Enhancing PID Control Robustness in CSTRs: A Hybrid Approach to Tuning Under External Disturbances with GA, PSO, and Machine Learning

doi:10.21203/rs.3.rs-5150659/v1

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Enhancing PID Control Robustness in CSTRs: A Hybrid Approach to Tuning Under External Disturbances with GA, PSO, and Machine Learning

https://doi.org/10.21203/rs.3.rs-5150659/v1

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The optimization of PID controllers for Continuous Stirred Tank Reactors (CSTRs) is critical for ensuring stable and efficient chemical processing under varying operational conditions and external disturbances. This study presents a novel approach that integrates advanced tuning techniques, including Genetic Algorithms (GA), Particle Swarm Optimization (PSO), and their hybrid combinations with a Machine Learning (ML) surrogate model, to improve PID controller performance in disturbed environments. A unique evolutionary framework is employed, where populations of both controllers and plants are co-evolved to handle the most challenging plant models. An adversarial testing approach is utilized to evaluate the best-tuned controller against the four most difficult CSTR plants, with disturbances such as feed concentration and temperature fluctuations. The results demonstrate that both GA and PSO, when enhanced with the ML surrogate model, effectively tune PID controllers to manage disturbances, with the GA tuned controller achieving faster convergence and the PSO tuned controller showing greater robustness. Additionally, the hybrid ML surrogate model significantly improved control performance and disturbance rejection. The findings highlight the ability of the ML_GA and ML_PSO controllers to maintain stability and accuracy across a range of challenging conditions, providing a robust solution for optimizing control systems in nonlinear dynamic environments. This research contributes to the field of process control, showcasing the potential of combining evolutionary algorithms with machine learning surrogate models for adaptive, resilient control strategies in complex chemical systems.

Artificial Intelligence and Machine Learning

Machine Learning (ML)

Genetic Algorithms (GA)

Particle Swarm Optimization (PSO)

Robust PID Controller Robust PID Controller

Adversarial nonlinear process

Continuous Stirred Tank Reactor (CSTR) Plant

The evolution of control systems has led to the development of increasingly sophisticated methodologies for optimizing system performance, particularly in nonlinear process control. Among these, Proportional-Integral-Derivative (PID) controllers remain one of the most widely used control strategies in industrial applications due to their simplicity and effectiveness [1]. However, the complexities inherent in nonlinear systems often present challenges in selecting the optimal PID parameters—proportional (Kp), integral (Ki), and derivative (Kd) [2]. Traditional tuning methods, such as Ziegler-Nichols or manual tuning, frequently struggle to address the dynamic, nonlinear characteristics of modern control systems, often leading to suboptimal performance [3].

In this context, Genetic Algorithms (GAs) have emerged as a powerful optimization tool for nonlinear PID controllers [15, 23]. Rooted in the principles of natural selection and genetics, GAs use a population-based search strategy to iteratively evolve solutions toward optimality. By encoding potential PID parameter sets as chromosomes, GAs can explore a vast search space and effectively identify parameter values that minimize key performance indices, such as the Integral of Time-weighted Absolute Error (ITAE) or the Integral of Squared Error (ISE) [4]. This approach offers a significant improvement over traditional tuning methods by enabling more efficient exploration and less susceptibility to local minima.

Integrating GAs with nonlinear PID tuning not only enhances the precision of controller parameters but also provides a more robust solution that is less prone to common optimization pitfalls [15, 23, 26, 27]. As demand grows for efficient and adaptive control systems, the synergy between GAs and PID control strategies presents promising avenues for improved system performance. This approach allows for adaptability to varying operating conditions while reducing the time and expertise required for manual tuning efforts [5].

Moreover, while traditional PID controllers are known for their robustness in a wide range of industrial systems, they often struggle with complexities such as disturbances, time delays, and fluctuating set points in nonlinear environments. In such cases, precise tuning of parameters (Kp, Ki, and Kd) is critical to maintaining stability and optimizing performance [6, 10, 15, 24, 25, 26, 27]. GAs offer a dynamic and adaptive tuning mechanism that ensures higher resilience against these challenges.

Historically, PID tuning methods such as Ziegler-Nichols have been widely used to tune these parameters, but these methods often rely heavily on trial-and-error and require a detailed understanding of system dynamics [7]. While heuristic approaches may yield satisfactory results in some situations, they frequently fall short in more complex, nonlinear systems where dynamic behaviors like saturation, dead zones, or time-varying processes are present [8]. Additionally, systems exposed to stochastic disturbances further complicate the challenge of tuning PID controllers effectively [9].

PID controllers, irrespective of how they are tuned, have inherent limitations in predicting future system behavior, particularly in systems with unknown dynamics or significant disturbances. To address these shortcomings, researchers have increasingly turned to machine learning (ML) techniques to enhance control systems. Machine learning models, especially those designed for regression tasks, such as Gradient Boosting Regressors (GBR), can be trained on historical data to predict future system outputs [11]. By integrating ML-based predictions with traditional control approaches, controllers can more effectively anticipate future system behavior, leading to improvements in both control accuracy and performance.

In this study, we propose a hybrid approach that integrates Genetic Algorithm (GA) optimization and Particle Swarm Optimization (PSO) with Machine Learning techniques to enhance the performance of PID controllers in Continuous Stirred Tank Reactors (CSTRs).

The ML surrogate model adjusts the PID controller gains, this means instead of relying purely on traditional control theory for tuning the PID (Proportional-Integral-Derivative) gains, we leverage machine learning to predict or adjust these gains dynamically based on the current state of the system. The GA and PSO are utilized to optimize the PID gains (Kp, Ki, and Kd), while the ML predicts future system outputs based on historical control data. Additionally, we employ an adversarial evaluation framework wherein the best-tuned PID controller is assessed against a set of five poorly performing plant models. The use of an adversarial evaluation framework is intended to add depth to the testing, ensuring that the controllers are robust against poor-performing models. This collaborative approach enables effective adaptation of the PID controller to complex, nonlinear dynamics and external disturbances, ultimately improving control precision and system stability. By testing the robustness of the optimized controllers under challenging conditions, we aim to provide valuable insights into the reliability and performance of PID control strategies in dynamic chemical systems.

We apply this hybrid control strategy to a nonlinear plant with time varying set points and stochastic disturbances, demonstrating its effectiveness in minimizing error and improving response time when compared to traditional PID tuning methods. Our results indicate that the integration of evolutionary optimization and machine learning prediction holds great potential for advancing the performance of industrial control systems.

The design and optimization of PID controllers have made significant strides with the adoption of modern techniques, particularly in the context of nonlinear systems. Genetic Algorithms (GAs) and Neural Networks (NNs) have been widely employed to enhance PID control. In addition, hybrid approaches that integrate GAs with Fuzzy Logic and Neural Networks have shown considerable promise in improving controller performance for both linear and nonlinear systems [25, 26, 27].

Genetic Algorithms (GAs) have gained widespread acceptance for designing and optimizing PID controllers. In linear systems, GAs provide a systematic approach to tuning PID gains, outperforming traditional methods such as Ziegler-Nichols. The research presented in [12], applied GAs to optimize PID controllers for linear systems, achieving improved transient responses and reduced overshoot. Similar work was presented for a variations of controller design utilizing different hybrid GA techniques by a number of researchers [25, 26, 27].

In nonlinear systems, GAs have proven particularly effective due to their capability to handle complex, non-convex optimization problems. Ahmad et al. [13] demonstrated that GA-optimized PID controllers performed better in terms of robustness and stability compared to conventional tuning methods in nonlinear control applications. Furthermore, the application of GAs in designing gain-scheduled controllers for nonlinear systems has been explored, as in the work presented in[25, 26, 27]. [14, 26, 27], where GAs were utilized to design gain-scheduled PID controllers for a nonlinear concentration control system. This approach allowed for better management of the plant’s nonlinear dynamics by optimizing PID gains across various operating points.

In response to design limitations, Machine Learning (ML) has emerged as a powerful tool for enhancing the design and performance of PID controllers. ML, with its ability to learn patterns from data, offers a more adaptive and intelligent approach to PID tuning and control [15]. By leveraging historical data or real-time system feedback, ML algorithms can predict optimal PID parameters (Kp, Ki, and Kd) for a given set of conditions, eliminating the need for manual tuning and reducing reliance on system-specific knowledge [16].

Machine Learning can be integrated into PID controller design in various ways. One of the most common approaches is using supervised learning algorithms, such as Gradient Boosting, Support Vector Machines, or Neural Networks, to model the system’s behavior and predict the ideal PID gains based on historical control data [11]. This allows the controller to adapt dynamically to changes in the system, ensuring better performance in the face of uncertainties, disturbances, or time-varying parameters. Reinforcement learning, another branch of ML, can be employed to train controllers by learning directly from the control environment. This technique enables the PID controller to optimize its actions based on the rewards from its performance, making it highly effective in complex, nonlinear, or stochastic systems [17].

The integration of ML into PID controller design marks a significant departure from traditional control paradigms. It enables systems to self-tune, learn from data, and adjust in real-time, ensuring optimal performance across a wide range of operating conditions. As the demand for more intelligent, adaptive control systems grows, Machine Learning offers a promising pathway to enhance the flexibility, precision, and robustness of PID controllers in both linear and nonlinear applications [18].

control system is designed to regulate the behavior of dynamic systems to achieve specific objectives. It comprises key components such as sensors, actuators, controllers, and feedback loops, working together to adjust the system’s behavior in real time based on the difference between the desired output (setpoint) and the actual system output see Fig. 1. These systems are essential in various industries, ranging from aerospace to manufacturing, where precision and automation are necessary.

Control systems are essential in managing the behavior of dynamic systems across various engineering, automation, and technological applications. They can be broadly divided into linear and nonlinear control systems, and further classified into discrete-time and continuous-time systems. Additionally, there are specialized methodologies such as time series control systems. This section explores these concepts in detail, focusing on the mathematical frameworks and the design and application of Proportional-Integral-Derivative (PID) controllers.

Linear control systems operate using linear differential equations and adhere to the principle of superposition. These systems can be represented using state-space or transfer function models.

The State-Space Representation is considered as Linear Time-Invariant (LTI) system in state-space form which is described as

$$\:\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:1$$

$$\:y\left(t\right)=Cx\left(t\right)+Du\left(t\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:2$$

where, $\:x\left(t\right)$ is the state vector, $\:u(t$) is the input vector, $\:y\left(t\right)$ is the output vector, $\:A$ is the system matrix, $\:B$ is the input matrix, $\:C$ is the output matrix, and $\:D$ is the feedforward matrix.

The transfer function $\:H\left(s\right)$ in the Laplace domain relates the output to the input:

$$\:H\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{{b}_{m}{s}^{m}+{b}_{m-1}{s}^{m-1}+\dots\:+{b}_{0}}{{s}^{n}+{a}_{n-1}{s}^{n-1}+\dots\:+{a}_{0}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:3$$

This function is crucial for analyzing system dynamics, designing controllers, and assessing performance metrics. The stability of a linear control system is determined by the characteristic polynomial derived from the system matrix $\:A$. For stability, all roots (or poles) of this polynomial must have negative real parts. Methods like the Routh-Hurwitz criterion and Nyquist stability criterion are used for stability analysis.

Nonlinear control systems involve nonlinear relationships between inputs and outputs, complicating their analysis. Nonlinearities can arise from factors such as saturations and time-varying system parameters. A nonlinear system is expressed in state-space form as

$$\:x\left(t\right)=f(x\left(t\right),u\left(t\right)),\:y\left(t\right)=h\left(t\right),u\left(t\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:4$$

where, $\:f$ and $\:h$ represent nonlinear functions.

In nonlinear systems, stability is often evaluated using Lyapunov’s direct method. A Lyapunov function $\:V\left(x\right)$ that is positive definite and its time derivative negative definite ensures stability near the equilibrium point.

There are several control techniques for nonlinear systems, they include feedback linearization, sliding mode, and backstepping control.

The Feedback Linearization is a technique used to convert a nonlinear system into an equivalent linear system through state feedback. It is primarily used for systems where the nonlinearities can be systematically cancelled out. Feedback Linearization relies on the concept that many nonlinear systems can be transformed into linear systems through appropriate feedback control. The idea is to design a control input that cancels out the nonlinearities in the system dynamics. Consider a nonlinear system described by:

$$\:\dot{x}=f\left(x\right)+g\left(x\right)u\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:5$$

$$\:y=h\left(x\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:6$$

where $\:xϵ{\mathbb{R}}^{n}$ s the state vector, $\:u\in\:\mathbb{R}$ is the control input, $\:f\left(x\right),\:g\left(x\right)$, and $\:h\left(x\right)$ are nonlinear functions.

To achieve linearization, we need to find a control law $\:u$ such that the resulting system behaves like a linear system. The control law is usually designed in the form

$$\:u=\alpha\:\left(x\right)+\beta\:\left(x\right)\bullet\:v\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:7$$

Where $\:v$ is a new control input to be designed and $\:\alpha\:\left(x\right)$ and $\:\beta\:\left(x\right)$ are functions chosen to cancel out the nonlinear terms. Substituting $\:u$ into the system equations and simplifying should yield a linear system in $\:v$. The most common control strategy used in industry is Proportional-Integral-Derivative (PID) control. A PID controller adjusts the control input based on three terms: the proportional term, which responds to the current error between the setpoint and actual output; the integral term, which accounts for the accumulation of past errors to eliminate steady-state error; and the derivative term, which predicts future errors based on the rate of change, improving system stability and reducing overshoot. The control law for a PID controller is typically expressed as

$$\:u\left(t\right)={K}_{p}e\left(t\right)+{K}_{i}\left(et\right)){\int\:}_{0}^{t}e\left(\tau\:\right)d\tau\:+{K}_{d}\frac{de\left(t\right)}{dt}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:8$$

where $\:u\left(t\right)$ is the control input, $\:e\left(t\right)$ is the error, and $\:{K}_{p}$, $\:{K}_{i}$, and $\:{K}_{d}$ are the proportional, integral, and derivative gains, respectively.

Tuning the PID controller to achieve optimal performance involves selecting appropriate values for $\:{K}_{p}$, $\:{K}_{i}$, and $\:{K}_{d}$. Several methods exist for tuning these parameters, such as the Ziegler-Nichols method [3], which is a heuristic approach based on the system’s response to disturbances. More advanced methods, such as optimization algorithms like Particle Swarm Optimization (PSO) or Genetic Algorithms (GA) [15], can also be employed to minimize performance criteria like the Integral of Squared Error (ISE), which quantifies the overall deviation from the desired output over time.

Control systems are widely applied in various industries. In aerospace, they are used in autopilots and flight control systems, which adjust an aircraft's flight path and stability during different flight phases. According to the work presented in [19], modern flight control systems are essential for improving both stability and handling of aircraft under various flight conditions. In automotive systems, control strategies manage engine performance, anti-lock braking systems (ABS), and cruise control. An in-depth study [20] outlines how advanced control systems are used to optimize vehicle dynamics, including powertrain and braking systems. Chemical processing plants use control systems to regulate variables such as temperature, pressure, and flow rates in reactors and distillation columns, as highlighted in the work presented in [21], where control systems ensure consistent product quality and safety. In robotics, control systems are critical for the precise movement of robotic arms and motors, enabling tasks such as assembly and welding. The work presented in [22] discusses how control theory is applied in industrial robotics to improve accuracy and efficiency in various.

In conclusion, control systems are fundamental to modern technology, allowing for the precise regulation of dynamic systems. Whether through simple PID controllers or more advanced strategies like MPC, these systems ensure stable, efficient, and reliable operation across a wide range of applications. Their ability to handle disturbances and changing conditions makes them indispensable in fields ranging from industrial automation to aerospace engineering.

3.1. Gain-Scheduled PID Controllers

Gain scheduling is an adaptive control strategy designed for systems whose behavior varies significantly under different operating conditions, such as changes in temperature, speed, or other state variables. In these dynamic environments, the system's characteristics can shift, making a fixed-gain controller, like a traditional PID controller, less effective. Gain-scheduled PID controllers address this issue by adjusting the controller gains in real-time based on the current operating conditions or system state.

Gain scheduling involves pre-defining and storing various sets of controller parameters (gains) for specific operating conditions to ensure stability and maintain desired performance across a wide range of operational points. This technique is especially useful for nonlinear systems, where a single fixed-gain controller may fail to provide satisfactory performance across all conditions.

In a gain-scheduled PID system, the operating conditions are first identified and characterized, with each operating point having its unique system dynamics and performance criteria. A scheduling variable, often a measurable system state like temperature or speed, is selected to adjust the PID controller’s gains accordingly. The proportional (Kp), integral (Ki), and derivative (Kd) gains are then modeled as functions of this scheduling variable, ensuring the controller adapts to changes in the system.

In a gain-scheduled PID controller the first step is system characterization, where the system's behavior is analyzed over the entire range of operating conditions to identify changes in dynamics. This process often involves collecting data to observe how the system's response shifts with different operating points. By understanding the system's dynamic behavior under different conditions, will help to tailor the control strategy accordingly.

In controller tuning at key points, a traditional PID controller is tuned for each critical operating point. The goal is to determine the optimal values for the proportional $\:{K}_{p}$, integral $\:{K}_{i}$, and derivative $\:{K}_{d}$ gains. These tuned gains form the foundation for constructing the gain-scheduled model, allowing the controller to adjust to various operating conditions effectively.

Once the PID gains are tuned at key points, the next step is gain function modeling. This involves creating smooth, continuous functions that relate the controller gains to the scheduling variable. These functions are typically modeled using interpolation techniques like linear interpolation, or more advanced methods such as polynomial fitting or splines. The resulting gain functions ensure that the controller smoothly adjusts the PID parameters based on the current operating conditions.

GAs are optimization algorithms inspired by the process of natural selection and biological evolution. Introduced by John Holland in the 1970s, GAs have become a powerful tool for solving complex optimization problems where traditional methods struggle, particularly in non-linear, multi-objective, and non-convex scenarios. The algorithm evolves a population of candidate solutions (chromosomes) to a problem by applying operators analogous to biological evolution: selection, crossover (recombination), and mutation.

The essential components and steps involved in a Genetic Algorithm include Chromosome Representation, Initialization, Fitness Function, Selection, Crossover (Recombination), Mutation, Replacement, Termination.

The chromosome represents a possible solution to the problem. Each chromosome consists of a string of genes, and each gene corresponds to a parameter in the solution. For example, in the context of PID controller optimization, a chromosome might represent the gains $\:{K}_{p}$, $\:{K}_{p}$, and $\:{K}_{p}$. A chromosome $\:c$ for a PID controller be represented as a vector of three genes:

$$\:c=[{K}_{p},\:{K}_{p},{K}_{d}]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:9$$

These genes are encoded in as Binary encoding where each parameter is represented by a binary string. A population of $\:N$ chromosomes is initialized randomly within a specified range for each gene. The population size $\:N$ influences the diversity of the search space and the computational efficiency of the algorithm. Each chromosome in the population is initialized as

$$\:C=\:[{K}_{p},\:{K}_{p},{K}_{d}],\:i=1,\:2,\:\dots\:,\:N\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:10$$

where $\:{K}_{p}$, $\:{K}_{p},{K}_{d}$ are randomly selected values within the given bounds for each parameter.

The fitness function evaluates how well a given chromosome solves the problem. In a GA, the fitness function serves as the objective function, guiding the selection process. For example, in PID tuning, the fitness could be based on minimizing the Integral of Square Error (ISE), Integral of Absolute Error (IAE), or other performance indices. For a PID controller, the fitness function $\:f\left(c\right)$ might be defined as

$$\:f\left(c\right)={\int\:}_{0}^{T}e(t{)}^{2}dt\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:11$$

where $\:e\left(t\right)$ is the error between the desired setpoint and the system output at time $\:e\left(t\right)$, and $\:T$ is the total simulation time. The goal is to minimize $\:f\left(c\right)$.

The selection is the process of choosing chromosomes from the current population to participate in reproduction. Chromosomes with higher fitness values are more likely to be selected, mimicking the survival of the fittest principle. In this work the roulette wheel selection is used, in this method the probability of selecting a chromosome $\:C$ is proportional to its fitness $\:f\left({C}_{i}\right)$. The probability $\:P\left({C}_{i}\right)$ is given by:

$$\:P\left({C}_{i}\right)=\frac{f\left({C}_{i}\right)}{\sum\:_{j=1}^{N}f\left({C}_{j}\right)}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:12$$

The Crossover combines two parent chromosomes to produce offspring. This is achieved by exchanging genetic material between two solutions to explore new regions of the solution space. A Single-point crossover is employed in this study, where a crossover point is selected at random, and the genes beyond that point are swapped between the two parents. If $\:{C}_{1}=[{K}_{p1}$, $\:{K}_{p1},{K}_{d1}]$ and $\:{C}_{2}=[{K}_{p2}$, $\:{K}_{p2},{K}_{d2}]$ are two parents, the offspring after a single-point crossover at gene 2 would be:

$$\:Ofsprin{g}_{1}=[{K}_{p1},\:{K}_{p2},{K}_{d2}],\:Ofsprin{g}_{2}=[{K}_{p2},\:{K}_{p1},{K}_{d1}]$$

Mutation introduces diversity into the population by randomly altering some genes in the offspring. This process helps prevent the algorithm from getting stuck in local optima by exploring new areas of the search space. The mutation operation is applied with a small probability $\:{P}_{m}$ to each gene.

If the gene $\:{K}_{p}$ undergoes mutation, its value is adjusted by adding a small random perturbation:

$$\:{K}_{p}^{{\prime\:}}={K}_{p}+\delta\:,\:\delta \sim N\left(0,\sigma\:\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:13$$

where $\:\delta\:$ is a random value drawn from a normal distribution with mean $\:0$ and standard deviation $\:\sigma\:$.

The offspring replace some or all of the parent population. Replacement strategies vary, with some algorithms allowing offspring to replace only the least fit parents, while others replace the entire population. The new generation is evaluated using the fitness function, and the process repeats.

The algorithm iterates through the selection, crossover, mutation, and replacement steps until a termination condition is met. Figure 2, illustrate the GA process, while Fig. 3, illustrates the crossover process.

PSO is an effective optimization technique that can be applied to tune the PID controller for the CSTR system. The technique is capable of handling nonlinear, complex systems like the CSTR, where traditional tuning methods (such as Ziegler-Nichols) might struggle to provide optimal performance across a wide range of operating conditions.

In PSO, a group of particles (candidate solutions) moves through the solution space to find the optimal solution. Each particle adjusts its position based on its own experience and the experience of its neighbors. For PID tuning, the particles represent different sets of PID gains $\:{K}_{p},{K}_{i},\:\text{a}\text{n}\text{d}\:{K}_{d}$.

The process stars by an initialization of a swarm of particles, each with a random position $\:{K}_{p},{K}_{i},\:\:{K}_{d}$and velocity. Just like the GA, we define a fitness function to evaluate each particle’s performance. The same error-based functions ISE is emplyed.

Each particle adjusts its velocity based on its personal best position $\:{P}_{best}$ and the global best position $\:{g}_{best}$ found by any particle in the swarm. The velocity update rule is:

$$\:{v}_{i}\left(t+1\right)=w{v}_{i}\left(t\right)+{c}_{1}{r}_{1}\left({P}_{best}-{K}_{i}\left(t\right)\right)+{c}_{2}{r}_{2}\left({g}_{best}-{K}_{i}\left(t\right)\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:14$$

where: $\:w$ is the inertia weight (controls the influence of the previous velocity), $\:{c}_{1}$ and $\:{c}_{2}$ are cognitive and social coefficients, respectively, and $\:{r}_{1}$ and $\:{r}_{2}$are random numbers between 0 and 1.

The position of each particle is updated using its velocity:

$$\:{K}_{i}\left(t+1\right)={K}_{i}\left(t\right)+{v}_{i}\left(t+1\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:15$$

The process repeats until the particles converge. In this case each particle represents a set of PID gains:

$$\:{X}^{\left(i\right)}=\left({K}_{p}^{\left(i\right)},{K}_{i}^{\left(i\right)},{K}_{d}^{\left(i\right)}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:16$$

The fitness of each particle is calculated as the error over time:

$$\:f\left({X}^{\left(i\right)}\right)={\int\:}_{0}^{T}{e\left(t\right)}^{2}dt\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:17$$

Both GA and PSO are effective for PID tuning in nonlinear systems like the CSTR, offering adaptive optimization over a range of operating conditions.

A CSTR is a reactor where reactants are continuously fed, mixed, and reacted, with products simultaneously removed. The core challenge is maintaining control over the reactor's temperature and reactant concentrations, as these factors directly influence the reaction rate and product quality. Given that reaction rates and heat generation can vary significantly based on temperature and concentration, fixed-gain PID controllers may struggle to provide optimal performance under all operating conditions.

The temperature and reactant concentration in a CSTR can vary dramatically depending on factors such as heat removal rate, the inflow concentration of reactants, and the cooling jacket temperature. Gain scheduling allows for the adaptation of controller gains to the reactor’s dynamic behavior at different operating points, thus ensuring stable and efficient control.

The behavior of a CSTR can be described using mass and energy balance equations. Consider a first-order exothermic reaction $\:A\to\:B$ occurring in the reactor. The key variables in this case are $\:{C}_{A}$ concentration of reactant $\:A\:(mol/L)$, $\:T$ temperature of the reactor ($\:K$), $\:{T}_{c}$ Temperature of the cooling jacket (K), $\:F$ flow rate of the feed stream ($\:L/min$), and $\:V$ volume of the reactor ($\:L$). The differential equations governing concentration and temperature are concentration balance and energy balance. The Concentration Balance is represented by

$$\:\frac{d{C}_{A}}{dt}=\frac{F}{V}\left({C}_{A0}-{C}_{A}\right)-{k}_{0}{e}^{-\frac{{E}_{a}}{RT}}{C}_{A}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:18$$

where, $\:{C}_{A0}$ is the inlet concentration of $\:A\:(mol/L)$, $\:{k}_{0}$ is the pre-exponential factor ($\:1/min$), $\:{E}_{a}$ is the activation energy ($\:J/mol$), $\:R$ is the universal gas constant ($\:J/mol·K$), T is the reactor temperature $\:\left(K\right)$. The Energy Balance equation is represented by

$$\:\frac{dT}{dt}=\frac{E}{V}\left({T}_{0}-T\right)+\frac{-\varDelta\:H}{\rho\:{C}_{p}}{k}_{0}{e}^{-\frac{{E}_{a}}{RT}}{C}_{A}-\frac{UA}{\rho\:{C}_{p}V}\left(T-{T}_{c}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:19$$

where $\:{T}_{0}$ is the inlet temperature (K), $\:\varDelta\:H$ is the heat of reaction $\:(J/mol$), $\:\rho\:$ is the density of the mixture ($\:kg/m³$), $\:{C}_{p}$ is the specific heat capacity ($\:J/kg·K$), $\:U$ is the heat transfer coefficient ($\:W/m²K$), $\:A$ is the heat transfer area ($\:m²$), and $\:{T}_{c}$ is the cooling jacket temperature (K). The primary control objectives are to maintain the reactor temperature $\:T$ at a setpoint by adjusting the cooling jacket temperature $\:{T}_{c}$ and to regulate the concentration $\:{C}_{A}$ of the reactant to meet product quality specifications.

6.1. CSTR Gain-Scheduled PID Controller Design

Given the nonlinear behavior of the CSTR, especially under varying reaction temperatures and concentrations, it is necessary to define critical operating points (different reactor temperatures and concentrations) and design PID controllers for each. The PID gains $\:{K}_{p}$, $\:{K}_{p},{\:\text{a}\text{n}\text{d}\:K}_{d}$ are then scheduled based on the current operating conditions.

It starts by identify operating points, such as critical operating points may include low, medium, and high temperatures (300 K, 350 K, and 400 K). Then it is followed by tuning PID at Each Point. The gain scheduling in this study utilizes Linear interpolation technique to represented is used to define the PID gains as smooth functions of reactor temperature or concentration such as

$$\:{K}_{p}\left(T\right),{K}_{i}\left(T\right),{K}_{d}\left(T\right)$$

The gains are scheduled based on the concentration of the reactant $\:{C}_{A}$. The gain-scheduled PID control law is expressed as

$$\:u\left(t\right)={K}_{p}\left(T\left(t\right)\right)e\left(t\right)+{K}_{i}\left(T\left(t\right)\right){\int\:}_{0}^{t}e\left(\tau\:\right)+{K}_{d}\left(T\left(t\right)\right)\frac{de\left(t\right)}{dt}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:20$$

where: $\:e\left(t\right)$ = $\:{T}_{setpoint}$ represents the error between the desired and actual reactor temperature, $\:{K}_{p}\left(T\left(t\right)\right),\:{K}_{i}\left(T\left(t\right)\right),\:and\:{K}_{d}\left(T\left(t\right)\right)$ are the temperature-dependent PID gains.

This study proposes integrating Machine Learning (ML) into the optimization process of PID controllers using Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). The aim is to enhance the performance, speed up optimization, and improve the control system's robustness. By combining ML with GA and PSO, the method introduces adaptability, allowing the system to learn from past optimizations and generalize the PID tuning process for a wide range of operating conditions.

GA optimizes PID gains by evolving candidate solutions over generations, but its computational expense arises from evaluating each candidate via simulations of the continuous stirred tank reactor (CSTR) model. To address this, surrogate models (which are also known as meta-models or response surfaces) are introduced to approximate the performance of each PID controller candidate. These models are trained using ML, utilizing neural networks, on previously evaluated PID candidates. Once trained, GA can predict fitness values for new candidates using the surrogate model, reducing the need for full simulations.

In cases where the CSTR undergoes changes or new operating conditions arise, the standard GA would require re-optimization from scratch. However, ML enables the GA to utilize knowledge gained from previous optimizations, learning patterns in PID gains across different operating conditions such as temperature and reactant concentration. This reduces the search space and accelerates convergence. Furthermore, ML can dynamically adjust GA's mutation and crossover rates based on population diversity or previous generations' outcomes, making the process more efficient. The ML model $\:\widehat{f}\left(P\right)$ is trained to approximate the true fitness function $\:f\left(P\right)$, where:

$$\:\widehat{f}\left({P}^{\left(i\right)}\right)\approx\:f\left({P}^{\left(i\right)}\right)={\int\:}_{0}^{T}{e\left(t\right)}^{3}dt\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:21$$

where $\:{P}^{\left(i\right)}=({K}_{p}^{\left(i\right)},{K}_{i}^{\left(i\right)},{K}_{d}^{\left(i\right)})$.

In this case, the optimization is performed as

$$\:{P}_{new}^{\left(i\right)}=GA\left({P}_{old}^{\left(i\right)},\widehat{f}\left({P}^{\left(i\right)}\right)\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:22$$

The GA uses the ML-predicted fitness $\:\widehat{f}$ instead of performing full simulations to guide optimization. ML also enhance the PSO performance by improving the swarm’s ability to explore and exploit the search space more intelligently. Similar to GA, PSO benefits from surrogate models that predict candidate PID gains’ performance without running full simulations. Additionally, ML can prevent premature convergence and accelerate optimization by dynamically adjusting the swarm’s behavior.

In classical PSO, particle velocities are updated using fixed cognitive and social parameters. However, in different regions of the search space, varying exploration strategies may be needed. A neural network can be used to dynamically adjust these parameters based on the current particle and swarm states. The velocity update rule:

$$\:{v}_{i}\left(t+1\right)=w{v}_{i}\left(t\right)+{c}_{1}{r}_{1}\left({P}_{best}-{K}_{i}\left(t\right)\right)+{c}_{2}{r}_{2}\left({g}_{best}-{K}_{i}\left(t\right)\right)\:\:\:\:\:\:\:\:\:\:\:23$$

Equation (23) is modified as

$$\:{v}_{i}\left(t+1\right)=w{v}_{i}\left(t\right)+{c}_{1}\left(t\right){r}_{1}\left({P}_{best}-{K}_{i}\left(t\right)\right)+{c}_{2}\left(t\right){r}_{2}\left({g}_{best}-{K}_{i}\left(t\right)\right)\:\:\:\:\:\:\:\:\:\:24$$

where $\:{c}_{1}\left(t\right)$ and $\:{c}_{2}\left(t\right)$ are learned by a neural network that adapts to the swarm’s dynamics.

The ML model for particle movement uses the particle's current position $\:{X}_{i}\left(t\right)=({K}_{p},{K}_{i},{K}_{d})$ to adjust the velocity $\:{v}_{i}(t+1)$ and optimize the negative of the fitness function:

$$\:r\left(t\right)=-f\left({X}_{i}\left(t\right)\right)=-{\int\:}_{0}^{T}{e\left(t\right)}^{2}dt\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:25$$

The ML is further employed to learn gain-scheduling rules. Neural networks can be trained to map operating conditions (e.g., temperature $\:T$, concentration $\:{C}_{A}$ to optimal PID gains $\:({K}_{p},{K}_{i},{K}_{d})$.. Data is collected by running GA or PSO under varying conditions and recording the optimal gains. The trained ML model can then predict the best gains for new operating conditions:

$$\:\left[{K}_{p},{K}_{i},{K}_{d}\right]=ML\:Model\left(T,{C}_{A}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:26\:\:$$

The ML model may be a neural network, decision tree, or other regression method that maps operating conditions to optimal PID gains. By incorporating ML into GA and PSO, the system becomes more adaptable and efficient in handling complex, nonlinear dynamics, such as those in a CSTR system. This significantly improves the overall performance and reduces computational overhead in PID tuning.

In this approach, neural-based machine learning (ML) is used to optimize PID controller parameters ($\:{K}_{p},{K}_{i},{K}_{d})$ that can handle the worst-case conditions of a Continuous Stirred Tank Reactor (CSTR) system. The worst-case scenario involves system parameters such as reaction rates, inflow concentrations, and heat removal rates that make the CSTR difficult to control. These parameters include the reaction rate constant $\:{k}_{0}$, activation energy $\:{E}_{a}$, heat transfer coefficient $\:U$, and inflow concentration $\:{C}_{A}^{0}$. The optimization problem is structured as

$$\:\underset{{K}_{p},{K}_{i},{K}_{d}}{\text{min}}\underset{{\theta\:}_{{CSTR}}}{\text{max}}J({K}_{p},{K}_{i},{K}_{d},\:{\theta\:}_{CSTR})\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:27$$

where $\:J({K}_{p},{K}_{i},{K}_{d},\:{\theta\:}_{CSTR})$ is the cost function based on an error metric, such as the Integral of Square Error (ISE). The goal is to find the PID parameters that minimize error while accounting for the most challenging CSTR parameters $\:{\theta\:}_{CSTR}$, including $\:{k}_{0},{E}_{a},U$, which could make the system difficult to control.

In this study, neural networks are employed to enhance both the optimization of PID gains and the identification of worst-case CSTR parameters. The process involves two neural models, the first Neural Model for Worst-Case CSTR Parameter Prediction: This neural network is trained to predict the worst-case CSTR system parameters $\:{\theta\:}_{CSTR}$that maximize the error for a given set of PID gains. The goal is to use the model to efficiently approximate which combination of CSTR parameters leads to the most difficult-to-control dynamics.

The second Neural Model for PID Optimization: Another neural network is used to optimize the PID parameters ($\:{K}_{p},{K}_{i},{K}_{d}$). This network learns the relationship between PID gains and system performance by analyzing data from previous optimizations. The aim is to generalize this knowledge, allowing the network to suggest optimal PID settings for minimizing the error, even under adversarial CSTR conditions. The optimization problem is then tackled as

$$\:\underset{({K}_{p},{K}_{i},{K}_{d})}{\text{min}}\underset{{\theta\:}_{{CSTR}}}{\text{max}}\widehat{J}({K}_{p},{K}_{i},{K}_{d},{\theta\:}_{CSTR})\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:28$$

where $\:\widehat{J}({K}_{p},{K}_{i},{K}_{d},{\theta\:}_{CSTR})$ is the neural network approximation of the true cost function $\:J({K}_{p},{K}_{i},{K}_{d},{\theta\:}_{CSTR})$. This neural-based surrogate model reduces the need for computationally expensive simulations by predicting system behavior and performance for both PID and CSTR parameter combinations.

The optimization process in the GA-based approach, each individual in the population is represented as a vector that includes both the PID gains and CSTR parameters:

$$\:{P}^{\left(i\right)}=\left({K}_{p}^{\left(i\right)},{K}_{i}^{\left(i\right)},{K}_{d}^{\left(i\right)},{\theta\:}_{CSTR}^{\left(i\right)}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:29$$

A neural network surrogate model is trained to predict the fitness value for each individual, estimating the combined effect of the PID gains and worst-case CSTR parameters. The network accelerates the GA by providing fast fitness evaluations:

$$\:f\left({P}^{\left(i\right)}\right)=\widehat{J}\left({K}_{p}^{\left(i\right)},{K}_{i}^{\left(i\right)},{K}_{d}^{\left(i\right)},{\theta\:}_{CSTR}^{\left(i\right)}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:30$$

This reduces the need to run full simulations for each candidate solution, enabling the GA to evolve PID gains and CSTR parameters more efficiently. The crossover, mutation, and selection processes focus on minimizing the error for the worst-case CSTR parameters predicted by the neural network.

In the PSO, each particle represents both the PID gains and the CSTR parameters:

$$\:{X}^{\left(i\right)}=\left({K}_{p}^{\left(i\right)},{K}_{i}^{\left(i\right)},{K}_{d}^{\left(i\right)},{\theta\:}_{CSTR}^{\left(i\right)}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:31$$

The particle swarm is guided by neural-based predictions of both the worst-case CSTR parameters and the corresponding optimal PID gains. The fitness function for PSO is evaluated using the neural model:

$$\:f\left({X}^{\left(i\right)}\right)=\widehat{J}\left({K}_{p},{K}_{i},{K}_{d},{\theta\:}_{CSTR}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:32$$

The velocity and position updates adjust the particles based on the neural network’s predictions, helping the swarm converge faster to both the best PID parameters and the most challenging CSTR conditions.

The neural networks are trained using data from multiple optimization runs where GA or PSO is used to tune the PID parameters under various CSTR conditions. The data consists of input-output pairs: operating conditions $\:(T,{C}_{A}^{0},{k}_{0},{E}_{a})$ and corresponding optimal PID gains $\:({K}_{p},{K}_{i},{K}_{d})$.

The neural network learns the mapping:

$$\:\left[{K}_{p},{K}_{i},{K}_{d}\right]=ML\:Model\:\left(T,{C}_{A}^{0},{k}_{0},{E}_{a}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:33$$

This allows the model to predict the optimal PID gains based on given CSTR parameters and vice versa.

Once trained, the model generalizes the relationship between operating conditions and optimal PID gains, enabling the optimizer to efficiently explore the parameter space. It also suggests worst-case scenarios that push the PID controller to its limits.

Adversarial ML techniques are applied to actively generate worst-case CSTR scenarios that stress-test the PID controller. The neural network model is trained to identify which CSTR parameters $\:{\theta\:}_{CSTR}$ lead to the worst performance for a given PID controller. These adversarial scenarios are then introduced during the optimization process, helping to discover more robust PID settings.

By leveraging neural-based ML, the optimization process becomes more efficient and adaptable. The system learns from past optimization runs and applies this knowledge to reduce the computational effort required for evaluating new scenarios. This makes the approach ideal for handling the complex, nonlinear dynamics typical of CSTR systems.

In the adversarial optimization case, the goal is to find the best PID controller parameters ($\:{K}_{p},{K}_{i},{K}_{d})$ that perform well under the worst-case CSTR system parameters, such as the worst combination of reaction rates, inflow concentrations, or heat removal rates. The system parameters to be tuned include variables such as the reaction rate constant $\:{k}_{0}$, activation energy $\:{E}_{a}$, heat transfer coefficient $\:U$, and other parameters that affect the CSTR’s dynamics. To achieve this goal, the optimization problem is formed as using Eq. 27. The inner maximization problem finds the worst-case CSTR parameters $\:{\theta\:}_{CERT}$, which make the system behave in a way that is hardest to control. The outer minimization problem seeks the optimal PID gains ($\:{K}_{p},{K}_{i},{K}_{d}$) that minimize the error in controlling the worst-case CSTR plant.

In this study both GA and PSO are used simultaneously optimize both the PID gains and the CSTR system parameters. One optimization loop focuses on finding the worst-case CSTR parameters, while the other loop finds the best PID controller parameters.

In the GA case the Population Representation is achieved by representing each individual in the population is represented as a vector consisting of both PID gains and CSTR parameters as illustrated in Eq. (27). The PID parameters $\:{(K}_{p},{K}_{i},{K}_{d})$ are optimized in the outer loop to minimize the error, while the CSTR parameters $\:{\theta\:}_{CSTR}$ are optimized in the inner loop to maximize the error. The fitness function evaluates both how well the PID controller performs and how adversarial the CSTR system behaves:

$$\:f\left({P}^{\left(i\right)}\right)=\underset{{K}_{p}^{\left(i\right)},{K}_{i}^{\left(i\right)},{K}_{d}^{\left(i\right)}}{\text{min}}\underset{{\theta\:}_{{CSTR}}^{\left(i\right)}}{\text{max}}{\int\:}_{0}^{T}{e\left(t\right)}^{2}dt\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:34$$

where $\:e\left(t\right)={T}_{setpoint}-T\left(t\right)$ is the error between the desired and actual reactor temperature.

The evolutionary process for both PID gains and CSTR parameters undergo crossover, mutation, and selection to evolve toward the optimal solution. The PID gains evolve to minimize the error, while the CSTR parameters evolve to make the system as difficult as possible to control.

The PSO particle representation of each particle represents both the PID controller gains and the CSTR system parameters are illustrated in Eq. (31). The velocity and position updates aim to find the worst-case CSTR parameters $\:{\theta\:}_{CSTR}$ that maximize the error and the best PID parameters ($\:{K}_{p},{K}_{i},{K}_{d}$) that minimize the error.

The position of each particle is updated based on both individual best (PID and CSTR) and global best (worst-case CSTR and best PID) solutions. The fitness function for this arrangement is given by Eq. (32). The dual optimization problem involves, minimizing the outer loop of the cost function to achieve an optimal PID parameters $\:{(K}_{p},{K}_{i},{K}_{d})$, while the inner loop maximises the cost function by finding the worst case CSRT parameters $\:{\theta\:}_{CSTR}$. This result of the min-max formulation is given as in Eq. (28).

The ML will significantly improve the adversarial optimization process through the use of surrogate model that evaluate both the CSTR system and the PID controller over a wide range of parameters can be computationally expensive. The surrogate model will be trained using Neural Networks to approximate the cost function $\:J({K}_{p},{K}_{i},{K}_{d},\:{\theta\:}_{CSTR})$, reducing the need for full evaluations. The surrogate model can predict the performance of candidate PID and CSTR parameter combinations, allowing GA or PSO to explore the parameter space more efficiently. The surrogate model $\:\widehat{j}$ approximates the true cost function as

$$\:\widehat{J}\left({K}_{p},{K}_{i},{K}_{d},\:{\theta\:}_{CSTR}\right)\approx\:J\left({K}_{p},{K}_{i},{K}_{d},\:{\theta\:}_{CSTR}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:35$$

In this case each adversarial optimization episode can be treated as a trial in which an RL agent learns the optimal strategy for adjusting the PID gains to minimize the error under varying worst-case conditions. The RL agent can be trained using a reward signal based on the performance of the PID controller for each worst-case CSTR scenario as

$$\:r\left(t\right)=\underset{{\theta\:}_{{CSTR}}}{\text{max}}J({K}_{p},{K}_{i},{K}_{d},\:{\theta\:}_{CSTR})\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:36$$

The adversarial ML techniques is used to actively generate adversarial samples of worst case CSTR scenarios that push the PID controller to its limits. The adversarial ML model can be trained to learn which CSTR parameters $\:{\theta\:}_{CSTR}$ result in the worst performance for any given PID controller. This adversarial model can then suggest worst-case scenarios during the optimization process to help find a more robust PID controller.

In this study, we explored the implementation of both a Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) for optimizing the PID controller of a Continuous Stirred Tank Reactor (CSTR) plant model. Both GA and PSO were further enhanced with a Machine Learning (ML) surrogate model for dynamic gain adjustment and were used to control the CSTR plant under added disturbances. A detailed explanation of the methods has been provided in earlier sections. In this section, we present the results, focusing on the performance and effectiveness in controlling the CSTR plant under disturbances, followed by its performance in adversarial conditions, where the best-tuned PID controller is applied to manage the five most challenging CSTR plants.

Figure 4, illustrate a visual image of the ability of GA tune a PID controller to control the performance of CSRT plant.

From Fig. 4, it can be seen that the GA-tuned controller performed very well, achieving steady state within 1 second with near-zero overshoot.

Figure 5 illustrates the ability of Particle Swarm Optimization (PSO) to tune a PID controller for managing the performance of the CSTR plant. As shown in the Fig. 5, the PSO-tuned controller also performed exceptionally well, reaching steady state in less than 1 second (0.5 seconds) with near-zero overshoot, making it significantly faster than the GA-tuned controller.

Figure 6 illustrates the ability of the Genetic Algorithm (GA) to tune a PID controller for managing the performance of a Continuous Stirred Tank Reactor (CSTR) plant with added disturbance. From the Fig. 6, it can be observed that the GA-tuned controller is capable of bringing the CSTR plant under control to a certain extent. Despite the initial disturbance, the controller gradually adjusts and stabilizes the system. The results indicate that while the GA requires some time to fully stabilize the controller, it successfully adapts to the disturbance. However, it is evident that the presence of disturbance significantly impacts the controller's performance.

Figure 7 illustrates the ability of the Particle Swarm Optimization (PSO to tune a PID controller for managing the performance of a Continuous Stirred Tank Reactor (CSTR) plant with added disturbance. From the Fig. 7, it can be observed that the PSO-tuned controller is incapable of bringing the CSTR plant under control.

It is evident that the presence of disturbance significantly impacts the controller's performance. Therefore, a second test was conducted with reduced plant disturbance. The results of this test are illustrated in Fig. 8 and Fig. 9 for the GA-tuned and PSO-tuned controllers, respectively.

From Fig. 8 and Fig. 9, it can be observed that both the GA-tuned and PSO-tuned controllers achieved similar, acceptable control once the disturbance was reduced by nearly 60% of the original level. This suggests that, in this scenario, the GA-tuned controller demonstrates greater robustness compared to the PSO-tuned controller when handling the disturbed CSTR plant.

Figure 10 illustrates the performance of the hybrid Machine Learning (ML) model integrated with both the GA and PSO-tuned controllers. As shown in the Fig. 10, the performance of both controllers has improved, demonstrating a better convergence rate compared to the results achieved without ML integration. This highlights the effectiveness of using ML to enhance the tuning process and overall controller performance.

The results in Fig. 10 demonstrate that the hybrid ML controller significantly enhanced the ability to manage aggressive plant disturbances by effectively suppressing randomly introduced disturbances, resulting in improved control performance and faster convergence. In the upcoming test, the hybrid ML controller, optimized using both GA and PSO, will be evaluated under adversarial conditions. The best-tuned controller will be tested against the worst four CSTR plants, each with added disturbances, to assess its robustness and effectiveness.

In the adversarial approach, the population of plants is optimized using a maximizing cost function to identify the most challenging plants, while the population of controllers uses a minimizing cost function to effectively control these difficult CSTR plants. This procedure effectively searches the plant population for the highest-cost plant and then tunes the controller for optimal performance with that plant. The test results, illustrated in Fig. 11, show the performance of the best-tuned controller against the four most challenging plants. As shown in Fig. 11, the controller successfully manages all four plants, indicating that it can be considered a robust controller, capable of handling various disturbances and uncertainties.

It is expected that the ML_PSO controller will achieve similar results to those illustrated in Fig. 11, demonstrating its ability to effectively manage the CSTR plant under varying levels of disturbance and uncertainty. The enhanced performance from the ML integration should enable the PSO-tuned controller to converge rapidly while maintaining robust control, comparable to the results previously observed.

In this study, we proposed an evolutionary technique within a novel paradigm to address the challenge of designing robust hybrid ML_GA and ML_PSO PID controllers in the time domain for plants with prescribed parameter uncertainties. The evolutionary approach involved generating two separate populations: one representing controllers and the other representing plants. The plant population, containing plants with prescribed uncertainties, was evolved to include the most difficult plants to control, while the controller population was evolved to effectively manage these challenging plants.

We successfully implemented both Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) for optimizing the PID controller of a Continuous Stirred Tank Reactor (CSTR) plant model, enhancing both techniques with a Machine Learning (ML) surrogate model for dynamic gain adjustment. The results demonstrate that GA and PSO were effective in managing the CSTR plant under disturbances, with PSO showing faster convergence and GA displaying greater robustness in handling more difficult scenarios.

The integration of the hybrid ML model further improved controller performance, providing superior handling of aggressive disturbances and faster system stabilization. The hybrid ML controller was tested under adversarial conditions and demonstrated its robustness by controlling the four most difficult CSTR plants.

Overall, the results indicate that the combination of GA/PSO with ML provides a robust and adaptive control strategy capable of handling varying levels of disturbance and uncertainty in CSTR plant models. This hybrid approach offers a promising solution for optimizing control systems in dynamic and unpredictable environments, improving system resilience and control effectiveness.

Author Contributions: The sole author handled the conceptualization, methodology, analysis, experiments, testing, and writing of the manuscript.

Funding: Not applicable (No Funding).

Ethical Declarations: All methods were conducted in accordance with relevant guidelines and regulations.

Informed Consent Statement: Not applicable.

Data availability: Not applicable, as the data used in the manuscript are synthetic and randomly generated.

Conflict of interest: The author declares no conflict of interest.

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Enhancing PID Control Robustness in CSTRs: A Hybrid Approach to Tuning Under External Disturbances with GA, PSO, and Machine Learning

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Abstract

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1. Introduction

2. Related Work

3. Control System

3.1. Gain-Scheduled PID Controllers

4. Genetic Algorithms (GAs)

5. Particle Swarm Optimization (PSO)

6. Continuous Stirred Tank Reactor (CSTR)

6.1. CSTR Gain-Scheduled PID Controller Design

7. Proposed Method

8. Results and discussion

9. Conclusion

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References

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