabilisable linear approximations can be stabilised by~linear feedback. The so-called theorem of the stability at linear approximation provides an estimate of the region of attraction via Lyapunov's method. The estimate can be increased by applying additional linear damping provided by an LgV controller. In this study, we quantify the increase by a single parameter, namely the gain of the LgV controller. In particular, we provide a necessary and sufficient condition for the increase, and we give bounds for the maximal achievable increase.