Bridging direct & indirect data-driven control formulations via regularizations and relaxations

Abstract

We discuss connections between sequential system identification and control for linear time-invariant systems, which we term indirect data-driven control, as well as a direct datadriven control approach seeking an optimal decision compatible with recorded data assembled in a Hankel matrix and robustified through suitable regularizations. We formulate these two problems in the language of behavioral systems theory and parametric mathematical programs, and we bridge them through a multicriteria formulation trading off system identification and control objectives. We illustrate our results with two methods from subspace identification and control: namely, subspace predictive control and low-rank approximation which constrain trajectories to be consistent with a non-parametric predictor derived from (respectively, the column span of) a data Hankel matrix. In both cases we conclude that direct and regularized data-driven control can be derived as convex relaxation of the indirect approach, and the regularizations account for an implicit identification step. Our analysis further reveals a novel regularizer and sheds light on the remarkable performance of direct methods on nonlinear systems.

Data-Driven Nearly Optimal Control for Constrained Nonlinear Systems

“…, L− T +1}, in the constraints. Here, we describe a convex-concave procedure from [24] that can be iteratively employed to solve it 3 . We first describe the bilinear terms with new variables r i,j = α i α j , which we employ in the constraints in (P2) to make them all become affine.…”

“…(Population control): We consider a population control problem introduced in [20, Example 1] evolving in continuous time. For the horizon T = 20, we use a first-order 3 Local optimal solutions to bilinear programs can also be found using the OPTI Toolbox in MATLAB.…”

“…Indirect methods identify system models from data prior to proceeding to the synthesis of model-based controllers, while direct approaches bypass the intermediate modeling step and construct controllers directly from data. A diverse range of factors, including the complexity of the plant, the cost and practicality of performing system identification, and the amount and quality of the available data, play a key role in the suitability and performance of each of these approaches, see e.g., [2], [3]. The direct data-driven approach has been particularly fruitful for linear systems, where tools from behavioral theory [4] have allowed to express the system trajectories in terms of sufficiently-rich data.…”

Data-driven Distributed and Localized Model Predictive Control

“…Specifically, we highlight the work in De Persis and Tesi (2019), which applies behavioral systems theory to parametrize systems from past trajectories. 1 This idea has then given rise to several different data-enabled Model Predictive Control (MPC) approaches (Dörfler et al, 2021b;Coulson et al, 2021;Berberich et al, 2020;Xue and Matni, 2021). However, these approaches require gathering past trajectories of the global system, which hinders their scalability and challenges their applicability in the distributed setting.…”

Learning Controllers from Data via Approximate Nonlinearity Cancellation

“…This idea is suggested by a regularization procedure in which the hard constraint of the first approach, corresponding to an exact nonlinearity cancellation, is lifted to an objective function, corresponding to an approximate nonlinearity cancellation. (In different contexts, this "lifting" idea has been pursued in [26], [27], [28]). In general the design based on an approximate nonlinearity cancellation does not return globally stabilizing controllers, whence the need to explicitly characterize the region of attraction of the closed-loop system.…”