Integrated Modeling and Intelligent Control Methods of Grinding Process

Abstract

The grinding process is a typical complex nonlinear multivariable process with strongly coupling and large time delays. Based on the data-driven modeling theory, the integrated modeling and intelligent control method of grinding process is carried out in the paper, which includes the soft-sensor model of economic and technique indexes, the optimized set-point model utilizing case-based reasoning, and the self-tuning PID decoupling controller. For forecasting the key technology indicators (grinding granularity and mill discharge rate of grinding process), an adaptive soft-sensor modeling method based on wavelet neural network optimized by the improved shuffled frog leaping algorithm (ISFLA) is proposed. Then, a set point optimization control strategy of grinding process based on case-based reasoning (CBR) method is adopted to obtain the optimized velocity set-point of ore feed and pump water feed in the grinding process controlled loops. Finally, a self-tuning PID decoupling controller optimized is used to control the grinding process. Simulation results and industrial application experiments clearly show the feasibility and effectiveness of control methods and satisfy the real-time control requirements of the grinding process.

1. Introduction

Grinding process has complex production technique and many influencing factors, such as the characteristics of the ore fed into the circuit (ore hardness, particle size distribution, mineral composition, or flow velocity), the flow velocity of water fed into the loops, and the changes of the cyclone feed ore. Grinding process is a serious nonlinear, strong coupling, and large time delay industrial production process. Obtaining the optimal control results by the traditional control method is difficult. Scholars at home and abroad have carried out many advanced control strategies for the grinding process, such as fuzzy control [1–3], neural network control [4], soft sensor modeling [5–8], and other advanced control technology [9–12]. Reference [3] proposed a multivariable fuzzy supervisory control method composed by the fuzzy supervisor, loop precedent set-point model, and the particle size soft-sensor model. Reference [4] studied the grinding process with non-linear, multivariable, time varying parameters, boundary conditions, and fluctuations complex features and proposed an integrated intelligent model for dynamic simulating, of the grinding and classification process.

Because of the limitations of the industrial field conditions and a lack of mature detectors, the internal parameters (particle size and grinding mills discharging rate) of the grinding process is difficult to obtain the real-time quality closed-loop control directly. The soft-sensing technology can effectively solve the predictive problem of the online measurement of the quality indices. Therefore, the soft-sensor model according to the auxiliary variables can be set up in order to achieve the particle size and grinding mills discharging rate for the real-time forecasting and monitoring, which has great significance on improving the grinding process stability and energy conservation. Domestic scholars have proposed many soft-sensor models, such as neural network model [5–7] and the case-based reasoning technology [8]. Combining the actual working conditions of the grinding classification process of [5] proposed a RBFNN-based particle size soft-sensor model. Reference [6] introduced a grinding size neural network soft-sensor model and adopts the real-coded genetic algorithm for training multi-layer neural network. Reference [7] put forward a multiple neural network soft sensor model of the grinding roughness on the basis that multiple models can improve the overall prediction accuracy and robustness. Reference [8] adopted the case-based reasoning (CBR) technology for predicting the key process indices of the grinding process. These algorithms do not effectively settle off the online correction of the soft-sensor model.

Aiming at the grinding industrial process, the integrated automation control system is proposed, which includes the economic and technical indices soft sensor model, the set-point optimized model based on the case-based reasoning method, and the self-tuning PID decoupling controller. Simulation and experimental results show the feasibility and effectiveness of the proposed control method for meeting the real-time control requirements of the grinding production process. The paper is organized as follows. In Section 2, intelligent control strategy of grinding process is introduced. An adaptive soft-sensor modeling of grinding process based on SFLA-WNN is presented in Section 3. In Section 4, the optimized set-point model utilizing case-based reasoning is summarized. In Section 5, the design of self-tuning PID decoupling controller of grinding process is introduced in detail. Finally, the conclusion illustrates the last part.

2. Intelligent Control Strategy of Grinding Process

2.1. Technique Flowchart of Grinding Process

Grinding process is the sequel of the ore crushing process, whose purpose is to produce useful components of the ore to reach all or most of the monomer separation, while avoiding excessive wear phenomenon and achieving the particle size requirements for sorting operations. A typical grinding and classification process is shown in Figure 1.

Figure 1 

Technique flowchart of grinding process.

Grinding process is a complex controlled object. There are many factors to influence this process, such as the milling discharge ratio 𝑌₁, milling granularity 𝑌₂, the milling ore feed velocity 𝑈₁ and the pump water feed velocity 𝑈₂, water amount of ore feed 𝐴, new ore feed 𝐵, suboverflow concentration 𝐶, milling current 𝐷, and classifier current 𝐸. 𝑉₁ and 𝑉₂ represent the sand return and water resupply.

2.2. Intelligent Control Strategy of Grinding Process

The block diagram of the data-driven integrated modeling and intelligent control strategy of the grinding process is shown in Figure 2 [11].

Figure 2 

System configuration of the integrated modeling and intelligent control methods of grinding process.

The integrated modeling and intelligent control system of grinding process includes the adaptive wavelet neural network soft-sensor model of economic and technique indexes, the optimized set-point model utilizing case-based reasoning technology, and the self-tuning PID decoupling controller based on the ISFLA. Firstly, the milling granularity and the discharge ratio predicted by the soft-sensor model are named as the input parameters of the set-point model. Then, through the case-based reasoning, the milling ore feed ratio and the water feed velocity of the pump pool are optimized. Finally, the self-tuning PID decoupling controller is adopted to achieve the optimized control on the milling discharge ratio and milling granularity ultimately.

3. Soft-Sensor Modeling of Grinding Process

3.1. Structure of Soft-Sensor Model

The structure of the proposed wavelet neural network soft-sensor model optimized by the improved SFLA is shown in Figure 3 [13], seen from Figure 3, 𝐴 is the water amount of ore feed, 𝐵 is the new ore feed, 𝐶 is the concentration of sub-overflow, 𝐷 is the milling current and 𝐸 is the grading machine power. For the key process indicators of grinding process (feedback grinding granularity and the discharge rate), the two multi-input single-output wavelet neural network soft-sensor model is set up. (1) Input variables are 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and the previous moment of grinding granularity. Grinding granularity is output for the feedback. (2) Input variables are 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and the previous moment milling discharge ratio. The discharge ratio is output for the feedback. The differences between the predictive values and the actual values are used to optimize the parameters of wavelet neural network through the improved shuffled frog leaping algorithm.

Figure 3 

Soft-sensor model structure of grinding process.

Considering a multi-input single-output (MISO) system, the training sample set can be expressed as 𝐷={𝑌,𝑋_𝑖∣𝑖=1,2,…,𝑚}. 𝑌 is the output variable. 𝑋_𝑖 represents the 𝑖th input vector and can be expressed as 𝑋_𝑖=[𝑥_1𝑖,𝑥_2𝑖,…,𝑥_𝑛𝑖]^′ (𝑛 is the number of samples in the training, set and 𝑚 is the number of input variables). Soft-sensing modeling requires a datum set from the normal conditions as the modeling data. Assume that the system has 𝑚 process variable and 𝑛 data vectors composing the test sample datum matrix 𝑋∈𝑅^𝑛×𝑚. In order to avoid the different dimensions of the process variables affecting the results and obtain the easy mathematical treatment, it is necessary to normalize the datum. Set 𝜇 is the mean vector of 𝑋, and 𝜎 is the standard deviation vector of 𝑋. So, the normalized process variable is expressed as follows: 𝑋=𝑋−𝜇𝜎. (1)

The input vector 𝑋 of the training samples is fed into the wavelet neural network to predict the output 𝑌. The root mean square error (RMSE) is selected as the fitness of the WNN soft-sensor model: RMSE=∑^𝑛_𝑘=1𝑌_𝑘−𝑌^∗_𝑘²𝑛. (2)

3.2. Wavelet Neural Network

Wavelet neural network (WNN) is similar to BP neural network with the same topology, which adopts the wavelet base function as the transfer function of hidden layer nodes [14]. Its structure is shown in Figure 4.

Figure 4 

Structure of wavelet neural network.

In Figure 4, 𝑥₁,𝑥₂,…,𝑥_𝑘 is the input parameters of the wavelet neural network, 𝑌₁,𝑌₂,…,𝑌_𝑚 is the prediction output of the wavelet neural network, and 𝜔_𝑖𝑗 and 𝜔_𝑗𝑘 are the weights of the wavelet neural network. When the input signal sequence is 𝑥_𝑖 (𝑖=1,2,…,𝑘), the output of the hidden layer is calculated as follows: ℎ𝑗=ℎ_𝑗∑^𝑘_𝑖=1𝜔_𝑖𝑗𝑥_𝑖−𝑏_𝑗𝑎_𝑗,𝑗=1,2,…,𝑙, (3) where ℎ(𝑗) is the 𝑗th node output of the hidden layer, 𝜔_𝑖𝑗 is the connection weights between input layers and hidden layers, 𝑏_𝑗 is the translation factor of the wavelet base function ℎ_𝑗, 𝑎_𝑗 is the stretching factor of the wavelet basis function ℎ_𝑗, and ℎ_𝑗 is the wavelet function. The morlet wavelet function is adopted in this paper, which is represented as follows: 𝑦=cos(1.75𝑥)𝑒^−𝑥2/2. (4)

The parameters of output layers of the wavelet neural network are calculated as 𝑦(𝑘)=^𝑙_𝑖=1𝜔_𝑖𝑘ℎ(𝑖),𝑘=1,2,…,𝑑, (5) where 𝜔_𝑖𝑘 is weight for the hidden layer to output layer, ℎ(𝑖) is the 𝑖th output in the hidden layer, 𝑙 is the number of the hidden layer nodes, and 𝑑 is the number of the input layer nodes.

Standard wavelet neural network uses the gradient descent method to train the structural parameters. But the inherent characteristics of gradient descent method make the WNN training process convergence slow, easy to fall into local minimum, and easily lead to oscillation effect [15]. Therefore, the paper adopts the improved shuffled frog-leap algorithm to train WNN.

3.3. Improved Shuffled Frog Leaping Algorithm

Shuffled frog leap algorithm [16] (SFLA) is a population-based heuristic cooperative swarm intelligent search algorithm. SFLA adopts the metaheuristic algorithm based on swarm intelligence to solve the combinatorial optimization problems, which is based on the meme evolution of the individuals in the population and global information exchange of the memes. SFLA combines the advantages of the genetic-based memetic algorithm (MA) and particle swarm optimization (PSO) with foraging behaviors of the population, such as simple concept, few parameters, quick calculation speed, global optimization capability, easy to implement features [17]. SFLA has been successfully applied in many fields, such as water network optimization problems [16], placement sequence optimization [18, 19], flow shop scheduling problem [20], clustering [21], and so forth.

SFLA is an evolutionary computation algorithm combining deterministic method and stochastic method. Deterministic algorithm can make effective use of strategic information to guide the search response and the random element to ensure the flexibility and robustness of the algorithm searching patterns. The SFLA is described in detail as follows. First, an initial population of 𝑁 frogs 𝑃={𝑋₁,𝑋₂,…,𝑋_𝑁} is created randomly. For 𝑆-dimensional problems (𝑆 variables), the position of a frog 𝑖 in the search space is represented as 𝑋_𝑖=[𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑆]. After the initial population is created, the individuals are sorted in a descending order according to their fitness. Then, the entire population is divided into 𝑚 memeplexes, each containing 𝑛 frogs (i.e., 𝑁=𝑚×𝑛), in such a way that the first frog belongs to the first memeplex, the second frog goes to the second memeplexe, the 𝑚th frog goes to the 𝑚th memeplex, and the (𝑚+1)th frog goes back to the first memeplex, so forth. Let 𝑀^𝑘 is the set of frogs in the 𝑘th memeplex, this dividing process can be described by the following expression: 𝑀^𝑘=𝑋_{𝑘+𝑚(𝑙−1)}∈𝑃∣1≤𝑙≤𝑛,(1≤𝑘≤𝑚). (6)

In each memeplex, the frogs with the best fitness and worst fitness are identified as 𝑋_𝑏 and 𝑋_𝑤. The frog with the global best fitness in the population is identified as 𝑋_𝑔. Then, the local searching is carried out in each memeplex; that is to say, the worst frog 𝑋_𝑤 leaps towards the best frog 𝑋_𝑏 according to the original frog leaping rules (shown in the Figure 5(a)) described as follows: 𝐷=𝑟⋅𝑋_𝑏(𝑡)−𝑋_𝑤(𝑡),𝑋^′_𝑤(𝑡)=𝑋_𝑤(𝑡)+𝐷,‖𝐷‖≤𝐷_max, (7) where 𝑟 is a random number between 0 and 1 and 𝐷_max is the maximum allowed change of frog’s position in one jump. If the new frog 𝑋^′_𝑤 is better than the original frog 𝑋_𝑤, it replaces the worst frog. Otherwise, 𝑋_𝑏 is replaced by 𝑋_𝑔 and the local search is carried out again according to formula (7). If no improvement is achieved in this case, the worst frog is deleted and a new frog is randomly generated to replace the worst frog 𝑋_𝑤. The local search continues for a predefined number of memetic evolutionary steps 𝐿_max within each memeplex, and then the whole population is mixed together in the shuffling process. The local evolution and global shuffling continue until convergence iteration number 𝐺_max arrives.

(a)

(b)

(c)

Figure 5 

Frog leaping rules.

3.4. Improved Frog Leaping Rule

During the natural memetic evolution of the frogs, the worse frog is affected by the better frog to leap for the better one in order to get more food. According to the above description of the initial frog leaping rule (shown in the Figure 5(a)), the likely position of the worst frog is limited to the line segment between the current value and the position of the best frog. So this frog-leaping rule limits the search scope of memetic evolution which not only reduces the convergence velocity but also easily leads to the premature convergence. A modified shuffled frog leaping algorithm [22] based on a new frog leaping rule (shown in Figure 5(b)) can be expressed as follows: 𝐷=𝑟⋅𝑐⋅𝑋_𝑏−𝑋_𝑤𝑟+𝑊,𝑊=₁𝑤_1,max,𝑟₂𝑤_2,max,…,𝑟_𝑆𝑤_𝑆,max^𝑇,𝑋^′_𝑤=𝑋_𝑤+𝐷,if‖𝐷‖≤𝐷_max,𝑋_𝑤+𝐷𝐷^𝑇𝐷𝐷_max,if‖𝐷‖>𝐷_max, (8)(9) where 𝑟 is a random number of [0,10], 𝑐 is a constant of [1,2], 𝑟_𝑖 (1≤𝑖≤𝑆) is a random number of [−1,1], and 𝑤_𝑖,max (1≤𝑖≤𝑆) is the maximum perceptual and the movement uncertainty of the 𝑖th search space.

This frog leaping rule increases the algorithm search scope in a certain degree. Combined with the characteristics of SFLA, the paper puts forward a new frog leaping rule (shown in the Figure 5(c)) described as follows: 𝑋^′_𝑏=𝑋_𝑏+𝑟₁⋅𝑟_max𝑒^𝑗𝜃,𝐷=𝑟₂⋅𝑋^′_𝑏−𝑋_𝑤, (10) where 𝜃 is a random angle of [0,360^∘], 𝑟_max is the maximum local search radius, 𝑟₁ and 𝑟₂ are random numbers of [0,1], and 𝑟_max defines the maximum perceptual surrounding the local optimization value of the frog memetic groups. 𝜃, 𝑟₁, and 𝑟₂ decide the uncertainty of frog leaping. The position vector is still updated by using formula (9).

3.5. Algorithm Procedure of Optimization of WNN Soft-Sensor Model Based on ISFLA

Two wavelet neural network soft-sensor models optimized by the improved SFLA are set up in the paper for predicting the grinding granularity and grinding discharge ratio. The algorithm procedure of ISFLA-based WNN soft-sensor model is shown in Figure 6.

Figure 6 

Algorithm procedure of optimization of WNN soft-sensor model based on ISFLA.

Combined with the proposed new frog leaping rule, the algorithmic procedure of the ISFLA-based wavelet neural network training is described as follows.

Step 1 (initialize the SFLA parameters). Initialize the frog population size 𝑁, the search space dimension 𝑆, the number of meme groups is 𝑚 (each meme group contains 𝑛 frogs) (𝑁=𝑚×𝑛), the allowed frog leaping maximum step 𝐷_max, the local search number 𝐿_max and the global hybrid iteration number 𝐺_max, and maximum local search radius 𝑟_max.

Step 2 (frog population creation). Randomly initial the population of 𝑁 frogs 𝑃={𝑋₁(𝑡),…,𝑋_𝑘(𝑡),…,𝑋_𝑁(𝑡)} (𝑘=1,…,𝑁). Set the iteration counter 𝑡=0. Each frog 𝑋_𝑘(𝑡) is set as the structure parameters of the wavelet neural network soft-sensor model (wavelet stretch factor 𝑎_𝑘, translation factor 𝑏_𝑘, and the network connection weights 𝜔_𝑖𝑗 and 𝜔_𝑗𝑘, 𝑖=1,…,𝑘, 𝑗=1,…,𝑙, 𝑘=1,…,𝑑). Then, the training sample datum is fed into the wavelet neural network to carry out the precedent calculation according to the formula (3)–(5). Each individual’s fitness value 𝐹_𝑘(𝑡)=𝐹(𝑋_𝑘(𝑡)) is calculated according to the formula (2) after the simulation. Finally, the frogs are sorted in a descending order according to their fitness. The outcome is stored with the style 𝑈_𝑘(𝑡)={𝑋_𝑘(𝑡),𝐹_𝑘(𝑡)}. The global best frog in the frog population is identified as 𝑋_𝑔(𝑡)=𝑈₁(𝑡).

Step 3 (memeplex creation). The 𝑈 is divided into the 𝑚 memeplex 𝑌₁(𝑡),𝑌₂(𝑡),…,𝑌_𝑚(𝑡) (𝑗=1,…,𝑚) according to formula (6). Each memeplex includes 𝑛 frogs. The frogs with the best fitness and worst fitness in the memeplex are identified as 𝑋^𝑗_𝑏(𝑡) and 𝑋^𝑗_𝑤(𝑡).

Step 4 (memeplex evolution). The frog in the memeplex 𝑌_𝑗(𝑡) (𝑗=1,…,𝑚) is assigned a probability value 𝑃_𝑗𝑘=2(𝑛+1−𝑘)/(𝑛(𝑛+1)), (𝑘=1,…,𝑛). Set the random value 𝑟∈[𝑃_𝑗1,𝑃_𝑗𝑛]. If 𝑃_𝑗𝑘<𝑟, the 𝑘th frog in the 𝑗th memeplex evolves in accordance with formula (10), and the objective function value of the new frog is calculated. If 𝑃_𝑗𝑘>𝑟, the evolution will be given up.
If the frog doesn’t achieve the meme evolution, 𝑋_𝑗𝑏(𝑡) is substituted by 𝑋_𝑔(𝑡) to carry out the local search again. If no improvement is achieved, a new frog is created randomly to substitute the 𝑋^𝑗_𝑤(𝑡).

Step 5 (local depth search). The frog in the meme group 𝑌_𝑗(𝑡) (𝑗=1,…,𝑚) carries out Step 4 for 𝑖 times to get the meme group 𝑌₁(𝑡)^′,𝑌₂(𝑡)^′,…,𝑌_𝑚(𝑡)^′.

Step 6 (memeplex shuffled). The frogs in the iterated memeplex 𝑌₁(𝑡)^′,𝑌₂(𝑡)^′,…,𝑌_𝑚(𝑡)^′ are mixed together in the shuffling process and identified as (𝑡+1)={𝑌₁(𝑡)^′,𝑌₂(𝑡)^′,…,𝑌_𝑚(𝑡)^′}. In 𝑈(𝑡+1), the frog in the objective function value according to ascending sort will be recorded as the best group of frogs 𝑋_𝑔(𝑡+1)=𝑈₁(𝑡+1).

Step 7 (test the algorithm termination conditon). 𝑡=𝑡+1; if 𝑡<𝐺_max, go to Step 3. Otherwise output the best frog.

3.6. Adaptive Revision of Soft-Sensor Model Based on Model Migration Method

The model migration method [23, 24] based on process similarity is shown in the Figure 7, which is based on the well-established model to develop a new model of the similar process by adopting few datum.

Figure 7 

Basic principles block diagram of PMBPS.

Due to the fluctuations in ore grade and other working conditions of the grinding process, the current soft-sensor results are no longer accurate so that the soft-sensor model must be adaptive corrected. At the moment, a small amount of datum may be adopted to set up a new soft-sensor model based on the model migration (linear correction and planning) from the original soft-sensor model to be adapted to the new working conditions. In this paper, the migration modeling method based on the input-output correction programming method is adopted, whose basic principle is shown in Figure 8.

Figure 8 

IOSBC migration modeling principles chart.

Assume the original soft-sensor model: 𝑌_base=𝑓𝑋_base, (11) where 𝑋_base and 𝑌_base are the input and output of the original model.

Through the output space migration and plan a new model is obtained as follows 𝑌_new=𝑆₀𝑓𝑆_𝐼𝑋_new+𝐵_𝐼+𝐵₀, (12) where 𝑆₀ and 𝐵₀ is the scale factor and offset parameters of output space in the original model.

Then, the input space is shifted and revised. The input 𝑋_new of the new model can be obtained by the input bias correction of the original model input 𝑋_base, which is described as follows 𝑋_base=𝑆_𝐼𝑋_new+𝐵_𝐼, (13) where 𝑆_𝐼 and 𝐵_𝐼 are the scale factor and offset parameters of the input space, respectively.

Therefore, the new model is obtained by the input-output offset correction of the original model, which is described as follows: 𝑌_new=𝑆₀𝑓𝑆_𝐼𝑋_new+𝐵_𝐼+𝐵₀. (14)

New sample datum can be used to train the correct parameters: Min𝐽𝑆₀,𝐵₀,𝑆_𝐼,𝐵_𝐼=𝑒𝑒^𝑇st.𝑒_𝑖=𝑦_𝑖−𝑆₀𝑓𝑆_𝐼𝑋_new,𝑖+𝐵_𝐼+𝐵₀, (15) where 𝑦_𝑖 is the 𝑖th observation data of the new process, 𝜀 represents the prediction error between the measurement and the prediction value of the new model. The dimension of the identified amendments and planning parameters is determined by the input-output space dimension.

3.7. Simulation Results

Aiming at the grinding and classification process, the grinding granularity and grinding discharge ratio soft-sensor model is set up based on the wavelet neural network. Firstly, the input-output data set is shown in Table 1 in order to train and test the ISFLA-based WNN soft-sensor model. The precedent 260 group data comes from the same working condition. The later 40 group data comes from another dynamic working condition due to the variation of the ore feed grade in order to verify the adaptive performance of the oft-sensor model. The first 200-group data was used to train the wavelet neural network by the ISFLA and gradient descent method. The later 100-group data was adopted to carry out the soft-sensor model validation. The predictive results of the validation data by the proposed soft-sensor model illustrated in Figures 9 and 10.

Table 1 

Input data set of forecasted grinding granularity.

(a) Predictive Output

(b) Predictive error

Figure 9 

Predictive output of grinding granularity.

(a) Predictive Output

(b) Predictive error

Figure 10 

Predictive output of mill discharge rate.

Usually the average relative variance (ARV) [1] is adopted to measure the difference between the predicted value and the measured value, which is defined as follows: ∑ARV=^𝑁𝑖=1[𝑥(𝑖)−𝑥(𝑖)]²∑^𝑁_𝑖=1[𝑥(𝑖)−𝑥(𝑖)]², (16) where 𝑁 is the number of comparative data, 𝑥(𝑖) is the measurement value, 𝑥 is the average of the measurement values, and 𝑥(𝑖) is the predictive value. Obviously, the smaller the average relative variance, the better the predictive performance. ARV=0 means that the model has an ideal prediction. ARV=1 indicates that the model only obtains the average prediction results. The contrast results of the AVR values under the WNN soft-sensor model and the ISFLA-based WNN soft-sensor model are listed in Table 2.

Table 2 

Predictive AVR.

As seen from Figures 9 and 10 and Table 2, the WNN adaptive soft-sensor model optimized by the improved shuffled frog-leaping algorithm (ISFLA) of the grinding process for predicting the key technique indicators (grinding granularity and milling discharging ratio) has higher prediction accuracy and generalization ability than those of the standard wavelet neural network soft-sensor model. The proposed ISFLA can effectively adjust the structure parameters of the WNN soft-sensor model. On the other hand, when the working condition of the grinding process changes, the soft-sensor model can be corrected adaptively based on the model migration strategy, which results in the more accurate predictions.

4. Set-Point Optimization of Grinding Process Based on Case Reasoning

4.1. Basic Flowchart of Case-Based Reasoning

The general procedure of the case-based reasoning process includes retrieve-reuse-revise-retain. In the CBR process, the case retrieval is the core of CBR technology, which directly determines the speed and accuracy of decision making. The basic procedure of the case-based reasoning technology [25, 26] is shown in Figure 11.

Figure 11 

Basic flowchart of case-based reasoning.

The case-based reasoning process is mainly divided into four basic steps [27]: (1) case retrieval: by a series of searching and similarity calculation, the most similar case with the current problem is found in the case database. (2) Case reuse: compare the differences between the source case and the target case. The solution case recognized by the user will be submitted to the user, and the effect of its application will be observed. (3) Case revision: the solution strategy of the retrieval case is adjusted by combining the effect of case reuse and the current issue in order to fit the current problem. (4) Case storage: the current issue is resolved and stored in the case database for the future use.

4.2. Set-Point Optimization Strategy of Grinding Process

Grinding process is a complex nonlinear industrial controlled object. Combining the real problems that exist in grinding process control with the theory of case-based reasoning, the basic procedure of the set point optimization strategy is shown in Figure 12 [25]. By carrying out a comprehensive analysis and case-based reasoning for the complex process, the intelligent set-point of the grinding feed ratio and pump water supply ratio are obtained in an optimized manner.

Figure 12 

Diagram of the grinding intelligent set-point control based on case-based reasoning.

The basic procedure is described as follows. Firstly, the working conditions, the process indicators, and the process datum are dealt with for the case reasoning. Then, the case retrieval and case matching are carried out for obtaining the matched case. If the matched case is not obtained, the new case will appear and be studied and stored into the database. Thirdly, the matched case will be reused and corrected. Finally, maintain the case database, output the results, and store the datum.

4.3. Case Description

The most commonly used knowledge representation methods have production rules, semantic networks, frames, decision trees, predicate logic and fuzzy relations, and so forth. In theory, the form that knowledge is represented by in the case is not a new knowledge representation method, but it is an abstract knowledge representation based on the past ones, which means that the case is a logical concept. The case must be based on the existing variety knowledge representation methods. That is to say that almost all the existing knowledge representation methods can be used as the implementation of the case representation. A typical case generating process is essentially refinement process of case databases. It represents a large number of similar cases and experiences in common and can reduce not only the retrieval process in the selected set of objects but also other parts of the analog process the workload.

The case model in the CBR process is described as 𝐶_𝑘=[𝑇_𝑘,𝐹_𝑘,𝐽_𝑘]. 𝐶_𝑘 means there are 𝑘 cases in total, 𝑘=(1,2,…,𝑛). 𝑇_𝑘 represents the time at which the case occurs. 𝐹_𝑘=(𝑓_1𝑘,𝑓_2𝑘,…,𝑓_5𝑘) expresses the characteristics of what 𝐶_𝑘 describes. 𝑓_1𝑘 is the working conditions of industrial process, 𝑓_2𝑘 is the process indicators, 𝑓_3𝑘 is the grinding ore feed ratio, 𝑓_4𝑘 is the pump water feed velocity, and 𝑓_5𝑘 denotes the feedback amount. 𝐽_𝑘=(𝑗₁,𝑗₂,…,𝑗_𝑛) expresses the characteristics solutions of the case 𝐶_𝑘.

4.4. Case Retrieval and Matching

Case matching and case retrieval are important steps in the case-based reasoning process and the key of the information extraction from the case databases. In general, the case matching strategy includes the serial and parallel search methods. In the serial search process, the cases are organized with the hierarchical manner. The top-down refinement layer by layer retrieval approach is adopted, which means the more down the layer, the higher the similarity. The parallel searching strategy weakens the level features among the cases. The retrieve method is to return to the most similar case by retrieving many cases. The commonly used search strategies have nearest neighbor strategy, inductive reasoning strategy, and knowledge guidance strategy.

If the current working condition is 𝑁, the similarity degree between the description features 𝑓_𝑖 (𝑖=1,…,𝑛) of 𝑁, and the description features 𝑓_𝑖𝑘 of the case is described as follows: sim𝑓_𝑖,𝑓_𝑖𝑘=1,𝑓_𝑖=𝑓_𝑖𝑘,0,𝑓_𝑖≠𝑓_𝑖𝑘. (17)

The similarity function between 𝑁 and 𝐶_𝑘 is described as follows: sim𝑁,𝐶_𝑘=^𝑛_𝑖=1𝑐_𝑖,sim𝑓_𝑖,𝑓_𝑖𝑘, (18) where 𝑐_𝑖 is the feature weight.

So the static similarity threshold adopted in the paper is described as follows: sim_𝑘𝑗=𝑋_𝑘𝑗,sim_max≥𝑋_𝑘𝑗,sim_max,sim_max<𝑋_𝑘𝑗, (19) where 𝑋_𝑘𝑗 is 0.9 and sim_𝑘𝑗 is similarity threshold.

4.5. Case Reuse

In the actual production process, because the case library does not have a case fully matching with the current work under normal circumstances, the retrieved solution parameters matching the working conditions are not directly selected as the control parameters of the current conditions. Therefore, the similar case solution retrieved must be reused. That is to say that the CBR system will adjust the retrieved case solution according to the specific circumstances of the new case to get the solution of the input new case. The adjustment strategy uses the existing process knowledge to obtain the current working parameters based on the differences between the input case working conditions and the retrieval case working conditions. The SSTD-based case reuse strategy is described as follows: 𝑄𝑄=^′,sim_max=1∨sim_𝑘𝑗=sim_max,_𝐻𝑘−1sim_𝑘×𝐽_{𝑘𝐻𝑘−1}sim_𝑘,Other, (20) where sim_𝑘 is the similarity property value of the case and 𝑄^′ is the solution of the maximum sim_max.

4.6. Case Correction

In order to prove the effectiveness of case reuse, the case must be amended in the new implementation process in order to form the effective feedback. Usually the case model obtained by the case-based reasoning can be directly applied to the current working conditions. However, due to some differences between the current working conditions and the retrieved cases in some characteristics, the retrieved cases cannot be directly used in the current conditions. The retrieved case must be amended to adapt to the current issue.

The main contents of the case correction mainly include the amendments of the case features and structures. If sim_max<𝐸_𝑖, 𝐸_𝑖 is identified by the experts, the case 𝐶_𝑘^′=[𝑇_𝑘^′,𝐹_𝑘^′,𝐽_𝑘^′] is added into the new case database in order to make the case database be constantly updated.

4.7. Case Maintenance

Case storage strategy is to store the new cases and their solutions into case database according to a certain strategy. The case storage is the base of the case library. By doing so, the case database keeps growing and expanding to increase the searching scope of the case database. At the same time, the maintenance of case database has become an essential work. In the case of storage, the new case 𝐶^′_𝑘=[𝑇^′_𝑘,𝐹^′_𝑘,𝐽^′_𝑘] very similar to the case in the database, that is to say the similarity is 1, is not stored. This is to simplify the case database and reduce the maintenance time.

5. PID Decoupling Controller Based on ISFLA

The paper mainly studies the relationship between the input variables (grinding ore feed ratio and pump water feed velocity) and output variables (grinding granularity and grinding discharge ratio). Through experiments, the dynamic process model of the grinding circuit includes the ball milling mechanistic model, based on material balance, the empirical model of hydrocyclones, the pump pool hybrid model based on the mechanistic model and empirical model. Through the step response of the grinding process, the system transfer function model [22] is described in formula (21): 𝑦₁𝑦₂=−0.425𝑒^−1.52𝑠()11.7𝑠+10.105247.1𝑠+111.5𝑠+12.9775.5𝑠+11.063𝑒^−2.26𝑠𝑢2.5𝑠+1¹𝑢₂. (21)

The mathematical model of the grinding process described in formula (21) is decoupled by the diagonal matrix decoupling method. The two control variables are the grinding ore feed ratio 𝑈₁ and pump water feed velocity 𝑈₂. The two controlled variables are the overflow mass fraction 𝑌₁ and the grinding discharge ratio 𝑌₂. The structure of the parameters self-tuning multivariable PID decoupling controller optimized by the ISFLA is shown in Figure 13 [28], which is composed of PID controller and decoupling compensator based on diagonal decoupling method. The parameters of the PID controller are optimized by the improved shuffled frog leaping algorithm.

Figure 13 

Self-tuning PID Decoupling controller.

5.1. Diagonal Matrix Decoupling

Diagonal matrix decoupling procedure is described as follows [29]: 𝐷₁₂(𝑠)𝑊₁₁(𝑠)+𝐷₂₂(𝑠)𝑊₁₂(𝑠)𝐷=0,₁₁(𝑠)𝑊₂₁(𝑠)+𝐷₂₁(𝑠)𝑊₂₂(𝑠)=0. (22)

Additionally, 𝑀_𝑐2≠0 and 𝑀_𝑐1≠0. The system may be achieved by decoupling. By using the diagonal matrix decoupling strategy to obtain the two single-input single-output (SISO) systems, 𝑌₁𝑌₂𝑊^′₁₁00𝑊^′₂₂=𝑈₁𝑈₂ (23) must be satisfied.

For meeting 𝑊^′₂₁=𝑊^′₁₂=0, the system must satisfy𝑊^′₁₁00𝑊^′₂₂=𝑊₁₁(𝑠)𝑊₁₂(𝑠)𝑊₂₁(𝑠)𝑊₂₂(𝑠)𝐷₁₁(𝑠)𝐷₁₂(𝑠)𝐷₂₁(𝑠)𝐷₂₂(𝑠)=𝑊₁₁(𝑠)𝐷₁₁(𝑠)+𝑊₁₂(𝑠)𝐷₂₁(𝑠)𝑊₁₁(𝑠)𝐷₁₂(𝑠)+𝑊₁₂(𝑠)𝐷₂₂(𝑠)𝑊₂₁(𝑠)𝐷₁₁(𝑠)+𝑊₂₂(𝑠)𝐷₂₁(𝑠)𝑊₂₁(𝑠)𝐷₁₂(𝑠)+𝑊₂₂(𝑠)𝐷₂₂(𝑠). (24)

So 𝑊^′₁₁=𝑊₁₁𝐷₁₁+𝑊₁₂𝐷₂₁,𝑊^′₂₂=𝑊₂₁𝐷₁₂+𝑊₂₂𝐷₂₂,𝑊^′₁₁=0.425⋅11.7𝑠+11.063𝑒2.5𝑠+1^−3.78𝑠+4.95𝑠+0.1052⋅11.5𝑠+12.977,𝑊5.5𝑠+1_′22=1.063⋅12.5𝑠+1𝑒11.7𝑠+1^−3.78𝑠+2.977⋅5.5𝑠+14.95𝑠+0.105.4.89𝑠+0.43 (25)(26)(27)(28)

So obtain two SISO systems: 𝑌₁=𝑊^′₁₁𝑈₁ and 𝑌₂=𝑊^′₂₂𝑈₂. By decoupling, two separate single-input single-output channels are created to make the direct control of the milling ore feed ratio to milling discharge ratio and the pump water feed velocity to the grinding granularity.

5.2. PID Controller Optimization Based on ISFLA

PID control is a regulator based on the linear combination of the proportion, differential, and integral of the bias, whose transfer function is described as follows [30]: 𝑢(𝑡)=𝐾_𝑝𝑒(𝑡)+1𝑇_𝑖^𝑡₀𝑒(𝑡)𝑇𝑑𝑡+^𝑑𝑑𝑒(𝑡),𝑑𝑡 (29) where 𝐾_𝑖=𝐾_𝑝/𝑇_𝑖, 𝐾_𝑑=𝐾_𝑝𝑇_𝑑, and 𝑒(𝑡) are the feedback bias.

Firstly, the Z-N method is adopted to determine the initial parameters of PID controller. So the two groups of PID controller parameters (𝐾_𝑝, 𝐾_𝑖, and 𝐾_𝑑) are (6, 35, and 0.1) and (4, 23.2, and 1), respectively. Then with the PID controller parameters achieved by Z-N method as the center, the searching space of the PID parameters is expanded, which is described as follows: (1−𝛼)𝐾^′_𝑝≤𝐾_𝑝≤(1+𝛼)𝐾^′_𝑝,(1−𝛼)𝐾^′_𝑖≤𝐾_𝑖≤(1+𝛼)𝐾^′_𝑖,(1−𝛼)𝐾^′_𝑑≤𝐾_𝑑≤(1+𝛼)𝐾^′_𝑑. (30)

Four performance indicators IAE, ITAE, ISE, and ITSE are described in (31). The best performance index is selected through simulation comparison to determine the PID parameters. The target of selecting the performance index is to obtain the best control performance, such as increasing response speed, decreasing overshoot of the PID controller, and so forth 𝐽₁=IAE=^∞₀|𝑒(𝑡𝐽)|𝑑𝑡,₂=ITAE=^∞₀𝑡|𝑒(𝑡𝐽)|𝑑𝑡,₃=ISE=^∞₀𝑒(𝑡)²𝐽𝑑𝑡,₄=ITSE=^∞₀𝑡²𝑒(𝑡)²𝑑𝑡. (31)

The original two-input-two-output (TITO) system is decoupled into two independent single-input-single-output (SISO) channels by adopting diagonal matrix decoupling methods. The parameters of the PID controller are optimized by the proposed ISFLA.

5.3. Simulation Research and Industrial Application Experiments

The simulation parameters are selected as follows: frog population size 𝑁=50, search space dimension 𝑆=3, the number of memeplex 𝑚=5, the number of frogs in each memeplex 𝑛=10, the allowed leaping maximum distance 𝐷_max=0.01, the number of local search 𝐿_max=10, and overall hybrid iteration number 𝐺_max=100. Aiming at two SISO models (denoted as M1 and M2) described in Formula (27)-(28), the parameters of the PID controller are tuned by the ZN method and ISFLA under four fitness functions (ISE, IAE, ITAE, and ISTE). By comparing the simulation results under four different fitness functions, the optimization effects can be obtained. The output response curves of M1 and M2 are shown in Figures 14 and 15.

Figure 14 

Output response curves of M1.

Figure 15 

Output response curves of M2.

For model M1 and M2, the parameters of the PID controller tuned by a variety of methods are shown in Tables 3 and 4. The performance indexes of the step response, such as overshoot, rise time, and settling time, are adopted to carry out comparison with the different PID controllers, which are illustrated in Tables 5 and 6. As seen from the above simulation results, the PID controllers optimized by the ISFLA under four fitness functions (ISE, IAE, ITAE, and ISTE) achieve better performances than the conventional ZN method, while the PID controller under the fitness function ITAE obtains more superior performances.

Table 3 

PID controller parameters of M1.

Table 4 

PID controller parameters of M2.

Table 5 

Output response performance index of M1.

Table 6 

Output response performance index of M2.

Under the premise that the other process variables remain unchanged, the grinding ore feed ratio and pump water feed velocity before and after optimization have a different influence on the performance indexes of the grinding process. Because the ultimate impact factors on the economic efficiency of grinding process are the concentrate grade and tailings grade. So the industrial application experiments are carried out under the proposed data-driven integrated modeling and intelligent control method in the grinding process. The technique indexes controlled scopes are described as follows. Concentrate grade 𝐽 and tailings grade 𝑊 are 66%≤𝐽≤71.5% and 𝑊≤28%, respectively. The target is to increase the concentrate grade and reduce the tailing grade as much as possible. As the concentrate grade and tailings grade can directly determine the effect of optimized controller, the experiments results are shown in Figures 16–19, which includes 200 groups datum (concentrate grades and tailings grades before and after optimization controller).

Figure 16 

Concentrate grade before intelligent optimization.

Figure 17 

Concentrate grade after intelligent optimization.

Figure 18 

Tailings grade before intelligent optimization.

Figure 19 

Tailings of grade after intelligent optimization.

As seen from Figures 16 and 17, the performances under the intelligent optimized controller are better than those under the conventional controller, such as the lower fluctuate of the concentrate grade and the higher concentrate grade, which indicates that the proposed data-driven integrated modeling and intelligent control strategy helps to increase the product quality. As seen from Figures 18 and 19, the performances under the intelligent optimized controller are better than those under the conventional controller, such as the lower fluctuate of the tailings grade and the lower tailings grade, which indicates that the proposed data-driven integrated modeling and intelligent control strategy helps to increase the resources utilization rate.

6. Conclusions

For the grinding process, a complex industrial controlled object, an integrated automation, and control system are researched in detail, which includes the economic and technical indicators of soft-sensor model, the set-point optimized model based on the case-based reasoning methods, and the self-tuning PID decoupling controller. Simulation and industrial experimental results show that the proposed data-driven integrated modeling and intelligent control methods have a better feasibility and effectiveness to meet the real-time control requirements of the grinding production process.

Acknowledgments

This work is partially supported by the Program for China Postdoctoral Science Foundation (Grant no. 20110491510), the Program for Liaoning Excellent Talents in University (Grant no. LJQ2011027), and the Program for Research Special Foundation of University of Science and Technology of Liaoning (Grant no. 2011zx10).