Learning Controllers from Data via Approximate Nonlinearity Cancellation

Abstract

We introduce a method to deal with the data-driven control design of nonlinear systems. We derive conditions to design controllers via (approximate) nonlinearity cancellation. These conditions take the compact form of data-dependent semidefinite programs. The method returns controllers that can be certified to stabilize the system even when data are perturbed and disturbances affect the dynamics of the system during the execution of the control task, in which case an estimate of the robustly positively invariant set is provided. I. INTRODUCTIONA Utomating the control design process is important to cope with complex dynamical plants whose dynamics is poorly known. Data-driven control is a notable example of such an automated synthesis. Namely, data-driven control refers to the procedure of designing controllers for an unknown system starting solely from measurements collected from the plant and some priors about the plant itself (linear vs. nonlinear parametrization, nature of the noise, etc.). In this paper we study the problem of designing controllers for nonlinear systems from data.Related literature. System identification followed by control design for the identified system is a classical way to indirectly perform data-driven control [1]. By direct data-driven control instead it is meant a procedure in which no intermediate step of identifying the system model is taken, earlier examples being the iterative feedback tuning (IFT) [2], and the virtual reference feedback tuning (VRFT) [3]. Recent times have seen a renewed interest in direct data-driven control, viewed as compact data-dependent conditions which, once verified, automatically return controllers without explicitly identifying the plant. One of the focus points in these data-driven control results is how to deal with perturbations and noise affecting the data and the resulting noise-induced uncertainty. Assuming a process noise with bounded ∞ norm, [4] defines a set of system's matrices pairs consistent with the data and, using an extended Farkas' lemma, derives conditions under which stability of all systems in the set hold. These conditions can be checked using polynomial optimization techniques.The papers [5], [6] highlight the relevance of a result in [7], about representing the behavior of a linear time-invariant system via a single input-output trajectory, and use this result to develop data-enabled, rather than model-based, predictive control, providing probabilistic guarantees on performance for systems subject to stochastic disturbances.C. De Persis and M. Rotulo are with ENTEG and the J.

Data-Driven Stabilizer Design and Closed-Loop Analysis of General Nonlinear Systems Via Taylor's Expansion

“…It is a generalization of the previous result [6,Theorem 6] where the firstorder approximation and a linear controller is considered. Compared to results that relies on specific choices of the basis functions, such as [17], Theorem 4 synthesizes a datadriven controller using Taylor polynomials as basis functions. The approach for the control of polynomial systems in Theorem 4 is an alternative to the results presented in [9] and [25], where [9] handles the additional noisy term in a different way, and [25] searches for a Lyapunov function without restricting to a special form.…”

“…The estimated RoA gives insights to the closed-loop system under the designed data-driven controller, and is relevant for both theoretical and practical purposes. Note that alternatively, the RoA can be estimated based on the bound on the remainder via numerical methods such as the one used in [17]. Simulations results on the inverted pendulum show the applicability of the design and estimation approach.…”

“…For instance, under suitable conditions, some nonlinear systems can be lifted into polynomial systems in an extended state for control, such as the results shown in [15, Section IV] and [16,Section 3.2]. Using knowledge of the basis functions, [17] designed data-driven controllers by (approximate) cancellation of the nonlinearity. When the system nonlinearities cannot be expressed as combination of known functions, [17] presented data-driven local stabilization results by choosing basis functions carefully such that the neglected nonlinearities are small in a known set of the state.…”

“…Using knowledge of the basis functions, [17] designed data-driven controllers by (approximate) cancellation of the nonlinearity. When the system nonlinearities cannot be expressed as combination of known functions, [17] presented data-driven local stabilization results by choosing basis functions carefully such that the neglected nonlinearities are small in a known set of the state. On the other hand, if the knowledge on the basis functions is not available, approximations of the nonlinear systems are often involved.…”

“…On the other hand, the focus of this paper is on general nonlinear systems, while the previous works [9] and [25] dealt with only polynomial systems. In comparison with [17] where the influence of the remaining nonlinearity is attenuated by careful choices of the basis functions, this work achieves the objective by using Taylor's polynomials as basis functions and conduct the experiment closed to the equilibrium. It is noted that, the SOS technique based data-driven controller design for discrete-time nonlinear systems has not appeared in any our previous works.…”