Multivariable PI for Packed-Bed Ethanol Reactors: A Davison–Moore–Penrose Approach

We address regulation of an ethanol-to-hydrocarbon packed-bed reactor (HZSM-5) with input redundancy by combining a centralized multivariable PI controller with Davison-style shaping and a Moore–Penrose control-allocation layer. A one-cell finite-volume model is derived from axial mass and energy balances with Danckwerts boundary conditions and used to identify a rectangular, highly anisotropic steady-state gain matrix G for the two controlled variables—bed temperature T and in-bed concentration C_i—and three manipulated inputs (superficial velocity μ, inlet temperature T₀, and coolant temperature T_c). Because G is ill-conditioned, the allocator employs the pseudoinverse (with Tikhonov regularization and physical scaling) to distribute the PI demand among actuators, while setpoint prefiltering limits proportional kick and back-calculation anti-windup preserves bias-free recovery under amplitude/rate limits. The numerical allocation clarifies actuator roles: T_c provides dominant thermal authority for T, μ primarily shapes residence time for C_i, and T₀ acts as a trim to reduce interaction. Closed-loop simulations show fast, well-damped tracking and zero steady-state error for reference changes. Under constraints, the augmented loop degrades gracefully without offset. Against a temperature-centric SISO PID baseline, the proposed design markedly reduces peaking (start-up overshoot ≈40% with PID) and ringing, with lower total actuator variation. The results position centralized PI with pseudoinverse allocation as an implementation-ready, interpretable alternative to MPC for over-actuated biofuel reactors, offering robust, bias-free performance at low computational cost.