A Novel Chaotic Particle Swarm-Optimized Backpropagation Neural Network PID Controller for Indoor Carbon Dioxide Control

Abstract

In the continuously evolving landscape of novel smart control strategies, optimization techniques play a crucial role in achieving precise control of indoor air quality. This study aims to enhance indoor air quality by precisely regulating carbon dioxide (CO₂) levels through an optimized control system. Prioritizing fast response, short settling time, and minimal overshoot is essential to ensure accurate control. To achieve this goal, chaos optimization is applied. By using the global search capability of the chaos particle swarm optimization (CPSO) algorithm, the initial weights connecting the input layer to the hidden layer and the hidden layer to the output layer of the backpropagation neural network (BPNN) are continuously optimized. The optimized weights are then applied to the BPNN, which employs its self-learning capability to calculate the output error of each neuronal layer, progressing from the output layer backward. Based on these errors, the weights are adjusted accordingly, ultimately tuning the proportional–integral–derivative (PID) controller to its optimal parameters. When comparing simulation results, it is evident that, compared to the baseline method, the enhanced Chaos Particle Swarm Optimization Backpropagation Neural Network PID (CPSO-BPNN-PID) controller proposed in this study exhibits the shortest settling time, approximately 0.125 s, with a peak value of 1, a peak time of 0.2 s, and zero overshoot, demonstrating exceptional control performance. The novelty of this control algorithm lies in the integration of four distinct technologies—chaos optimization, particle swarm optimization (PSO), BPNN, and PID controller—into a novel controller for precise regulation of indoor CO₂ concentration.

1. Introduction

As pillars of modern building technology, indoor environmental control systems are ubiquitous across various building types [1], providing occupants with a comfortable living environment and ensuring high-quality air [2]. However, with the proliferation of new synthetic materials and chemical products in daily life, indoor air quality issues have gained increasing prominence [3]. These factors have led to a surge in indoor air pollutants, posing a potential health hazard to occupants [4].

As modern lifestyles evolve, individuals now spend approximately 90% of their time indoors, rendering Indoor Air Quality (IAQ) a paramount indicator of daily life comfort [5]. To effectively assess and enhance IAQ, scientists have identified several air quality indicators, among which CO₂ concentration emerges as a pivotal reference metric. When indoor CO₂ concentration is excessively high, it can have a negative impact on learning and work efficiency. In contrast, by ventilating frequently, the indoor CO₂ concentration can be maintained at approximately 400 ppm, which is similar to the CO₂ concentration in outdoor air. This well-ventilated environment can significantly enhance learning and work efficiency, even up to 15% [6]. According to Settimo et al. [7], the indoor CO₂ intervention threshold in most regions is 1000 ppm. Exceeding this indicates potential adverse effects on occupant comfort and health. Therefore, maintaining indoor CO₂ concentrations at or below outdoor levels is imperative for ensuring fresh indoor air.

To accomplish this objective, sophisticated control strategies are essential to bolster control performance [8]. Commonplace control strategies, including PID controllers, are widely utilized, but they frequently encounter obstacles due to the exactness of the building model and the calibration of parameters. When dealing with complex systems, the efficacy of traditional PID controllers can be significantly compromised, potentially leading to system instability.

To surmount the limitations inherent in traditional control methods, the emergence of intelligent control technology in recent years has presented novel solutions for managing intricate systems [9]. This technology exhibits substantial advantages, especially when tackling complex nonlinear time-delay systems [10] and uncertainties [11]. By amalgamating traditional and intelligent control techniques [12], innovative and adaptable control strategies can be formulated [13].

The focus of this study is to regulate indoor carbon dioxide concentrations within IAQ. Our objective is to create a cutting-edge intelligent solution that enhances speed, achieves a short settling time, and maintains minimal overshoot.

The following sections of our paper are organized in this manner: An extensive overview of the relevant literature is presented in Section 2. Section 3 introduces a novel error backpropagation neural network proportional–integral–derivative adaptive controller, optimized using the chaos particle swarm optimization algorithm. In Section 4, the results obtained from the experiments are showcased, and a comprehensive analysis of multiple performance indicators is provided. In Section 5, the research findings are summarized. The main contributions of this work are as follows:

(1)

A novel neural network PID intelligent controller based on CPSO algorithm optimization is proposed, termed the CPSO-BPNN-PID controller. The novelty of this controller resides in the integration of four distinct technologies: chaos theory, particle swarm optimization, BPNN, and PID control.

(2)

The controller was applied to the regulation of indoor CO₂ concentration. Experimental results indicate that the CPSO-BPNN-PID controller outperforms both the BPNN-PID controller and the conventional PID controller in terms of response speed, adjustment time, and reduced overshoot.

2. Related Works

In pursuit of enhancing IAQ and optimizing how well the system responds quickly, maintains stability, and avoids excessive overshoot, researchers have investigated and implemented various control methods. Traditional techniques like PID control and optimal control have gained popularity in IAQ control systems due to their stability and reliability, with PID control being favored for its simplicity and practicality.

Concurrently, intelligent control technology is evolving and finding applications in IAQ control systems, presenting novel solutions for managing intricate nonlinear systems [14,15]. Technologies such as fuzzy logic control, neural networks, genetic algorithms (GA), and evolutionary algorithms have emerged as focal points of research [16,17]. The novel PID controller based on fuzzy control and neural network optimization is widely used due to its excellent control performance [18,19]. For instance, Uskenbayeva et al. [20] proposed a PID regulator design using backpropagation neural networks. The experiments manipulated PID regulators via neural networks and applied the system to interpret model results. The controller effectively maintained the desired indoor air quality level. Similarly, Aziz et al. [21] utilized Firefly Algorithm and PSO algorithms to refine the PID control mechanism for regulating Air Handling Unit temperature, addressing temperature control issues in building subsystems. The results indicated improved temperature regulation and occupant comfort compared to baseline methods.

Despite achieving commendable control effects in indoor environmental systems and enhancing IAQ, these control methods still face challenges. One significant hurdle is the time-consuming task of setting optimal parameters. Additionally, there is room for further refinement in algorithm optimization techniques. To tackle these obstacles, researchers are proactively exploring new control strategies that integrate neural networks and optimization algorithms, aiming to capitalize on their respective strengths [22].

Within intelligent optimization algorithms, the PSO algorithm frequently finds application in neural network training. For instance, Chaturvedi et al. [23] proposed a novel neural network PID controller based on PSO for reactor temperature control. The results indicate that compared to the baseline method, this controller reduces overshoot by 23.13% and decreases rise time by 0.1283 s. Wang et al. [24] proposed a conductivity control system using PSO-optimized BPNN-PID. The PSO algorithm optimizes BP network weights to adjust PID parameters. The verified model showed minimal conductivity fluctuations (0.003–0.119 mS/cm) with low overshoot (0.3–0.14%). This approach outperforms traditional PID and BPNN-PID controllers. However, the PSO algorithm has limitations, such as the absence of dynamic speed adjustment, which can predispose it to local optima and reduced convergence accuracy. To overcome these limitations, various improvements have been introduced, including improved PSO (IPSO), genetic algorithm-particle swarm optimization (GA-PSO), and CPSO.

The IPSO algorithm incorporates a novel particle update approach based on the PSO algorithm, blending individual and group learning. This enhances the versatility and efficacy of particles during the search process. Shao et al. [25] presented a novel transformer fault diagnosis model integrating the PSO algorithm and a radial basis function (RBF) network, incorporating the Adam optimization algorithm into the PSO algorithm. The research results indicate that the diagnostic accuracy and stability of this model are both superior to the baseline model.

The GA-PSO algorithm is a hybrid optimization approach that fuses the GA with the PSO algorithm. Gupta et al. [26] employed feature-level fusion and a GA-PSO hybrid optimization neural network for breast cancer identification. The results indicated that this technology aided in addressing scalability issues in GA and local optimization challenges in PSO, achieving a 99.72% accuracy in image classification (normal, benign, malignant), and demonstrating faster convergence and superior performance.

The CPSO algorithm utilizes chaotic mapping to produce random number sequences, replacing the pseudo-random number sequences in traditional PSO, thereby augmenting the global search capability of the PSO algorithm. Tang et al. [27] introduced a CPSO-based radial basis function-backpropagation (RBF-BP) model to enhance the accuracy of reactor temperature predictions. Experimental data revealed that the model achieved a root mean square error of 17.3%, an average absolute error of 11.4%, and a fitting value of 99.791%. Compared to the baseline models, this model demonstrated superior prediction accuracy and efficiency, thus confirming its efficacy and practicality.

3. Research Methodology

3.1. Proportional-Integral-Derivative Controller

The traditional PID controller operates based on the PID control algorithm. Its input consists of the system’s deviation, while its output is the control variable. Refer to Figure 1, which provides a visual illustration of the PID controller’s structure [28]. The PID controller functions as follows: Initially, the proportional component generates a control output, u(t), based on the deviation, e(t), between the system’s actual output value, y(t), and the expected value, r(t). Subsequently, the integral component integrates this deviation to eliminate any static errors. Lastly, the differential component compensates for the rate of change in deviation, thereby mitigating dynamic errors.

The expression of the control quantity is defined as Equation (1). The transfer function expression is defined as Equation (2).

$$u(t)=K_p[e(t)+{\displaystyle \frac{1}{T_i}}{\displaystyle {\int }_0^{t}e(t)dt+T_d\frac{de(t)}{dt}}]$$

(1)

$$G(s)={\displaystyle \frac{U(s)}{E(s)}}=K_p[1+{\displaystyle \frac{1}{T_{i}S}}+T_{d}S]$$

(2)

In Equations (1) and (2), K_p is the proportional coefficient; T_i is the integral time constant; T_d is a differential time constant.

3.2. Backpropagation Neural Network-Based PID Controller

The BPNN-PID controller seamlessly integrates the adaptive learning capabilities of a BPNN with the reliable stability inherent in a PID controller. This hybrid control strategy significantly enhances the effectiveness of the control system, particularly when addressing issues such as nonlinearity and time delays. As clearly illustrated in Figure 2, the BPNN is employed to approximate nonlinearities within the control system or to model its complex, unexplained dynamics. In this configuration, the BPNN takes the current status of the control system as input, analyzes it, leverages its predictive capabilities to compute any necessary adjustments, and then feeds these adjustments into the PID controller for a final output. This correction primarily fine-tunes the proportional, integral, and differential components of the PID controller, thereby enabling the control system to more effectively address intricate issues.

The algorithm flow of the BPNN-PID controller is as follows [20]:

(1)

Establish a BPNN by determining its network structure, setting appropriate values for the learning rate and inertia factor (also known as momentum), and initializing the weights for each neural network layer.

(2)

Calculate error(k) by comparing the sampled reference input rin(k) with the actual output yout(k).

(3)

Calculate the input and output values for each layer of neurons within the neural network. More specifically, the output generated by the final layer corresponds to the parameters of the PID controller.

(4)

By employing the incremental digital PID control algorithm, the output u(k) of the PID controller is derived.

(5)

Carry out neural network learning, adjust weights $w_{ij}^{(2)}(k)$ and $w_{lj}^{(3)}(k)$ online, and realize self-tuning of PID control parameters $k_p$, $k_i$, and $k_d$. Specifically, $w_{ij}^{(2)}(k)$ is the weight between the ith neuron in the hidden layer and the jth neuron in the input layer, and $w_{lj}^{(3)}(k)$ is the weight between the first neuron in the output layer and the ith neuron in the hidden layer.

(6)

Set k = k + 1 and return to the first step. The block diagram of the BPNN-PID algorithm is shown in Figure 3.

3.3. Chaotic Particle Swarm Optimization-Improved Backpropagation Neural Networks for PID Controller

3.3.1. Chaotic Particle Swarm Optimization Algorithm

In nonlinear systems, chaos is a common phenomenon of motion, and its mathematical form can be shown in Equation (3) [29]:

$$\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^{T}f(x_t)}={\displaystyle \int dxF(x)}f(x)$$

(3)

The distribution function is given by Equation (4).

$$F(x)={\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^T\delta (x-x_n)}{|}_{T\to \infty }$$

(4)

Chaos exhibits characteristics such as ergodicity, randomness, and regularity, making it highly advantageous in addressing optimization problems. Utilizing chaotic variables for optimal searches proves superior to blind random searches, effectively preventing algorithms from getting stuck in local minima. Among the various chaotic search methods, chaotic mapping algorithms occupy a pivotal position. Standard chaotic mapping algorithms encompass logistic mapping, tent mapping, and Lorenz mapping, among others. In this paper, we employ the logistic mapping algorithm, whose mathematical expression is presented in Equation (5):

$$y_j^{i+1}=\mu (1-y_j^i)y_j^i(j=1,2,\dots ,m)$$

(5)

In Equation (5), $y\in [0,1]$, $\mu$ is the chaotic coefficient.

By integrating chaos theory within the iterative sequence of the PSO method, the CPSO algorithm comes into being. This CPSO algorithm leverages the ergodicity of chaos to refine the best positions searched by the entire particle swarm. To achieve this, chaotic sequences are generated based on the current best position, and these sequences are subsequently used to replace the previous particles.

The CPSO algorithm performs optimization according to the following process:

(1)

Calculate the positions and velocities of every particle in the swarm.

(2)

Evaluate the fitness of each particle and determine the best position: $x_{best}^i=(x_{best,1}^i,x_{best,2}^i,\dots ,x_{best,m}^i)$. Let the variable i represent the number of iterations. Using Equation (6), map the chosen position to the {0, 1} field. Subsequently, compute the chaotic variable for the next iteration using Equation (5) and perform M iterations. Afterward, utilizing Equation (7), remap the chosen location into the original solution space to acquire a feasible solution from the chaotic sequence.

$$y_j^i={\displaystyle \frac{x_{best,j}^i-x_{\min }^i}{x_{\max }^i-x_{\min }^i}}(j=1,2,\dots ,m)$$

(6)

$$x_{best,j}^{i+1}=x_{\min }^i+(x_{\max }^i-x_{\min }^i)y_j^{i+1}$$

(7)

(3)

Assess the fitness of the feasible solution and choose the best among them.

(4)

From the swarm, randomly pick out a particle and replace it with the best feasible solution found.

(5)

Update the velocity and position of each particle, repeat the loop, and return to the second step until the termination condition is satisfied.

3.3.2. CPSO-Enhanced Backpropagation Neural Networks for PID Controller

The improved CPSO-BPNN-PID controller consists of two primary components: a BPNN controller enhanced by the CPSO algorithm and a conventional PID controller. Refer to Figure 4 for a visual representation of the controller’s structure. The fundamental principle behind this hybrid controller lies in leveraging the global search capabilities of the CPSO algorithm. This algorithm continuously refines the initial weights connecting the input, hidden, and output layers of the BPNN. Once optimized, these weights are relayed to the BPNN. Leveraging the self-learning capabilities of the BPNN, the system then calculates the output error for each neuronal layer, progressing from the output layer backward. Based on these errors, the weights are adjusted to attain optimal PID controller parameters, ultimately delivering the desired control performance.

The design process of the improved CPSO-BPNN-PID controller algorithm is as follows:

(1)

Establish the BPNN with a neural network architecture of 4-5-3 and set the appropriate values for the learning rate and inertia factor.

(2)

Initialize the input and output weight matrices of the BPNN. Use the rand function to generate random numbers in the range of [–1, 1]. The input weight matrix for the hidden layer neurons, denoted as wi, is randomly initialized to dimensions of 5 × 4, and the output weight matrix for the hidden layer neurons, denoted as wo, is randomly initialized to dimensions of 3 × 5.

(3)

Initialize the particle swarm and set the learning factors c1, c2, population size, maximum number of iterations, and particle search space dimension.

(4)

Assess the fitness of each particle by calculating the sum of squared errors, denoted as tol_fitness, which serves as the fitness function for the optimization process.

$$tol\_fitness=tol\_fitness+abs(error{(k)}^2)$$

(8)

In Equation (8), error(k) is the system error.

(5)

Set the initial values for both the individual and global optima.

(6)

Update particle position and velocity. Adjust the speed and position based on Equations (9) and (10) in each iteration.

$$v_{ij}(t+1)=w\cdot v_{ij}(t)+c_1{r}_1(t)[p_{ij}(t)-x_{ij}(t)]+c_2{r}_2(t)[p_{gi}(t)-x_{ij}(t)]$$

(9)

$$x_{ij}(t+1)=x_{ij}(t)+v_{ij}(t+1)$$

(10)

where the inertia coefficient w adopts the linear weight decreasing method, and its expression is shown in Equation (11).

$$w=w_{\max }-{\displaystyle \frac{(w_{\max }-w_{\min })\cdot t}{T_{\max }}}$$

(11)

In Equation (11), $T_{\max }$ is the maximum evolution algebra, $w_{\max }$ is the maximum inertia weight, $w_{\min }$ is the minimum inertia weight, and t is the number of iterations.

(7)

Evaluate the fitness of each particle’s current position in comparison to its historical optimal fitness. If the current fitness exceeds the previous best, update the individual’s optimal position to the current one. Additionally, compare the fitness of the best-performing particle in the current iteration to the global optimal fitness. If it is better, adjust the global optimal position accordingly.

(8)

The optimal position in particle swarm optimization is further optimized through chaos. The entire particle swarm’s searched optimal position is mapped to the logistic equation and undergoes M iterations to produce a chaotic sequence. This chaotic sequence is then remapped back into the original solution space, and the fitness of feasible solutions stemming from these chaotic variables is evaluated to determine the most optimal chaotic solution. The optimal position particle in the generated chaotic sequence is randomly replaced by a particle in the current particle swarm.

(9)

Assess whether the stopping criteria have been fulfilled, specifically whether the prescribed maximum iteration count has been attained. If these conditions are met, the optimization process concludes, yielding the initial weights for the BPNN that have been refined through the CPSO algorithm, denoted as wi_Init and wo_Init. If not, the iterative refinement process persists.

(10)

During the training process of BPNN, specific activation functions are employed at different network layers. For the transition from the input layer to the hidden layer, the sigmoid function is utilized, enabling nonlinear transformations. To ensure non-negative outputs, a modified version of the sigmoid function, designed to produce only non-negative values, is used for the connection between the hidden layer and the output layer. The input-output relationship for each layer can be described as follows [24]:

The input to the jth neuron in the input layer is denoted as xj, and its corresponding output is presented in Equation (12).

$$O_j^{(1)}=xj,j=1,2,3,4$$

(12)

The input of the ith neuron in the hidden layer is shown in Equation (13):

$$ne{t}_i^{(2)}(k)={\displaystyle \sum _{j=1}^4{w}_{ij}^{(2)}}{O}_j^{(1)},i=1,2,3,4$$

(13)

where $w_{ij}^{(2)}$ represents the weight connecting the ith neuron in the hidden layer and the jth neuron in the input layer.

Equation (14) illustrates the output of the jth neuron in the hidden layer.

$$O_i^{(2)}=f(ne{t}_i^2(k))$$

(14)

in which f () serves as the activation function, as shown in Equation (15):

$$f(x)={\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}}}$$

(15)

The incoming signal to the lth neuron in the output layer is shown in Formula (16):

$$ne{t}_l^{(3)}(k)={\displaystyle \sum _{j=1}^5{w}_{lj}^{(3)}}{O}_j^{(2)},i=1,2,3$$

(16)

where $w_{lj}^{(3)}$ represents the connection weight linking the initial neuron of the output layer with the ith neuron in the hidden layer.

The output from the first neuron in the output layer is shown in Equation (17):

$$O_l^{(3)}=g(ne{t}_l^{(3)}(k))$$

(17)

where $O_1^{(3)}=k_p$, $O_2^{(3)}=k_i$, and $O_3^{(3)}=k_d$. Since $k_p$, $k_i$, and $k_d$ are positive values, a non-negative activation function is employed, and its formulation is presented in Equation (18):

$$g(x)={\displaystyle \frac{e^x}{e^x+e^{-x}}}$$

(18)

(11)

The error backpropagation process starts at the output layer. It systematically calculates the output error for each neuron layer and then fine-tunes the weights according to these errors. This process utilizes the gradient descent method. Once specific criteria are fulfilled, the BPNN’s self-learning journey concludes, and the optimal control parameters $k_p$, $k_i$, and $k_d$ are ascertained. If these criteria remain unmet, the optimization persists.

(12)

PID control. Calculate u(k) and yout(k) according to Equations (19) and (20). The flowchart of the CPSO-BPNN-PID algorithm is shown in Figure 5.

$$u(k)=k_p\cdot x1+k_i\cdot x2+k_d\cdot x3$$

(19)

$$yout(k)=-d2\cdot y1-d3\cdot y2+n2\cdot u(k)$$

(20)

4. Simulation Results and Analysis

4.1. Simulation Result

To assess the efficacy of the enhanced CPSO-BPNN-PID controller, this study focuses on the indoor CO₂ concentration model [28,30]. The mathematical expression of the model is shown in Formula (21):

$$y(k)={\displaystyle \frac{a(k)\cdot y(k-1)}{1+y^2(k-1)}}+u(k-1)+0.2u(k-2)$$

(21)

In Formula (21), a(k) is a time-varying parameter, and its expression is as shown in Formula (22):

$$a(k)=1.4(1-0.9e^{-0.3k})$$

(22)

The BPNN is configured with a 4-5-3 structure, a learning rate of 0.30, and an inertia factor of 0.05. Additionally, the particle swarm optimization is set with a swarm size of 20, a maximum iteration count of 50, a chaotic coefficient of 4, and learning factors c1 and c2 both equal to 1.5. The particle positions are constrained within the range of [−5, 5], and velocities are limited to the range of [−5, 5]. Furthermore, a linear decreasing weight strategy is employed, with Wmax set to 0.9 and Wmin set to 0.4.

The system’s output response curve, depicted in Figure 6, is generated using a unit step input signal.

4.2. Performance Indicators

In the control system, the step response test is often employed for assessing the system’s performance. This is because the step signal represents an instantaneous change in the time domain and contains a wide spectrum of frequencies in the frequency domain. The system needs to respond and adapt quickly to such a rapid change. Generally, if the system performs well under this signal, it can be expected to exhibit good performance when handling other types of signals as well. Typically, the merits of a control algorithm are assessed primarily based on the stability, accuracy, and responsiveness of the system when the algorithm is in effect. Therefore, this paper adopts the following performance indicators:

(1)

Peak $y_{\max }$ arefers to the value at the highest point of the system response curve.

(2)

Peak time $t_p$ refers to the time from the initial state of the system response time to the maximum value for the first time.

(3)

Adjustment time $t_s$ refers to the time required for system response time from initial state to steady state.

(4)

Maximum overshoot $\sigma \%={\displaystyle \frac{y_{\max }-y_{\infty }}{y\infty }}\times 100\%$, where $y_{\infty }$ is the final value.

Based on the step response curve depicted in Figure 6, the calculation of various indices is presented in

Table 1. Performance indicators.

Controller Name	${\mathit{y}}_{\mathbf{max}}$	${\mathit{t}}_{\mathit{p}}$	${\mathit{t}}_{\mathit{s}}$	$\mathit{\sigma }\%$
PID	1.13	0.08	0.21	13
BPNN-PID	1	0.37	0.21	0
CPSO-BPNN-PID	1	0.2	0.125	0

.

4.3. Analysis of Results

Based on the comprehensive analysis of Figure 6 and Table 1, it can be concluded that under traditional PID control, the peak value of the system response curve reaches 1.13, exceeding the given set point of 1. However, while the peak time is relatively short, the system takes a considerable amount of time to reach a steady state, with an adjustment time of 0.21 s; there is a noticeable overshoot of 13%, indicating that the control effect of the traditional PID controller is less than ideal. Under the action of the BPNN-PID controller, although the system response curve exhibits no overshoot, the adjustment time is slightly longer than desired; this suggests room for further improvement in the control effect. In contrast, the stable output value of the system response curve—which refers to the steady-state output value after any transients have settled—under the improved CPSO-BPNN-PID controller closely matches the set point. The system demonstrates a rapid response, short adjustment time, and complete elimination of overshoot. It attains a steady state in just 0.125 s, showcasing an excellent control effect. By comparing the step response curves and analyzing key performance indicators, it becomes evident that the improved CPSO-BPNN-PID controller possesses distinct advantages; its performance metrics surpass those of both the traditional PID and BPNN-PID controllers, resulting in a superior control outcome.

The particle fitness curve is displayed in Figure 7.

Additionally, Figure 8 illustrates the adaptive tuning curves for the K_p, K_i, and K_d parameters of the BPNN-PID controller, while Figure 9 depicts the adaptive tuning curves for the K_p, K_i, and K_d parameters of the CPSO-BPNN-PID controller.

One can observe from Figure 7 that as the number of iterations increases, the fitness of the particles declines steeply, indicating a fast convergence of the CPSO algorithm. After approximately eight iterations, the fitness value stabilizes at approximately 0.0142. At this point, the parameters $K_p$, $K_i$, and $K_d$ of the improved CPSO-BPNN-PID controller, along with its performance index, stabilize and continue to approach their optimal values. Figure 9 shows that the optimal parameters for the improved CPSO-BPNN-PID controller are $K_p$ = 0.2, $K_i$ = 0.03, and $K_d$ = 0.05. In contrast, Figure 8 presents the BPNN-PID controller parameters as $K_p$ = 0.092, $K_i$ = 0.058, and $K_d$ = 0.0258, indicating a potential for further optimization of these parameters. Based on a comprehensive analysis incorporating Figure 6 and Table 1, it is concluded that the CPSO algorithm effectively performs global optimization of the initial weights of the BPNN, thereby enhancing the performance of the BPNN-PID controller. Additionally, it is verified that the improved CPSO-BPNN-PID controller demonstrates a superior control effect when dealing with complex nonlinear time-delay systems.

5. Conclusions

In this paper, an enhanced CPSO-BPNN-PID controller is proposed, which integrates the global search capability of the CPSO algorithm, the self-learning ability of BPNN, and the stability and practicality of the PID algorithm. This controller incorporates chaos theory into the iterative process of the PSO algorithm. Specifically, the Logistic map is utilized to generate chaotic sequences and map them onto the original solution space, identifying feasible solutions within the chaotic sequences. The best feasible solution is then selected based on its fitness to randomly replace a particle in the current swarm. Through continuous iteration, the optimal initial weights connecting the input layer to the hidden layer and from the hidden layer to the output layer of the BPNN are obtained. This approach eliminates the need for random initial weights, addressing the limitations of BPNN, namely its slow convergence and susceptibility to local optima.

The study focuses on controlling indoor CO₂ concentration. By comparing the simulation results and performance indicators among the PID controller, the BPNN-PID controller, and the proposed CPSO-BPNN-PID controller, it reveals that the proposed controller exhibits significant advantages, with a fast response, short adjustment time, and no overshoot. Although the proposed controller has achieved relatively positive results in multiple aspects, it must be acknowledged that the control algorithm still has certain limitations in optimizing peak time. This is because the control algorithm focuses more on overall optimization and higher accuracy. Therefore, how to effectively shorten the peak time while maintaining these advantages will be the focus of future research. Furthermore, in recent years, the application of PSO variant algorithms, such as fractional PSO, fractional comprehensive learning PSO, and so on, has become a new trend. These algorithms have shown potential in optimizing performance and improving computational efficiency. In future research, further exploration will be conducted on the combination of these algorithms and neural networks to further expand and optimize control strategies.