Search

The numerical minimization of the spectral abscissa function of linear time-invariant time-delay systems, an established approach to compute stabilizing controllers with a fixed structure, often gives rise to minima characterized by active characteristic roots with multiplicity higher than one. At the same time, recent theoretical results reveal situations where the so-called multiplicity induced dominancy property holds, i.e. a sufficiently high multiplicity implies that the root is dominant. Using an integrative approach, combining analytical characterizations, computation of characteristic roots and numerical optimization, a complete characterization of the stabilizability of second-order systems with input delays is provided, for both state feedback and delayed output feedback. The level sets of the minimal achievable spectral abscissa are also characterized. These results shed light on the complex relations between (configurations involving) multiple roots, the property of being dominant roots and the property of corresponding to (local/global) minimizers of the spectral abscissa function. [ABSTRACT FROM AUTHOR]

Qiao et al.21 developed adaptive PID controllers for trajectory tracking of robotic manipulators with uncertainties, which consist of a standard PID controller augmented by an adaptation law. Özbay and Gündes4 addressed the problems of pole placement and delay margin by the design of low-order controllers, including specifically PID controllers. Shi et al.11 considered a so-called dynamic equational PID controller, which augments the derivative control in the PID controller to high-order differentials. For well over a century, from the enabling of early governors to a standard tool in process control, PID (proportional-integral-derivative) control stood out as the most favored method for its unparalleled simplicity, ease of implementation, and cost-effectiveness, and indeed has been an enduring gift to feedback control. [Extracted from the article]

This article focuses on the root behavior of the characteristic function of some LTI SISO systems controlled by a PD‐control when a delay‐difference operator is used to approximate the derivative action. We will focus on the cases when the corresponding stability problem is improperly posed for "small" delay values. In this context, we will express the solutions of the corresponding characteristic function as power series and exploit their structure in deriving the asymptotic behavior of the characteristic roots. This analysis will allow detecting and explicitly characterizing the cases when delay‐difference approximations lead to improperly posed stability problems. Furthermore, in the case when the derivative approximation guarantees the properly posedness of the closed‐loop system, the robustness of the scheme with respect to the delay margin of the delay‐difference approximation and control gains parameters are explicitly addressed. More precisely, upper bounds on the delays as well as the robustness of the corresponding controller are computed. Some illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]

This paper presents the analysis of the stability properties of proportional-integralderivative (PID) controllers for dynamical systems with multiple state delays, focusing on the mathematical characterization of the potential sensitivity of stability with respect to infinitesimal parametric perturbations. These perturbations originate, for instance, from neglecting feedback delay, a finite-difference approximation of the derivative action, or neglecting fast dynamics. It is also shown that adding a low-pass filter, which might be necessary to reduce sensitivity to high-frequency noise, under certain conditions on the derivative feedback gain may destabilize the closed loop even for arbitrarily large cutoff frequencies. The analysis of these potential sensitivity problems leads us to the introduction of a "robustified" notion of stability called strong stability, inspired by the corresponding notion for neutral functional differential equations. We prove that strong stability can be achieved by adding a low-pass filter with a sufficiently large cutoff frequency to the control loop on the condition that the filter itself does not destabilize the nominal closed loop system. Throughout the paper, the theoretical results are illustrated by examples that can be analyzed analytically, including, among others, a third-order unstable system for which both proportional and derivative control action are necessary for achieving stability, while the regions in the gain parameter-space for stability and strong stability are not identical. Besides the analysis of strong stability, a computational procedure is provided for designing strongly stabilizing PID controllers. Computational case studies illustrating this design procedure complete the presentation. [ABSTRACT FROM AUTHOR]

This article addresses the boundary control problem of the transport equation. Namely, we propose a control method, which is merely a delayed output feedback relying on a partial pole placement idea, that consists in assigning an appropriate exponential decay rate to the closed‐loop system's solution. The proposed control structure appearing in the transport boundary, which has proven its effectiveness in controlling finite dimensional systems, consists of an autoregressive relation linking the transport equation's input and output. The obtained result provides an analytical lower bound for the solution's exponential decay. The contribution is concluded with a numeric discussion on the concrete implementation of the method with a special emphasis on the delay seen as a control parameter. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we study the behavior of characteristic roots of a class of strictly proper LTI SISO systems that are subject to a PD controller for which the derivative action has been implemented by means of a delay‐difference approximation scheme that uses multiple delays. To such an end, the improperly‐posed case is discussed in depth, that is, the situations where "small delays" induce instability in the closed‐loop system. The effect of adding more delays on the improperly‐posed case is analyzed and a methodology for determining the set of parameters that produces such a situation for a given number of delays is derived. Some illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the International Journal of Robust & Nonlinear Control on the topic of time-delay systems. It provides a historical overview of functional differential equations and highlights the contributions of Nikolay Nikolayevich Krasovskii and Aristide Halanay to the field. The special issue includes 21 research papers covering stability, robustness, observer theory, predictors, and delay-based approaches to partial differential equation models. Each paper is briefly described, covering topics such as stabilization of uncertain systems, sampled-data control for nonlinear systems, observer theory for differential delay systems, sensor fault estimation, and control approaches for vehicular platoons. The summaries aim to provide library patrons with a quick understanding of the papers' contents and their relevance to their research interests. [Extracted from the article]

In this work, we show that the coexistence of the maximal number of real spectral values of generic single-delay retarded second-order differential equations guarantees the realness of the rightmost spectral value. From a control theory standpoint, this entails that a delayed proportional-derivative (PD) controller can stabilize a delayed second-order differential equation. By assigning the maximum number of negative roots to the corresponding characteristic function (a quasipolynomial), we establish the conditions for asymptotic stability. If the assigned real spectral values are uniformly distributed, we specify a necessary and sufficient condition for the rightmost root to be negative, thus guaranteeing the exponential decay rate of the system's solutions. We illustrate the proposed design methodology in the delayed PD control of the damped harmonic oscillator. It is worth mentioning that this work represents a natural continuation of Amrane et al. (2018) and Bedouhene et al. (2020), addressing the problem of coexisting real spectral values for linear dynamical systems including delays in their models. [ABSTRACT FROM AUTHOR]

In this paper, which is a direct continuation and generalization of the recent works by the authors [17, 35], we show the validity of the generic multiplicity-induced-dominancy property for a general class of linear functional differential equations with a single delay, including the retarded as well as the neutral cases. The result is based on an appropriate integral representation of the corresponding characteristic quasipolynomial functions involving some appropriate degenerate hypergeometric functions. [ABSTRACT FROM AUTHOR]

This paper concerns some spectral properties of the scalar dynamical system defined by a linear delay-differential equation with two positive delays. More precisely, the existing links between the delays and the maximal multiplicity of the characteristic roots are explored, as well as the dominance of such roots compared with the spectrum localization. As a by-product of the analysis, the pole placement issue is revisited with more emphasis on the role of the delays as control parameters in defining a partial pole placement guaranteeing the closed-loop stability with an appropriate decay rate of the corresponding dynamical system. [ABSTRACT FROM AUTHOR]

We study a class of continuous-time nonlinear systems with discrete measurements, model uncertainty, and sensor noise. We provide an estimator of the state for which the observation error enjoys a variant of the exponential input-to-state stability property with respect to the model uncertainty and sensor noise. A valuable novel feature is that the overshoot term in this stability estimate only involves a recent history of uncertainty values. Also, the rate of exponential convergence can be made arbitrarily large by reducing the supremum of the sampling intervals. Our proof uses a recently developed trajectory based approach. We illustrate our work using a model for a pendulum whose suspension point is subjected to an unknown time-varying bounded horizontal oscillation. [ABSTRACT FROM AUTHOR]

This paper focuses on the location of the non-asymptotic zeros of Whittaker and Kummer confluent hypergeometric functions. Based on a technique proposed by E. Hille for the analysis of solutions of some second-order ordinary differential equations, we characterize the sign of the real part of zeros of Whittaker and Kummer functions and provide estimates on the regions of the complex plane where those zeros can be located. Our main result is a correction of a previous statement by G. E. Tsvetkov whose propagation has induced mistakes in the literature. In particular, we review some results of E. B. Saff and R. S. Varga on the error of Padé's rational approximation of the exponential function, which are based on the latter. [ABSTRACT FROM AUTHOR]

In this article we consider the γ‐stabilization of nth‐order linear time‐invariant dynamical systems using multiplicity‐induced‐dominancy‐based controller design in the presence of delays in the input or the output channels. A sufficient condition is given for the dominancy of a real root with multiplicity at least n+1 and at least n using an integral factorization of the corresponding characteristic function. A necessary condition for γ‐stabilizability is analyzed utilizing the property that the derivative of a γ‐stable quasipolynomial is also γ‐stable under certain conditions. Sufficient and necessary conditions are given for systems with real‐rooted open‐loop characteristic function: the delay intervals are determined where the conditions for dominancy and γ‐stabilizability are satisfied. The efficiency of the proposed controller design is shown in the case of a multilink inverted pendulum. [ABSTRACT FROM AUTHOR]

• Sensing and computational framework for estimating the nature of the drilled rock. • Force/velocity on bit computed using top-drive measurements. • Three algorithms: Seismic-While-Drilling, direct estimation, Neural networks. • Validation on simulations against a reliable model. • Extension to non-linear side forces, coupled axial-torsional oscillations. In this paper, we consider the axial motion of a directional multi-sectional drill-string. The drill-string dynamics are represented by a distributed dynamical model (wave equations) coupled with an ordinary differential equation at the downhole boundary (bit-rock interaction). The interaction between the drill-bit and rock can introduce severe vibrations in the drill-string and result in safety and performance issues. Consequently, the performance of drilling is interwoven with our knowledge of the subsurface. To address these problems, we propose a sensing and computational framework for estimating the drill-string dynamics and the specific intrinsic energy of the rocks while drilling. By exploiting the derived models' particular structure, we combine the drill-string dynamics modeling with top-drive hook-load (force) and hook-speed (velocity) measurements to estimate the force-on-bit without requiring the knowledge of the sub-surface. Then, we record and model the seismic radiation patterns of drill-bit rock interactions near the surface (i.e., seismic while drilling). Such an idea allows deriving an appropriate estimation of the rocks' intrinsic energy while drilling. We introduce two alternative rock property estimation algorithms based on direct parameter estimation and machine learning concepts to complete the analysis. The different approaches are tested and validated in simulations. We discuss their respective advantages and drawbacks. Finally, we show how to extend our methodologies in the presence of non-linear Coulomb friction terms and of coupled axial–torsional oscillations. [ABSTRACT FROM AUTHOR]

In this paper, the output feedback stabilization of an underactuated chain of scalar hyperbolic systems coupled at one end with a finite-dimensional system is addressed. As encountered in practical applications, only the first PDE subsystem is actuated. The measurements are collocated with the actuation. The proposed approach uses a recursive dynamics interconnection framework. More precisely, we first estimate (delayed) values of the states at each subsystem's boundaries. Then, we design a state-predictor that gives access to the boundary states' present and future values. It becomes finally possible to design a state-observer for the entire system. Combining this observer with a stabilizing state-feedback law, we can recursively design an output-feedback control law that stabilizes the whole chain. Some illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]

The paper presents a controller design for systems suffering from multi-harmonic periodic disturbance and substantial input time-delay. It forms an alternative approach to Repetitive Control where the goal is to stabilize a close-loop that encapsulates an explicit time-delay model of the periodic signal. The proposed controller design is based on the Internal Model Control (IMC) framework, and it consists of the inverse system model and a tuneable distributed delay with an overall length related to the period of the disturbance. The properness of the controller can be ensured by utilizing a low-pass filter, however, such a component is shown to be unnecessary when the relative order of the system model is one. This fact makes the alternative approach especially suitable for systems approximated by a first-order model with input time-delay, leading to a straightforward controller design thanks to its simple structure and attainable conditions. Stability of the configuration is guaranteed by an ideal IMC framework. For further performance and robustness requirements for the non-ideal case the tuning of the controller is posed as a weighted- H ∞ optimization problem where frequency-, spectral- and time-domain requirements are formulated as constraints. The overall control design is experimentally verified on a laboratory setup that has high-order dynamics approximated by a first-order model with input delay. [ABSTRACT FROM AUTHOR]

The PID control is favored in controlling industrial processes for its ease of implementation. In this paper we present an analytical study, with an aim at stability robustness, tracking performance, and pole placement of first-order unstable plants under PID control. We employ a multiplicity-induced dominancy (MID) strategy to design and tune PID controller gains, which is shown to possess desirable properties that ensure specified decay rate of the closed-loop response and the stability robustness of the closed-loop system in spite of variations in the delay parameter. We also study the tradeoff between delay robustness and tracking performance of PID controllers. We show that under the constraint of steady-state tracking, the maximal delay robustness range can be computed by solving a unimodal pseudo-concave optimization problem. [ABSTRACT FROM AUTHOR]