On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems

For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in time-forward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects.Comment: 33 pages, 8 figures, "This is a pre-print of an article published in 'Nonlinear Dynamics'. The final authenticated version is available online at https://doi.org/10.1007/s11071-020-05620-8

Stability Criteria for Time-Delay Systems from an Insightful Perspective on the Characteristic Equation

This note provides a delay-dependent and a necessary and sufficient delay-independent stability criterion for linear autonomous continuous-time systems with a discrete delay. We take a simple perspective on the two-variable formulation of the characteristic equation, which leads to the following advantageous but not widespread delay-independent criterion: the sum of the coefficient matrix of the delay-free term and the elementwise unitarily rotated matrix of the delay term must remain Hurwitz for all rotation angles. A graphical test for the latter is shown to require no more than three lines of code. Concerning delay-independent stability, our main contribution is to extend this sufficient criterion to a necessary and sufficient one. Concerning delay-dependent stability, the focus is on the critical delay that bounds the initial delay interval of stability. We formulate a constrained minimization problem that gives the exact value of this critical delay. The taken perspective is especially insightful in terms of how the coefficient matrices may look like, and, for scalar systems, the delay-dependent stability chart becomes obvious at a first glance. The presented criteria are complementary to the well-known frequency-sweeping test, which results from another of three possible perspectives on the two-variable criterion. Besides of a unified treatment of these different perspectives, the note also discusses corollaries of the one taken, which include Mori\u27s famous criterion

Methodik zur Integration von Vorwissen in die Modellbildung

Das Buch zeigt, wie Vorwissen über Eigenschaften dynamischer Systeme und über Funktionen in die mathematische Modellbildung integriert werden kann. Hierzu wird im ersten Teil der Arbeit das verbale Vorwissen mathematisch formuliert. Der zweite Teil beschreibt vier Zugängen, um die entstehenden restringierten Probleme zu lösen. Zahlreiche Beispiele, Tabellen und Zusammenstellungen vervollständigen das Buch

Methodik zur Integration von Vorwissen in die Modellbildung

This book describes how prior knowledge about dynamical systems and functions can be integrated in mathematical modelling. The first part comprises the transformation of the known properties into a mathematical model and the second part explains four approaches for solving the resulting constrained optimization problems. Numerous examples, tables and compilations complete the book

Characterization of Aggregated Building Heating, Ventilation, and Air Conditioning Load as a Flexibility Service Using Gray‐Box Modeling

Integrating large amounts of volatile renewable power into the electricity grid requires ancillary services (ASs) from multiple providers including flexible demand. These should be comparable by uniform and efficiently evaluable performance criteria. The objective is to characterize the technical flexibility of aggregated building heating, ventilation, and air conditioning (HVAC) under different operating conditions. New bounds of flexible power and holding durations, accordingly pay-back power and recovery times, and ramping rates are derived, using a new gray-box model of stochastically actuated aggregations of thermostatically controlled loads (TCLs) that can serve as well for load control. New closed formulas of the expected switching temperatures are derived using survival processes and hazard functions. This ex-ante characterization enables fast decision tools for AS feasibility testing and planning by demand aggregators, as it neither relies on simulation or optimization, nor on the identification and clustering of unit-level parameters. The estimates are explored in a sensitivity study of urban-level heat pump heating with respect to six key input factors. A case study using dynamic regulation signals from Pennsylvania–New Jersey–Maryland (PJM) demonstrates the benefit, in terms of tracking precision, of the refined energy measures over pure energy or power capacity bounds

A Mode-Based Averaged Power Converter Model for Large Transients

Power converters employ high-frequency switching between multiple switch states, each of which causes the system to exhibit a different dynamic behavior. Averaged models are a common simplification used for describing the behavior in one or two specific cycles of switch states, also called operating modes. In this context, we propose extending the method of Sun et al. (2001), which allows averaging in two operating modes, to a converter model with four operating modes. We show in simulations that our model results in a reasonable approximation of the true moving average of the original switching converter model during large transients that pass through multiple operating modes

Characteristics-based Simulink Implementation of First-order Quasilinear Partial Differential Equations

The paper deals with solving first-order quasilinear partial differential equations in an online simulation environment, such as Simulink, utilizing the well-known and well-recommended method of characteristics. Compared to the commonly applied space discretization methods on static grids, the characteristics-based approach provides better numerical stability. Simulink subsystem implementing the method of characteristics is developed. It employs Simulink’s built-in solver and its zero-crossing detection algorithm to perform simultaneous integration of a pool of characteristics as well as to create new characteristics dynamically and discard the old ones. Numerical accuracy of the solution thus obtained is established. The subsystem has been tested on a full-state feedback example and produced better results than the space discretization-based “method of lines”. The implementation is available for download and can be used in a wide range of model

Characteristics-based Simulink implementation of first-order quasilinear partial differential equations

The paper deals with solving first-order quasilinear partial differential equations in an online simulation environment, such as Simulink, utilizing the well-known and well-recommended method of characteristics. Compared to the commonly applied space discretization methods on static grids, the characteristics-based approach provides better numerical stability. Simulink subsystem implementing the method of characteristics is developed. It employs Simulink's built-in solver and its zero-crossing detection algorithm to perform simultaneous integration of a pool of characteristics as well as to create new characteristics dynamically and discard the old ones. Numerical accuracy of the solution thus obtained is established. The subsystem has been tested on a full-state feedback example and produced better results than the space discretization-based "method of lines". The implementation is available for download and can be used in a wide range of models.Comment: Abridged and updated conference version. Accepted to SIMULTECH 202