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Multi-objective fractional-order particle swarm optimization algorithms for data processing of multi-wavelength pyrometer

Mei Liang, Yongsheng Wang, and Changhui Wang

Open Access Open Access

Abstract

This paper focuses the algorithm of the true temperature inversion for high-temperature targets with unknown emissivity by transforming multi-spectral true temperature inversion into multi-objective minimum optimization. Two improved fractional-order particle swarm optimizations (IFOPSO), high-order nonlinear time-varying inertia weight (Hntiw) IFOPSO and global-local best values (Glbest) IFOPSO, are proposed to obtain the true temperature by solving the multi-objective minimum optimization. Combining the inherent advantages of fractional-order calculus to jump out of the local extreme value, the Hntiw IFOPSO algorithm is proposed by replacing the linear time-varying inertia weights with nonlinear functions related to the total number of iterations and the current number of iterations. The Glbest IFOPSO algorithm is designed by using the global local optimal inertia weight and acceleration constant to update the particle velocity and position values, which improves the multi-objective optimization ability and the accuracy of the true temperature inversion. The effectiveness of the proposed methods is verified by the simulation with typical spectral emissivity models and the measured data from rocket tail flame.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Multi-spectral thermometry is a measurement method to obtain the true temperature of the targets through the inversion of radiation data from the multi-spectral channels [1–5]. As the powerful technology to measure the high-temperature targets with the advantages of fast response, wide application range, strong anti-interference and non-contact, multi-spectral radiation temperature measurement technology is widely used in military equipment research and development, steel manufacturing, glass manufacturing, and so on, such as rocket nozzle temperature measurement, midcourse flight trajectory missile surface temperature measurement, and high-temperature molten metal measurement [6–11].

In the actual process of radiation temperature measurement, the spectral emissivity is transient and changeable, which is affected by the time, measuring position and temperature [12,13]. There will be a large error of true temperature inversion if the spectral emissivity model does not conform to the actual spectral emissivity [14]. The secondary measurement method without limiting the functional relationship between emissivity and wavelength is proposed in [15], but the emissivity is required to have an approximately linear relationship with temperature at the selected wavelength. By using the emissivity range and iteration cutoff conditions, an improved secondary measurement method is designed in [16], in which the initial temperature can be any positive number. To obtain the effective search range of emissivity in each iteration, the emissivity range constraint algorithm is proposed by using the relationship between emissivity deviation and true temperature [17]. To realize the effective sample screening, a sample screening method for emissivity is proposed in [18], which greatly reduces the calculation time.

Multi-spectral radiation thermometry is a method to solve the equation of true temperature inversion. In fact, there is a $n$-order equation established by using the measurement voltages from the n spectral channels of the multi-spectral pyrometer. Since the true temperature T and n spectral emissivities are unknown, there are n equations with $n + 1$ unknown parameters, which is difficult to obtain the unique and optimal solution of true temperature [19–21]. In [22], the gradient projection algorithm and the internal penalty function algorithm are applied to the true temperature inversion of pyrometers. In [23], a multi-objective minimum optimization method is established, and the objective function is solved by quadratic programming method to complete the true temperature inversion. Compared with the objective function with least variance, the multi-objective function optimization is more comprehensive, which requires multiple objectives to be satisfied simultaneously. However, the multi-objective optimization solution subject to many objective functions and constraints is difficult to achieve the optimization of all objectives, which is very likely to make other targets the worst and fall into the local optimal solution when one target reaches the optimal in the process of optimization.

Particle swarm optimization (PSO) is a dynamic optimization algorithm to simulate the evolution of individual biological populations over a long period of time, which is suitable for solving multi-objective extremum optimization problems. By introducing the fractional-order calculus, the fractional-order particle swarm optimization (FOPSO) can better grasp the balance between global search and local search, in which the particles can better explore the whole search space [24]. In [25], an improved fractional-order particle swarm optimization (IFOPSO) is proposed to address the shortcomings of traditional PSO with low search accuracy in high-dimensional space and easy to fall into local extreme values. In [26], a fractional-order velocity updating formula is presented in the PSO, in which an adaptive variable related to evolution state is proposed. In [27], a new IFOPSO by setting the inertia weight factor to linear decline is designed to improve the optimization performance.

Based on the above analysis, the inversion problem of the true temperature for the high-temperature target with unknown emissivity is transformed into a multi-objective minimum optimization. By designing the high-order nonlinear inertia weight function, the inertia weight function and acceleration coefficient function based on the local and global optimal values, the high-order nonlinear time-varying inertia weight (Hntiw) IFOPSO and global-local best values (Glbest) IFOPSO are proposed respectively. Compared with the PSO in [28], the accuracy of true temperature inversion is improved by using the proposed algorithms.

2. Algorithm principle

2.1 Multi-spectral radiation thermometry reference temperature model

The measurement principle of the multi-wavelength pyrometer is shown in Fig. 1. The radiation information of the measured target reaches the combined dispersion prism through the pyrometer main objective lens and collimating objective lens in turn, and is dispersed according to different wavelengths to form a spectrum. The spectrum reaches the detector array through the camera obscura objective and reflector, and outputs the current signal through photoelectric conversion. The current passes through a current-voltage converter and amplifier to output the voltage signals. According to the principle that the theoretical true temperature of different spectral channels is equal, the optimal objective function can be designed. Combined with the emissivity range of the measured target, the voltage signal and wavelengths of the spectral channels are brought into the optimal solution algorithm, and the true temperature can be obtained through cyclic iterative calculation.

 figure: Fig. 1.

Fig. 1. Measurement principle of the multi-wavelength pyrometer

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Assuming that there are n channels in the actual temperature measurement, the voltage signal collected by the $i - \textrm{th}$ channel is:

$${V_i} = {A_{{\lambda _i}}} \cdot \varepsilon ({\lambda _i},T) \cdot \frac{1}{{\lambda _i^5({e^{\frac{{{C_2}}}{{{\lambda _i}T}}}} - 1)}},i = 1,2, \cdots ,n$$
where ${A_{{\lambda _i}}}$ is a calibration factor dependent on the wavelength, which is related to the sensitivity of the detector, the geometric structure, the absorption through the window, and the first radiation constant. $\varepsilon ({\lambda _i},T)$ is the target emissivity at true temperature T; ${\lambda _i}$ represents the wavelength of the $i - \textrm{th}$ channel; ${C_2}$ is the second radiation constant.

According to Wien's formula, the Eq. (1) can be simplified as:

$${V_i} = {A_{{\lambda _i}}} \cdot \varepsilon ({\lambda _i},T) \cdot \lambda _i^{ - 5} \cdot {e^{ - \frac{{{C_2}}}{{{\lambda _i}T}}}},i = 1,2, \cdots ,n$$

When the blackbody reference temperature is ${T_R}$, the emissivity of the blackbody satisfies $\varepsilon ({\lambda _i},T) = 1$, and the output voltage signal ${V_R}$ from the $i - \textrm{th}$ channel in Eq. (2) is:

$${V_R} = {A_{{\lambda _i}}} \cdot \lambda _i^{ - 5} \cdot {e^{ - \frac{{{C_2}}}{{{\lambda _i}{T_R}}}}},i = 1,2, \cdots ,n$$

From Eqs. (2) and (3), we can get:

$$\frac{{{V_i}}}{{{V_R}}} = \varepsilon ({\lambda _i},T) \cdot {e^{ - \frac{{{C_2}}}{{{\lambda _i}T}}}} \cdot {e^{\frac{{{C_2}}}{{{\lambda _i}{T_R}}}}}$$

Before the true temperature measurement, the output ${V_R}$ of each channel should be measured under stable conditions of the reference temperature ${T_R}$ in Eq. (4), which ensures the calculation accuracy of the target true temperature ${T_i}$ and emissivity $\varepsilon ({\lambda _i},T)$.

2.2 True Temperature Inversion Optimization Problem

In order to describe the temperature in different spectral channels, calculating the logarithm on both sides of the Eq. (4) obtains

$$\ln (\frac{{{V_i}}}{{{V_R}}}) - \frac{{{C_2}}}{{{\lambda _i}{T_R}}} ={-} \frac{{{C_2}}}{{{\lambda _i}{T_i}}} + \ln \varepsilon ({\lambda _i},{T_i})$$
where $\varepsilon ({\lambda _i},{T_i})$ represents the emissivity with the temperature ${T_i}$ from the $i - \textrm{th}$ channel.

Then, one can get:

$${T_i} = \frac{1}{{\left( {\ln \varepsilon ({\lambda_i},{T_i}) + \frac{{{C_2}}}{{{\lambda_i}{T_R}}} - \ln (\frac{{{V_i}}}{{{V_R}}})} \right) \cdot \frac{{{\lambda _i}}}{{{C_2}}}}}$$

According to Eq. (6), the ideal inversion temperature for each spectral channel should be the same and equal to the true temperature if $\varepsilon ({\lambda _i},{T_i})$ is known. The number of the spectral channels in this paper is $n = 8$, then the objective functions by using the absolute values of the inverse temperature differences from the interval spectral channels can be obtained as

$$\begin{array}{l} {F_1} = |{{T_1} - {T_3}} |\\ {F_2} = |{{T_2} - {T_4}} |\\ {F_3} = |{{T_3} - {T_5}} |\\ {F_4} = |{{T_4} - {T_6}} |\\ {F_5} = |{{T_5} - {T_7}} |\\ {F_6} = |{{T_6} - {T_8}} |\end{array}$$

Using the average value of the absolute value of the temperature difference of the interval spectral channel, another objective function is constructed as

$${F_7} = \frac{1}{6}({|{{T_1} - {T_3}} |+ |{{T_2} - {T_4}} |+ |{{T_3} - {T_5}} |+ |{{T_4} - {T_6}} |+ |{{T_5} - {T_7}} |+ |{{T_6} - {T_8}} |} )$$

The absolute value of the difference between the sum of the true temperature of the middle four channels and the sum of the true temperature of the remaining four channels can also constitute an objective function, that is

$${F_8} = \frac{1}{4}\left|{\sum\limits_{i = 3}^6 {|{{T_i}} |} - \left( {\sum\limits_{i = 1}^2 {|{{T_i}} |} + \sum\limits_{i = 7}^8 {|{{T_i}} |} } \right)} \right|$$

The smaller the absolute deviation, the smaller the dispersion of the mean value of a single measurement, and the higher the reliability of the data measurement. Therefore, an objective function by using the sum of the absolute value deviations through the eight channels can be

$${F_9} = \sqrt {\frac{{\sum\limits_{i = 1}^8 {|{{T_i} - E({{T_i}} )} |} }}{8}}$$
where $E({T_i}) = \frac{1}{n}\sum\limits_{i = 1}^n {{T_i}} $ is the average of all channels.

By minimizing the deviation, (8), (9), and (10) are transformed into constrained optimization problems to approach zero infinitely. Since the range of emissivity is $[0,1]$, i.e. $0 \le \varepsilon ({\lambda _i},{T_i}) \le 1$, the multi-objective constrained optimization by defining $\varepsilon ({\lambda _i},{T_i}) = {x_i}$ is obtained as:

$$\left\{ {\begin{array}{{c}} {\min [{{F_1}({{x_i}} ),{F_2}({{x_i}} ), \cdots ,{F_9}({{x_i}} )} ]}\\ {0 \le {x_i} \le 1,i = 1,2, \cdots ,8.} \end{array}} \right.$$
where (11) a multi-objective constrained optimization problem, which describes the difference of inversion temperature in different spectral channels. In order to reduce this difference and obtain the best inversion temperature, the Eq. (11) can be solved by IFOPSO.

3. Improved fractional-order particle swarm optimization

3.1 Fractional-order particle swarm optimization

The PSO searches for the optimal solution by updating the velocity and position of the particle. Suppose that there is a population with the particles number $M\;(M \in {N^\ast })$, ${x_{ml}} = ({x_{m1}},{x_{m2}}, \cdots ,{x_{me}})$, ${x_{ml}} \in (0,1)$, $l \in [{1,e} ]$, $e \in {N^\ast }$, ${x_{ml}}$ represents the position of the $m\textrm{ - th}$ particle in the vector space. ${v_{ml}} = ({v_{m1}},{v_{m2}}, \cdots ,{v_{me}})$, ${v_{ml}} \in ( - 0.1,0.1)$, $l \in [{1,e} ]$, ${v_{ml}}$ represents the velocity of the $m\textrm{ - th}$ particle. ${g_{ml}} = ({g_{m1}},{g_{m2}}, \cdots ,{g_{me}})$, ${g_{ml}} \in N$, $l \in [{1,e} ]$, ${g_{ml}}$ represents the optimal position of the $m\textrm{ - th}$ particle. ${q_m} = ({q_1},{q_2}, \cdots ,{q_M})$, ${q_m} \in N$, $m \in [{1,M} ]$, ${q_m}$ represents the optimal position of the particle swarm. The formula for updating the velocity and position of particles is presented as follows:

$${v_{ml}}(t + 1) = \omega {v_{ml}}(t) + {c_1}{r_1}({g_{ml}}(t) - {x_{ml}}(t)) + {c_2}{r_2}({q_m}(t) - {x_{ml}}(t))$$
$${x_{ml}}(t + 1) = {x_{ml}}(t) + {v_{ml}}(t + 1)$$
where ${c_1},{c_2} \in [{0.1,2} ]$ are the learning factors. ${r_1},{r_2} \in [0,1]$ are the random numbers. $\omega $ is the inertia weight factor. t represents the number of current iterations. ${g_{ml}}$ is the individual extreme value. ${q_m}$ is the global extreme value.

By using the historical information of variables, fractional-order calculus can better capture the long-term dependence of variables, and has better genetic adaptability to global optimization. The definition of the fractional-order calculus by Grunwald-Letnikov [28]:

$${D^\alpha }[{g(s )} ]= \mathop {\lim }\limits_{h \to 0} \left( {\frac{1}{{{h^\alpha }}}\sum\limits_{k = 0}^{\left[ {\frac{{{a_2} - {a_1}}}{h}} \right]} {\frac{{{{({ - 1} )}^k}\Gamma ({\alpha + 1} )g({s - kh} )}}{{\Gamma ({k + 1} )\Gamma ({\alpha - k + 1} )}}} } \right)$$
where $\alpha $ is the fractional order, and $g(s )$ is a continuous function of s, $s \in [{{a_1},{a_2}} ],{a_1} < {a_2} \in R$. h is the step size, and $\Gamma (n )= \int_0^\infty {{e^{ - t}}{t^{n - 1}}dt}$ is the Gamma function.

Fractional-order differentiation in (14) is an infinite sequence, and the following approximate formula by discretization to facilitate calculation can be obtained as:

$${D^\alpha }[{g(s )} ]= \frac{1}{{{{(P )}^\alpha }}}\sum\limits_{k = 0}^r {\frac{{{{({ - 1} )}^k}\Gamma ({\alpha + 1} )g({s - kP} )}}{{\Gamma ({k + 1} )\Gamma ({\alpha - k + 1} )}}}$$
where P is the sampling period and $r \in {N^\ast }$ is the truncation order.

With the sampling period $P = 1$ from (15), one can obtain:

$$\begin{aligned} {D^\alpha }[{v({t + 1} )} ]&= v({t + 1} )- \alpha v(t )- \frac{1}{2}\alpha ({1 - \alpha } )v({t - 1} )\\ &- \frac{1}{6}\alpha ({1 - \alpha } )({2 - \alpha } )v({t - 2} )\\ &- \frac{1}{{24}}\alpha ({1 - \alpha } )({2 - \alpha } )({3 - \alpha } )v({t - 3} )- \ldots \end{aligned}$$

The difference transformation of (12) yields:

$$\begin{aligned} {v_{ml}}({t + 1} )- {v_{ml}}(t )&= ({\omega - 1} ){v_{ml}}(t )\\ &+ {c_1}{r_1}({{g_{ml}}(t )- {x_{ml}}(t )} )+ {c_2}{r_2}({{q_m}(t )- {x_{ml}}(t )} )\end{aligned}$$
where ${v_{ml}}({t + 1} )- {v_{ml}}(t )$ is the discrete form with $\alpha = 1$, $P = 1$ and $\omega = 1$. Then, the left side of Eq. (17) can be transformed into:
$${D^\alpha }[{{v_{ml}}({t + 1} )} ]= {c_1}{r_1}({{g_{ml}}(t )- {x_{ml}}(t )} )+ {c_2}{r_2}({{q_m}(t )- {x_{ml}}(t )} )$$

By taking the first 4 terms of fractional-order calculus in (16), the updating formula of the FOPSO velocity from (18) can be obtained as:

$$\begin{aligned} {v_{ml}}({t + 1} )&= \alpha {v_{ml}}(t )+ \frac{{\alpha ({1 - \alpha } )}}{2}{v_{ml}}({t - 1} )\\ &+ \frac{{\alpha ({1 - \alpha } )({2 - \alpha } )}}{6}{v_{ml}}({t - 2} )\\ &+ \frac{{\alpha ({1 - \alpha } )({2 - \alpha } )({3 - \alpha } )}}{{24}}{v_{ml}}({t - 3} )\\ &+ {c_1}{r_1}({{g_{ml}}(t )- {x_{ml}}(t )} )+ {c_2}{r_2}({{q_m}(t )- {x_{ml}}(t )} )\end{aligned}$$
where $\alpha $ can be solved by:
$$\alpha = 0.9 - 0.5 \times \frac{t}{{maxgen}}$$
where $maxgen$ represents the total number of iterations.

3.2 High-order nonlinear time-varying inertial weight improved fractional-order particle swarm optimization

In this subsection, the Hntiw IFOPSO algorithm is proposed by introducing high-order term adapting to the complex dynamic environments.

The inertia weight function is designed as:

$$\omega ({t + 1} )= {\omega _{max}} - ({{\omega_{max}} - {\omega_{\min }}} ){\left( {\frac{t}{{maxgen}}} \right)^\psi }$$
where ${\omega _{max}}$ and ${\omega _{\min }}$ are the upper and lower bounds of inertia weights, respectively. $\psi $ represents the linear time-varying parameter of the inertia weight.

The learning factor functions are constructed as:

$${c_1}({t + 1} )= {c_{1i}} - ({{c_{1i}} - {c_{1f}}} ){\left( {\frac{t}{{maxgen}}} \right)^\beta }$$
$${c_2}({t + 1} )= {c_{2i}} - ({{c_{2i}} - {c_{2f}}} ){\left( {\frac{t}{{maxgen}}} \right)^\gamma }$$
where ${c_{1i}}$, ${c_{1f}}$, ${c_{2i}}$ and ${c_{2f}}$ are the initial and final values of the learning factor ${c_1}$ and ${c_2}$, respectively. $\beta $ and $\gamma $ are the linear time-varying parameters of ${c_1}$ and ${c_2}$, respectively.

Due to the contribution made by the learning factors ${c_1}$ and ${c_2}$ are not obvious in the process of particle movement, the two parameters are set as constant values, namely $\beta = \gamma = 0$ and ${c_{1i}} = {c_{1f}} = {c_{2i}} = {c_{2f}} = 2$. Therefore, the value of the learning factor ${c_1},{c_2}$ can be obtained as ${\textrm{c}_1} = {c_2} = 2$.

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From equations (16), (17) and(21), the particle velocity update formula of the Hntiw IFOPSO algorithm can be obtained as:

$$\begin{aligned} {v_{ml}}(t + 1) &= \left( {\left( {{\omega_{\max }} - ({{\omega_{\max }} - {\omega_{\min }}} ){{\left( {\frac{t}{{\max gen}}} \right)}^\psi }} \right) - 1 - \alpha } \right){v_{ml}}(t)\\ &+ \frac{{\alpha (1 - \alpha )}}{2}{v_{ml}}(t - 1) + \frac{{\alpha (1 - \alpha )(2 - \alpha )}}{6}{v_{ml}}(t - 2)\\ &+ \frac{{\alpha (1 - \alpha )(2 - \alpha )(3 - \alpha )}}{{24}}{v_{ml}}(t - 3) + {c_1}{r_1}({g_{ml}}(t) - {x_{ml}}(t))\\ &+ {c_2}{r_2}({q_m}(t) - {x_{ml}}(t)) \end{aligned}$$

In fact, the searchability and convergence performance of Hntiw IFOPSO algorithm can be enhanced by introducing high-order term and nonlinear factor mechanism, so that it can optimize high-order nonlinear objective function efficiently. The pseudo-code of the Hntiw IFOPSO is shown in Algorithm 1, and the flow chart is shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Flowchart of Hntiw IFOPSO algorithm

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3.3 Global local best values improved fractional-order particle swarm optimization

This subsection proposes the Glbest IFOPSO by using a new inertia weight function. Different from the traditional FOPSO algorithm, the inertia weight is defined as a functional equation with locally optimal value $gFbest$ and global optimal value $qFbest$, i.e.

$${\omega _m} = \left( {1.1 - \frac{{qFbest}}{{gFbes{t_ - }avg}}} \right)$$
where $gFbes{t_ - }avg = \frac{1}{m}\sum\limits_{k = 1}^m {qFbest(k )}$ is the average value of the local optimal value.

In addition, a new acceleration coefficient function ${C_m}$, which is also defined as a functional equation of local optimal value $gFbest$ and global optimal value $qFbest$, is designed as:

$${C_m} = \left( {1 + \frac{{qFbest}}{{gFbes{t_m}}}} \right)$$

Let ${c_1} = {c_2} = {C_m}$, and Eq. (17) becomes

$${v_{ml}}({t + 1} )- {v_{ml}}(t )= ({\omega - 1} ){v_{ml}}(t )+ {C_m}{r_1}({{g_{ml}}(t )+ {q_m}(t )- 2{x_{ml}}(t )} )$$

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According to equations (16), (25) and(27), the speed update formula of the Glbest IFOPSO algorithm can be obtained as:

$$\begin{aligned} {v_{ml}}(t + 1) &= \left( {\left( {1.1 - \frac{{mqFbest}}{{\sum\limits_{k = 1}^m {qFbest(k )} }}} \right) - 1 - \alpha } \right){v_{ml}}(t) + \frac{{\alpha (1 - \alpha )}}{2}{v_{ml}}(t - 1)\\ &+ \frac{{\alpha (1 - \alpha )(2 - \alpha )}}{6}{v_{ml}}(t - 2)\\ &+ \frac{{\alpha (1 - \alpha )(2 - \alpha )(3 - \alpha )}}{{24}}{v_{ml}}(t - 3)\\ &+ \left( {1 + \frac{{qFbest}}{{gFbes{t_m}}}} \right){r_1}({g_{ml}}(t) + {q_m}(t) - 2{x_{ml}}(t)) \end{aligned}$$

The global optimal value from Glbest IFOPSO is the optimal position of all particles in the whole population, and the local optimal value is the best of all the locations searched by the particle. In each iteration, the velocity and position of the particles are updated, while the local optimal values and global optimal positions are updated based on the current position and the optimal position of the surrounding particles. Through the guidance of global and local optimal values, the particles gradually tend to the optimal position, avoiding the problem of local optimal solution. The pseudo-code is shown in Algorithm 2, and the flow chart of the Glbest IFOPSO algorithm is shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Flowchart of Glbest IFOPSO algorithm

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4. Simulation verification

To verify the accuracy of the proposed algorithms, six targets (A∼F) are simulated by Hntiw IFOPSO, Glbest IFOPSO and FOPSO. The six targets represent six typical emissivity trends (as shown in Fig. 4, * represents emissivity target values, $\lozenge$, $\vartriangle$, and $\triangleright$ lines represent emissivity trends). The true temperature is 1800K and the blackbody reference temperature is 1600 K. The eight effective wavelengths are 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, and 1.1 µm, respectively. For Glbest IFOPSO, Hntiw IFOPSO and FOPSO, the initial particle number is set to 10, and iteration number is set to 200. (Simulation environment: Matlab R2021b; Intel(R) Core(TM) i7-8700 CPU @ 3.20 GHz)

 figure: Fig. 4.

Fig. 4. Variation trend of emissivity with wavelength

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Figure 4 shows that all the three algorithms can well follow the target value changing with the wavelength. However, the Glbest IFOPSO is better comparing other two optimization algorithms, which changes with wavelength more accurately and accords with the real distribution of emissivity.

Table 1 shows that the absolute relative error of the true temperature by Glbest IFOPSO is less than 0.15%, by Hntiw IFOPSO is less than 0.32%, and by FOPSO is less than 0.61%. The inversion time of Glbest IFOPSO is less than 0.2 s, IFOPSO is less than 0.28 s, and FOPSO is less than 0.56 s. It can be seen that the accuracy of the three algorithms is Glbest IFOPSO, Hntiw IFOPSO and FOPSO in order from high to low. The inversion time of the three algorithms is less than 1 s, but the calculation speed of Glbest IFOPSO and Hntiw IFOPSO is slightly higher than that of FOPSO.

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Table 1. Temperature simulation results with 1800K true temperature

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The inversion temperature, relative error and inversion time of Glbest IFOPSO and Hntiw IFOPSO running for 10 times are shown in Table 2 and Table 3, respectively. The maximum relative error, minimum relative error and average error of Glbest IFOPSO and Hntiw IFOPSO running for 10 times are shown in Table 4. It can be seen that the maximum relative error of Glbest IFOPSO is within 0.6%, and the maximum relative error of Hntiw IFOPSO is within 1.1%.

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Table 2. Simulation results of Glbest IFOPSO running for 10 times

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Table 3. Simulation results of Hntiw IFOPSO running for 10 times

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Table 4. Comparison of the relative error and average error

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In summary, the accuracy and inversion speed of Glbest IFOPSO and Hntiw IFOPSO are higher than FOPSO, and have better stability under repeated operation.

5. Verification with measured data

The rocket engine nozzle temperature data provided from Ref. [15] (Table 5 and Table 6) is employed to verify the application ability of the proposed algorithms. The temperature inversion results are shown in Table 7.

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Table 5. Effective wavelength of the pyrometer and output value at reference temperature (${\textrm{T}_\textrm{R}}\textrm{ = 2252 K}$)

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Table 6. Measured data of solid rocket engine nozzles

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Table 7. Temperature inversion results

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Setting the rocket engine design temperature to 2490 K, the Glbest FOPSO, Hntiw IFOPSO and FOPSO are used for temperature inversion in the same experimental environment, respectively. The results are shown in Table 7, in which the absolute value of relative error of temperature inversion by Glbest IFOPSO is less than 0.325%, by Hntiw IFOPSO is less than 0.639%, and by FOPSO is less than 0.948%.

The inversion results of Glbest IFOPSO and Hntiw IFOPSO running for 10 times by using measurement data are shown in Table 8 and Table 9, respectively. It can be seen from Table 8 that the maximum relative error of Glbest IFOPSO is 0.54%, and the inversion time is all within 1 s. As can be seen from Table 9, the maximum relative error of Hntiw IFOPSO is 0.99%, and the inversion time is also within 1 s. Therefore, both Glbest IFOPSO and Hntiw IFOPSO have better accuracy, inversion speed and stability in practical applications, and Glbest IFOPSO is superior to Hntiw IFOPSO in accuracy.

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Table 8. The inversion results of Glbest IFOPSO by using measurement data

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Table 9. The inversion results of Hntiw IFOPSO by using measurement data

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6. Conclusion

The true temperature inversion problem is transformed into a multi-objective optimization problem in this paper, and two improved particle swarm optimization algorithms Glbest IFOPSO and Hntiw IFOPSO are proposed to solve the optimization and obtain well temperature inversion accuracy. In Hntiw IFOPSO, the linear time-varying inertia weight is replaced by a nonlinear function related to the total number of iterations and the current number of iterations. The Glbest IFOPSO replaces the linear time-varying inertia weight and acceleration constant with the inertia weight and acceleration coefficient expressed by local and global optimal values. Simulation results verify the effectiveness and inversion accuracy of the proposed methods, and the measured data of rocket tail flame verify the practicability of the algorithms.

Funding

Yantai University 2023 Graduate Science and Technology Innovation Project of China (GGIFYTU2348); National Natural Science Foundation of China (62205280).

Acknowledgments

This paper was supported in part by the National Natural Science Foundation of China (62205280), and Yantai University 2023 Graduate Science and Technology Innovation Project of China (GGIFYTU2348).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (4)
Fig. 1.
Fig. 1. Measurement principle of the multi-wavelength pyrometer

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Fig. 2.
Fig. 2. Flowchart of Hntiw IFOPSO algorithm

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Fig. 3.
Fig. 3. Flowchart of Glbest IFOPSO algorithm

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Fig. 4.
Fig. 4. Variation trend of emissivity with wavelength

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Tables (9)

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Table 1. Temperature simulation results with 1800K true temperature

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Table 2. Simulation results of Glbest IFOPSO running for 10 times

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Table 3. Simulation results of Hntiw IFOPSO running for 10 times

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Table 4. Comparison of the relative error and average error

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Table 5. Effective wavelength of the pyrometer and output value at reference temperature ( T R  = 2252 K )

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Table 6. Measured data of solid rocket engine nozzles

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Table 7. Temperature inversion results

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Table 8. The inversion results of Glbest IFOPSO by using measurement data

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Table 9. The inversion results of Hntiw IFOPSO by using measurement data

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Equations (28)

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V i = A λ i ⋅ ε ( λ i , T ) ⋅ 1 λ i 5 ( e C 2 λ i T − 1 ) , i = 1 , 2 , ⋯ , n
V i = A λ i ⋅ ε ( λ i , T ) ⋅ λ i − 5 ⋅ e − C 2 λ i T , i = 1 , 2 , ⋯ , n
V R = A λ i ⋅ λ i − 5 ⋅ e − C 2 λ i T R , i = 1 , 2 , ⋯ , n
V i V R = ε ( λ i , T ) ⋅ e − C 2 λ i T ⋅ e C 2 λ i T R
ln ⁡ ( V i V R ) − C 2 λ i T R = − C 2 λ i T i + ln ⁡ ε ( λ i , T i )
T i = 1 ( ln ⁡ ε ( λ i , T i ) + C 2 λ i T R − ln ⁡ ( V i V R ) ) ⋅ λ i C 2
F 1 = | T 1 − T 3 | F 2 = | T 2 − T 4 | F 3 = | T 3 − T 5 | F 4 = | T 4 − T 6 | F 5 = | T 5 − T 7 | F 6 = | T 6 − T 8 |
F 7 = 1 6 ( | T 1 − T 3 | + | T 2 − T 4 | + | T 3 − T 5 | + | T 4 − T 6 | + | T 5 − T 7 | + | T 6 − T 8 | )
F 8 = 1 4 | ∑ i = 3 6 | T i | − ( ∑ i = 1 2 | T i | + ∑ i = 7 8 | T i | ) |
F 9 = ∑ i = 1 8 | T i − E ( T i ) | 8
{ min [ F 1 ( x i ) , F 2 ( x i ) , ⋯ , F 9 ( x i ) ] 0 ≤ x i ≤ 1 , i = 1 , 2 , ⋯ , 8.
v m l ( t + 1 ) = ω v m l ( t ) + c 1 r 1 ( g m l ( t ) − x m l ( t ) ) + c 2 r 2 ( q m ( t ) − x m l ( t ) )
x m l ( t + 1 ) = x m l ( t ) + v m l ( t + 1 )
D α [ g ( s ) ] = lim h → 0 ⁡ ( 1 h α ∑ k = 0 [ a 2 − a 1 h ] ( − 1 ) k Γ ( α + 1 ) g ( s − k h ) Γ ( k + 1 ) Γ ( α − k + 1 ) )
D α [ g ( s ) ] = 1 ( P ) α ∑ k = 0 r ( − 1 ) k Γ ( α + 1 ) g ( s − k P ) Γ ( k + 1 ) Γ ( α − k + 1 )
D α [ v ( t + 1 ) ] = v ( t + 1 ) − α v ( t ) − 1 2 α ( 1 − α ) v ( t − 1 ) − 1 6 α ( 1 − α ) ( 2 − α ) v ( t − 2 ) − 1 24 α ( 1 − α ) ( 2 − α ) ( 3 − α ) v ( t − 3 ) − …
v m l ( t + 1 ) − v m l ( t ) = ( ω − 1 ) v m l ( t ) + c 1 r 1 ( g m l ( t ) − x m l ( t ) ) + c 2 r 2 ( q m ( t ) − x m l ( t ) )
D α [ v m l ( t + 1 ) ] = c 1 r 1 ( g m l ( t ) − x m l ( t ) ) + c 2 r 2 ( q m ( t ) − x m l ( t ) )
v m l ( t + 1 ) = α v m l ( t ) + α ( 1 − α ) 2 v m l ( t − 1 ) + α ( 1 − α ) ( 2 − α ) 6 v m l ( t − 2 ) + α ( 1 − α ) ( 2 − α ) ( 3 − α ) 24 v m l ( t − 3 ) + c 1 r 1 ( g m l ( t ) − x m l ( t ) ) + c 2 r 2 ( q m ( t ) − x m l ( t ) )
α = 0.9 − 0.5 × t m a x g e n
ω ( t + 1 ) = ω m a x − ( ω m a x − ω min ) ( t m a x g e n ) ψ
c 1 ( t + 1 ) = c 1 i − ( c 1 i − c 1 f ) ( t m a x g e n ) β
c 2 ( t + 1 ) = c 2 i − ( c 2 i − c 2 f ) ( t m a x g e n ) γ
v m l ( t + 1 ) = ( ( ω max − ( ω max − ω min ) ( t max g e n ) ψ ) − 1 − α ) v m l ( t ) + α ( 1 − α ) 2 v m l ( t − 1 ) + α ( 1 − α ) ( 2 − α ) 6 v m l ( t − 2 ) + α ( 1 − α ) ( 2 − α ) ( 3 − α ) 24 v m l ( t − 3 ) + c 1 r 1 ( g m l ( t ) − x m l ( t ) ) + c 2 r 2 ( q m ( t ) − x m l ( t ) )
ω m = ( 1.1 − q F b e s t g F b e s t − a v g )
C m = ( 1 + q F b e s t g F b e s t m )
v m l ( t + 1 ) − v m l ( t ) = ( ω − 1 ) v m l ( t ) + C m r 1 ( g m l ( t ) + q m ( t ) − 2 x m l ( t ) )
v m l ( t + 1 ) = ( ( 1.1 − m q F b e s t ∑ k = 1 m q F b e s t ( k ) ) − 1 − α ) v m l ( t ) + α ( 1 − α ) 2 v m l ( t − 1 ) + α ( 1 − α ) ( 2 − α ) 6 v m l ( t − 2 ) + α ( 1 − α ) ( 2 − α ) ( 3 − α ) 24 v m l ( t − 3 ) + ( 1 + q F b e s t g F b e s t m ) r 1 ( g m l ( t ) + q m ( t ) − 2 x m l ( t ) )
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