Data-driven nonlinear state observation for controlled systems: A kernel method and its analysis

This work proposes a data-driven state observation algorithm for nonlinear dynamical systems, when the true state trajectory is not measurable and hence the states information needs to be reconstructed from input and output measurements. Such a reduction is formed by kernel canonical correlation analysis (KCCA), which (i) implicitly maps the available input-output data into a higher-dimensional feature space, namely the reproducing kernel Hilbert space (RKHS); (ii) finds a projection of the past input-output data and a projection of the future input-output data with maximal correlation; and (iii) identifies the projected inputs and outputs, namely the canonical variates, as the observed states. We adopt a least squares support vector machine (LS-SVM) formulation for KCCA, which imposes regularization on the vectors that specify the projections and is amenable to convex optimization. We prove theoretically that, based on the statistical consistency of KCCA, the observed states determined by the proposed state observer has a guaranteed correlativity with the actual states (when properly transformed). Furthermore, such observed states, when supplemented with the information of succeeding inputs, can be used to predict the succeeding outputs with guaranteed upper bound on the prediction error. Case studies are performed on two numerical examples and an exothermic continuously stirred tank reactor (CSTR).