Reachability Analysis of Hybrid Automata with Clocked Linear Dynamics

Abstract

Disributed control systems often exhibit aperiodic sampling behavior due to varying communication delays and execution times. In such cases traditional analysis methods fall short because the functional and temporal behaviors need to be analyzed simultaneously. Therefore such systems are often modeled by Hybrid Automata (HA) with clock and non-clock variables, and verified using reachability analysis. However, modern reachability tools introduce a large overapproximation error because non-clock variables, as well as clock variables, are equally treated by the algorithm.In this paper we present a reachability algorithm which exploits the explicit separation of clock and non-clock variables in the Hybrid Automata with Clocked Linear Dynamics (HA-CLD) subclass, as well as restricting that guard and invariant constraints can only be specified in the HA-CLD model for clock variables. These properties of HA-CLD allow independent computation of tight reachable set overapproximations, in the form of flow-pipes, of clock and nonclock variables. A computationaly efficient and tight intersection operation is obtained by relating segments of the clock flow-pipe with segments of the non-clock flow-pipe.We demonstrate the effectiveness of our approach using two benchmarks, in the context of verifying stability. From the results it can be concluded that our reachability approach obtains significantly tighter results with an up-to 65 times smaller run-time compared to the start-of-the-art model checker SpaceEx.

Dynamics-aware subspace identification for decomposed aggregation in the reachability analysis of hybrid automata

“…To evaluate our approach, we use the implementation of the reachability algorithm as described in [11], because it allows computing the reachable set of clock and other variables independently. Thus it is suitable for the analysis of our models.…”

“…However instead of the standard convex hull algorithm used in their implementation, we use our own decomposed convex hull and subspace identification algorithm. Specifically, at line 12 of Algorithm 2 in [11], the operation Conv is replaced by dconv(•) in equation ( 2), and preceded by optimBasis from Algorithm 2, or by PCA as defined in equation ( 5…”

“…This makes sense, because a box is simply a zonotope with 𝑉 being an orthogonal matrix. Then for any matrix 𝐺 ∈ R 𝑛×𝑛 , vol(Box𝐼 (𝐺𝑋𝑈 )) can be derived using equations ( 9), (10), and (11) as the function: 𝑔(𝑈, 𝐺, 𝑋) = prod(abs(𝐺𝑈 ) abs(𝑑 − 𝑋 )), (12) where abs(•) is the element-wise or component-wise absolute value function of a matrix or vector, respectively, and prod(•) is the product of components of a vector. Thus the optimization problem ( 7) is reformulated to the equivalent simplified problem:…”

“…We believe that these models demonstrate the importance of good aggregation in reachability, since it can greatly affect the accuracy of the reachability algorithm, an observation we demonstrate in our results. In fact, as shown in [4,10,11], state-of-the-art tools such as SpaceEx and Flow* consistently fail to analyze such models.…”

Efficient reachability analysis of parametric linear hybrid systems with time-triggered transitions

“…Separation of clock and non-clock variables during the analysis is implemented in HyDra, which is based on the HyPro library [21], but the algorithmic concepts have not been published. In [16] the authors propose a custom algorithm to analyze hybrid automata with clocked linear dynamics based on polytopes in vertex representation for the non-clock variables. This representation has the advantage that clustering (a technique to mitigate complexity explosion in reachability analysis [10]) can be applied to the vertices directly.…”