Decentralized Adaptive Generalized-Minimum-Variance Control of Large-Scale Interconnected Multivariable Hammerstein Systems

This paper presents a decentralized adaptive generalized-minimum-variance (GMV) control framework for large-scale stochastic nonlinear systems composed of interconnected multi-input multi-output (MIMO) Hammerstein subsystems with unknown time-varying parameters. Each subsystem consists of a coupled multivariable static nonlinearity represented on a known invertible basis, followed by a matrix-polynomial dynamic block affected by colored noise and delayed input–output interconnections. The proposed scheme estimates only identifiable composite Hammerstein parameters through a decentralized recursive extended least-squares algorithm with forgetting, thereby avoiding the non-unique separation of nonlinear and linear gains. A constructive matrix Diophantine identity is established to derive an optimal multi-step predictor, leading to a GMV control law expressed as a multivariable polynomial equation in the current input. Sufficient conditions for real solvability, mean-square boundedness, and near-optimal adaptive tracking are provided using Hadamard–Lévy global-diffeomorphism, minimum-phase, small-gain, persistent-excitation, strict-positive-realness, and convex-projection arguments, and the implemented controller—inexact Newton solver with fallback and persistent dither—is itself covered by the analysis. The analysis further shows that delayed interconnections become measurable and can be exactly compensated, while robustness to basis under-modeling is explicitly quantified. Simulation results on an interconnected two-subsystem MIMO Hammerstein process with coupled cubic nonlinearities, colored noise, delayed interactions, and time-varying parameters—run in the forgetting-factor regime required by the theory, with measured persistent excitation and complete solver diagnostics—demonstrate operational-noise-floor tracking and a 2.3-fold mean-RMSE reduction relative to the strongest linear-MIMO surrogate, while a channel-wise SISO Hammerstein design fails structurally and a feedback-linearization controller with exactly known nonlinearity offers no advantage. The study further demonstrates scalability on a chain of four subsystems with size-independent per-subsystem computational cost, validates a physically motivated interconnected coupled-tank network with progressive-valve nonlinearities, and confirms agreement between the observed stability limits and the predicted small-gain boundary.