Keywords: Lyapunov Methods, Computational Efficiency, Stability
Abstract: Recent efforts in the control community focused on developing methods to guarantee closed-loop stability of systems controlled by multilayer perceptron (MLP)-based policies. However, little attention has been paid to the computational demand of such controllers due to frequent network evaluations. In this paper, we address this challenge by proposing a dynamic Event-Triggering Mechanism (ETM) to reduce the computational burden. Specifically, we focus on the stabilization of discrete-time Lur’e systems with input saturation. The proposed strategy reduces the evaluation frequency of the layers of the neural controller while preserving stability guarantees. The ETM is constructed using Linear Matrix Inequality (LMI)-based conditions, which leverage the known properties of activation functions and employ Finsler’s lemma to reduce conservativeness. Numerical results demonstrate the effectiveness of the proposed method, achieving significant computational savings compared to state-of-the-art solutions.

Keywords: Lyapunov Methods, Dissipativity, Stability
Abstract: This paper proposes a novel framework for cascading supply rates to simultaneously certify positivity and stability globally of dynamical systems with interior equilibria by composing single Lyapunov functions. This paper employs asymmetrically scaled sectorial (ASSEC) supply rates introduced recently, which are associated with storage functions defined on semi-infinite spaces specifying system positivity. A strong benefit of the single Lyapunov function approach is the capability of maintaining the positive space restriction automatically without adding conservativeness. In contrast to gain-type dissipativity, the total supply rate of a cascade connection is not smaller than a feedback connection. Due to the space asymmetry reflected in the supply rates, end nodes need to be processed separately from non-end nodes. Inspired by this, this paper proposes a unique idea of combining feedback and cascade techniques for dealing with interconnections which cannot be solved by a typical feedback technique alone.

Keywords: Lyapunov Methods, Dynamical Systems Techniques
Abstract: For a stochastic approximation of a differential inclusion, results are given on almost sure boundedness of the approximate solutions and on their convergence to the chain recurrent part of the global attractor. The former involve bounding the stochastic approximation's variance using a Lyapunov-like function. The novelty of the latter is in the use of Lyapunov-like characterizations of Morse decompositions of the attractor.

Keywords: Lyapunov Methods, Feasibility and Stability Issues, Numerical Methods
Abstract: Discrete-time Control Barrier Functions (DTCBFs) are commonly used in the literature as a powerful tool for synthesizing control policies that guarantee the safety of discrete-time dynamical systems. However, the systematic synthesis of DTCBFs in a computationally efficient way is at present an important open problem. This article proposes a novel alternating-descent approach based on Sum-of-Squares programming to synthesize quadratic DTCBFs and corresponding polynomial control policies for discrete-time control-affine polynomial systems with semi-algebraic safe sets. We demonstrate the proposed method on a numerical case study.

Keywords: Switching Control, Lyapunov Stability Methods, Input-To-State Stability
Abstract: The notion of incremental input-to-state stability provides a powerful tool for the analysis and design of nonlinear systems. In this paper, we present conditions under which a class of discontinuous piecewise linear systems is incrementally input-to-state stable. The conditions arise from a common quadratic incremental Lyapunov function approach, and are formulated in terms of numerically tractable matrix inequalities. We derive an insightful frequency-domain counterpart of the matrix inequalities, which reveals that, essentially, the class of piecewise linear systems that is considered in this paper is incrementally input-to-state stable, if the underlying linear subsystems possess a strong passivity property.

Keywords: Nonlinear Cooperative Control, Lyapunov Methods, Stabilization
Abstract: This paper addresses the leader-following consensus problem for networked wave PDEs with a switching topology. Initially, a relatively ideal scenario in which the induced digraph remains weakly connected is considered. Subsequently, we extend the result to a more practical scenario, where the weak connectivity of the interaction topology is guaranteed only during certain disconnected time intervals, resulting from communication limitations among agents. Building on these two settings, a boundary control protocol is proposed, and a systematic analysis on consensus and convergence rates for both situations is presented. Numerical simulations validate the theoretical results.

Keywords: Lyapunov Methods, Bifurcation and Chaos, Dynamical Systems Techniques
Abstract: Unlike Hopf-bifurcations in deterministic systems were a pair of complex conjugate eigenvalues traverse the imaginary axis simultaneously, stochastic Hopf-bifurcations exhibit a sequential crossing of the eigenvalues. Thereby, instead of a limit cycle, a chaotic attractor is established that governs the dynamics during and after the change of stability. Exploiting novel simulation methods based on a cohomolgy between flows generated by stochastic and random ordinary differential equations we numerically study phenomenologically the generation and development of this chaotic attractor for the stochastic Duffing-van der Pol oscillator.

Keywords: Lyapunov Methods, Lyapunov Stability Methods, Stability
Abstract: Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. We consider the problem of data-based approximation of a given Lyapunov function using the principal eigenfunctions of the Koopman operator without explicitly computing the operator itself. We demonstrate the effectiveness of our algorithm through numerical examples.

Keywords: Lyapunov Methods, Stability, Dynamical Systems Techniques
Abstract: Switched systems are a family of dynamical systems where a switching rule indicates which system is "switched on". This rule can be dependent on time and/or position in the state space. Stability of switched systems is a property that is often investigated using the existence of one (or multiple) Lyapunov function(s). We develop an algorithm using a scattered approximation method to construct a Lyapunov function for switched systems, accompanied with stability results. The construction adapts a previous method for autonomous ODEs that uses meshfree collocation and quadratic programming.

Keywords: Lyapunov Methods, Stability, Dynamical Systems Techniques
Abstract: Lyapunov functions are an important tool for identifying the basin of attraction for attractors. Specifically, the connected component of a sublevel set of a Lyapunov function containing the attractor is a subset of its basin of attraction. One approach to construct a Lyapunov function is the continuous piecewise affine (CPA) method, which defines a Lyapunov function that is affine on each simplex of a predetermined triangulation. Our objective is to develop an algorithm to identify the largest connected sublevel set of a CPA Lyapunov function. A previous method has been developed for an equilibrium. However, CPA Lyapunov functions often fail to have a negative orbital derivative in a neighborhood mathcal{F} of the equilibrium. The objective of this paper is to develop an algorithm that identifies the largest subset of the basin of attraction, represented by mathcal{L}^{sup}_{V}, for such a CPA Lyapunov function ( V ) which has negative orbital derivative outside of mathcal{F}. We assume that a sublevel set (mathcal{L}^{inf}_{V}) which contains mathcal{F} is given, and our objective is to extend it to the larger sublevel set (mathcal{L}^{sup}_{V}).

Keywords: Lyapunov Stability Methods, Dynamical Systems Techniques, Lyapunov Methods
Abstract: One of the prominent approaches to the problem of Lyapunov function construction in autonomous ordinary differential equations is the continuous piecewise affine (CPA) method. This method uses linear programming to compute a CPA Lyapunov function that is defined over a specified simplicial complex in a compact subset of the basin of attraction. In this paper, we propose the use of single hidden layer neural networks to learn CPA Lyapunov functions with gradient descent. We demonstrate the approach numerically and provide an algorithm to verify that the learned CPA neural network is a valid Lyapunov function.

Keywords: Numerical Methods, Lyapunov Methods, Optimization and Scheduling
Abstract: Finite-time stability (FTS) of a differential equation guarantees that solutions reach a given equilibrium point in finite time, where the time of convergence depends on the initial state of the system. For traditional stability notions such as exponential stability, the convex optimization framework of Sum-of-Squares (SoS) enables computation of polynomial Lyapunov functions to certify stability. However, finite-time stable systems are characterized by non- Lipschitz, non-polynomial vector fields, rendering standard SoS methods inapplicable. To this end, we show that computation of a non-polynomial Lyapunov function certifying finite-time stability can be reformulated as feasibility of a set of polynomial inequalities under a particular transformation. As a result, SoS can be utilized to verify FTS and obtain a bound on the settling time. Numerical examples are used to demonstrate the accuracy of the conditions in both certifying finite-time stability and bounding the settling time.

Keywords: Stabilization, Applications of Observer Design, Robustness
Abstract: In this paper it is shown how a method for asymptotic stabilization of a multivariable nonlinear system via state-feedback suggested by Liberzon in the early 2000s can be enhanced so as to obtain a robust output-feedback controller, if the latter satisfies the invertibility condition introduced by Hirschorn in the late 1970s.

Keywords: Variable Structure Control and Sliding Mode, Disturbance Atténuation, Robustness
Abstract: In this paper a low-chattering modification of first-order dynamic output-feedback controllers based on a linear approximation close to the control goal is proposed. Stability is proven for the application to super-twisting control in absence of disturbances and unmodeled dynamics. Based on the method of harmonic balance a tuning method for the modification is determined that avoids chattering in presence of first-order unmodeled dynamics. Robust gain and frequency margins are derived for the proposed controller. Finally, the results are verified in simulations.

Keywords: Robustness, Lyapunov Methods, Disturbance Atténuation
Abstract: This paper presents a safe controller synthesis of discrete-time stochastic systems using Control Barrier Functions (CBFs). The proposed condition allows the design of a safe controller synthesis that ensures system safety while avoiding the conservative bounds of safe probabilities. In particular, this study focuses on the design of CBFs that provide flexibility in the choice of functions to obtain tighter bounds on the safe probabilities. Numerical examples demonstrate the effectiveness of the approach.

Keywords: Robustness, Lyapunov Stability Methods
Abstract: This paper presents a prescribed time controller formulation for a class of nonlinear systems having uncertainty in their dynamical model parameters and subject to bounded disturbances. An adaptive prescribed time controller is proposed in order to compensate parametric uncertainty and ensure that the error signals converge precisely to the origin exactly at the prescribed time. The convergence and boundedness of the closed-loop system are ensured via Lyapunov-based arguments. In order to test the effectiveness of the controller, extensive simulation studies are presented.

Keywords: Robustness, Model Based Control, Automotive Systems Marine Systems
Abstract: Autonomous systems often have the capability to anticipate disturbances, such as upcoming terrain or road conditions for mobile agents, through sensing or predictive technologies known as preview or lookahead. Since preview is often only available for a limited horizon, our approach is designed to be robust to unpreviewed disturbances beyond the horizon in addition to unpreviewable disturbances/uncertainties. Moreover, our approach presents a closed-form solution that explicitly incorporates input constraints; thus, our approach is guaranteed to maintain recursive feasibility. This approach improves on existing control barrier function designs by reducing conservatism yet ensuring safety in environments with unpreviewable disturbances, limited preview capability, and input delays.

Keywords: System Structure Identification, Robustness, Stabilization
Abstract: The problem of controlling nonlinear systems with unknown model is considered. A testing and design process that leads to the construction of robust state-feedback controllers for such systems is described. The framework presented in the note is based on the use of bang-bang controllers, and it applies to a wide range of nonlinear systems.

Keywords: Nonlinear Cooperative Control, Optimization and Scheduling, Lyapunov Methods
Abstract: This paper studies distributed feedback optimization for linear uncertain multiagent systems subject to a class of external disturbances with unknown frequencies. Specifically, the objective is to regulate the outputs of the multiagent system to a common value that minimizes a prescribed global cost function, a sum of local cost functions. A crucial strategy is to develop a distributed optimizer that generates reference signals for the low-level decentralized tracking controllers. For the control synthesis, we use the normal form to derive an internal-model-based control law for the reference tracking and disturbance rejection, and use an adaptive control technique to handle the unknown frequencies of the disturbances. The coupling between the optimizer and the physical control systems is dealt with by a composite Lyapunov function. It is shown that the proposed solution guarantees the boundedness of the closed-loop signals and ensures that the output of each agent converges to the desired minimizer for any initial state. A numerical example demonstrates the effectiveness of the proposed method.

Keywords: Switching Control, Lyapunov Methods, State Estimation and Applications
Abstract: This paper addresses the task of reference tracking for a class of switching affine systems. In contrast to existing work which uses common Lyapunov functions to ensure asymptotic tracking, the method proposed here is based on multiple Lyapunov functions to guarantee convergence. The underlying principle is further modified to obtain less conservative convergence conditions for state observers applied to the same class of switching systems. The advantage of multiple over common Lyapunov functions in reference tracking is also illustrated by a numeric example.

Keywords: Lyapunov Methods, Robotics, Robustness
Abstract: In this work, a position-constrained smooth robust controller formulation for a class of nonlinear systems, known as Euler--Lagrange systems, have been proposed. Specifically, the proposed nonlinear PI type controller ensures that, when the initial position tracking error signal starts below a pre--defined threshold, the tracking error signal will stay inside this region and asymptotically converge around the origin despite the presence of uncertainties, ensuring practical tracking of error signal without overshooting. Contrary to most robust controllers in the related literature, the proposed robust controller is differentiable, therefore the output of the controller is “smoother” compared to its counterparts. This smoothness property of the controller reduces the risk of stress on the mechanical system hence, improves the lifetime of the controlled system. The theoretical work is backed up by experimental studies performed on a two--link planar robot.

Keywords: Lyapunov Methods, Robustness, Mechatronic Systems
Abstract: Electro-hydraulic systems are an essential part of modern industry. Nonetheless, these systems have input constraints and must be controlled under model uncertainties and disturbances using a saturated control signal. In this study, a saturation compensation scheme is developed using an adaptive fuzzy logic based methodology via the use of Lyapunov based synthesis and analysis method. The proposed methodology ensures uniform ultimate boundedness of the error signals despite the uncertainties in the system dynamics. The effectiveness of the proposed methodology is demonstrated through numerical simulations.

Keywords: Passivity, Stabilization, Lyapunov Methods
Abstract: This study proposes an adaptive controller based on kinetic-potential energy shaping (KPES) for mechanical systems with uncertain parameters. The KPES method is a recently proposed approach to obtain a strict Lyapunov function. Since this method requires sufficient plant information, it cannot be applied when the system contains unknown parameters. We replace the unknown parameters in the input of KPES with their estimation and update it appropriately, thereby proposing an adaptive controller. Since the proposed method is based on a technique that provides a strict Lyapunov function, steady-state errors do not occur regardless of the estimates. The effectiveness of the proposed method is confirmed through numerical examples.

Keywords: Lyapunov Methods, Passivity, Robustness
Abstract: The update law in adaptive control schemes can be extended to include feedthrough of an error term. This reduces undesired oscillations of the calculated weights. When the σ-modification is used for achieving robustness against unstructured uncertainty, the gain of the feedthrough in the update law cannot be chosen arbitrarily without losing the guarantee for closed-loop stability provided by a Lyapunov-based analysis. Compared to our previous result, we show stability of the closed loop for a larger parameter-range for the gain of the feedthrough in the update law. This parameter-range includes a configuration for which the influence of the integration in the update law diminishes over time, i.e. for which the adaptation for large times is governed solely by the feedthrough in the update law. By initializing at zero, this allows for removing the integration from the update law, resulting in a static update law. For the purely linear case, the adaptation acts like a disturbance observer. Frequency-domain analysis of the closed loop with a second order plant shows that removing the integration from the update law with σ-modification and feedthrough affects how precisely disturbances in the low-frequency band are observed. If the damping injected into the adaptation process by the σ-modification exceeds certain bounds, then the precision is increased by using the static update law.

Keywords: Lyapunov Methods, Feasibility and Stability Issues, Robustness
Abstract: This paper addresses the critical challenge of developing data-driven certificates for the stability and safety of unmodeled dynamical systems by leveraging a tree data structure and an upper bound of the system's Lipschitz constant. Previously, an invariant set was synthesized by iteratively expanding an initial invariant set. In contrast, this work iteratively prunes the constraint set to synthesize an invariant set -- eliminating the need for a known, initial invariant set. Furthermore, we provide stability assurances by characterizing the asymptotic stability of the system relative to an invariant approximation of the minimal positive invariant set through synthesis of a discontinuous piecewise affine Lyapunov function over the computed invariant set. The proposed method takes inspiration from subdivision techniques and requires no prior system knowledge beyond Lipschitz continuity.

Keywords: Lyapunov Methods, Numerical Methods, Switching Control
Abstract: We present a new method to compute the minimum average dwell-time needed to assert the global exponential stability of the equilibrium at the origin for a switched linear system. The method attempts to compute compatible Lyapunov functions for the individual subsystems of the switched system using linear programming or linear matrix inequalities optimization problems. We test the method on four examples to demonstrate its applicability.

Keywords: Lyapunov Stability Methods, Lyapunov Methods, Stability
Abstract: Dayawansa and Martin proved in 1999 that locally Lipschitz continuous and homogenous Lyapunov functions for a switched linear systems can be smoothed to C^infty Lyapunov functions retaining the homogeneity. Their proof used some rather advanced concepts in differential geometry. In this paper we give a more elementary proof and, additionally, show that our smooth Lyapunov function and its orbital derivatives approximate the original Lyapunov function and its orbital derivatives arbitrary close and that the smoothing technique preserves symmetry of the Lyapunov functions. These additional properties of the smooth Lyapunov function are useful, for example, when studying numerical methods to compute Lyapunov functions. Finally, our proof works for switched nonlinear systems, provided the individual subsystems have globally Lipschitz continuous right-hand sides.

Keywords: Stabilization, Lyapunov Methods
Abstract: We present an approach to finding globally stabilizing feedback controllers to the equation dot{phi} = f(phi) +g(phi)u, where f,g are (positively) homogeneous functions of degree d_f and d_g, respectfully. We show that there exists a globally stabilizing homogeneous feedback controller of degree d_f-d_g iff there exists a globally stabilizing homogeneous neural network feedback controller of degree d_f-d_g and there exists a homogeneous neural network global control Lyapunov function. We then provide numerical examples of finding these neural network controllers (and control Lyapunov functions). In all examples we use satisfiability modulo theories (SMT) solvers to prove that the found controllers are globally stabilizing.

Keywords: Lyapunov Stability Methods, Adaptive Observers, Parameter Estimation
Abstract: We revisit the problem of designing a strict Lyapunov function for systems appearing in model-reference adaptive control. Even though there exist well-established conditions for uniform exponential stability (for the linear case) and uniform global asymptotic stability (for the nonlinear case) in the literature, and some works provide strict Lyapunov functions, most of them are based on the assumption that the regressor is smooth and its derivative admits a (known) uniform bound. Our main result consists in relaxing this assumption, which is notably useful in tracking control tasks, where it is required to track arbitrarily rapid reference trajectories. Our main statement is formulated for linear time-varying systems with regressors of dimension one, but we show how it applies in the analysis of nonlinear systems; notably in adaptive control and adaptive observer design.

Keywords: Passivity, Control of Sampled Data Systems
Abstract: In this paper, our goal is to provide a sampled-data control scheme to achieve input consensus with average output regulation for a system under unknown constant disturbance. In the continuous-time case, a controller solving the problem is given for a Krasovskii or shifted passive system. We show that its adequate temporal discretization also achieves our control objective for a Krasovskii or shifted passive discrete-time system.

Keywords: Observer and Filter Design By Observer Error Linearization, Applications of Observer Design, Observability and Observer Design
Abstract: In this paper, the idea of using approximate discretization of continuous-time nonlinear systems to get an exact linear (or state affine) representation up to input/output injection suitable for estimation problems is considered. This provides a simple practical observer approach, only subject to the discrete approximation. Some classes of systems for which the technique can be applied are exhibited, and illustrations are provided via a couple of concrete examples, either in hydroelectric production on the one hand, or in pipeline water transport on the other hand.

Keywords: Optimal Control, Computational Efficiency
Abstract: This paper presents a real-time nonlinear quadratic trajectory tracking control (NLQTC) method for polynomial nonlinear state-space systems. Building on prior regulator designs, a stacked feedback control structure is developed, incorporating an additional term to enable trajectory tracking. The proposed approach guarantees fixed computational costs per control cycle, making it suitable for real-time applications. Validation on a nonlinear mass-spring-damper system demonstrates effective trajectory tracking. Comparative simulations against nonlinear model predictive control (NMPC) show that NLQTC achieves equivalent tracking accuracy while significantly improving computational efficiency.

Keywords: Small Gain Theorems, Discrete Events, Input-To-State Stability
Abstract: Characterizations of the input-to-state stability (ISS) of an infinite-dimensional discrete-time system are investigated. In particular, the ISS is equivalent to UBIBS and UAG and equivalent to UBIBS and ULIM. Afterwards, we consider a network of two interconnected systems and examine a small-gain condition for the ISS of the whole system. The proof of our small-gain result is different from the finite-dimensional case as it uses a different (from finite- dimensional case) characterization of the ISS property.

Keywords: Model Based Control, Quantized Feedback and Feedback with Communication Constraints, Optimal Control
Abstract: This paper presents an innovative approach to optimal input quantization for control design with saturation for a Lotka-Volterra system with harvest. First a passivity- based state feedback control design is carried out showing the local input-to-state stability of the desired operation point against quantization errors using a Lyapunov-based analysis and design approach. The associated input quantization problem is then formulated as a multi- objective mixed-integer optimal control problem, namely keeping the number of quantization levels as low as possible while ensuring fast convergence with as few changes in the input signal as possible. In particular, the paper explores particle swarm optimization techniques to find a robust quantization with respect to initial conditions provided a given probability density distribution of the initial conditions and using the passivity-based controller as initial guess for the optimizer. The design methodology is illustrated using numerical simulations.

Keywords: Dynamical Systems Techniques, Bifurcation and Chaos
Abstract: In this study, we numerically explore the dynamics of the nonlinear Helmholtz oscillator subject to fractional-order damping. To conduct our numerical analysis, we utilize the Grünwald-Letnikov fractional derivative algorithm for numerical simulations. Our focus is on understanding how the inclusion of a fractional derivative in the dissipative term influences system behavior as a function of the parameter alpha. We find that alpha plays a pivotal role in either triggering or suppressing chaotic motions. On the other hand, to regulate particle confinement within the well, we implement a phase control technique within a parametric term, a method frequently employed in practical applications. In the standard (non-fractional) scenario, prior research has established that an optimal phase difference of approximately phi_{OPT} approx pi effectively prevents particle escape. Here, we extend this investigation by considering both the phase difference phi and the fractional parameter alpha as control variables, identifying the conditions under which particles escape. Our findings highlight the robustness of phase control, as the optimal phase difference phi even when varying the fractional parameter alpha. Finally, we present preliminary numerical results on phase-controlled escapes in cases of non-resonant frequencies, where omega_{2}/omega_{1} neq 1.

Keywords: Lyapunov Stability Methods, Lyapunov Methods, Biological and Biomedical Systems
Abstract: Generalized Ribosome Flow Models (GRFMs) are nonlinear compartmental systems which can be used to model the flow of objects (e.g., ribosomes, vehicles) in a networked environment. Contraction analysis is a fundamental tool in examining the long term qualitative behaviour of dynamical systems. However, computing appropriate contraction metrics is known to be challenging even in low dimensional cases. In this paper, the contractivity of GRFMs is studied. It is known that the Jacobian of strongly connected GRFMs with in- and/or outflows from/to the environment is point-wise diagonally stable in the state space. This motivates us to search the contraction metric in a state-dependent positive diagonal form. Therefore, we propose a computational approach based on convex optimization to compute a contraction metrics. The contraction inequality is reformulated as a sufficient polytopic linear matrix inequality (LMI). To reduce the conservatism of the LMI, we follow a so-called S-variable approach. Namely, we introduce slack variables accompanied by affine matrix functions, called annihilators decoding the algebraic coupling between the nonlinear terms in the state equations. The results are illustrated on a 4-dimensional GRFM.

Keywords: Stabilization, Lyapunov Methods
Abstract: In nonlinear control theory, systems that do not satisfy Brockett's theorem are extremely important because they cannot be stabilized by linear state-feedback laws alone. For such systems, feedback laws that include discontinuous or time-varying elements are generally constructed, while the systematic design of such laws is still a challenging issue. As a third method, we consider the use of stabilization by noise, a method that involves the artificial application of white noises. However, since the resulting systems are represented by stochastic differential equations, the asymptotic stability for ordinary differential equations is not achieved. Therefore, in this paper, we define stochastic stability for a neighborhood of the origin in a form similar to practical stability and introduce a minimum probability that the target subset is invariant, thereby constructing a quantitative evaluation criterion for stability. Then, we derive a state-feedback law based on stabilization by noise for a chained system, which is a typical example that do not satisfy Brockett's theorem.

Keywords: Adaptive Observers, Lyapunov Stability Methods
Abstract: Dynamic consensus pertains to the case in which a group of interconnected systems adopt a consensual dynamic behaviour; this covers in particular the more common consensus problem in which case all the systems admit a stable common equilibrium point. In this article we tackle the problem of dynamic consensus for systems under measurement bias. The latter may result, e.g., from sensor faults or attacks, but it is assumed that bias is constant. Under this condition we design a nonlinear adaptive estimator that successfully estimates the bias, so the controller ensures dynamic consensus. Our results are designed for linear controllable systems, but the class of systems that we consider is broader than in the available literature since, in particular, it covers systems with singular matrices (e.g., chains of integrators). The only assumption on the graph topology is that it is directed and contains a spanning tree. For illustration purpose, we address a case-study of formation control of a group of satellites, whose orbital relative motion is modelled as per Clohessy-Wiltshire equations, under the assumption that the measurements are biased.

Keywords: Smart Structures, Robotics, Mechatronic Systems
Abstract: Flexible manipulators have become more relevant because of the push towards lightweight robotics to reduce energy consumption and larger work spaces. This paper proposes a decoupled position and vibration control strategy for a flexible manipulator in robotics, where the manipulator's position is controlled by an active feedback loop of the joint angle while a passive control strategy does the vibration control. This passive control strategy involves a piezoelectric layer bonded to the flexible manipulator with its electrodes shunted by an impedance, a strategy not yet explored for flexible manipulators. The investigated shunt circuits are the linear and nonlinear oscillator circuits. No feedback loop or sensor is required for the vibration control, as the piezoelectric layer and impedance are self-sensing. It is shown that the position and vibration control strategies can be decoupled, where the manipulator is assumed to be rigid for the position control and flexible for the vibration control. The purpose of this strategy is to have a robust position and vibration control, regardless of changes in the manipulator's flexible behavior. The global asymptotic stability of the combined control strategy is studied. Furthermore, the control performance of the flexible manipulator for different payloads is investigated, and it is found that a nonlinear shunt circuit is more robust.

Keywords: Switching Control, Stabilization, Lyapunov Methods
Abstract: We address the stabilisation problem for the interconnection obtained by a linear system cascaded with a hybrid system by developing a forwarding method. We first propose a hybrid version of a (dual) Sylvester equation, the solution of which is characterised by an explicit expression. Then, we show that the solution can be used to transform the cascade interconnection into an upper-triangular form, which can be stabilised by a state-feedback controller. The stability of the hybrid system after transformation is proved via Lyapunov theory. We finally illustrate the results by means of a numerical example.

Keywords: Switching Control, Stabilization, Lyapunov Methods
Abstract: In this paper, we propose a stabilizing control approach for interconnected Switched Affine Systems (SASs). Over decades, several methodologies have been presented in the literature for the control of individual SASs, but not much attention has been paid to the topic of systems represented by an interconnection of different SASs. In the present paper, we focus on a network characterized by SASs coupled by a linear system. It is shown here that the proposed control design approach has attractive numerical properties and that the obtained controller has a decentralized structure, meaning that only local measurements on each SAS are necessary. The effectiveness of the proposed technique is illustrated by simulation with an example of interconnected DC/DC power converters.

Keywords: Stability, Dynamical Systems Techniques, Lyapunov Stability Methods
Abstract: This paper introduces a novel approach to evaluating the asymptotic stability of equilibrium points in both continuous-time (CT) and discrete-time (DT) nonlinear autonomous systems. By utilizing indirect Lyapunov methods and linearizing system dynamics through Jacobian matrices, the methodology replaces traditional semi-definite programming (SDP) techniques with computationally efficient linear programming (LP) conditions. This substitution substantially lowers the computational burden, including time and memory usage, particularly for high-dimensional systems. The stability criteria are developed using matrix transformations and leveraging the structural characteristics of the system, improving scalability. Several examples demonstrated the computational efficiency of the proposed approach compared to the existing SDP-based criteria, particularly for high-dimensional systems.

Keywords: Anti-Control and Synchronization, Dynamical Systems Techniques, Complex Network
Abstract: It is well-known that some chaotic oscillators are impossible to synchronize by using diffusive couplings when the `wrong output' is measured in the systems and the resulting coupling signal is applied at the `wrong input'. Here, by means of an example, we discuss a potential solution for this problem, which is based on the design of a vanishing dynamic perturbation. Specifically, we consider a triplet of identical Rössler oscillators, for which we show that it is impossible to induce synchronization via diffusive couplings. Then, a first-order dynamic perturbation is designed such that the systems are enforced to synchronize. Once synchronization is achieved, the effect of the dynamic perturbation vanishes.The obtained results are numerically illustrated and experimentally validated using electronic circuits.

Keywords: Dynamical Systems Techniques, Biological and Biomedical Systems, Anti-Control and Synchronization
Abstract: This article introduces the problem of robust event disturbance rejection. Inspired by the design principle of linear output regulation, a control structure based on excitable systems is proposed. Unlike the linear case, contraction of the closed-loop system must be enforced through specific input signals. This induced contraction enables a steady-state analysis similar to the linear case. Thanks to the excitable nature of the systems, the focus shifts from precise trajectory tracking to the regulation of discrete events, such as spikes. The study emphasizes rejecting events rather than trajectories and demonstrates the robustness of the approach, even under mismatches between the controller and the exosystem. This work is a first step towards developing a design principle for event regulation.

Keywords: Lyapunov Methods, Dissipativity, Dynamical Systems Techniques
Abstract: This paper proposes asymmetric dissipativity tools for feedback connections of supply rates to synthesize dynamical systems with interior equilibria on semi-infinite state spaces. The idea is to simultaneously establish positivity and stability of a dynamical system by explicitly constructing a single storage function which is a Lyapunov function defined exclusively on the positive orthant. The tools developed recently for asymmetrically scaled sectorial supply rates are extended to a generalized framework covering a larger class of systems, including saturation nonlinearities. This paper overcomes fundamental difficulties of asymmetry that does not arise in popular supply rates, which leads us to systematic and simple tools for users. Their effectiveness and usefulness are illustrated through an example in biology. This paper proposes asymmetric dissipativity tools for feedback connections of supply rates for asymmetric storage functions to synthesize dynamical systems with interior equilibria on semi-infinite state spaces. The idea is to simultaneously establish positivity and stability of a dynamical system by explicitly constructing a single storage function which is a Lyapunov function defined exclusively on the positive orthant. The tools developed recently for asymmetrically scaled sectorial supply rates are extended to a generalized framework covering a larger class of systems, including saturation nonlinearities. This paper shows unique techniques to overcome fundamental difficulties of asymmetry that do not arise in the proofs for popular supply rates, which successfully leads us to systematic and simple tools for users. Their effectiveness and usefulness are illustrated through an example in biology.

Keywords: Geometric Methods, Dynamical Systems Techniques
Abstract: Recently it has been shown that the property of forward-flatness for discrete-time systems, which is a generalization of static feedback linearizability and a special case of a more general concept of flatness, can be checked by two different geometric tests. One is based on unique sequences of involutive distributions, while the other is based on a unique sequence of integrable codistributions. In this paper, the relation between these sequences is discussed and it is shown that the tests are in fact dual. The presented results are illustrated by an academic example.

Keywords: Dynamical Systems Techniques, Hybrid Nonlinear Systems, Discrete Events
Abstract: Stochastic hybrid systems are dynamic systems that undergo both random continuous-time flows and random discrete jumps. Depending on how randomness is introduced into the continuous dynamics, discrete transitions, or both, stochastic hybrid systems exhibit distinct characteristics. This paper investigates the role of uncertainties in the interplay between continuous flows and discrete jumps by studying probability density propagation. Specifically, we formulate stochastic Koopman/Frobenius-Perron operators for three types of one-dimensional stochastic hybrid systems to uncover their unique dynamic characteristics and verify them using Monte Carlo simulations.

Keywords: Small Gain Theorems, Input-To-State Stability, Dynamical Systems Techniques
Abstract: This paper introduces a small-gain-based method for the construction of barrier functions for a class of large-scale nonlinear systems subject to external inputs. The generation of input-to-state safe (ISSf) barrier function for the total system is based on the ISSf-barrier functions of individual subsystems, under cyclic-small-gain conditions. A numerical example is provided to show the effectiveness of our proposed methodology based on the integration of the cyclic-small-gain and ISSf-barrier function techniques.

Keywords: Complex Network, Stability
Abstract: We develop a compartmental model that describes the evolution of the popularity of various channels hosted on a digital platform. Free users on the platform pick a channel based on its popularity, and after interacting with it, may go back to the platform list or may get redirected/recommended to another channel. These channels could be YouTube channels or Twitch streamers. All these channels are competitive, as time spent by users on one channel implies less time spent on another. However, in practice, it has been observed that many channels support and collaborate with one another, i.e., redirect users to other channels. In this work, we first analyze the compartmental model and then try to explain this seemingly irrational behavior using game theory.

Keywords: Control of Sampled Data Systems, Stability, Model Based Control
Abstract: Co-design of safety controllers are constructed upon a novel compensation scheme between sampling period and control gain for a class of nonlinear planar system to relax the requirement of sufficiently small sampling period imposed in previous studies. Under the co-designed and sampled-data controllers, not only transient safety for system state in stabilization problem is guaranteed but also flexibility of selecting different sampling periods are further enhanced which is much preferred in the digital implementation of obtained controllers. Numerical studies conducted under different sampling periods using Matlab illustrate effectiveness of our co-designed and sampled-data safety controllers.

Keywords: Stability, Lyapunov Methods
Abstract: Determination of Lyapunov functions has been one of the most fundamental challenges in the analysis and control of nonlinear systems. There has been significant recent interest in using neural networks to compute Lyapunov functions. Often, the loss function is modified to encourage the Lyapunov conditions, and the architecture of the neural network is also adapted to satisfy certain Lyapunov conditions by design. However, it remains unclear whether all conditions can be strictly verified and whether such modifications are necessary. In this paper, we investigate this question. We show that, under the usual assumption that the equilibrium point is exponentially stable, a standard feedforward neural network trained using the usual Lyapunov conditions can be easily modified after training to become strictly verifiable, e.g., using satisfiability theory modulo solvers. We demonstrate the results with numerical examples.

Keywords: Stability, Input-To-State Stability
Abstract: Achieving desired set-point tracking is a fundamental challenge in control theory. Within this context, low-gain integral control methods stand out as an attractive design tool in many applications thanks to their effective and straightforward implementation. We present here a generalization of existing low-gain integral control techniques, which allows for the plant to exhibit complex behavior. For instance, we allow the plant to exhibit limit cycles. In these settings, we guarantee average set-point tracking, in place of ``standard'' set-point tracking. Our result is unifying in the sense that we recover known results from the literature, and we provide several generalizations. We showcase our theoretical results through a numerical example, where we show how to control the average output of a nonlinear oscillator.

Keywords: Stability, Lyapunov Methods, Power Systems
Abstract: We estimate the lock-in domain of the origin of a current control system which is used in common DC/AC inverter designs. The system is a cascade connection of a 4-dimensional linear system (current controller, CC) followed by a two-dimensional nonlinear system (phase-locked loop, PLL). For the PLL, we construct a Lyapunov function via numerical approximation of its level curves. In combination with the quadratic Lyapunov function of the CC, it forms a vector Lyapunov function (VLF) for the overall system. A forward-invariant set of the VLF is found via numerical application of the comparison principle. By LaSalle's invariance principle, convergence to the origin is established.

Keywords: Nonlinear Model Predictive Control Theory and Applications, Stabilization, Aerospace
Abstract: This paper introduces a novel cascade control structure with formal guarantees of uniform almost global asymptotic stability for the state tracking error dynamics of a quadcopter. The proposed approach employs a model predictive control strategy for the outer loop, explicitly addressing the non-zero total thrust constraint. The outer-loop controller generates an acceleration reference, which is then converted into attitude, angular velocity and acceleration references, subsequently tracked by a nonlinear inner-loop controller. A numerical case study validates the proposed scheme, highlighting its ability to track fast trajectories with small error.