A Nonlinear Functional Analysis Framework for Stability Optimization of Dynamic Control Systems
Keywords:
Nonlinear functional analysis, fixed point theorems, krasnoselskii's theorem, stability optimization, dynamic control systems, Lyapunov functional, robotic manipulator control.Abstract
A nonlinear functional approach is proposed for investigating the existence, uniqueness, and optimal stability of a large class of nonlinear dynamical control systems modeled as an operator equation in Banach space. The state trajectory of the system is modeled as a mild solution to an abstract semilinear evolution equation. The solution operator of the closed loop system is considered as composed of two operators, one being a contracting operator and the other being compact as defined by Krasnoselskii. The results are based on the Banach contraction principle, which is applied for proving the existence and uniqueness of the mild solution, whereas the existence is proved using a fixed-point technique similar to Krasnoselskii. An explicit exponential decay bound for the closed-loop trajectory is then obtained by combining the weighted-norm stability theorem with the fixed-point formulation, and the feedback gain operator is optimised over an admissible compact convex set in order to maximize the guaranteed decay rate, which is shown to exist by a Weierstrass-type compactness argument. The structure of the theorems is explicitly checked in the case of the two-link rigid robotic manipulator trajectory-tracking problem, and the obtained exponential stability and near-optimal gain selection is numerically demonstrated against a PID controller. These findings contribute to the current stability investigation of nonlinear and fractional order dynamical systems using fixed point theory by developing a realistic and application-oriented methodology for stability optimization.
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