Computing bisimulation functions using SOS optimization and δ-decidability over the reals
Computing bisimulation functions using SOS optimization and δ-decidability over the reals
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使用 SOS 优化和实数上的 δ 可判定性计算互模拟函数
DOI：
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发表时间：
2015
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通讯作者：
Patricia M. LoRusso
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作者：
M. Cecchini;Eric H. Rubin;Gideon M. Blumenthal;Kassa Ayalew;Howard A. Burris;M. Russell;Hildy Dillon;H. Lyerly;Gregory H. Reaman;S. Boerner;Patricia M. LoRusso
提出了一个基于平方和（SOS）优化和实数δ-可判定性的自动化框架BFComp，用于计算表征动力系统输入输出稳定性（IOS）的互模拟函数（BFs）. BF是类李雅普诺夫函数，其沿着给定系统对的轨迹沿着衰减，并且可以用于建立输出相对于有界输入偏差的稳定性。除了建立IOS之外，BFComp还旨在尽可能提供系统之间平方输出误差的严格界限。为此，两个SOS优化配方：SOSP 1，它强制执行的输入空间上的离散化网格的衰减要求，和SOSP 2，它涵盖了输入空间穷尽。首先尝试SOSP 2，并且如果得到的误差界限不令人满意，则使用SOSP 1来计算候选BF（CBF）。然后将BF的衰减要求编码为δ-可判定公式，并使用dReal工具在CBF的水平集上进行验证。如果dReal产生一个包含违反衰减要求的状态和输入的反例，则使用这对向量来细化输入空间网格，并迭代SOSP 1。通过计算BF呼吁一个小增益定理，BFComp框架可以用来表明，一个反馈组成的系统的子系统可以被替换-有界误差-由一个近似等效的抽象，从而使近似模型阶动力系统的减少。我们说明了一个典型的心脏细胞模型上的BFComp的效用，表明四变量马尔可夫模型的缓慢激活钾电流IKs可以安全地取代一个变量的霍奇金-赫胥黎型近似。
We present BFComp, an automated framework based on Sum-Of-Squares (SOS) optimization and δ-decidability over the reals, to compute Bisimulation Functions (BFs) that characterize Input-to-Output Stability (IOS) of dynamical systems. BFs are Lyapunov-like functions that decay along the trajectories of a given pair of systems, and can be used to establish the stability of the outputs with respect to bounded input deviations. In addition to establishing IOS, BFComp is designed to provide tight bounds on the squared output errors between systems whenever possible. For this purpose, two SOS optimization formulations are employed: SOSP 1, which enforces the decay requirements on a discretized grid over the input space, and SOSP 2, which covers the input space exhaustively. SOSP 2 is attempted first, and if the resulting error bounds are not satisfactory, SOSP 1 is used to compute a Candidate BF (CBF). The decay requirement for the BFs is then encoded as a δ-decidable formula and validated over a level set of the CBF using the dReal tool. If dReal produces a counterexample containing the states and inputs where the decay requirement is violated, this pair of vectors is used to refine the input-space grid and SOSP 1 is iterated. By computing BFs that appeal to a small-gain theorem, the BFComp framework can be used to show that a subsystem of a feedback-composed system can be replaced--with bounded error--by an approximately equivalent abstraction, thereby enabling approximate model-order reduction of dynamical systems. We illustrate the utility of BFComp on a canonical cardiac-cell model, showing that the four-variable Markovian model for the slowly activating Potassium current IKs can be safely replaced by a one-variable Hodgkin-Huxley-type approximation.