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This book mostly results from a selection of papers presented during the 11th IFAC (International Federation of Automatic Control) Workshop on Time-Delay Systems, which took place in Grenoble, France, February 4 - 6, 2013. During this event, 37 papers were presented. Taking into account the reviewers' evaluation and the papers' presentation the best papers have been selected and collected into the present volume. The authors of 13 selected papers were invited to participate to this book and provided a more detailed and improved version of the conference paper. To enrich the book, three more chapters have been included from specialists on time-delay systems who presented their work during the 52nd IEEE Conference on Decision and Control, which held in December 10 - 13, 2013, at Florence, Italy. The content of the book is divided into four main parts as follows: Modeling, Stability analysis, Stabilization and control, and Input-delay systems. Focusing on various topics of time-delay systems, this book will be interesting for researchers and graduate students working on control and system theory. .

Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the 1D heat equation, under the assumption that one of the observation or control operators is bounded. This paper suggests a constructive method for such controllers for 1D parabolic PDEs with both (observation and control) operators being unbounded. We consider the Kuramoto-Sivashinsky equation (KSE) under either boundary or in-domain point measurement and boundary actuation. We employ a modal decomposition approach via dynamic extension, using eigenfunctions of a Sturm-Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer dimension $N$ may be larger than this number. We suggest a direct Lyapunov approach to the full-order closed-loop system, which results in an LMI whose elements and dimension depend on $N$. The value of $N$ and the decay rate are obtained from the LMI. We extend our approach to internal stabilization with guaranteed $L^2$-gain and input-to-state stabilization. We prove two crucial properties of the derived LMIs. First, We prove that the LMIs are always feasible provided $N$ and the $L^2$ or ISS gains are large enough, thereby obtaining guarantees for our approach. Moreover, for the case of stabilization, we show that feasibility of the LMI for some $N$ implies its feasibility for $N+1$ (i.e., enlarging $N$ in the LMI cannot deteriorate the resulting decay rate of the closed-loop system). Numerical examples demonstrate the efficiency of the method.

International audience; In this paper, input-to-state stability and stabilization conditions in time-delay generalized Persidskii systems are studied. These conditions are formulated in terms of linear matrix inequalities, which may depend on delay values, and be local or global in state space. Numerical examples of opinion dynamics and the Lotka-Volterra model illustrate the efficiency of the proposed results.

This paper addresses sampled-data control of 2D Kuramoto-Sivashinsky equation over a rectangular domain. We suggest to divide the 2D rectangular into N sub-domains, where sensors provide spatially averaged or point state measurements to be transmitted through communication network to the controller. Note that differently from 2D heat equation, here we manage with sampled-data control under point measurements. We design a regionally stabilizing controller applied through distributed in space characteristic functions. Sufficient conditions ensuring regional stability of the closed-loop system are established in terms of linear matrix inequalities (LMIs). By solving these LMIs, we find an estimate on the set of initial conditions starting from which the state trajectories of the system are exponentially converging to zero. A numerical example demonstrates the efficiency of the results.

We develop a PDE-based approach to multi-agent deployment where each agent measures its relative position to only one neighbor. First, we show that such systems can be modeled by a first-order hyperbolic partial differential equation (PDE) whose L2-stability implies the stability of the multi-agent system for a large enough number of agents. Then, we show that PDE modelling helps to construct a Lyapunov function for the multi-agent system using spatial discretisation. Then, we use the PDE model to estimate the leader input delay preserving the stability.

In this paper, we study the digital implementation of derivative-dependent control for consensus of stochastic multi-agent systems. The consensus controllers that depend on the output and its derivatives are approximated as delayed sampled-data controllers. First, we consider the nth-order stochastic multi-agent systems. Second, we consider PID control of the second-order stochastic multi-agent systems. For the consensus analysis, we propose novel Lyapunov-Krasovskii functionals to derive linear matrix inequalities (LMIs) that allow to find admissible sampling period. The efficiency of the presented approach is illustrated by numerical examples.

In the present paper, we consider network-based control of linear systems with state multiplicative noise. For the sensor-controller network, Round-Robin (RR), try-once-discard (TOD) and independent and identically-distributed (i.i.d) protocols are proposed to orchestrate the measurement transmission from multiple sensor nodes. By using the time-delay approach to networked control systems, sufficient conditions for the exponential mean-square stabilization are obtained in terms of linear matrix inequalities (LMIs). Moreover, a discrete-time dynamic event-trigger is suggested to reduce the number of sent control signals from controller to actuator. An air vehicle system from the literature illustrates the efficiency of the presented approach.

A novel control law is proposed to attenuate the influence of bounded disturbances and measurement noises for plants with vector output and sector-bounded nonlinearities. The control law is based on the estimation of measurement noises. Differently from the existing results, the ultimate bound of the closed-loop system depends only on one component of the noise vector (as well as on the disturbance). The proposed control law is extended to systems with uncertain input and output delays. The input-to-state stability conditions are given in terms of matrix inequalities. The efficiency and advantages of the results over the existing methods are demonstrated by numerical examples.

This article proposes a decentralized predictor-based output feedback to compensate for large delays in large-scale interconnected systems under the premise that the interconnections between subsystems are not strong. The full state of each subsystem is assumed to be unmeasurable and the observer-based output feedback is employed. Two methods are used to tackle the large delays: the backstepping-based partial differential equation (PDE) approach and the reduction-based ordinary differential equation (ODE) approach. The PDE method is used for continuous-time control and manages with larger delays, whereas the ODE method is applicable to sampled-data implementation under discrete-time measurement.

We consider delayed boundary stabilization of a one-dimensional reaction–diffusion equation under boundary delayed measurements. We design an observer-based control law via the modal decomposition approach. The observer is governed by a partial differential equation, which leads to separation of the observer and the controller design. We suggest a network-based implementation of the controller in the presence of two networks: from sensor to controller, and from the controller to actuator. To reduce the workload of the second network, we suggest a novel switching-based dynamic event-triggering mechanism. We extend the results to the vector case and illustrate their efficiency by a numerical example.

Static output-feedback stabilization of systems with relative degree $n$ with matched disturbances is considered. Assuming that the system is controllable, a static output-feedback sliding mode controller is designed, where the output derivatives up to the order $(n-1)$ are approximated by using the current and the delayed values of the output. Numerical examples illustrate the efficiency of the method.

Event-triggered control (ETC) is becoming increasingly popular in load frequency control (LFC) operation in power systems. So far, the existing literature has primarily focused on load disturbance rejection in face of communication delay without considering much on networked-induced measurement disturbances and messages sent. In this paper, we study the LFC operation as a problem of networked control system and propose, for the first time, an ETC based switching approach for frequency regulation. We derive delay-dependent conditions to obtain the lower bound on inter-event time during event-trigger mechanism which assures the robust stability under networked induced time-varying delays and measurement disturbance and reduction in the amount of sent measurements. The effectiveness of the proposed approach is verified via numerical examples based on single-, two-area power systems and single-area with electric vehicle (EV) participation.

In this paper, we design a novel observer for a class of semilinear heat one dimmensional (1-D) equations under the delayed and sampled point measurements. The main novelty is that the delay is arbitrary. To handle any arbitrary delay, the observer is constituted of a chain of subobservers. Each subobserver handles a fraction of the considered delay. The resulting estimation error system is shown to be exponentially stable under a sufficient number of subobservers is used. The stability analysis is based on a specific Lyapunov–Krasovskii functional and the stability conditions are expressed in terms of linear matrix inequalities (LMIs).

In this paper, we propose a state-dependent switching controller for multiple-input multiple-output (MIMO) bilinear systems with constant delays in both the state and the input. The control input is assumed to be restricted to take only a finite number of values. The stability analysis of the closed loop is based on a Lyapunov–Krasovskii functional, and the design is reduced to solve a system of linear matrix inequalities. The controller can be designed by considering (state) delay-dependent or delay-independent conditions.

The objective of the present paper is finite-dimensional observer-based control of 1-D linear heat equation with constructive and easily implementable design conditions. We propose a modal decomposition approach in the cases of bounded observation and control operators (i.e, non-local sensing and actuation). The dimension of the controller is equal to the number of modes which decay slower than a given decay rate δ > 0. The observer may have a larger dimension N. The observer and controller gains are found separately of each other. We suggest a direct Lyapunov approach to the full-order closed-loop system and provide linear matrix inequalities (LMIs) for finding N and the exponential decay rate of the closed-loop system. Different from some existing qualitative methods, we prove that the LMIs are always feasible for large enough N leading to easily verifiable conditions. A numerical example demonstrates the efficiency of our method that gives non-conservative bounds on N and δ.

A majority of existing literature on time-delay systems focus on the robust stability of a single plant with respect to a “small” delay. This paper proposes a decentralized predictor-based feedback to compensate large delays for large-scale interconnected systems. The full-state of each subsystem is assumed to be unmeasurable and the observer-based output feedback is employed. Two methods are used to tackle the large delays: the backstepping-based partial differential equation (PDE) method is employed for continuous-time control, which derives simpler linear matrix inequality (LMI) conditions and manages with larger delays, whereas the reduction-based ordinary differential equation (ODE) method is applied to sampled-data implementation under continuous-time measurement. Instead of treating the large-scale systems as a whole, a decentralized Lyapunov-Krasovskii method is presented to guarantee the exponential stability of the large-scale systems under decentralized predictors.

In this paper, we study the digital implementation of derivative-dependent control for consensus of the nth-order stochastic multi-agent systems. The consensus controllers that depend on the output and its derivatives up to the order n - 1 are approximated as delayed sampled-data controllers. For the consensus analysis, we propose novel Lyapunov-Krasovskii functionals to derive linear matrix inequalities (LMIs) that allow to find admissible sampling period. The efficiency of the presented approach is illustrated by numerical examples.

The work addresses constrained control of Burgers’ equation by using distributed in space point actuation and measurements. An observer-based distributed-in-domain point control law is suggested to stabilize the system, where the controller employs averaged values of the observer. Constructive conditions are derived to ensure that the resulting closed-loop system is regionally exponentially stable. A numerical example demonstrates the efficiency of the results.

A novel control law is proposed to attenuate the influence of input and output disturbances for systems with vector output and sector bounded nonlinearities. The control law is based on estimation of the disturbance in the output. Differently from the existing results, the ultimate bound of the closed-loop system depends on only one component of the output disturbance vector (as well as, on the input disturbance). The results are formulated in terms of LMIs. The efficiency and advantages of the results over the existing methods are demonstrated by numerical examples.

We study stability of linear systems with fast time-varying coefficients. The classical averaging method guarantees the stability of such systems for small enough values of parameter provided the corresponding averaged system is stable. However, it is difficult to find an upper bound on the small parameter by using classical tools for asymptotic analysis. In this paper we introduce an efficient constructive method for finding an upper bound on the value of the small parameter that guarantees a desired exponential decay rate. We transform the system to a model with time-delays of the length of the small parameter. The resulting time-delay system is a perturbation of the averaged LTI system which is assumed to be exponentially stable. The stability of the time-delay system guarantees the stability of the original one. We construct an appropriate Lyapunov functional for finding sufficient stability conditions in the form of linear matrix inequalities (LMIs). The upper bound on the small parameter that preserves the exponential stability is found from LMIs. Two numerical examples (stabilization by vibrational control and by time-dependent switching) illustrate the efficiency of the method.

Deployment of a second-order nonlinear multi agent system over a desired open smooth curve in 2D or 3D space is considered. We assume that the agents have access to their velocities and to the local information of the desired curve and their displacements with respect to their closest neighbors, whereas in addition a leader is able to measure his absolute position. We assume that a small number of leaders transmit their measurements to other agents through a communication network. We take into account the following network imperfections: the variable sampling, transmission delay and quantization. We propose a static output-feedback controller and model the resulting closed-loop system as a disturbed (due to quantization) nonlinear damped wave equation with delayed point state measurements, where the state is the relative position of the agents with respect to the desired curve. To manage with the open curve we consider Neumann boundary conditions. We derive linear matrix inequalities (LMIs) that guarantee the input-to-state stability (ISS) of the system. The advantage of our approach is in the simplicity of the control law and the conditions. Numerical example illustrates the efficiency of the method.

This paper studies sampled-data implementation of event-triggered PI control for time-delay systems with parametric uncertainties. The systems are given by continuous-time linear systems with parameter uncertainty polytopes. We propose an event-triggered PI controller, in which the controller transmits its signal to the actuator when its relative value goes beyond a threshold. A state-space formulation of the Smith predictor is used to compensate the time-delay. An asymptotic stability condition is derived in the form of LMIs using a Lyapunov-Krasovskii functional. Numerical examples illustrate that our proposed controller reduces the communication load without performance degradation and despite plant uncertainties.

International audience; Constant power loads (CPLs) in power systems have a destabilizing effect that gives rise to significant oscillations or to network collapse, motivating the development of new methods to analyse their effect in AC and DC power systems. A sine qua non condition for this analysis is the availability of a suitable mathematical model for the CPL. In the case of DC systems power is simply the product of voltage and current, hence a CPL corresponds to a first-third quadrant hyperbola in the loads voltagecurrent plane. The same approach is applicable for balanced three-phase systems that, after a rotation of the coordinates to the synchronous frame, can be treated as a DC system. Modelling CPLs for single-phase (or unbalanced poly-phase) AC systems, on the other hand, is a largely unexplored area because in AC systems (active and reactive) power involves the integration in a finite window of the product of the voltage and current signals. In this paper we propose a simple dynamic model of a CPL that is suitable for the analysis of single-phase AC systems. We give conditions on the tuning gains of the model that guarantee the CPL behaviour is effectively captured.

We construct finite-dimensional observers for a one-dimensional reaction-diffusion system with boundary measurements subject to time-delays and data sampling. The system has a finite number of unstable modes approximated by a Luenberger-type observer. The remaining modes vanish exponentially. For a given reaction coefficient, we show how many modes one should use to achieve a desired rate of convergence. The finite-dimensional part is analyzed using appropriate Lyapunov-Krasovskii functionals that lead to linear matrix inequalitie (LMI)-based convergence conditions feasible for small enough time-delay and sampling period. The LMIs can be used to find appropriate injection gains.

The problem of existence of a Lyapunov-Krasovskii functional (LKF) for nonlinear neutral type time-delay systems is revisited considering the uniform stability analysis and the LKF in a Sobolev space of absolutely continuous functions with bounded derivatives.

We study global finite-dimensional observer-based stabilization of a semilinear 1D heat equation with globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We propose a controller that is based on a nonlinear finite-dimensional Luenberger observer. Our Lypunov $H^1$-stability analysis leads to LMIs, which are shown to be feasible for a large enough observer dimension and small enough Lipschitz constant. Next, we consider the case of a constant input delay $r>0$. To compensate the delay, we introduce a chain of $M$ sub-predictors that leads to a nonlinear closed-loop ODE system, coupled with nonlinear infinite-dimensional tail ODEs. We provide LMIs for $H^1$-stability and prove that for any $r>0$, the LMIs are feasible provided $M$ and $N$ are large enough and the Lipschitz constant is small enough. Numerical examples demonstrate the efficiency of the proposed approach.

We consider an LTI system of relative degree r ≥ 2 that can be stabilized using r − 1 output derivatives. The derivatives are approximated by finite differences leading to a time-delayed feedback. We present a new method of designing and analyzing such feedback under continuous-time and sampled measurements. This method admits essentially larger time-delay/sampling period compared to the existing results and, for the first time, allows to use consecutively sampled measurements in the sampled-data case. The main idea is to present the difference between the derivative and its approximation in a convenient integral form. The kernel of this integral is hard to express explicitly but we show that it satisfies certain properties. These properties are employed to construct the Lyapunov–Krasovskii functional that leads to LMI-based stability conditions. If the derivative-dependent control exponentially stabilizes the system, then its time-delayed approximation stabilizes the system with the same decay rate provided the time-delay (for continuous-time measurements) or the sampling period (for sampled measurements) are small enough.

We study constant input delay compensation by using finite-dimensional observer-based controllers in the case of the 1D heat equation. We consider Neumann actuation with nonlocal measurement and employ modal decomposition with $N+1$ modes in the observer. We introduce a chain of $M$ sub-predictors that leads to a closed-loop ODE system coupled with infinite-dimensional tail. Given an input delay $r$ , we present LMI stability conditions for finding $M$ and $N$ and the resulting exponential decay rate and prove that the LMIs are always feasible for any $r$ . We also consider a classical observer-based predictor and show that the corresponding LMI stability conditions are feasible for any $r$ provided $N$ is large enough. A numerical example demonstrates that the classical predictor leads to a lower-dimensional observer. However, it is known to be hard for implementation due to the distributed input signal.

This paper addresses and solves the secondary voltage regulation control problem in inverter-based islanded Microgrids (MGs) via a fully distributed delayed sampled-data PID controller, whose derivative action is approximated using finite difference. By choosing a small enough sampling period and leveraging artificial delays approach, the proposed strategy ensures the secondary voltage regulation with closed-loop performances similar to ones achievable via a continuous-time PID controller, but with a significant reduction of the communication burden, while improving the efficiency of the entire MG. Exponential stability of the closed-loop MG network is analytically proved via Lyapunov-Krasovskii theory and the derived sampling-dependent stability conditions are expressed as a set of LMIs, whose solution allows finding the weighted L2 gain. Finally, a detailed simulation analysis confirms the effectiveness and the robustness of the proposed approach.

International audience; The distributed estimation problem is solved for continuous-time observer nodes that obtain real-time measurements but communicate with their neighbors over a communication network. To this end, the digital communication between the observer nodes is modeled by the time-delay approach where variable sampling intervals, transmission delays and packet dropouts are taken into account. An LMI for the design of the observer gains is derived using Halanay’s inequality, the feasibility of which guarantees exponential stability with a selected convergence rate up to a maximum total delay. A comparison of the maximal delay on a numerical example shows the advantage of a distributed observer over a centralized one.

International audience; This work addresses distributed event-triggered control law of 1-D nonlinear Korteweg-de Vries (KdV) equation posed on a bounded domain. Such a system, in a continuous framework, is exponentially stabilizable by a linear state feedback as a source term. Here we consider the situation where the feedback is sampled in time and piecewise averaged in space, and an event-triggering mechanism is designed to maintain stability of this infinite dimensional system. Both well-posedness of the closed-loop system and avoiding the Zeno behaviour issues are addressed. Sufficient LMI-based conditions are constructed to guarantee the regional exponential stability. A numerical example illustrates the efficiency of the method.

International audience; In this paper, we present improved results on observer design for 1D heat equation. We first introduce an observer under delayed spatially point measurements that leads to an error heat equation with time-delay. Inspired by recent developments in the area of delayed ODEs, we propose novel Lyapunov functionals based on the Legendre polynomials. Then, new Bessel-Legendre (BL) inequalities are provided to derive sufficient stability conditions in the form of linear matrix inequalities (LMIs) that are parameterized by the degree of the polynomials. Finally, a numerical example illustrates the efficiency of the results that allow to enlarge the value of delay preserving the stability by more than 20%.

The increasing ease of obtaining and processing data together with the growth in system complexity has sparked the interest in moving from conventional model-based control design towards data-driven concepts. Since in many engineering applications time delays naturally arise and are often a source of instability, we contribute to the data-driven control field by introducing data-based formulas for state feedback control design in linear discrete-time time-delay systems with uncertain delays. With the proposed approach, the problems of system stabilization as well as of guaranteed cost and $H_{\infty}$ control design are treated in a unified manner. Extensions to determine the system delays and to ensure robustness in the event of noisy data are also provided., 16 pages, 4 figures, 2 tables

International audience; The problem of output robust adaptive stabilization for a class of Lipschitz nonlinear systems is studied under assumption that the measurements are available with a constant bias. The state reconstruction is avoided by using delayed values of the output in the feedback and adaptation laws. The control and adaptation gains can be selected as a solution of the proposed linear matrix inequalities (LMIs). The efficiency of the presented approach is demonstrated on a nonlinear pendulum through simulations.

[EN] This paper deals with asymptotic stabilisation of a class of nonlinear input-delayed systems via dynamic output feedback in the presence of disturbances. The proposed strategy has the structure of an observer-based control law, in which the observer estimates and predicts both the plant state and the external disturbance. A nominal delay value is assumed to be known and stability conditions in terms of linear matrix inequalities are derived for fast-varying delay uncertainties. Asymptotic stability is achieved if the disturbance or the time delay is constant. The controller design problem is also addressed and a numerical example with an unstable system is provided to illustrate the usefulness of the proposed strategy., This work was partially supported by: Ministerio de Economía y Competitividad, Spain (TIN2017-86520-C3-1-R); Universitat Politècnica de València (FPI-UPV 2014 PhD Grant); and Israel Science Foundation (Grant No. 1128/14).

International audience; The conditions of existence of a Lyapunov-Krasovskii functional (LKF) for nonlinear input-to-state stable (ISS) neutral type systems are proposed. The system under consideration depends nonlinearly on the delayed state and the delayed state derivative, and satisfies the conditions for the existence and uniqueness of the solutions. The LKF and the system properties are defined in a Sobolev space of absolutely continuous functions with bounded derivatives.

International audience; An output robust adaptive control is designed for a class of Lipschitz nonlinear systems under assumption that the measurements are available with a constant bias and the state equations linearly parameterized by unknown parameters and external disturbances. A dynamic state reconstruction (synthesis of an observer) is avoided by using delayed values of the output in the feedback and adaptation laws. The analysis of robust stability for the resulted time-delay system is performed by using the Lyapunov-Krasovskii approach. The control and adaptation gains can be selected as a solution of the proposed linear matrix inequalities. This research is motivated by a nonlinear pendulum control problem, and the efficacy of the developed control is demonstrated on this application through experiments.

International audience; The problems of state estimation and observer-based control for heat non-homogeneous equations under distributed in space point measurements are considered. First, an interval observer is designed in the form of Partial Differential Equations (PDEs), without Galerkin projection. Second, conditions of boundedness of the interval observer solutions with non-zero boundary conditions and measurement noise are proposed. Third, the obtained interval estimates are used to design a dynamic output-feedback stabilizing controller. The advantages of the PDE-based interval observer over the approximation-based one are clearly shown in the numerical example.

Recently finite-dimensional observer-based controllers were introduced for the 1D heat equation, where at least one of the observation or control operators is bounded. In this paper, for the first time, we manage with such controllers for the 1D heat equation with both operators being unbounded. We consider Dirichlet actuation and point measurement and use a modal decomposition approach via dynamic extension. We suggest a direct Lyapunov approach to the full-order closed-loop system, where the finite-dimensional state is coupled with the infinite-dimensional tail of the state Fourier expansion, and provide LMIs for finding the controller dimension and the resulting exponential decay rate. We further study sampled-data implementation of the controller under sampled-data measurement. We use Wirtinger-based, discontinuous in time, Lyapunov functionals which compensate sampling in the finite-dimensional state. To compensate sampling in the infinite-dimensional tail, we use a novel form of Halanay's inequality, which is appropriate for Lyapunov functions with jump discontinuities that do not grow in the jumps. Numerical examples demonstrate the efficiency of the method.

International audience; A new theory of homogeneity for neutral type systems with application to fast stabilization of the 2nd-order integrator is proposed. It is assumed that only the position is available for measurements, and the designed feedback uses the output and its delayed values without an estimation of velocity. It is shown that by selecting the closed-loop system to be homogeneous with negative or positive degree, it is possible to accelerate the rate of convergence in the system at the price of a small steady-state error. Robustness of the developed stabilization strategy with respect to exogenous perturbations is investigated. The efficiency of the proposed control is demonstrated in simulations.

Recently, finite-dimensional observer-based controllers were introduced for 1D parabolic PDEs via the modal decomposition method. In the present paper we suggest a sampled-data implementation of a finite-dimensional boundary controller for 1D parabolic PDEs under discrete-time point measurement. We consider the heat equation under boundary actuation and point (either in-domain or boundary) measurement. In order to manage with point measurement, we employ dynamic extension and prove H 1 -stability. Due to dynamic extension, which leads to proportional–integral controller, we suggest a sampled-data implementation of the controller via a generalized hold device. We take into account the quantization effect that leads to a disturbed closed-loop system and input-to-state stability (ISS) analysis. We use Wirtinger-based piecewise continuous in time Lyapunov functionals which compensate sampling in the finite-dimensional state and lead to the simplest efficient stability conditions for ODEs. To compensate sampling in the infinite-dimensional tail, we introduce a novel form of Halanay’s inequality for ISS, which is appropriate for functions with jump discontinuities that do not grow in the jumps. Numerical examples demonstrate the efficiency of our method.