Towards a Constructive Framework for Stabilization and Control of Nonlinear Systems: Passivity and Immersion (P\&I) Approach

The varied and complex dynamics of real-world systems challenge the formulation of a systematic strategy for designing a stabilizing feedback law. Rather than taking a universal approach, the control strategies developed thus far to handle this problem are specific to the inherent structure of the system under consideration. Therefore, this paper attempts to develop a generalized theory for the design of the stabilizing feedback law wherever possible for a general class of systems, including the systems in standard structured and unstructured forms discussed in the existing literature. The theory behind this general controller design theory 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. Because 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) approach". Furthermore, being a constructive methodology, the various worked examples in this paper exemplify and demonstrate the various design paradigms such as Backstepping, Incremental Backstepping, Forwarding, I&I, and the techniques based on the generation of Control Contraction Metrics (CCM) can be unified in the P&I methodology.