On Controllability and Persistency of Excitation in Data-Driven Control: Extensions of Willems’ Fundamental Lemma
On Controllability and Persistency of Excitation in Data-Driven Control: Extensions of Willems’ Fundamental Lemma
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关于数据驱动控制中激励的可控性和持续性：威廉斯基本引理的扩展
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发表时间：
2021
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通讯作者：
Behçet Açikmese
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作者：
Yue Yu;S. Talebi;H. V. Waarde;U. Topcu;M. Mesbahi;Behçet Açikmese
Willems的基本引理断言，一个线性时不变系统的所有轨迹都可以从有限个测量轨迹中得到，假设可控性和激励条件保持不变。我们证明这两个条件可以放宽。首先，我们证明了可控性条件可以被一个条件的可控子空间，不可观测的子空间，和一定的子空间相关联的测量轨迹。其次，我们证明了如果某个极小多项式的次数是紧有界的，则激励的持续性要求可以放宽。我们的研究结果表明，数据驱动的预测控制使用在线数据是等效的模型预测控制，即使是不可控的系统。此外，我们的研究结果显着减少识别同质多智能体系统所需的数据量。
Willems’ fundamental lemma asserts that all trajectories of a linear time-invariant system can be obtained from a finite number of measured ones, assuming that controllability and a persistency of excitation condition hold. We show that these two conditions can be relaxed. First, we prove that the controllability condition can be replaced by a condition on the controllable subspace, unobservable subspace, and a certain subspace associated with the measured trajectories. Second, we prove that the persistency of excitation requirement can be relaxed if the degree of a certain minimal polynomial is tightly bounded. Our results show that data-driven predictive control using online data is equivalent to model predictive control, even for uncontrollable systems. Moreover, our results significantly reduce the amount of data needed in identifying homogeneous multi-agent systems.