Jian-xin ZHOU, Jian-xiong HUANG, Zhao LI

(College of Electrical Engineering， North China University of Science and Technology， Tangshan 063000， China)

Abstract: The traditional control method could accurately decouple the strip flatness and gauge complex system owning to the characteristics of heavy delay, nonlinear and strong coupling, which could led to lower control precision. Based on immune mechanism, an improved particle swarm optimization algorithm was developed, and the weights of PID neural network (PIDNN) were optimized by using this algorithm to form a new PIDNN controller. Two new PIDNN controllers were adopted to control the complex system of strip flatness and gauge to reduce the coupling effect of system. The simulation results showed that the present algorithm has more obvious advantages than that of the previous PIDNN decoupling control in both dynamic and static aspects. This study could provide some references for the decoupling problem in the control field.

Key words: Decoupling control, PID neural network, Immune genetic algorithm, Strip flatness, Gauge

1 Introduction

In the productive process of sheet and strip, strip flatness and gauge are two important parameters in industrial control system. In the early stage, these two parameters are generally distinguished and studied independently[1]. In fact, the strip flatness and gauge complex system should be considered as a comprehensive multivariable control system. The complex real-time system exhibits lower control effect owing to the nonlinear characteristics, heavy delay and strong coupling. Therefore, it is urgent to develop applicable control theory and means to overcome this situation.

On basis of practices, the traditional feed-forward decoupling control method was not applicable to the adjustment of system model parameters, and its implementation was more complicated[2-3]. Aiming at cold continues rolling system, the multivariable control system of strip flatness and gauge on basis of fuzzy RBF neural network could meet the requirements of disturbance rejection and self-adaption[4]. However, the decoupling control system of strip flatness and gauge of cold continuous rolling plate, based on neural network with time delay and PID, still had good robustness when external disturbances and model parameters were uncertain[5]. At present, although the above methods could effectively solve the coupling problem of strip flatness and gauge system, there are still greater improvement room in control error accuracy and response speed.

Based on the PID neural network algorithm, the immune particle swarm optimization algorithm was improved in this study, and the algorithm was used to optimize the initial weights of PIDNN. In addition, the superiorities of this algorithm has been verified by the simulation results.

2 Strip flatness and gauge model

Based on many formulas such as spring formula, plate shape formula, plastic formula, etc., the system model of strip flatness and gauge could be constructed[6].

1) The equation of thickness increment was as follows.

(1)

2) The equation of strip flatness incremental was as follows.

(2)

Where, ΔSwas the variation of roll gap; ΔFwas the variation of bending force of working roll;CFandCPwere the bending stiffness coefficient and the mill longitudinal stiffness coefficient, respectively.Hd=Hc-Hθ，HcandHθwere the middle thickness and edge thickness of incoming materia, respectively.Hwas the average thickness of incoming material;hwas the average thickness after rolling;kPwas the transverse stiffness coefficient of the mill;Qwas the rolling plasticity factor;KFwas the lateral bending roll stiffness coefficient; Δσ0and Δσ1were the changes of the transverse tensile stress difference between the inlet and outlet, respectively;Ewas the modulus of elasticity of rolled piece.

The integrated system modeling of strip flatness and gauge could be expressed by the following matrix equations.

(3)

The detection instruments in the system of strip flatness and gauge have a pure lag time constant, which can be regarded as a delay linkeτs. At the same time, the first inertial linkG1(s) andG2(s) approximated the hydraulic bending system and hydraulic pressing system of working roll, respectively.

The coupling model of the integrated system was shown in Fig.1. Where,u1andu2were the working roll bending force adjustment and the press down adjustment amount, respectively; A was the difference in tensile stress between plates, B was the thickness after rolling.

Fig.1 Strip flatness and gauge system coupling model

3 Improved immune particle swarm optimization (PSO) algorithm

PSO algorithm has been proposed since 1995 by American scholars Kennedy J. and Eberhar R.C., after they were inspired by the birds’ forage activities. PSO algorithm has received widespread attentions because of its simple concept, easy implementation and fast searching ability. However, with the development of basic PSO algorithm, the search speed becomes more slower in later stage, and particle swarm could easily show strong tendency, thus it will fall into local optimization[7].

3.1 Explore behaviors of particles

Suppose at a D-dimensional search space,xidrepresent the position point of theith particle in thedth dimensional space, with corresponding velocity ofvidafter the two optimal values were found, velocity and location of particles will be updated by Eq.4 and Eq.5.

c2r2(pg-xid)

(4)

(5)

An appropriate inertia weight was capable of improving the performance of algorithm and enhancing its optimization ability, as well as reduce iteration times. For better effects of PSO, the inertia weight strategy was optimized in this study through nonlinear dynamic strategy as the following formula[8].

(6)

Where,w(t) was the current inertia weight;wendwas inertia termination weight;wstartwas defined as initial inertia weight.Kwas the control factor, usually,K=3. Therefore, the position updating equations of particle swarm optimization algorithm were as follows.

c1r1(pi-xid)+c2r2(pg-xid)

(7)

(8)

3.2 Immune information transacting mechanism

In the immune particle swarm optimization algorithm on basis of immune selection, the immune information processing mechanism was applied. The problem to be solved was regarded as an antigen, and each antibody corresponding to each antigen was the solution of the problem. At the same time, each antibody was a single particle in the particle swarm. The affinity between antibody and antigen was solved by the fitness value of PSO algorithm, and it reflected the satisfaction degree of fitness values to objective function and constraint conditions. The differences among particles, i.e. the diversity of the particle swarm, were represented through the affinity between antibodies[9]. Benefited from immune memory function and self-regulation function, the diversity of particle swarm was effectively guaranteed, thus it will enhance the convergence performance of the algorithm.

4 PIDNN decoupling control systembased on improved PSO optimization

4.1 Structure and algorithm of PIDNN control system

PID neural network was a multi-layer forward neural network, which selected the input and output functions of hidden layer neurons according to PID regular pattern, making them changed scale elements, integral elements and differential elements with the functions of proportion, integration and differentiation, respectively. The structure of PIDNN for anN-input-N-output multivariable system was 2N×3N×N, and network were formed in the way thatNsame neural subsets in parallel[10]. The structure of PIDNN strip flatness and gauge decoupling controller was showed in Fig.2.

Fig.2 PIDNN strip flatness and gauge decoupling controller

Where,r1andr2were the system settings, representing the set values of the strip flatness and gauge;v1andv2were the output values of PID neural network;Y1andY2were the output values of this system. The output values of PID neural network decoupling control system was affected by system set values and actual output values. In the PID neural network, the hidden layer contained three neurons, and the sequence of input and output parameters was as following: proportion (P), integral (I) and differential (D). The hidden layer and output layer of sub-network were cross-coupled with each other. The output layer of network was capable of completing integration of PID control rate to form the control input of objects, thus solved the decoupling and control problem of multi-variable system.

PIDNN could achieve decoupling control due to its nonlinear mapping ability, parallel crossover structure and PID processing ability of hidden layer neurons. In order to complete decoupling control of the system, PID neural network took the objective function as the minimum objective, and adjusted the network weight autonomously through training. At any sampling time, the calculation model of PIDNN was as follows.

(1) Input layer

The input and output formulas of the layer neurons were as followis.

xsi(k)=Xsi(k)

(9)

Where,xsiwas the output value of input layer, whileXsiwas the input value of input layer.srepresented the number of parallel sub-networks,s=1,2,3,…,n;i=1,2.

(2) Hidden layer

The input values for each neuron in this layer were the same, and the function was as follows.

(10)

The output equation of each neuron in this layer was as follows.

Proportion neurons:

(11)

Integral neurons:

(12)

Differential neurons:

(13)

Where,netsjwas the input value of hidden layer.wijwas the connection weight between input layer and hidden layer for each sub-network.usjwas the output value of hidden layer. Usually,j=1,2,3;s=1,2,…,n.

(3) Output layer

The input of this layer was the weighted sum of output values of all the neurons in the hidden layer, and the function was as follows.

(14)

The input and output functions of this layer were as follows.

(15)

Where,uhwas the output value of neuron in output layer.wjkwas the connection weights between hidden layer and output layer.yhwas the input value of neuron in output layer.hwas the number of output neurons. Usually,h=1,2,3,…,n.

4.2 Training PIDNN weights based on improved immunity PSO

The algorithm flow of PIDNN decoupling controller on basis of the improved PSO was as follows.

Step1Initializing particle swarm, i.e., the velocity and location of N connection weights of PIDNN were initialized. The individual extremum and global extremum of each particle in the initialization group could be obtained by calculating the fitness value of these particles.

Step2Producing an immune vaccine was equivalent to use the best fitness value as the immune vaccine.

Step3The velocity vector and location vector of each particle were updated according to Eq.4 and Eq.5.

Step4The fitness value was calculated according to Eq. 16, and was compared with. Meanwhile, needs to be updated with the increasing fitness value.

(16)

Step5The adaptive values ofMparticles that randomly formed in each iteration were solved.

Step6From the combination ofN+Mparticles,Nsuitable particles were determined by a concentration selection mechanism.

Step7The fitness values of particles after inoculation was calculated using the R particles in the immunization group generated in step (2). If the fitness was less than the original fitness, the original fitness will be retained. Otherwise, the inoculated particles would be treated as new particles and replaced the original particles. In this way, a new generation group would be formed.

Step8Updating the fitness of new generation population, followed by update of individual extremum and global extremum.

Step9The steps of (2)-(8) were repeated until the termination condition was met. Subsequently, the weight coefficients (and) of PIDNN decoupling controller were output.

5 Simulation results

The simulation data were as follows.

KF=8 000 kN/mm,KP=80 000 kN/mm

CP=4 700 kN/mm,CF=8000 kN/mm

Q=6 765 kN/mm,E=206 kN/mm

Q=4.25 mm,h=3.29 mm,K1=1,K2=1,T1=0.01 s,T2=0.01 s, the sampling timeT=0.02 s， delay time constantτ=0.04 s. The disturbance variableη1andη2were not be considered. The desired output: the strip shape and thickness after rolling wereσ1=0.5 MPa andh=0 mm, respectively. Subsequently, the above data were introduced into the model of strip flatness and gauge integrated system.

When the traditional PIDNN decoupling control method and the PIDDNN decoupling control method based on the improved PSO algorithm were adopted, the system response curves were obtained as shown in Fig.3 and Fig.4.

Fig.3 Response curves of traditional PIDNN decoupling control

Fig.4 Response curves of PIDNN decoupling control based on improved PSO algorithm

Simulation results showed that when the set value of system was suddenly changed, the other set value could be restored to the original state after a short period of fluctuation, which indicate that both two methods could achieve decoupling control. Compared Fig.3 with Fig.4, it could be observed that the improved method could quickly approach the system set value. At the same time, the overshoot was reduced correspondingly. This result showed that PID neural network decoupling control based on the improved immunity particle swarm optimization algorithm was effective.

6 Conclusions

In this study, a PIDNN decoupling controller based on improved immune particle swarm optimization algorithm was proposed according to the nonlinear and strong coupling characteristics of strip flatness and gauge system. In order to replace traditional PID neural network backpropagation algorithm, the improved immune particle swarm optimization algorithm was used to adjust the weights between the neurons in the PID neural network. The simulation results indicated that the control system had better control performance, and the effectiveness of the proposed control strategy for the nonlinear system has been verified.