Heat Management in Batch Reactors Using Flatness-Based Neural Predictive Control

In the industrial processes, the rapid progression led to increasing demands for automation and precision in complex systems. Accurate temperature control in batch reactors ensures dependable reaction kinetics, product quality, and process safety. Even minor temperature deviations can lead to undesired by-products, reduced yield, or even hazardous conditions. Given batch reactor dynamics' strong nonlinearities and time-varying nature, achieving precise temperature tracking remains a key challenge in industrial applications. Batch reactors are widely used in pharmaceuticals, polymer manufacturing, and specialty chemicals, where reaction conditions must be precisely controlled to ensure product quality and safety.

For example, in pharmaceutical synthesis, small temperature deviations can change reaction pathways, leading to the formation of undesired impurities that compromise drug efficacy. Similarly, maintaining an optimal temperature profile in polymerization processes is essential to control molecular weight distribution and polymer properties, directly impacting material performance. Beyond quality, temperature fluctuations can also pose significant safety risks, particularly in exothermic reactions, where an uncontrolled temperature rise may trigger runaway reactions or even thermal decomposition of reactants. This is a major concern in fine and petrochemical industries, where heat buildup can lead to reactor damage or hazardous conditions. Therefore, achieving precise and reliable temperature control is not just an academic challenge but a crucial requirement for industrial-scale batch processes.

Computationally efficient and accurate tracking of nonlinear systems is crucial due to the current advancement and sophisticated technologies that industries are adopting [1]. Traditional control methods are inadequate for real-time performances in the face of highly nonlinear systems [2]. An accurate and reliable system model is required to capture the inherent nonlinearities and enhance the control algorithms' effectiveness to optimize production in modern-day industries [3]. While using model-based control in linear setup offers simplicity and reasonable computational requirements, the latter falls short in handling highly nonlinear industrial systems. Meanwhile, leveraging accurate nonlinear models enables more advanced control strategies to ensure the design objectives [4]. Using a nonlinear model in an optimal control problem presents notable challenges and downfalls, such as computational and model complexity [5]. Consequently, various control strategies have been developed to address challenges related to computational complexity and the structural properties of the system.

Controlling chemical reactors primarily focuses on regulating process variables such as temperature, concentration, and pressure to ensure the quality of the product and operation safety [6]. The nonlinear, time-varying dynamics with complex reaction pathways of the reactors demand advanced trajectory tracking and optimization control strategies. In the literature, advanced control methods have been applied to deal with the issues of operating chemical reactors. Abdullah and Christofides [7] suggested integrating MPC and sparse identification of nonlinear dynamics SINDy, a recent identification technique to control and model nonlinear chemical process systems. In the study of Yadav et al. [8], data-driven modeling of a pilot plant BR was conducted, and the article used NMPC to track the temperature. Meanwhile, Obando et al. [9] used a dual-mode based sliding mode control SMC for temperature control for nonlinear chemical processes. The structural properties of chemical reactors were investigated by Nguyen et al. [10], a structural method based on the thermodynamics extended model that resulted in a control strategy to stabilize the dynamics of the reaction system. Linear algebra was used to design a controller based on the integral of desired closed-loop behavior in the study of Sardella et al. [11], with an application to regulation and trajectory tracking in a chemical process. Emphasizing the current state of advanced chemical and materials process control, Dubljevic [12] called for closing the large gap between developed novel process control theories and their industrial applications.

Model predictive control (MPC) is a powerful strategy in process control applications, providing significant rewards over traditional control methods such as PID controllers [13]. It was observed by Samad et al. [14] that MPC is more impactful than other control technologies, such as nonlinear and robust control, with a future impact of 85%. MPC utilizes an explicit process model to optimize control actions and predict future responses, enabling it to deal with constraints effectively. In the process industry, MPC aligns with process control initiatives, facilitating real-time implementation and monitoring, which maintain process integrity according to reference [15]. While MPC offers excellent performance with its linear replica, it faces several challenges, such as the need for precise system models, difficulty selecting its weight factors, and computational complexity [16]. NMPC differs from MPC in a fundamental concept, which is the uncalled-for step of model linearization. It utilizes a nonlinear model of the system and performs discretization, making predictions and controlling decisions based on this approach. Nonlinear solvers are employed to find the optimal control moves to satisfy an objective function, according to Li et al. [17]. Despite the advantages of NMPC over linear MPC, the former faces significant drawbacks that have limited its applications to a great extent. One of the most significant challenges practitioners encounter when implementing NMPC in real-time is its high computational demand. Different approaches were used to aid NMPC in overcoming its computational complexity; a successive online linearization technique was suggested by Patne et al. [18]. Linear MPC is an available method for most process control problems. As an alternative method to NMPC, it requires linearization of the nonlinear model and, hence, solving a convex optimization problem. However, Linear MPC suffers from complications in its performance due to the linearization process.

Researchers have advocated combining feedback linearization with MPC to address nonlinear system challenges. However, a major drawback of this approach lies in canceling nonlinear terms, which introduces significant robustness issues, even when an uncertainty model is incorporated into the control scheme. Additionally, the integration of MPC with feedback linearization (FBL) struggles to manage known input time delays effectively [19]. Despite these limitations, the method benefits from requiring the optimization problem to be solved via quadratic programming (QP), a computationally efficient process.

To overcome the robustness challenges associated with FBL, Flatness-Based Feedforward Linearization (FFL) emerges as an alternative. This approach addresses parametric uncertainties that can cause pole-zero cancellation inaccuracies, offering improved control design robustness. In their latest publication [20], the authors stated that most oversimplified systems models are differentially flat. Moreover, the ability to track an open-loop trajectory was established using flatness. Flatness is a decisive property that can facilitate the design of a control law to derive a linear or nonlinear system to the required objectives. Recent publications have focused on using the flatness property in many fields of control applications, such as aerospace [21, 22], industrial, and processes [23, 24], robotics, and mechatronics [25, 26]. Flatness can be used to design a flatness-based control in successive loops for subsystems [27, 28], or with other control techniques depending on the objective of the required design. In differentially flat systems, all system variables can be expressed in terms of flat output and a finite number of its time derivatives. This parameterization leads to reforming the system structure to find a global linearization without solving nonlinear differential equations. Differentially flat systems are the ones whose states and inputs are parametrized by flat outputs and a finite number of their time derivatives. Differential flatness (DF) input-output representation can fully capture the system's dynamic behavior in reduced, decoupled behavior, simplifying the trajectory generation problem. The flatness-based approach separates the nonlinear model into linear dynamics in Brunovsky form and nonlinear transformation [29]. A system is differentially flat if the states and the input can be parametrized according to a flat output and a finite number of time derivatives. Using any differential equation, these derivatives must be independent and unrelated to each other [30]. This method transforms the nonlinear system into an equivalent linear representation within the flat space. As a result, the system's dynamics are reformulated, yielding a linearized system in the transformed domain. This linear system facilitates using linear MPC instead of NMPC and enjoys utilizing quadratic programs for optimal control and linear control theory. Accordingly, a computationally efficient technique for the numerical solution of the optimal control problem can be obtained. By merging tools from the nonlinear control theory, the dimension of the original control problem is lowered, and the computational time is significantly enhanced [31]. On the other hand, the nonlinear transformation resulting from flatness can be learned using machine learning to close the feedback loop of the control system successfully. Machine learning techniques provide enhanced trajectory tracking when facing unpredictable operational issues [32, 33]. Nevertheless, learning the entire system dynamics is computationally expensive, and researchers adopted these approaches in the absence of an accurate model of the systems. Moreover, when using a data-driven model of the systems, the neural network fully engages with the dynamic system, which increases the execution time, as in the study of Shettigar et al. [34].

Our proposed methodology combines MPC with flatness feedforward linearization to improve the batch reactor's robustness property and tracking performance. The conventional dynamic model remains a realistic representation of the batch reactor; meanwhile, learning all the dynamics from the start would be inefficient. A more realistic scenario is to enhance the existing model and train the NN to learn the inverse dynamics. Integrating neural networks (NNs) with control strategies, such as adaptive and predictive control, is well-established to address system uncertainties and imperfections. Once trained, the NN plays a critical role in refining inputs to the control strategy, thereby enhancing overall system performance and robustness, as in the study of Li et al. [35]. Using a standard feedforward neural network (FNN) architecture, where layers are fully connected, ensures convergence to the desired values.

In our approach, we propose an efficient controller for temperature control of the batch reactor and attain a reliable tracking performance. The contributions of the article can be listed as follows:

•A novel control architecture where FMPC is used to couple flatness feedforward linearization with MPC in a closed form to establish a reliable tracking performance for the temperature profile of the batch reactor’s heating phase.

•A feedforward neural network (FNN) is incorporated to enhance the performance of the proposed control strategy by learning the inverse dynamics introduced by flatness-based feedforward linearization. Consequently, the combined FNN with flatness-based feedforward linearization and FMPC enhances robustness and significantly improves tracking performance.

The proposed strategy achieved efficient execution time, leveraging the DF property of the batch reactor. It enables a direct generation of optimal trajectories and significantly reduces the computational burdens typically associated with NMPC.

This paper is organized as follows: Section 2 presents the dynamic model of the batch reactor. Background knowledge in Section 3. Section 4 presents the formulation of the proposed control strategy with the enhancement via a neural network. Section 5 displays the simulation results validating the proposed approach.

3.1 Flatness

DF was introduced by M. Fliess and his colleagues and gained increasing attention in nonlinear control. Flatness is a natural idea associated with under-determined systems represented by differential equations. It signifies the possibility of representing the system’s variables in terms of a finite set of free variables. A single input nonlinear system is said to be differentially flat if there exists a differential function of the state and a finite number of its time derivatives, called the flat output. As a structural property of systems, flatness allows for establishing the features required for designing feedback controller techniques, such as backstepping, passivity, and feedback linearization.

We consider a SISO system of the general form:

$\dot{x}=f(x, u), x \in R^n, u \in R$          (2)

where, $f=\left(f_1, \ldots, f_n\right)$ is a smooth function of $x$ and $y$ and the rank of the Jacobian matrix is maximal. 

A nonlinear system is considered differentially flat if there exist. $\zeta(t) \in R^m$, with differential independent components, and the following conditions are satisfied according to Fliess et al. [30]: 

$\zeta=\Lambda\left(x, u, \ldots, u^{(\alpha)}\right)$          (3)

$x=\Phi\left(\zeta, \zeta^{\dot{}}, \ldots, \zeta^{(\rho-1)}\right)$          (4)

$u=\Psi^{-1}\left(\zeta, \zeta^{\dot{}}, \ldots, \zeta^{(\rho)}\right)$          (5)

where, $\Lambda, \Phi$ and $\Psi^{-1}$ are smooth functions in their domains, $\alpha$ and $\rho$ are the maximum derivative order of the $\zeta$ and the control input $u$ respectively. $\zeta=\left[\zeta_1, \ldots, \zeta_m\right]^T$ is called flat output.

Systems that are classified as differentially flat can mathematically be represented by Brunovsky form, the states of which are called flat states: 

$z:=\left[\zeta_1, \zeta_1, \ldots, \zeta_1^{\left(\rho_1-1\right)}, \ldots, \zeta_m, \ldots, \zeta_1^{\left(\rho_m-1\right)}\right]$          (6)

where, $\rho_i$ is the highest-order derivative of the flat output $\zeta_i$ as in Eq. (5). The state transformation between the flat state $z$ and the system state $x$, which can be obtained by differentiating Eq. (3) and utilizing Eq. (4).

The nonlinear system can be put in the standard form:

$\zeta_i^{\left(\rho_i\right)}=\alpha_i\left(\zeta, \dot{\zeta}_{,}, \ldots, \zeta^{(\rho-1)}, u, \dot{u}, \ldots, u^{\left(\sigma_i\right)}\right):=v_i$          (7)

where, $\alpha_i, i=1, \ldots, m$ is a smooth function generated by the transformation. It’s worth noting that $\sigma_i$ represents the highest-order derivative of the control input $u$ which yields after $\rho_i$ times differentiation of the flat output $\zeta_i$ in Eq. (3).

The flat input is referred to as "v"  and can be defined as: 

$v:=\left[v_1, v_2, \ldots, v_m\right]^T$          (8)

Refreezing Eq. (7) and applying (6) and (8), we get:

$\dot{z}=A z+B v$          (9)

$v=\Psi\left(z, u, \dot{u}, \ldots, u^{(\sigma)}\right)$          (10)

where, $\sigma=\max \sigma_i$ and Eq. (9) is denoted as the linear flat model. The new definitions allow rephrasing Eq. (5) as follows: 

$u=\Psi^{-1}(z, v)$          (11)

Up to this point, the fundamental difference between flatness feedforward linearization and feedback linearization can be explained by the following theorem: 

Theorem 1 [37]: Consider a desired path in the flat space for the flat output $\zeta_d$, with the corresponding desired flat state $z_d$ and desired flat input $\mathrm{v}_d$. Applying the nominal control $\boldsymbol{u}=\Psi^{-1}\left(\mathbf{z}_d, \mathrm{v}_d\right)$ to a differentially flat system, given $\zeta_d$, and initial conditions as $\boldsymbol{z}(0)=\boldsymbol{z}_d(0)$, leads to a linear system (9) based on a change of coordinates. Noting that the variables $\boldsymbol{z}_d$, can be obtained from Eq. (6) by replacing the desired flat output and $\mathrm{v}_d$ are obtained from (7) by replacing the desired flat output.

Theorem 1 can enable controllers or trajectory generators to deal only with the linear flat model and produce the appropriate flat state and input as the output. Then, these outputs are fed through the inverse term (11) to offset the nonlinear term (10). 

3.2 Model predictive control

Linear model predictive control (LMPC) is conveniently used with a linearized system as a control strategy for modifying the behavior of systems with linear approximation. LMPC is an optimization-based, predictive framework that belongs to the family of MPC. The critical aspect of LMPC is the use of a linear model in the prediction of the future behavior of the system over a finite prediction horizon. The optimization problem is dealt with at each time step to minimize a quadratic cost function subject to constraints. 

Consider a discrete linear-time invariant system:

$x_{k+1}=A x_k+B u_k$          (12)

where, $k$ is the discrete-time instant, $\mathbf{x}_{\mathrm{k}} \in \mathbb{X} \subseteq \mathbb{R}^n$ is the state vector, $\mathbf{u}_k \in \mathbb{U} \subseteq \mathbb{R}^m$ is the control input vector, $A, B$ are appropriate size system matrices. $\mathbb{X}, \mathbb{U}$ are the state and control input constraint set, which are usually represented as linear inequality:

$\begin{aligned} & X=\left\{x \in R^n: F_x \leq g_x\right\} \\ & U=\left\{u \in R^m: F_u \leq g_u\right\}\end{aligned}$          (13)

The objective function is the balancing factor between tracking performance and the control input energy. It is responsible for minimizing the differences between the trajectories of the desired reference and states of the systems. Meanwhile, it penalizes the excessive control input signal using a quadratic cost function:

$J_k=\sum_{k=0}^{N_p} x_k^T Q x_k+\sum_{k=0}^{N_c} u_k^T R u_k$          (14)

where, $Q$ and $R$ are the weighting matrices, $N_p$, $N_c$ are the prediction and control horizon, respectively.

Dealing with operational constraints is an essential feature of MPC, which can include limits on state variables and control inputs. The constraints can be incorporated directly into the formulation of the optimization problem to ensure the reliability of the operation under strict requirements.

QP is used to solve the optimization problem in LMPC due to the linear model and quadratic cost formulation. This feature is considered the main advantage of using LMPC due to the moderate computational efforts and the ability to solve in real time using standard optimization algorithms. This algorithm starts by describing the predicted state, which is a function of the current state and the input sequence:

$X_k=A_k x_k+B_U U_k$          (15)

The predicted state can be represented by the current state and control input. The cost function in Eq. (14) is subjected to the predictive state equation:

$J_k=X_K^T Q_x X_k+U_k^T R_U U_k$          (16)

Finally, by defining the appropriate-sized matrices to be incorporated into the prediction equations, the state constraints in Eq. (13) can be represented in terms of the current state and control input vectors as:

$F_x X_k \leq g_x F_u U_k \leq g_u$          (17)

A single decision vector that combines the state and control input vector can be formed to represent the new cost function:

$\begin{gathered}\min _z \mathbf{z}^T H \mathbf{z} \\ S . \text { to: } F z \leq g \\ F_{e q} z=g_{e q}\end{gathered}$          (18)

Standard numerical optimization algorithms such as Newton's or the steepest-descent method can be used to solve QP problems. The optimization problem is solved at each time instant $k$ and only the first element of the control vector is applied.

This section presents the results of implementing the flatness-based model predictive control FMPC control strategy for the nonlinear BR system. A nonlinear MPC control strategy was also implemented to compare the results and assess the performance of FMPC. The results are accessed based on system performance, trajectory tracking precision, constrained control effort, and computational efficiency.

5.1 Temperature tracking response with FMPC

The presented figures provide valuable insights into the dynamics of the nonlinear BR under FMPC, highlighting its effectiveness in managing nonlinearity, achieving real-time feasibility, and ensuring system stability.

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Figure 4. FMPC temperature tracking response

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Figure 5. FMPC execution time analysis

Figure 4 illustrates the temperature tracking performance of the FMPC control strategy. The desired temperature trajectory is plotted together with the actual reactor temperature. Regarding tracking accuracy, FMPC demonstrates excellent tracking performance, which is evident by the close alignment between the actual and reference trajectories. Meanwhile, the controller ensured minimal overshoot and maintained stability.

FMPC execution time is depicted in Figure 5, a crucial metric in assessing the real-time feasibility of the controller. The relative reliability of execution time across simulation steps clearly indicates computational efficiency. This consistency is credited to the use of the flatness property of the BR, which simplifies the generation of feasible trajectories and reduces the computational complexity. The response has occasional peaks of 0.065 seconds at the highest, attributed to specific points where rapid changes in system dynamics occur. The average execution time is still way beyond the sampling time of 0.05, which was selected for the simulation of the BR, demonstrating the practicality of FMPC in real-time control applications.

5.2 Temperature tracking response with NMPC

The following figure illustrates the tracking performance of the NMPC applied to the nonlinear BR. Plotting the desired temperature trajectory with the actual temperature achieved utilizing FMPC.

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Figure 6. NMPC temperature tracking response

NMPC established high tracking accuracy when admitted to follow the nonlinear dynamics of BR, as shown in Figure 6. The discrepancies during the transient time are minimal due to NMPC’s ability to manage changes in reference trajectory. The formulation of NMPC allows for explicitly handling BR system nonlinearity, reflected by the close alignment between actual and desired trajectories.

The response, with a peak execution time of 0.14, is almost three times the sampling period used in the simulation, which was selected as 0.5 seconds. Solving an optimization problem at each step period, NMPC ensures optimal control actions tailored to the BR nonlinear dynamics.

Although NMPC achieves excellent tracking, its computational demands can be significantly higher than FMPC. Figure 7 clearly demonstrates NMPC computational demands as its operation requires solving a nonlinear optimization problem. It is evident that the trade-off between tracking accuracy and computational efficiency is a critical consideration.

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Figure 7. NMPC execution time analysis

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Figure 8. FMPC vs. NMPC constrained control signal comparison

The comparison between FMPC and NMPC-constrained control is illustrated in Figure 8. The constraints are imposed to ensure the physical limitations of the BR system are respected. The key observation is that FMPC and NMPC effectively maintain the control signal within the required boundaries. The FMPC control signal shows smoother variation than the NMPC control signal. NMPC exhibits more oscillatory behavior as compared to FMPC during transient time, which reflects the reactive nature of solving optimization problems dynamically. NMPC tends to respond more aggressively as the BR requires fast corrective action. Meanwhile, FMPC’s smoother response prioritizes computational efficiency and stability.

5.3 Temperature tracking with NN-FMPC

The tracking performance after integrating the Feedforward Neural Network FNN is investigated in this section. Utilizing NN enhances FMPC by supplying an optimized or adaptive new reference trajectory as an input to FMPC. The key observation is the achievement of improved tracking accuracy compared to standard FMPC. The computational time has no significant impact as the NN is trained in advance and deployed efficiently. The execution time is still below the sampling period, which ensures real-time feasibility. Table 2 compares the three controllers in terms of their mean and max deviation.  

Table 2. A comparison of control and temperature deviations

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Figure 9. NN-FMPC vs. NMPC temperature tracking response

As depicted in Figure 9, the integration of NN into FMPC significantly reduces the gap in the tracking performance between FMPC and NMPC. When real-time implementation and energy efficiency are prioritized, NN-FMPC emerges as a superior option. A quantitative comparison between the three control approaches is presented in Table 3.

Table 3. A comparison between the three control approaches