A Novel Chaotic Particle Swarm-Optimized Backpropagation Neural Network PID Controller for Indoor Carbon Dioxide Control
Abstract
:1. Introduction
- (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
3. Research Methodology
3.1. Proportional-Integral-Derivative Controller
3.2. Backpropagation Neural Network-Based PID Controller
- (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 and online, and realize self-tuning of PID control parameters , , and . Specifically, is the weight between the ith neuron in the hidden layer and the jth neuron in the input layer, and 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
- (1)
- Calculate the positions and velocities of every particle in the swarm.
- (2)
- Evaluate the fitness of each particle and determine the best position: . 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.
- (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
- (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.
- (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.
- (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]:
- (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 , , and 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.
4. Simulation Results and Analysis
4.1. Simulation Result
4.2. Performance Indicators
- (1)
- Peak arefers to the value at the highest point of the system response curve.
- (2)
- Peak time refers to the time from the initial state of the system response time to the maximum value for the first time.
- (3)
- Adjustment time refers to the time required for system response time from initial state to steady state.
- (4)
- Maximum overshoot , where is the final value.
4.3. Analysis of Results
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Controller Name | ||||
---|---|---|---|---|
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 |
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Zhang, S.; Li, H.; Chang, Y. A Novel Chaotic Particle Swarm-Optimized Backpropagation Neural Network PID Controller for Indoor Carbon Dioxide Control. Processes 2024, 12, 1785. https://doi.org/10.3390/pr12091785
Zhang S, Li H, Chang Y. A Novel Chaotic Particle Swarm-Optimized Backpropagation Neural Network PID Controller for Indoor Carbon Dioxide Control. Processes. 2024; 12(9):1785. https://doi.org/10.3390/pr12091785
Chicago/Turabian StyleZhang, Suli, Hui Li, and Yiting Chang. 2024. "A Novel Chaotic Particle Swarm-Optimized Backpropagation Neural Network PID Controller for Indoor Carbon Dioxide Control" Processes 12, no. 9: 1785. https://doi.org/10.3390/pr12091785