This paper investigates target tracking and spatiotemporal chaos suppression for a fractional Rössler reaction–diffusion partial differential equation (PDE) subject to simultaneous process and measurement noise. The distributed system is represented using a finite-difference spatial discretization and a finite-memory Grünwald–Letnikov approximation of the time-fractional dynamics. Because an arbitrary prescribed spatial profile is generally not a natural equilibrium of the uncontrolled reaction–diffusion system, a steady feedforward term is introduced to render the target invariant under the nominal controlled model. A nonlinear model predictive control (NMPC) strategy implemented in CasADi then computes a bounded corrective action around this feedforward input using a filtered measurement history. Under a common process- and measurement noise realization, the proposed feedforward-assisted fractional NMPC reduces the final normalized rootmean-square (RMS) tracking error from 2.7293 in the uncontrolled case to 2.1028 × 10⁻², corresponding to a 99.23% reduction. Controlled benchmark comparisons give final RMS errors of 3.7991 × 10⁻² for standard NMPC without feedforward, 2.2018 × 10⁻² for feedforward plus linear quadratic regulation, and 2.2212 × 10⁻² when the NMPC predictor neglects fractional memory. The proposed method therefore provides the smallest final tracking error among the considered controlled configurations while requiring comparable total control energy. A sweep over α ∈ [0.80, 1.00] shows that, for the present parameterization and finite-memory approximation, increasing α toward the integer-order limit increases both the tracking error and the corrective control effort. A ±10% mismatch in the Rössler parameter c degrades the tracking accuracy but preserves practical bounded regulation, with the corrective input returning well below its admissible bound after the initial transient. A negative tail-mean finite-time largest Lyapunov exponent (FTLE) provides an additional indication of local finite-time trajectory contraction. The FTLE is interpreted strictly as a numerical diagnostic rather than as a proof of global, asymptotic, or stochastic stability.