We consider discrete-time plants that interact with their controllers via fixed discrete alphabets. For this class of systems, and in the absence of exogenous inputs, we propose a general, conceptual procedure for constructing a sequence of finite state approximate models starting from finite length sequences of input and output signal pairs. We explicitly derive conditions under which the proposed construct, used in conjunction with a particular generalized structure, satisfies desirable properties of @math approximations thereby leading to nominal deterministic finite state machine models that can be used in certified-by- design controller synthesis. We also show that the cardinality of the minimal disturbance alphabet that can be used in this setting equals that of the sensor output alphabet. Finally, we show that the proposed construct satisfies a relevant semi-completeness property.
 Abstract The reduction of dynamic systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an “exact” manner–as is the case with mechanical systems with symmetry–which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction , in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamic system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e. when it is behaviourally similar to a dynamic system on a lower dimensional space. These concepts are illustrated on a series of examples. This paper presents a new algorithm for generating finite-state automata models of hybrid systems in which the continuous-state dynamics can be switched when the continuous-state trajectory generates threshold events. The automata state transitions correspond to threshold events and the automata states correspond to portions of the threshold surfaces in the continuous state space. The hybrid system dynamics are approximated by the automata models in the sense that the languages of threshold event sequences generated by the automata contain the threshold event language for the hybrid system. Properties of the algorithm for constructing and refining the approximating automata are demonstrated and the application of approximating automata for system verification is illustrated for a switching controller for an inverted pendulum. Relationships to other approaches to hybrid system synthesis and verification are also discussed. We consider the following problem: given a linear system and a linear temporal logic (LTL) formula over a set of linear predicates in its state variables, find a feedback control law with polyhedral bounds and a set of initial states so that all trajectories of the closed loop system satisfy the formula. Our solution to this problem consists of three main steps. First, we partition the state space in accordance with the predicates in the formula, and construct a transition system over the partition quotient, which captures our capability of designing controllers. Second, using a procedure resembling model checking, we determine runs of the transition system satisfying the formula. Third, we generate the control strategy. Illustrative examples are included.  The paper concerns the synthesis of supervisory controllers for a class of continuous-time hybrid systems with discrete-valued input signals that select differential inclusions for continuous-valued state trajectories and event-valued output signals generated by threshold crossings in the continuous state space, the supervisor is allowed to switch the input signal value when threshold events are observed. The objective is to synthesize a nonblocking supervisor such that the set of possible sequences of control and threshold event pairs for the closed-loop system lies between given upper and lower bounds in the sense of set containment. We show how this problem can be converted into a supervisor synthesis problem for a standard controlled discrete-event system (DES). A finite representation may not exist for the exact DES model of the hybrid system, however. To circumvent this difficulty, we present an algorithm for constructing finite-state Muller automata that accept outer approximations to the exact controlled threshold-event language, and we demonstrate that supervisors that solve the synthesis problem for the approximating automata achieve the control specifications when applied to the original hybrid system.