# A PSO-Based Recurrent Closed-Loop Optimization Method for Multiple Controller Single-Output Thermal Engineering Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- A controller parameter optimization approach based on closed-loop identification and simulation is proposed for thermal engineering processes. It takes advantage of field data in history database to identify precise dynamic model for optimizing practicable controller parameters.
- (2)
- A recurrent optimization working flow is proposed for tuning multiple controllers and single output system. Comparing to the simultaneous optimization on all controller parameters, it can better avoid unreasonable or infeasible optimization solution due to couplings among multiple controllers in a single-output system.
- (3)
- Considering multiple disturbances in thermal engineering system, the intelligence computing is considered for the optimization on controller parameter rather than classical tuning methods. The IC-based optimization can find improved practicable controller parameters through operating simulation model with field data inputs.
- (4)
- Due to efficient optimization ability on multi-dimension real-number problems, the canonical PSO algorithm is adopted for PID parameter optimizing in the approach.
- (5)
- For an integrated consideration of control performance and control energy cost, the multi-objective fitness function is adopted in PSO.

## 2. Methodology

#### 2.1. Control Loop Simulation

#### 2.2. Closed-Loop Identification on Controlled Plant

**Principle**

**1.**

**Principle**

**2.**

- (1)
- The controlled process has long time delay or large inertia
- (2)
- There are disturbances in the feedback channel and the inputs and outputs of a plant are measurable.
- (3)
- The data is sampled from setpoint perturbation processes.

**Principle**

**3.**

_{t}denotes the output at time instant t which is the linear combination of the previous output sequence y

_{t-i}with coefficient a

_{i}and the sum of n

_{u}- input linear combination. The pth input linear combination is the sum of inputs u

_{p}

_{,t-j-dp}, with coefficient b

_{p,j}, where u

_{p}

_{,t-j-dp}is the value of the pth input variable at time instant (t-j-d

_{p}), m

_{p}is the inertia order of the pth input variable, d

_{p}is the time delay of the pth input variable and n

_{u}is the number of input variables. n is the order of output variable, and m

_{p}is often taken the same as n in identification. The format of Model (1) implicates the long time delay denoted by d

_{p}or large inertia denoted by m

_{p}which are required by closed-loop identifiability as stated in Principle 2. When identifying model (1), the structural parameters of order and time delay need to be determined first and then the coefficients.

#### 2.2.1. Auto-Selection of Model Order and Time Delay

_{p}corresponding to the maximum of R

_{p}indicates the pure time delay from the pth input to the output, denoted as d

_{p,s}.

_{u}) input time series should be right shifted by d

_{p,s}-sampling points according to its output sequence. Second, a sufficient high-order MISO ARX model, e.g., 15th-order, need to be identified on the shifted input and output time series. Then the parameter estimation method like the recursive least squares (RLS) is adopted to estimate the coefficients of the sufficient high-order MISO ARX model for determining its proper reduced order. Third, using the identity impulse signal to excite the identified sufficient high-order MISO ARX model, one can get the impulse response sequence and then construct the following Hankel matrix.

_{i}(k) denotes the impulse response value from the ith input to the output at time instant k, l belongs to integer and is greater than the model order; letting k = 0, one makes the singular value decomposition on H(0) and has $H(0)={U}_{0}\Lambda {V}_{0}^{T}$,where matrix ${U}_{0}\in {R}^{l\times l}$ and ${V}_{0}\in {R}^{({n}_{u}\times l)\times ({n}_{u}\times l)}$ are orthogonal, $\Lambda =\left(\begin{array}{cccc}{\lambda}_{1}& & & \\ & \ddots & & \\ & & {\lambda}_{r}& \\ & & & 0\end{array}\right)\in {R}^{l\times (l\times {n}_{u})}$ has the singular values ${\lambda}_{i}>0,i=1,\cdots r$ in descending order as its diagonal entries and the rank of H(0) is r(≤l). The index of the sharply decreasing place in its diagonal value sequence can be taken as the estimation value of actual model order.

#### 2.2.2. Model Parameter Estimation

_{t}the actual measurement value of output at time instant t; the observed value vector of inputs and output is denoted as ${\varphi}_{t}=[-{y}_{t-1},-{y}_{t-2},\dots ,-{y}_{t-n},{u}_{1,t-1},{u}_{1,t-2},\dots ,{u}_{1,t-n},\dots ,{u}_{p,t-1},{u}_{p,t-2},\dots ,{u}_{p,t-n}],t=1,2,\dots ,S$, S denotes the number of samples; The parameter vector, $\mathit{\theta}={[{a}_{1},{a}_{2},\dots ,{a}_{n},{b}_{1,1},{b}_{1,2},\dots ,{b}_{1,n},\dots ,{b}_{p,1},{b}_{p,2},\dots ,{b}_{p,n}]}^{T}$ has its estimation of the previous moment denoted as ${\widehat{\theta}}_{t-1}$. Its initial value is set as ${\widehat{\theta}}_{0}={10}^{-6}{1}_{n\times (p+1)}$; the initial value of the estimated error covariance matrix P

_{t}is set as P

_{0}=10

^{6}I

_{n × (p+1)}; K

_{t}denotes the gain matrix. Formula (5) updates the estimations on K

_{t}, $\theta $ and P

_{t}at every sampling time with observed value vector ${\varphi}_{t}$ under sliding window.

#### 2.3. PSO for Controller-Parameter

#### 2.3.1. Canonical PSO

**x**

_{j}(t), j = 1, d, and d denotes the number of the parameters to be optimized. Based on stochastic and multipoint search, each particle has its own best individual position which takes the one of the minimal fitness value among the individual search records, and the best position of the swarm is the one of the minimum fitness value in the swarm′s search history. In the search process of a particle swarm, thee particles are distributed randomly in the search space initially and then each particle is oriented by its velocity. The initial velocity is random and then updated iteratively by Equation (6). Each particle position is change by Equation (7).

_{1}, c

_{2}are called the learning factor, rand (0,1) denotes random real numbers in (0,1), p

_{j}(t) is the jth particle′s best individual position after t times of iteration, and p

_{g}(t) is the swarm’s best position after t times of iteration [18]. Equation (6) is composed of an original velocity term and an individual best term and a global best term. Figure 2 shows that the velocity is updated using the total vector of the three terms in Equation (6) and then change its position. V

_{jBest}and V

_{gBest}in Figure 2 denote the second and third terms in Equation (6), respectively.

#### 2.3.2. Multi-Objective PSO for PID

_{p}is the proportional gain, k

_{i}is the integral gain, k

_{d}is the differential gain and k

_{a}is the filter gain, all in positive value. Then in PSO for optimizing these gains, the jth particle position vector after t times of iteration can be denoted by

_{p}is the change of proportional gain, Δk

_{i}is the change of integral gain, Δk

_{d}is the change of differential gain, and Δk

_{a}is the change of filter gain.

_{i,j}is the jth sampled value of the ith control variable, C

_{u}is the number of controllers, w

_{1}, w

_{2}, w

_{3}are the weights representing control performance preference. Moreover, since there are a lot of combinations of the proportional gain, integral gain, differential gain and the filter gain which can achieve best control performance, the mere optimization on fitness function J

_{1}cannot guarantee PID parameters to converge quickly, thus another objective function for optimizing the decision vector x

_{j}(t) to approach some optimal value is given by.

_{4}, w

_{5}, w

_{6}are the weights for suppressing the deviation from the optimal value $\left[\begin{array}{ccc}{k}_{p}^{*}& {k}_{i}^{*}& {\left({k}_{d}/{k}_{a}\right)}^{*}\end{array}\right]$ of the controller parameters. Big value of these parameters easily leads to fluctuating regulation and costs more energy on actuator actions or being very sensitive to noise, thus small values of $\begin{array}{ccc}{k}_{p}^{*},& {k}_{i}^{*},& {\left({k}_{d}/{k}_{a}\right)}^{*}\end{array}$ are preferred in (12) when energy cost, fluctuations of regulation process and actuator actuators and noise sensitivity are considered.

_{1}and J

_{2}may be in conflict with each other when performing the optimization, it is proposed to minimize the maximum deviation of the two objective functions to their optima, instead of directly minimizing the sum of weighted objective function J

_{1}and J

_{2}. The multi-objective fitness function is given by

_{1}and β

_{2}are the preference coefficients on objective J

_{1}and J

_{2}, respectively.

#### 2.4. Iterative-Tuning Monitor

- A closed-loop control simulation model for the thermal engineering process to be optimized should be built first. The utilization of actually applied control algorithm, the sampling data of input variables imported from historical database at big data platform and the accurate dynamic model of the controlled process ensures a high-approximation simulation from the model to the actual process. In the simulation, manipulated variables and controlled variables are produced by the simulation control algorithm and model.
- MISO ARX models for describing the controlled-process dynamics are identified on historical data. The chosen identification data segment includes the samples of manipulated variable, input disturbances and controlled variable. It should keep the identifiable conditions which have been discussed in Section 2.2. By applying the identification algorithms described in Section 2.2.1 and Section 2.2.2, the MISO ARX model of the controlled process can be obtained and imported into the simulation loop. Then an approximate simulation loop of the process is set up.
- The PSO-RCO method optimizes the controllers of a MISO thermal process one by one in several optimization circulations to approach the optimal controller parameters. When the ith (I = 1, 2, n)controller is selected for optimization, the controller parameters like the proportional gain, integral time and derivative gain, are coded into the particles of the PSO algorithm with the initial value of the previous value plusing a random bias. After running the simulation model for enough steps, each particle’s fitness which represents the performance of corresponding controller is evaluated on the fitness function. After enough generations of particle update, the particle swarm converges to the optimal fitness and controller parameters. The optimization result is the new parameters of the ith controller in place of the old ones.
- If the number of controllers in control loop, denoted by n, is larger than 1 and the order of the current optimized controller i is smaller than n, the next controller to be set is the (i + 1)th and let i = i + 1. Then the operations in (3) run again for optimizing the (i + 1)th controller parameters.
- When i reaches n, one round of optimization on multiple controller parameters for the single-output thermal process is done. If the rounds of recurrent parameter-tuning is less than the set number, the next round of optimization is initiated again, starting with the 1st controller and then proceeds to the nth through optimizing process (3). When the rounds of optimization is reached, the whole controller-parameter optimization is finished and the optimized controller parameters are set into the simulation model for verifications. After comparing the optimized control performance with the pre-optimized, the improved controller parameters can be applied to field control system or be a guidance for controller parameter tuning.

## 3. Verification

#### 3.1. Model Identification

_{1}denotes the steam temperature at the A-side desuperheater outlet, u

_{1,1}denotes the power, u

_{1,2}denotes the opening of A-side desuperheater valve, and u

_{1,3}denotes the steam temperature at the A-side desuperheater inlet.

_{2}denotes the steam temperature at the B-side desuperheater outlet, u

_{2,1}denotes the power, u

_{2,2}denotes the opening of B-side desuperheater valve, and u

_{2,3}denotes the steam temperature at the B-side desuperheater inlet.

_{3}denotes the superheated steam temperature at the outlet, u

_{3,1}denotes the steam temperature at the A-side desuperheater outlet, u

_{3,2}denotes the power, and u

_{3,3}denotes the steam temperature at the A-side desuperheater outlet. Since the models for A-side and B-side are in series with the model for lag sector, u

_{3,1}is equal to y

_{1}and u

_{3,1}to y

_{2}.

#### 3.2. Optimization on Controller Parameters

_{p}is the proportional gain, k

_{i}is the integral gain, k

_{d}is the differential gain, k

_{a}is the filter gain and k

_{1}is the total gain. The PSO algorithm is taken to optimize k

_{p}, k

_{i}, k

_{d}, k

_{a}for the A-side and B-side desuperheater controllers, k

_{1}is set as 0.2, and the parameters in objective function (11) and (12) are set as L = 800, w

_{1}= 3, w

_{2}= 0.02, w

_{3}= 0.001, w

_{4}= 50, w

_{5}= 60, w

_{6}= 70.

_{1}= β

_{2}= 1 in fitness function (13), and the initial value of the jth particle position vector is generated randomly by

_{1}= 1, β

_{2}= 2 in fitness function (13), and the initial value of the jth particle position vector is generated randomly around the optimization result of the previous round, i.e.,

#### 3.3. Comparisons with Other Methods

_{u}= 2. Through many trials with different weights of the optimization, it seems that some parameters often reach their limits before convergence, like letting w

_{1}= 3, w

_{2,1}= 0.05, w

_{2,2}= 0.02, w

_{3,1}= 0.001, w

_{3,2}= 0.001 and having the optimization result including boundary values listed in Table 3. The boundary gains of controller B lead to dramatical movements of desuperheater valve B as shown in Figure 14 and Figure 15 shows that the action of desuperheater valve A is much violent than that from the recurrent multi-objective optimization as shown in Figure 10, due to its higher proportional gain.

_{1}= 3, w

_{2}= 0.02, w

_{3}= 0.001, k

_{1}= 0.2. Through 2 rounds of the working flow in Figure 3 and 15-generation updates for each round, the best one group of optimized controller parameters among several results was obtained and listed in Table 4. Comparing with the results from the multi-objective PSO in Table 1 and Table 2, the proportional gains of both sides of controllers are of higher value. Therefore, the controller outputs are more violent as shown in Figure 16 and Figure 17 compared to Figure 10 and Figure 11 resulted from the proposed method. The varying of fitness values of function (11) for the desuperheater controllers A and B are shown in Figure 18 and Figure 19, respectively. In the figures, the circle mark with letter “a” denotes the endpoint of the 1st round of parameter optimization and “b” denotes the startpoint of the 2nd round of optimization for each controller. Figure 19 shows that the fitness value of optimizing desuperheater controller B cannot converge within predetermined 15 generations.

_{2}in optimization. Actually, there are many groups of control parameter combinations in the searching space resulting in acceptable J

_{1}(possibly not the minimum) in term of practical application. But the higher controller gains represent more energy cost by actuators and more fluctuations in regulating process.

## 4. Discussions

**Remark**

**1.**

**Remark**

**2.**

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Abbreviation | Terms |

PSO-RCO | PSO-based recurrent closed-loop optimization method |

PID | Proportional-Integral-Derivative |

IC | intelligent computing |

PSO | particle swarm optimization |

MISO | multiple-input-and-single-output |

RLS | recursive least squares |

ARX | AutoRegressive eXogenous |

SVD | singular value decomposition |

RMSE | Root-mean-square error |

## Nomenclature

Symbols | Meanings |

y_{t} | output variable value at time instant t |

a_{i} | coefficient of the output variable y_{t-i} |

n | order of output variable |

u_{p,t} | value of the pth input variable at time instant t |

d_{p} | time delay of the pth input variable |

b_{p,j} | coefficient of the input variable u_{p}_{,t-j-dp} |

m_{p} | order of the pth input variable |

n_{u} | number of input variables |

S | number of samples for RLS |

K_{p} | maximum of time delay from the pth input to the output |

d_{p,s} | estimated pure time delay from the pth input to the output |

h_{i}(k) | impulse response from the ith input to the output at time instant k |

l | a given integer number |

${R}^{l\times l}$ | l × l real matrix space |

$H(k)$ | Hankel matrix of time instant k |

${U}_{0},{V}_{0}$ | orthogonal matrices |

${\lambda}_{i}$ | the ith diagonal entry of matrix $\Lambda \in {R}^{l\times (l\times {n}_{u})}$ |

r | rank of Hankel matrix |

${\varphi}_{t}$ | observed value vector of inputs and output at time instant t |

$\mathit{\theta}$ | parameter vector |

${\widehat{\theta}}_{t-1}$ | estimation on $\mathit{\theta}$ at time instant t − 1 |

P_{t} | error covariance matrix of RLS |

K_{t} | gain matrix of RLS |

${W}_{PID}(s)$ | transfer function of PID controller |

k_{p} | proportional gain of transfer function |

k_{i} | integral gain of transfer function |

k_{d} | differential gain of transfer function |

k_{a} | filter gain of transfer function |

${x}_{j}(t)$ | the ith particle position vector after t times of iteration |

N | number of the particles in a swarm |

G | generations of PSO |

${v}_{j}(t)$ | the jth particle speed vector after t times of iteration |

Δk_{p} | the change of proportional gain of transfer function |

Δk_{i} | the change of integral gain of transfer function |

Δk_{d} | the change of differential gain of transfer function |

Δk_{a} | the change of filter gain of transfer function |

J | fitness function of PSO |

L | number of simulation epochs |

r | setpoint value |

C_{u} | number of controllers |

w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{6} | weights representing control performance preference |

${G}_{i,j}(z)$ | Z-transfer function from the jth input to the ith output |

s | Laplace transformation operator |

z | Z transformation operator |

## References

- Xu, S.; Hashimoto, S.; Jiang, W. Pole-Zero cancellation method for multi input multi output (MIMO) temperature control in heating process system. Processes
**2019**, 7, 497. [Google Scholar] [CrossRef] - Åström, K.J.; Hägglund, T. Advanced PID Control; ISA-The Instrumentation, Systems and Automation Society: Research Triangle Park, NC, USA, 2006. [Google Scholar]
- Nisi, K.; Nagaraj, B.; Agalya, A. Tuning of a PID controller using evolutionary multi objective optimization methodologies and application to the pulp and paper industry. Int. J. Mach. Learn. Cybern.
**2019**, 10, 2015–2025. [Google Scholar] [CrossRef] - Gani, M.M.; Islam, M.S.; Ullah, M.A. Optimal PID tuning for controlling the temperature of electric furnace by genetic algorithm. SN Appl. Sci.
**2019**, 1, 880. [Google Scholar] [CrossRef] - Boukhalfa, G.; Belkacem, S.; Chikhi, A.; Benaggoune, S. Genetic algorithm and particle swarm optimization tuned fuzzy PID controller on direct torque control of dual star induction motor. J. Cent. South Univ.
**2019**, 26, 1886–1896. [Google Scholar] [CrossRef] - Lapa, K.; Cpalka, K.; Przybyl, A. Genetic programming algorithm for designing of control systems. Inf. Technol. Contyrol.
**2018**, 47, 668–683. [Google Scholar] - Fister, D.; Fister, I.; Fister, I.; Safaric, R. Parameter tuning of PID controller with reactive nature-inspired algorithms. Robot. Auton. Syst.
**2016**, 84, 64–75. [Google Scholar] [CrossRef] - Kiam, A.H.; Chong, G.; Yun, L. PID control system analysis, design, and technology. IEEE Trans. Control Syst. Technol.
**2005**, 13, 559–576. [Google Scholar] [CrossRef] - Dimeo, R.; Lee, K.Y. Boiler-Turbine Control System Design Using a Genetic Algorithm. IEEE Trans. Energy Conv.
**1995**, 10, 752–759. [Google Scholar] [CrossRef] - Sun, L.; Hua, Q.S.; Shen, J.; Xue, Y.L.; Li, D.H.; Lee, K.Y. Multi-objective optimization for advanced superheater steam temperature control in a 300 MW power plant. Appl. Energy
**2017**, 208, 592–606. [Google Scholar] [CrossRef] - Liang, G.; Li, W.; Li, Z.J. Control of superheated steam temperature in large-capacity generation units based on active disturbance rejection method and distributed control system. Control Eng. Prac.
**2013**, 21, 268–285. [Google Scholar] [CrossRef] - Zhang, J.H.; Zhou, S.Q.; Ren, M.F.; Yue, H. Adaptive neural network cascade control system with entropy-based design. IET Control Theor. Appl.
**2016**, 10, 1151–1160. [Google Scholar] [CrossRef] - Moelbak, T. Advanced control of superheater steam temperatures—An evaluation based on practical applications. Control Eng. Prac.
**1999**, 7, 1–10. [Google Scholar] [CrossRef] - Zhang, X.T.; Ni, W.D.; Li, Z.; Zhang, S. Identifiability of building thermal system models using on-line data. J. Tsinghua Univ. Sci. Tech.
**2004**, 44, 1544–1547. [Google Scholar] - Rad, A.B.; Lo, W.L.; Tsang, K.M. Simultaneous online identification of rational dynamics and time delay: A correlation based approach. IEEE Trans. Control Syst. Technol.
**2003**, 11, 957–959. [Google Scholar] [CrossRef] - Yin, H.H.; Zhu, Z.F.; Ding, F. Model order determination using the Hankel matrix of impulse responses. Appl. Math. Lett.
**2011**, 24, 797–802. [Google Scholar] [CrossRef][Green Version] - Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the 1995 IEEE International Conference on Neural Networks (ICNN 95), Perth, Australia, 27 November–1 December 1995; Volumes 1–6, pp. 1942–1948. [Google Scholar]
- Taherkhani, M.; Safabakhsh, R. A novel stability-based adaptive inertia weight for particle swarm optimization. Appl. Soft Comput.
**2016**, 38, 281–295. [Google Scholar] [CrossRef] - Fu, H.; Pan, L.; Xue, Y.L.; Sun, L.; Zheng, S. Cascaded PI Controller Tuning for Power Plant Superheated Steam Temperature based on Multi-Objective Optimization. IFAC-PapersOnLine
**2017**, 50, 3227–3231. [Google Scholar] [CrossRef] - Zhu, J.L.; Ge, Z.Q.; Song, Z.H.; Gao, F.R. Review and big data perspectives on robust data mining approaches for industrial process modeling with outliers and missing data. Ann. Rev. Control
**2018**, 46, 107–133. [Google Scholar] [CrossRef] - Sengupta, S.; Basak, S.; Peters, R.A., II. Particle Swarm Optimization: A Survey of Historical and Recent Developments with Hybridization Perspectives. Mach. Learn. Knowl. Extr.
**2019**, 1, 157–191. [Google Scholar] [CrossRef]

**Figure 1.**Particle Swarm Optimization (PSO)-based recurrent closed-loop parameter-optimization method.

**Figure 3.**The working flow of PSO-based recurrent closed-loop optimization method (PSO-RCO) algorithm.

Times | k_{p} | k_{i} | k_{d} | k_{a} |
---|---|---|---|---|

1 | 10.065 | 1.436 | 0.839 | 15.097 |

2 | 10.072 | 1.456 | 0.839 | 15.106 |

3 | 10.088 | 1.444 | 0.841 | 15.132 |

4 | 10.037 | 1.480 | 0.836 | 15.055 |

5 | 10.048 | 1.457 | 0.837 | 15.072 |

Mean | 10.062 | 1.4546 | 0.8384 | 15.0924 |

Standard deviation | 0.02 | 0.0167 | 0.0019 | 0.03 |

Times | k_{p} | k_{i} | k_{d} | k_{a} |
---|---|---|---|---|

1 | 1.296 | 2.359 | 0.108 | 1.944 |

2 | 0.413 | 5.860 | 0.034 | 0.620 |

3 | 0.392 | 6.965 | 0.033 | 0.588 |

4 | 0.567 | 5.447 | 0.047 | 0.850 |

5 | 1.178 | 2.447 | 0.098 | 1.766 |

Mean | 0.7692 | 4.6156 | 0.0640 | 1.1536 |

Standard deviation | 0.4344 | 2.0949 | 0.0362 | 0.6513 |

Controller | k_{1} | k_{p} | k_{i} | k_{d} | k_{a} |
---|---|---|---|---|---|

A | 0.2 | 17.2029 | 0 | 7.7268 | 49.6785 |

B | 0.2 | 30 | 20 | 6.1738 | 37.3926 |

Controller | k_{1} | k_{p} | k_{i} | k_{d} | k_{a} |
---|---|---|---|---|---|

A | 0.2 | 12.6868 | 1.057 | 1.9453 | 12.4708 |

B | 0.2 | 13.2124 | 2.1215 | 2.7846 | 15.1559 |

Methods | Non-Recurrent Single-Objective PSO | Recurrent Single-Objective PSO | Recurrent Multi-Objective PSO |
---|---|---|---|

Control RMSE (°C) | 1.5964 | 0.7989 | 1.06 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, X.; Pan, L.
A PSO-Based Recurrent Closed-Loop Optimization Method for Multiple Controller Single-Output Thermal Engineering Systems. *Processes* **2019**, *7*, 784.
https://doi.org/10.3390/pr7110784

**AMA Style**

Liu X, Pan L.
A PSO-Based Recurrent Closed-Loop Optimization Method for Multiple Controller Single-Output Thermal Engineering Systems. *Processes*. 2019; 7(11):784.
https://doi.org/10.3390/pr7110784

**Chicago/Turabian Style**

Liu, Xingjian, and Lei Pan.
2019. "A PSO-Based Recurrent Closed-Loop Optimization Method for Multiple Controller Single-Output Thermal Engineering Systems" *Processes* 7, no. 11: 784.
https://doi.org/10.3390/pr7110784