Data-driven control design in the Loewner framework: Dealing with stability and noise

Abstract

The L-DDC (Loewner Data Driven Control) algorithm is a data-driven controller design method based on frequency-domain input-output data. The identification of the plant is skipped and the controller is designed directly from the measurements using the Loewner approach, known for model approximation and reduction. However, in the L-DDC method, the identified controller is not guaranteed to be stable and the effect of noise on the identified controller is unknown. In this article, we ensure the stability of the controller and propose a solution to deal with noisy data. The method is validated on a numerical example.

From Reference Model Selection to Controller Validation: Application to Loewner Data-Driven Control

“…The L-DDC algorithm, introduced in [3], is based on frequency-domain data {ω i , P (ıω i )} The Loewner framework allows to identify an interpolating model K of the ideal controller K in the second step, see [9]. In order to avoid the compensation of the plant's RHP zeros, the L-DDC alorithm has been modified in [10]: the interpolating model K is projected on RH ∞ , using the technique presented in [11]. It corresponds to step 3 of Algorithm 1.…”

“…By construction of M f , the ideal controller itself is also stable, see (1). Finally, since the L-DDC algorithm [10] enforces the stability of the identified controller K, ∆ is stable. The sufficient condition (11) is obtained by applying the small-gain theorem, see [15].…”

“…To avoid the compensation of instabilities in the openloop, a stable controller K s ∈ RH ∞ is obtained in [10], see Algorithm 1. In this case, enforcing the stability of the controller degrades the closed-loop performances.…”

“…Now that an achievable reference model M f is built, the controller K can be designed using the L-DDC algorithm [10]. Since the identified model may be a reduced-order one of the ideal controller K , there is no guarantee in the design process that K actually stabilizes the plant internally.…”

“…Real Since the plant is stable but non-minimum phase, this application enters in case 2 from Section IV. B z is defined as in (10) and an achievable reference model is therefore M f = M B z . The identification of the controller is shown on Figure 4a.…”