Executive Summary : | Contraction theory is a strong mathematical framework for analyzing the stability and robustness of dynamical systems. The basic idea behind contraction theory is to study how the distances between trajectories of a dynamical system evolve over time. A system is said to be contracting if the distances between nearby trajectories decrease exponentially fast. Contraction is a desirable property for control systems, as it implies stability and robustness to perturbations. Contraction theory provides tools for analyzing the stability and robustness of nonlinear systems, including systems with input constraints and uncertainties. It has been applied to a wide range of applications, including robotics, opinion dynamics, economics, biological systems, synchronization problems, frequency estimation and saturated control design for nonlinear systems etc. Contraction analysis is one of the approaches for ascertaining incremental stability. However, in the context of systems equipped with multiple equilibrium points, the usual contraction analysis (which will be referred to as 1-contraction from now onward) fails to comment on the convergence of trajectories. This has been sparsely studied in the literature. In the first phase of the study, we will develop a systematic analytical technique that establishes the strong relationship between higher-order contraction (k-contraction, where k is greater than 1) and its usefulness in the analysis of the global behavior of a nonlinear dynamical system with multiple equilibrium points. Following are the objectives of this proposal. Objective-1: To study the stability of a complex nonlinear networked system that has multiple equilibria. The generalized contraction theory (k-contraction) will be used to investigate sufficient conditions under which the system is guaranteed to converge to a steady state or limit cycle. * Objective-2: To exploit Contraction theory for the robustness analysis of such nonlinear systems. This can help to determine the degree to which a complex nonlinear networked system can tolerate perturbations, changes in its input, and parameter variations. * Objective-3: To verify and validate the proposed theoretical results on various benchmark problems such as opinion dynamics, power-flow networks, recurrent neural networks which are equipped with multiple equilibria. Overall, the application of contraction theory to complex nonlinear networked systems can provide valuable insights into the system's behavior and help guide the design of effective control strategies. |