Formal methods refer to rigorous, mathematical approaches to system development and have played a key role in establishing the correctness of safety-critical systems. The main building blocks of formal methods are models and specifications, which are analogous to behaviors and requirements in system design and give us the means to verify and synthesize system behaviors with formal guarantees. This monograph provides a survey of the current state of the art on applications of formal methods in the autonomous systems domain. We consider correct-by-construction synthesis under various formulations, including closed systems, reactive, and probabilistic settings. Beyond synthesizing systems in known environments, we address the concept of uncertainty and bound the behavior of systems that employ learning using formal methods. Further, we examine the synthesis of systems with monitoring, a mitigation technique for ensuring that once a system deviates from expected behavior, it knows a way of returning to normalcy. We also show how to overcome some limitations of formal methods themselves with learning. We conclude with future directions for formal methods in reinforcement learning, uncertainty, privacy, explainability of formal methods, and regulation and certification.

This monograph examines in mathematical terms an open question in neuroscience on the function of the cerebellum, a major brain region involved in regulation of the motor systems, speech, emotion, and other cognitive functions of the body. Reasoning from the perspective of control theory, we make a hypothesis that the primary function of the cerebellum is disturbance rejection of exogenous reference and disturbance signals. This brings to the fore the internal model principle of control theory: that any good controller must include a model of its environment. The monograph is structured around a pursuit of the validity of this hypothesis. Given the system level architecture and the measurement structure of the cerebellum, is disturbance rejection mathematically feasible? Second, is a disturbance rejection interpretation consistent with experiments? Specifically we investigate the possibility that the cerebellum provides adaptive internal models of signals generated by the environment. After a brief historical overview of computational theories of cerebellar function and of the relevant parts of control theory in the area of regulator theory, we carry out a more or less chronological review of subjects in control theory that impinge on our investigation. We begin with classical regulator theory with its characteristic features of a known plant and known frequency content of disturbance and reference signals. We then pause to examine classical regulator theory from the perspective of modeling the cerebellum, finding that several aspects are not well suited to this endeavor. Foremost is the unrealistic assumption that the plant and exosystem parameters are apriori known. More subtle issues arise from the fact that classical regulator theory developed in a setting where output and error measurements are regarded as persistent, an assumption not valid in the brain. Next, we review adaptive control theory, organized in terms of error models. The main control theoretic tools appear in Section 5, as a synthesis of classical regulator theory and adaptive control, where we present several adaptive internal model designs. To test our hypothesis on cerebellar function, we apply adaptive internal model designs to several motor systems regulated by the cerebellum. These include the slow eye movement systems: the vestibulo-ocular reflex, gaze holding, smooth pursuit, and the optokinetic system. We also study discrete time behaviors regulated by the cerebellum: the saccadic eye movement system and, more generally, visuomotor adaptation. The results from these modeling studies suggest that an interpretation of cerebellar function in terms of disturbance rejection is compelling, with the potential to provide a unifying framework to explain how the cerebellum can contribute to so many different systems in the body. The monograph concludes with suggestions for future research directions.

Multi-robot systems have huge potential for practical applications, which include sensor networks, area surveillance, environment mapping, and so forth. In many applications, cooperative coordination of the robots plays a central role. There are various types of coordination tasks such as consensus, formation, coverage, and pursuit. Most developments of control methods have been taken place for each task individually so far. The purpose of this monograph is to provide a systematic design method applicable to a wide range of coordination tasks for multi-robot systems. The features of the monograph are two-fold: (i) The coordination problem is described in a unified way instead of handling various problems individually, and (ii) a complete solution to this problem is provided in a compact way by using the tools of “group” and “graph” theories efficiently. As for item (i), it is shown that various coordination tasks can be formulated as a generalized coordination problem, where each robot should converge to some desired configuration set under the given information network topology among robots. In this problem, the solvability (i.e., whether robots can achieve the given coordination task or not) fully depends on the characteristics of both the desired configuration set and the network topology. Therefore, concerning item (ii), it is clarified when the generalized coordination problem can be solved in terms of the desired configuration set and the network topology. Furthermore, it is shown how to design a controller which achieves the given configuration task. In particular, the case where each robot can get only local information (e.g., relative position between two robots) is discussed.

System resilience captures the ability of the system to withstand a major disruption within acceptable performance degradation and to recover within an acceptable time frame. In this monograph we consider two possible sources of major disruptions, i.e., component faults and cyber intrusions. A component fault is an indigenous activity that renders unavailability or inaccessibility of certain functions within a component, either permanently or temporarily. It typically generates safety and performance concerns. Cyber intrusion on the other hand is an exogenous activity that tampers privacy, confidentiality, availability, or integrity of the system. These two sources are not always independent from each other. For example, a cyber intrusion may trigger a component fault, whereas a component fault may open a door for cyber intrusion, e.g., by keeping it undetected. For cyber intrusion, we will focus on opacity, which describes the system’s ability to hide certain secrets from an external observer (or eavesdropper), and sensor and actuator attacks that exploit the system’s existing controller to generate undesirable behaviours. In this monograph, we provide a detailed account of most recent research outcomes on fault diagnosis, opacity analysis and enhancement, and cyber security analysis and enforcement, within suitable discrete event system modelling frameworks. In each case, we describe basic problem statements and key concepts, and then point out the key challenges in each research area. After that, we present a thorough review of state-of-the-art techniques, and discuss their advantages and disadvantages. Finally, we highlight key research directions for further exploration.

This monograph presents some existing and new results on analysis and design of finite-time and fixed-time converging systems. Two main groups of approaches for analysis/synthesis of this kind of convergence are considered: based on Lyapunov functions and the theory of homogeneous systems. The focus is put on the dynamics described by ordinary differential equations, time-delay models and partial differential equations. Some popular control and estimation algorithms, which possess accelerated converge rates, are reviewed. The issues of discretization of finite-/fixed-time converging systems are discussed.

In recent years, Economic Model Predictive Control (EMPC) has received considerable attention of many research groups. The present tutorial survey summarizes state-of-the-art approaches in EMPC. In this context EMPC is to be understood as receding-horizon optimal control with a stage cost that does not simply penalize the distance to a desired equilibrium but encodes more sophisticated economic objectives. This survey provides a comprehensive overview of EMPC stability results: with and without terminal constraints, with and without dissipativity assumptions, with averaged constraints, formulations with multiple objectives and generalized terminal constraints as well as Lyapunov-based approaches.