Formulas for Data-Driven Control: Stabilization, Optimality, and Robustness

Abstract

In a paper by Willems and coauthors it was shown that persistently exciting data can be used to represent the inputoutput behavior of a linear system. Based on this fundamental result, we derive a parametrization of linear feedback systems that paves the way to solve important control problems using data-dependent Linear Matrix Inequalities only. The result is remarkable in that no explicit system's matrices identification is required. The examples of control problems we solve include the state and output feedback stabilization, and the linear quadratic regulation problem. We also discuss robustness to noise-corrupted measurements and show how the approach can be used to stabilize unstable equilibria of nonlinear systems.

Data Informativity: A New Perspective on Data-Driven Analysis and Control

“…Remark 18. To the best of our knowledge, LMI conditions for data-driven stabilization were first studied in [40]. In fact, the linear matrix inequality (20) is the same as that of [40,Theorem 3].…”

Willems’ Fundamental Lemma for State-Space Systems and Its Extension to Multiple Datasets

“…In [9,Thm. 4] an attractive design procedure is introduced to obtain K directly from input/state data.…”

“…has full row rank by Theorem 1(i). Subsequently, K is found by solving a semidefinite program involving the data x [0,T ] and u [0,T −1] alone; see [9,Eq. 27].…”

“…29] to obtain P + and K from input/state data. Both semidefinite programs of [9] and [19] are applicable to this example, but we will follow the method of [19] since it involves less decision variables, c.f. [19,Remark 31].…”

“…The starting point in this paper, however, is a reformulation of the lemma in terms of state-space systems. Such a version of Willems' fundamental lemma for state-space systems has appeared before in [9,Lem. 2] and [14,Thm.…”