The ILAHS (inhomogeneous linear algebraic hybrid system) is a kind of classic hybrid system. For the purpose of optimizing the design of ILAHS, one important strategy is to introduce equivalence to reduce the states. Recent advances in the hybrid system indicate that approximate trace equivalence can further simplify the design of ILAHS. To address this issue, the paper first introduces the trajectory metric d t r j for measuring the deviation of two hybrid systems’ behaviors. Given a deviation ε ≥ 0 , the original ILAHS of H 1 can be transformed to the approximate ILAHS of H 2 , then in trace equivalence semantics, H 2 is further reduced to H 3 with the same functions, and hence H 1 is ε -approximate trace equivalent to H 3 . In particular, ε = 0 is a traditional trace equivalence. We implement an approach based on RealRootClassification to determine the approximation between the ILAHSs. The paper also shows that the existing approaches are only special cases of our method. Finally, we illustrate the effectiveness and practicality of our method on an example.
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