Passivity and Immersion (P&I) Approach With Gaussian Process for Stabilization and Control of Nonlinear Systems

Abstract

The virtual derivatives computation and successive derivations of virtual inputs in an adaptive backstepping controller cause the explosion of complexity. Moreover, the feedback linearization has poor robustness features and necessitates exact estimation of the feedback control law’s coefficients. Due to measurement noise, the model-based estimation techniques for identifying uncertainties result in inaccurate gradient and Hessian calculations. Such limitations lead to model and measurement uncertainties that prevent effective stabilization and control of nonlinear systems. Machine learning-based data-driven approaches offer effective tools for identifying dynamical systems and uncertainties with minimal prior knowledge of the model structure. Therefore, the contribution of this research is two-fold: First, the general controller design theory is proposed which utilizes the idea of an invariant target manifold giving rise to a non-degenerate two form, through which the definition of certain passive outputs and storage functions leads to a generation of control law for stabilizing the system. Since the above concepts are linked with the Immersion and Invariance (I&I) design policy and the passivity theory of controller design, the proposed methodology is labeled as the “Passivity and Immersion (P&I) based approach”. Second, the proposed P&I approach is integrated with a Bayesian nonparametric approach, particularly the Gaussian Process for stabilization and control of the partially unknown nonlinear systems. The effectiveness of the proposed methodologies has been evaluated on an inverted pendulum using MATLAB in the presence of input-output uncertainties.

Keywords

• Feedback linearizable structure
• Gaussian process regression
• immersion and invariance
• stabilization and control
• uncertainties