Research ArticleParameter Optimization on FNN/PID CompoundController for a Three-Axis Inertially Stabilized Platform forAerial Remote Sensing Applications

Xiangyang Zhou ,1 Hao Gao,1 Yuan Jia,1 Lingling Li,1 Libo Zhao ,2 and Ruifang Yu 3

1School of Instrumentation Science and Opto-Electronics Engineering, Beihang University (BUAA), Beijing 100191, China2State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China3Institute of Geophysics, China Earthquake Administration, Beijing 100081, China

Correspondence should be addressed to Ruifang Yu; yrfang126@126.com

Received 16 April 2018; Revised 31 August 2018; Accepted 27 September 2018; Published 26 March 2019

Guest Editor: Pasquale Imperatore

Copyright © 2019 Xiangyang Zhou et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

This paper presents a composite parameter optimization method based on the chaos particle swarm optimization and the backpropagation algorithms for a fuzzy neural network/proportion integration differentiation compound controller, which is appliedfor an aerial inertially stabilized platform for aerial remote sensing applications. Firstly, a compound controller combining boththe adaptive fuzzy neural network and traditional PID control methods is developed to deal with the contradiction between thecontrol precision and robustness due to disturbances. Then, on the basis of both the chaos particle swarm optimization and theback propagation compound algorithms, the parameters of the fuzzy neural network/PID compound controller are optimizedoffline and fine-tuned online, respectively. In this way, the compound controller can achieve good adaptive convergence so as toget high stabilization precision under the multisource dynamic disturbance environment. To verify the method, the simulationsare carried out. The results show that the composite parameter optimization method can effectively enhance the convergence ofthe controller, by which the stabilization precision and disturbance rejection capability of the proposed fuzzy neuralnetwork/PID compound controller are improved obviously.

1. Introduction

Aerial remote sensing has an increasing attention in envi-ronmental applications: disaster monitoring, intelligent agri-culture, pollution detection, etc. Natural disasters appear incharacteristics of high frequency and intensity and oftenresult in huge losses in their area of destruction [1]. There-fore, gathering information and continuously monitoringthe affected areas are crucial to assess the damage and speedup the recovery process [2]. Remote sensing technologies cantake a significant place for decision-makers for the calcula-tion and estimation of the environment impacts [3]. On theother hand, precision agriculture includes various technolo-gies that allow agricultural professionals to use informationmanagement tools to optimize agriculture production [4].

The agricultural practices, planting patterns, the stage ofgrowth in the vegetation, soil composition, and humidityare important factors that affect the present-day visibility ofburied structures such as crop or soil marks [5]. There aregreat challenges for accurate predictive mapping at regionalscales for an agroecosystem [6]. Remote sensing technologiesoffer opportunities to break down the silos between energy,water, and resource management through cheaper, auto-mated, and high spatiotemporal resolution data collection.Remote sensing via aerial (i.e., manned or unmanned) vehi-cles generally allows for more detailed spatial resolution thansatellite measurements [7].

Inertial stabilized platform (ISP) is a key componentfor an aerial remote sensing system, which is mainly usedto hold and control the line of sight (LOS) of the imaging

HindawiJournal of SensorsVolume 2019, Article ID 5067081, 15 pageshttps://doi.org/10.1155/2019/5067081

sensors keeping steady in an inertial space [8–13]. For ahigh-resolution aerial remote sensing system, it is crucialto isolate the attitude changes of aircraft in three axes andto reject the multisource disturbances inside or outside ofthe aircraft body in real time. The first fundamental objectiveof an ISP is to help the imaging sensors to obtain high-qualityimages of the target or target region. Therefore, the most crit-ical performancemetric for an ISP is the disturbance rejection.

Many different control methods with high accuracy andstability are developed through suppressing various distur-bances. In [14], a dual-rate-loop control method based on thedisturbance observer (DOB) of angular acceleration is pro-posed to improve the control accuracy and stabilization ofthe ISP. In [15], a self-adaptiveonline genetic algorithmtuningis proposed to optimize the proportion integrationdifferentia-tion (PID) parameters of the ISP, which improve the systemcontrol precision and stability and response speed. In [16], aself-tuning fuzzy/PID control strategy is proposed to improvethe dynamic performance of the ISP. In [17], an automatic dis-turbance rejection controller (ADRC) is proposed to solve theissues such as system model uncertainty and measurementnoise in a three-axial ISP control system. In [18], a methodcombining a Kalman filter and a disturbance observer isput forward to improve the inertial stabilization per-formance of an aerial photoelectric platform. In [19], acompound control strategy combining the extended distur-bance observer (EDO) and continuous robust integral of thesign of error (RISE) is proposed to improve the stability pre-cision of an ISP. In [20], an integrated control method usingboth feed-forward control and disturbance observer isdesigned to improve the stabilization precision of the ISP.

As an intelligent control method, the fuzzy control is anon-open-loop control system that is based on fuzzy logicinference. It is especially suitable for the control of nonlinear,time-varying, and delay systems [21]. The control perfor-mance of the system depends on the parameter setting, so itis not easy to achieve the desired control effect [22]. Althoughthe PID regulator can get higher steady-state accuracyand dynamic characteristics, the parameter tuning is diffi-cult. The determination of the conventional PID controllerparameter tuning is based on obtaining the mathematicalmodel of controlled objects and the rules, which is difficultto adapt to complex control systems [23]. Comparatively,the fuzzy control method is a kind controller of language,which can reflect the approximate optimal control behaviorof controller and have strong robustness and stability toadapt to different object controls [24]. Therefore, the methodcombining both the adaptive fuzzy and the traditionalPID control methods should be developed to solve the con-tradiction between the control precision and robustnesson disturbances.

However, the method of fuzzy controller essentially isa nonlinear controller whose control algorithm is basedon intuition and experience on the plant; it does not haveany automatic learning capabilities to handle the un-certainty. It is well known that the adaptive neural net-work (NN) control has a learning capability and hasbeen considered as a powerful tool to identify any nonlin-ear function to any desired accuracy in control and

applications for nonlinear systems [25]. Therefore,although it is a particularly difficult problem for the fuzzysystem to determine the membership function, the input-output of the NN can approximate any function. So inthe fuzzy system design, it can take advantage of the learn-ing ability of the NN and operation by adjusting theweight membership functions in learning [26]. However,the parameters of the existing fuzzy neural network(FNN)/PID controller are large and the initial value has agreat influence on the convergence of the controller; it is dif-ficult to find a good initial value of parameters in the practicalapplication to the ISP to get a good control effect. Therefore,it is difficult to obtain more suitable initial values of parame-ters by the ordinary trial and error method. So it is necessaryto investigate the parameter optimization algorithms toobtain better parameters for the system.

The particle swarm optimization (PSO) is a compu-tational intelligence-oriented, stochastic, population-basedglobal optimization technique [27]. It is concerned withthe elementary algorithm, which has the characteristics ofsimplicity, easy implementation, and few parameters tobe adjusted [28]. However, the PSO seems to be sensitiveto the tuning of its parameters [29]. These advantages leadPSO to be applied broadly to different areas. For the basicPSO, the result easily falls into the local optimum withrandom initial choice, because of the nonuniform distribu-tion of initial particles, which will weaken the globalsearch ability of the PSO. Once getting trapped in localoptimum during the process of optimization, it is veryeasy to cause all particles to stagnate in the extreme valuepoint [30]. Therefore, when the algorithm runs into pre-maturity, the random perturbation strategy is adopted forthe best individual and the randomly selected individualto help them being out of the local minimum [31]. Sincethe PSO has the drawback of stopping optimizing whenreaching a near-optimal solution [27], the chaos mecha-nism is proposed to help the PSO to optimize the search-ing result. Thus, an improved algorithm based on themechanism of chaos and the PSO, i.e., the chaos particleswarm optimization (CPSO) algorithm, is proposed toadjust the parameters offline. In [32], the chaos searchingis proposed for global optimization problems and parame-ter inversion of the nonlinear sun shadow model, whichcan improve the computing accuracy and computing effi-ciency of the global optimization problems. In [33], thechaos is applied to avoid the untimely aggregation of par-ticle swarm and improve the mean best of the algorithmand the success rate of search. In [34], a chaotic searchingis applied to improve the global search performance, andthe applying results show that the CPSO algorithm is veryefficient at solving global optimization problems and is agood approach for reliability analysis. Furthermore, theback propagation (BP) algorithm is used to adjust onlineto obtain the optimized parameters. In this way, the con-trol system can achieve better control results.

In this paper, to improve the ability of disturbance rejec-tion of an aerial ISP, an FNN/PID compound control schemeis first designed. Then, a composite parameter optimizationmethod based on the CPSO and BP algorithms is proposed

2 Journal of Sensors

to improve the adaptive convergence performance of thecompound controller. To verify the method, the simulationsare carried out.

2. Background

2.1. Aerial Remote Sensing System. Figure 1 shows a sche-matic diagram to illustrate the important effect of theISP on improving the image quality in an aerial remotesensing system for aerial remote sensing applications.Due to the serious influences caused by disturbances aris-ing from diverse sources, including inside or outside of theaviation platform, it becomes very difficult to keep theLOS steady, particularly for the case of the jitter of threeangular attitudes of an aircraft. Thus, the case of unidealimages replacing the ideal images will occur. So thehigh-precision ISP, which is typically mounted on a movableplatform, is indispensable to isolate disturbances derivedfrom diverse sources [9, 12].

Figure 2 shows the schematic diagram of an aerial remotesensing system. Generally, an aerial remote sensing systemconsists of four main components: a three-axis ISP, a remotesensing sensor, a position and orientation system (POS), andanaviationplatform.When theaviationplatformrotatesor jit-ters, the control system of three-axis ISP gets thehigh-precision attitude reference information measured bythe POS and then routinely controls the LOS of the

imaging sensor to achieve accurate pointing and stabilizingrelative to ground level and flight track. The POS, which ismainly composed of the inertial measurement unit (IMU),the GPS receiving antenna, and the data processing system,is used to measure the minor angular movement of the ima-ging sensor.

2.2. Operating Principle of Three-Axis ISP System. Figure 3shows the schematic diagram of the three-axis ISP’s prin-ciple. We can see that the ISP consists of three gimbals,which are azimuth gimbal (A-gimbal), pitch gimbal(P-gimbal), and roll gimbal (R-gimbal). Among them, theA-gimbal is assembled on the P-gimbal and can rotatearound the Za axis. Likewise, the P-gimbal is assembledon the R-gimbal and can rotate around the Xp axis. TheR-gimbal is assembled on the basement and can rotatearound the Yr axis.

From Figure 3, we can see the relationships between thethree gimbals: Gp, Gr , and Ga, respectively, which stand forthe rate gyro that measures the inertial angular rate of P-gim-bal, R-gimbal, and A-gimbal. Er , Ep, and Ea, respectively,stand for the photoelectric encoder which measures relativeangular between gimbals. Mr , Mp, and Ma, respectively,stand for the gimbal servo motor which drives R-gimbal,P-gimbal, and A-gimbal to keep these three gimbals steadyin an inertial space.

2.3. Three Closed-Loop Compound Control Scheme. Figure 4shows the block diagram of the traditional three-loop con-trol system for ISP. Conventional stabilization techniquesemploy rate gyros, rate integrating gyros, or rate sensors tosense rate disturbances about the LOS. In Figure 4, theblocks of G-pos, G-spe, and G-cur separately represent thecontrollers in the position loop, speed loop, and currentloop; the PWM block represents the power amplificationused for the current amplifier to drive the torque motor; Lrepresents the inductance of a torque motor, and R repre-sents the resistance; Kt represents the torque coefficient ofthe motor, and N is the transition ratio from the torquemotor to the gimbals; Jm represents the moment of inertia

Jitter and vibrationof actual motion

Ideal motion trajectoryImage distortion Ideal image

Figure 1: Schematic diagram: effect of the ISP on improving the image quality in an aerial remote sensing system for aerial remotesensing applications.

POS

GPS

IMU

Angular velocityAngular acceleration

Position velocity

Angular velocityAttitude

Aircaft floor

Aerial remoteISP

Camera

Figure 2: Schematic diagram of an aerial remote sensing system [44].

3Journal of Sensors

of the motor, and J l represents the moment of inertia of thegimbals along the rotation axis.

3. Design of the FNN/PIDCompound Controller

In the FNN/PID control method, the input interface has twonodes, i.e., the error (e) and change of the error (ec), respec-tively. The role of the fuzzification layer is to make the inputof a reasonable fuzzy segmentation, in which the number ofnodes is equal to the number of variables [29]:

f1 i = X = x1,… , xn , n = 2, 1

where X stands for the domain.When the method is applied to the ISP, the fuzzy lan-

guage of each input variable is divided into seven seg-ments. According to the working principle of the ISP,different Gauss functions and bell functions are used torepresent the different fuzzy subsets which are expressed

as follows:

f 2 i, j = μAij = exp− Xi − cij

2

σ2ij

, i = 1, 2, j = 2, 3, 4, 5, 6,

2

f i, j = μAij =1

1 + Xi − cij /aij2σ2i j

, i = 1, 2, j = 1, 7,

3

where Xi stands for the input variable; c and σ stand forthe activation function centers and widths, respectively;and a stands for the fuzzy subsets corresponding to fuzzyvariables. For the different fuzzy subsets, the differentmembership functions should be chosen. In (2), i.e., theGaussian function’s marginal value and middle value areclose to 0 and 1, respectively, which is suitable for thefuzzy subset with a large membership degree of intermedi-ate element. In (3), i.e., the bell function’s marginal value

Flightdirection

Imagingsensor

LOSPitchgimbal

Azimuthgimbal Pitch gear-link

Azimuth gear-link

XY

Z

Ax

Gy

Ey

My

Y

Ay

Rollgimbal

Roll gear-link

Z

GX

EX

MX

X

GZ

MZ

EZ

POS

Figure 3: Schematic diagram of the three-axis ISP’s principle.

− − −

Commandangular

G-pos G-spe G-cur PWM

Current loopHall sensor

Gyro/coder differential

POS/coder

Stabilization loop

Tracking loop

1Ls + R

Kt

Mm

Md

N

+1

Angular rateoutput

1/s

Angularoutput

(N2Jms + J

1)s

Figure 4: A block diagram of traditional three-loop control system for ISP [44].

4 Journal of Sensors

is close to 1, which is suitable for the fuzzy subset with alarge membership degree of the boundary element [35].

Different from the general FNN/PID control algorithmthat sets up a relatively simple activation function, theimproved methods of membership function parameters arediverse from each other. This control method absorbs theexperience of fuzzy/PID control in the membership functiondesign. Therefore, it is more suitable for the system charac-teristics of the ISP, and its parameters are no longer updatedand adjusted in the BP algorithm. Thus, the blindness ofparameter updating is avoided, which further results in ashort computing time and a good control effect.

Corresponding to the ISP system, the number of thenodes of the fuzzy rule layer is 49, and the method of fuzzyinference is as follows [36]:

f3 j =N

j=1f2 i, j = μA1k x1 ∗ μA2k x2 ,

N =n

i=1Ni,

4

where k is the number of fuzzy rules.For the output layer, it can be obtained by the output of

the upper layer and the connection weight, as shown as fol-lows [32]:

f4 j =w ⋅ f3 = 〠N

j=1w i, j ⋅ f3 j , 5

where w stands for the connection weight matrix.The output of the improved FNN/PID controller is used

as the compensation of the constant PID parameters, asshown as

Kp = Kp0 + FNN ΔKp ,Ki = Ki0 + FNN ΔKi ,Kd = Kd0 + FNN ΔKd ,

6

where Kp0, Ki0, and Kd0 represent the initial parametersof the fuzzy/PID controller and FNN ΔKp , FNN ΔKi ,and FNN ΔKd represent the outputs of the improvedFNN/PID controller.

So the output of the controller is expressed as follows:

u k == kp e k − e k − 1 + ki 〠k

i=1e i + kd e k − e k − 1

7

The improved FNN/PID controller controls the ISP byadding the PID value of real-time adjustment and the fixedPID value. In this case, the FNN can only deal with the smallchange which reduces the dependence on the initial valueand makes the adjustment time shorter. Because the wholesystem is not entirely dependent on the output value of the

adaptive adjustment part, it can guarantee the stability andavoid the divergence.

The improved FNN/PID controller only needs to deter-mine the weights of the controller, since it uses fuzzy/PIDcontroller’s fuzzy subset number, membership function,quantization factor, and constant PID parameter value. Sincethe output of the method is small, the effect of the initial valueof the weight coefficient on its output is limited. Thus, thesuperior control effect can be obtained by only the randomlyobtained parameters. Figure 5 shows the schematic diagramof the FNN/PID compound controller structure.

4. Composite Parameter Optimization Based onCPSO and BP Algorithms

The FNN/PID controller inherits the advantages of fuzzycontrol in acquiring knowledge, which has the ability of theneural network to approach any nonlinear function at thesame time. However, the parameters of this method are largeand the initial value has a great influence on the convergenceof the controller; it is difficult to find a good initial value ofparameters in the practical application for the ISP to get agood control effect. The general FNN/PID controller is diffi-cult choosing the initial value which influences the improve-ment of the control accuracy and the convergence.

If the control level of the FNN/PID controller is expectedto improve, configuring a series of parameters which aremore appropriate is necessary. Thus, the parameter opti-mization problem is the key to the control effect of thecontroller. The structure of the controller is a forwardneural network, which can adapt to the control requestof the controlled object by adjusting the weight of the net-work itself. At the initialization stage of the algorithm, theposition of the particle is initialized by chaos. The weightcoefficients are encoded as vectors and expressed as Para.The number of particle swarm optimization is selected asN . Chaos initialization is applied to randomly produce n-dimensional vectors z1 = z11, z12,⋯, z1n , which is formedfrom zi+1j = μzij 1 − zij j = 1, 2,⋯, n, i = 1, 2,⋯,N − 1 andcarried to the range of optimized variable xij = aj + bj − ajzij j = 1, 2,⋯, n, i = 1, 2,⋯,N − 1 as the position of initial-ized particle swarm. In addition, the other system parame-ters, including acceleration constants, the maximum inertiaweight, and the minimum inertia weight, are assigned tothe numerical value based on other academic papers, inwhich the values of c1, c2, wp max, and wp min are 2, 2, 1.2,and 0.4, respectively.

4.1. Particle Swarm Optimization Algorithm (PSO). If theneural network is used as the controller in the control system,the astringency of the training algorithm depends largely onthe choice of the initial weights of the network and the gen-eral approach is cut-and-trial. However, it is difficult toachieve for complex problems. It will directly lead to the poorparameter setting for the controller and poor control quality.Therefore, it is very important to optimize the weights of thenetwork by using the optimization algorithm. For the offlineoptimization of the controller parameters, an improved

5Journal of Sensors

algorithm based on the mechanism of chaos and the PSO, i.e.,the CPSO, is proposed.

The PSO adopts the velocity-position model and sets thereasonable inertia weight to balance the global and localsearch to make the algorithm easier to converge to the opti-mal or optimal solution. The standard PSO algorithm formu-las are shown as (8) [26].

Vi =wp ∗ Vi + c1 ∗ rand ∗ pbest − Posi+ c2 ∗ rand ∗ gbest − Posi ,

Pos = Posi + Vi,8

where i = 1, 2,… , n; n stands for the total number of particlesin the population; Vi stands for the moving rate; wp standsfor the inertia weight whose range of value usually is0.4~1.2; Posi stands for the position of the particle; pbest isthe location of the best solution in iteration; gbest standsfor the location of global best solution; rand stands for arandom number between 0 and 1; and c1 and c2 stand forthe acceleration constants. The values of c1 and c2 are 2, whichare determined on the basis of the existing research [37].The position and velocity of a particle in an n-dimensionalspace can be expressed as Posi = Pos1, Pos2,⋯, PosN andVi = v1, v2,⋯, vN , respectively. The fitness function is cal-culated by the method of self-defined objective function.The fitness value of each particle in each iteration is calcu-lated according to the demand. The optimal value of eachparticle which is searched by itself currently, and the optimalvalue in the current population should be stored for dynam-ically adjusting as experience. According to the characteris-tics of the determining effect of the system, the fitnessfunction associated with the time is required. So the integralof the absolute value of error multiply time is adopted as the

criterion which is also called the ITAE index. The system canobtain the advantages such as fast, smooth, and small over-shoot under the ITAE index. Its expression is as follows:

J =t

0t e t dt 9

The inertia weight is generally linear decreasing weightalgorithm as shown in the following formula:

wp iter =wp max −wp max −wp minIter max ∗ iter , 10

where Iter max stands for the largest evolutionary algebra, iterstands for the algebra, wp max and wp min stands for the maxi-mum inertia weight and the minimum inertia weight, respec-tively. The introduction of wp significantly improves theperformanceof thePSOalgorithmsuch as adjusting the searchability of particles in global and local. Also, the introduction ofwp offsets the problem of the standard PSO algorithm andmakes it apply to more practical problems [31]. Figure 6 illus-trates the overall view of particle swarm optimization.

In accordance with the above algorithm, the particle con-tinuously updates its velocity and position in the preset solu-tion space and ultimately converges to a suboptimal oroptimal location.

4.2. Offline Tuning Based on the CPSO. The chaos is a univer-sal phenomenon in a nonlinear system. The chaos phenom-enon has stochastic property, ergodicity, and regularity. Inthe optimization area, the ergodic property can be used asan optimization mechanism to escape from local optimums.The chaos has been a kind of novel global optimization tech-nique. People pay much attention to the research of the

x

1𝜇

NB NM NS ZO PS PM PB

Controllableparameters PID

ISP

de/dt

𝛥Kp 𝛥K

i𝛥K

d

e

Node of fuzzy rules

Weights matrix w

Membership function

ec

Online updatingof BP algorithm

u

y

Quantization factor

Scaling factor

r

×

−

Kp = K

p0 + FNN (𝛥Kp)

Ki = K

i0 + FNN (𝛥Ki)

Kd = K

d0 + FNN (𝛥Kd)

Figure 5: Schematic diagram of the FNN/PID compound controller structure.

6 Journal of Sensors

optimization method based on the chaotic search [29, 31, 36,38]. At present, some scholars have applied it to the optimi-zation of neural network weights and achieved good results.

Based on the three inherent properties of the chaos,including stochastic property, ergodicity, and regularity[39], the new superior individuals are reproduced by chaoticsearching on the current global best individuals. For theregularity and ergodicity property, the chaos searching cantraverse all states without repeating within a certain range.For the stochastic property applied to selection of individual,a stochastic selected individual from the current populationis replaced by the new superior individual. The particleswarm optimization-embedded chaotic search quickens theevolution process and improves the abilities to seek the globalexcellent result and convergence speed and accuracy.

The chaotic motion is usually generated by a logistic mapwhich is illustrated as follows:

Xc n + 1 = μXc n 1 − Xc n , n = 0, 1, 2,… ,M 0 < Xc 0 < 1 ,

11

where μ stands for the control parameter whose range ofvalue is (0, 4). The value of the chaotic control parameter μis larger, the chaotic degree is higher, and the populationstructure has suffered more destructive. In the running pro-cess of the CPSO algorithm, the control parameters shouldbe dynamically reduced or increased on the basis of the con-vergence of the population, which can reduce the structural

damage to the population and help population escape fromlocal optimizations. In engineering applications and aca-demic studies of CPSO, the value range of the chaotic controlparameter μ is usually from 0 to 4 [40]. The chaotic motion isvery sensitive to the initial value selection, and the differentinitial values will be different.

The basic principle of the CPSO algorithm is that chaosinitialization is adopted to improve individual quality andchaos perturbation is utilized to avoid the search beingtrapped in local optimum [31]. The process of the CPSO isconducted as follows:

Step 1. Encode the optimized parameter. The weight coeffi-cients are encoded as vectors and expressed as Para. Thenumber of particle swarm optimization is selected as N . Ran-domly produce n-dimensional vectors as z1 = z11, z12,⋯,z1n . Initialize the particle’s position with logistic chaos map-ping as xij = aj + bj − aj zij j = 1, 2,⋯, n, i = 1, 2,⋯,N − 1which is formed from zi+1j = μzij 1 − zij j = 1, 2,⋯, n, i =1, 2,⋯,N − 1 which is carried to the range of optimizedvariable.

Step 2. Initialize the system parameters, including c1 = c2 = 2,wp max = 1 2, and wp min = 0 4.

Step 3. In chaos-aided search, this paper sets appropriateiteration times λco and small value ξco as triggers for sharptuning of chaos. When the CPSO search accuracy is lessthan ξco in the λco iterations, save the current parameter toPnow , and assign it to the initial value of the chaotic searchP0, P0 = Pnow . Randomly initialize the chaotic variable E inthe range of (0,0.5) (the logistic map is symmetric in therange of (0,1), and the dimension is consistent with Pnow .Define i = 0. Use the following two equations to generatenew parameter values [30]:

P0 i + 1 = P0 i + αco 2Xc i − 1 ,Xc i + 1 = 4Xc i 1 − Xc i , i = i + 1,

12

where αco is the search radius and the system can traversethe search in a larger range by adjusting the value of αco.However, it is more time-consuming. Through this step, itis expected that the particle velocity and particle positionhave a proper update value. So the role of αco is onlysharp tuning and generally takes smaller values. If a bettervalue is obtained which can be saved as Pb, otherwise, αcoshould be adjusted.

Chaos phenomena widely existing in nonlinear systemshave stochastic property, ergodicity, and regularity, whichare widely applied in chaotic search to obtain optimizedsolution [41]. Chaos has also been used to optimize theweight of a neural network. The chaotic motion is mainlygenerated by the logistic map. The traversal trajectory isdirectly affected by the initial value. A slightly differentinitial value will directly cause the variations of the tra-versal trajectory. Therefore, the appropriate initial value

Initializes the size of the group,the random position of particles

and the initial velocity

Calculated fitness valuefor each particle

Is current fitness valuebetter than Pbest?

Assign currentfitness as new Pbest

Adjust the velocity and positionof the particle according to the

formula 8 and 9

Keep previousPbest

End

YesNo

Yes

Target or maxiumepochs reached

No

Figure 6: Overall view of particle swarm optimization.

7Journal of Sensors

of initial parameters has a significant impact on the reducetime-consuming. So we have chosen the appropriate initialvalue for relevant parameters and initialized the positionof the particle by chaos before running the fuzzy neural net-work/PID compound algorithm. In this way, a PC com-puter can meet the computation.

Step 4. The chaos strategy is enlargement as the end of thePSO algorithm. In the early stage of search, PSO tends toconverge faster, but in the later stage, it is easy to betrapped by local optimizations. Chaos could escape fromlocal optimizations and approach the global optimizations.In the CPSO algorithm, the chaotic method is applied to

Offline optimizationresults

Initialization ofweight parameter

Modelinitialization

Giving input to calculateobject’s output

Object inputs and outputs areused as network inputs

Calculate hidden layerand output layer output

Adjust the network weightparameter to minimize the

deviation between theoutput of the minimizedobject and the output of

the output layer

Update input/outputweight parameters

Arrivetermination condition

N

Y

Figure 7: Flowchart of online adjustment of the BP algorithm.

O X

YZ

Rollgear train

RollDC motor

Pitchrotation axis

Pitchgeartrain

Pitch DC motor

Rollrotation

axis

Figure 8: Three-dimensional CAD model for a three-axis ISP and its gimbal-transmission system.

8 Journal of Sensors

randomly generate particles in the initialization stage.Based on the standard PSO, the chaotic search strategy isapplied in two stages, which assists the PSO in searchingoptimizations in the first stage, extends their search scopesat the beginning of the second stage, and avoids gettinginto local optimizations, escape from local optimizations,and approach the global optimizations at end of second stage[42]. The chaos optimization is very time-consuming, and ifit is applied in a large scale, so the last step is auxiliary in theglobal scope. This step is the real search. With the aid of

chaos, the CPSO searches for a set of parameter values whichare denoted as Pf . Use the following two equations to opti-mize again:

Pf i + 1 = Pf i + βco 2Xc i − 1 ,Xc i + 1 = 4Xc i 1 − Xc i , i = i + 1

13

Traverse smaller range with βco as search radius, and thehigher value is used as the final parameter of the controller

Gimbal

x channel Motor

1Out1

ddthaFf

Friction subsystem

0.195

0.00104s + 1

Transfer Fcn5

s + 1

10s + 1

Transfer Fcn4

num(s)

den(s)

Transfer Fcn31

1.912s+2.7225

Transfer Fcn10

controlvari

To workspace

t4.mat

To File5

t3.mat

To File4

signal.mat

To File3

yout.mat

To File1

Step

Signalgenerator2

Scope8Scope5

Scope4

Scope3Scope1

Scope

Saturation

Product

Mux

Mux

fnncS

Level-2 MATLABS-function

1s

Integrator1

1s

Integrator-K-

Gain6

6.6

Gain4

w0

Gain

-K-

FOGX

du/dt

Derivative2

du/dt

Derivative

1

Constant

Clock1

Clock

|u|

Abs

wifx

wifx

thxthx

1Out1

Signalgenerator2

Product

fnncS

Level-2 MATLABS-function

1s

Integrator1

w0

Gain

1

Constant

Clock1

Clock

|u|

AbsOutput the ITEA of thesystem to the optimizer

FNN/PID controller

Obtain parametersfrom the optimizer

Figure 9: Simulink simulation diagram of the CPSO offline for the FNN/PID compound controller.

9Journal of Sensors

saved as Pfinal. If the accuracy is up to standard, the optimiza-tion is over. Otherwise, parameters should be recalculated byadjusting βco.

λco can refer to the total number of iterations of theCPSO algorithm. Its interval does not need to be toodense for designing to assist particles out of the localsmall. ξco generally refers to smaller values of the fitnessfunction in the possible range. When the fitness value isdifficult to be better, the particles can be adjusted by cha-otic motion in order to obtain new searching ability. αcoand βco play an important role in supporting and shouldensure that the results do not exceed the solution spaceof the optimization variables in order to avoid waste ofcomputing resources. At the same time, the value of themshould be ensured small to guarantee the chaos optimiza-tion is kept in a small range.

4.3. Online Tuning Based on the BP Algorithm. The BP algo-rithm is used to adjust the online simulation when theparameters of the offline suboptimal controller are obtained.The results of offline are close to the optimal values, becauseof the BP algorithm which is adjusted on the basis of the ini-tial parameters. Therefore, it is necessary to obtain better ini-tial suboptimal parameters in order to ensure real-timeonline adjustment. The online BP algorithm adjustment isadjusted according to the following formulas [38]:

c n + 1 = c n + xite ∗ ∂E∂c

+ η ∗ Δc n ,

σ n + 1 = σ n + xite ∗ ∂E∂σ

+ η ∗ Δσ n ,

wall n + 1 =wall n + xite∗ ∂E∂wall

+ η∗Δwall n ,

14

where wall is the weight matrix of the BP network; xite and ηstand for learning factor and momentum factor, respectively;and E stands for the mean squared error (MSE) calculatedwith the function as E = 1/2 rin k − yout k 2 [43]. Theprocessing is shown in Figure 7.

5. Simulation Model

Figure 8 shows the three-dimensional CAD model for athree-axis ISP and its gimbal transmission system. Figure 9

shows the simulation diagram of the CPSO offline for theFNN/PID compound controller. The simulation model isbuilt to simulate the whole ISP system under friction distur-bance which represents the effect of the main disturbance oncontrol precision. When the PSO and the CPSO are used tooptimize the initial value of the weight coefficient offline,the optimization program should be written in the file asthe optimizer. The controller outputs the parameters to theblock diagram with the program to constantly update thevalue of w0. It also outputs the value of time multiplied bythe integral of the error absolute value to the optimizer.The controller determines the control effect and then calcu-lates the optimization algorithm. In addition, all chosenparameters for the model have been gathered in Table 1 forquick and easy reference.

6. Results and Analysis

6.1. FNN/PID Controller Based on the CPSO. In this paper,the RMS is the abbreviation of the “root mean square”error of the angles in a period of time, which is an errorresult, calculated from the angle values shown inFigure 10. In Figures 11 and 12, the horizontal axis repre-sents the iteration times in the optimization process thatare dimensionless, and the vertical axis represents the fitnessvalues which are the effect of parameter optimization thatare also dimensionless.

As it is seen in Figure 10, after the initial value of theweight coefficient of the FNN/PID controller is optimizedby the CPSO algorithm, step response overcomes the oscil-lation problem held by the trial and error method. Theovershoot in the CPSO method is reduced from 1.3° RMSto 0.07° RMS with a great decline extent of 94.62%. How-ever, the stability of the system is decreased somewhatwhich is vibrated in a small error range of 0–0.01° RMS.For the whole range, the RMS error between 0 and 30 sec-onds is 0.1453° RMS.

As shown in Figure 11, the optimum individual fitness islarger at the previous stage which proves that the controlmethod is not easy to converge. In addition, the change ofthe optimum individual fitness is small at the end of theiteration whose optimal value at the end of the iterationis 0.5718.

Table 1: Chosen parameters for the model.

Parameter name Parameter symbol Value

The error input interface in FNN/PID control method e [−48, 48]The error change of input interface in FNN/PID control method ec [−4, 4]

The scaling factors of the fuzzy/PID controller Kp0, Ki0, Kd0 1, 0.005, 0.05

Chaotic control parameter μ [0, 4]

Acceleration constants c1, c2 2

Inertia weight wp [0.4, 1.2]

Iteration times λco 100

10 Journal of Sensors

The parameter optimization method improves theperformance of the system by traversing all parameters tooptimize the parameters. However, it is fundamentally basedon the principle of random, and the result can only beguaranteed to be suboptimal. So the method needs furtherimprovement which has not obtained ideal results whileusing in the ISP. In addition, the optimization algorithmhas a long design cycle, and the optimization process istime-consuming. Therefore, it is desirable to design aFNN/PID control method which can achieve a very goodcontrol effect without optimization.

6.2. FNN/PID Controller Based on the Composite ParameterOptimization. In this paper, the numerical results on thestabilization precision obtained three different methods,including the proportion integration differentiation (PID),the fuzzy neural network (FNN)/PID compound controller

based on trial and error method, and the FFN/PID basedon chaos particle swarm optimization (CPSO) and theback propagation (BP) algorithms, responding to the stepinput, are compared together. Compared with the PID,from 0 s to 30 s, the error of the stabilization precision ofthe FFN/PID based on the trial and error method is0.0432°, which is decreased up to 53.6% than that of thePID (which is 0.0931°). The errors of the stabilization preci-sion of the FFN/PID based on CPSO and the BP algorithmsare 0.0214°, which is decreased up to 77% than that of thePID. Compared with the FFN/PID based on the trial anderror method, the errors of the stabilization precision of theFFN/PID based on CPSO and the BP algorithms havedecreased up to 50.5%. From above, it can be concluded thatthe FNN/PID compound controller can achieve the high sta-bilization precision with good disturbance rejection ability.-Table 2 shows the numerical results on the stabilizationprecision obtained by three different methods responded tothe step input.

Figure 12(a) shows that the control effect has been con-siderably improved which demonstrate the effectiveness ofthe improved FFN/PID approach. And then, the optimalindividual fitness in the optimization process is shown inFigure 12(b); the control system is indeed able to find bet-ter individual fitness values with the increasing of itera-tions. In addition, the difference in the magnitude of thefitness values is always small and the variation of optimumfitness is less than 0.0001°. Therefore, it can be shown thatthe improved FNN/PID control algorithm has less depen-dence on the initial value of the weight coefficients and iseasy to obtain good control results. Based the analysis oftheory and simulation, the feedback response times ofthe speed loop and position loop of FNN/PID control sys-tem are 0.617 s and 1.376 s, respectively.

7. Conclusion

In this paper, to improve the convergence of the fuzzy neuralnetwork (FNN)/proportion integration differentiation (PID)compound controller applied for an aerial inertially stabilizedplatform, a composite parameter optimization method isproposed. Based on both the chaos particle swarm optimiza-tion (CPSO) and the back propagation (BP) algorithms, thecontroller parameters are optimized offline and fine-tunedonline together. In this way, the FNN/PID compound con-troller can realize excellent adaptive convergence so as tohigh stabilization precision under multisource dynamic dis-turbances. To verify the method, the simulations are carriedout. The main conclusions are as follows:

(1) The results show that depending on the proposedcomposite parameter optimization method, theFNN/PID compound controller can reach good abil-ity in self-learning and self-adaptation, by which thehigh stabilization precision with good disturbancerejection ability is achieved

(2) Compared with the PID, the FFN/PID methodshave excellent stabilization precision and the

Time of iteration

Optimal individual adaptive value

Fitn

ess v

alue

9

8

7

6

5

4

3

2

1

00 20 40 60 80 100

Figure 11: Curves of optimal individual fitness change of optimizedparticle.

Time (s)

Angl

e (°)

0 5 10 15 20 25 30

1

0.8

0.6

0.4

0.2

0

Step inputCPSO based FNN/PID

Figure 10: Step response of the FNN/PID controller.

11Journal of Sensors

disturbance rejection ability. Furthermore, com-pared with the FFN/PID based on the trial anderror method, the FFN/PID based on the compositeparameter optimization method is more prominent,by which the stabilization precision is improved upto 50.5% than the former

(3) The CPSO algorithm has strong global search capa-bility, which could escape from local optimizations

and approach the global optimizations, by whichthe FNN/PID compound controller can realizeexcellent adaptive convergence

Nomenclature

BP: Back propagationc1, c2: Acceleration constants

1

0.8

1.15

1.1

1.05

1

0.95

0.9

1 2 3 4

Step inputCPSO&BP-based FNN/PID

5 6

0.6

0.4Ang

le (°

)

0.2

0

−0.20 5 10 15

Time (s)20 25 30

(a)

0.043184

0.043182

0.04318

0.043178

0.043176

0.043174

0.043172

0.04317

0.0431680 10 20 30 40 50

Time of iteration60 70 80 90 100

Ada

ptiv

e val

ue

Optimal individual adaptive value

(b)

Figure 12: Optimal result map of weight coefficient matrix of the FFN/PID controller. (a) Response curve of the FFN/PID control to the stepinput. (b) Optimization process curve of optimal individual adaptive value.

Table 2: The numerical results on the stabilization precision obtained three different methods responding to the step input.

Control methods 0–30sImprovement (%)FFN/PID vs. PID

Improvement (%) FFN/PID with optimizedparameters vs. FFN/PID with unoptimized parameters

PID/RMS (°) 0.0931 — —

FFN/PID based on trial and error method/RMS (°) 0.0432 53.6 —

FFN/PID based on composite parameteroptimization method/RMS (°)

0.0214 77.0 50.5

12 Journal of Sensors

c, σ: The centers and widths of activa-tion function

CPSO: Chaos particle swarmoptimization

e: The error input interface in theFNN/PID control method

Ex, Ey, Ez: Photoelectric encoders installedon R-gimbal, P-gimbal, andA-gimbal

Er , Ep, Ea: Photoelectric encoder measuringrelative angular between gimbals

ec: Theerrorchangeof input interfacein the FNN/PID control method

FNN: Fuzzy neural networkGp, Gr, Ga: Rate gyros measuring the iner-

tial angular rates of P-gimbal,R-gimbal, and A-gimbal

G-pos, G-spe, and G-cur: The controllers in the positionloop, speed loop, and current loop

ISP: Inertially stabilized platformPID: Proportion integration

differentiationJ l: The moment of inertia of the

gimbals along the rotation axisJm: The moment of inertia of the

motork: Constant of exponential reach-

ing law representing reachingspeed

Kp0, Ki0, Kd0: The initial parameters of the fuz-zy/PID controller

Kt : The torquecoefficientof themotorkT : Torque coefficient of motorL: Inductance of torque motorMr, Mp, Ma: Gimbal servo motors of R-gim-

bal, P-gimbal, and A-gimbalN : Transmission ratioPosi: The position of a particleR: Resistance of torque motorVi: The velocity of a particlew: Connection weight matrixwall: The weight matrix of the BP

networkwp: Inertia weightwp max,wp min: The maximum inertia weight and

the minimum inertia weightXi: The input variable of Gaussian

function and bell-shapedfunction

μ: The control parameter of thechaotic motion

αco: Search radiusλco: Iteration timesωs: Critical Stribeck speed.

Data Availability

The data used to support the findings of this study areincluded in the article.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

This project is supported in part by the National NaturalScience Foundation of China (Grant nos. 51775017 and51375036), by the Beijing Natural Science Foundation (Grantno. 3182021), and by the Open Research Fund of the StateKey Laboratory for Manufacturing Systems Engineering(sklms2018005).

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