2023 | Buch

# Controlling Delayed Dynamics

## Advances in Theory, Methods and Applications

herausgegeben von: Dimitri Breda

Verlag: Springer International Publishing

Buchreihe: CISM International Centre for Mechanical Sciences

2023 | Buch

herausgegeben von: Dimitri Breda

Verlag: Springer International Publishing

Buchreihe: CISM International Centre for Mechanical Sciences

This book gathers contributions on analytical, numerical, and application aspects of time-delay systems, under the paradigm of control theory, and discusses recent advances in these different contexts, also highlighting the interdisciplinary connections. The book will serve as a useful tool for graduate students and researchers in the fields of dynamical systems, automatic control, numerical methods, and functional analysis.

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Abstract

In the first part of this chapter we review the recently developed theory of twin semigroups and norming dual pairs in the light of neutral delay equations. In the second part we extend the perturbation theory for twin semigroups to include time-dependent perturbations. Finally we apply this newly developed theory to neutral periodic delay equations.

Abstract

In the first part of this chapter we recall the notion of a characteristic matrix function for classes of operators as introduced in Kaashoek and Verduyn Lunel (2023). The characteristic matrix function completely describes the spectral properties of the corresponding operator. In the second part we show that the period map or monodromy operator associated with a periodic neutral delay equation has a characteristic matrix function. We end this chapter with a number of illustrative examples of periodic neutral delay equations for which we can compute the characteristic matrix function explicitly.

Abstract

Delay equations generate dynamical systems on infinite-dimensional state spaces. Their stability analysis is not immediate and reduction to finite dimension is often the only chance. Numerical collocation via pseudospectral techniques recently emerged as an efficient solution. In this part we analyze the application of these methods to discretize the infinitesimal generator of the semigroup of solution operators associated to the system. The focus is on both local stability of equilibria and general bifurcation analysis of nonlinear problems, for either delay differential and renewal equations.

Abstract

Delay equations generate dynamical systems on infinite-dimensional state spaces. Their stability analysis is not immediate and reduction to finite dimension is often the only chance. Numerical collocation via pseudospectral techniques recently emerged as an efficient solution. In this part we analyze the application of these methods to discretize the evolution family associated to linear problems. The focus is on local stability of either equilibria and periodic orbits as well as on generic nonautonomous systems, for either delay differential and renewal equations.

Abstract

This chapter addresses the stability analysis of linear dynamical systems represented by delay differential equations with a focus on the effects induced by the delay, seen as a parameter, on the dynamical behavior. More precisely, we propose a frequency-sweeping framework for treating the problem, and the stability problem is reformulated in terms of properties of frequency-sweeping curves. The presentation is teaching-oriented and focuses more on discussing the main ideas of the method and their illustration through appropriate examples and less on explicit proofs of the results. Some applications from Life Sciences complete the presentation.

Abstract

This chapter briefly presents some “user-friendly” methods and techniques (frequency-domain approaches) for the analysis and control of linear dynamical systems in presence of delays. The presentation is as simple as possible, focusing more on the main intuitive (algebraic, geometric) ideas to develop theoretical results, and their potential use in practical applications. To fix better the ideas, scalar and second-order examples are largely discussed. Next, a particular attention will be paid to the existing links between the maximal allowable multiplicity of the characteristic roots and the spectral abscissa of the dynamical system. The underlying property—multiplicity induced dominancy— will be particularly useful in constructing low-complexity controllers by partial pole placement. Such an idea is particularly exploited in vibration control.

Abstract

This chapter presents a dynamical systems point of view of the study of systems with delays. The focus is on how advanced tools from bifurcation theory, as implemented for example in the package DDE-BIFTOOL, can be applied to the study of delay differential equations (DDEs) arising in applications, including those that feature state-dependent delays. We discuss the present capabilities of the most recent release of DDE-BIFTOOL. They include the numerical continuation of steady states, periodic orbits and their bifurcations of codimension one, as well as the detection of certain bifurcations of codimension two and the calculation of their normal forms. Two longer case studies, of a conceptual DDE model for the El Niño phenomenon and of a prototypical scalar DDE with two state-dependent feedback terms, demonstrate what kind of insights can be obtained in this way.

Abstract

We present an overview of control design methods for linear time-delay systems, which are grounded in matrix theory and numerical linear algebra techniques, such as eigenvalue computations, solving Lyapunov matrix equations, eigenvalue perturbation theory and eigenvalue optimization. The methods are particularly suitable for the design of structured controllers, as they rely on a direct optimization of stability, robustness and performance indicators as a function of controller or design parameters. Several illustrations complete the presentation.

Abstract

This chapter discusses a scalable controller synthesis method for networked systems with a large number of identical subsystems based on the \(\mathcal {H}_{\infty }\)-norm control framework. The dynamics of the individual subsystems are described by identical linear time-invariant delay differential equations and the effect of transport and communication delay is explicitly taken into account. The presented method is based on the result that, under a particular assumption on the graph describing the interconnections between the subsystems, the \(\mathcal {H}_{\infty }\)-norm of the overall system is upper bounded by the robust \(\mathcal {H}_{\infty }\)-norm of a single subsystem with an additional uncertainty. This chapter will therefore briefly discuss a recently developed method to compute this last quantity. The resulting controller is then obtained by directly minimizing this upper bound in the controller parameters.

Abstract

Two basic models of machine tool vibrations are presented. First, a simple model of orthogonal turning process is discussed where material is removed from the rotating workpiece by a tool. Vibrations of the tool are copied on the workpiece’s surface and, after one revolution, the tool cuts this wavy surface. This phenomenon is called surface regeneration and the equations governing the vibrations are delay differential equations where the time delay is equal to the rotation period of the workpiece. Then, the mechanical model of milling operation is presented. Here the surface regeneration effect is combined by the parametric forcing of the entering and exiting cutting teeth. The governing equation is hence a time-periodic delay differential equation where the time delay and the principal period are both equal to the tooth passing period. Stability diagrams in the plane of the technological parameters are constructed for both turning and milling operations.

Abstract

Mechanical models of two human balancing tasks, quiet standing and stick balancing on the fingertip, are discussed with special attention to the reaction time delay. As in many control systems, time delay sets limitation to performance during balancing tasks. Human subjects cannot balance an arbitrarily short stick on the fingertip, because a short stick falls faster than the time needed to make a corrective motion. Also, increased reaction time delay is one of the main reasons of instability during quiet standing, which can lead to falls especially among the elderly. The governing equation of the two models are second-order delay differential equations. In this chapter, stabilizability issues are discussed in terms of the critical delay for different feedback concepts, such as proportional-derivative, proportional-derivative-acceleration and predictor feedback.

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