Data-Driven Gaussian Process-Based Model Predictive Control for a Nonlinear Gas–Liquid System

Abstract

This study develops a data-driven Gaussian process-based model predictive control (GP-MPC) method for an interacting nonlinear gas–liquid system. Three independent Gaussian process models with Matérn 5/2 kernels are trained to predict the gas flowrate, gas-side pressure and liquid-side pressure from the states, two equivalent pressure commands and pressure-command increments. The mean predictions from the three GP models are recursively propagated within the MPC formulation, which accounts for flow tracking, pressure imbalance, control variation, terminal tracking error, convergence, input constraints and online flowrate bias correction. The GP models and MPC optimization are executed on a Raspberry Pi 3, while a physics-based gas–liquid plant model is implemented in MATLAB R2026a/Simulink-Simscape and connected through TCP/IP in a hardware-in-the-loop configuration with a sampling period of 0.1 s. All three GP models achieved validation R² scores above 0.99998, while the gas-flow model achieved an RMSE of 0.2447 L/min over a 50-step recursive prediction horizon. At references of 8.3, 16.6 and 25 L/min, settling times ranged from 2.5 to 6.8 s, with steady-state errors below 0.121% and overshoots below 0.285%. Under continuous bidirectional reference changes, the controller achieved an RMSE of 0.654 L/min and an MAE of 0.091 L/min, with settling times between 1.9 and 3.7 s.

1. Introduction

Gas–liquid transport in membrane-based contacting systems occurs across a defined interface between the two phases, allowing gas–liquid mass transfer without direct dispersion of one phase into the other. Maintaining an appropriate pressure difference across this interface is an important operating requirement because the pressure balance determines the location and stability of the gas–liquid interface within a porous membrane. For hydrophobic membranes, excessive liquid-side pressure relative to the gas side can exceed the breakthrough pressure and promote liquid penetration or membrane wetting [1]. Polytetrafluoroethylene (PTFE) is widely used as a membrane material because of its intrinsic hydrophobicity and high chemical stability, while its porous structure and membrane geometry also influence mass-transfer resistance and gas-transfer performance [2,3]. These coupled transport and pressure-balancing requirements create a control problem in which gas-flow regulation must be coordinated with the pressures on both sides of the interface while respecting operating constraints.

Model-based nonlinear control methods have been investigated for systems involving uncertain and time-varying dynamics. Phan et al. developed an adaptive sliding-mode controller for a blower-based breathing simulator with unknown and time-varying airway resistance and compliance [4]. Truong et al. subsequently combined an identified second-order dynamic model, a disturbance observer and sliding-mode control to improve tracking performance in a ventilation blower-based system under external disturbances [5]. In a wider scope of applications, Le et al. developed a discrete-time model-reference adaptive sliding-mode controller to cope with model uncertainties and disturbances [6]. These studies show how adaptive control and sliding-mode control may be used to increase robustness to parameter uncertainties and disturbances. However, their approach has been centered on tracking and disturbance rejection of particular plants. Conversely, the gas–liquid system considered here involves the simultaneous control of two pressure setpoints, prediction of several coupled process variables, and satisfaction of both input and differential pressure constraints. These requirements are in support of an optimization-based control approach that can handle the multivariable interactions and constraints. Model predictive control (MPC) is ideal for such applications because it predicts future system behavior, optimizes control moves over a finite horizon, and considers the effects of multivariable interactions and constraints. Grancharova et al. developed an explicit MPC strategy for a two-input two-output laboratory gas–liquid separation plant, in which the optimal control law was approximated offline through orthogonal search-tree partitioning to reduce online computation [7]. For the same class of plant, Likar and Kocijan exploited the different bandwidths of the level and pressure control loops to formulate a single-input single-output pressure control problem and implemented predictive control using a Gaussian process (GP) model [8]. Pedersen et al. subsequently applied nonlinear MPC to underbalanced drilling, where pressure and flow variables must be coordinated while satisfying process-dependent operating requirements [9]. These studies demonstrate the suitability of predictive control for interacting fluid systems and exhibit that controller performance depends strongly on the accuracy and computational tractability of the prediction model.

Constructing a sufficiently accurate first-principles model for a nonlinear gas–liquid system can require detailed descriptions of fluid properties, pressure losses, wall and interfacial interactions, and flow-regime-dependent closure relations [10,11]. Inaccurate closure relations and model simplifications can consequently degrade multi-step predictions and closed-loop control performance. Data-driven system identification provides an alternative by learning nonlinear dynamic relationships directly from representative input–output measurement [12]. Among the available learning methods, GP regression provides a flexible nonparametric framework for learning nonlinear mappings while producing both a predictive mean and a variance that represents model confidence [13,14]. Kocijan et al. established the use of GP models for nonlinear dynamic-system identification and showed that dynamic transitions can be learned from lagged measurements and manipulated inputs without specifying an explicit analytical process model [13]. GP models were subsequently incorporated into predictive control, where recursive predictions were used to optimize future control actions [15]. These properties make GP regression particularly attractive for interaction systems in which the dominant nonlinear relationships can be observed from data but are difficult to derive accurately from physical principles. The integration of GP models with MPC has since developed in several directions. Hewing et al. used the nominal model with GP-learned residual dynamics and developed the idea of a cautious MPC controller that considers predicted uncertainty to formulate the chance constraints [16]. Bradford et al. put forward a data-driven stochastic MPC strategy where the GP samples are generated offline in order to compute constraint back-offs, thus alleviating the online computational burden of dealing with uncertainty [17]. In order to safely explore, Koller et al. included statistically reliable GP confidence intervals within a learning-based MPC architecture [18]. For model mismatch problems, Maiworm et al. came up with an online learning-based MPC approach with evolving GP models that includes criteria for recursive satisfaction of constraints and input-to-state stability [19]. Overall, learning-based MPC research has concentrated on the development of prediction models, the management of constraints, online adaptation, and control reliability versus learning efficiency [20]. Practical implementation has also been investigated by recent research. Learning-based GP-MPC was implemented in embedded hardware for mobile robots [21]. On the other hand, L4acados is a new interface to integrate learning-based residual models such as GPs in real-time nonlinear optimal-control solvers [22].

Nevertheless, GP-MPC algorithms remain computationally demanding because online optimization may require repeated evaluation of GP predictions and their associated quantities [23]. Challenges include scalable GP inference, efficient multi-step prediction, uncertainty handling, online adaptation, and reliable constraint treatment [24,25], particularly when the controller is executed with a short sampling period on computationally limited hardware. Representative embedded GP-MPC studies have primarily focused on robotic and autonomous platforms, such as miniature race cars [26] and unmanned quadrotors [27], while recent frameworks such as L4acados facilitate the integration of learning-based models into real-time nonlinear optimal-control solvers [22]. In the gas–liquid application of Likar and Kocijan [8], GP-based predictive control was formulated for a single pressure output and a single manipulated input. In contrast, the present study considers a multivariable gas–liquid system with three predicted process variables, gas flowrate, gas-side pressure, and liquid-side pressure, and two coordinated equivalent pressure commands. The complete GP posterior-mean inference and MPC optimization procedure is executed directly on a Raspberry Pi 3 within a TCP/IP-based HIL configuration operating at a 0.1 s sampling period. Thus, the implementation-oriented distinction of the present study lies in combining multivariable GP-based prediction, coordinated two-input MPC, and low-cost embedded execution for an interacting gas–liquid process.

This study addresses this implementation-oriented gap by developing a data-driven GP-MPC framework for an interacting nonlinear gas–liquid system. The system state consists of the gas flowrate, the gas-side pressure and the liquid-side pressure, while the manipulated variables include the equivalent gas- and liquid-side pressure commands. Predictions for the gas flowrate, gas-side pressure, and liquid-side pressure for the next sampling time are obtained by training three separate GP models. Each model uses a regression input vector comprising the current gas flowrate, gas-side pressure, liquid-side pressure, the two applied pressure commands, and their respective increments, thereby capturing both the current operating condition and the effects of changes in the pressure commands. GP predictive means are propagated recursively during the MPC prediction horizon for computation of coordinated pressure commands for the gas and liquid sides. The optimization objective minimizes gas-flow tracking errors, pressure balance errors, changes in control effort, terminal tracking accuracy, and reference convergence. At the same time, both the pressure command and its increment values are subject to hard limits, while the flowrate bias correction procedure is used to reduce the influence of local GP model errors on the process. In particular, since the designed controller is based on the GP posterior mean as the nominal model of the system transition dynamics, this approach is clearly differentiated from the stochastic GP-MPC where the predictive variance is propagated to guarantee the probabilistic constraints.

With this setup, the GP models and the MPC optimizations are performed on the Raspberry Pi 3 while the nonlinear gas–liquid process is modeled using MATLAB/Simulink-Simscape. The two systems exchange data on states, control input, reference input and optimized control command using TCP/IP communication protocol, thus forming a HIL testing facility with a sampling period of 0.1 s. With this setup, it is possible to test the suggested control algorithm on the physical hardware using a simulated plant; hence, the results obtained constitute a hardware-in-the-loop verification process but not a real experiment performed on the physical gas–liquid system. The model’s accuracy is checked using both one-step-ahead validation and recursive multi-step prediction based on an independent time-ordered test dataset. The performance of the closed-loop system is later investigated at fixed gas flow references and with continuous two-way step changes, taking into consideration tracking error, transient response, and pressure difference.

The principal contributions of this study are summarized as follows:

• A fully data-driven dynamic model comprising three independent GP predictors is developed for the simultaneous prediction of gas flowrate, gas-side pressure and liquid-side pressure. The regression structure combines measured states; two applied pressure commands and pressure increments and its predictive capability are evaluated using both one-step and recursive multi-step tests.
• A multivariable GP-MPC structure is formulated to regulate gas flow through coordinated gas-side and liquid-side pressure commands. The proposed controller recursively propagates GP predictive means, considers input-magnitude and input-rate bounds, regulates the pressure difference between the gas and liquid sides and incorporates online flowrate bias correction.
• The complete GP inference and MPC optimization procedure is deployed on a Raspberry Pi 3 and evaluated in a TCP/IP-based HIL configuration with a sampling period of 0.1 s. Fixed-reference and stepwise-reference scenarios are used to assess tracking accuracy, transient performance, pressure imbalance, and embedded computational feasibility. The 0.1 s value denotes the HIL sampling interval, while computational execution time is evaluated separately from the recorded GP-MPC optimization timing.

2. Materials and Methods

The dual-phase fluidic system consists of a gas flow channel enclosed within a polytetrafluoroethylene (PTFE) tube (Quickun, Ningbo, Zhejiang, China), which is immersed in a surrounding liquid medium. The gas is supplied from a pressurized gas container and regulated by a motorized back-pressure regulator (BPR1) manufactured by Tianjin Tianfei High-Tech Valve Co., Ltd., Tianjin, China. The gas pressure is measured by a pressure transmitter (PT1) from MEASUREMAN, Fuzhou, Jiangxi, China, and a pressure gauge (PG1) from SCTSNSR, Wenzhou, Zhejiang, China. A check valve is installed to ensure unidirectional gas flow toward the membrane module. After entering the PTFE tube, the gas flows through the membrane module and exits toward the downstream line, where it passes through an air dryer (ADRY) from SMC Corporation, Tokyo, Japan and a mass flow controller (MFC) from Alicat Scientific, Tucson, Arizona, USA for flowrate regulation and measurement. On the liquid side, water is drawn from a water tank by a pump and delivered to the outer side of the PTFE tube through a check valve. The liquid pressure is monitored by a pressure transmitter (PT2) from the same manufacturer as PT1 and a pressure gauge (PG2) supplied by the same manufacturer as PG1. The liquid flow and pressure are regulated by a motorized back-pressure regulator (BPR2) provided by the same manufacturer as BPR1.

Let q_g,k, p_g,k and p_l,k denote the gas flowrate, gas-side pressure and liquid-side pressure at time step k, respectively. In the reduced-order HIL model, the manipulated input vector is defined as

$${\mathit{u}}_k\triangleq {\left[u_{g,k},u_{l,k}\right]}^T,1\le u_{g,k},u_{l,k}\le 25\mathrm{bar}\left(\mathrm{a}\right),$$

(1)

where u_g,k and u_l,k are the equivalent absolute-pressure commands applied to the gas-side and liquid-side controlled pressure sources, respectively. They do not represent motor voltage or valve-opening ratio in the present HIL implementation. The motor and actuator dynamics of the physical back-pressure regulators are omitted from the reduced-order plant model; therefore, no calibration relationship between BPR voltage or valve opening and the equivalent pressure commands is employed in this study. The primary control objectives are (i) to regulate the gas flowrate to the desired reference and (ii) to regulate the pressure difference between the gas and liquid sides.

The pressure difference Δp across the tube is defined as

Δp_k ≜ p_g,k − p_l,k,

(2)

where p_g,k and p_l,k denote the gas-side and liquid-side absolute pressures, respectively, and are expressed in bar(a). The differential pressure is expressed in bar. This pressure difference plays a dominant role in governing gas transport through the PTFE interface.

The dynamics of the dual-phase fluidic system are characterized by the complex interaction between the gas pressure, liquid pressure and effluent mass flowrate. The dual-phase fluidic system is modeled within a discrete-time state-space framework where state vector transitions are governed by an unknown nonlinear mapping to be identified. The state vector is defined as

$${\mathit{x}}_k\triangleq {\left[\begin{array}{ccc}{p}_{g,k}& p_{l,k}& q_{g,k}\end{array}\right]}^T\in {\mathbb{R}}^3,$$

(3)

and the system dynamics are described by

x_k+1 = f(x_k,u_k) + ϵ_k,

(4)

where x_k+1 ∈ ℝ³ is the state at the subsequent time step, f(∙) is the unknown nonlinear transition function and $ϵ~\mathcal{N}(0,{\sigma }_n^2)$ represents the additive Gaussian white noise, with σ_n as the measurement noise standard deviation.

As the mass flowrate q_g is a measureable process variable depending on the internal pressure of the gas and the dynamics of the mass flow controller, it is considered an endogenous state variable and not an exogenous control input. Due to the strong coupling between gas and liquid dynamics and the high sensitivity of liquid pressure to changes in the applied pressure commands, the system exhibits fast transient behavior that cannot be captured solely by the instantaneous input values. In particular, rapid pressure variations can occur even for small changes in the equivalent pressure commands. To represent this transient input effect, the regression model is augmented with pressure-command increments. The input increment is defined as

Δu_k ≜ u_k − u_k−1.

(5)

Accordingly, the regression vector is constructed as

$${\mathit{z}}_k={\left[\begin{array}{ccc}{\mathit{x}}_k& {\mathit{u}}_k& \Delta {\mathit{u}}_k\end{array}\right]}^T\in {\mathbb{R}}^7,$$

(6)

and the system dynamics (4) can be equivalently written as

x_k+1 = f(z_k) + ϵ_k.

(7)

The nonlinear mapping f(⋅) defines a transformation f: ℝ⁷ → ℝ³. In practice, each component of the state transition is modeled independently using a scalar Gaussian process:

f_s(z) ~ $𝒢$$𝒫$(0,k(z,z′)), s = 1, 2, 3,

(8)

where each mapping f_s: ℝ⁷ → ℝ represents the transition function of the s-th state component.

A Matérn 5/2 kernel is selected to capture non-smooth variations in pressure and flow associated with nonlinear gas–liquid interactions and rapid changes in the equivalent pressure commands, and is defined as

$$k\left(\mathit{z},{\mathit{z}}^{\prime }\right)={\sigma }_f^2\left(1+{\displaystyle \frac{\sqrt{5}r}{l}}+{\displaystyle \frac{5r^2}{3l^2}}\right)\exp \left(-{\displaystyle \frac{\sqrt{5}r}{l}}\right),$$

(9)

where r≜‖z − z′‖ denotes the Euclidean distance between input pairs, ${\sigma }_f^2$ represents the signal variance and l is the characteristic length scale. The Matérn 5/2 kernel provides sufficient flexibility to represent non-smooth transient behavior associated with pressure-command changes and nonlinear gas–liquid interactions while maintaining numerical stability.

Given a training dataset ${\mathcal{D}}_s={\left\{\left({\mathit{z}}_j,y_j\right)\right\}}_{j=1}^n,$ where n denotes the number of training samples and j indexes the data points, the predictive distribution of the s-th state component at a test input z_∗ follows a Gaussian distribution with mean:

$${\mu }_s\left({\mathit{z}}_{\ast }\right)=\mathit{K}\left({\mathit{z}}_{\ast },\mathit{Z}\right){\left[\mathit{K}\left(\mathit{Z},\mathit{Z}\right)+{\sigma }_n^2\mathit{I}\right]}^{-1}{\mathit{y}}_{\mathit{s}},$$

(10)

where K(z_∗, Z) is the covariance vector between a new test and the matrix of training inputs, K(Z,Z) is the covariance matrix of the training regression inputs and y_s ∈ ℝⁿ is the vector of observations corresponding to the s-th state component.

The uncertainty of the prediction is quantified by the predictive variance

$${\sigma }_s^2\left({\mathbf{z}}_{\ast }\right)=k\left({\mathit{z}}_{\ast },{\mathit{z}}_{\ast }\right)-\mathit{K}\left({\mathit{z}}_{\ast },\mathit{Z}\right){\left[\mathit{K}\left(\mathit{Z},\mathit{Z}\right)+{\sigma }_n^2\mathit{I}\right]}^{-1}\mathit{K}(\mathit{Z},{\mathit{z}}_{\ast }),$$

(11)

where Z ∈ ℝ^n×7 denotes the matrix of regression inputs, with each row corresponding to one sample z_j. In the present framework, the GP posterior mean μ_s(z_∗) provides the nominal state-transition model used within the MPC optimization, whereas the predictive variance ${\sigma }_s^2\left({\mathit{z}}_{\ast }\right)$ is retained only as an indicator of model confidence during GP evaluation. The predictive variance is not recursively propagated over the MPC prediction horizon and does not directly enter the objective function or constraint formulation.

The optimal hyperparameters θ_s = {l_s, σ_f,s, σ_n,s} are identified by maximizing the log-marginal likelihood

$$\log p\left({\mathit{y}}_s|\mathit{Z},{\theta }_s\right)=-{\displaystyle \frac{1}{2}}{\mathit{y}}_{\mathrm{s}}^T{\left[{\mathit{K}}_y\right]}^{-1}{\mathit{y}}_s-{\displaystyle \frac{1}{2}}\log \left|{\mathit{K}}_y\right|-{\displaystyle \frac{n}{2}}\log 2\pi ,$$

(12)

where ${\mathit{K}}_y\triangleq \mathit{K}\left(\mathit{Z},\mathit{Z}\right)+{\sigma }_n^2\mathit{I}$ denotes the covariance matrix of the observed noisy targets.

The identification procedure makes it possible for the dual-phase fluidic system to adapt to its particular nonlinearities without having to use an explicit physical model. The use of the log-marginal likelihood for hyperparameter tuning is justified due to the natural capacity of this approach to provide the automatic Occam’s razor. The marginal likelihood integrates over all possible latent functions, effectively penalizing models that are overly complex or insufficiently flexible. The first term of the log-marginal likelihood, the data-fit term $-{\displaystyle \frac{1}{2}}{\mathit{y}}_s^T{\left[{\mathit{K}}_y\right]}^{-1}{\mathit{y}}_{\mathrm{s}},$ encourages the model to accurately capture the observed state transitions of the dual-phase fluidic system. Conversely, the complexity penalty term $-{\displaystyle \frac{1}{2}}\log \left|{\mathit{K}}_y\right|$ prevents overfitting by favoring smoother and simpler covariance structures. In doing so, the optimal length scales and noise variances are determined such that the GP can support prediction for unseen samples within the represented operating domain. Maximization of the log-marginal likelihood provides a principled data-driven procedure for selecting the GP hyperparameters by balancing data fit and model complexity, thereby reducing the need for manual kernel-parameter tuning. However, this procedure does not by itself guarantee robustness to measurement uncertainty or unobserved operating conditions.

The identification process is executed using a structured dataset generated from the reduced-order MATLAB/Simulink-Simscape model of the dual-phase fluidic system. The training matrix is constructed from n simulation-generated samples, where each row represents a feature vector at time k with a dimension of d encompassing the state variables and control inputs as defined in the previous section. To ensure numerical stability and facilitate optimization of the marginal likelihood, the input and output features are standardized prior to GP training using scaling parameters determined from the training subset. The systematic implementation of the system identification for the dual-phase fluidic system is outlined in the procedure in Algorithm 1.

Algorithm 1: Gaussian process system identification.

Input: Simulation-generated time-series dataset $\left\{p_g\left(k\right),p_l\left(k\right),q_g\left(k\right),u_g\left(k\right),u_l\left(k\right){\}}_{k=0}^{T_{\mathit{data}}}\right.$ Output: Trained GP models $\{f_s(\cdot ){\}}_{s=1}^3$

for k = 0 to Tdata do (where Tdata denotes the length of the collected time-series data)

- Construct state vector: ${\mathit{x}}_k={\left[\begin{array}{ccc}{p}_{g,k}& p_{l,k}& q_{g,k}\end{array}\right]}^T;$

- Construct input vector: ${\mathit{u}}_k={\left[\begin{array}{cc}{u}_{g,k}& u_{l,k}\end{array}\right]}^T;$

if k > 0 then

Compute input increment: Δuk = uk − uk−1;

else

Set Δuk = 0;

end if

Form ${\mathit{z}}_j={\left[\begin{array}{ccc}{\mathit{x}}_k^T& {\mathit{u}}_k^T& \Delta {\mathit{u}}_k^T\end{array}\right]}^T;$

end for

for k = 0 to Tdata − 1 do

Build training pair: (zk, xk+1);

end for

- Normalize input and output data;

- Select the Matérn 5/2 kernel;

for s = 1 to 3 do

- Construct scalar training dataset: ${\mathcal{D}}_s={\left\{\left({\mathit{z}}_j,y_{s,j}\right)\right\}}_{j=1}^n;$

- Optimize hyperparameters: θs by maximizing the log-marginal likelihood;

- Train GP model for the s-th state component;

end for

- Validate the trained models using unseen data;

- Return trained GP models.

For the present dual-phase fluidic model, the training subset is limited to a compact number of samples to balance GP predictive performance with the computational and memory constraints of Raspberry Pi 3 implementation. The identified GP models are evaluated using validation and independent time-ordered test data generated from the same MATLAB/Simulink-Simscape plant model. Therefore, the reported prediction and control performance should be interpreted as simulation-based HIL verification and not as evidence of robustness or generalization on the actual physical gas–liquid apparatus.

3. Gaussian Process-Based Model Predictive Control

The control strategy for the dual-phase fluidic system is formulated as a deterministic nonlinear model predictive control (MPC) scheme using the previously identified Gaussian process (GP) models as data-driven state-transition models. Although each GP provides both a posterior mean and predictive variance, only the posterior mean is recursively propagated within the present MPC formulation. Predictive uncertainty is not propagated over the prediction horizon and is not incorporated into the objective function or constraints. This design choice is adopted to limit the online computational burden associated with uncertainty propagation and chance-constrained reformulation, while targeting embedded execution on the Raspberry Pi 3 within the 0.1 s HIL sampling interval. The objective of the controller is to determine an optimal sequence of gas-side and liquid-side pressure-command increments that minimizes the MPC objective function over a finite prediction horizon.

The GP model provides a one-step prediction of the system dynamics using the predictive mean:

x_k+1 = μ(z_k).

(13)

For multi-step prediction over a finite horizon N, the GP model is recursively applied. Starting from the current measured state x_k, the predicted state sequence is generated as

$$\left\{\begin{array}{l}{\widehat{\mathit{x}}}_{k|k}={\mathit{x}}_k,\\ {\widehat{\mathit{x}}}_{k+j+1|k}=\mu \left({\mathit{z}}_{k+j}\right),j=0,1,\dots ,N-1,\end{array}\right.$$

(14)

For the MPC tracking objective, the three-dimensional predicted state is mapped to a two-dimensional controlled-output vector ${\mathit{y}}_{k+j\mid k}={\left[q_{g,k+j\mid k},\Delta p_{k+j\mid k}\right]}^T={\left[q_{g,k+j\mid k},p_{g,k+j\mid k}-p_{l,k+j\mid k}\right]}^T$. The corresponding reference vector is defined as ${\mathit{r}}_{k+j}={\left[q_{g,\mathrm{ref},k+j},0\right]}^T$, where the second component represents the desired gas–liquid pressure balance. For reproducibility, the objective function is expressed below in the component-wise form implemented in the online controller. Defining the normalized gas-flow tracking error as ${\bar{e}}_{q,j}=\left({\widehat{q}}_{g,k+j|k}^{\mathrm{corr}}-q_{g,\mathrm{ref},k+j}\right)/s_q$ and the normalized differential-pressure term as ${\bar{e}}_{\Delta p,j}=\left({\widehat{p}}_{g,k+j|k}-{\widehat{p}}_{l,k+j|k}\right)/s_{\Delta p}$, the optimization objective at each control update is

$$\begin{array}{c}{J}_k=w_q{\displaystyle \sum _{j=1}^{N_p}}{\bar{e}}_{q,j}^2+w_{\Delta p}{\displaystyle \sum _{j=1}^{N_p}}{\bar{e}}_{\Delta p,j}^2+w_T{\bar{e}}_{q,N_p}^2\\ +w_{\Delta u_g}{\displaystyle \sum _{j=0}^{N_c-1}}{\left({\displaystyle \frac{\Delta u_{g,k+j|k}}{\Delta u_{g,\max }}}\right)}^2+w_{\Delta u_l}{\displaystyle \sum _{j=0}^{N_c-1}}{\left({\displaystyle \frac{\Delta u_{l,k+j|k}}{\Delta u_{l,\max }}}\right)}^2\\ +w_{\mathrm{cross}}\parallel {\stackrel{\mathbf{‾}}{\mathbf{v}}}_{\mathrm{cross}}{\parallel }_2^2+w_{\mathrm{mono}}\parallel {\stackrel{\mathbf{‾}}{\mathbf{v}}}_{\mathrm{mono}}{\parallel }_2^2+w_{\mathrm{contr}}{\bar{v}}_{\mathrm{contr}}^2.\end{array}$$

(15)

The implemented prediction and control horizons are N_p = 3 and N_c = 2, respectively. The weighting coefficients are w_q = 50, w_Δp = 5, $w_{\Delta u_g}=2$, $w_{\Delta u_l}=1$, w_T = 100, w_cross = 500, w_mono = 200, and w_contr = 300. No separate terminal weighting matrix P is used; instead, the terminal gas-flow tracking error is weighted directly by w_T. These parameters were empirically selected to balance gas-flow tracking, differential-pressure regulation, convergence behavior, and smooth pressure-command variation and are not claimed to represent a globally optimal tuning.

Let e₀ = q_g(k) − q_g,ref,k+1, $e_j={\widehat{q}}_{g,k+j|k}^{\mathrm{corr}}-q_{g,\mathrm{ref},k+j}$, and σ = sign(e₀). For |e₀| > 0.05 L/min, the convergence violations are defined as v_cross,j = max(−σe_j − 0.03,0), v_mono,j = max[σ(e_j − e_j−1) − 0.01,0], and $v_{\mathrm{contr}}=\max \left(\sigma e_{N_p}-0.97\left|e_0\right|-0.01,0\right)$. These terms are normalized by s_q before entering Equation (15). When |e₀| ≤ 0.05 L/min, the convergence penalties are set to zero.

$$\begin{array}{cc}\hfill 1& \le u_{g,k+j|k}\le 25 \mathrm{bar}\left(\mathrm{a}\right),\hfill \\ \hfill 1& \le u_{l,k+j|k}\le 25 \mathrm{bar}\left(\mathrm{a}\right),\hfill \\ \hfill \left|\Delta u_{g,k+j|k}\right|& \le 0.015 \mathrm{bar},\hfill \\ \hfill \left|\Delta u_{l,k+j|k}\right|& \le 0.010 \mathrm{bar},\hfill \\ \hfill \left|{\widehat{p}}_{g,k+j|k}-{\widehat{p}}_{l,k+j|k}\right|& \le 0.85 \mathrm{bar}.\hfill \end{array}$$

(16)

The manipulated inputs u_g and u_l are equivalent absolute-pressure commands. Their increments are optimized over the control horizon N_c = 2; for the remaining prediction steps, Δu_g = Δu_l = 0, such that the pressure commands are held constant. These constraints therefore represent pressure-command magnitude, pressure-command rate, and differential-pressure limits rather than motor-voltage or valve-opening limits of the physical BPR actuators. The differential-pressure limit is imposed on the nominal GP-mean predicted trajectory and should not be interpreted as an experimentally validated material-failure threshold. Since predictive uncertainty is not propagated within the present deterministic MPC formulation, nominal predicted constraint satisfaction does not constitute a probabilistic guarantee of plant-side constraint satisfaction.

The online numerical optimization procedure combines candidate screening with Sequential Least Squares Programming (SLSQP) refinement implemented through scipy.optimize.minimize. At each control update, the shifted solution from the preceding update and a hold-input candidate are evaluated first. If neither candidate satisfies the hard constraints, a deterministic recovery search using a two-level candidate grid is activated. The highest-ranked candidate is then used to initialize one SLSQP refinement. No additional random candidates are used in the reported configuration. SLSQP is configured with maxiter = 8 and ftol = 1 × 10⁻⁴, while hard feasibility is independently checked using a tolerance of 1 × 10⁻⁶. At each control update, only the first optimized pressure-command pair is applied according to the receding-horizon principle.

The operational procedure of the GP-MPC follows a receding-horizon strategy, where the optimization problem is solved at each sampling instance based on the current state feedback. The systematic execution of the control law for the dual-phase fluidic system is delineated in Algorithm 2.

Algorithm 2: GP-based model predictive control execution.

1. Initialize the prediction horizon Np, control horizon Nc, MPC weights and constraints, and the pre-trained GP models.

2. At each sampling time k:

3. Measure the current state vector ${\mathbf{x}}_k={\left[\begin{array}{ccc}{p}_{g,k}& p_{l,k}& q_{g,k}\end{array}\right]}^T$ from the PT1, PT2 and MFC sensors.

4. Formulate the nonlinear programming problem using the GP predictive mean μ(f∗) as the recursive transition model for j = 0, …, N − 1.

5. Solve the optimization problem to find the optimal control sequence $u_{k:k+N-1}^{\ast }={\left\{u_{g,j}^{\ast },u_{l,j}^{\ast }\right\}}_{j=0}^{N-1}$ that minimizes the cost function J.

6. Apply only the first optimized equivalent pressure-command pair $u_k^{\ast }={\left[\begin{array}{cc}{u}_{g,0}^{\ast }& u_{l,0}^{\ast }\end{array}\right]}^T$ to the controlled pressure sources BPR1_Gas and BPR2_Liquid.

7. Discard the remaining sequence $u_{k+1:k+N-1}^{\ast }$ and wait for the next sampling interval Ts.

8. Shift the horizon and repeat the process from Step 2 with updated measurements.

The recursive prediction in Step 4 implies that the predicted state at each future step is obtained by evaluating the GP posterior-mean model using the state predicted at the preceding step and the corresponding candidate control input. This iterative propagation allows the controller to anticipate the coupling effects and the delayed pressure response within the dual-phase fluidic system. By applying only the immediate control action and re-optimizing after updated measurements become available, the receding-horizon procedure can reduce the influence of modeling discrepancies and support gas-flow tracking while regulating the pressure differential toward the prescribed operating range.

4. Hardware-in-the-Loop Simulation Results

4.1. HIL Setup and GP Model Development

4.1.1. Simulink-Simscape Plant Model

To evaluate the GP-MPC controller before implementation on the experimental rig, a hardware-in-the-loop (HIL) setup is implemented. The nonlinear gas–liquid dynamics are simulated in MATLAB/Simulink-Simscape, while the GP models and MPC optimizer run on a Raspberry Pi 3 acting as the target hardware. This HIL setup is used to assess prediction accuracy, flow tracking performance, pressure constraint adherence, execution speed and the dynamic effects of TCP/IP communication. Compared with full simulation on a single computer, the HIL configuration better reflects real-world deployment conditions while reducing risks and costs relative to immediate direct testing on the actual physical gas–liquid system.

Based on the HIL architecture, a reduced-order physical model of the gas–liquid system is constructed in MATLAB/Simulink-Simscape according to the schematic structure shown in Figure 1. The model retains key components directly governing the dynamics to be identified and controlled, including the gas line, liquid line, pressure generation sources, gas pressure sensor (PT1), liquid pressure sensor (PT2) and mass flow controller/sensor (MFC). Auxiliary devices beyond the scope of this study are omitted or represented by equivalent boundary conditions. This order reduction lowers the computational complexity of the HIL model while preserving the nonlinear interactions among gas pressure, liquid pressure and gas flowrate.

Figure 1. Schematic diagram of the dual-phase fluidic system.

Figure 2 presents the overall structure of the Simulink-Simscape model, organized into functional regions comprising the physical plant model, the NCS nonlinear interaction simulation block, the TCP/IP communication interface and auxiliary blocks for different operating modes. Within the physical plant model region, the gas line consists of the controlled pressure source (BPR1_Gas), the pressure sensor (PT1_GasPressure), the throttling element MFC, the flowrate sensor (MFC_GasFlowSensor) and the boundary condition atmosphere. The liquid line comprises the controlled pressure source (BPR2_Liquid), the sensor (PT2_LiquidPressure), the tank (LiquidTank) and the annulus (GlassChamber_LiquidShell) representing the liquid volume surrounding the PTFE tube. Physical quantities are converted from the Simscape domain to the Simulink signal domain and logged at discrete sampling instant to form raw signals (q_g_raw, p_g_raw, p_l_raw). The three raw signals are subsequently routed into the NCS Simulation region.

Figure 2. Simulink-Simscape model of the gas–liquid plant and TCP/IP-based HIL architecture.

The NCS block incorporates the additional effects of differential pressure and nonlinear coupling interactions between the two phases, thereby generating measurement signals (q_g_meas, p_g_meas and p_l_meas). These signals serve as feedback states of the plant and are transmitted to monitoring blocks and the data transmission interface. The Measurement Bounds Check region is utilized to monitor pressure commands, measured states and achievable flow ranges, thereby verifying compliance with established input pressure limits and differential pressure bounds.

Compared with the physical rig, the HIL model omits the dynamic characteristics of the motor and actuator mechanisms of the BPR valves. Instead of issuing motor-voltage, position or valve-opening commands, the GP-MPC controller generates two equivalent absolute-pressure commands, u_g and u_l, expressed in bar(a). These commands are held over each sampling interval and applied directly to the controlled pressure sources BPR1_Gas and BPR2_Liquid. Accordingly, no voltage-to-opening, opening-to-pressure, or other BPR actuator calibration relationship is used in the present reduced-order HIL model. This substitution enables the study to focus on the coupled gas–liquid dynamics and GP-MPC performance while leaving actuator-level dynamics and calibration outside the scope of the present HIL investigation.

The same plant model is used across three operating modes by altering command sources and enabling corresponding signal paths, while maintaining the physical structure of the dual-phase system. In data acquisition mode, the block (Random_u) utilizes the RBC function to randomly generate two commands (u_g and u_l) within the allowable pressure domain. These commands are applied to the plant to excite nonlinear characteristics and cross-coupling interactions between the two branches. At each sampling instant, the dataset-comprising sample index, measured states (q_g_meas, p_g_meas and p_l_meas) and applied commands (u_g and u_l) are transmitted via the TCP/IP interface to be stored for GP model identification. In operating domain identification mode, the random command source is replaced by coordinated sweep signals for both gas and liquid sides. Both pressure commands are swept within the range of 0 ÷ 25 bar, while responses q_g, p_g and p_l are recorded. After filtering out points violating pressure limits, the feasible gas-flow range is identified, serving as the basis for selecting reference values (q_g_reference) in closed-loop control scenarios. In online control mode, the TCP/IP Network Interface region forms a closed loop between the Simulink-Simscape model and the Raspberry Pi 3. At each sampling instant, Simulink multiplexes the data vector [seq, q_g_meas, p_g_meas, p_l_meas, u_g_applied, u_l_applied, q_g_reference]. Here, seq denotes a monotonically increasing sequence number used to verify the ordering and continuity of the transmitted samples. The data vector is then transmitted to the Raspberry Pi through the TCP/IP Client Send block. The Raspberry Pi employs the trained GP models to solve the MPC optimization problem and returns two next control commands via the TCP/IP Client Receive block. These commands are held and applied to the physical model at the subsequent time step. The sequence of data acquisition, transmission, optimization and control command generation is repeated at each sampling step according to the receding-horizon strategy.

4.1.2. Randomized Input Generation and Training Data Collection

The block (Random_u) utilizes the RBC function to generate two randomized pressure commands u_g_cmd and u_l_cmd for collecting system identification data at a sampling time of 0.1 s. Both commands are expressed as equivalent absolute-pressure commands in bar(a) and are generated within the allowable interval of 1 − 25 bar(a), consistent with the manipulated-input definition in Equation (1). The pressure-command range is divided into several operating regions, and a target pressure-command pair is randomly selected within each region to improve data coverage. Each selected pair is maintained for a predefined time interval before the next pair is applied, thereby generating a diverse excitation dataset distributed across the entire operating range for training the GP models.

Figure 3 shows the input signals and measured responses obtained over 15,000 s of simulation. The commands (u_g_applied and u_l_applied) continuously explore different pressure levels across nearly the entire range of 0 ÷ 25 bar, rather than remaining confined to a narrow interval. Corresponding to this excitation, the pressures (p_g_meas and p_l_meas) vary over a wide band while respecting set pressure limits. The flowrate (q_g_meas) also exhibits significant variation depending on the pressure state of both branches, providing coupled response data across diverse operating conditions. The signals (q_g_meas, p_g_meas, p_l_meas, u_g_applied, u_l_applied) are subsequently transmitted to Raspberry Pi and logged in CSV files to build the training dataset for the GP models.

Figure 3. Randomized equivalent pressure commands and measured responses for GP training data generation.

4.1.3. Raspberry Pi and TCP/IP Communication Configuration

Configured as a TCP server, the Raspberry Pi 3 receives data streamed from the Simulink model via the TCP/IP Client Send block. The data acquisition routine is built in Python using the TCP/IPv4 protocol, listening for incoming connections on port 5000. The Raspberry Pi software environment used for the online GP-MPC implementation consisted of Python 3.11.2, NumPy 2.4.6, SciPy 1.17.1, and scikit-learn 1.9.0. Upon establishing the connection between Simulink and Raspberry Pi, data from the HIL model are continuously streamed to the acquisition program. At each sampling instant, Simulink transmits seven signals, [seq, q_g_meas, p_g_meas, p_l_meas, u_g_applied, u_l_applied, q_g_reference], encoded as 64-bit floating-point values in big-endian byte order. Upon receiving and decoding the incoming data stream, the Raspberry Pi verifies the validity of each sample before logging into a CSV file for offline processing and GP model training.

4.1.4. Dataset Preprocessing and Partitioning

Following data acquisition, the CSV file is inspected for mandatory field integrity, and all required entries are cast to numeric types. Rows containing invalid values, NaN, or Inf are removed, together with consecutive duplicate samples, and the sequence index is checked for monotonicity. The resulting dataset contains 150,001 valid time-ordered samples and is partitioned chronologically, without shuffling, into contiguous training, validation, and test blocks using a 70%/15%/15% split, corresponding to 105,000, 22,500, and 22,501 raw samples, respectively. Splitting is performed before constructing the GP prediction pairs so that no transition crosses a train–validation or validation–test boundary. Within each block, the pressure-command increments are calculated in physical engineering units as ∆u_g,k = u_g,k − u_g,k−1 and ∆u_l,k = u_l,k − u_l,k−1. After removing the first sample required for increment calculation and the last sample for which the next-step target is unavailable, the resulting datasets contain 104,998 training pairs, 22,498 validation pairs, and 22,499 test pairs. Auxiliary fields and differential pressure are retained as analytical metadata but excluded from the GP input features. The resulting datasets are exported to gp_train.csv, gp_val.csv, and gp_test.csv. Standardization is subsequently performed using a StandardScaler fitted only to the training subset.

4.1.5. GP Model Training and Validation

Model training is conducted on the Raspberry Pi under memory-constrained execution settings. To reduce RAM usage, gp_train.csv and gp_val.csv are processed in chunks of 25,000 rows. From the 104,998 available training pairs, 1200 pairs are selected without replacement using a random seed of 42, while 5000 pairs are selected from the 22,498 available validation pairs using a seed of 1042. The three GP models use random states of 42, 43, and 44 for the gas-flow, gas-side pressure, and liquid-side pressure models, respectively. This random subsampling is performed only after the chronological train–validation–test partition has been established; therefore, it does not mix samples across temporal partitions or alter the k → k + 1 relationship within each transition pair. The independent test dataset is not subsampled and all 22,499 test pairs are retained for model evaluation. The input features are normalized using a StandardScaler fitted on the training subset and subsequently applied to the validation set. Likewise, each output target variable is individually standardized prior to model fitting. Three independent models, the gas flowrate model (GP_qg), the gas-side pressure model (GP_pg) and the liquid-side pressure model (GP_pl), are trained sequentially to predict the gas flowrate (q_g,k+1), gas-side pressure (p_g,k+1) and liquid-side pressure (p_l,k+1) at the next sampling instant, respectively. Each GP model uses a composite kernel formed by the product of a Constant Kernel and a Matérn 5/2 kernel with Automatic Relevance Determination (ARD) length scales. A White Kernel is added to represent observation noise. The kernel hyperparameters are optimized by maximizing the log-marginal likelihood using the L-BFGS-B algorithm. After each model is trained, temporary memory is released before training the next model to reduce RAM usage on the Raspberry Pi. One-step-ahead prediction is evaluated on the validation set in batches of 250 samples. Reported evaluation metrics include R², RMSE, MAE, mean error, maximum absolute error, average prediction standard deviation and empirical coverage probability within the 95% confidence interval. Operational parameters such as training time, validation prediction runtime, average per-sample prediction latency, and final log-marginal likelihood values are systematically logged. Finally, the three models alongside their input/output scalers are exported to gp_qg.joblib, gp_pg.joblib, and gp_pl.joblib, while a consolidated training report containing configuration details, dataset sizes, and validation metrics is saved in training_metrics.json.

Table 1 shows that all three GP models achieve high one-step-ahead prediction accuracy, with R² values exceeding 0.99998. Values marked with an asterisk (*) were rounded to 1.000000 from 0.9999999. The gas-flow model (GP_qg) yields an RMSE of 0.035226 L/min and an empirical coverage probability of 94.94%, closely matching the nominal 95% target. The gas-side pressure model (GP_pg) and the liquid-side pressure model (GP_pl) exhibit low prediction errors, with empirical coverage probabilities of 100.00% and 99.90%, respectively. The gas-side flowrate model (GP_qg) requires the longest training duration, followed by the gas-side and liquid-side pressures (GP_pg and GP_pl). Overall, these metrics indicate that the three models effectively capture the one-step dynamics of the system across the validation dataset.

Table 1. One-step validation performance of the trained GP Models.

GP Model	Predicted Output	RMSE	MAE	R2	95% Coverage	Training Time
GP_qg	qg(k + 1), L/min	0.035226	0.007488	0.9999899	94.94%	1006.107 s
GP_pg	pg(k + 1), bar(a)	0.000212	0.000143	1.000000 *	100.00%	654.655 s
GP_pl	pl(k + 1), bar(a)	0.003979	0.000214	0.9999996	99.90%	298.208 s

Table 2 indicates that prediction errors for the gas-flow pressure model (GP_qg) gradually increase over longer prediction horizons due to recursive error accumulation. However, even at a horizon of 50 steps, the RMSE remains as low as 0.244702 L/min with an R² of 0.9995136. Meanwhile, the gas-side and liquid-side pressures (GP_pg and GP_pl) maintain remarkably low errors and R² values above 0.99999 across all evaluated horizons. Values marked with an asterisk (*) were rounded to 1.000000 from 0.9999999. Compared to the zero-order hold baseline, the GP models reduce RMSE by 82.752% to 99.939%, demonstrating strong multi-step predictive performance on the independent test dataset.

Table 2. Recursive multi-step prediction performance on the independent test set.

Horizon	GP_qg RMSE/R2	GP_pg RMSE/R2	GP_pl RMSE/R2	qg ipmr	pg ipmr	pl ipmr
1	0.049754/0.9999798	0.000243/1.000000 *	0.000194/1.000000 *	82.75%	99.80%	99.84%
5	0.109221/0.9999012	0.000915/1.0000000	0.016074/0.9999928	91.97%	99.85%	97.30%
10	0.145657/0.9998271	0.001483/0.9999999	0.015969/0.9999929	94.59%	99.88%	98.65%
20	0.168838/0.9997688	0.002403/0.9999998	0.015823/0.9999930	96.35%	99.89%	99.30%
50	0.244702/0.9995136	0.003115/0.9999997	0.015374/0.9999932	97.35%	99.94%	99.70%

4.1.6. Online GP-MPC

In the online control mode, the three trained GP models and their corresponding scalers are loaded onto the Raspberry Pi and converted into fast predictors to reduce computational cost. The training-dependent GP quantities are precomputed when the models are loaded, while online posterior-mean prediction still requires kernel evaluations against all retained training samples. Consequently, for the fixed regression-input dimension used in this study, the online GP inference cost increases approximately linearly with the retained training-set size. At each control interval, the current measured states and applied control inputs are used to initialize the prediction process. At each step of the prediction horizon, the GP input vector is constructed from the predicted states, control inputs and corresponding input increments. The posterior means of the three GP models are then used to predict q_g, p_g and p_l at the next step. This procedure is recursively repeated to generate the predicted state trajectory over the entire prediction horizon. The GP input vector excludes the reference gas flowrate q_g_reference, which serves solely as the setpoint in the MPC optimization problem. In addition, the difference between the measured q_g and its previous one-step prediction is used to update a bias correction term, thereby reducing the influence of local model errors on flowrate tracking performance. The online predictor is configured with a prediction horizon of N_p = 3 and a control horizon of N_c = 2. The first two input increments are treated as decision variables, while the control inputs are held constant beyond the control horizon. For each candidate control sequence, the three GP models are recursively evaluated to predict the system response while checking the constraints on u_g, u_l, input variations and the pressure difference between p_g and p_l. Feasible predicted trajectories are evaluated using an objective function that considers the q_g tracking error, pressure difference, control variation, terminal error and convergence behavior. The control sequence with the minimum objective value is then selected to update the control inputs for the next sampling interval.

The online gas-flow bias correction is implemented additively to the GP gas-flow prediction rather than as an additional MPC cost term. The bias estimator uses an update coefficient of α = 0.02, with both the innovation and bias limited to ±1 L/min. Bias updating begins after 30 control packets and is enabled only when the gas-flow tracking error is within 2 L/min. Computational timing was evaluated from the recorded HIL execution data rather than inferred from the HIL sampling interval. Across 7501 control packets, the mean and maximum MPC solution times were t_solve,mean = 0.1045 s and t_solve,max = 1.1616 s, respectively. These measurements show that the present Raspberry Pi 3 implementation does not consistently satisfy a 0.1 s wall-clock computational deadline. Accordingly, T_s = 0.1 s is interpreted as the HIL sampling interval rather than as a demonstrated 10 Hz real-time execution guarantee. Separate TCP/IP transmission latency was not instrumented in the recorded logger, and no per-cycle socket timeout was configured; therefore, independent communication-latency and timeout values are not reported.

4.1.7. Determination of the Feasible Gas-Flow Range

The feasible gas-flow range of the model is identified using an input sweep configuration. In this test, pressure commands u_g and u_l are swept in a coordinated fashion across the allowable operating envelope, while maintaining absolute pressure bounds and the differential pressure constraint |p_g − p_l| ≤ 0.85 bar. The simulation duration is set to 200 s and the q_g_meas response is recorded throughout.

Figure 4 illustrates the variation in q_g_meas during the pressure-command sweep. The gas flowrate increases as the pressure commands approach the upper region of their specified range, reaches its maximum near the midpoint of the sweep and then decreases as the inputs return toward their lower limits. The maximum value is determined over the entire simulation interval from 0 ÷ 200 s, whereas the minimum value is determined along the descending branch after the maximum to exclude the influence of the initial conditions. The results show that q_g_meas reaches a maximum of 88.915 L/min at t = 100.1 s and a minimum of 0.540 L/min at t = 200.0 s, corresponding to a variation range of 88.374 L/min. Therefore, the interval of 0.540 ÷ 88.915 L/min is considered the observed flowrate range of the model under the applied pressure-command sweep and specified pressure limits.

Figure 4. Observed range of the measured gas flowrate during the pressure-command scan.

4.2. Closed-Loop GP-MPC Performance Evaluation Under HIL

4.2.1. Scenario 1: Evaluation of Tracking Capability at Fixed Reference Flowrates

The first scenario is designed to evaluate the gas-flow tracking capability of the GP-MPC controller at distinct operating points within the HIL setup. In experimental studies on gas–liquid two-phase flows, Radhi et al. investigated gas–water systems at gas flowrate levels of 8.3, 16.6, and 25 L/min [28]. Based on these benchmarks, these three flowrates are selected as constant setpoints in this simulation scenario. Three independent tests are conducted, where the reference flowrate remains constant throughout each run. The identical GP-MPC configuration and operational constraints detailed in previous sections are applied across all cases to assess and compare tracking performance across operating regimes.

Figure 5, Figure 6 and Figure 7 illustrate the closed-loop responses of the physics-based gas–liquid plant model implemented in MATLAB/Simulink-Simscape under GP-MPC regulation for fixed reference flowrates of 8.3, 16.6 and 25 L/min, while Table 3 summarizes key quantitative performance metrics. Overall, GP-MPC effectively drives the gas flowrate q_g from the initial zero state to the setpoint, maintaining tight stability around the specified reference flowrates across all tested operating points. Following the transient phase, the responses remain well within a ±5% error band around the reference without noticeable steady-state oscillations. These results confirm that the integrated GP model provides reliable predictions, enabling the MPC controller to calculate accurate control signals across diverse operating conditions.

Figure 5. Closed-loop GP-MPC response for a gas-flow reference of 8.3 L/min.

Figure 6. Closed-loop GP-MPC response for a gas-flow reference of 16.6 L/min.

Figure 7. Closed-loop GP-MPC response for a gas-flow reference of 25 L/min.

Table 3. Summary of GP-MPC performance at fixed gas flowrate references.

qg_ref (L/min)	RMSE (L/min)	ess (%)	ts (s)	Overshoot (%)	|Δp|max (bar)
8.3	0.498	0.121	2.5	0.144	0.982
16.6	1.102	0.053	3.9	0.065	0.982
25	1.909	0.03	6.8	0.285	0.982

Performance differences across the three cases become clearer through the metrics summarized in Table 3. Settling time increases from 2.5 s at reference flowrate of 8.3 L/min to 3.9 s at 16.6 L/min and 6.8 s at 25 L/min. This trend indicates that higher flowrate targets requiring larger state transitions from q_g = 0 demand longer settling periods for the controller to bring the system into steady state. However, longer settling times have a little effect on steady-state accuracy with relative steady-state errors of 0.121%, 0.053% and 0.030%, respectively. Furthermore, overshooting across all three operating points remains negligible (0.144%, 0.065% and 0.285%). Thus, despite a slower response at 25 L/min, there is no significant trade-off between convergence speed and steady-state precision within the tested operating range.

Table 3 shows that the overall RMSE increases with the reference flowrate across the three tested operating points of 8.3, 16.6, and 25 L/min. This increase has little effect on steady-state tracking performance. Because the RMSE is calculated over the entire simulation duration, it includes the startup transitions from 0 L/min to 8.3, 16.6 and 25 L/min. At higher reference flowrate, the initial gas flowrate tracking error is larger and the settling interval becomes longer, resulting in a greater contribution of the transient response to the accumulated squared error. This interpretation is consistent with the small steady-state errors reported in Table 3. Therefore, the overall RMSE reflects both transient tracking performance and steady-state accuracy rather than steady-state precision alone.

Figure 5, Figure 6 and Figure 7 evaluate the system behavior with respect to the differential-pressure operating constraint, |∆p| ≤ 0.85 bar. During startup initialization, all three fixed-reference cases exhibit a brief peak absolute differential pressure of approximately 0.982 bar. This peak exceeds the 0.85-bar MPC operating constraint by approximately 0.132 bar and the 0.90-bar evaluation threshold by approximately 0.082 bar. The 0.90-bar threshold is exceeded for four consecutive samples at the 8.3 L/min reference and five consecutive samples at the 16.6 and 25 L/min references, corresponding to approximately 0.4 s and 0.5 s, respectively, at the 0.1 s sampling interval. After the startup transient, |∆p| decreases below the 0.85-bar operating constraint and remains within this limit for the remainder of each fixed-reference test. The nearly identical startup peaks observed across the three reference cases indicate that this transient is primarily associated with the initial conditions and controller initialization rather than the reference-flow magnitude. Because the differential pressure constraint is enforced on the nominal trajectory predicted using the GP posterior mean, this plant-side startup violation also shows that nominal predicted constraint satisfaction does not guarantee satisfaction of the same bound by the HIL plant.

4.2.2. Scenario 2: Evaluation of Tracking Performance Under Stepwise Reference Changes

Following the fixed setpoint evaluations in Scenario 1, Scenario 2 investigates the dynamic tracking response of the GP-MPC controller under stepwise setpoint changes during continuous operation. Unlike Scenario 1, this test is conducted continuously without resetting system states during setpoint transitions. The reference flowrates of 8.3, 16.6 and 25 L/min are used for consistency. Initially, the reference flowrate is set at 8.3 L/min for 150 s, then stepped up to 16.6 L/min (150 ÷ 300 s) and 25 L/min (300 ÷ 450 s). In the second half of the run, setpoints are stepped down to 16.6 L/min (450 ÷ 600 s) and back to 8.3 L/min (600 ÷ 750 s). This setup generates four consecutive step transitions (two step-ups and two step-downs), with each operating point maintained for 150 s. This simulation is used to assess transient response, settling time, tracking error, adaptation to bidirectional reference changes and satisfaction of the operating constraints.

Closed-loop responses of the physics-based gas–liquid plant model implemented in MATLAB/Simulink-Simscape during the stepwise test are shown in Figure 8, with segment-by-segment quantitative metrics summarized in Table 4. The GP-MPC scheme exhibits consistent tracking across both upward and downward setpoint transitions, achieving an overall 750 s RMSE of 0.654 L/min and MAE of 0.091 L/min. After each setpoint change, the gas flowrate converges rapidly to the target value without lingering oscillations. For the four transitions, 5% settling times are 2.3, 3.7, 3.5 and 1.9 s, respectively, while steady-state errors across all segments remain within 0.026 ÷ 0.121%. The transient overshoot or undershoot values for four changes are 0.061%, 0.406%, 0.312% and 1.069%. Directional asymmetry is observed at the 16.6 L/min setpoint: stepping up from 8.3 L/min requires 2.3 s to settle, whereas stepping down from 25 L/min requires 3.5 s. However, steady-state errors remain virtually identical (0.053% and 0.056%), indicating that transition direction primarily influences transient speed rather than steady-state precision. For the differential-pressure response, the maximum value of |∆p| = 0.9823 bar occurs during the initial startup period, exceeding both the 0.85-bar MPC operating constraint and the 0.9-bar evaluation threshold. No further violations of the 0.85-bar MPC operating constraints are observed during the subsequent step transitions at 150, 300, 450, and 600 s, where the corresponding |∆p_max| values are 0.731, 0.564, 0.593, and 0.744 bar, respectively.

Figure 8. Closed-loop response of the GP-MPC under stepwise gas flow reference changes.

Table 4. GP-MPC performance under stepwise gas flow reference changes.

Segment	qg_ref (L/min)	RMSE (L/min)	ess (%)	ts (s)	Peak Deviation (%)	|Δp|max (bar)
1	8.3	0.704	0.121	2.5	0.144	0.9823
2	16.6	0.491	0.053	2.3	0.061	0.7310
3	25.0	0.757	0.026	3.7	0.406	0.5640
4	16.6	0.711	0.056	3.5	0.312	0.5930
5	8.3	0.567	0.121	1.9	1.069	0.7440

5. Discussion

This study developed a fully data-driven GP-MPC framework for coordinated gas flowrate and pressure regulation in an interacting nonlinear gas–liquid system. Three independent GP models with Matérn 5/2 kernels were trained to predict the gas flowrate, gas-side pressure and liquid-side pressure from the current states, two equivalent pressure commands and their increments. The GP mean predictions were recursively propagated within the MPC formulation, which incorporated gas flowrate tracking, differential-pressure regulation, control variation, terminal tracking error, input constraints and online flowrate bias correction. The complete GP inference and optimization procedure was executed on a Raspberry Pi 3 and connected through TCP/IP to a MATLAB/Simulink-Simscape plant model with a sampling period of 0.1 s.

The model validation confirmed that all three GP models achieved one-step-ahead values above 0.99998. The gas flowrate model retained an RMSE of 0.2447 L/min over a 50-step recursive prediction horizon, indicating that the learned dynamics remained sufficiently accurate for short horizon online prediction. At fixed reference flowrates of 8.3, 16.6 and 25 L/min, the GP-MPC controller achieved settling times ranging from 2.5 to 6.8 s, relative steady-state errors below 0.121% and overshoots below 0.285%. During continuous bidirectional reference changes, the controller achieved an overall RMSE of 0.654 L/min and MAE of 0.091 L/min, with settling times between 1.9 and 3.7 s. A brief differential-pressure constraint violation occurred during startup, reaching approximately 0.982 bar. After this startup transient, no further violations of the 0.85-bar MPC operating constraint were observed during the subsequent reference transitions. This startup violation indicates that enforcing the constraint on the nominal GP-mean prediction does not guarantee plant-side constraint satisfaction.

Overall, these findings demonstrate the computational execution of the multivariable GP-MPC framework on Raspberry Pi 3 hardware within the present HIL configuration. They should not be interpreted as validation of practical deployment or performance on the physical gas–liquid apparatus. The gas–liquid plant is simulated rather than physical, the equivalent pressure commands omit the actuator dynamics of the motorized back-pressure regulators, and the controller propagates only the GP predictive mean without accounting for predictive uncertainty. Consequently, the present results do not provide probabilistic constraint guarantees or establish performance under physical experimental conditions. Future work will therefore investigate improved startup initialization and reference shaping, soft-constraint formulations, and uncertainty-aware or chance-constrained MPC formulations, together with incorporation of actuator dynamics and experimental validation on the physical gas–liquid system. Additional validation procedures, including residual autocorrelation analysis, repeated training under different random seeds, and comparative statistical testing, will also be conducted to further assess model consistency, robustness, and statistical significance.