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This paper describes an approach to the control of continuous systems through the use of symbolic models describing the system behavior only at a finite number of points in the state space. These symbolic models can be seen as abstract representations of the continuous dynamics enabling the use of algorithmic controller design methods. We identify a class of linear control systems for which the loss of information incurred by working with symbolic subsystems can be compensated by feedback. We also show how to transform symbolic controllers designed for a symbolic subsystem mb controllers for the original system. The resulting controllers combine symbolic controller dynamics with continuous feedback control laws and can thus be seen as hybrid systems. Furthermore, if the symbolic controller already accounts for software/hardware requirements, the hybrid controller is guaranteed to enforce the desired specifications by construction thereby reducing the need for formal verification. [ABSTRACT FROM AUTHOR]

We present an abstraction and refinement methodology for the automated controller synthesis to enforce general predefined specifications. The designed controllers require quantized (or symbolic) state information only and can be interfaced with the system via a static quantizer. Both features are particularly important with regard to any practical implementation of the designed controllers and, as we prove, are characterized by the existence of a feedback refinement relation between plant and abstraction. Feedback refinement relations are a novel concept introduced in this paper. Our work builds on a general notion of system with set-valued dynamics and possibly non-deterministic quantizers to permit the synthesis of controllers that robustly, and provably, enforce the specification in the presence of various types of uncertainties and disturbances. We identify a class of abstractions that is canonical in a well-defined sense, and provide a method to efficiently compute canonical abstractions. We demonstrate the practicality of our approach on two examples. [ABSTRACT FROM AUTHOR]

The problem of efficiently steering dynamical systems by generating input plans is considered. Plans are considered which consist of finite-length words constructed on an alphabet of input symbols, which could be, e.g., transmitted through a limited capacity channel to a remote system, where they can be decoded in suitable control actions. Efficiency is considered in terms of the computational complexity of plans, and in terms of their description length (in number of bits). We show that, by suitable choice of the control encoding, finite plans can be efficiently built for a wide class of dynamical systems, computing arbitrarily close approximations of a desired equilibrium in polynomial time. The paper also investigates how the efficiency of planning is affected by the choice of inputs, and provides some results as to optimal performance in terms of accuracy and range. [ABSTRACT FROM AUTHOR]

This note presents an approach that integrates elements from grammatical inference and game theory to address the problem of supervising finite-state transition systems operating in adversarial, partially known, rule-governed environments. The combined formulation produces controllers which guarantee that a transition system satisfies a task specification in the form of a logical formula, if and only 1) the true model of the environment is in the class of models inferable from positive data presentation (observations), 2) a characteristic sample of the environment's behavior is observed, and 3) the task specification is satisfiable given the capabilities described by the abstractions of the system and its environment. [ABSTRACT FROM AUTHOR]

This article presents a symbolic control approach to the design of distributed safety controllers for a class of continuous-time nonlinear systems. More precisely, we consider systems made of components where each component is equipped with a sampled-data controller with its own sampling period, resulting globally in a distributed multiperiodic sampled-data system. Moreover, controllers receive partial information on the state of the other components. We propose a component-based approach to controller synthesis, which relies on the use of abstractions and continuous-time assume-guarantee contracts. The abstractions describe the dynamics of the system from the point of view of each component based on the information structure, whereas assume-guarantee contracts specify guarantees that a component must satisfy if assumptions on the other components are met. We show that our approach makes it possible to decompose a global safety control problem into local ones that can be solved independently. We then show how symbolic control techniques can be used to synthesize controllers that enforce the local control objectives. Illustrative applications in building automation and vehicle platooning are shown. [ABSTRACT FROM AUTHOR]

We present novel results on the solution of a class of leavable, undiscounted optimal control problems in the minimax sense for nonlinear, continuous-state, discrete-time plants. The problem class includes entry-(exit-)time problems as well as minimum-time, pursuit-evasion, and reach-avoid games as special cases. We utilize auxiliary optimal control problems (“abstractions”) to compute both upper bounds of the value function, i.e., of the achievable closed-loop performance, and symbolic feedback controllers realizing those bounds. The abstractions are obtained from discretizing the problem data, and we prove that the computed bounds and the performance of the symbolic controllers converge to the value function as the discretization parameters approach zero. In particular, if the optimal control problem is solvable on some compact subset of the state space, and if the discretization parameters are sufficiently small, then we obtain a symbolic feedback controller solving the problem on that subset. These results do not assume the continuity of the value function or any problem data, and they fully apply in the presence of hard state and control constraints. [ABSTRACT FROM AUTHOR]

The practical impact of abstraction-based controller synthesis methods is currently limited by the immense computational effort for obtaining abstractions. In this note we focus on a recently proposed method to compute abstractions whose state space is a cover of the state space of the plant by congruent hyper-intervals. The problem of how to choose the size of the hyper-intervals so as to obtain computable and useful abstractions is unsolved. This note provides a twofold contribution towards a solution. Firstly, we present a functional to predict the computational effort for the abstraction to be computed. Secondly, we propose a method for choosing the aspect ratio of the hyper-intervals when their volume is fixed. More precisely, we propose to choose the aspect ratio so as to minimize a predicted number of transitions of the abstraction to be computed, in order to reduce the computational effort. To this end, we derive a functional to predict the number of transitions in dependence of the aspect ratio. The functional is to be minimized subject to suitable constraints. We characterize the unique solvability of the respective optimization problem and prove that it transforms, under appropriate assumptions, into an equivalent convex problem with strictly convex objective. The latter problem can then be globally solved using standard numerical methods. We demonstrate our approach on an example. [ABSTRACT FROM AUTHOR]

A class of hybrid systems with convergent continuous dynamics is abstracted to a special type of finite automata which operate on an infinite alphabet. The abstraction is predicate-based, enabled by the convergence properties of the continuous dynamics of the hybrid system, and encompasses existing low-level controllers rather than replacing them during synthesis. The abstract models are finite yet capable of storing and manipulating continuous data expressing attributes of the concrete hybrid model. The latter is shown to weakly simulate its discrete abstraction, and thus behaviors planned using the abstraction are always implementable on the concrete system. A case study illustrates how this methodology can be put into practice, orchestrating a temporal sequence of continuous controllers that allows the hybrid system to achieve a performance objective. [ABSTRACT FROM PUBLISHER]

Sufficiently accurate finite state models, also called symbolic models or discrete abstractions, allow one to apply fully automated methods, originally developed for purely discrete systems, to formally reason about continuous and hybrid systems and to design finite state controllers that provably enforce predefined specifications. We present a novel algorithm to compute such finite state models for nonlinear discrete-time and sampled systems which depends on quantizing the state space using polyhedral cells, embedding these cells into suitable supersets whose attainable sets are convex, and over-approximating attainable sets by intersections of supporting half-spaces. We prove a novel recursive description of these half-spaces and propose an iterative procedure to compute them efficiently. We also provide new sufficient conditions for the convexity of attainable sets which imply the existence of the aforementioned embeddings of quantizer cells. Our method yields highly accurate abstractions and applies to nonlinear systems under mild assumptions, which reduce to sufficient smoothness in the case of sampled systems. Its practicability in the design of discrete controllers for nonlinear continuous plants under state and control constraints is demonstrated by an example. [ABSTRACT FROM AUTHOR]

We discuss the generation of symbolic feedback con- trol sequences for navigating a sparsely-described and uncertain environment, together with the problem of sensing landmarks sufficiently well to make feedback meaningful. We explore the use of a symbolic control approach for mitigating the lack of a detailed map of the environment and for reducing the complexity associated with finding control laws which steer a control system between distant locations. Under our language-based approach, control inputs take the form of symbolic strings. The decision process that generates those strings is guided by estimates of the vehicle's location within a set of important landmarks and by the statistical effectiveness of each string. This arrangement, and in particular the symbolic nature of the control set, allows us to formulate and solve a class of optimal navigation problems which would be exceedingly difficult to handle if approached at the level of sensors and actuators. Our approach is illustrated in a series of numerical indoor navigation experiments. [ABSTRACT FROM AUTHOR]