Willems' fundamental lemma asserts that all trajectories of a linear time-invariant (LTI) system can be obtained from a finite number of measured ones, assuming 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 reduced if the degree of certain minimal polynomial is known or tightly bounded. Our results shows 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.
Recently, data-driven methods for control of dynamic systems have received considerable attention in system theory and machine learning as they provide a mechanism for feedback synthesis from the observed time-series data. However learning, say through direct policy updates, often requires assumptions such as knowing a priori that the initial policy (gain) is stabilizing, e.g., when the open-loop system is stable. In this paper, we examine online regulation of (possibly unstable) partially unknown linear systems with no a priori assumptions on the initial controller. First, we introduce and characterize the notion of "regularizability" for linear systems that gauges the capacity of a system to be regulated in finite-time in contrast to its asymptotic behaviour (commonly characterized by stabilizability/controllability). Next, having access only to the input matrix, we propose the Data-Guided Regulation (DGR) synthesis that-as its name suggests-regulates the underlying states while also generating informative data that can subsequently be used for data-driven stabilization or system identification (sysID). The analysis is also related in spirit, to the spectrum and the "instability number" of the underlying linear system, a novel geometric property studied in this work. We further elucidate our results by considering special structures for system parameters as well as boosting the performance of the algorithm via a rank-one matrix update using the discrete nature of data collection in the problem setup. Finally, we demonstrate the utility of the proposed approach via an example involving direct (online) regulation of the X-29 aircraft.
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