Application of improved PSO-based to neural network control system of parallel mechanism

Chang-jian WANG，Peng WANG（School of Mechanical Engineering，Yangtze University，Jingzhou 434000，China）

Application of improved PSO-based to neural network control system of parallel mechanism

Chang-jian WANG，Peng WANG*
（School of Mechanical Engineering，Yangtze University，Jingzhou 434000，China）

As the traditional PID neural network could not effectively control the real-time nonlinear multivariable system，this paper proposed a new type of multivariable adaptive PID neural network controller.This control system could put out feedback and activation feedback，with the function of proportion，integration and differentiation.We used the Particle Swarm Algorithm which is based on the solution space division to optimize the parameters of the controller.It also could eliminate effect of initial values on the accuracy of the controller and can be applied to the parallel mechanism control system.As the simulation results shown，controller had higher precision，better robustness and adaptability.This research provided a theoretical basis for the optimization design and performance analysis of the parallel mechanism.

PID neural network，Parallel mechanism，Improved particle swarm algorithm

Hydromechatronics Engineering

http：//jdy.qks.cqut.edu.cn

E-mail：jdygcyw@126.com

1 Introduction

Comparing to the series robot，the parallel robot has high stiffness，strong bearing capacity，high precision and compact structure.It could be suitable for some applications in machining，aircraft manufacturing，and health care which have small work space and large load strength.

With the rapid development of computer technology and artificial intelligence，people integrate mathematical models and operating experience into the computer in order to control the entire mechanical system［1］. It is very difficult to establish an accurate model of control system of parallel mechanism due to typically nonlinear multivariable systems and uncertainties and other factors outside interference.So PID control and its combination with other control theory can be used to solve such problems and favored by the majority of researchers.Neural network has powerful computing，strong robust，high fault tolerance and self-learning that can be approached to continuous linear function，but the presence of slow learning speed，many input parameters and poor dynamic performance make it not easy to achieve in reality.To solve this problem，the advantages of PID controller and neural network were combined and then a new neural network PID controller（PIDNNC）was brought up that it had robustness，high control accuracy and could overcome the above drawbacks［2］.This paper introduced an improved Particle Swarm Optimization（PSO）based on the solution space and put it into PIDNNC in order to figure out the local minima due to the gradient descent method of adjusting the weights and thresholds.This method not only improved the learning speed and convergence rate，but also obtained a better accuracy，sta-bility and convergence.Furthermore it provided a theoretical basis for improving control precision of parallel mechanism.

2 Improved particle swarm optimization

PSO is an optimized algorithm which makes use of group collaboration to achieve the global intelligent research.PSO imitate the process of bird population prey：each particle in the PSO“flights”towards to the optimal direction based on a search of all the particles and their own experience［3-4］.Firstly，PSO initializes group of particles N，and then finds the optimal solution through an iterative process.The extreme speed and position of particles are updated by tracking the personal best position and global best position in each iteration.Update formula is as follows：

Vi：evolution of the ithparticle velocity；Xi：position of the ithparticle；pBest［i］：the“best”position of the ithparticle；g：the“best”position in group；w：inertia weigh；c1，c2：acceleration factor；rand（t）：random function，generate［0，1］of the random number.

The solution space is derived from a term of linear algebra which is defined as follows：if ξ1，ξ2，…，ξnare N solutions of homogeneous linear equations，then any linear combination of their c1ξ1+c2ξ2+…+cnξnis also the solution vector of homogeneous linear equations.The collection of all the solutions formed a vector space，which is called solution space［5］.It can be divided precisely and refined PSO.But one of the most critical factors is how to determine the extreme value area p，when p is stability as well as other areas are basically stable.How to divide solution space is as follows.

1）Initialize attribute of particles，such as equally spaced and speed distribution；

2）Record test value and solutions of statistics for each particle，all the particles are ranked according to the initialized attribute，and then identify the most value area and extreme area of every range and calculate the probability p；

3）If the probability p is stable，output the value；otherwise return to（1）with doubling the size of particle swarm.

Determination of the probability p is that it assumed p1，…，pnafter n times testing，if p satisfy i≥n/2 andwe define p is stable，moreover β≤0.001 could meet test requirements.

In the researching process of particle swarm，every particle is constantly pursuing the known optimal position.But it also could cause other particles chasing the local extremes when it becomes the temporary optimal location，hence the whole population into this local extreme.Therefore，to solve this problem of blind search，we first divided the solution space of all the particle swarm into several regions；if there was only one extreme value in a certain area，then blind search could work：the particles could be automatically tend to it.Space can be divided equally，randomly or with the graphics division（such as triangles，squares）；each interval was an independent group of small particles.Interval extreme could be found in each area with performance standards，and then compare each interval extreme；finally，find a whole range extreme position for an optimum solution.

3 Controller design of parallel mechanism

3.1 Multivariable control PIDNNC

PID control，produced in the early 20th century，has dominated the field of automatic control，depending on its simple structure，good stability and flexible handling.Neural network with its own self-correction and adaptive capacity has been widely adopted in different situations.A new controller PIDNNC，in which structural features and control laws were effectively combined，is shown in Fig.1.There are input section ej（j=1，2，…，s），one output section and three hidden layers；hidden layer，input and output end existed recursive feedback loop；there is a linear activation function in the hidden layer and output layer［6］.In the controller，the first node a1of hidden layer contained a dynamic output feedback and record function which could feedback the weighting sum to node n1，while the second node a2does not had feedback；the third node a3has active feedback and it delayed with minus units of output after weighting sum of nodes n3and regarded it as a new input to n3.

As is shown in the Fig.1，PIDNNC is negative feedback loop；input is rj（k）（j=1，2…s），output is yj（k）（j=1，2，…，s），system output error is ej（k）= rj（k）-yj（k）.The controller’s output at time k in hidden layer is αi（k）（i=1，2，3）

Network terminal output：

Compared to formula（1）（2）（3）to（5），（1）presents the integral feature like the PID control.（2）is of proportion character which has activated feedback and（3）reflects the differential aspect.Unlike previous PID neural network，which is caused by controller that contains the output feedback and self-feedback network hybrid recursive composition，PIDNNC is designed convenient，simple structure and the determined number of nodes in the hidden layer.In addition，three sets of hidden layer weights w1j，w2j，w3j（j=1，2，…，s）are similar to the proportion，integration and differential that make physical meaning of parameters relatively clear.Multivariable controller is designed according to the complexity of the object and this process is more convenient than conventional PID.

Fig.1 ControIIer of PIDNNC

In the design process of the controller，we need to determine the number of input layer，hidden layer and output layer first，and then to adjust the network weights w1j，w2j，w3j（j=1，2，…，s）and output weights w1（k），w2（k），w3（k）to obtain better properties neural network.According to（4），the characteristics of PIDNNC are determined by the weights of hidden layer while the rule of output layer is summation which function is linear.Therefore，in order to decrease training time and study design，the output weights wi（i=1，2，3）is set to 1，and learning optimization parameters to wij（i=1，2，3；j=1，2，…，s）.

In this article，we made the improved PSO into m file using Matlab，and then optimized the objective function with Sim function.Firstly，the initial values of parameters were entered into the parameter matrix X，then system block diagram was built by Simulink and saved to mdl format.Finally，used the Sim function to write the objective function program and optimized it with m file.

3.2 Controller of parallel mechanism

Fig.2 displays the 3-TPT parallel mechanism，which is composed of fixed platform，moving platform，driven rod and connecting rod.Both moving platform and fixed platform are equilateral triangle，each drive rod is connected to parallel mechanism with Hooke joint，so as the moving platform and fixed platform. Three drive rods are driven by servo motor and adjust the position of movable platform by changing the length.They withstand external forces and torque［7］.

Fig.2 3-TPT paraIIeI mechanisms

Degree of freedom can be deduced by KutzbachGrable：

F：DOF；n：the number of component；g：kinematic pair；fi：the relative freedom of kinematic pair of i-th.

In the parallel mechanism，n=8，g=9，each Hooke joint has 2 rotational DOF and each moving pair has 1 DOF，socording to（6）F=3，the DOF of 3-TPT parallel mechanism is 3.

In this paper，the model of parallel mechanism was established by Simulink in SimMechanics simulation and integrated to the control system［8］.The model of PIDNNC is shown below.

Fig.3 ModeI of PIDNNC

Fig.4 System simuIation diagram

4 Improved PSO algorithm steps

When PSO optimized to PIDNNC，the objective function of the controller is fitness function；to search the optimal position by improved PSO is to minimize mean square error，fitness function is as follow：

Where，l：sampled data；s：the number of input node；rj（k）-yj（k）：output error.Optimization steps are as follows.

1）PIDNNC controller and particle swarm initialization parameter is set according to the number of input layer neurons of controlled object［9］，hidden layer nodes are set 3（Kp，Ki，Kd）；initialized the population of position and velocity，set test number M and divided entire population into n subintervals.

2）Put the values of Kp，Ki，Kdobtained by using conventional calculation as an initial value of hidden layer weights wij（0），then set output layer weights wi=1（i=1，2，3），computing u（0）［10］.

3）Calculated a1（k），a2（k），a3（k）and output u（k）；set k=k+1，return to recalculate until the output meet accuracy requirements.

5 Simulations

Set parameters of 3-TPT parallel mechanism：R= 600 mm，r=200 mm；the size of provision population is 200，maximum number of iteration is 200，acceleration factor c1=2，c2=2，maximum speed v=0.2，inertia weight w=0.8.

This paper performed a contrast experiment between traditional PID and PIDNNC optimized by improved PSO.Figure 5 is an improved PSO evolutionary curve，it can be seen that the it converges very fast early，later to slow down when search the optimal solution.This method could solve effectively the problem of local convergence.And in Fig.6，under the signal control of sine wane，PIDNNC optimized by PSO can adjust three output parameters online，accuracy and systematics error are improved and displacement is better.

Fig.5 PSO evoIution curve

Fig.6 Contrast curve of dispIacement and controI error

6 Conclusions

This paper introduced a new PSO based on divided solution space and put it into the design of a new multivariate controller PIDNNC which effectively solved the problem of multivariable nonlinear systems of traditional PID neural network.Hidden layer of the controller had the effect of proportional，integral and derivative at same time it had better stability，accuracy and robustness.Taking the improved PSO to optimized neural network system overcomed the problem of local minimum caused by the use of gradient descent，this made selection and learning of neural network more simpler，convergence more faster and looking for solutions more accurately.

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基于改進的PSO在并聯(lián)機構(gòu)神經(jīng)網(wǎng)絡(luò)控制系統(tǒng)中的應(yīng)用

王長建，王 鵬*
長江大學(xué)機械工程學(xué)院，湖北荊州 434000

針對傳統(tǒng)PID神經(jīng)網(wǎng)絡(luò)不能實時有效地控制非線性多變量系統(tǒng)的問題，設(shè)計了一種新型多變量自適應(yīng)PID神經(jīng)網(wǎng)絡(luò)控制器。該控制器的隱含層帶有輸出反饋和激活反饋，實現(xiàn)了比例、微分和積分功能。利用一種基于解空間劃分的改進粒子群算法對控制器參數(shù)進行優(yōu)化，消除了初始值對控制器準(zhǔn)確性的影響，并將控制器應(yīng)用于并聯(lián)機構(gòu)控制中。由仿真結(jié)果可知：控制器控制精度高，魯棒性和自適應(yīng)性較強。這一研究為并聯(lián)機構(gòu)的精準(zhǔn)控制和優(yōu)化設(shè)計提供了理論基礎(chǔ)。

PID神經(jīng)網(wǎng)絡(luò)；并聯(lián)機構(gòu)；改進PSO算法

10.3969/j.issn.1001-3881.2015.12.010Document code：A

TH165+.2

1 July 2014；revised 17 February 2015；accepted 5 March 2015

Chang-jian WANG，Professor.E-mail：wangchangjian2468@ 163.com

*Corresponding author：Peng WANG，Master.

E-mail：47361222@qq.com