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Contraction theory formulates the analysis of nonlinear systems in terms ofJacobian matrices. Although this provides the potential to develop a linearmatrix inequality (LMI) framework for nonlinear control design, conditions areimposed not on controllers but on their partial derivatives, which makescontrol design challenging. In this paper, we illustrate this so-calledintegrability problem can be solved by a non-standard use of Gaussian processregression (GPR) for parameterizing controllers and then establish an LMIframework of contraction-based control design for nonlinear discrete-timesystems, as an easy-to-implement tool. Later on, we consider the case where thedrift vector fields are unknown and employ GPR for functional fitting as itsstandard use. GPR describes learning errors in terms of probability, and thuswe further discuss how to incorporate stochastic learning errors into theproposed LMI framework.

An LMI Framework for Contraction-based Nonlinear Control Design by Derivatives of Gaussian Process Regression