Auswahl der wissenschaftlichen Literatur zum Thema „Multiplicity-Induced-Dominancy“

This work revisits recent results on maximal multiplicity induced-dominancy for spectral values in reduced-order time-delay systems and extends it to the general class of second-order retarded differential equations. A parametric multiplicity-induced-dominancy property is characterized, allowing to a delayed stabilizing design with reduced complexity. As a matter of fact, the approach is merely a delayed-output-feedback where the candidates’ delays and gains result from the manifold defining the maximal multiplicity of a real spectral value, then, the dominancy is shown using the argument principle. Sensitivity of the control design with respect to the parameters uncertainties/variation is discussed. Various reduced order examples illustrate the applicative perspectives of the approach.

AbstractThis paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration (PDA) feedback, which can be used to understand human balance in quiet standing. The closed-loop system is described by a neutral delay differential equation (NDDE). The optimal feedback gains (OFGs) that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4. Such a property is called multiplicity-induced dominancy of time-delay systems, and has been discussed intensively by many authors for retarded delay differential equations (RDDEs). This paper shows that multiplicity-induced dominancy can be achieved in NDDEs. In addition, the OFGs are delay-dependent, and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays. Thus, the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains. The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.

The current work exploits the effects of time-delays on the stability of unmanned aerial vehicles s). In this regard, the main contribution is a symbolic/numeric application of the multiplicity-induced-dominancy property in the control of unmanned aerial vehicles rotorcrafts featuring time-delays. The multiplicity-induced-dominancy property is considered to address two of the most representative aerial robotic platforms: a classical quadrotor vehicle and a quadrotor vehicle endowed with tilting rotors. The aforementioned property leads to an effective delayed feedback control design (multiplicity-induced-dominancy tuning criteria), allowing the system to meet prescribed behavior conditions based on the placement of the rightmost root of the corresponding closed-loop characteristic function/quasipolynomial. Lastly, the results of detailed numerical simulations, including the linear and non-linear dynamics of the vehicle, are presented and discussed to validate the proposal.

Single and double inverted pendulum systems subjected to delayed state feedback are analyzed in terms of stabilizability. The maximum (critical) delay that allows a stable closed-loop system is determined via the multiplicity-induced-dominancy property of the characteristic roots, that is the dominant (rightmost) roots are associated with higher multiplicity under certain conditions of the system parameters. Other methods such as tracking the changes of the D-curves with increasing delay and the Walton–Marshall method are also demonstrated for the example of the single pendulum. For the double inverted pendulum subjected to full state feedback, the number of control gains is four, and application of numerical methods requires therefore high computational effort (i.e. optimization in a four-dimensional space). It is shown that, with the multiplicity-induced-dominancy–based approach, the critical delay and the associated control gains can be determined directly using the characteristic equation and its derivatives.

Une des questions d'intérêt pour les systèmes linéaires à retard est de déterminer les conditions sur les paramètres de l'équation qui garantissent la stabilité exponentielle des solutions. En général, c'est un véritable défi d'établir des conditions sur les paramètres du système afin de garantir une telle stabilité. L'une des approches efficaces dans l'analyse de stabilité des systèmes à retard est l'approche fréquentielle. Dans le domaine de Laplace, l'analyse de stabilité revient à étudier la distribution des racines des fonctions quasipolynomiales caractéristiques. Une fois la stabilité d'un système à retard prouvée, il est important de caractériser le taux de décroissance exponentielle des solutions de ces systèmes. Dans le domaine fréquentiel, ce taux de décroissance correspond à la valeur spectrale dominante. Des travaux récents ont mis en évidence le lien entre la multiplicité maximale et les racines dominantes. En effet, les conditions pour qu'une racine multiple donnée soit dominante sont étudiées, cette propriété est connue sous le nom de Multiplicity-Induced-Dominancy (MID). Dans cette thèse, trois sujets liés à la propriété MID sont étudiés. Premièrement, l'effet des racines multiples avec des multiplicités admissibles présentant, sous des conditions appropriées, la validité de la propriété MID pour les équations différentielles neutres du second ordre avec un seul retard est exploré. La stabilisation de l'oscillateur classique bénéficie des résultats obtenus. Deuxièmement, les effets des retards sur la stabilité des véhicules aériens sans pilote (UAV) sont exploités. À cet égard, une application symbolique/numérique de la propriété MID dans le contrôle des drones à rotor avec des retards est fournie. Enfin, la stabilisation d'une balance à roulettes par le biais de la propriété MID est considérée
One of the questions of ongoing interest for linear time-delay systems is to determine conditions on the equation's parameters that guarantee the exponential stability of solutions. In general, it is quite a challenge to establish conditions on the parameters of the system in order to guarantee such a stability. One of the effective approaches in the stability analysis of time-delay systems is the frequency domain approach. In the Laplace domain, the stability analysis amounts to study the distribution of characteristic quasipolynomial functions' roots. Once the stability of a delay system has been proven, it is important to characterize the exponential decay rate of the solutions of such systems. In the frequency domain, this decay rate corresponds to the dominant spectral value. Recent works emphasized the link between maximal multiplicity and dominant roots. Indeed, conditions for a given multiple root to be dominant are investigated, this property is known as Multiplicity-Induced-Dominancy (MID). In this dissertation, three topics related to the MID property are investigated. Firstly, the effect of multiple roots with admissible multiplicities exhibiting, under appropriate conditions, the validity of the MID property for second-order neutral time-delay differential equations with a single delay is explored. The stabilization of the classical oscillator benefits from the obtained results. Secondly, the effects of time-delays on the stability of Unmanned Aerial Vehicles (UAVs) is exploited. In this regard, a symbolic/numeric application of the MID property in the control of UAV rotorcrafts featuring time-delays is provided. Lastly, the stabilization of a rolling balance board by means of the MID property is considered