Model-free Learning of Regions of Attraction via Recurrent Sets
Model-free Learning of Regions of Attraction via Recurrent Sets
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DOI：
10.1109/cdc51059.2022.9993280
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发表时间：
2022-04
期刊：
影响因子：
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通讯作者：
Yue Shen;Maxim Bichuch;Enrique Mallada
中科院分区：
文献类型：
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作者：
Yue Shen;Maxim Bichuch;Enrique Mallada
我们考虑在没有显式动力学模型的情况下，学习渐近稳定平衡点的吸引区(ROA)的内逼近问题。与其利用具有有界不确定性的近似模型来寻找ROA中包含的(稳健的)不变集，我们建议学习满足更宽松的包容概念的集合，即递归。我们定义了一个集是τ-循环的(Rep.K-递归)如果在集合内开始的每个轨迹在至多τ秒后返回到它(分别K步)。我们证明了在较弱的假设下，包含稳定平衡点的τ-递归集一定是其ROA的子集。然后，我们利用这一特性来开发算法，该算法使用通过采样有限长度轨迹获得的递归的反例来计算ROA的内近似。我们的算法按顺序处理样本，这使得它们即使在最初的离线训练阶段之后也可以继续执行。此外，我们还给出了算法所用反例个数的上界，并且几乎可以保证算法的收敛。
We consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. Rather than leveraging approximate models with bounded uncertainty to find a (robust) invariant set contained in the ROA, we propose to learn sets that satisfy a more relaxed notion of containment known as recurrence. We define a set to be τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set, returns to it after at most τ seconds (resp. k steps). We show that under mild assumptions a τ-recurrent set containing a stable equilibrium must be a subset of its ROA. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allows them to continue being executed even after an initial offline training stage. We further provide an upper bound on the number of counter-examples used by the algorithm, and almost sure convergence guarantees.