Model Predictive Control versus Decentralised PID for the Wood-Berry Distillation Column: A Simulation Study

This dissertation asks a question every instrumentation student meets but few answer rigorously: on a strongly interacting two-input two-output column, when does model predictive control actually beat well-tuned decentralised PID — and by how much? The plant is the Wood–Berry binary distillation column, the classic published benchmark whose transfer-function model, disturbance model and interaction structure have anchored multivariable-control literature for fifty years.

The comparison is run as a controlled simulation study with a declared tuning protocol: both controllers are tuned to a comparable closed-loop speed on the dominant loop, then evaluated over identical servo and regulatory test scenarios on the same plant discretisation, at the same sampling time, with the same disturbance realisations and noise seeds. The challengers are (i) two single-loop PI controllers tuned by Skogestad's SIMC rules on effective models, with and without a static decoupler chosen by relative gain array (RGA) analysis, and (ii) a centralised linear MPC built with do-mpc, an industry-standard Python toolbox on the CasADi optimisation backend, over a grid of prediction horizons, control horizons and weight ratios.

Every run reports IAE, ISE and ITAE per output, overshoot, settling time, control moves and constraint violations, each averaged over repeated noise realisations. Robustness is tested by re-running the winning designs on perturbed plants with gain and delay errors. The work is pure simulation — no hardware — and produces a paper-style results section for the VTU M.Tech 2022 Project Work Phase-2 (22SXX41) dissertation, with all numbers coming from your own runs.

On an interacting 2x2 distillation column, a control engineer must choose between two philosophies: decentralised PI loops — with or without decoupling — and centralised MPC. Vendor material and textbook examples suggest MPC wins, but the comparisons that reach students usually skip the details that decide the outcome: the two controllers are tuned to different speeds, scenarios are not repeated, noise is ignored, constraint handling is not scored, and robustness to model error is never tested. As a result the real size of the MPC advantage, and the conditions under which it appears or disappears, are unknown to the engineer making the choice.

This study poses the problem precisely: under one declared tuning protocol with equalised closed-loop speed, identical scenarios, repeated noise realisations and scored constraint handling, how large is the measured performance difference between centralised MPC and SIMC-tuned decentralised PI control — with and without static decoupling — on the Wood–Berry column, in nominal and perturbed-plant conditions? The answer must be reproducible from the protocol alone, so that another student can rerun it and challenge any number.

Abstract

Distillation columns are the canonical multivariable-control problem: reflux and steam both affect both product compositions, through opposite signs, long and unequal dead times. This dissertation quantifies, under one declared tuning protocol, the performance gap between a centralised linear MPC and decentralised PI control with optional static decoupling on the Wood–Berry benchmark. Performance is measured with integral error criteria, overshoot, settling time, control effort and constraint behaviour across servo, regulatory and combined scenarios, and the conclusions are stress-tested on perturbed plants.

Introduction

The Wood–Berry column (Wood and Berry, 1973) separates a binary mixture, with the distillate composition controlled by reflux and the bottoms composition by steam. Its published model — first-order-plus-dead-time transfer functions with an interacting, opposite-sign off-diagonal — is small enough to simulate exactly, yet hard enough that naive single-loop control suffers from loop interaction. That combination makes it the standard yardstick for multivariable strategies from decoupling to MPC.

Existing System

• Decentralised PI control remains the industrial default: cheap, transparent, maintainable, and understood by every plant instrument engineer.
• Static or dynamic decouplers reduce interaction but depend on an accurate steady-state model; when the plant drifts, decoupling can amplify rather than reduce interaction.
• MPC has dominated advanced-process-control practice in refineries since the 1980s: it handles constraints, dead time and multivariable interaction in one formulation, at the cost of model maintenance and tuning expertise.

Literature Gap

• Comparative MPC–PID studies often tune the two controllers to different effective speeds, making the integral-error comparison meaningless.
• Many papers report one nominal run per controller, without repeats, so nothing is known about variability.
• Gap: a single, fully declared protocol — equalised closed-loop bandwidth, fixed scenarios, fixed seeds, repeated runs, constraint-aware metrics — on the Wood–Berry benchmark, with a horizon-and-weight grid for MPC, SIMC-based PI tuning, RGA-driven pairing and decoupling ablation, and an explicit robustness study at ±20% gain and +50% delay mismatch.

Proposed Work

• Exact discretisation of the published continuous model at a 1-minute sampling time; noise-free and noisy measurement variants.
• PI designs by SIMC rules on effective first-order-plus-dead-time models; static decoupler from the inverse steady-state gain matrix; RGA computed to justify pairing.
• Centralised MPC with quadratic cost, input-rate and amplitude constraints, via do-mpc; grid over prediction horizon, control horizon and Q/R weight ratio.
• Scorecard across scenarios, repeats with different seeds, and a perturbed-plant robustness section.

Feasibility

Everything is open-source Python on a laptop: do-mpc and CasADi are free, the plant is a handful of transfer functions, and each 300-minute simulation at 1-minute steps runs in seconds even with the MPC optimiser in the loop. The effort goes into protocol discipline and analysis — exactly the depth an M.Tech dissertation should demonstrate.

In scope

• The published Wood–Berry 2x2 model and its feed-composition disturbance model, simulated in continuous-equivalent form at Ts = 1 min over 300-minute scenarios.
• Decentralised PI control (SIMC tuning), static decoupling, and centralised linear MPC with quadratic cost and input constraints.
• Servo tests (step on each setpoint, then simultaneous), regulatory tests (feed-composition disturbance step), and combined tests; noise-free and noisy variants with repeated seeds.
• Nominal and perturbed-plant (±20% steady-state gain, +50% delays) robustness runs.

Out of scope

• Nonlinear or first-principles column modelling — the published linear benchmark is used as-is.
• Identification from data: the model is given, not estimated (plant-model mismatch is imposed artificially for robustness tests).
• Economic optimisation, real-time optimisation layers and distillate/bottoms product-value trade-offs.
• Hardware or pilot-plant experiments; actuator saturation dynamics beyond magnitude and rate limits; asynchronous multirate control.
• Nonlinear MPC, stochastic MPC and machine-learning-based control — noted as future work.

The research design is a controlled comparative simulation experiment: one plant, one protocol, several controllers, many repeated runs. The protocol is declared before any result is seen, in the discipline of the Research Methodology & IPR course (22RMI16), and never changed afterwards.

Declared protocol. (1) Plant: Wood–Berry transfer functions discretised exactly (zero-order hold) at Ts = 1 min; scenario length 300 min. (2) Fair tuning: both PI and MPC are first tuned to a dominant-loop closed-loop bandwidth of 0.05 rad/min on loop 1 — PI by SIMC rules with the tuning parameter adjusted to hit it, MPC by moving the closed-loop prediction-horizon/weight combination until its nominal step response matches — after which each may be refined only inside its declared grid. (3) MPC grid: prediction horizon Np ∈ {10, 20, 40}, control horizon Nc ∈ {2, 5, 8}, Q/R weight ratio ∈ {1, 10, 100}; best combination by IAE on the tuning scenario, reported with the full grid table. (4) Scenarios: S1 setpoint step on loop 1; S2 setpoint step on loop 2; S3 simultaneous steps; R1 feed-composition disturbance step using the published disturbance model; C1 combined S1+R1. (5) Repeats: every scenario run five times with different noise seeds (σ = 0.2 mol fraction % on measured outputs); mean and standard deviation reported. (6) Metrics: IAE, ISE, ITAE per output; overshoot %; 2% settling time; total |Δu|; constraint violations counted, never clipped silently. (7) Robustness: winning designs re-run on plants with gains ×(0.8, 1.2) and delays ×1.5.

Analysis plan: pairwise comparisons use the repeated seeds as blocks — for each metric, mean ± standard deviation across seeds and the sign test across scenario families, so a claim like "MPC cuts ITAE by half" must hold in direction across every seed before it is printed.

The simulation framework is a closed loop around the discretised plant, with two alternative controller branches and one shared measurement and logging path, so that every controller sees exactly the same plant, noise and scenario timing.

flowchart TD
  SC["Scenario generator:<br/>setpoint and disturbance schedules"] --> PL
  PL["Plant: discretised Wood-Berry<br/>2x2 model + disturbance"] --> ME["Measurement block:<br/>noise, sampling Ts = 1 min"]
  ME --> SW{"Controller switch"}
  SW -->|"Branch A"| PID["Two SIMC-tuned PI loops<br/>with optional static decoupler"]
  SW -->|"Branch B"| MPC["Centralised MPC:<br/>quadratic cost, constrained QP via do-mpc"]
  PID --> ACT["Actuator block:<br/>amplitude and rate limits"]
  MPC --> ACT
  ACT --> PL
  ME --> LOG[("Run log: y, u, d,<br/>violations, seed")]
  LOG --> AN["Analysis: IAE, ISE, ITAE,<br/>overshoot, settling, plots"]

The MPC branch inside do-mpc has its own structure worth drawing, because examiners ask where the optimiser sits.

flowchart LR
  M["Linear plant model<br/>(discrete state space)"] --> O["Optimal-control problem:<br/>quadratic cost + constraints"]
  S["Current state estimate"] --> O
  O --> C["CasADi / IPOPT solver"]
  C --> U["Optimal move u(0)"]
  U --> P["Plant input"]
  P --> M

The design keeps plant, controllers, scenarios and metrics in separate modules with frozen interfaces: a robustness run changes only the plant module; a tuning study changes only the controller module. That separation is what makes the comparison fair and the code checkable line by line during the viva.

• Robust MPC formulations: replace the nominal design with a tube or multi-scenario MPC and measure whether the robustness degradation shrinks enough to justify the larger problem.
• Dynamic decoupling and interaction measures: extend the ablation with dynamic decouplers and with the dynamic RGA (DRGA) and relative disturbed gain analysis at frequency, deepening the interaction story.
• Moving horizon estimation: add MHE for state estimation when only noisy composition measurements are available, closing the estimation loop around the same plant.
• Economic layer: place a steady-state economic optimiser above the MPC and study profit versus tracking trade-offs under disturbance.
• Hardware validation: port the winning discrete designs to a temperature-control rig or a level-flow pilot loop in the department lab, closing the gap between benchmark simulation and real plant behaviour.