Data-Driven Control of Nonlinear Systems: Beyond Polynomial Dynamics

Abstract

In this paper, we present a data-driven controller design method for continuous-time nonlinear systems with rational system dynamics, using no model knowledge but only measured data affected by noise. We first extend recent results on data-driven control for linear time-invariant systems by presenting a purely data-driven representation of unknown nonlinear systems with rational dynamics. By applying robust control techniques to this parametrization, we obtain sum-of-squares based criteria for designing controllers with closed-loop robust stability guarantees for all continuoustime systems with rational system dynamics which are consistent with the measured data and the assumed noise bound. Finally, we apply the developed approach to a numerical example.

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.…”

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

“…Put differently, the indices q and s (and, thus, r = n − q) are invariants of the original system (1) preserved in the quasi-Weierstraß form. Similarly to before, we consider the full and manifest behavior as well as their restrictions to finite time intervals for system (5). Specifically, we denote these behaviors by…”

“…Proof: We consider the corresponding trajectories (z, u, y), (z, ũ, ỹ) ∈ B f [0, q + s − 2] of the equivalent system (5), that is z = P −1 x, z = P −1 x. According to (6) we have…”

“…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].…”

Data-Based System Analysis and Control of Flat Nonlinear Systems

“…There, input-affine continuous-time systems were addressed assuming constant intersampling behavior of the states under high enough sampling rate. Other data-driven stabilization techniques for some classes of nonlinear systems appeared in [18]- [20]. In this paper, we consider the wide class of flat nonlinear systems [21], also referred to as difference flat nonlinear systems in discrete time [22].…”