Keywords: Nonlinear dynamics, Biology, Stability
Abstract: We consider a neural field model consisting of a network of Wilson-Cowan nodes. Each node has an excitatory population and an inhibitory population with homeostatic adjustment of the inhibitory to excitatory coupling strength. The nodes are connected via through distributed, time delayed coupling between their excitatory populations. We show that this excitatory coupling can induce multiple Hopf bifurcations and determine how the connectivity matrix influences the phase-locked oscillations produced in these bifurcations. We show that interaction between different Hopf bifurcations can lead to desynchronization and more complex solutions, such as intermittent synchronization. We study the effect of different time distributions on these results.

Keywords: Nonlinear dynamics, Biology, Stability
Abstract: The initiation and regeneration of pulsatile activity is a ubiquitous feature observed in excitable systems with delayed feedback. In neuronal systems, persistence of such pulsatile activity, in the form of trains of action potentials (or spikes), is linked to working memory and is generated via a combination of the intrinsic excitability properties of individual neurons and the dynamics of synaptic connections between cells. In previous work, we showed using a combination of numerical simulation, bifurcation theory and closed loop patch clamp (dynamic clamp) experiments the ability of an isolated, self-coupled neuron to `store' distinct, repetitive spiking patterns akin to memories. The former used a simplified description of the synaptic dynamics that ignored the effects of synaptic plasticity, which have been shown to be a key contributing factor to memory formation and consolidation in neuronal networks. Here, we extend the previous study to account for two mechanisms of short term plasticity. The first describes short-term depression of the strength of the self-coupling (autapse) via neurotransmitter depletion; the second captures short-term facilitation due to increases in Ca2+ concentrations associated with spiking behaviour. We explore the role played by the relative timescales of the depression and facilitation compared to the neuron's refractory time and the delay time in the existence and stability of different spiking solutions.

Keywords: Approximation techniques and numerical methods, Infinite dimensional systems, Biology
Abstract: Reproduction numbers play a fundamental role in population dynamics, as they can be used to address control strategies. In this talk, we focus on age-structured models, that can be formulated as integro-partial differential equations with nonlocal boundary conditions or, equivalently, as delay equations via integration along characteristics. For these models, reproduction numbers are characterized as the spectral radius of compact infinite-dimensional operators, which are obtained by suitably splitting the linearization of the model around an equilibrium into birth and transition. We present a general numerical method to approximate the reproduction numbers for a large class of age-structured population models with finite age span. We allow for complete flexibility in the choice of the birth and transition processes by reformulating the problem in a space of absolutely continuous functions via integration of the age-state, so that processes described by boundary conditions can be seen as perturbations of an operator with trivial domain, and pointwise evaluation is well-defined. Then, we discretize the birth and transition operators via pseudospectral collocation and we approximate the reproduction numbers through the spectral radius of matrices. Numerical results attesting the validity of the approach and applications to epidemiological models are discussed.

Keywords: Nonlinear dynamics, Stability, Biology
Abstract: We study an extension of a compartmental SEIR model by including a TTI strategy (test, trace, isolate). Here, infected individuals are identified with some testing rate and then together with their contacts are quarantined for a fixed period of time, resulting in a system of delay differential equations. We carry out combined analytical and numerical investigations of the dynamics.

Keywords: Stability, Biology, Infinite dimensional systems
Abstract: We give new conditions for the asymptotic stability of the zero solution of non-autonomous linear scalar delay differential equations with non-negative coefficients. Furthermore, sufficient conditions for the divergence of nontrivial solutions are also given. The results naturally generalize to equations with multiple delays with non-negative coefficients. We illustrate with specific examples how our theorem is different from each of the seven other criteria previously found in the literature.

Keywords: Nonlinear dynamics, Infinite dimensional systems, Biology
Abstract: We investigate the dynamics of neural fields on a spherical domain, incorporating gap junctions and transmission delays within a two-population model of excitatory and inhibitory neurons. By introducing diffusion terms to represent gap junctions, the work explores their interplay with synaptic delays, analysing the impact on pattern formation and oscillatory behaviour. Utilizing equivariant bifurcation theory for Hopf bifurcations, the analysis predicts the emergence and stability of periodic orbits with spherical symmetry. Numerical simulations support the theoretical findings, revealing that enhanced diffusion in excitatory neurons can stabilize higher-order oscillatory modes, offering new insights into neural field dynamics.

Keywords: Manufacturing, Robotics, Filtering and estimation
Abstract: In-process measurement of cutting forces is strategic in robotic milling with the aim of increasing the performances of the cutting process in terms of productivity, dimensional accuracy and machined surface quality, even when machining light alloys, e.g. for aerospace applications. This target is very challenging because every dynamometer transmissibility depends on the robotic arm configuration, which is continuously varying during the process. In this work the new concepts of Universal Dynamometer and Universal inverse Filter will be implemented for a robotic milling application and tested through a preliminary modal analysis, showing the potential of such approach for a correct force reconstruction under these challenging conditions.

Keywords: Regenerative chatter, Stability, Modeling and identification
Abstract: Milling processes can experience harmful chatter vibrations that limit productivity and surface quality. Variable pitch milling tools can improve the stability properties of machining processes. In this paper, a hardware-in-the-loop experimental setup is used to test the chatter suppression capability of a variable pitch cutter, which is modeled by multiple-delayed oscillators.

Keywords: Numerical simulation, Manufacturing
Abstract: Studying stability, steady-state solutions, and transient behavior in stochastic turning models presents significant numerical challenges. One of the popular methods is statistically evaluating numerous trajectories. However, we propose a faster, more precise direct integration approach focusing on the mean square of the process to assess variation around the average solution.

Keywords: Regenerative chatter, Manufacturing, Nonlinear dynamics
Abstract: Undesired chatter vibrations are still a challenging problem affecting most industrial machining operations. Linear stability analysis of milling dynamics is generally preferred for a fast calculation of the stability lobes diagram, which are useful for process optimization. Nevertheless, this approach may lead to important errors with respect to the more accurate but also more time consuming non-linear time domain simulations of milling dynamics. This is especially true in peripheral milling, where the contact between each active tooth and the workpiece is highly interrupted. In this work, a breakthrough non-linear analytical model of tooth-workpiece contact will be introduced, which will provide a more accurate representation of the effective perturbation of milling dynamics with respect to the steady-state periodic solution.

Keywords: Approximation techniques and numerical methods, Numerical simulation, Manufacturing
Abstract: A numerical method with linear time complexity is presented for solving time-periodic delay differential equations with external forcing in order to compute the Floquet multipliers and the corresponding periodic orbits. The solution approach is based on the system's direct integration and the affine mapping's numerical eigenvalue calculation.

Keywords: Modeling and identification, Stability, Regenerative chatter
Abstract: This study introduces an experimental approach for identifying chatter vibrations that arise during typical milling operations. The core concept involves capturing the dominant spectral characteristics of the transient vibrations generated during milling, offering insight into its dynamic behaviour and delivering a quantitative measure of stability.

Keywords: Stability, Robustness, Nonlinear dynamics
Abstract: We characterize input-to-state stability (ISS) for nonlinear time-delay systems (TDS) in terms of stability and attractivity properties for systems with inputs. Using the specific structure of TDS, we obtain much tighter results than those possible for general infinite-dimensional systems. We underline the differences in the ISS characterizations for TDS with those for finite-dimensional systems.

Keywords: Decentralized and distributed control, Infinite dimensional systems, Lyapunov functionals and Lyapunov matrices
Abstract: This paper explores an extension of sampled-data distributed control method for coupled nonlinear fractional order reaction-advection-diffusion (FRAD) systems with time varying delays. Herein, the sampled-data stabilization is under investigation by time-delayed sampled-data controller and sampled-data in space and time controller, respectively. Lyapunov-Krasovskii functional and fractional derivative's property are used to make the controlled system asymptotically stable for the case of time-delayed sampled-data control strategy. The obtained sufficient conditions are related to delay and space interval. Using Jensen's inequality and Lyapunov-Krasovskii functional, the sufficient conditions of asymptotic stability for the case of sampled-data in space and time control strategy are also derived, which are dependent on delay, space interval and time interval.

Keywords: Stability, Networked and multi-agent systems
Abstract: In epidemic networks, delays can stem from intra-population factors, like recovery time, and inter-population factors, such as transition times. These delays can destabilize the network. This study investigates the stability of a linear epidemic Susceptible-Infectious-Susceptible (SIS) network dynamics with two delays, to explain the impact of epidemic rates on stability maps.

Keywords: Stability, Traffic flow analysis and control, Nonlinear dynamics
Abstract: We consider the stability of the non-zero equilibrium state for the viscous Burgers equation with a delay effect. The linear stability is analyzed by using the characteristic equation of the corresponding eigenvalue problem. If our equation does not have a delay effect, the characteristic equation is given by a polynomial equation. On the other hand, if our equation has a delay effect, the characteristic equation becomes a transcendental equation, and it is difficult to analyze it. In this situation, we apply the useful known result concerned with the characteristic equation for the ordinary delay differential equations and try to get the sharp stability condition for the viscous Burgers equation with delay.

Keywords: Stability, Infinite dimensional systems, Control of linear systems
Abstract: We study the stability of a linear system with periodic delay that is rapidly varying around a nominal value. We are inspired by [1], where the stabilizing effect of the rapidly-varying periodic delay on the system stability was qualitatively studied via analysis of a suitable comparison system with distributed delays, but without providing explicit upper bounds on the delay period which guarantees stability of the original system. Here we give explicit bounds on the delay period under which the exponential stability of the original system is preserved. We find these bounds via a state transformation that explicitly relates the original and comparison systems and by assuming that we know an exponential bound on the norm of fundamental matrix of the comparison system.

Keywords: Stability, Control of linear systems, Robustness
Abstract: A general class of linear time-invariant (LTI) systems with two independent time delays is considered in the characterization of the stability switching curves (SSCs). We show that the exact frequency sweeping intervals of the crossing frequency set (CFS) of this class of systems is determined by the Jacobian discriminant through the turning point theory. Moreover, we prove the invariance of root crossing directions over the imaginary axis at frequencies within the same CFS interval, with directions computed by the determinant of the Jacobian. These results valid for two delay problems are paralleled with the single delay treatment by Walton and Marshall (1987). A benchmark example is provided to demonstrate the effectiveness of the proposed approach.

Keywords: Stability, Control of linear systems, PID control
Abstract: This note focuses on the design of a delay-based proportional integral (PI) control scheme, proposed as an alternative to the classical PI controller. By adopting a geometric approach, we establish tuning rules to achieve closed-loop stability, ensure a specific decay rate, and assess its fragility. Specifically, through the appropriate design of the delay parameter, the proposed controller facilitates improvement in settling time and reduction of overshoot, while maintaining control efforts lower than its classical counterpart. We present several numerical examples to illustrate the effectiveness of the proposed scheme.

Keywords: Spectrum computation and optimization, Infinite dimensional systems, Control of linear systems
Abstract: Systems governed by delay differential equations are distinguished by having infinite-dimensional dynamics. Moreover, in numerous applications, an accurate description leads to time-periodic models. To address the stabilization problem using a periodic output feedback controller, we minimize the spectral radius of the discretized monodromy operator. We propose two approaches for the controller design to impose smoothness on the feedback gain. In our first approach, we limit the total variation of the periodic feedback gain and impose approximate periodicity of the controller gain by adding regularization terms to the cost function. Conversely, in the second approach, the feedback gain is constrained to be characterized by a finite number of harmonics of its Fourier series representation.

Keywords: Control of linear systems, Infinite dimensional systems
Abstract: The talk deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time t is obtained as a linear combination of the control evaluated at time t and of the state evaluated at finitely many previous instants of time t-Lambda_1,dots,t-Lambda_N. Based on the realization theory developed by Y.~Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in L^q spaces, q in [1, +infty). We also provide a necessary and sufficient condition for L^1 exact controllability, which can be seen as the closure of the L^1 approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by dmax{Lambda_1,dots,Lambda_N}, where d is the dimension of the state space.

Keywords: Stability, Control of linear systems, Infinite dimensional systems
Abstract: The stability of a linear time-delay system can be characterized through the roots of its characteristic function, which is a quasipolynomial. Studying the location of the roots of a quasipolynomial in terms of its coefficients is a nontrivial question which has motivated several recent works. In particular, some works have highlighted a property known as multiplicity-induced-dominancy (MID): for some families of quasipolynomials, a root of maximal multiplicity is necessarily the rightmost one in the spectrum. In this contribution, we provide a new characterization of the characteristic quasipolynomial of a time-delay system with a single delay in the presence of a root of maximal multiplicity, showing that this function can be expressed by means of iterated integrals of an exponential kernel.

Keywords: Stability, Infinite dimensional systems, Control of linear systems
Abstract: This paper focuses on the output feedback stabilization of a chain of integrators using delay-difference operators to approximate derivative actions. The primary contribution of this study is the explicit computation of an upper bound for the time-delay, guaranteeing that the closed-loop system, employing the delay-difference approximation of the derivative action, maintains (exponential) stability. Illustrative numerical examples complete the presentation to demonstrate the efficiency of the proposed methodology.

Keywords: Stability, Control of linear systems, Spectrum computation and optimization
Abstract: In recent work, the multiplicity-induced-dominancy (MID) property for single delay time-delay systems has been fully characterized in the over-order case, that is when the multiplicity of the spectral abscissa exceeds the system's order. In this note, despite the fact that this assumption is not met, we provide an analytical proof for the MID. The coexistence of real spectral values is the main ingredient. The obtained result is illustrated through the stabilization of an unstable second-order plant by a delayed PD controller.

Keywords: Robust control, PID control, Control of linear systems
Abstract: Application of Robust Control Toolbox for Time Delay Systems implemented in the Matlab system to the oscillating plant with uncertain time delay and astatism using the D-K iteration and algebraic approach. The algebraic approach combines the structured singular value, algebraic theory and algorithm of global optimization solving remaining issues in structured singular value framework. The algorithm for global optimization can be alternated with direct search methods such as Nelder-Mead simplex method giving solutions for problems with one local extreme. As a global optimization method, Differential Migration is used proving to be reliable in solving this type of problems. The D-K iteration represents a standard method in the structured singular value theory. The results obtained from the D-K iteration are compared with the algebraic approach.

Keywords: Prediction based control, Robustness
Abstract: It is demonstrated that the recently proposed closed-loop robust predictor (see Nikiforov and Gerasimov (2023)) provides also safe numerical implementation of the state prediction-based controller used in the problem of finite spectrum assignment. This result is rigorously proved and illustrated by simulation results.

Keywords: Prediction based control, Adaptive control
Abstract: The design of an adaptive Smith predictor controller for control of industrial hydraulic processes subjected to the simultaneous effect of load disturbances and time-varying time-delay is developed in this paper. In order to improve the performance of the Smith predictor, an adaptive block is introduced into the control structure to estimate and update the current value of the time-varying time-delay. Furthermore, a disturbance compensator is introduced to reject the effect of load disturbances. Simulations of the control system are carried out with the proposed controller and with a classical Smith predictor. The comparison of the obtained results shows the higher performance of our proposed controller, both in rejecting load disturbances and in maintaining the closed-loop stability when the time-delay is time-varying

Keywords: Mechatronics, Control of linear systems, Modeling and identification
Abstract: Time delays in control systems are often present due to inherent delays of the instrumentation and signal processing computations. Among the proposed solutions for time-delay compensation in linear control systems, an important role is played by the Smith Predictor scheme, as it allows the design of the controller for reference tracking without considering time delay. However, the Smith Predictor presents poor disturbance rejection capabilities, in particular when the system to be controlled has an integral action, e.g. as typical of motion control systems. This paper illustrates the ineffectiveness of the Smith Predictor when applied on an linear time-delay system with integral action and provides an analytical expression of the steady-state error when a step disturbance acts on the control signal.

Keywords: PID control, Modeling and identification, Mechatronics
Abstract: A methodology for tuning the digital proportional-integral-retarded (PIR) controllers via the dominant pole placement approach is presented in this paper. Based on the desired performance criteria for the closed-loop system, the dominant poles are positioned, while the remaining poles are allocated to specific locations, ensuring they are sufficiently distant from the dominant poles. To achieve this, a desired polynomial is formulated, and the coefficient equalization method is applied. The presented approach is illustrated through an industrial application employing Kalman Filter (KF) based online system identification. The results show that both the PIR controller itself and the online system identification-based tuning methods exhibit satisfactory performance across various load characteristics, thereby affirming their validity and effectiveness for utilization in industrial control systems with delays.

Keywords: Physics, Nonlinear dynamics, Approximation techniques and numerical methods
Abstract: In earlier work, we modeled the dynamics of a fission process of a mass splitting into two equal parts, as a simple toy model for an exploding star. In this contribution, we generalize the process for fission with different masses involved, given the initial energetic event. In a post-Newtonian approximation (where the fields are weak and velocities small) to general relativity (GR) the problem reduces to coupled equations with state-dependent delay. Moreover, this state-dependency is only implicitly given. We review some relevant results in the theory of systems with state-dependent delays and provide the solution to the fission problem via a perturbation expansion. Defining escape as sustained separation, it is discovered that with delayed gravity, transported at the speed of light, the escape occurs for higher velocities than predicted in Newtonian physics (where gravity acts instantaneously).

Keywords: PID control, Control of linear systems, Robust control
Abstract: During the tutorial, participants will explore the innovative web-based design tool PID Hinf Designer, which can analyse, design, and tune various controllers, including PID, PR, and advanced ones like the Smith predictor. The tool employs an analytical computational method to determine the boundary of Hinf region in the parameter plane of the controller. This boundary corresponds to the specified Hinf performance and robustness constraint. A key advantage of this approach is its ability to address multi-criteria and multiple models problems. The tool is accessible at www.pidlab.com.

Keywords: Nonlinear dynamics, Stability, Numerical simulation
Abstract: A novel approach to the practical study of delay equations consists in reformulating them as abstract differential equations, applying a discretization in order to obtain a system of ordinary differential equations (ODEs) and using available numerical methods for ODEs, such as the MATLAB-based MatCont package for the stability and bifurcation analysis. In order to make the approach more accessible, we added to MatCont's graphical interface an import utility for delay differential and renewal equations with constant finite discrete and distributed delays. The analysis of the resulting system of ODEs can be performed as usual in MatCont.