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.

Adaptive Estimation and Control With Online Data Memory: A Historical Perspective

“…The forgetting-data MRE with a HOT was also applied to improve the transient response of MRAC for a class of discrete LTI systems in [34]. A passivity and immersion-based modified gradient estimator extended by forgetting-data MRE was proposed to improve transient response and convergence of parameter estimation in [35].…”

A New Perspective of Accelerated Gradient Methods: The Controlled Invariant Manifold Approach

“…The optimization problem in (1) is geometrically visualized as the manifold stabilization problem to design an algorithm by adding the virtual control term u. This input u is evaluated through the procedure of passivity and immersion (P&I) approach [21] (The step-wise procedure is overviewed in Appendix) such that the trajectories converge exponentially to the equilibrium with a suitable choice of an implicit manifold.…”

“…The above-defined control law ensures the GAS equilibrium point of the system to zero/origin. Remark 12: The mathematical details & related theoremsproofs of the P&I approach can be found in [21].…”

“…P&I is proposed for control and stabilization, and the concept is extended to manifold stabilization. Recently the approach has been applied to parameter estimation [18] by visualizing the estimation problem as a manifold stabilization problem, and the accurate estimation of the unknown parameters is accomplished via a control perspective. P&I is an efficient and structured formulation that is useful to solve any problem by visualizing it as a manifold stabilization problem.…”

The sensitivity demonstration and propagation of hyper-geometric soliton waves in plasma physics of Kairat-II equation

“…Technological developments have opened up new avenues for mathematicians working in the applied sciences, greatly contributing to this growing interest [9]. The increasing importance of nonlinear PDEs in contemporary scientific research is highlighted by the convergence of mathematical analysis, computational capabilities, and physical applications [10]. The study of optical solitons in nonlinear materials is one particular field of intense research interest.…”