This paper offers a synthetic overview of, and original contributions to, the use of logics and formal methods in the analysis of hybrid systems

. In this paper, we formulate and robustly solve a quite general class of hybrid controller synthesis problems. The type of controller we investigate is the switching control mechanism of a hybrid automaton (via guard and mode invariant sets), and the robustness result is with respect to variations in the right hand sides of the dierential equations that depend continuously on a parameter. We present a novel methodology for controller design and synthesis which uses modal logic as a formalism for reasoning about sets of plant states, and various operators on sets arising from the dierential equations and from metric tolerance relations on the state space. 1 Introduction In general terms, a hybrid system H can be said to satisfy a performance speci cation robustly if every system H 0 in some nominated variation class around H also satis es that speci cation. Likewise, a synthesis procedure for a class of control problems can be called robust if the nominal closed-loop hyb...

Abstract: In this chapter, a class of discrete time uncertain linear hybrid systems, affected by both parameter variations and exterior disturbances, is considered. The main question is whether there exists a controller such that the closed loop system exhibits desired behavior under dynamic uncertainty and exterior disturbances. The notion of attainability is introduced to refer to the specified behavior that can be forced to the plant by a control mechanism. We give a method for attainability checking that employs the predecessor operator and backward reachability analysis, and a procedure for controller design that uses finite automata and linear programming techniques. Finally, Networked Control Systems (NCS) are proposed as a promising application area of the results and tools developed here, and the ultimate boundedness control problem for the NCS with uncertain delay, package dropout and quantization effects is formulated as a regulation problem for an uncertain hybrid system.

In this paper, we formulate and solve a quite general class of hybrid controller synthesis problems. The plant consists of a finite number of continuous systems, given by di#erential equations over a common state space; the controller steers the plant state by determining when to discretely switch between the various di#erential equations; and the closed-loop trajectories correspond to those of (a subclass of) the widely accepted hybrid automaton model. In addition to the well-studied classes of safety (reachability or invariance) and liveness (non-blocking and non-Zeno) performance specifications, we deal with a general class of active behavioural specifications, requiring that trajectories traverse in prescribed sequences through the blocks of a given finite partition of the plant state space. We give an abstract algorithm which solves this controller synthesis problem for arbitrary di#erential equations with unique solutions, and give an analytic proof of finite terminat...