From electrical switched networks to hybrid automata

In this paper, we propose a novel symbolic approach to automatically synthesize a Hybrid Automaton (HA) from a switched electrical network. The input network consists of a set of physical components interconnected according to some reconfigurable network topology. The underlying model defines a local dynamics for each component in terms of a Differential-Algebraic Equation (DAE), and a set of network topologies by means of discrete switches. Each switch configuration induces a different topology, where the behavior of the system is a Hybrid Differential- Algebraic Equations. Two relevant problems for these networks are validation and reformulation. The first consists of determining if the network admits an Ordinary Differential Equations (ODE) that describes its dynamics; the second consists of obtaining such ODE from the initial DAE. This step is a key enabler to use existing formal verification tools that can cope with ODEs but not with DAEs. Since the number of network topologies is exponential in the number of switches, first, we propose a technique based on Satisfiability Modulo Theories (SMT) that can solve the validation problem symbolically, avoiding the explicit enumeration of the topologies. Then, we show an SMT-based algorithm that reformulates the network into a symbolic HA. The algorithm avoids to explicitly enumerate the topologies clustering them by equivalent continuous dynamics. We implemented the approach with several optimizations and we compared it with the explicit enumeration of configurations. The results demonstrate the scalability of our technique.