Abstract
Motivated by the goal of having a building block in the direct design of data-driven controllers for nonlinear systems, we show how, for an unknown discrete-time bilinear system, the data collected in an offline open-loop experiment enable us to design a feedback controller and provide a guaranteed under-approximation of its basin of attraction. Both can be obtained by solving a linear matrix inequality for a fixed scalar parameter, and possibly iterating on different values of that parameter. The results of this data-based approach are compared with the ideal case when the model is known perfectly.
Highlights
Direct data-driven control design aims at obtaining a control law based only on input-output data collected from the system during an experiment, thereby avoiding altogether the identification of a model of the system from the data
We focus here on discrete-time bilinear systems since the data in (2) are samples obtained from experiments
The matrix inequality (16) of Lemma 4, contains products of decision variables and inverses of decision variables. We address this in the proposition, which obtains a matrix inequality that is as close as possible to an linear matrix inequalities (LMI) and expresses explicitly the matrix inequality in terms of the available data
Summary
Direct data-driven control design aims at obtaining a control law based only on input-output data collected from the system during an experiment, thereby avoiding altogether the identification of a model of the system from the data. To address the intrinsic difficulty of dealing with the control design of unknown nonlinear systems, a natural approach is to reduce their complexity by considering the system evolution along a given Lyapunov function. This classical control theoretic analysis is enhanced by nonparametric regression methods from machine learning to cope with the large uncertainty in the model [4] and is performed using a sufficiently dense set of samples taken from the system. The approach of [35, 17] to reduce the complexity of controlling unknown nonlinear systems consists of considering systems with a well-defined relative degree, in such a way that the uncertainty only appears in the form of two Lie derivatives of the output
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