Symmetric Adaptive Tracking Control of Stochastic High-Order Delay Systems with Stochastic Inverse Dynamics and Unknown Time-Varying Powers

We study practical output tracking for stochastic high-order systems with unknown time-varying powers, uncertain positive control coefficients, time-varying delays, and stochastic inverse dynamics. Relying on the symmetric property of signed-power operation, an adaptive backstepping design with symmetric error handling is combined with a Lyapunov–Krasovskii functional to uniformly cope with both positive and negative tracking errors. Under explicit growth, regularity, and parameter-drift hypotheses, the closed-loop solution is global, its signals are bounded in probability, and the tracking error has an ultimate mean-square bound. A reproducible single-stage example implements the actual adaptive law with an unknown time-varying input power. The conclusion is practical tracking; neither almost-sure uniform boundedness nor convergence of the error to zero is asserted.