Keywords: Approximation techniques and numerical methods, Infinite dimensional systems, Robust control
Abstract: We reformulate the Lanczos tau method for the discretization of time-delay systems in terms of a pencil of operators, allowing for new insights into this approach. As a first main result, we show that, for the choice of a shifted Legendre basis, this method is equivalent to Padé approximation in the frequency domain. We illustrate that Lanczos tau methods straightforwardly give rise to sparse, self-nesting discretizations. Equivalence is also demonstrated with pseudospectral collocation, where the non-zero collocation points are chosen as the zeroes of orthogonal polynomials. The importance of such a choice manifests itself in the approximation of the H^2-norm, where, under mild conditions, super-geometric convergence is observed and, for a special case, super convergence is proved; both of which are significantly faster than the algebraic convergence reported in previous work.

Keywords: Infinite dimensional systems, Nonlinear dynamics, Stability
Abstract: We consider nonlinear delay differential and renewal equations with infinite delay and introduce a unifying abstract differential equation capturing both. Using the pseudospectral method, the abstract differential equation is approximated with a finite-dimensional ODE. We discuss the one-to-one correspondence of equilibria between the original and the approximating system, and the convergence of the characteristic roots as the dimension of the approximation increases, when the collocation nodes are chosen as the zeros or the extrema of the Laguerre polynomials suitably scaled. The approximating system is based on the differentiation matrix associated with the nodes, and our results provide an exact characterisation of its spectrum. The theoretical results are illustrated with numerical tests.

Keywords: Approximation techniques and numerical methods, Numerical simulation, Infinite dimensional systems
Abstract: We propose the use of exponential Runge-Kutta methods for the time integration of delay differential equations. The approach is based on their reformulation as abstract differential equations and the reduction of the latter to finite-dimensional systems of ordinary differential equations via pseudospectral discretization. We substantiate our results by means of some illustrative numerical simulations with EXPINT, a MATLAB package for exponential integrators.

Keywords: Spectrum computation and optimization, Approximation techniques and numerical methods
Abstract: Quasi-Polynomial mapping-based Rootfinder (QPmR) has been designed as a user-friendly Matlab tool for computing and analyzing spectra of Quasi-polynomial roots located in a closed region of the complex plane. As the QPmR uses mapping-based technique for root location, its applicability has some limits concerning the quasi-polynomial order and structural complexity. Within the presented QPmR update, numerical efficiency and memory effectiveness have been improved to increase speed and enhance reliability of computations. Next to Matlab, the updated QPmR algorithm is now also available as an open-source Python package. The use and efficiency of QPmR is demonstrated on examples.

Keywords: Numerical simulation, Infinite dimensional systems, Stability
Abstract: We consider the propagation, along the solution of a linear scalar of perturbations in the initial values. We analyze this propagation by looking at the relative error of the perturbation, rather than the absolute error. Our interested is in the long-time behavior of the relative error, quite different from the long-time behavior of the absolute error.

Keywords: Vibration control, Infinite dimensional systems, Stability
Abstract: We present a methodology for designing a dynamic controller with delayed output feedback for achieving non-collocated vibration suppression with a focus on the multi-frequency case. To synthesize the delay-based controller, we first remodel the system of equations as a delay-differential algebraic equation (DDAE) in such a way that existing tools for design of a static output feedback controller can be easily adapted. The problem of achieving non-collocated vibration suppression with sufficient damping is formulated as a constrained optimization problem of minimizing the spectral abscissa in the presence of zero-location constraints, with the constraints exhibiting polynomial dependence on its parameters. We transform the problem into an unconstrained one using elimination, following which we solve the resulting non-convex, non-smooth optimization problem.

Keywords: Vibration control, Control of linear systems
Abstract: We propose an active vibration absorption technique termed fractional-order delayed resonator (FODR) and address the associated control design, analysis, and optimization problems. The FODR utilizes the fractional calculus theory to provide additional degrees of freedom for control, enabling improved flexibility and robustness in suppressing vibrations. Parameterized control formulas are proposed, followed by a detailed analysis regarding the control parameters. Additionally, we introduce the ‘order scheduling’ tuning concept, which treats the order as a tunable parameter to further enhance the control flexibility. The results demonstrate that this powerful technique effectively widens feasible frequency bands, improves vibration control speed, and enhances stability robustness.

Keywords: Vibration control, Modeling and identification, Control of linear systems
Abstract: This work explains the generalization of active absorbers using the delayed resonator principle towards multi-degree-of-freedom mechanical structures. Several types of planar and spatial active absorbers (multidimensional delayed resonators) are presented both in terms of theoretical analysis and through the introduction of several demonstrators. The specific problems of these devices are also analyzed, in particular the influence of passive resistances and the possibilities of their active elimination. The main challenges to be solved arise from handling simultaneously several actuators with delayed feedback, three for planar and six for spatial vibration absorption. The challenging analysis and synthesis are performed in both the frequency and spectral domain.

Keywords: Control of linear systems, Mechatronics
Abstract: Recently, the problem of non-collocated vibration absorption by the delayed resonator was introduced by Olgac and Jenkins. It was shown that for a serially composed structure, the harmonic perturbation force excites a so-called resonating substructure, which, in fact, absorbs the vibration. However, such a resonating substructure, tuned to be marginally stable by the resonator's design, may have a strong destabilizing effect on the overall system. To extend the applicable frequency range and achieve a balanced force and potential energy distribution across the structure and thus lower the risk of fatigue, we apply a simultaneous optimization of the system structural parameters and control design of the delayed resonator.

Keywords: Vibration control, Nonlinear dynamics, Numerical simulation
Abstract: We propose a sophisticated vibration suppression technique termed the Quasi-Zero-Stiffness Delayed Resonator (QZS-DR) and delve into the associated challenges of control design, analysis, and optimization. This paper introduces the concept of a delay-independent resonant frequency, providing a theoretical explanation of the interplay between system stiffness and the resonator's resonant frequency. Utilizing the unique high-static-low-dynamic stiffness properties of quasi-zero-stiffness structures, the QZS-DR significantly reduces the dynamic stiffness of the absorber system. This reduction facilitates the expansion of the absorber’s low-frequency absorption band. Our results demonstrate that the QZS-DR notably enhances the vibration absorption capabilities of the delayed absorber across low-frequency ranges, proving its substantial applicability in sectors prone to low-frequency vibrations such as automotive seating, marine vessels, and structural engineering.

Keywords: Stability, Lyapunov functionals and Lyapunov matrices, Robustness
Abstract: We show that the existence of a Lyapunov-Krasovskii functional (LKF) with a point-wise dissipation suffices for input-to-state stability, provided that uniform global stability can also be ensured using the same LKF. To prove this result, we develop a stability theory, in which the behavior of solutions is not assessed through the classical norm but rather through a specific LKF, which may provide significantly tighter estimates.

Keywords: Observer design, Lyapunov functionals and Lyapunov matrices
Abstract: We provide a new framework for the analysis of observability and detectability properties of linear time-delay systems based on Lyapunov-Krasovskii functionals of complete type. Moreover, formulae for the observer construction are given. At the basis of the obtained results is the representation of complete type functionals via a bounded operator in a Hilbert space.

Keywords: Lyapunov functionals and Lyapunov matrices, Approximation techniques and numerical methods, Robustness
Abstract: Lyapunov-Krasovskii functionals of robust type have recently been introduced for the stability analysis in time-delay systems. They are inspired by the wide-spread Lyapunov-Krasovskii functionals of complete type, but aim at improved robustness results. The present paper recaps the most important aspects and proposes a numerical method to solve the defining equation of these functionals numerically. The numerical method relies on a Legendre-tau-based ODE approximation of the time-delay system and has recently been shown to yield convincing results for the finite-dimensional approximation of Lyapunov-Krasovskii functionals of complete type. In the latter case, a Lyapunov equation has to be solved. In the case of Lyapunov-Krasovskii functionals of robust type, an algebraic Riccati equation has to be solved instead.

Keywords: Lyapunov functionals and Lyapunov matrices, Stability
Abstract: By the Lyapunov Stability Test, we refer to a previously developed criterion for the exponential stability of linear time-invariant delay systems. This criterion requires the validation of a special matrix, expressed in terms of a discrete set of values of the delay Lyapunov matrix, for positive definiteness. The dimension of this matrix is determined by the system parameters and deduced from some conservative estimates. In this paper, we have significantly reduced the conservatism. As a result, we obtained a dimension that depends cubically on the delay value, whereas previously this dependence was exponential.