Sample-efficient reinforcement learning for nonlinear process control with safety constraints

Introduction

Advanced process control has been shaped by model predictive control because MPC can handle multivariable dynamics, constraints, and economic tradeoffs in a form that is acceptable to plant engineers [1,2]. The strength of MPC is also its limitation: performance depends on the fidelity of the model, the disturbance description, the chosen objective, and the conservatism used to enforce constraints. Learning-based MPC addresses part of this problem by updating models and uncertainty descriptions from data while retaining an optimisation-based safety structure [3,4].

Reinforcement learning offers a complementary route. A policy can adapt to nonlinear plant behaviour, unmeasured disturbances, changing economic incentives, and long-horizon consequences that are awkward to encode in a fixed setpoint controller. Reviews and demonstrations in process systems have shown growing interest in RL for continuous chemical processes, batch operations, and self-driving process concepts [9,10,11,18,20]. However, ordinary trial-and-error exploration is poorly matched to process industries. A few unsafe actions can trigger product losses, relief-system events, catalyst damage, or shutdowns. Sample efficiency and constraint satisfaction are therefore not convenience features; they are the central research problem.

Several control and RL traditions address this problem from different sides. Constrained Markov decision processes provide a formal language for policies under expected cumulative constraints [7]. Control barrier functions and Lyapunov-barrier methods provide state-space conditions for safety-critical systems [6,17]. Predictive safety filters and learning-based MPC certify proposed actions by solving a constrained finite-horizon optimisation problem before the action is applied [3,4,5]. Recent process-control work has adapted these ideas to nonlinear optimal control and constrained RL, including model-based safe RL, invariant-set-enhanced learning, and constraint-tightening approaches [12,13,14,15,16].

This article reports a safe-RL study for a nonlinear process-control benchmark. The contribution is not a new universal RL algorithm. It is a carefully bounded architecture that combines familiar process-control tools with data-driven adaptation. The controller is deliberately conservative during early learning, becomes less conservative only when the residual model earns it through validated predictions, and always passes policy actions through a safety filter. We compare this architecture with PID control, tracking MPC, nominal economic NMPC, unconstrained actor-critic learning, and two ablations to quantify where the sample efficiency and safety margins come from.

Control problem

The plant is represented as a discrete-time nonlinear system x(k+1) = f(x(k), u(k), d(k)), sampled every 60 s. The measured state vector contains reactor temperature, reactor composition estimates, separator pressure, separator liquid level, recycle composition, coolant inlet temperature, two delayed analyser measurements, and four filtered actuator positions. The manipulated variables are fresh-feed rate, coolant valve opening, separator heat duty, and recycle purge fraction. The disturbance vector includes feed composition, feed temperature, ambient coolant temperature, and analyser bias.

The economic objective rewards product rate and purity while penalising utility consumption, purge losses, and aggressive actuator movement. Unlike tracking control, the optimal operating point moves with feed composition and energy price. The controller must respect hard input bounds and hard state constraints, including reactor temperature below 389 deg C, separator pressure below 2.7 MPa, coolant valve opening between 8% and 95%, purge fraction below 0.28, and product impurity below 1.2 mol%. Soft penalties are used for efficiency and smoothness, but not for the hard safety limits.

We formulate the learning problem as a constrained discounted control problem. The policy proposes continuous actions, but the applied action is the result of a safety projection. The reward is economic, while the constraint functions are physical margins. This differs from many constrained-MDP treatments in which violations are controlled only in expectation [7]. For process safety, a small expected violation probability may still be unacceptable. The proposed controller therefore uses explicit pointwise certification for the next applied action and a finite-horizon feasible backup plan.

All controller tuning was fixed before the final comparison runs. The RL policy was allowed to learn from closed-loop trajectories, but the safety limits, actuator-rate limits, prediction horizon, and fallback controller were not adjusted after seeing the test results. This separation is important because otherwise a safe-RL study can quietly become a manual tuning exercise.

Nonlinear reactor-separator benchmark

The benchmark combines an exothermic continuous stirred-tank reactor with a flash separator and material recycle. The structure is intentionally reminiscent of plant-wide challenge problems such as the Tennessee Eastman process, which remains influential for process monitoring and control studies [21,22]. Our model is smaller than Tennessee Eastman but retains the features that matter for learning: nonlinear reaction kinetics, recycle delay, constrained inventories, interacting energy and composition dynamics, and an economic optimum close to active constraints.

The nominal model used by the controllers differs from the plant model used for evaluation. Activation energy, heat-transfer coefficient, vapour-liquid equilibrium constants, and analyser delay are perturbed by 3-12% in the plant. The model mismatch is large enough that a nominal economic NMPC drifts toward conservative constraint margins, but not so large that model-based control becomes invalid. This setting reflects the real use case for learning: the first-principles model is useful, but incomplete.

The plant has two operating regions. In the low-throughput region, thermal constraints are inactive and product impurity is controlled mainly by purge fraction. In the high-throughput region, reactor temperature and separator pressure become active, while recycle composition creates a delayed coupling between feed changes and impurity. The economic optimum for the nominal model lies 2.5 deg C below the reactor-temperature limit and 0.06 MPa below the pressure limit. A small modelling error is therefore enough to make unconstrained exploration unsafe.

A steady-state estimator provides state estimates from noisy measurements. The estimator is shared by all controllers to avoid giving the learning controller an information advantage. Measurement noise corresponds to 0.15 deg C for temperature, 0.004 MPa for pressure, and 0.025 mol% for product impurity. The delayed analyser updates every three samples, so the policy must act under partial information. A two-layer gated recurrent unit was tested, but the final controller used a feedforward policy with a six-sample state-history embedding because it was easier to certify in the safety filter.

Offline data and warm start

The initial dataset contained 24 closed-loop trajectories generated by a conservative tracking MPC under feed-composition steps, energy-price changes, and analyser-bias perturbations. Each trajectory lasted 90 min, giving 2160 one-minute transitions. The dataset was deliberately modest. It is smaller than what would normally be available from a simulator campaign, but it is closer to the amount of plant-equivalent data that might be collected during commissioning or a dedicated optimisation trial.

The policy was warm-started by behaviour cloning the MPC actions and then by offline critic training on the economic reward. Behaviour cloning prevents the actor from proposing obviously unreasonable actions during the first online episodes. The critic was not allowed to extrapolate reward values outside the convex hull of the initial state-action data; outside that region, the value target was clipped toward the conservative MPC return. This simple restriction was less ambitious than modern offline RL, but it avoided optimistic value estimates that would push the policy toward unvisited high-throughput states.

A residual dynamics model was trained on the difference between the nominal process model and observed one-step transitions. The residual model used an ensemble of five probabilistic neural networks with diagonal covariance output. Gaussian-process approaches have a long history in data-efficient control and robotics [8], but the ensemble was faster for repeated safety-filter calls in this medium-dimensional process model. The ensemble variance was used for constraint tightening rather than as a full Bayesian posterior.

The warm-start produced a safe but conservative policy. Its economic performance was only 1.1% better than tracking MPC in offline evaluation, and the safety filter modified 32% of proposed actions. This is the desired starting point. Early online learning should begin near a known safe controller, not near an exploratory policy that relies on the safety layer to reject most actions.

Safe reinforcement-learning architecture

The learning controller has three parts. First, an actor network proposes a continuous action from the current state-history embedding and the current economic weights. Second, a critic estimates the economic return under the applied action distribution. Third, a predictive safety filter either accepts the proposed action or computes the nearest feasible alternative. The actor is trained on the action that it proposed, while the critic is trained on the action that was actually applied; this distinction is important when the safety filter intervenes.

The actor and critic use two hidden layers of 128 units with layer normalisation and smooth activation functions. The policy output is scaled to actuator limits and then rate-limited before entering the safety filter. Training uses replay data from all online episodes plus the initial MPC dataset. Recent process-control RL studies have shown that direct deep RL can work on nonlinear processes, but can require many trials or careful shaping [11,20]. We reduce that burden by keeping the actor close to the MPC warm start until the residual model has enough data to justify lower constraint tightening.

The RL update uses a constrained actor-critic loss. The economic critic estimates discounted profit, while separate margin critics estimate expected distance to the reactor-temperature, pressure, and impurity constraints. These margin critics do not certify safety; they guide the policy toward regions where the safety filter is less likely to intervene. The certified safety guarantee is provided only by the optimisation-based filter. This split reflects the lesson from constrained model-free process optimisation: expected constraint penalties can guide learning, but hard process constraints require a separate enforcement mechanism [14].

Policy updates are trust-region limited. After each episode, the actor is updated for at most 40 gradient steps, and the average change in proposed action over a validation set must remain below 3% of actuator span. If the validation set contains a state where the proposed action would be rejected by the safety filter with less than 0.5 deg C or 0.01 MPa margin, the update is rolled back. This rule is heuristic, but it made learning smoother and reduced the number of filter interventions.

Predictive safety filter

The safety filter solves a finite-horizon nonlinear program at every sample. Its decision variables are the applied action at the current sample and a backup input sequence over a 20 min horizon. The objective minimises the weighted distance between the policy proposal and the applied action, plus a small penalty on backup-control effort. The constraints enforce tightened state and input limits under the nominal model plus the learned residual mean. Constraint tightening is proportional to ensemble uncertainty and to the local sensitivity of each constraint margin.

If the nonlinear program is feasible, the first action is applied. If it is infeasible, the controller applies the first action of the conservative tracking MPC and logs a safety-filter failure. During the final comparison runs, no such infeasibility occurred for the full controller. The filter modified 18% of proposed actions during the first six online episodes and 6% during the last six episodes. The declining intervention rate indicates that the actor learned to propose actions that were both profitable and certifiable.

The filter follows the same philosophy as predictive safety filters for learning-based control [5] and learning-based MPC with robust safety guarantees [3,4]. It also overlaps with control-barrier-function approaches in the sense that both enforce forward-looking safety constraints [6]. We chose a predictive filter rather than a pure barrier-function quadratic program because the process has delayed composition measurements and active recycle dynamics. A myopic barrier condition was too conservative near high-throughput operation.

The filter is not a proof of global safety. It depends on model validity over the prediction horizon, bounded disturbances, solver convergence, and the existence of a feasible backup plan. We therefore report empirical safety margins and intervention rates alongside economic performance. Recent process-specific safe-RL work similarly emphasises that invariant sets, constraint tightening, and Lyapunov-barrier certificates must be interpreted relative to their modelling assumptions [13,15,16,17].

Experimental protocol and baselines

The online learning campaign consisted of 18 episodes, each 90 min long, with randomised feed-composition disturbances and energy-price profiles. The first six episodes used full constraint tightening, the next six allowed tightening to shrink as residual-model validation error decreased, and the final six used fixed validation-calibrated tightening. The policy was evaluated without exploration noise after every third episode on eight held-out disturbance scenarios.

Five baselines were used. The first was a tuned PID and override-logic controller representing conventional operation. The second was the conservative tracking MPC used to generate the warm-start data. The third was a nominal economic NMPC using the same nominal model as the safety filter, but without online residual learning. The fourth was an unconstrained actor-critic policy with the same architecture and warm start but no safety filter. The fifth was an oracle economic NMPC using the true plant model; it is not implementable, but provides an upper bound on the achievable economic objective.

Two ablations were also tested. The no-residual ablation retained the safety filter but used only the nominal model, with fixed conservative tightening. The no-margin-critic ablation retained the residual model and safety filter but removed the learned margin critics from the actor loss. These ablations separate sample efficiency from certification. A controller can be safe because the filter is conservative, but not economically useful; conversely, a controller can learn a profitable policy that depends too heavily on the filter to repair unsafe proposals.

Performance metrics were average economic reward, product impurity, utility use, number and severity of hard constraint violations, minimum safety margin, safety-filter intervention rate, and number of plant-equivalent transitions required to reach a target reward. The target reward was defined as 90% of the gap between conservative tracking MPC and oracle economic NMPC. Reporting this sample-efficiency metric avoids overstating the benefit of small late-stage reward changes.

Closed-loop performance

The full safe-RL controller reached the target reward after 11 online episodes, corresponding to 990 plant transitions or 16.5 h of plant-equivalent interaction. By episode 18, it improved the economic objective by 7.6% relative to tracking MPC and by 4.1% relative to nominal economic NMPC. The oracle economic NMPC remained better, with an additional 2.9% economic improvement. This ranking is plausible: learning compensates for model mismatch and conservative tuning, but it does not exceed a controller with the true plant model.

The improvement came mainly from operating closer to the reactor-temperature and separator-pressure constraints during high-price intervals, while increasing purge fraction earlier during feed-composition disturbances. The policy learned to anticipate recycle-composition drift and avoid late corrective cooling that wastes energy. This behaviour was visible in the state trajectories: temperature rose smoothly toward the tightened limit, separator pressure stayed below its active bound, and impurity excursions were shorter than under tracking MPC.

The conservative tracking MPC had the lowest intervention risk but left large margins: median reactor-temperature margin was 7.4 deg C and median pressure margin was 0.19 MPa. The safe-RL controller reduced these margins to 3.1 deg C and 0.08 MPa while still avoiding hard violations. Nominal economic NMPC occasionally approached within 0.6 deg C of the true temperature limit under model mismatch, although it did not violate the limit in the held-out scenarios. The unconstrained actor-critic produced five hard violations in 18 online trials, including two reactor-temperature excursions above the limit and three impurity-limit violations.

Compared with recent process-RL reports, the main difference is not headline reward but interaction budget. Reviews of RL for process control have warned that many published controllers rely on extensive simulator interaction or weakly constrained exploration [9,10,18]. Here, most of the learning signal is extracted from a small number of closed-loop episodes because the actor begins near a working MPC policy, the residual model targets only model mismatch, and unsafe exploration is screened before action application.

Constraint handling and safety margins

No hard input or state constraint violations occurred during online training with the full controller. The smallest reactor-temperature margin was 1.9 deg C, the smallest separator-pressure margin was 0.027 MPa, and the maximum product impurity reached 1.13 mol% against a 1.2 mol% limit. The tightest impurity events occurred after upward feed-composition steps, where analyser delay made the state estimate temporarily optimistic. In those cases, the safety filter increased purge fraction and reduced feed before the actor update had learned the disturbance pattern.

The safety filter modified actions most often in the first third of training. Early interventions primarily reduced feed-rate increases proposed by the actor during high-price intervals. Later interventions were smaller and mostly adjusted coolant valve movement to satisfy rate constraints. The average Euclidean action correction decreased from 5.8% of actuator span in episodes 1-6 to 1.6% in episodes 13-18. This decline matters because a policy that is always repaired by the filter is hard to interpret and can cause unnecessary actuator movement.

The no-residual ablation was safe but slower to learn. It reached the target reward after 17 episodes and ended 2.3% below the full controller. Fixed conservative tightening kept the policy away from profitable high-throughput states even after online data showed that the nominal model was pessimistic in parts of the operating envelope. The no-margin-critic ablation reached similar final reward but required more filter interventions, suggesting that the margin critics improve policy shaping rather than formal safety.

The no-filter ablation illustrates the central point. Warm-starting from MPC did not make unconstrained actor-critic learning safe. Exploration noise and critic error were enough to produce unsafe actions near active constraints. This agrees with safe-RL process-control studies showing that model-based stability guarantees, invariant sets, or explicit constraint handling are needed before RL policies are credible for nonlinear constrained operation [12,13,15,16,17].

Sample efficiency

The full method used 2160 offline transitions and 1620 online transitions. In contrast, a model-free actor-critic trained from the same warm start but without residual-model rollouts required more than 7200 online transitions in simulation to reach the target reward, and still violated constraints when deployed without the filter. The difference is not surprising: the residual model supplies short-horizon gradients and synthetic rollouts in the part of the state-action space where the safety filter permits exploration.

We limited synthetic rollouts to five steps. Longer model rollouts initially improved apparent critic accuracy but produced policies that exploited residual-model errors near the pressure constraint. This is a familiar model-based RL problem: sample efficiency improves when the model is useful, but model bias can steer the policy toward false optima. Short rollouts and validation-based rollback provided a practical compromise.

Offline data quality was more important than data volume. A warm start built from random feasible actions contained the same number of transitions as the MPC dataset but required 15 episodes to match the episode-6 reward of the MPC-warm-started controller. The random-feasible dataset covered more state-action volume, yet it did not teach the actor how an experienced process controller balances inventory, temperature, and impurity. Control-informed RL methods make a similar argument: process knowledge and controller structure can reduce the burden on generic function approximation [19].

We also tested reducing the initial dataset by half. With 12 MPC trajectories, the controller still avoided hard violations but required 14 episodes to reach the target reward. With six trajectories, the residual-model uncertainty remained high and the safety filter became too conservative, producing only a 3.2% improvement over tracking MPC after 18 episodes. This gives a practical lower bound: sample-efficient does not mean data-free.

Robustness tests

Robustness was assessed on 40 disturbance scenarios not used during online learning. The scenarios included larger feed-temperature steps, slow analyser drift, a 7% heat-transfer loss, and energy-price sequences outside the training range. The full controller maintained all hard constraints in 39 scenarios. In the remaining scenario, the safety filter declared infeasibility after a simultaneous heat-transfer loss and feed-composition step; the fallback tracking MPC reduced throughput, and no hard state constraint was violated. We count this as an operational failure but not a safety violation.

Economic performance degraded gracefully. Under heat-transfer loss, the full controller retained a 4.8% advantage over tracking MPC, compared with 7.6% in nominal tests. Under analyser drift, the advantage fell to 3.9% because the estimator became conservative around product impurity. The nominal economic NMPC lost more performance under the same drift because it trusted the biased composition estimate and then backed away sharply after delayed analyser correction.

The safety filter solved within the 60 s sample interval in every reported run. Median solution time was 0.42 s on a workstation CPU, and the 99th percentile was 2.8 s. The actor and critic evaluation cost was negligible relative to the nonlinear program. This is important for process applications: an elegant learning algorithm is not useful if its online certification problem cannot meet the controller sample time.

Solver failures were simulated by forcing the filter to return no solution in 1% of samples. The fallback controller handled isolated failures without violations, but clustered failures reduced economic reward sharply. A real deployment would therefore need watchdog logic, solver health monitoring, and an auditable handover to conventional control. These implementation details are as important as the RL update itself.

Discussion

The results suggest a realistic role for reinforcement learning in nonlinear process control. RL is useful here because it learns residual economic behaviour and disturbance responses that the nominal MPC model misses. It is credible because the learned policy is not trusted blindly. Every action remains subject to an optimisation-based safety check, and the policy is penalised when it repeatedly proposes actions that require repair.

The architecture sits between pure model-free constrained RL and classical MPC. Model-free constrained RL has flexibility but poor plant-sample efficiency and weak pointwise safety guarantees. Classical MPC has strong constraint handling but can be conservative under model mismatch. The hybrid controller uses MPC rollouts to initialise behaviour, residual learning to adapt, actor-critic updates to improve economics, and a predictive filter to maintain feasible operation. This combination is consistent with the direction of learning-based MPC and safe process RL rather than a departure from it [3,5,13,15,16].

There are several limitations. First, the plant is a benchmark, not a refinery or pharmaceutical unit. Its nonlinearities and constraints are realistic enough to stress the controller, but it does not include valve stiction, operator interventions, maintenance constraints, or unmodelled safety-instrumented-system logic. Second, the residual-model uncertainty is calibrated on the explored operating envelope. If the plant enters a new regime, the filter may become either unsafe or overly conservative. Third, the economic objective is known and scalar. Multi-product plants with changing schedules would require a higher-level planning layer.

A further limitation is certification. The safety filter provides a finite-horizon, model-based certificate, not an absolute guarantee. The distinction matters in publications and in deployment. A hard guarantee would require stronger assumptions on model error, disturbance bounds, solver convergence, estimator error, and backup feasibility than we can justify. We therefore recommend reporting constraint margins, intervention rates, infeasibility events, and fallback behaviour whenever safe RL is proposed for process control.

The strongest practical lesson is that safety and sample efficiency are coupled. The safety filter narrows the exploration set, making learning easier. The residual model reduces unnecessary conservatism, making the safety filter less restrictive. The warm-start policy reduces early interventions, improving data quality. Removing any one of these pieces either slows learning or shifts risk back onto the plant.

Conclusion

We developed and tested a sample-efficient safe reinforcement-learning controller for a nonlinear reactor-separator process with active safety constraints. The controller combines MPC warm-start data, residual dynamics learning, constrained actor-critic updates, and a predictive safety filter. In plant-equivalent tests, it improved economic performance relative to conservative tracking MPC and nominal economic NMPC while avoiding hard state and input constraint violations during online learning.

The study supports a conservative interpretation of RL for process control. RL can be valuable as an adaptive economic layer, especially when the nominal model is imperfect and plant samples are expensive. It should remain inside a control architecture that includes explicit constraints, uncertainty handling, fallback operation, and transparent safety diagnostics. The useful question is not whether RL can replace MPC, but how learning can make constrained control less conservative without asking the plant to absorb unsafe exploration.

Data and code availability

The supplementary archive contains the reactor-separator model, nominal and perturbed plant parameters, controller configuration files, MPC warm-start trajectories, online training logs, safety-filter intervention records, trained policy checkpoints, and analysis scripts. Random seeds and disturbance profiles are included for all reported runs. No proprietary industrial plant data or unpublished third-party datasets were used.

References

1. Qin, S. J. & Badgwell, T. A. A survey of industrial model predictive control technology. Control Eng. Pract. 11, 733-764 (2003).
2. Mayne, D. Q., Rawlings, J. B., Rao, C. V. & Scokaert, P. O. M. Constrained model predictive control: Stability and optimality. Automatica 36, 789-814 (2000).
3. Hewing, L., Wabersich, K. P., Menner, M. & Zeilinger, M. N. Learning-based model predictive control: Toward safe learning in control. Annu. Rev. Control Robot. Auton. Syst. 3, 269-296 (2020).
4. Aswani, A., Gonzalez, H., Sastry, S. S. & Tomlin, C. Provably safe and robust learning-based model predictive control. Automatica 49, 1216-1226 (2013).
5. Wabersich, K. P. & Zeilinger, M. N. A predictive safety filter for learning-based control of constrained nonlinear dynamical systems. Automatica 129, 109597 (2021).
6. Ames, A. D., Xu, X., Grizzle, J. W. & Tabuada, P. Control barrier function based quadratic programs for safety critical systems. IEEE Trans. Autom. Control 62, 3861-3876 (2017).
7. Altman, E. Constrained Markov Decision Processes. Chapman and Hall/CRC (1999).
8. Deisenroth, M. P., Fox, D. & Rasmussen, C. E. Gaussian processes for data-efficient learning in robotics and control. IEEE Trans. Pattern Anal. Mach. Intell. 37, 408-423 (2015).
9. Shin, J., Badgwell, T. A., Liu, K.-H. & Lee, J. H. Reinforcement learning: Overview of recent progress and implications for process control. Comput. Chem. Eng. 127, 282-294 (2019).
10. Nian, R., Liu, J. & Huang, B. A review on reinforcement learning: Introduction and applications in industrial process control. Comput. Chem. Eng. 139, 106886 (2020).
11. Spielberg, S., Tulsyan, A., Lawrence, N. P., Loewen, P. D. & Gopaluni, R. B. Toward self-driving processes: A deep reinforcement learning approach to control. AIChE J. 65, e16689 (2019).
12. Kim, Y. & Lee, J. M. Model-based reinforcement learning for nonlinear optimal control with practical asymptotic stability guarantees. AIChE J. 66, e16544 (2020).
13. Kim, Y. & Kim, J. W. Safe model-based reinforcement learning for nonlinear optimal control with state and input constraints. AIChE J. 68, e17601 (2022).
14. Pan, E., Petsagkourakis, P., Mowbray, M., Zhang, D. & del Rio-Chanona, E. A. Constrained model-free reinforcement learning for process optimization. Comput. Chem. Eng. 154, 107462 (2021).
15. Bo, S., Agyeman, B. T., Yin, X. & Liu, J. Control invariant set enhanced safe reinforcement learning: Improved sampling efficiency, guaranteed stability and robustness. Comput. Chem. Eng. 179, 108413 (2023).
16. Kim, Y. & Oh, T. H. Model-based safe reinforcement learning for nonlinear systems under uncertainty with constraints tightening approach. Comput. Chem. Eng. 183, 108601 (2024).
17. Wang, Y. & Wu, Z. Control Lyapunov-barrier function-based safe reinforcement learning for nonlinear optimal control. AIChE J. 70, e18306 (2024).
18. Park, J., Jung, H., Kim, J. W. & Lee, J. M. Reinforcement learning for process control: Review and benchmark problems. Int. J. Control Autom. Syst. 23, 1-40 (2025).
19. Bloor, M., Ahmed, A., Kotecha, N., Mercangoz, M., Tsay, C. & del Rio-Chanona, E. A. Control-informed reinforcement learning for chemical processes. Ind. Eng. Chem. Res. 64, 4966-4978 (2025).
20. Bao, Y., Zhu, Y. & Qian, F. A deep reinforcement learning approach to improve the learning performance in process control. Ind. Eng. Chem. Res. 60, 5504-5515 (2021).
21. Downs, J. J. & Vogel, E. F. A plant-wide industrial process control problem. Comput. Chem. Eng. 17, 245-255 (1993).
22. Ricker, N. L. & Lee, J. H. Nonlinear model predictive control of the Tennessee Eastman challenge process. Comput. Chem. Eng. 19, 961-981 (1995).