混成系统稳定性分析的代数化与机械化及应用

With the rapid development of information sciences, hybrid systems are paid more attention to. They are a class of dynamical systems that involve both continuous states and discrete events. Among numerous research directions on hybrid systems, safety verification and stability analysis are their two core research directions. We have already obtained quite a few outstanding achievements on these two research directions. For example, we directly handled safety problem of nonlinear hybrid systems by constructing and then solving nonlinear reachability constraints, which broke through the limitations of other theories that mainly use linearization for approximation; we analyzed asymptotic stability of switched hybrid systems by applying real root classification in computer algebra to compute multiple Lyapunov functions, which makes our approach more efficient than quantifier elimination based method, linear matrix inequalities based method and sum of squares based method. Some representative results have been published in top-ranking international conferences such as ISSAC, CAV and HSCC and top-ranking international journals such as ACM TECS and SICON. Based on our obtained achievements, this project will deeply focus on stability analysis of hybrid systems. That is, under the framework of Lyapunov stability theory, practical stability theory and bifurcation theory, we hope to further analyze the algebraic characters of stability for hybrid systems, realize the characterizations of algebraic properties for each theory, and then construct their corresponding semi-algebraic systems; Afterward, combining existing theories and methods in computer algebra and semi-definite problems solving, we will provide new and efficient algorithms for automatically computing mutliple Lyapunov functions and regions of attraction and then analyzing the stability of hybrid systems; Finally, we will carry out case studies.

随着信息科学的快速发展，混成系统备受关注。从数学上看，它是一类连续状态和离散事件并存的动力系统，其核心问题是安全性验证与稳定性分析。我们在这两个问题上都取得了若干突破。例如，我们曾通过构造并求解非线性可达约束条件，对非线性混成系统的安全性验证进行了直接处理，突破了其他理论主要采用线性近似处理的局限性；我们还曾利用实根分类来计算多重Lyapunov 函数，进而分析了切换式混成系统的渐近稳定性，其效率明显优于量词消去法、线性矩阵不等式法、平方和分解法等。部分成果发表在ISSAC、CAV、HSCC、ACM 汇刊等一流的会议和期刊上。 在Lyapunov 稳定性、实用稳定性以及分支等理论框架下，本课题将进一步分析混成系统稳定性的代数特征并构造相应理论下的半代数系统；结合计算机代数与正定问题求解上的现有理论与方法，提出新的高效算法来机械化地实现多重Lyapunov函数与吸引域的计算，并进行实例研究。

随着信息科学的快速发展，混成系统研究备受关注。在国家自然科学基金项目的资助下，本项目主要围绕混成系统稳定性分析的代数化与机械化及应用展开研究，在SIAM期刊、IEEE汇刊、NAHS等国际权威期刊以及CAV、AAAI、CDC等国际顶级会议上共发表标注国家自然科学基金项目资助的学术论文25篇，完满地完成了申请书里设定的预期目标。.本项目的主要学术成果可归纳如下：.1).基于实根分类，提出了高效判定连续时间混成系统稳定性的理论与方法，该方法与国际上现行的量词消去法、线性矩阵不等式法、平方和分解法相比，具有明显的优越性；.2).提出了多步多重类Lyapunov函数的概念，该概念拓展了经典的多重Lyapunov 函数的概念以及之后的有限步Lyapunov函数的概念，并依此建立了一个放缩的关于离散切换非线性系统在给定切换率下渐近稳定的判定法则；.3).基于（多重）类Lyapunov函数，建立了可迭代内估计吸引盆的理论体系并给予了算法实现，与Automatic 2008，IEEE TAC 2008，AIAA JGCD 2011，ISA Transactions 2014，IJRNC 2015等期刊上的方法比较，我们的估计效果更好；.4).利用不变集对同步态进行了数学刻画，并借鉴多重Lyapunov函数思想给出了复杂网络同步的更一般的判定准则，特别地，针对多项式系统及一类特殊的非多项式系统，我们能够将同步判据问题转化为平方和优化问题并借助半定规划类软件进行求解，从而在实际操作中具有更强的可行性。.项目负责人项目负责人获2014年度国家优秀青年科学基金项目，积极参与国内外学术合作与交流，并指导博士生8名（已毕业3名，在读5名）、硕士生5名（毕业2名，在读3名）。同时，本项目严格按照《资助计划书》规定的经费预算支出，所有支出范围都严格遵循《国家自然科学基金项目经费管理办法》的规定。

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