A Novel Approach for Solving the BMI Problem in Barrier Certificates Generation

Abstract

Barrier certificates generation is widely used in verifying safety properties of hybrid systems because of the relatively low computational complexity it costs. Under sum of squares (SOS) relaxation, the problem of barrier certificate generation is equivalent to that of solving a bilinear matrix inequality (BMI) with a particular type. The paper reveals the special feature of the problem, and adopts it to build a novel computational method. The proposed method introduces a sequential iterative scheme that is able to find analytical solutions, rather than the nonlinear solving procedure to produce numerical solutions used by general BMI solvers and thus is more efficient than them. In addition, different from popular LMI solving based methods, it does not make the verification conditions more conservative, and thus reduces the risk of missing feasible solutions. Benefitting from these two appealing features, it can produce barrier certificates not amenable to existing methods, which is supported by a complexity analysis as well as the experiment on some benchmarks.

Synthesizing Invariant Barrier Certificates via Difference-of-Convex Programming

“…We present a weak completeness result of our method, in the sense that a barrier certificate is guaranteed to be found (under some mild assumptions) whenever there exists an inductive invariant (in the form of a given template) that suffices to certify the system's safety. A similar result on completeness is previously provided only by symbolic approaches, yet to the best of our knowledge, not by methods base on numerical constraint solving, e.g., [4,60,61]. Experiments on a collection of examples suggested that our invariant barriercertificate condition recognizes more barrier certificates than existing conditions, and that our DCP-based algorithm is more efficient than directly solving the BMIs via off-the-shelf solvers.…”

“…The comparison in Table 1 suggests that (1) Our invariant barrier-certificate condition recognizes more barrier certificates than the original (more conservative) condition as implemented in SOSTOOLS. In particular, the lie-high-order example does admit an inductive invariant in the form of the given template, but none of the existing barrier-certificate conditions [4,60,63] -concerning Lie derivatives only up to the first order-recognizes it, since we have L 1 f B(x) = 0 We remark that symbolic methods based on, e.g., quantifier elimination [36], can hardly deal with any of the examples listed in Table 1 due to the prohibitively high computation complexity. Moreover, it would be desirable to pursue a comparison with the augmented Lagrangian method for solving BMIs as proposed in [4], which unfortunately is not yet possible due to the unavailability of the implementation thereof.…”

“…Moreover, it would be desirable to pursue a comparison with the augmented Lagrangian method for solving BMIs as proposed in [4], which unfortunately is not yet possible due to the unavailability of the implementation thereof. We will discuss crucial differences to [4] in Sect. 8.…”

“…Notice that the consecution constraint in Definition 4 involves Lie derivatives of orders up to N B,f ∈ N + , as is the case in Theorem 1. Our invariant barriercertificate condition hence generalizes existing conditions on barrier certificates, e.g., [4,60,63], which consider Lie derivatives only up to the first order.…”

“…Therefore, non-convex conditions were suggested [60], for which the synthesis problem can be reduced to BMI problems solvable via customized schemes, e.g., the augmented Lagrangian method [4] and the alternating minimization algorithm [63]. Our synthesis techniques also exploit a BMI reduction, with three crucial differences: (1) our invariant barriercertificate condition is equivalent to the inductive invariant condition in the sense of Theorem 4, and thus is less conservative than all the aforementioned conditions which consider Lie derivatives only up to the first order; (2) our DCP-based techniques for solving BMIs naturally inherit appealing results on convergence and (weak) completeness, which are not (and can hardly be) provided by the approaches in [4,60,63]; (3) our DCP-based iterative procedure visits only feasible solutions to the original BMI problem, and hence whenever a solution that induces a non-negative objective value is found, we can safely terminate the algorithm and claim a feasible solution to the original BMI problem, which may yield a valid barrier certificate. This is not the case for the approaches in [4,60,63].…”