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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]

The article presents an overview on several studies published within the June 2006 issue of "IEEE Transactions on Automatic Control" in the U.S. The study conducted by Sean B. Andersson and Dimitrios Hristu presents a method for generating sequences of symbolic feedback control laws for mobile robots navigating in unknown environments. Another study conducted by L.C.G.J.M. Habets and colleagues investigates the problem of synthesizing control laws for piecewise-affine hybrid systems on simplices. Furthermore, an approach to self-assembly and self-organization is proposed and investigated based on graph grammars by Eric Klavins and colleagues.

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]