Fundamental Lemma for Data-Driven Analysis of Linear Parameter-Varying Systems

Abstract

In data-driven analysis and control, the so-called Fundamental Lemma by Willems et al. has gained a lot of interest in recent years. Using behavioural system theory, the Fundamental Lemma shows that the full system behaviour of a Linear Time-Invariant (LTI) system can be characterised by a single sequence of data of the system, as long as the input is persistently exciting. In this work, we aim to generalize this LTI result to Linear Parameter-Varying (LPV) systems. Based on the behavioural framework for LPV systems, we prove that one can obtain a result similar to Willems'. This implies that the result is also applicable to nonlinear system behaviour that can be captured with an LPV representation. We show the applicability of our result by connecting it to earlier works on data-driven analysis and control for LPV systems.

Linear tracking MPC for nonlinear systems Part II: The data-driven case

“…Although different successful applications to complex nonlinear systems have been reported in the literature, see, e.g., [11], [12], providing theoretical guarantees of data-driven MPC for nonlinear systems remains a widely open research problem. The literature contains various extensions and variations of [2] for specific classes of nonlinear systems such as Hammerstein and Wiener systems [13], Volterra systems [14], polynomial systems [15], [16], systems with rational dynamics [17], flat systems [18], and linear parameter-varying systems [19]. However, all of these works assume that the system is linearly parametrized in known basis functions, which restricts their practical applicability.…”

Behavioral systems theory in data-driven analysis, signal processing, and control

“…The fundamental lemma has been generalized for uncontrollable systems (Mishra et al, 2020;Yu et al, 2021), data consisting of multiple trajectories (van Waarde et al, 2020b), other matrix structures (Coulson et al, 2020), as well as the following model classes: affine (Berberich et al, 2021c), linear parameter-varying (Verhoek et al, 2021), flat systems (Alsalti et al, 2021), finite impulse response Volterra (Rueda-Escobedo and Schiffer, 2020), and Wiener-Hammerstein (Berberich and Allgöwer, 2020). Other works extending the fundamental lemma to nonlinear systems use the Koopman operator .…”

Willems' fundamental lemma for linear descriptor systems and its use for data-driven output-feedback MPC

“…Recently, data-driven control-and in particular Willems' fundamental lemma [1]-is subject to substantial research interest. This includes non-parametric system representations for deterministic discrete-time linear time-invariant (LTI) systems [2] and linear parameter-varying (LPV) systems [3], stochastic LTI systems [4], as well as extensions to polynomial and non-polynomial nonlinear systems [5], [6]. These non-parametric representations enable system identification [7], control design [8], and also the implementation of predictive control [9], [10].…”