基于符号数值混合计算的混成系统Lyapunov函数构造  

COMPUTING THE VERIFIED LYAPUNOV FUNCTIONS OF HYBRID SYSTEMS BY SYMBOLIC-NUMERIC METHOD

作　　者：林望[1,2] 吴敏[2] 杨争峰[2] 曾振柄[2] 

机构地区：[1]温州大学数学与信息科学学院,温州325035 [2]华东师范大学上海市高可信计算重点实验室,上海200062

出　　处：《系统科学与数学》2012年第5期610-625,共16页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金项目(10901055;10801052;91018012;61021004);973项目(2011CB302802);浙江省教育厅科研项目(Y201120383);华东师范大学创新基金项目(78210043)

摘　　要：基于平方和松弛和有理向量恢复,提出了一种符号数值混合计算方法来构造多项式Lyapunov函数以判定非线性混成系统的稳定性,首先,为Lyapunov函数预定一个给定次数的多项式模板,则Lyapunov函数构造问题可转化为相应的带参数的多项式优化问题,然后运用平方和松弛方法求得一个近似的数值多项式Lyapunov函数,再应用高斯-牛顿精化和有理向量恢复将数值多项式转化为验证的有理多项式Lyapunov函数.In this paper, we present a symbolic-numeric hybrid method, based on Sum-of- Squares （SOS） relaxation and rational vector recovery, to compute an verified Lyapunov func- tion for analyzing the stability of nonlinear hybrid systems. At first, finding Lyapunov functions of hybrid systems can be converted into the constrained polynomial optimization problem with parameters. SOS relaxation method is then used to compute approximate Lyapunov functions with floating point coefficients. And Gauss-Newton refinement and rational vector recovery are applied on the approximate polynomials to obtain candidates, whose coefficients are rational numbers. The existences of SOS representation is used to verify the polynomial which exactly satisfies the conditions of Lyapunov functions. In the end, several examples are given to show that our method can successfully yield Lyapunov functions with rational coefficients.

关 键 词：混成系统 LYAPUNOV函数 平方和松弛 半正定规划 

分 类 号：O174.4[理学—数学]