Fixed-point mapping and stability analysis for adaptive decision equilibrium in autonomous cyber-physical engineering systems

Abstract

One of the challenges in designing Cyber-Physical Engineering Systems is to demonstrate how the decentralized physical behavior and cyber-decisions of the system are able to reach a distinct compatible stable state with continuous nonlinear interplay between the two. This paper proposes the joint evolution of the two physical and decision-state trajectories as an operator in a Banach space of continuous functions, which leads to the formulation of the problem of finding an adaptive decision equilibrium as a fixed-point problem. The existence and uniqueness of the equilibrium is based on the Banach Contraction Principle and the global Lipschitz condition, but the result is later extended to non-expansive decision mappings on compact convex feasible domains using the Schauder fixed point theorem. In the process, we use the Lyapunov-Krasovskii functional based on the enlarged state of the physical and cyber errors to obtain sufficient conditions for local exponential stability, which can also be extended to Input-to-State Stability (ISS) when bounded external disturbances are taken into account. Furthermore, a clear K and KL type bounds on the state trajectories were empirically established through the numerical simulation of four adaptive gain setups: the stability margin rises non-monotonically from σ = 0.380 (with Γ = 0.5·I) to σ = 0.105 in an oscillatory-stable setting (with Γ = 1.2·I), and enters an unstable zone of the system (λmin(Q) = −0.24) at Γ = 2.5·I when the Lipschitz constant equals 11.30. The average stability margin equals 0.209 among the three stable setups, whereas the admissible contraction horizon T < 1/LF varies between 0.194 and 0.806 second. In conclusion, it can be stated that moderate values of adaptive gain are conducive to the fastest convergence speeds, while great gain values yield an unstable state, which once again confirms the effectiveness of the fixed-point approach and the Lyapunov-Krasovskii framework of stability.