Data-Driven Tests for Controllability

Abstract

The fundamental lemma due to Willems et al. "A note on persistency of excitation," Syst. Control Lett., vol. 54, no. 4, pp. 325-329, 2005 plays an important role in system identification and data-driven control. One of the assumptions for the fundamental lemma is that the underlying linear timeinvariant system is controllable. In this paper, the fundamental lemma is extended to address system identification for uncontrollable systems. Then, a data-driven algebraic test is derived to check whether the underlying system is controllable or not. An algorithm based on the singular value decomposition of a Hankel matrix constructed from the data is provided to implement the developed test. The algorithm has cubic computational cost. Examples are given to illustrate the theoretical results.

On Controllability and Persistency of Excitation in Data-Driven Control: Extensions of Willems’ Fundamental Lemma

“…Second, we show that the order of persistent excitation required by Willems' fundamental lemma can be reduced from n`L to δ min `L, where δ min is the degree of the minimal polynomial of the system matrix A. Our first result completes those presented in [18], [19] by showing exactly which trajectories are parameterizable by a finite number of measured ones for an arbitrary LTI systems. Furthermore, this result show that data-driven predictive control using online data is equivalent to model predictive control, not only for controllable systems, as shown in [12], [13], but also for uncontrollable systems.…”

“…Recent results on system identification has shown that, assuming sufficient persistency of excitation, the linear combinations of a finite number of measured input-output trajectories contain any trajectory whose initial state is in the controllable subspace [18], [19]. As we show subsequently, such results only partially answer the first question above: trajectories with initial state outside the controllable subspace can also be contained in the said linear combinations.…”

Data-Driven Controllability Analysis and Stabilization for Linear Descriptor Systems

“…Remark 2: If r(E) = r(M ) = n, then the system (1) is equivalent to a normal system x k+1 = E −1 Ax k + E −1 Bu k , whose controllability analysis and stabilization have been well studied in [4], [11] and [26]. Thus, we will mainly deal with the singular case, i.e., r(E) < n in the rest of this paper.…”

“…In order to improve the numerical feasibility of datasets and deal with missing data, the work in [6] extended the fundamental lemma to multiple datasets. Subsequent works have extended the DDC method in [4] to various scenarios, such as robust control [7], [8], [9] and LQR [10] in the presence of noisy data, data-based controllability tests [11], linear time-varying systems [12], switched linear systems [13] and linear delay systems [14]. Moreover, model predictive control was considered in [15] and [16].…”

A Data-Driven Convex Programming Approach to Worst-Case Robust Tracking Controller Design

“…This has motivated a renewed interest in system analysis and control design methods relying on finite-length data sequences [1][2][3][4]. Several recent works propose to use raw measurements for representing discrete-time systems, and solving system analysis and control design problems [1,2,[5][6][7][8][9][10][11][12][13][14][15][16]. As mentioned in [1], the main feature of these approaches is to bypass explicit system identification that is usually required in standard control design.…”