Approximate Bisimulation and Discretization of Hybrid CSP

Abstract

Hybrid Communicating Sequential Processes (HCSP) is a powerful formal modeling language for hybrid systems, which is an extension of CSP by introducing differential equations for modeling continuous evolution and interrupts for modeling interaction between continuous and discrete dynamics. In this paper, we investigate the semantic foundation for HCSP from an operational point of view by proposing the notion of approximate bisimulation, which provides an appropriate criterion to characterize the equivalence between HCSP processes with continuous and discrete behaviour. We give an algorithm to determine whether two HCSP processes are approximately bisimilar. In addition, based on which, we propose an approach on how to discretize HCSP, i.e., given an HCSP process A, we construct another HCSP process B which does not contain any continuous dynamics such that A and B are approximately bisimilar with given precisions. This provides a rigorous way to transform a verified control model to a correct program model, which fills the gap in the design of embedded systems.

Synthesizing SystemC Code from Delay Hybrid CSP

“…To better understand dHCSP, we introduce delay behavior to the water tank system considered in [4,24]. Example 1.…”

“…The process on generating code from dHCSP is similar to that from HCSP [24], consisting of two phases: (1) discretization of the dHCSP model; (2) code generation from the discretized dHCSP model to SystemC.…”

“…Based on the above fact, we can define a set of rules to discretize a given dHCSP process S and obtain a discrete dHCSP process D h,ε (S) such that they are (h, ε)-approximately bisimilar on [0, T ], for given h, ε and T . The rule for the discretization of DDE is given below, and other rules are same as the ones for HCSP presented in [24]. N (B, ε) at the next discretized step; then, if both N (B, ε) and N (B, ε) still hold, but the time has already reached the upper bound T , the process behaves like stop, which indicates that the behavior after T will not be concerned.…”

“…4.2 is approximately bisimilar to S, we need to put some extra conditions on S, i.e., requiring it to be robustly safe. The condition is similar to that in [24]. We define the (− )-neighbourhood like the -neighbourhood, i.e., for a set φ ⊆ R n and ≥ 0,…”

“…The proof of the the existence of h such that the global discretized error constraint is satisfied, i.e., after the execution of P and D h,ε (P ) the global error is D 2 h for some constant D 2 , is given by structural induction on P . Since rules of discretization for constructors in dHCSP are closely similar to those in HCSP, except a slight difference for terms containing DDEs, so we only illustrate the proofs for these kinds of dHCSP processes here, and proofs for other cases can refer to [24].…”