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Computing Controlled Invariant Sets from Data Using Convex Optimization
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2020-09-16 , DOI: 10.1137/19m1305835
Milan Korda

SIAM Journal on Control and Optimization, Volume 58, Issue 5, Page 2871-2899, January 2020.
This work presents a data-driven method for approximation of the maximum positively invariant (MPI) set and the maximum controlled invariant (MCI) set for nonlinear dynamical systems. The method only requires knowledge of a finite collection of one-step transitions of the discrete-time dynamics, without the requirement of segments of trajectories or the control inputs that effected the transitions to be available. The approach uses a novel characterization of the MPI and MCI sets as the solution to an infinite-dimensional linear programming (LP) problem in the space of continuous functions, with the optimum being attained by a (Lipschitz) continuous function under mild assumptions. The infinite-dimensional LP is then approximated by restricting the decision variable to a finite-dimensional subspace and by imposing the nonnegativity constraint of this LP only on the available data samples. This leads to a single finite-dimensional LP that can be easily solved using off-the-shelf solvers. We analyze the convergence rate and sample complexity, proving probabilistic as well as hard guarantees on the volume error of the approximations. The approach is very general, requiring minimal underlying assumptions. In particular, the dynamics is not required to be polynomial or even continuous (forgoing some of the theoretical results). Detailed numerical examples up to state-space dimension 10 with code available online demonstrate the method.


中文翻译:

使用凸优化从数据计算受控不变集

SIAM控制与优化杂志,第58卷,第5期,第2871-2899页,2020年1月。
这项工作提出了一种数据驱动的方法,用于近似非线性动态系统的最大正不变(MPI)集和最大控制不变(MCI)集。该方法仅需要离散时间动力学的一步过渡的有限集合的知识,而无需轨迹段或影响过渡的控制输入可用。该方法使用MPI和MCI集的新颖特征作为连续函数空间中无限维线性规划(LP)问题的解决方案,并且在温和的假设下,通过(Lipschitz)连续函数可以达到最佳效果。然后,通过将决策变量限制为有限维子空间,并将此LP的非负约束仅施加在可用数据样本上,来近似无限维LP。这导致可以使用现成的求解器轻松地求解单个有限维LP。我们分析了收敛速度和样本复杂性,证明了对近似值的体积误差的概率性和严格保证。该方法非常通用,需要最少的基本假设。特别地,动力学不需要是多项式甚至是连续的(放弃了一些理论结果)。直到状态空间维度10的详细数字示例以及在线可用的代码就演示了该方法。这导致可以使用现成的求解器轻松地求解单个有限维LP。我们分析了收敛速度和样本复杂度,证明了对近似值的体积误差的概率性和严格保证。该方法非常通用,需要最少的基本假设。特别地,动力学不需要是多项式甚至是连续的(放弃了一些理论结果)。直到状态空间维度10的详细数字示例以及在线可用的代码就演示了该方法。这导致可以使用现成的求解器轻松地求解单个有限维LP。我们分析了收敛速度和样本复杂度,证明了对近似值的体积误差的概率性和严格保证。该方法非常通用,需要最少的基本假设。特别地,动力学不需要是多项式甚至是连续的(放弃了一些理论结果)。直到状态空间维度10的详细数字示例以及在线可用的代码就演示了该方法。动力学不需要是多项式甚至是连续的(放弃了一些理论结果)。直到状态空间维度10的详细数字示例以及在线可用的代码就演示了该方法。动力学不需要是多项式甚至是连续的(放弃了一些理论结果)。直到状态空间维度10的详细数字示例以及在线可用的代码就演示了该方法。
更新日期:2020-09-18
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