Sensors and Materials, Vol. 35, No. 10 (2023) 4585–4596
MYU Tokyo

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S & M 3416

An Enhanced Quantum Genetic Algorithm and Its Applicationin the Health Monitoring of a Rocket Engine
Hao Xiang*
Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing,
Yulin Normal University, Yulin 537000, China(Received June 5, 2023; accepted October 3, 2023)

Keywords: quantum genetic algorithm, simulated annealing algorithm, LSSVR, fault prediction, liquidfuel rocket engine

In this study, the characteristics of different intelligent algorithms, from the simulated
annealing algorithm (SA) and genetic algorithm to the quantum genetic algorithm, are analyzed.
By utilizing the variety of the population and the rapidity of the convergence of the real doublechain coding objective gradient quantum genetic algorithm, this algorithm is fused with SA; theresulting real double-chain coding objective gradient quantum genetic simulated annealing
algorithm is proposed. As the performance of the least squares support vector regression(LSSVR) is very sensitive to its key parameters, the proposed algorithm is applied to optimize
these parameters to improve the generalization ability of LSSVR. Following that, a new hybrid
nonparametric regression prediction model is put forward. The health monitoring of a liquid-fuelrocket engine is a very important topic, and all types of sensors are used to monitor the factorsthat can affect the engine thrust from different aspects. The relationship between the thrust andthese factors is nonlinear and difficult to express using some formulas. The proposed model isused in the fault prediction of liquid-fuel rocket engine thrust, and the simulation results showthat the average relative error is determined to be 0.37% using LSSVR and 0.2977% using the
proposed model. Thus, this model is applicable to small samples, nonlinearity, and highdimensions of failure prediction, and is worth promoting to a certain extent.

1.

Introduction

At present, the artificial neural network is widely applied in engineering practices, and mostof the neural network models use the back propagation (BP) network and its variations.(1–7) The
notable feature of the artificial neural network is that it achieves highly complex nonlinear
mapping through its own learning, so it has been widely applied in various fields such asfunction approximation and pattern recognition.(8–10) Although a multilayer neural network can
be applied in the fields of linear and nonlinear systems for approximating any function, an
artificial neural network does not always become solvable because the optimization process is
closely related to the selected initial points. On the other hand, the actual research objects are*Corresponding author: e-mail: 15971536347@163.com

https://doi.org/10.18494/SAM4523

ISSN 0914-4935 © MYU K.K.
https://myukk.org/

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often dynamically changing, and it is difficult to grasp the characteristics of these samples;
subject to the limited samples, the effect of artificial neural networks in practical applications isaffected. The support vector machine (SVM), which was proposed by Vapnik as a new machinelearning algorithm,(11,12) was developed to provide a new approach to solve these problems. Thisalgorithm is based on the rigorous statistical learning theory, and the actual risk is obtained byusing the structural risk minimization criterion. SVM can effectively improve its generalization
ability, solve the problems of small sample, nonlinearity, and high dimension in the previous
learning methods, and overcome shortcomings of neural network learning methods such as thedifficulty in determining the network structure, slow convergence, ease of falling into local
minima, overlearning and underlearning, and the need for a large number of data samples during
training. Thus, SVM has widespread applications.(13–19) However, the performance of SVM isvery sensitive to its key parameters, so different intelligent algorithms were used(20–23) tooptimize these parameters.
The genetic algorithm (GA) has a strong search capability as well as good global optimization
performance, so combining GA with BP can yield the global optimum and improve the accuracyof the result.(24,25) However, GA emphasizes the evolutionary relationship between two
generations, and its mutation could lose the best solution. In SA, the state between two
temperatures is irrelevant. In theory, any Markov chain of temperatures gradually reaches thestationary distribution, that is, the process from one state to another with increasing number of
iterations is independent of the starting state, and the probability of each state in the Markov
chain finally obeys a stationary distribution. Combining the characteristics of GA and SA, Xing
and Xie presented the genetic simulated annealing (GSA) algorithm.(26)With the acceleration of the quantum algorithm, a quantum evolutionary algorithm
integrating the quantum algorithm with the genetic algorithm was proposed. Narayanan and
Moore presented the quantum genetic algorithm (QGA) in 1996 and successfully solved the TSP
problem by using it.(27) Han and Kim combined the concepts of the quantum bit and quantum
gate in the evolutionary algorithm in 2000 and proposed the genetic quantum algorithm.(28)
Because of using the quantum superposition state in QGA, the variety and convergence of the
population in QGA are better than those in the common genetic algorithm. However, most of theexisting quantum genetic algorithms adopt the binary coding method based on quantum bit
measurement.(27–29) This method, which is used in numerical optimization problems, requires
frequent coding and decoding, thereby increasing the computation amount. Therefore, the real
double-chain coding objective gradient quantum genetic algorithm (DCQGA) was proposed by
Li and Li.(30) In the quantum physical universe, the coherent superposition of quantum states in
two-particle systems is usually known as entanglement. When a certain performance of aquantum system is measured, the measurement will cause the coherence of the quantum system
to be destroyed, so that it is converted from the superposition state to a basic state. This process
is called the collapse of the quantum state. After the collapse of the astrophysical universe, it
underwent a long and slow process of evolution. In this study, the real double-chain coding
objective gradient quantum genetic simulated annealing algorithm (DCQGSAA) based on
DCQGA is proposed. Then, we used it to optimize the parameters of LSSVR and proposed a
mixture model based on DCQGSAA and LSSVR.

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The establishment of an accurate and reliable fault prediction model for liquid-fuel rocket
engine thrust is of great significance for the health monitoring and fault diagnosis of liquid-fuelrocket engines.(31) All types of sensors are used to monitor the factors that can affect the engine
thrust from different aspects. The relationship between the thrust and these factors is nonlinear
and difficult to express using some formulas. Consequently, the above presented model isadopted for the failure prediction of liquid-fuel rocket engine thrust. The prediction results showthat this model works well.

2.

DCQGA

DCQGA is presented in Ref. 30. Xiang and Wang(32) used this algorithm to optimize theweights and thresholds of the BP neural network that was applied in the field of intelligent fault
diagnosis.
2.1

Describing the problem of continuous optimization
The common continuous optimization problem is expressed asmin f ( x ) = f ( x1 , , xn )
1, 2, , n,s.t. ai ≤ xi ≤ bi ; i =(1)where x ∈  n. To evaluate the quality of the approximate solution, the fitness function is
specified asfit=( x) cmax − f (x),(2)

where cmax is a proper constant.
2.2

Double-chain coding plan

In DCQGA, the probability amplitude of the quantum bit is encoded. The double-chain
coding plan is defined as cos ( t ) cos ( t )  cos ( t )i1i2inpi = 
 sin ( ti1 ) sin ( ti 2 )  sin ( tin )



,



(3)

where tij = 2π × rand, with rand being a random number from 0 to 1, i = 1, 2, ..., m, j = 1, 2, ..., n,m the population size, and n the number of quantum bits. In DCQGA, the probability amplitudeof each quantum bit is considered as two balanced pieces of genes, and each chromosome
contains two balanced gene chains representing two different optimization solutions.

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Sensors and Materials, Vol. 35, No. 10 (2023)

Solution space transformation

Each chromosome in the group contains the probability amplitudes with 2n quantum bits,
and these probability amplitudes are mapped from the n-dimensional unit space of In = [−1, 1]n to
the solution space of the optimization problem by linear transformation. Assuming that the i-th
quantum bit of chromosome pj is [αij, βij]T, the corresponding solution space is

(

) (

(

) (

)

X ic j =0.5 ⋅ bi 1 + α i j + ai (1 − α i j )  ,


X is

j

)

=0.5 ⋅ bi 1 + βi j + ai 1 − βi j  .



(4)

In the above formula, the probability amplitude αij of the quantum state |0> is related to Xicj; theprobability amplitude βij of the quantum state |1> is related to Xisj; i = 1, 2, ..., n, j = 1, 2, ..., m.
2.4

Angle direction for quantum rotation gate

In DCQGA, the quantum rotation gate, which is used to update the phase of the quantum bit,is defined ascos (∆θ ) − sin (∆θ ) U (∆θ ) = sin (∆θ ) cos (∆θ )  .



(5)

The updating way is described ascos (∆θ ) − sin (∆θ )  cos (t )  cos (t + ∆θ )  sin (∆θ ) cos (∆θ )   sin (t )  =  sin (t + ∆θ )  .


 


(6)

To define the direction of the rotation angle Δθ, the principle is depicted asA=α 0 α1.
β 0 β1(7)

If A ≠ 0, the direction is set as −sgn(A). If A = 0, the direction can be positive or negative.
2.5

Size of rotation angle in quantum rotation gate

The step size function of the rotation angle, using the gradient’s definition, is defined as( ) ∇f x j − ∇f
ijmin 
∆θij = − sgn( A) ⋅ ∆θ0 ⋅ exp  −
,
∇
f
−
∇
f
jmax
jmin





(8)

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( )

where Δθ 0 is the original number of iterations, ∇f xi j is the gradient with the evaluation
function f(x) at point xi j, and ∇f jmax and ∇f jmin are defined as ∂f ( x )
1 , , ∂f ( xm )  ,∇f jmax =max j∂xm j  ∂x1 ∂f ( x )
1 , , ∂f ( xm )  .
∇f jmin =min j∂xm j  ∂x1(9)Here, xi j (i = 1, 2, ..., m; j = 1, 2, ..., n) indicates the j-th component of the vector, m is the
population size, and n is the dimension number of the space.
2.6

Mutation procedure

While chromosomes are mutated by applying a quantum non-gate and randomly choosing achromosome based on their mutation probability, quantum non-gates are changed at a numberof quantum bits, which are selected randomly; then, these two probability amplitudes of thequantum bit are exchanged, causing the two chains of chromosomes to mutate.

3.

DCQGSAA

3.1

Algorithm theory

SA can take on the ability of a sudden jump with a certain probability to effectively avoid the
local minimum solutions in the search process. This algorithm receives a good solution as well
as a worse solution when using a particular probability in the annealing process. This probability
is affected by the temperature parameter, and it will decrease along with decreasing temperature.
Therefore, SA is combined with DCQGA to complement their advantages. In DCQGSAA,
during each loop, SA is implemented after DCQGA has been implemented.
3.2

Algorithm process

Step 1:	Initialize the population. The original population comprises m chromosomes, which are
generated by Eq. (3). Determine the original step size of the rotation angle with θ 0 andthe mutation probability pm.
Step 2:	Determine the initial temperature t.
Step 3:	
Using the following formula, determine the fitness value of each chromosome pi at thecurrent temperature.

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TF ( pi ) = Ne(f ( p0 ) − f ( pi ) ) / t∑ e ( ( ) ( ))− f p0 − f pi / t

(10)

i =1

 ere, p0 is the chromosome corresponding to the optimal solution X0 up to now, f(‧) is theH
objective function being optimized, and N is the number of chromosomes.
Step 4:	Use the roulette strategy to ascertain some chromosome substitute p0’ relating to the so
far global optimal solution from all pi.
Step 5:	
Transform the solution space. The approximate solution expressed by each chromosome
is transformed from the unit space to the solution space described by Eq. (1), and thefitness of each chromosome is estimated using Eq. (2). Set the current optimal solution
as X 0 and its related chromosome as p0, and set the optimal solution so far as X0 and its
corresponding chromosome as p0′ .
		 When fit ( X 0 ) > fit ( X 0 ), p0′ = p0.
Step 6:	Determine the rotation angle size of each quantum bit for each chromosome in the
population space using Eq. (8). Their quantum bits are changed by the quantum rotationgate.
Step 7:	Mutate each chromosome using the quantum non-gate in accordance with the mutation
probability.
Step 8:	Implement the annealing temperature operation.
Step 9:	Return to Step 2. The program is not terminated until the convergence condition or the
maximum number of iterations is satisfied.
The initial temperature and annealing method affect the accuracy of the algorithm to acertain extent, and they are given as tk+1 = λtk and t0 = f(pg)/ln5, where pg is the chromosome
corresponding to the optimal solution in the initial population and λ is the annealing constant.

4.

Mixture Model

The performance of LSSVR is associated with the type of kernel function, the corresponding
kernel parameters, and the penalty coefficient C, among which the type of kernel function has
little effect on LSSVR, but the latter two items have a considerable effect on the performance of
LSSVR. The commonly used kernel functions are, for example, the polynomial function andradial basis function (RBF). Among them, the polynomial function has two adjustableparameters, but RBF has only one. Since choosing the proper kernel function helps to reduce thecomputational duty, RBF is commonly used. In addition, the polynomial kernel function only
maps from the original space to a finite high-dimensional feature space, but RBF can map fromthe original space to an infinite high-dimensional feature space. Thus, we selected RBF toestablish the prediction model.

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RBF is expressed as x−x 2i,K ( x,=xi ) exp  −
2


2σ



(11)

where σ is the width of RBF. There is no uniform method for properly selecting C and σ, but thecommonly used method is cross-validation. It can be seen from Refs. 33 and 34 that the crossvalidation method cannot result in a more accurate performance of LSSVR than those using
intelligent algorithms; thus, in this study, we used the intelligent algorithm DCQGA. The
flowsheet of the proposed model is shown in Fig. 1.

Fig. 1.

Flowsheet of proposed model.

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5.

Application

5.1

Setup of experiment plan

The thrust of a liquid-fuel rocket engine is an important indicator of its health, and by
predicting its changes and comparing the predicted value with its threshold, we can predictwhether the engine will fail at some time. Under the given working conditions of the engine
system, the thrust F is closely related to parameters such as the oxidant flow m o, the combustion
flow m f , the pressure of the combustion chamber pc, and the sampling time t. There exist highcomplexity and nonlinearity between each parameter and thrust, and this relationship can besummarized as the function F = f(m o, m f , pc, t,...). Therefore, F can be predicted using theseparameters. Thus, it can be seen that the characteristics of the liquid-fuel rocket engine fault are
the high nonlinearity between the causes and the result, the high dimension of experiment
samples, and the small size of samples. However, the use of LSSVR alleviates such problems.
Therefore, this mixture model can be used to establish the failure prediction model of the liquidfuel rocket engine.
The training and testing samples from the trial running data shown in Table 1 are scaled.(31)
From Table 1, the total number of samples is 25 and the sampling time is 0.1 s. We used the first
Table 1
Trial running data.
Sampling time t(s)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4

Combustion flow (m f )Oxidant flow (m o)0.31790.3931
0.57800.65320.60120.6012
0.67630.77460.85550.90170.9306
0.9526
0.96650.97690.98270.98840.94420.99650.99770.99881.00001.00001.00001.00001.00000.40110.4661
0.53390.59350.56580.5572
0.63740.67210.82380.87530.91490.94200.95930.97020.98100.9864
0.99190.99570.99730.99951.00001.00001.00001.00001.0000Pressure of
combustion chamber(pc)
0.00980.05590.22380.46430.61700.70490.78320.84470.89510.9301
0.95800.97900.9874
0.99300.9944
0.9951
0.99720.99861.00001.00001.00001.00001.00001.00001.0000Thrust (F)0.00000.04960.14180.32620.41130.46810.56740.6596
0.77300.84400.88650.91630.93620.95040.9574
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4593

13 rows of the table as the training samples and the remaining 12 rows as the testing samples todetermine the effect of prediction and validate the generalization capability of the model. The
independent variables of the function F = f(m o, m f , pc, t,...) were used as inputs in the model andF as the output of the model. Thus, this model can use multiple parameters to predict the thrust.
5.2

Simulation experiment

In this experiment, the parameters of DCQGSAA are given as follows: the population size of
20, the number of iterations of 100, the mutation probability of 0.05, the step size of rotation
angle of 0.001*pi, the initial temperature determined by t0 = f(pg) /ln5, and the annealing
constant of 0.7. Since the performance of LSSVR is extremely sensitive to the parameters of σ2
and C, their ranges are related to the data distribution of the research object. According to Refs.
33–39, the range of σ 2 values is determined to be between 0.01 and 1000, and the range of C
values is determined to be between 0.01 and 10000.
To enable LSSVR to have good generalization performance, we used DCQGSAA to globally
optimize the parameters of σ 2 and C. First, the training samples were used to train LSSVR; then,
the test samples were input to the trained LSSVR to obtain predicted values; the negative valueof the root mean squared error (RMSE) between the predicted values of the test samples andtheir real values, −RMSE, was used as the fitness function. Clearly, the larger the fitness
function, the higher the generalization ability of LSSVR. Then, LSSVR uses the best σ 2 and C totrain the training samples and obtain the fitted values on these samples. At the same time, theamount of time required for this training process is recorded; then, the test samples are input into
the trained LSSVR with the best σ 2 and C to obtain predicted values of the test samples.
For the failure prediction of liquid-fuel rocket engine thrust, we used the proposed model;
then, we obtained the fitted curve of the training samples, the forecast values and forecast curveof the test samples, and the changing curve of the fitness function. These simulation results areshown in Figs. 2–4.
Figure 2 shows that after using DCQGSAA to optimize the key parameters of LSSVR, theoptimized LSSVR better fit the training samples. Figure 3 shows that the optimized LSSVR canalso better predict the values of the testing samples, which indicates that the optimized LSSVR
has good generalization performance. Figure 4 shows that when using DCQGSAA to optimize
the key parameters of LSSVR, after 26 iterations, the fitness function becomes stable.
As seen from Table 2, when using DCQGSAA, the simulation results show that the averagerelative error (ARE) on the testing samples is 0.2977%; when not using any intelligent algorithm,
this error is 0.37%.
As seen from the above, when adopting DCQGSAA to optimize the parameters of LSSVR,
the number of iterations is very small and reduced; moreover, from the simulation results, wefind that ARE on the testing samples decreases. Therefore, the proposed mixture model iseffective.

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Fig. 2. (Color online) Fitted curve of the training
samples using DCQGSAA.

Fig. 4.

Fig. 3. (Color online) Predicted curve of the test
samples using DCQGSAA.

(Color online) Changing curve of fitness function using DCQGSAA.

Table 2Estimation results of test samples.
No.
1
2
3
4
5
6
7
8
9
10
11
12
ARE (%)
Training time (s)Real value
0.95040.9574
0.97160.97870.98580.99290.99861.00001.00001.00001.00001.0000Prediction value
Prediction value of LSSVR(using DCQGSAA)(Ref. 31)0.95700.95950.9681
0.9694
0.97590.97450.97480.98350.98990.98850.9941
0.9931
0.99700.9976
0.99881.00130.99971.00241.0002
1.00241.00011.00240.99951.00240.29770.37
00.0006(Simulation result using MATLAB)

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6.

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Conclusion

In this study, after analyzing the characteristics of DCQGA, we combined DCQGA with SA
and then proposed DCQGSAA. As the performance of LSSVR is extremely sensitive to its key
parameters, we used DCQGSAA to optimize these parameters and established the hybrid
nonparametric regression prediction model. This is of considerable significance to the healthmonitoring of a liquid-fuel rocket engine, and all types of sensors are used to monitor the factorsthat can affect the work condition of the engine from different aspects. The relationships
between the thrust and these factors are nonlinear and difficult to express by some formulas.
The fault characteristics of a liquid-fuel rocket engine show that its thrust is closely associatedwith various factors, and the fault samples are small and multidimensional. Thus, we used the
proposed model to predict the thrust of a liquid-fuel rocket engine. The simulation results denotethat the proposed model is effective for small and multidimensional samples, and thus, it is worth
promoting.

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About the Authors
Hao Xiang received his B.S. and M.S. degrees from Henan Polytechnic University, China, in2000 and 2004, respectively, and his Ph.D. degree from Nanjing University of Science andTechnology, China, in 2013. He currently works in Yulin Normal University, China. His researchinterests are in AI, PHM, and industrial big data processing.