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## Über dieses Buch

This volume collects contributions related to selected presentations from the 12th IFAC Workshop on Time Delay Systems, Ann Arbor, June 28-30, 2015. The included papers present novel techniques and new results of delayed dynamical systems. The topical spectrum covers control theory, numerical analysis, engineering and biological applications as well as experiments and case studies. The target audience primarily comprises research experts in the field of time delay systems, but the book may also be beneficial for graduate students alike.

## Inhaltsverzeichnis

### Sampled-Data Stabilization of Nonlinear Delay Systems with a Compact Absorbing Set and State Measurement

Abstract
We present a methodology for the global sampled-data stabilization of systems with a compact absorbing set and input/measurement delays. The methodology is based on a numerical prediction scheme, which is combined with a projection of the state measurement on an appropriate sphere. The stabilization is robust to perturbations of the sampling schedule and is robust with respect to measurement noise. The obtained results are novel even for the delay-free case.
Iasson Karafyllis, Miroslav Krstic

### Parametric Transfer Matrices for Sampled-Data Control Systems with Linear Continuous Periodic Process and Control Delay

Abstract
For the class of multi-input multi-output systems being composed of a linear continuous periodic (LCP) process, pure delay, and a digital controller, the paper provides closed expressions for the parametric transfer matrix (PTM), even for the difficult, but practically important case, when the external excitations act on continuous system parts. In the same way as ordinary transfer matrices in the linear time-invariant (LTI) case, the PTM for LCP systems is a fundamental concept for analysis and design of those systems. The properties of the constructed parametric transfer matrices as functions of the real parameter and the complex variable are investigated. These properties are similar to those from ordinary transfer matrices, so that the PTM after some modifications, can be applied with similar tools. Moreover, formulae are derived that are appropriate for the practical computation of the PTM. An example demonstrates, how the formulae can be handled.
Bernhard P. Lampe, Efim N. Rosenwasser

### An SL/QP Algorithm for Minimizing the Spectral Abscissa of Time Delay Systems

Abstract
We consider a problem of eigenvalue optimization, in particular finding a local minimizer of the spectral abscissa—the value of a parameter that results in the smallest value of the largest real part of the spectrum of a system. This is an important problem for the stabilization of control systems, but it is difficult to solve because the underlying objective function is typically nonconvex, nonsmooth, and non-Lipschitz. We present an expanded sequential linear and quadratic programming algorithm that solves a series of linear or quadratic subproblems formed by linearizing, with respect to the parameters, a set of right-most eigenvalues at each point as well as historical information at nearby points. We present a comparison of the performance of this algorithm with the state of the art in the field.
Vyacheslav Kungurtsev, Wim Michiels, Moritz Diehl

### The Principle of “Borrowed Feedback” and Application to Control and Observation for Systems with Implicit State Dependent Delay

Abstract
This chapter develops a general principle of “borrowed feedback” for linear systems with input delay, and in its dual form of observation with delayed measurements. This is of interest in problems where the external interaction with the system incurs a delay. We first focus on the simplest case of constant delay. This is then extended to the case where the delay may be varying. One application is the control of systems over a network. However, in this case the delay cannot be considered constant, but depends on a congestion state of the network. It is shown that if the causality constraint, which imposes an upper bound on the rate of change of the delay, does not hold the design method may fail to give consistent results.
Erik I. Verriest

### A New Delay-Independent Stability Test for LTI Systems with Single Delay

Abstract
A new method complying necessary and sufficient conditions to test delay-independent stability of the general linear time invariant (LTI) dynamics with single delay is presented. The method is based on investigating the location of zeros of an auxiliary characteristic polynomial obtained via Kronecker summation. The proposed approach enables to determine the exact regions of the unknown parameters, e.g., system and controller parameters, ensuring delay-independent stability.
Baran Alikoç, Ali Fuat Ergenç

### Numerical Stability Test of Linear Time-Delay Systems of Neutral Type

Abstract
This paper begins with a brief discussion on the root location of linear time-delay systems of neutral type, to show its peculiar properties compared with time-delay systems of retarded type. Then it introduces a numerical stability test that can be carried out using a rough integral evaluation. The upper limit of the testing integral is estimated in two ways: parameter dependent or parameter independent. As an application of the testing method, the paper shows how to calculate the rightmost characteristic root(s) of a neutral delay differential equation with multiple delays.
Qi Xu, Gabor Stepan, Zaihua Wang

### Utilizing Topological Data Analysis for Studying Signals of Time-Delay Systems

Abstract
This chapter describes a new approach for studying the stability of stochastic delay equations by investigating their time series using topological data analysis (TDA). The approach is illustrated utilizing two stochastic delay equations. The first model equation is the stochastic version of Hayes equation—a scalar autonomous delay equation—where the noise is an additive term. The second model equation is the stochastic version of Mathieu’s equation—a time-periodic delay equation. In the latter, noise is added via a multiplicative term in the time-periodic coefficient. The time series is generated using Euler–Maruyama method and a corresponding point cloud is obtained using the Takens’ embedding. The point cloud is then analyzed using a tool from TDA known as persistent homology. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems that are described using constant as well as time-periodic coefficients. The presented approach can be used for signals generated from both numerical simulation and experiments. It can be used as a tool to study the stability of stochastic delay equations for which there are currently a limited number of analysis tools.
Firas A. Khasawneh, Elizabeth Munch

### Stability and Control of Fractional Periodic Time-Delayed Systems

Abstract
In this chapter, two new methods are proposed to study the stability of linear fractional periodic time-delayed (FPTD) systems. First, the explicit harmonic balance (EHB) method is proposed to find necessary and sufficient conditions for fold, flip, and secondary Hopf transition curves in linear FPTD systems, from which the stability boundaries are obtained as a subset. Transition curves of the fractional damped delayed Mathieu equation are obtained by using the EHB method. Next, an approximated monodromy operator in a Banach space is defined for FPTD systems, which gives the linear map between two solutions. The fractional Chebyshev collocation (FCC) method is proposed to approximate this monodromy operator. The FCC method is outlined and illustrated with three practical problems including obtaining the parametric stability charts of the fractional Hayes equation and the fractional second-order system with delay, and designing an optimal linear periodic gain fractional delayed state feedback control for the damped delayed Mathieu equation.
Eric A. Butcher, Arman Dabiri, Morad Nazari

### Design of Imaginary Spectrum of LTI Systems with Delays to Manipulate Stability Regions

Abstract
This chapter is on the design problem of linear time-invariant (LTI) systems with delays. Our recent studies on the use of algebraic techniques, namely resultant and iterated discriminants operations, in connection with the well-known Rekasius transformation implemented on the system characteristic equation already revealed that it is indeed possible to compute the exact range of the imaginary spectrum of such systems. This know-how, which is the key toward understanding the stabilty/instability decomposition of the system, is utilized here to craft the imaginary spectrum of LTI systems with multiple delays, specifically with the aim to manipulate stability regions in a systematic manner in the delay parameter space.
Rifat Sipahi

### Algorithm for Robust Stability of Delayed Multi-Degree-of-Freedom Systems

Abstract
Computation of the stability limits of systems with time delay is essential in many research and industrial applications. Most of the computational methods consider the exact model of the system, and do not take into account the uncertainties. However, the stability charts are highly sensitive to the change of some input parameters, especially to time delays. An algorithm has been developed to determine the robust stability limits of delayed dynamical systems, which is not sensitive to the fluctuations of selected parameters in the dynamic system. The algorithm is combined with the efficient Multi-Dimensional Bisection Method. The single-degree-of-freedom delayed oscillator is investigated first and the resultant robust stability limits are compared to the derived analytical results. For multi-degree-of-freedom systems, the system of equations of the robust stability limits are modified with the aim to reduce the computational complexity. The method is tested for the 2-cutter turning system with process damping.
Daniel Bachrathy, Marta Janka Reith, Gabor Stepan

### Stability and Robustness Analysis of a Class of Cyclic Biological Systems

Abstract
In the first part of this chapter, we do a local stability and robustness analysis of a model of cyclic biological network representing GRNs (Gene Regulatory Networks) under positive feedback. We present conditions leading to bistable behavior. In the second part, we analyze the GRN under negative feedback. Our analysis depends on an extension of the so-called secant condition, which gives a local stability condition. We deduce conditions leading to global stability of the network.
Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu

### Transformations from Variable Delays to Constant Delays with Applications in Engineering and Biology

Abstract
The transformation from systems with time-varying delays or state-dependent delays to systems with constant delays is studied. The transformation exists if the delay is defined by a transport with a variable velocity over a constant distance. In fact, time-varying or state-dependent delays, which are generated by such a mechanism, are common in engineering and biology. We study two paradigmatic time delay systems in more detail. For metal cutting processes, or more precisely turning with spindle speed variation, we show that the analysis of the machine tool vibrations via constant delays in terms of the spindle rotation angle is more advantageous than the conventional analysis of the vibrations with time-varying delays in the time domain. In a second example, motivated by the McKendrick equation modeling structured populations, we show that systems with variable delays and the equivalent systems with constant delays are, in addition, equivalent to partial differential equations with moving and constant boundaries.

### Coupling Design for Consensus in Switching Topologies with Time-Varying Delays

Abstract
This chapter presents a method to design the coupling strengths between the agents in a network of multi-agent systems in order to enable the group to achieve consensus on a given variable of interest while subject to time-varying delays and switching topologies. The multiple time-varying delays are considered to be nonuniform and nondifferentiable acting in the agents’ control laws, and the topology switches are governed according to a continuous time Markov chain. The main result, formulated as conditions in the form of linear matrix inequalities, is obtained by analyzing the stability of an associated Markov jump linear system. The results are illustrated by a numerical example.
Heitor J. Savino, Fernando O. Souza, Luciano C. A. Pimenta

### Analysis of a Model-Free Predictor for Delay Compensation in Networked Systems

Abstract
One important challenge with networked systems is that communication delays can significantly deteriorate system performance. This chapter proposes a model-free predictor framework to compensate for communication delays and improve networked system performance, where the term “model-free” indicates that the predictor does not need to know the dynamic equations governing the system. The motivation to pursue a model-free approach lies in its robustness, ease of design, and implementation. The proposed predictor has a first-order time delay system structure with only one design parameter. Stability of the predictor is analyzed for constant delays and the range of the design parameter to guarantee a stable predictor is established as a function of the network time delay. Since ensuring stability does not necessarily guarantee a good performance, understanding when the predictor can perform well and what its limitations are also critical. To this end, a frequency-domain analysis is given, through which the relationship between the predictor design parameter, time delay, and steady-state performance is revealed. Fundamental limitations of the predictor at higher frequencies are laid out. Finally, this analysis is confirmed on a case study. The case study further allows for testing the transient performance of the predictor in closed loop with the networked system, and shows that the predictor holds significant potential to alleviate the negative impact of communication delays, even if its high frequency performance may be limited.
Xinyi Ge, Yingshi Zheng, Mark J. Brudnak, Paramsothy Jayakumar, Jeffrey L. Stein, Tulga Ersal

### Predictor Feedback for Extremum Seeking with Delays

Abstract
In this paper, we derive the design and analysis for scalar gradient extremum seeking (ES) subject to arbitrarily long input–output delays, by employing a predictor with a perturbation-based estimate of the Hessian. Exponential stability and convergence to a small neighborhood of the unknown extremum point can be guaranteed. This result is carried out using backstepping transformation and averaging in infinite dimensions. Generalization of the results for Newton-based ES is also indicated. Some simulation examples are presented to illustrate the performance of the delay-compensated ES control scheme.
Tiago Roux Oliveira, Miroslav Krstic

### Improving Stability Margins via Time-Delayed Vibration Control

Abstract
Time-delayed vibration control of a two degree-of-freedom mechanical system approximates state-derivative feedback and reduces sensitivity and improves the stability margins. Additional sensors are not required since state derivatives are approximated using available measurements and time delays. A Lambert W method based design approach is used to solve the resulting delay differential equations. Simulation results demonstrate excellent performance with improved stability margins over state feedback control only.
A. Galip Ulsoy

### A Delay-Based Spacing Policy for Vehicle Platooning: Analysis and Control

Abstract
Reducing inter-vehicular distances and the formation of groups of closely spaced vehicles have the potential to increase traffic flow, reduce congestion, and reduce fuel consumption. In this chapter, such vehicle platoons subject to a delay-based spacing policy are considered and the design of distributed controllers is pursued. Specifically, it is shown that the use of the delay-based spacing policy ensures that all vehicles in the platoon track the same velocity profile in the spatial domain, which offers advantages as road properties such as hills, bends, or road speed limits are specified in this domain. The proposed controller exploits delayed information about the preceding vehicle to achieve string-stable platoon behavior. In addition, a relaxation of the delay-based spacing policy is presented that exploits more information about the preceding vehicle. This extended delay-based spacing policy is shown to lead to improved platoon behavior. The results are illustrated by means of simulations.

### To Delay or Not to Delay—Stability of Connected Cruise Control

Abstract
The dynamics of connected vehicle systems are investigated where vehicles exchange information via wireless vehicle-to-vehicle (V2V) communication. In particular, connected cruise control (CCC) strategies are considered when using different delay configurations. Disturbance attenuation (string stability) along open chains is compared to the linear stability results using ring configuration. The results are summarized using stability diagrams that allow one to design the control gains for different delay values. Critical delay values are calculated and trade-offs between the different strategies are pointed out.
Jin I. Ge, Gábor Orosz, Dávid Hajdu, Tamás Insperger, Jeff Moehlis

### H-infinity Filtering for Cloud-Aided Semi-active Suspension with Delayed Information

Abstract
This chapter presents an $$H_{\infty }$$ filtering framework for cloud-aided semi-active suspension system with time-varying delays. In this system, road profile information is downloaded from a cloud database to facilitate onboard estimation of suspension states. Time-varying data transmission delays are considered and assumed to be bounded. A quarter-car linear suspension model is used and an $$H_{\infty }$$ filter is designed with both onboard sensor measurements and delayed road profile information from the cloud. The filter design procedure is designed based on linear matrix inequalities (LMIs). Numerical simulation results are reported that illustrate the fusion of cloud-based and onboard information that can be achieved in Vehicle-to-Cloud-to-Vehicle (V2C2V) implementation.
Zhaojian Li, Ilya Kolmanovsky, Ella Atkins, Jianbo Lu, Dimitar Filev

### Analysis of Time Delays in Quadrotor Systems and Design of Control

Abstract
In analyzing and designing control for unmanned aerial vehicles (UAVs), existence of transmission delays caused by wireless communication is one of the critical challenges. Estimation of the delays and analysis of their effects are not straightforward. A delay estimation method is introduced using transient responses of a quadrotor type of UAVs and analytical solutions of delay differential equations (DDEs). Experimental data sets in the time domain are compared to the predicted ones based on the analytical solutions of DDEs. The Lambert W function-based approach for first-order DDEs is used for the analysis. The dominant characteristic roots among an infinite number of roots are obtained in terms of coefficients and the delay. The effects of the time delay on the responses are analyzed via root locations. Based on the estimation result, proportional- plus-velocity controllers are proposed to improve transient altitude responses.
Stephen K. Armah, Sun Yi

### Experimental Validation of Robust Chatter Control for High-Speed Milling Processes

Abstract
This chapter presents results on the design and experimental implementation and testing of robust controllers for the high-speed milling process for the purpose of avoiding chatter vibrations. Chatter vibrations are intimately related to the delay nature of the cutting process inherent to milling and should be avoided to ensure a high product quality. A design approach based on $$\mu$$-synthesis is used to synthesize a controller that avoids chatter vibrations in the presence of model uncertainties and while respecting key performance specifications. The experimental validation of this controller on a benchmark setup, involving a spindle system including an active magnetic bearing, shows that chatter can be robustly avoided while significantly increasing the material removal rate, i.e., the productivity.
N. van de Wouw, N. J. M. van Dijk, A. Schiffler, H. Nijmeijer, E. Abele

### Time-Delay Identification for Linear Systems: A Practical Method Using the Frequency Response Function

Abstract
The present paper gives a comprehensive study on delay parameter identification in linear system with time-delayed control. For the cases when the state matrix is prior known and is prior unknown, identification algorithms are provided. For the former case, delay identifiability depends on the measurability of the outputs that serve as the delayed feedback; while for the latter case, the external input information, including the positions where the delayed control acts, is also a necessity. The algorithms are stated in a unified programming scheme because the main steps for the two cases are basically the same. To verify the algorithm, an experiment on an active vibration absorber and a simulation on an active truss are performed. The results show a good convergence and preciseness of the proposed algorithm.
Xiaoxu Zhang, Jian Xu

### Analysis of Thermoacoustic Instability: A Time-Delay System Approach

Abstract
This study is on the analysis of thermoacoustic instability on a Rijke tube. This phenomenon results from a coupling between the heat release rate fluctuations and acoustic pressure. The simplified dynamics is modeled as a linear time-invariant multiple time-delayed system of neutral type. The conditions leading to unstable operation are identified using the Cluster Treatment of Characteristic Roots (CTCR) paradigm. This method assesses the stability of time-delay systems exhaustively and non-conservatively in the space of system parameters. Several experimental tests are conducted on a laboratory scale Rijke tube setup, and their results are used to verify the analytical findings.
Umut Zalluhoglu, Nejat Olgac
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