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## Über dieses Buch

This edited monograph includes state-of-the-art contributions on continuous time dynamical networks with delays. The book is divided into four parts. The first part presents tools and methods for the analysis of time-delay systems with a particular attention on control problems of large scale or infinite-dimensional systems with delays. The second part of the book is dedicated to the use of time-delay models for the analysis and design of Networked Control Systems. The third part of the book focuses on the analysis and design of systems with asynchronous sampling intervals which occur in Networked Control Systems. The last part of the book exposes several contributions dealing with the design of cooperative control and observation laws for networked control systems. The target audience primarily comprises researchers and experts in the field of control theory, but the book may also be beneficial for graduate students.

## Inhaltsverzeichnis

### Chapter 1. On the Codimension of the Singularity at the Origin for Networked Delay Systems

Continuous-time dynamical networks with delays have a wide range of application fields from biology, economics to physics, and engineering sciences. Usually, two types of delays can occur in the communication network: internal delays (due to specific internal dynamics of a given node) and external delays (related to the communication process, due to the information transmission and processing). Besides, such delays can be totally different from one node to another. It is worth mentioning that the problem becomes more and more complicated when a huge number of delays has to be taken into account. In the case of constant delaysDelayconstant delay, this analysis relies much on the identification and the understanding of the spectral values behavior with respect to an appropriate set of parameters when crossing the imaginary axis. There are several approaches for identifying the imaginary crossing roots, though, to the best of the authors’ knowledge, the bound of the multiplicity of such roots has not been deeply investigated so far. This chapter provides an answer for this question in the case of time-delay systemsSystemtime-delay system, where the corresponding quasi-polynomial functionPolynomialsquasi-polynomial function has non-spare polynomials and no coupling delays. Furthermore, we will also show the link between this multiplicity problem and Vandermonde matricesMatrixVandermonde matrix, and give the upper bound for the multiplicity of an eigenvalue at the origin for such a time-delay systemSystemtime-delay system modeling network dynamics in the presence of time-delay.

Dina-Alina Irofti, Islam Boussaada, Silviu-Iulian Niculescu

### Chapter 2. Stability and Stabilization for Continuous-Time Difference Equations with Distributed Delay

Motivated by linear hyperbolic conservation lawsConservation law, we investigate in this chapter new conditions for stabilityStabilityasymptotic stability and stabilizationStabilization for linear continuous-time difference equations with distributed delayDelaydistributed delay. For this, we propose first a state-space realization of networks of linear hyperbolic conservation laws via continuous-time difference equations. Then, based on some recent works, we propose sufficient conditions for exponential stabilityStabilityexponential stability, which appear also to be necessary and sufficient in some particular cases. Then, the stabilization problem as well as the closed-loop performances are analyzed with constructive methods for state feedback synthesis.

Michael Di Loreto, Sérine Damak, Sabine Mondié

### Chapter 3. Model Reduction for Norm Approximation: An Application to Large-Scale Time-Delay Systems

The computation of $$\mathscr {H}_2$$H2 and $$\mathscr {H}_{2,\varOmega }$$H2,Ω norms for LTI Time-Delay SystemsSystemtime-delay system (TDS) are important challenging problems for which several solutions have been provided in the literature. Several of these approaches, however, cannot be applied to systems of large dimension because of the inherent poor scalability of the methods, e.g., LMIs or Lyapunov-based approaches. When it comes to the computation of frequency-limited norms, the problem tends to be even more difficult. In this chapter, a computationally feasible solution using $$\mathscr {H}_2$$H2 model reduction for TDS, based on the ideas provided in [3], is proposed. It is notably demonstrates on several examples that the proposed method is suitable for performing both accurate model reduction and norm estimation for large-scale TDS.

Igor Pontes Duff, Pierre Vuillemin, Charles Poussot-Vassal, Corentin Briat, Cédric Seren

### Chapter 4. General Formula for Event-Based Stabilization of Nonlinear Systems with Delays in the State

In this chapter, a universal formula is proposed for event-based stabilizationStabilization of nonlinear systems affine in the control and with delays in the state. The feedback is derived from the seminal law proposed by E. Sontag (1989) and then extended to event-based control of affine nonlinear undelayed systems. Under the assumption of the existence of a control Lyapunov–Krasovskii functional (CLKF), the proposal enables smooth (except at the origin) asymptotic stabilization while ensuring that the sampling intervals do not contract to zero. Global asymptotic stabilityStabilityasymptotic stability is obtained under the small control property assumption. Moreover, the control can be proved to be smooth anywhere under certain conditions. Simulation results highlight the ability of the proposed formula. The particular linear case is also discussed.

Sylvain Durand, Nicolas Marchand, J. Fermi Guerrero-Castellanos

### Chapter 5. Analysis of Bilinear Systems with Sampled-Data State Feedback

In this chapter we consider the stability analysis of bilinear systems controlled via a sampled-data state feedback controller. Sampling periods may be time-varyingSamplingtime-varying sampling and subject to uncertainties. The goal of this study is to find a constructive manner to estimate the maximum allowable sampling period (MASP)Samplingmaximum allowable sampling period that guarantees the local stability of the system. Stability criteria are proposed in terms of linear matrix inequalities (LMI).

Hassan Omran, Laurentiu Hetel, Jean-Pierre Richard, Françoise Lamnabhi-Lagarrigue

### Chapter 6. On the Stability Analysis of Sampled-Data Systems with Delays

Controlling a system through a network amounts to solve certain difficulties such as, among others, the consideration of aperiodic samplingSamplingtime-varying sampling schemes and (time-varying) delays. In most of the existing works, delays have been involved in the input channel through which the system is controlled, thereby delaying in a continuous way the control input computed by the controller. We consider here a different setup where the delay acts in a way that the current control input depends on past state samples, possibly including the current one, which is equivalent to considering a discrete-time delay, at the sample level, in the feedback loop. An approach based on the combination of a discrete-time Lyapunov–Krasovskii functionalLyapunov–Krasovskii Functional and a looped-functionalLooped-functional is proposed and used to obtain tailored stability conditions that explicitly consider the presence of delays and the aperiodic nature of the sampling events. The stability conditions are expressed in terms of linear matrix inequalities and the efficiency of the approach is illustrated on an academic example.

Alexandre Seuret, Corentin Briat

### Chapter 7. Output Feedback Event-Triggered Control

Event-triggered controlEvent-triggered control has been proposed as an alternative implementation to conventional time-triggered approach in order to reduce the amount of transmissions. The idea is to adapt transmissions to the state of the plant such that the loop is closed only when it is needed according to the stability or/and the performance requirements. Most of the existing event-triggered control strategies assume that the full state measurement is available. Unfortunately, this assumption is often not satisfied in practice. There is therefore a strong need for appropriate tools in the context of output feedback control. Most existing works on this topic focus on linear systems. The objective of this chapter is to first summarize our recent results on the case where the plant dynamics is nonlinear. The approach we follow is emulation as we first design a stabilizing output feedback law in the absence of sampling; then we consider the network and we synthesize the event-triggering condition. The latter combines techniques from event-triggered and time-triggered control. The results are then proved to be applicable to linear time-invariant (LTI) systems as a particular case. We then use these results as a starting point to elaborate a co-design method, which allows us to jointly construct the feedback law and the triggering condition for LTI systems where the problem is formulated in terms of linear matrix inequalities (LMI). We then exploit the flexibility of the method to maximize the guaranteed minimum amount of time between two transmissions. The results are illustrated on physical and numerical examples.

Mahmoud Abdelrahim, Romain Postoyan, Jamal Daafouz, Dragan Nešić

### Chapter 8. Stabilization by Quantized Delayed State Feedback

This chapter is devoted to the design of a static-state feedback controller for a linear system subject to saturatedSaturationquantizationQuantization and delay in the input. Due to quantization and saturation, we consider, for the closed-loop system, a weaker notion of stability, namely local ultimate boundedness. The closed-loop system is then modeled as a stable linear system subject to discontinuous perturbations. Then by coupling a certain Lyapunov–Krasovskii functionalLyapunov–Krasovskii Functional via S-procedure to adequate sector conditions, we derive sufficient conditions to ensure for the trajectories of the closed-loop system finite time convergence into a compact $$S_u$$Su surrounding the origin, from every initial condition belonging to a compact set $$S_0$$S0. Moreover, the size of the initial condition set $$S_0$$S0 and the ultimate set $$S_u$$Su are then optimized by solving a convex optimization problem over linear matrix inequality (LMI) constraints. Finally, an example extracted from the literature shows the effectiveness of the proposed methodology.

Francesco Ferrante, Frédéric Gouaisbaut, Sophie Tarbouriech

### Chapter 9. Discrete-Time Networked Control Under Scheduling Protocols

This chapter analyzes the exponential stabilityStabilityexponential stability of discrete-time networked control systems viaStabilitydelay-dependent condition delay-dependent Lyapunov-Krasovskii methodsLyapunov–Krasovskii Functional. The time-delay approach has been developed recently for the stabilization of continuous-time networked control systemsSystemnetworked control system under a Round-Robin protocolCommunication protocolsRound-Robin protocol and a weighted Try-Once-Discard protocolCommunication protocolsTry-Once-Discard, respectively. In the present chapter, the time-delay approach is extended to the stability analysis of discrete-time networked control systems under both these scheduling protocols. First, the closed-loop system is modeled as a discrete-time switched system with multiple and ordered time-varying delays under the Round-Robin protocol. Then, a discrete-time hybrid system modelSystemhybrid system for the closed-loop system is presented under these protocols. It contains time-varying delays in the continuous dynamics and in the reset conditions. The communication delaysDelaynetwork delay are allowed to be larger than the sampling intervals. Polytopic uncertainties in the system model can be easily included in our analysis. The efficiency of the time-delay approach is illustrated in an example of a cart-pendulum system.

Kun Liu, Emilia Fridman, Karl Henrik Johansson

### Chapter 10. Stabilization of Networked Control Systems with Hyper-Sampling Periods

This chapter considers the stabilization of Networked Control SystemsSystem networked control system (NCSs) under the hyper-sampling mode. Such a sampling mode, recently proposed in the literature, appears naturally in the scheduling policies of real-time systems under constrained (calculation and communication) resources. Meanwhile, as expected, the stabilization problemStabilization under the hyper-sampling mode is much more complicated than in the case of single-sampling mode. In this chapter, we propose a procedure to design the feedback gain matrix such that we can obtain a stabilizable region as large as possible. In the first step, we determine the stabilizable region under the single-sampling period. This step can be easily obtained by solving some linear matrix inequalities (LMIs) and from the result we may obtain a stabilizable region for the hyper-sampling period. Then, in the second step, we further detect the stabilizable region, based on the one found in the first step, by adjusting the feedback gain matrices based on the asymptotic behavior analysis. By this step, a larger stabilizable region may be found and this step can be used in an iterative manner. The proposed procedure will be illustrated by a numerical example. We can see from the example that the stabilizable region under the hyper-sampling period may lead to a smaller average sampling frequency (ASF) guaranteeing the stability of the NCS than the single-sampling period, i.e., less system resources are required by the hyper-sampling periodSamplinghyper-sampling.

Xu-Guang Li, Arben Çela, Silviu-Iulian Niculescu

### Chapter 11. Optimal Control Strategies for Load Carrying Drones

This chapter studies control strategies for load carrying drones. Load carrying drones not only have to fly in a cooperative way, but also are mechanically interconnected. Due to these characteristics, the control problem is an interesting and challenging issue to deal with. Throughout this chapter, a dynamic model based on first principle is developed. To that end, it is proposed to model this system as a ball and beam system lifted by two drones. Afterwards, different control techniques are implemented and compared by simulations. Specifically, linear-quadratic regulatorControlLQR (LQR) and model predictive control (MPC)ControlMPC are studied. Both control techniques belong to the optimal control methodology. This comparison is interesting since LQR permits to perform an optimal control law with short execution times, while MPC deals with physical constraints and predictions, being the execution time and the physical constraints important issues to handle in this kind of systems. Finally, simulation results and open issues are discussed.

Alicia Arce Rubio, Alexandre Seuret, Yassine Ariba, Alessio Mannisi

### Chapter 12. Delays in Distributed Estimation and Control over Communication Networks

This chapter introduces a distributed estimation and control technique with application to networked systems. The problem consists of monitoring and controlling a large-scale plant using a network of agentsAgent which collaborate exchanging information over an unreliable network. We propose an agent-based scheme based on an estimation structure that combines local measurements of the plant with remote information received from neighboring agents. We discuss the design of stabilizing distributed controllers and observers when the interagent communication is affected by delaysDelaynetwork delay and packet dropouts. Some simulations will be shown to illustrate the performance of this approach.

Pablo Millán, Luis Orihuela, Isabel Jurado

### Chapter 13. Design and Analysis of Reset Strategy for Consensus in Networks with Cluster Pattern

This chapter addresses the problem of consensusConsensus in networks partitioned in several disconnected clusters. Each cluster is represented by a fixed, directed, and strongly connected graphs. In order to enforce the consensus, we assume that each cluster poses a leader that can reset its state by taking into account other leaders state. First, we characterize the consensus value of this model. Second, we provide sufficient condition in LMI form for the stability of the consensus. Finally, we perform a decay rate analysis and design the interaction network of the leaders which allows to reach a prescribed consensus value.

Marcos Cesar Bragagnolo, Irinel-Constantin Morărescu, Jamal Daafouz, Pierre Riedinger

### Chapter 14. Synthesis of Distributed Control Laws for Multi-agent Systems Using Delayed Relative Information with LQR Performance

In this chapter, a multiagent systemSystemmultiagent system composed of linear identical dynamical agents is considered. The agents are assumed to share relative state information over a communication network. This exchange of relative information is assumed to be subject to delays. New methods to synthesize distributed state feedback control laws for the multiagent system, using delayed relative information along with local state information with guaranteed LQRControlLQR performance, are presented in this chapter. Two types of delays are considered in the relative information exchange: fixed and time-varying. Existing delay-dependent stability criteria are modified to incorporate LQR performance guarantees while retaining convex LMI representations to facilitate the synthesis of the control gains.

Paresh Deshpande, Prathyush P. Menon, Christopher Edwards

### Chapter 15. Topology Preservation for Multi-agent Networks: Design and Implementation

We consider a network of interconnected systems with discrete-time dynamics. Each system is called agent and we assume that two agents can interact as far as their states are close in a sense defined by an algebraic relation. In this work, we present several implementation strategies answering to different classical problems in multiagent systemsSystemmultiagent system. The primary goal of our methodology is to characterize the controllers that preserve a given interconnection subgraph that makes possible the global coordination. The second goal is to choose among these controllers those that ensure an agreement. This is done by solving a convex optimization problem associated to the minimization of a well-chosen cost function. Examples concerning full or partial consensus of agents with double-integrator dynamics illustrate the implementation of the proposed methodology.

Irinel-Constantin Morărescu, Mirko Fiacchini

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