The feedback control system is defined as the sampling of an output signal and feeding it back to the input, resulting in an error signal that drives the overall system. This dissertation focuses on the stabilization and performance of state feedback control systems. Chapters 3 and 4 focus on the feedback control protocol approaching in the multi-agents system. In particular, the global regulation of distributed optimization problems has been considered. Firstly, we propose a distributed optimization algorithm based on the proportional-integral control strategy and the exponential convergence rate has been delivered. Moreover, a decentralized mechanism has been equipped to the proposed optimization algorithm, which enables an arbitrarily chosen agent in the system can compute the value of the optimal solution by only using the successive local states. After this, we consider the cost function follows the restricted secant inequality. A dynamic event-triggered mechanism design has been proposed. By ensuring the global regulation of the distributed proportional-integral optimization algorithm, the dynamic event-triggered mechanism efficiently reduces the communication frequency among agents. Chapter 5 focuses on the feedback control protocol approaching the single-agent system. Specifically, we investigate the truncated predictor feedback control of the regulation of linear input-delayed systems. For the purpose of improving the closed-loop performance, we propose a design of the truncated predictor feedback method with time-varying feedback parameters and give the potential range of choosing the time-varying feedback parameters to replace the traditional constant low gain parameters.

Cyber-physical systems (CPSs) tightly integrate physical processes with computing and communication to autonomously interact with the surrounding environment.This enables emerging applications such as autonomous driving, coordinated flightof swarms of drones, or smart factories. However, current technology does notprovide the reliability and flexibility to realize those applications. Challenges arisefrom wireless communication between the agents and from the complexity of thesystem dynamics. In this thesis, we take on these challenges and present three maincontributions.We first consider imperfections inherent in wireless networks, such as communication delays and message losses, through a tight co-design. We tame the imperfectionsto the extent possible and address the remaining uncertainties with a suitable controldesign. That way, we can guarantee stability of the overall system and demonstratefeedback control over a wireless multi-hop network at update rates of 20-50 ms.If multiple agents use the same wireless network in a wireless CPS, limitedbandwidth is a particular challenge. In our second contribution, we present aframework that allows agents to predict their future communication needs. Thisallows the network to schedule resources to agents that are in need of communication.In this way, the limited resource communication can be used in an efficient manner.As a third contribution, to increase the flexibility of designs, we introduce machinelearning techniques. We present two different approaches. In the first approach,we enable systems to automatically learn their system dynamics in case the truedynamics diverge from the available model. Thus, we get rid of the assumption ofhaving an accurate system model available for all agents. In the second approach, wepropose a framework to directly learn actuation strategies that respect bandwidthconstraints. Such approaches are completely independent of a system model andstraightforwardly extend to nonlinear settings. Therefore, they are also suitable forapplications with complex system dynamics. / <p>QC 20190118</p>

In this dissertation, we study multi-robot coordination in the context of multi-target tracking. Specifically, we are interested in the coordination achieved by means of submodular function optimization. Submodularity encodes the diminishing returns property that arises in multi-robot coordination. For example, the marginal gain of assigning an additional robot to track the same target diminishes as the number of robots assigned increases. The advantage of formulating coordination problems as submodular optimization is that a simple, greedy algorithm is guaranteed to give a good performance. However, often this comes at the expense of unrealistic models and assumptions. For example, the standard formulation does not take into account the fact that robots may fail, either randomly or due to adversarial attacks. When operating in uncertain conditions, we typically seek to optimize the expected performance. However, this does not give any flexibility for a user to seek conservative or aggressive behaviors from the team of robots. Furthermore, most coordination algorithms force robots to communicate at each time step, even though they may not need to. Our goal in this dissertation is to overcome these limitations by devising coordination algorithms that are parsimonious in communication, allow a user to manage the risk of the robot performance, and are resilient to worst-case robot failures and attacks. In the first part of this dissertation, we focus on designing parsimonious communication strategies for target tracking. Specifically, we investigate the problem of determining when to communicate and who to communicate with. When the robots use range sensors, the tracking performance is a function of the relative positions of the robots and the targets. We propose a self-triggered communication strategy in which a robot communicates its own position with its neighbors only when a certain set of conditions are violated. We prove that this strategy converges to the optimal robot positions for tracking a single target and in practice, reduces the number of communication messages by 30%. When tracking multiple targets, we can reduce the communication by forming subsets of robots and assigning one subset to track a target. We investigate a number of measures for tracking quality based on the observability matrix and show which ones are submodular and which ones are not. For non-submodular measures, we show a greedy algorithm gives a 1/(n+1) approximation, if we restrict the subset to n robots. In optimizing submodular functions, a common assumption is that the function value is deterministic, which may not hold in practice. For example, the sensor performance may depend on environmental conditions which are not known exactly. In the second part of the dissertation, we design an algorithm for stochastic submodular optimization. The standard formulation for stochastic optimization optimizes the expected performance. However, the expectation is a risk-neutral measure. Instead, we optimize the Conditional Value-at-Risk (CVaR), which allows the user the flexibility of choosing a risk level. We present an algorithm, based on the greedy algorithm, and prove that its performance has bounded suboptimality and improves with running time. We also present an online version of the algorithm to adapt to real-time scenarios. In the third part of this dissertation, we focus on scenarios where a set of robots may fail naturally or due to adversarial attacks. Our objective is to track as many targets as possible, a submodular measure, assuming worst-case robot failures. We present both centralized and distributed resilient tracking algorithms to cope with centralized and distributed communication settings. We prove these algorithms give a constant-factor approximation of the optimal in polynomial running time. / Doctor of Philosophy / Today, robotics and autonomous systems have been increasingly used in various areas such as manufacturing, military, agriculture, medical sciences, and environmental monitoring. However, most of these systems are fragile and vulnerable to adversarial attacks and uncertain environmental conditions. In most cases, even if a part of the system fails, the entire system performance can be significantly undermined. As robots start to coexist with humans, we need algorithms that can be trusted under real-world, not just ideal conditions. Thus, this dissertation focuses on enabling security, trustworthiness, and long-term autonomy in robotics and autonomous systems. In particular, we devise coordination algorithms that are resilient to attacks, trustworthy in the face of the uncertain conditions, and allow the long-term operation of multi-robot systems. We evaluate our algorithms through extensive simulations and proof-of-concept experiments. Generally speaking, multi-robot systems form the "physical" layer of Cyber-Physical Sytems (CPS), the Internet of Things (IoT), and Smart City. Thus, our research can find applications in the areas of connected and autonomous vehicles, intelligent transportation, communications and sensor networks, and environmental monitoring in smart cities.

In modern cyber-physical systems (CPSs) where the control signals are generally transmitted via shared communication networks, there is a desire to balance the closed-loop control performance with the communication cost necessary to achieve it. In this context, aperiodic real-time scheduling of control tasks comes into being and has received increasing attention recently. It is well known that model predictive control (MPC) is currently widely utilized in industrial control systems and has greatly increased profits in comparison with the proportional integral-derivative (PID) control. As communication and networks play more and more important roles in modern society, there is a great trend to upgrade and transform traditional industrial systems into CPSs, which naturally requires extending conventional MPC to communication-efficient MPC to save network resources. Motivated by this fact, we in this thesis propose robust MPC and scheduling co-design algorithms to networked CPSs possibly affected by both parameter uncertainties and additive disturbances. In Chapter 2, a dynamic event-triggered robust tube-based MPC for constrained linear systems with additive disturbances is developed, where a time-varying pre-stabilizing gain is obtained by interpolating multiple static state feedbacks and the interpolating coefficient is determined via optimization at the time instants when the MPC-based control is triggered. The original constraints are properly tightened to achieve robust constraint optimization and a sequence of dynamic sets used to test events are derived according to the optimized coefficient. We theoretically show that the proposed algorithm is recursively feasible and the closed-loop system is input-to-state stable (ISS) in the attraction region. Numerical results are presented to verify the design. In Chapter 3, a self-triggered min-max MPC strategy is developed for constrained nonlinear systems subject to both parametric uncertainties and additive disturbances, where the robust constraint satisfaction is achieved by considering the worst case of all possible uncertainty realizations. First, we propose a new cost function that relaxes the penalty on the system state in a time period where the controller will not be invoked. With this cost function, the next triggering time instant can be obtained at current time instant by solving a min-max optimization problem where the maximum triggering period becomes a decision variable. The proposed strategy is proved to be input-to-state practical stable (ISpS) in the attraction region at triggering time instants under some standard assumptions. Extensions are made to linear systems with additive disturbances, for which the conditions reduce to a linear matrix inequality (LMI). Comprehensive numerical experiments are performed to verify the correctness of the theoretical results. / Graduate

Networked control over wireless sensor and actuator systems is of growing importancein many application domains. Energy and communication bandwidth are scarce resources in such systems. Despite that feedback control might only be needed occasionally, sensor and actuator communications are often periodic and with high frequency in today’s implementations. In this thesis, resource-constrained wireless networked control systems with an adaptive sampling period are considered. Our first contribution is a system architecture for aperiodic wireless networked control. As the underlying data transmission is performed over a shared wireless network, we identify scheduling policies and medium access controls that allow for an efficient implementation of sensor communication. We experimentally validate three proposed mechanisms and show that best performance is obtained by a hybrid scheme, combining the advantages of event- and self-triggered control as well as the possibilities provided by contention-based and contention-free medium accesscontrol. In the second contribution, we propose an event-triggered PI controller for wireless process control systems. A novel triggering mechanism which decides the transmission instants based on an estimate of the control signal is proposed. It addresses some side-effects that have been discovered in previous PI proposals, which trigger on the state of the process. Through simulations we demonstrate that the new PI controller provides setpoint tracking and disturbance rejection close to a periodic PI controller, while reducing the required network resources. The third contribution proposes a co-design of feedback controllers and wireless medium access. The co-design is formulated as a constrained optimization problem, whereby the objective function is the energy consumption of the network and the constraints are the packet loss probability and delay, which are derived from the performance requirements of the control systems. The design framework is illustrated in a numerical example. / QC 20111004

978-91-7729-579-2A multi-agent system consists of multiple agents cooperating to achieve a common objective through local interactions. An important problem is how to reduce the amount of information exchanged, since agents in practice only have limited energy and communication resources. In this thesis, we propose dynamic event-triggered control strategies to solve consensus and formation problems for multi-agent systems under such resource constraints. In the first part, we propose dynamic event-triggered control strategies to solve the average consensus problem for first-order continuous-time multi-agent systems. It is proven that the state of each agent converges exponentially to the average of all agents' initial states under the proposed triggering laws if and only if the underlying undirected graph is connected.In the second part, we study the consensus problem with input saturation over directed graphs. It is shown that the underlying directed graph having a directed spanning tree is a necessary and sufficient condition for achieving consensus. Moreover, in order to reduce the overall need of communication and system updates, we propose an event-triggered control strategy to solve this problem. It is shown that consensus is achieved, again, if and only if the underlying directed graph has a directed spanning tree.In the third part, dynamic event-triggered formation control with connectivity preservation is investigated. Single and double integrator dynamics are considered. All agents are shown to converge to the formation exponentially with connectivity preservation.The effectiveness of the theoretical results in the thesis is verified by several numerical examples. / <p>QC 20171025</p>

La commande à transmissions événementielles est une approche dans laquelle les instants de transmission sont définis selon un critère dépendant de l'état du système et non plus d'une horloge à l'instar des implantations périodiques. Dans cette thèse, nous nous concentrons sur la synthèse de telles lois de commande par retour de sortie. Les contributions sont les suivantes : (i) nous proposons une méthode de synthèse dite par émulation pour des systèmes non linéaires; (ii) nous présentons une méthode de synthèse jointe de la loi de commande et de la condition de déclenchement pour les systèmes linéaires; (iii) nous nous intéressons au cas de systèmes non linéaires singulièrement perturbés et nous construisons le contrôleur à partir d’approximation de la dynamique lente uniquement. / Event-triggered control is a sampling paradigm in which the sequence of transmission instants is determined based on the violation of a state-dependent criterion and not a time-driven clock. In this thesis, we deal with event-triggered output-based controllers to stabilize classes of nonlinear systems. The contributions of the presented material are threefold: (i) we stabilize a class of nonlinear systems by using an emulation-based approach; (ii) we develop a co-design procedure to simultaneously design the output feedback law and the event-triggering condition for linear systems; (iii) we propose stabilizing event-triggered controllers for nonlinear systems whose dynamics have two-time scales (in particular, we only rely on the knowledge of an approximate model of the slow dynamics)

Low-power wireless technology is a part and parcel of our daily life, shaping the way in which we behave, interact, and more generally live. The ubiquity of cheap, tiny, battery-powered devices augmented with sensing, actuation, and wireless communication capabilities has given rise to a ``smart" society, where people, machines, and objects are seamlessly interconnected, among themselves and with the environment. Behind the scenes, low-power wireless protocols are what enables and rules all interactions, organising these embedded devices into wireless networks, and orchestrating their communications. The recent years have witnessed a persistent increase in the pervasiveness and impact of low-power wireless. After having spawned a wide spectrum of powerful applications in the consumer domain, low-power wireless solutions are extending their influence over the industrial context, where their adoption as part of feedback control loops is envisioned to revolutionise the production process, paving the way for the Fourth Industrial Revolution. However, as the scale and relevance of low-power wireless systems continue to grow, so do the challenges posed to the communication substrates, required to satisfy ever more strict requirements in terms of reliability, responsiveness, and energy consumption. Harmonising these conflicting demands is far beyond what is enabled by current network stacks and control architectures; the need to timely bridge this gap has spurred a new wave of interest in low-power wireless networking, and directly motivated our work. In this thesis, we take on this challenge with a main conceptual and technical tool: concurrent transmissions (CTX), a technique that, by enforcing nodes to transmit concurrently, has been shown to unlock unprecedented fast, reliable, and energy efficient multi-hop communications in low-power wireless networks, opening new opportunities for protocol design. We first direct our research endeavour towards industrial applications, focusing on the popular IEEE 802.15.4 narrowband PHY layer, and advance the state of the art along two different directions: interference resilience and aperiodic wireless control. We tackle radio-frequency noise by extensively analysing, for the first time, the dependability of CTX under different types, intensities, and distributions of reproducible interference patterns, and by devising techniques to push it further. Specifically, we concentrate on CRYSTAL, a recently proposed communication protocol that relies on CTX to rapidly and dependably collect aperiodic traffic. By integrating channel hopping and noise detection in the protocol operation, we provide a novel communication stack capable of supporting aperiodic transmissions with near-perfect reliability and a per-mille radio duty cycle despite harsh external interference. These results lay the ground towards the exploitation of CTX for aperiodic wireless control; we explore this research direction by co-designing the Wireless Control Bus (WCB), our second contribution. WCB is a clean-slate CTX-based communication stack tailored to event-triggered control (ETC), an aperiodic control strategy holding the capability to significantly improve the efficiency of wireless control systems, but whose real-world impact has been hampered by the lack of appropriate networking support. Operating in conjunction with ETC, WCB timely and dynamically adapts the network operation to the control demands, unlocking an order-of-magnitude reduction in energy costs w.r.t. traditional periodic approaches while retaining the same control performance, therefore unleashing and concretely demonstrating the true ETC potential for the first time. Nevertheless, low-power wireless communications are rapidly evolving, and new radios striking novel trade-offs are emerging. Among these, in the second part of the thesis we focus on ultra-wideband (UWB). By providing hitherto missing networking primitives for multi-hop dissemination and collection over UWB, we shed light on the communication potentialities opened up by the high data throughput, clock precision, and noise resilience offered by this technology. Specifically, as a third contribution, we demonstrate that CTX not only can be successfully exploited for multi-hop UWB communications but, once embodied in a full-fledged system, provide reliability and energy performance akin to narrowband. Furthermore, the higher data rate and clock resolution of UWB chips unlock up to 80% latency reduction w.r.t. narrowband CTX, along with orders-of-magnitude improvements in network-wide time synchronization. These results showcase how UWB CTX could significantly benefit a multitude of applications, notably including low-power wireless control. With WEAVER, our last contribution, we make an additional step towards this direction, by supporting the key functionality of data collection with an ultra-fast convergecast stack for UWB. Challenging the internal mechanics of CTX, WEAVER interleaves data and acknowledgements flows in a single, self-terminating network-wide flood, enabling the concurrent collection of different packets from multiple senders with unprecedented latency, reliability, and energy efficiency. Overall, this thesis pushes forward the applicability and performance of low-power wireless, by contributing techniques and protocols to enhance the dependability, timeliness, energy efficiency, and interference resilience of this technology. Our research is characterized by a strong experimental slant, where the design of the systems we propose meets the reality of testbed experiments and evaluation. Via our open-source implementations, researchers and practitioners can directly use, extend, and build upon our contributions, fostering future work and research on the topic.

Tesis por compendio / [ES] Con esta tesis se pretende dar solución a algunos de los problemas más habituales que aparecen en los Sistemas de control basados en red (NCS) como son los retardos variables en el tiempo, las pérdidas y el desorden de paquetes, y la restricción de ancho de banda y de recursos computacionales y energéticos de los dispositivos que forman parte del sistema de control. Para ello se ha planteado la integración de técnicas de control multifrecuencial, de control basado en paquetes, de control basado en predictor y de control basado en eventos. Los diseños de control realizados se han simulado utilizando Matlab-Simulink y Truetime, se ha analizado su estabilidad mediante LMIs y QFT, y se han validado experimentalmente en un péndulo invertido, un robot cartesiano 3D y en robots móviles de bajo coste. El artículo 1 aborda el control basado en eventos, el cual minimiza el ancho de banda consumido en el NCS mediante un control basado en eventos periódicos y presenta un método para obtener sus parámetros óptimos para el sistema específico en que se utilice. Los artículos 2, 4 y 6 añaden el control basado en paquetes, así como el control multifrecuencia, que aborda problemas de falta de datos por bajo uso del sensor y los retardos, pérdidas y desórdenes de paquetes en la red. También afrontan, mediante tecnicas de predicción basadas en un filtro de Kalman multifrecuencia variable en el tiempo, los problemas de ruido y perturbaciones, así como la observación de los estados completos del sistema. El artículo 7 hace frente a un modelo no lineal que utiliza las anteriores soluciones junto con un filtro de Kalman extendido para presentar otro tipo de estructura para un vehículo autónomo que, gracias a la información futura obtenida mediante estas técnicas, puede realizar de forma remota tareas de alto nivel como es la toma de decisiones y la monitorización de variables. Los artículos 3 y 5, presentan una forma de obtener y analizar la respuesta en frecuencia de sistemas SISO multifrecuencia y estudian su comportamiento ante ciertas incertidumbres o problemas en la red haciendo uso de procedimientos QFT. / [CA] Amb aquesta tesi es pretén donar solució a alguns dels problemes més habituals que apareixen als Sistemes de Control Basats en xarxa (NCS) com son els retards d'accés i transferència variables en el temps, les pèrdues y desordenament de paquets, i la restricció d'ampli de banda així com de recursos computacionals i energètics dels dispositius que foment part del sistema de control. Per tal de resoldre'ls s'ha plantejat la integració de tècniques de control multifreqüencial, de control basat en paquets, de control basat en predictor i de control basat en events. Els dissenys de control realitzats s'han simulat fent ús de Matlab-Simulink i de TrueTime, s'ha analitzat la seua estabilitat mitjançant LMIs i QFT, i s'han validat experimentalment en un pèndul invertit, un robot cartesià 3D i en robots mòbils de baix cost. L'article 1 aborda el control basat en events, el qual minimitza l'ampli de banda consumit a l'NCS mitjançant un control basat en events periòdics i presenta un mètode per a obtindré els seus paràmetres òptims per al sistema específic en el qual s'utilitza. Els articles 2, 4 i 6 afegeixen el control basat en paquets, així com el control multifreqüència, que aborda problemes de falta de dades per el baix us del sensor i els retards, pèrdues i desordre de paquets en la xarxa. També afronten, mitjançant tècniques de predicció basades en un filtre de Kalman multifreqüència variable en el temps. Els problemes de soroll i pertorbacions, així com la observació dels estats complets del sistema. L'article 7 fa referència a un model no lineal que utilitza les anteriors solucions junt a un filtre de Kalman estès per a presentar altre tipus d'estructura per a un vehicle autònom que, gracies a la informació futura obtinguda mitjançant aquestes tècniques, pot realitzar de manera remota tasques d'alt nivell com son la presa de decisions i la monitorització de variables. Els articles 3 y 5 presenten la manera d'obtindre i analitzar la resposta en frequencia de sistemes SISO multifreqüència i estudien el seu comportament front a certes incerteses o problemes en la xarxa fent us de procediments QFT. / [EN] This thesis attempts to solve some of the most frequent issues that appear in Networked Control Systems (NCS), such as time-varying delays, packet losses and packet disorders and the bandwidth limitation. Other frequent problems are scarce computational and energy resources of the local system devices. Thus, it is proposed to integrate multirate control, packet-based control, predictor-based control and event-based control techniques. The control designs have been simulated using Matlab-Simulink and Truetime, the stability has been analysed by LMIs and QFT, and the experimental validation has been done on an inverted pendulum, a 3D cartesian robot and in low-cost mobile robots. Paper 1 addresses event-based control, which minimizes the bandwidth consumed in NCS through a periodic event-triggered control and presents a method to obtain the optimal parameters for the specific system used. Papers 2, 4 and 6 include packet-based control and multirate control, addressing problems such as network delays, packet dropouts and packet disorders, and the scarce data due to low sensor usage in order to save battery in sensing tasks and transmissions of the sensed data. Also addressed, is how despite the existence of measurement noise and disturbances, time-varying dual-rate Kalman filter based prediction techniques observe the complete state of the system. Paper 7 tackles a non-linear model that uses all the previous solutions together with an extended Kalman filter to present another type of structure for an autonomous vehicle that, due to future information obtained through these techniques, can remotely carry out high level tasks, such as decision making and monitoring of variables. Papers 3 and 5, present a method for obtaining and analyzing the SISO dual-rate frequency response and using QFT procedures to study its behavior when faced with specific uncertainties or network problems. / This work was supported by the Spanish Ministerio de Economía y Competitividad under Grant referenced TEC2012-31506. / Alcaina Acosta, JJ. (2020). Design and implementation of event-based multi-rate controllers for networked control systems [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/159884 / Compendio

L’objectif principal de cette thèse est d’étudier le problème du contrôle de suivi distribué pour les systèmes de formation de multi-robots à contrainte non holonomique. Ce contrôle vise à entrainer une équipe de robots mobile de type monocycle pour former une configuration de formation désirée avec son centroïde se déplaçant avec une autre trajectoire de référence dynamique et pouvant être spécifié par le leader virtuel ou humain. Le problème du contrôle de suivi a été résolu au cours de cette thèse en développant divers contrôleurs distribués pratiques avec la considération d’un taux de convergence plus rapide, une précision de contrôle plus élevée, une robustesse plus forte, une estimation du temps de convergence explicite et indépendante et moins de coût de communication et de consommation d’énergie. Dans la première partie de la thèse nous étudions d’abord au niveau du chapitre 2 la stabilité à temps fini pour les systèmes de formation de multi-robots. Une nouvelle classe de contrôleur à temps fini est proposée dans le chapitre 3, également appelé contrôleur à temps fixe. Nous étudions les systèmes dynamiques de suivi de formation de multi-robots non holonomiques dans le chapitre 4. Dans la deuxième partie, nous étudions d'abord le mécanisme de communication et de contrôle déclenché par l'événement sur les systèmes de suivi de la formation de multi-robots non-holonomes au chapitre 5. De plus, afin de développer un schéma d'implémentation numérique, nous proposons une autre classe de contrôleurs périodiques déclenchés par un événement basé sur un observateur à temps fixe dans le chapitre 6. / The main aim of this thesis is to study the distributed tracking control problem for the multi-robot formation systems with nonholonomic constraint, of which the control objective it to drive a team of unicycle-type mobile robots to form one desired formation configuration with its centroid moving along with another dynamic reference trajectory, which can be specified by the virtual leader or human. We consider several problems in this point, ranging from finite-time stability andfixed-time stability, event-triggered communication and control mechanism, kinematics and dynamics, continuous-time systems and hybrid systems. The tracking control problem has been solved in this thesis via developing diverse practical distributed controller with the consideration of faster convergence rate, higher control accuracy, stronger robustness, explicit and independent convergence time estimate, less communication cost and energy consumption.In the first part of the thesis, we first study the finite-time stability for the multi-robot formation systems in Chapter 2. To improve the pior results, a novel class of finite-time controller is further proposed in Chapter 3, which is also called fixed-time controller. The dynamics of nonholonomic multi-robot formation systems is considered in Chapter 4. In the second part, we first investigate the event-triggered communication and control mechanism on the nonholonomic multi-robot formation tracking systems in Chapter 5. Moreover, in order to develop a digital implement scheme, we propose another class of periodic event-triggered controller based on fixed-time observer in Chapter 6.