Generic asymptotic error estimates for the numerical simulation of hybrid systems

Metrization and Simulation of Controlled Hybrid Systems

“…The choice of metric for this controlled hybrid system dictates exactly the type of trajectories that can be faithfully simulated. For example, existing trajectory-space metrics impose at least unit distance between any states that reside in different discrete modes [1], [2], [3], [4]. As a result, simulation algorithms based on these metrics cannot provably approximate executions that undergo simultaneous discrete transitions (e.g.…”

“…Ames and Sastry [7] derive a category-theoretic colimit topology over the regularization proposed by Johansson et al [8] that relaxes domains at the guard sets. We propose a metric topology over the state space of controlled hybrid systems that connects disparate domains through the reset map, effectively metrizing the hybrifold and colimit R (1,3) R (0,2) R (2,3) (b) controlled hybrid system Fig. 1.…”

“…The execution x undergoes two discrete transitions simultaneously; the nearby executions y δ , z δ encounter different discrete transition sequences. Since R (1,3) • R (0,1) (0, 0) = R (2,3) • R (0,2) (0, 0) = (0, 0), either transition sequence may be chosen for x. topologies, and generalizing the phase space metric proposed by Schatzman for an impact oscillator [9]. In contrast, Tavernini [1] and Gobel and Teel [10] directly metrize the space of executions of hybrid systems; Gokhman [2] demonstrates the equivalence of the resulting topology with that generated by the Skorohod trajectory metric [11,Chapter 6]).…”

Topologies for hybrid solutions

“…In this section we follow [11] to briefly introduce the concept of a differential automaton, hybrid solutions to an initial value problem, and the Tavernini metric.…”

“…Numerical simulation and qualitative analysis of hybrid systems were carried out by Tavernini [9], who introduced the concept of a differential automaton and a metric on the space of solutions incorporating the uniform distances between extended continuous parts of the solutions and the distances between switching points (as well as the distances between the discrete states) (see [9], Section 4 [10,11]). …”