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 $d_{trj}$ for measuring the deviation of two hybrid systems’ behaviors. Given a deviation $\epsilon \ge 0$ , the original ILAHS of ${\mathcal{H}}_1$ can be transformed to the approximate ILAHS of ${\mathcal{H}}_2$ , then in trace equivalence semantics, ${\mathcal{H}}_2$ is further reduced to ${\mathcal{H}}_3$ with the same functions, and hence ${\mathcal{H}}_1$ is $\epsilon$ -approximate trace equivalent to ${\mathcal{H}}_3$ . In particular, $\epsilon =0$ 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.

In generating invariants for hybrid systems, a main source of intractability is that transition relations are first-order assertions over current-state variables and next-state variables, which doubles the number of system variables and introduces many more free variables. The more variables, the less tractability and, hence, solving the algebraic constraints on complete inductive conditions by a comprehensive Gröbner basis is very expensive. To address this issue, this paper presents a new, complete method, called the Citing Instances Method (CIM), which can eliminate the free variables and directly solve for the complete inductive conditions. An instance means the verification of a proposition after instantiating free variables to numbers. A lattice array is a key notion in this paper, which is essentially a finite set of instances. Verifying that a proposition holds over a Lattice Array suffices to prove that the proposition holds in general; this interesting feature inspires us to present CIM. On one hand, instead of computing a comprehensive Gröbner basis, CIM uses a Lattice Array to generate the constraints in parallel. On the other hand, we can make a clever use of the parallelism of CIM to start with some constraint equations which can be solved easily, in order to determine some parameters in an early state. These solved parameters benefit the solution of the rest of the constraint equations; this process is similar to the domino effect. Therefore, the constraint-solving tractability of the proposed method is strong. We show that some existing approaches are only special cases of our method. Moreover, it turns out CIM is more efficient than existing approaches under parallel circumstances. Some examples are presented to illustrate the practicality of our method.