Barrier Certificate Generation for Safety Verification of Continuous Systems for a Bounded Time

DOI：10.13328/j.cnki.jos.004986

 作者 单位 E-mail 甘庭 北京大学数学科学学院, 北京 100871 gant@pku.edu.cn,xbc@math.pku.edu.cn 夏壁灿 北京大学数学科学学院, 北京 100871

栅栏函数在连续系统验证方面有着广泛的应用,其主要想法在于:在可达集和非安全集之间寻找一个栅栏,从初始区域出发的路径不会越过这个栅栏,而非安全区域在栅栏的另外一端.这样,就可以通过寻找栅栏函数来验证一个系统的安全性.近年来,已有一些工作讨论连续系统在无界时间情况下的栅栏函数生成.但是对于有些系统,人们可能只关心其在有界时间内的安全性.因为在无界时间内不安全并不能说明在给定时间内也是不安全的,所以对于这类问题,无界时间栅栏函数方法并不适用.受无界时间栅栏函数方法的启发,针对有界时间的情况,给出有界时间栅栏函数生成方法.首先给出有界时间栅栏函数的一些充分条件,对于多项式系统,将多项式非负的条件做平方和松弛后利用平方和规划工具求解这些充分条件得到栅栏函数;对于初等系统(包含一些初等函数),先将该初等系统转化为一个多项式系统,然后求解对应多项式系统的栅栏函数.对一些无界时间不安全的实例,演示了该方法在验证有界时间安全性问题上的有效性.

Barrier certificates have been widely used in verification of continuous systems.The main idea is to find a barrier which separates the reachable set from the unsafe set such that all the trajectories starting from the initial set will never go across the barrier.Thus the system's safety can be guaranteed by constructing a barrier.In recent years, barrier certificates have been successfully used for verification of continuous systems with unbounded time.However sometimes the safety for bounded time needs to be addressed.Since a system is unsafe with unbounded time cannot imply it is also unsafe with a bounded time, the unbounded time barrier certificate method could fail to verify the safety with bounded time.In this paper, a method is presented to generate a bounded time barrier certificate for safety verification of continuous systems with bounded time.Some sufficient conditions for the bounded time barrier certificate are specified.If the continuous system is a polynomial system, relax all the conditions of positive semi-definite polynomial to the sum of squares(SOS) polynomial and then use semi-definite programming(SDP) to solve the conditions for a bounded time barrier certificate;if the continuous system is an elementary system(containing some elementary functions), transform it to a polynomial system approximately, and then solve the corresponding polynomial system for a bounded time barrier certificate.For some practical examples which are unsafe for unbounded time, the paper shows the effectiveness of the proposed method for generating bounded time barriers.
HTML  下载PDF全文  查看/发表评论  下载PDF阅读器