# A Global Optimization Algorithm for Intelligent Electromechanical Control System with Improved Filling Function

Authors: Shuguang Zhang; Kode Srividya; Ismail Kakaravada; Dimitrios A. Karras; Vishal Jagota; Inamul Hasan; Abdul Wahab Rahmani
Journal: Scientific Programming
 (2022)

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## Abstract

To improve the speed of global optimization algorithm, a class of global optimization algorithms for intelligent electromechanical control system with improved filling function is proposed. By attaching the intelligent managing system improving algorithm and the filling function procedure, the algorithm can stand out from the current particular optimal solution, avoid the phenomenon of falling into the local favorable solution in the process of algorithm iteration, make the algorithm find a better solution, and improve the efficiency of solving the multiextremum global improving problem. Multiextremum-seeking is an optimal control technique that works with unknown conditions while assuming that measurements of the plant’s input and output signals are accessible. The presented work is for an electromechanical system which will handle the low accuracy and untimely tendency of conventional systems which are used in various practical applications. Few learning algorithms have been developed to explicitly optimize mean average precision (MAP) due to computational constraints. The outcomes show that the convergence of the test functions F6 and F7 is not good when the MAPID algorithm is only used for optimization. The MAPID_FF algorithm not only ensures the convergence and optimization precision of the two test functions, but also reduces the optimization time compared with the filling function method. Compared with the filling function method, the improved algorithm has higher accuracy and faster speed, and it is not simple to fall into the local optimum, so the global optimal value is more accurate.


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## Body

## 1. Introduction

Since the 1990s, smart building technology has grown in popularity. This innovation serves a wide range of study fields, including space vehicles, aviation, train and car systems, robots, heavy machinery, and medical equipment. Noise and vibration are two examples of use. Common components include a host structure, actuators, and sensors, and a CPU that analyses data signals, a control law for changing the structure’s characteristics, and integrated power electron signals, a control law to adjust the structure’s features, and integrated power electronics [1, 2]. Optimization is a powerful tool to solve practical problems, which can be widely applied in many fields such as engineering, production, and science. At the early stage of the development of optimization theory, the research on it mainly focuses on the method and application of local optimization, and abundant theories and solutions have been obtained in the research of local optimization. However, with the rapid development of computing science and technology, more practical problem is modeled as an optimization model required to the global optimal solution rather than the local optimal value, which makes the global optimization become the hotspot in the academic field, and global optimization is becoming increasingly wide in the actual application and in many fields, such as molecular biology, neural network, and environmental engineering. It has been extended to many important fields such as economy, science, national defense. It is an indispensable numerical technique to calculate the global optimal solution of the corresponding optimization problem. Therefore, industrial departments, scientific research institutions, and government departments pay high attention to the research on the relevant theories and solving methods of global optimization problem, and the in-depth research on it has far-reaching application significance [3].Due to the low accuracy and untimely tendency of conventional particle swarm optimization, the reactive power optimization control of an electromechanical system based on fuzzy particle swarm optimization approach was proposed. Working within the constraints of the operating environment is the idea. The network device loss was reduced by changing the voltage and reactive transmission lines of the system, and a static reactive power optimization mathematical model of the electromechanical system was created. A control law for changing the structure’s characteristics and integrated power electronics for singnals was described in [4, 5]. The control system optimization algorithm uses the idea of “feedback” of the feedback control system. In the closed-loop control system, the output of the system is constantly approaching the set value of the system. If the iterative process of the optimization algorithm can be continuously approaching the global optimal, then the algorithm can successfully converge to the global optimal solution of the objective function. The controller of the feedback control system generally adopts the incremental PID control strategy. At this time, the three parameters kp, ki, and kd need to be set manually. Combining the neural network technology with the traditional PID control, it can solve the traditional PID controller to a certain extent, which makes it not easy for it to carry out online real-time parameter tuning and other defects, and give full play to the advantages of PID control [6]. The proportional-integral-derivative (PID) control approach is the most often used and is well known in industrial control. The attractiveness of PID controllers originates in part from their resilience in a wide range of operating settings, as well as their functional simplicity, which allows engineers to run them in a straightforward and basic manner [7, 8]. This forms the neural network PID, referred to as PIDNN. Inspired by the idea of particle swarm parallel computing and group learning, the optimization algorithm of PIDNN was improved, and the optimization algorithm of intelligent control system MPIDNN was proposed (multiple control systems compute in parallel and learn from each other, which has the characteristics of “intelligence”). Because the global minimum value of the objective function is not clear at the beginning of the algorithm design, it is difficult to give an appropriate system value. If the initial values of Kp, Ki, Kd, and other parameters are not appropriate, the algorithm may not converge to the global minimum of the objective function or the optimization accuracy of the algorithm is very low. The filling function method is a transformation function method, which can solve the problem of initial parameter setting and precocious convergence of the optimization algorithm of intelligent control system. Combining the optimization algorithm of intelligent control system with the filling function method can not only reduce the difficulty of parameter setting of the optimization algorithm of intelligent control system, but also shorten the optimization time compared with the filling function method alone [9]. In recent years, several diagnostic and prognostic models based on statistical, artificial intelligence (AI), and soft computing (SC) techniques have been proposed and get satisfactory results. In this work, we apply the DE to optimize the control parameter by reducing the mechanical friction of the worm gear and reducing the fault occurrence due to friction in the electromechanical control [10, 11]. PIDNN controllers have recently been one of the most preferred approaches for controlling complicated systems. Several robust and autotuning options for improving the PIDNN controller’s control and robust performance have been provided. An adaptive PIDNN controller was presented. To improve the convergence speed and prevent weights from being stuck in local optima, the neural network was initialized using the particle swarm optimization (PSO) technique. The utility of the proposed method is proved by actual evidence. The PSO approach, on the other hand, requires a long time to initialize the weights [12, 13]. Shi Dongwei et al. proposed an efficient global algorithm for solving quadratic programs with quadratic constraints. In this algorithm, we propose a new linearization method to solve the linear programming relaxation problem of quadratic programs with quadratic constraints. The proposed algorithm is integrated with the global optimal solution of the initial problem, and numerical experiments show the computational efficiency of the proposed algorithm [14]. Metaheuristics can help policymakers and decision-makers get the greatest results by lowering the cost function. In this research, an improved grasshopper optimization algorithm (GOA) was proposed with a novel starting approach to balance GOA's search capacity. The new method is known as the Improved Grasshopper Algorithm (IGOA). GOA is based on grasshopper swarms and mimics their natural behavior [15, 16]. Yue and Zhang proposed a GOA algorithm combining two strategies, namely, a new variant of PCA-GOA. First, using principal component analysis (PCA) strategies to obtain grasshoppers with small correlated variables can improve GOA’s development capacity. Then, a new inertial weight is proposed to balance exploration and exploitation in an intelligent way, which gives GOA a better search capability. In addition, the performance of PCA-GOA was evaluated by addressing a number of baseline functions. Experimental results show that PCA-GOA provides better results than basic GOA for most functions and other most advanced algorithms, which indicates the superiority of PCA-GOA [9].With its efficiency and simplicity, the grasshopper optimization algorithm (GOA), one of the most recent metaheuristic algorithms, has a wide range of applications. The fundamental GOA, on the other hand, offers plenty of space for development. As a result, a novel variation GOA algorithm is suggested that combines the two techniques, called PCA-GOA. To begin, the principal component analysis approach is used to get grasshoppers with minimum correlated variables, which can increase the GOA's exploitation capabilities [17, 18]. Based on this, the control system parameter adaptive adjustment and multisystem parallel computation are adopted to improve the control system optimization algorithm, and the global optimization algorithm of intelligent control system is proposed, which is called MAPID (Multisystem Adaptive EPID). In the algorithm of intelligent control system, the setting of the given value of the system is a troublesome problem. It is limited by the global minimum value of the unknown objective function, and it is difficult for the objective function with complex characteristics to use PID control strategy to achieve high precision optimization results.The presented work is a global optimization algorithm approach for electromechanical system to handle the low accuracy and untimely tendency of conventional systems which are used in various practical applications. It possesses higher accuracy and faster speed and therefore its global optimal value is more accurate. In addition, the setting of initial parameters in the control decision will affect the optimization result and the optimization speed. For this reason, the filling function method is introduced to improve the optimization algorithm of intelligent control system, which is called MAPID_FF. In the simulation experiment, the results of 7 standard test functions show that MAPID_FF is superior to MAPID in optimization precision and superior to the filling function method in optimization speed [19]. Therefore, machine learning is often used to improve ranked retrieval systems. Despite its widespread application in evaluating such systems, few learning algorithms have been developed to explicitly optimize mean average precision (MAP) due to computational constraints. Existing MAP optimization approaches either fail to find a globally optimal solution or are wasteful in terms of computation [20, 21].The practical application of the model has multidimensional aspects. These machines employ switches and relay circuitry to do sophisticated calculations. They may discover system faults and offer retry commands based on the programmed. Almost all moving equipment is powered by an electromechanical mechanism. These systems are found in the majority of electric motors, solenoids, and mechatronics. Most of the equipment we use in our everyday lives, from automotive electric windows and seats to dryers, rely on these systems. The novel approach of the algorithm is that it not only ensures the convergence and optimization precision of the two test functions, but also reduces the optimization time compared with the filling function method. Few learning algorithms have been developed to explicitly optimize mean average precision (MAP) due to computational constraints.


## 2. Research Methods

### 2.1. Global Optimization Algorithm of Intelligent Control System

#### 2.1.1. Iterate Formula Definition

The model of the single-loop closed-loop control system is shown in Figure1. The controlled object is the optimized objective function, R is the given value of the system, and the control strategy adopts PID algorithm. The global optimization problem is to solve the minf(x) problem, where x∈Rn. In order to simplify the design of the control strategy, the commonly used PID algorithm is adopted, and the parameter values in the PID algorithm adopt the method of self-adaptive adjustment. Therefore, this control strategy is called adaptive PID control strategy (APID).Figure 1
Single-loop control system model.Definition 1.
According to the APID algorithm, the iteration formula of the independent variablex in the iteration process is defined as shown in equations (1)∼(4):(1)Δ2Εk=Εk−2×Εk−1+Εk−2,ΔΕk=Εk−Εk−1,Εk=R−fk,xk+1=ω⋅xk+kp⋅ΔΕk+ki⋅Εk+kd⋅Δ2Εk.
Among them:kp, ki, kd are the parameters of PID algorithm, ω is the inertia coefficient, and these four parameters are vectors of the same dimension as the independent variable x [22].
To prevent the calculation of a single system from falling into the local minimum, the parallel calculation method of multiple control systems is adopted, inspired by the parallel calculation of particle swarm optimization algorithm. The model of multisystem parallel computing is shown in Figure2. The given value of each system is R, and the control strategy is PID algorithm, but the starting point of iterative calculation of each system is different. In each iteration, each system should not only learn from its own optimal value, but also learn from the optimal value of all systems.Figure 2
Multisystem parallel computing model.Definition 2.
After adopting the multisystem parallel computing method, equations (1)∼(4) are replaced by equations (5)∼(8):(2)xi,k+1=ωi⋅xi,k+kpi⋅ΔΕi,k+kii⋅Εi,kkdi⋅Δ2Εi,k+r1i⋅c1i⋅x_besti−xi,k+r2i⋅c2i⋅x_allbest−xi,k.Εi,k=R−fi,k,ΔΕi,k=Εi,k−Εi,k−1,Δ2Εi,k=Εi,k−2⋅Εi,k−1+Εi,k−2.
It is assumed that there aren independent control systems in the multisystem (n = 5 for the simulation parts); in the above formula, xi,k represents the value of the KTH iteration independent variable x of the ith system; r1, r2 are adjustable parameters and often take the constant of 0∼1; c1, c2 are vectors of the same dimension as the independent variable x, which are usually random values of -0.5∼0.5. X_best (I) is the best position of the ith system in the iteration process, and AA is the best position of all systems in the iteration process [23].


#### 2.1.2. Parameter Adjustment Mechanism

The optimization algorithm of the intelligent control system adopts APID control strategy, and the parameters of the strategy adopt the adaptive adjustment method. To ensure that equation (1) converges during iteration, Theorem 1 is given. To make the objective function f(x) converge in the iterative process, the adaptive adjustment formulas of ω, kp, ki, kd are shown in equations (9)-(12):(3)ωk+1=ωk−η⋅∂fx∂x⋅xk,(4)kpk+1=kpk−η⋅∂fx∂x⋅ΔΕk,(5)kik+1=kik−η⋅∂fx∂x⋅Εk,(6)kdk+1=kdk−η⋅∂fx∂x⋅Δ2ΕkProve that(7)∂fx∂t=∂fx∂x⋅dxdt=∂fx∂x⋅∂x∂ω⋅dωdt+∂x∂kp⋅dkpdt+∂x∂ki⋅dkidt+∂x∂kd⋅dkddt.Obviously,∂fx/∂t<0 ensures that the objective function converges during iteration. If any of the following equations (14)-(17) are true, ∂fx/∂t<0 must be true(8)dωdt=−∂fx∂x⋅∂x∂ω,(9)dkpdt=−∂fx∂x⋅∂x∂kp,(10)dkidt=−∂fx∂x⋅∂x∂ki,(11)dkddt=−∂fx∂x⋅∂x∂kd.Based on equation (1), we can get(12)∂x∂ω=x,∂x∂kp=ΔΕ,∂x∂ki=Ε,∂x∂kd=Δ2Ε.After discretization of equations (14)-(17):(13)Δω=−η⋅∂fx∂x⋅xk,(14)Δkp=−η⋅∂fx∂x⋅ΔΕk,(15)Δki=−η⋅∂fx∂x⋅Εk,(16)Δkd=−η⋅∂fx∂x⋅Δ2Εkwhere η is the iteration step size adjusted by parameters ω, kp, ki, kd. Based on equations (19)-(21), the adaptive adjustment formula of equations (19)-(21) as parameters can be obtained. Theorem 1 is proved [24].


#### 2.1.3. The Initial Position

There exists the problem of selecting the initial position of each system in multisystem parallel computing. The first step in adopting the Latin square method is to determine an appropriate Latin square. For example, if 5 systems are parallel and the independent variablex is of 10 dimensions, then the Latin square v5,10 is(17)v5,10=12345123452345123451345123451245123451235123451234.The initial positionoij can be calculated from (18), where x∈l,u, I represents the ith system, and n is the number of parallel systems.(18)oij=l+2υij−12nu−1.Let us sayx∈−10,10; then we have(19)o5,10=−8−4048−8−4048−4048−8−4048−8048−8−4048−8−448−8−4048−8−408−8−4048−8−404.Thus, there is a way to pick the initial position.In this paper, an optimization algorithm of intelligent control system (MAPID) is proposed.


### 2.2. A Class of Improved Filling Function Algorithm Construction

#### 2.2.1. Algorithm Design of a Class of Improved Filling Functions

According to the ideas of the filled function method and basic implementation steps, the algorithm in the feasible region is directly from any initial pointfx. In the process of minimising the objective function, find a local minimum point. As the dimension of optimization increases, so will the iteration algorithm and the number of function evaluations. This will increase the calculation amount of the algorithm, thus affecting the efficiency of the algorithm. Based on the idea of selecting a better initial point to improve the search ability of the algorithm, a new local search strategy is firstly introduced. This local search method defines the neighborhood of the current initial point, uniformly selects several points in the neighborhood, and then gets the probability value of each point in the neighborhood according to the calculation and normalizes the value. That is, the sum of the probability values of the points in the calculated neighborhood to be selected is 1; then the higher the value of the objective function is, the larger the corresponding calculated value is, and the higher the probability of this point to be selected is. This method is briefly known as PLSM (Probability-selected Local Search Method). Theoretically, the PLSM method may be used to screen out a better point near the current initial point as a new initial point minimization function, which can improve the speed of the algorithm in finding the local minimum [25]. The steps of PLSM method are shown in Table 1.Table 1
Implementation steps of PLSM method.
Input: select an initial pointxk at random in the feasible region X=a,b=x|a≤x≤b,a,b∈Rn and output local minimum point xk∗1. Uniformly selects∗n points in the neighborhood U=xk−b−a/m,xk+b−a/m of xk , put in the set Y=y1,y2,…,ysn, where, n is the dimension of the problem variable, s is the number of points selected in each dimension, and m>a, and 1 is a positive integer.2. Find the set y¯ of the points in Y with the highest probability of being selected:(1)To calculate Pyp=1−fyp/∑j=1snfyj/s,p=1,2,…,sn(2) 令 P¯=Pyp,Y¯=argmaxP¯ (3) Z=argmaxxkyeUxkpxk,py is the set of initial points. Select any zk0∈Z to minimize fx to obtain the local minimum point set Z′=zk∗ and local minimum sequence F=fzk∗ 4. Make fxk∗=minF; the local minimum point xk∗ is obtainedBased on Table1, PLSM, and new filled function (C2.5), design a new filled function with one parameter algorithm (A New Filled Function Algorithm: NFFA); the specific algorithm is described as follows: NFFA algorithm.I. Initialization Steps:(1)
Admissible error0<ε<1,e.g.ε:=1.0e−5 of local minimization algorithm.(2)
Makek=1,fxk−1∗=+∞.(3)
A large constant C is given as the upper bound on the number of iterationsk.(4)
Choose a positive real number R0, as large as possible.II. Main Steps:(1)
s∗n points within the feasible region X are uniformly selected to form Y=y1,y2,…ysn, according to the probability value Pyp,p∈1,2,…,sn of each point calculated in Table 1; the set Y¯=argmaxPypyp∈Y,p∈1,2,⋯,sn is obtained; pick any xk∈Y¯ as the starting point.(2)
Use PLSM to minimizefx from xk to obtain the local minimum point xk∗, turn 5°;(3)
Construct filling functionPx,xk∗ at xk∗.(4)
Letx=xk∗+ξi be the starting point, where ξ>0 is a small constant, ei is the unit coordinate vector, and i=1,2,…,2n, use PLSM to minimize Px,xk∗ and find a local minimum x′ of Px,xk∗, let xk+1=x,k=k+1, and then rotate by 2 degrees.(5)
(Stop criterion) ifk < C and it satisfies fxk∗−fxk−1∗≤ε , or k > C, the algorithm terminates; at this point, x∗=xk∗ is the global minimum of fx, and fx∗ is the global minimum; otherwise, make r≔10⋅r, increment: and turn 3°.Note: (1) In this paper,fx and Px,xk∗ are minimized. The selected local minimization method is “FMINCON” in MATLAB. (2) In 5°, when r is sufficiently large in theory, a solution that is better than the current local optimal solution can be found.


#### 2.2.2. Fill Function Method

The filling function method is a transformation function method, that is, using auxiliary functions from a certain point of the objective function to find another local minimum point better than the current point (the new local minimum point has a smaller objective function value). Therefore, the filling function method is actually a combination of local optimization algorithm and constructor method to achieve the purpose of global optimization. The realization process is mainly divided into three steps: firstly, the local optimization algorithm is used to find a local minimumX1 of the objective function; then, an auxiliary function C is constructed at point X1, and local optimization algorithm is used to conduct local optimization on function C with X1 as the initial point, and a local minimum point X2 of C is obtained. Finally, the local optimization algorithm is used to optimize the objective function with X2 as the initial point, and a new local minimum point X3, where fX3<fX1, is obtained. Repeat the above three steps until you find the global minimum of the objective function fx, as follows:(20)CX,a=−fX−Fx1⋅X−X12+12a,where fx is the objective function, X1 is a local minimum point of the objective function, which is a positive number, and C(X, a) is the filling function constructed at the point X1. Figure 3 reflects the characteristics of the constructed filling function. The objective function fx=xsinx,x1=4.9132 in the figure is constructed by taking a = 0.3 and a = 0.5, respectively. It is clear from Figure 3 that, at a = 0.3, the function value increases. At a = 0.5, the function value increases to the maximum and the curve gets almost flattened.Figure 3
Characteristic diagram of filling function C(x).


### 2.3. Improved Optimization Algorithm of Intelligent Control System

The filling function optimization method starts from a local minimum point of the objective function and gradually converges to the global minimum point of the objective function. In each step, the corresponding local optimization algorithm should be called twice (once for the objective function and once for the filling function). This may lead to a longer optimization time, especially for the multimodal objective function. The specific idea of the improved intelligent control system optimization algorithm is as follows: (1) Use the optimization algorithm of the intelligent control system to optimize the objective function, stop after reaching the set number of iterations, and write down the best advantage obtained by searching; (2) the optimal advantage obtained by the optimization algorithm of the intelligent control system is used as the starting point and the filling function method is used to optimize, to obtain the global minimum value of the objective function. Combining the optimization algorithm of intelligent control system with the filling function method can not only reduce the difficulty of parameter setting of the optimization algorithm of intelligent control system, but also shorten the optimization time compared with the filling function method alone.The concrete implementation steps of the improved optimization algorithm of the intelligent control system using the filling function method are as follows:Step 1.
Latin square is used to set the initial position ofX (5 systems are used in parallel in the experimental part), set the initial value of ω,kp,ki,kd, and determinate the number of iteration steps and the initial values of fx, error Ε,∂fx/∂x, f_best, and f_allbest.Step 2.
Use equation (5) to update the position of X and judge whether X has crossed the feasible region. If it has crossed the feasible region, the optimal position of the current five systems will be assigned to the current system X (that is, X = X_AllBest). This method has achieved very good results in the experiment.Step 3.
Update the objective functionfx, error Ε,∂fx/∂x, and f_best and determine the best of the five f_best values (that is, updates f_allbest).Step 4.
Adjust the value ofω,kp,ki,kd with equations (9)∼(12).Step 5.
Judge whether the number of iteration steps has been reached. If the number of iteration steps has been reached, the value of F_ALLBEST will be recorded as the optimization result of this part and transferred to Step6; otherwise, the value of F_ALLBEST will be transferred to Step 2.Step 6.
Set the initial value ofX0,a,ξ,Mt,fmin, where X0=x_allbest, A is a positive number, ξ is a number greater than 1, and Mt is the upper limit of M (fmin<f_allbest).Step 7.
M sets zero to enter the first stage of the filling function method. Local optimization algorithm was used (FMINUNC function in MATLAB optimization toolbox was used in the experiment part) to optimize the objective functionfx; the starting point is X0. Obtain the local minimum point X1; if fX1≤fmin, then assign fX1 to fmin; otherwise, the a=ξ⋅a.Step 8.
Enter the second stage of the fill function method. UseX1 to construct the filling function C.Step 9.
Generate an appropriate random numbere1,e2,…,en of the same dimension as X1, and use X1=x11,x12,…,x1n to construct X=x11+e1β,x12+e2β,…,x1n+enβ, where β is the linear search step size in the second stage. The search direction S is defined as X−X1/X−X1. Let M = M+1; take X1 as the starting point and S as the search direction to search for the optimal linear search of C function.Step 10.
Continue the linear search until you reach pointX.(A)
IfX crosses the feasible region, jump to Step 11.(B)
IffX<fX1, set X0=X ; then jump to Step 6; otherwise go to Step 9.(C)
IfX is a local minimum of function c, let X0=X jump to Step 6; otherwise go to Step 9.Step 11.
IfM<Mt, skip to Step 7; otherwise, the optimal point found is taken as the global minimum point of the objective function, and the algorithm stops. Steps 1∼5 are the implementation of the optimization algorithm of intelligent control system (MAPID), and Steps 6∼11 are the implementation of the filling function (FF) method.


## 2.1. Global Optimization Algorithm of Intelligent Control System

### 2.1.1. Iterate Formula Definition

The model of the single-loop closed-loop control system is shown in Figure1. The controlled object is the optimized objective function, R is the given value of the system, and the control strategy adopts PID algorithm. The global optimization problem is to solve the minf(x) problem, where x∈Rn. In order to simplify the design of the control strategy, the commonly used PID algorithm is adopted, and the parameter values in the PID algorithm adopt the method of self-adaptive adjustment. Therefore, this control strategy is called adaptive PID control strategy (APID).Figure 1
Single-loop control system model.Definition 1.
According to the APID algorithm, the iteration formula of the independent variablex in the iteration process is defined as shown in equations (1)∼(4):(1)Δ2Εk=Εk−2×Εk−1+Εk−2,ΔΕk=Εk−Εk−1,Εk=R−fk,xk+1=ω⋅xk+kp⋅ΔΕk+ki⋅Εk+kd⋅Δ2Εk.
Among them:kp, ki, kd are the parameters of PID algorithm, ω is the inertia coefficient, and these four parameters are vectors of the same dimension as the independent variable x [22].
To prevent the calculation of a single system from falling into the local minimum, the parallel calculation method of multiple control systems is adopted, inspired by the parallel calculation of particle swarm optimization algorithm. The model of multisystem parallel computing is shown in Figure2. The given value of each system is R, and the control strategy is PID algorithm, but the starting point of iterative calculation of each system is different. In each iteration, each system should not only learn from its own optimal value, but also learn from the optimal value of all systems.Figure 2
Multisystem parallel computing model.Definition 2.
After adopting the multisystem parallel computing method, equations (1)∼(4) are replaced by equations (5)∼(8):(2)xi,k+1=ωi⋅xi,k+kpi⋅ΔΕi,k+kii⋅Εi,kkdi⋅Δ2Εi,k+r1i⋅c1i⋅x_besti−xi,k+r2i⋅c2i⋅x_allbest−xi,k.Εi,k=R−fi,k,ΔΕi,k=Εi,k−Εi,k−1,Δ2Εi,k=Εi,k−2⋅Εi,k−1+Εi,k−2.
It is assumed that there aren independent control systems in the multisystem (n = 5 for the simulation parts); in the above formula, xi,k represents the value of the KTH iteration independent variable x of the ith system; r1, r2 are adjustable parameters and often take the constant of 0∼1; c1, c2 are vectors of the same dimension as the independent variable x, which are usually random values of -0.5∼0.5. X_best (I) is the best position of the ith system in the iteration process, and AA is the best position of all systems in the iteration process [23].


### 2.1.2. Parameter Adjustment Mechanism

The optimization algorithm of the intelligent control system adopts APID control strategy, and the parameters of the strategy adopt the adaptive adjustment method. To ensure that equation (1) converges during iteration, Theorem 1 is given. To make the objective function f(x) converge in the iterative process, the adaptive adjustment formulas of ω, kp, ki, kd are shown in equations (9)-(12):(3)ωk+1=ωk−η⋅∂fx∂x⋅xk,(4)kpk+1=kpk−η⋅∂fx∂x⋅ΔΕk,(5)kik+1=kik−η⋅∂fx∂x⋅Εk,(6)kdk+1=kdk−η⋅∂fx∂x⋅Δ2ΕkProve that(7)∂fx∂t=∂fx∂x⋅dxdt=∂fx∂x⋅∂x∂ω⋅dωdt+∂x∂kp⋅dkpdt+∂x∂ki⋅dkidt+∂x∂kd⋅dkddt.Obviously,∂fx/∂t<0 ensures that the objective function converges during iteration. If any of the following equations (14)-(17) are true, ∂fx/∂t<0 must be true(8)dωdt=−∂fx∂x⋅∂x∂ω,(9)dkpdt=−∂fx∂x⋅∂x∂kp,(10)dkidt=−∂fx∂x⋅∂x∂ki,(11)dkddt=−∂fx∂x⋅∂x∂kd.Based on equation (1), we can get(12)∂x∂ω=x,∂x∂kp=ΔΕ,∂x∂ki=Ε,∂x∂kd=Δ2Ε.After discretization of equations (14)-(17):(13)Δω=−η⋅∂fx∂x⋅xk,(14)Δkp=−η⋅∂fx∂x⋅ΔΕk,(15)Δki=−η⋅∂fx∂x⋅Εk,(16)Δkd=−η⋅∂fx∂x⋅Δ2Εkwhere η is the iteration step size adjusted by parameters ω, kp, ki, kd. Based on equations (19)-(21), the adaptive adjustment formula of equations (19)-(21) as parameters can be obtained. Theorem 1 is proved [24].


### 2.1.3. The Initial Position

There exists the problem of selecting the initial position of each system in multisystem parallel computing. The first step in adopting the Latin square method is to determine an appropriate Latin square. For example, if 5 systems are parallel and the independent variablex is of 10 dimensions, then the Latin square v5,10 is(17)v5,10=12345123452345123451345123451245123451235123451234.The initial positionoij can be calculated from (18), where x∈l,u, I represents the ith system, and n is the number of parallel systems.(18)oij=l+2υij−12nu−1.Let us sayx∈−10,10; then we have(19)o5,10=−8−4048−8−4048−4048−8−4048−8048−8−4048−8−448−8−4048−8−408−8−4048−8−404.Thus, there is a way to pick the initial position.In this paper, an optimization algorithm of intelligent control system (MAPID) is proposed.


## 2.1.1. Iterate Formula Definition

The model of the single-loop closed-loop control system is shown in Figure1. The controlled object is the optimized objective function, R is the given value of the system, and the control strategy adopts PID algorithm. The global optimization problem is to solve the minf(x) problem, where x∈Rn. In order to simplify the design of the control strategy, the commonly used PID algorithm is adopted, and the parameter values in the PID algorithm adopt the method of self-adaptive adjustment. Therefore, this control strategy is called adaptive PID control strategy (APID).Figure 1
Single-loop control system model.Definition 1.
According to the APID algorithm, the iteration formula of the independent variablex in the iteration process is defined as shown in equations (1)∼(4):(1)Δ2Εk=Εk−2×Εk−1+Εk−2,ΔΕk=Εk−Εk−1,Εk=R−fk,xk+1=ω⋅xk+kp⋅ΔΕk+ki⋅Εk+kd⋅Δ2Εk.
Among them:kp, ki, kd are the parameters of PID algorithm, ω is the inertia coefficient, and these four parameters are vectors of the same dimension as the independent variable x [22].
To prevent the calculation of a single system from falling into the local minimum, the parallel calculation method of multiple control systems is adopted, inspired by the parallel calculation of particle swarm optimization algorithm. The model of multisystem parallel computing is shown in Figure2. The given value of each system is R, and the control strategy is PID algorithm, but the starting point of iterative calculation of each system is different. In each iteration, each system should not only learn from its own optimal value, but also learn from the optimal value of all systems.Figure 2
Multisystem parallel computing model.Definition 2.
After adopting the multisystem parallel computing method, equations (1)∼(4) are replaced by equations (5)∼(8):(2)xi,k+1=ωi⋅xi,k+kpi⋅ΔΕi,k+kii⋅Εi,kkdi⋅Δ2Εi,k+r1i⋅c1i⋅x_besti−xi,k+r2i⋅c2i⋅x_allbest−xi,k.Εi,k=R−fi,k,ΔΕi,k=Εi,k−Εi,k−1,Δ2Εi,k=Εi,k−2⋅Εi,k−1+Εi,k−2.
It is assumed that there aren independent control systems in the multisystem (n = 5 for the simulation parts); in the above formula, xi,k represents the value of the KTH iteration independent variable x of the ith system; r1, r2 are adjustable parameters and often take the constant of 0∼1; c1, c2 are vectors of the same dimension as the independent variable x, which are usually random values of -0.5∼0.5. X_best (I) is the best position of the ith system in the iteration process, and AA is the best position of all systems in the iteration process [23].


## 2.1.2. Parameter Adjustment Mechanism

The optimization algorithm of the intelligent control system adopts APID control strategy, and the parameters of the strategy adopt the adaptive adjustment method. To ensure that equation (1) converges during iteration, Theorem 1 is given. To make the objective function f(x) converge in the iterative process, the adaptive adjustment formulas of ω, kp, ki, kd are shown in equations (9)-(12):(3)ωk+1=ωk−η⋅∂fx∂x⋅xk,(4)kpk+1=kpk−η⋅∂fx∂x⋅ΔΕk,(5)kik+1=kik−η⋅∂fx∂x⋅Εk,(6)kdk+1=kdk−η⋅∂fx∂x⋅Δ2ΕkProve that(7)∂fx∂t=∂fx∂x⋅dxdt=∂fx∂x⋅∂x∂ω⋅dωdt+∂x∂kp⋅dkpdt+∂x∂ki⋅dkidt+∂x∂kd⋅dkddt.Obviously,∂fx/∂t<0 ensures that the objective function converges during iteration. If any of the following equations (14)-(17) are true, ∂fx/∂t<0 must be true(8)dωdt=−∂fx∂x⋅∂x∂ω,(9)dkpdt=−∂fx∂x⋅∂x∂kp,(10)dkidt=−∂fx∂x⋅∂x∂ki,(11)dkddt=−∂fx∂x⋅∂x∂kd.Based on equation (1), we can get(12)∂x∂ω=x,∂x∂kp=ΔΕ,∂x∂ki=Ε,∂x∂kd=Δ2Ε.After discretization of equations (14)-(17):(13)Δω=−η⋅∂fx∂x⋅xk,(14)Δkp=−η⋅∂fx∂x⋅ΔΕk,(15)Δki=−η⋅∂fx∂x⋅Εk,(16)Δkd=−η⋅∂fx∂x⋅Δ2Εkwhere η is the iteration step size adjusted by parameters ω, kp, ki, kd. Based on equations (19)-(21), the adaptive adjustment formula of equations (19)-(21) as parameters can be obtained. Theorem 1 is proved [24].


## 2.1.3. The Initial Position

There exists the problem of selecting the initial position of each system in multisystem parallel computing. The first step in adopting the Latin square method is to determine an appropriate Latin square. For example, if 5 systems are parallel and the independent variablex is of 10 dimensions, then the Latin square v5,10 is(17)v5,10=12345123452345123451345123451245123451235123451234.The initial positionoij can be calculated from (18), where x∈l,u, I represents the ith system, and n is the number of parallel systems.(18)oij=l+2υij−12nu−1.Let us sayx∈−10,10; then we have(19)o5,10=−8−4048−8−4048−4048−8−4048−8048−8−4048−8−448−8−4048−8−408−8−4048−8−404.Thus, there is a way to pick the initial position.In this paper, an optimization algorithm of intelligent control system (MAPID) is proposed.


## 2.2. A Class of Improved Filling Function Algorithm Construction

### 2.2.1. Algorithm Design of a Class of Improved Filling Functions

According to the ideas of the filled function method and basic implementation steps, the algorithm in the feasible region is directly from any initial pointfx. In the process of minimising the objective function, find a local minimum point. As the dimension of optimization increases, so will the iteration algorithm and the number of function evaluations. This will increase the calculation amount of the algorithm, thus affecting the efficiency of the algorithm. Based on the idea of selecting a better initial point to improve the search ability of the algorithm, a new local search strategy is firstly introduced. This local search method defines the neighborhood of the current initial point, uniformly selects several points in the neighborhood, and then gets the probability value of each point in the neighborhood according to the calculation and normalizes the value. That is, the sum of the probability values of the points in the calculated neighborhood to be selected is 1; then the higher the value of the objective function is, the larger the corresponding calculated value is, and the higher the probability of this point to be selected is. This method is briefly known as PLSM (Probability-selected Local Search Method). Theoretically, the PLSM method may be used to screen out a better point near the current initial point as a new initial point minimization function, which can improve the speed of the algorithm in finding the local minimum [25]. The steps of PLSM method are shown in Table 1.Table 1
Implementation steps of PLSM method.
Input: select an initial pointxk at random in the feasible region X=a,b=x|a≤x≤b,a,b∈Rn and output local minimum point xk∗1. Uniformly selects∗n points in the neighborhood U=xk−b−a/m,xk+b−a/m of xk , put in the set Y=y1,y2,…,ysn, where, n is the dimension of the problem variable, s is the number of points selected in each dimension, and m>a, and 1 is a positive integer.2. Find the set y¯ of the points in Y with the highest probability of being selected:(1)To calculate Pyp=1−fyp/∑j=1snfyj/s,p=1,2,…,sn(2) 令 P¯=Pyp,Y¯=argmaxP¯ (3) Z=argmaxxkyeUxkpxk,py is the set of initial points. Select any zk0∈Z to minimize fx to obtain the local minimum point set Z′=zk∗ and local minimum sequence F=fzk∗ 4. Make fxk∗=minF; the local minimum point xk∗ is obtainedBased on Table1, PLSM, and new filled function (C2.5), design a new filled function with one parameter algorithm (A New Filled Function Algorithm: NFFA); the specific algorithm is described as follows: NFFA algorithm.I. Initialization Steps:(1)
Admissible error0<ε<1,e.g.ε:=1.0e−5 of local minimization algorithm.(2)
Makek=1,fxk−1∗=+∞.(3)
A large constant C is given as the upper bound on the number of iterationsk.(4)
Choose a positive real number R0, as large as possible.II. Main Steps:(1)
s∗n points within the feasible region X are uniformly selected to form Y=y1,y2,…ysn, according to the probability value Pyp,p∈1,2,…,sn of each point calculated in Table 1; the set Y¯=argmaxPypyp∈Y,p∈1,2,⋯,sn is obtained; pick any xk∈Y¯ as the starting point.(2)
Use PLSM to minimizefx from xk to obtain the local minimum point xk∗, turn 5°;(3)
Construct filling functionPx,xk∗ at xk∗.(4)
Letx=xk∗+ξi be the starting point, where ξ>0 is a small constant, ei is the unit coordinate vector, and i=1,2,…,2n, use PLSM to minimize Px,xk∗ and find a local minimum x′ of Px,xk∗, let xk+1=x,k=k+1, and then rotate by 2 degrees.(5)
(Stop criterion) ifk < C and it satisfies fxk∗−fxk−1∗≤ε , or k > C, the algorithm terminates; at this point, x∗=xk∗ is the global minimum of fx, and fx∗ is the global minimum; otherwise, make r≔10⋅r, increment: and turn 3°.Note: (1) In this paper,fx and Px,xk∗ are minimized. The selected local minimization method is “FMINCON” in MATLAB. (2) In 5°, when r is sufficiently large in theory, a solution that is better than the current local optimal solution can be found.


### 2.2.2. Fill Function Method

The filling function method is a transformation function method, that is, using auxiliary functions from a certain point of the objective function to find another local minimum point better than the current point (the new local minimum point has a smaller objective function value). Therefore, the filling function method is actually a combination of local optimization algorithm and constructor method to achieve the purpose of global optimization. The realization process is mainly divided into three steps: firstly, the local optimization algorithm is used to find a local minimumX1 of the objective function; then, an auxiliary function C is constructed at point X1, and local optimization algorithm is used to conduct local optimization on function C with X1 as the initial point, and a local minimum point X2 of C is obtained. Finally, the local optimization algorithm is used to optimize the objective function with X2 as the initial point, and a new local minimum point X3, where fX3<fX1, is obtained. Repeat the above three steps until you find the global minimum of the objective function fx, as follows:(20)CX,a=−fX−Fx1⋅X−X12+12a,where fx is the objective function, X1 is a local minimum point of the objective function, which is a positive number, and C(X, a) is the filling function constructed at the point X1. Figure 3 reflects the characteristics of the constructed filling function. The objective function fx=xsinx,x1=4.9132 in the figure is constructed by taking a = 0.3 and a = 0.5, respectively. It is clear from Figure 3 that, at a = 0.3, the function value increases. At a = 0.5, the function value increases to the maximum and the curve gets almost flattened.Figure 3
Characteristic diagram of filling function C(x).


## 2.2.1. Algorithm Design of a Class of Improved Filling Functions

According to the ideas of the filled function method and basic implementation steps, the algorithm in the feasible region is directly from any initial pointfx. In the process of minimising the objective function, find a local minimum point. As the dimension of optimization increases, so will the iteration algorithm and the number of function evaluations. This will increase the calculation amount of the algorithm, thus affecting the efficiency of the algorithm. Based on the idea of selecting a better initial point to improve the search ability of the algorithm, a new local search strategy is firstly introduced. This local search method defines the neighborhood of the current initial point, uniformly selects several points in the neighborhood, and then gets the probability value of each point in the neighborhood according to the calculation and normalizes the value. That is, the sum of the probability values of the points in the calculated neighborhood to be selected is 1; then the higher the value of the objective function is, the larger the corresponding calculated value is, and the higher the probability of this point to be selected is. This method is briefly known as PLSM (Probability-selected Local Search Method). Theoretically, the PLSM method may be used to screen out a better point near the current initial point as a new initial point minimization function, which can improve the speed of the algorithm in finding the local minimum [25]. The steps of PLSM method are shown in Table 1.Table 1
Implementation steps of PLSM method.
Input: select an initial pointxk at random in the feasible region X=a,b=x|a≤x≤b,a,b∈Rn and output local minimum point xk∗1. Uniformly selects∗n points in the neighborhood U=xk−b−a/m,xk+b−a/m of xk , put in the set Y=y1,y2,…,ysn, where, n is the dimension of the problem variable, s is the number of points selected in each dimension, and m>a, and 1 is a positive integer.2. Find the set y¯ of the points in Y with the highest probability of being selected:(1)To calculate Pyp=1−fyp/∑j=1snfyj/s,p=1,2,…,sn(2) 令 P¯=Pyp,Y¯=argmaxP¯ (3) Z=argmaxxkyeUxkpxk,py is the set of initial points. Select any zk0∈Z to minimize fx to obtain the local minimum point set Z′=zk∗ and local minimum sequence F=fzk∗ 4. Make fxk∗=minF; the local minimum point xk∗ is obtainedBased on Table1, PLSM, and new filled function (C2.5), design a new filled function with one parameter algorithm (A New Filled Function Algorithm: NFFA); the specific algorithm is described as follows: NFFA algorithm.I. Initialization Steps:(1)
Admissible error0<ε<1,e.g.ε:=1.0e−5 of local minimization algorithm.(2)
Makek=1,fxk−1∗=+∞.(3)
A large constant C is given as the upper bound on the number of iterationsk.(4)
Choose a positive real number R0, as large as possible.II. Main Steps:(1)
s∗n points within the feasible region X are uniformly selected to form Y=y1,y2,…ysn, according to the probability value Pyp,p∈1,2,…,sn of each point calculated in Table 1; the set Y¯=argmaxPypyp∈Y,p∈1,2,⋯,sn is obtained; pick any xk∈Y¯ as the starting point.(2)
Use PLSM to minimizefx from xk to obtain the local minimum point xk∗, turn 5°;(3)
Construct filling functionPx,xk∗ at xk∗.(4)
Letx=xk∗+ξi be the starting point, where ξ>0 is a small constant, ei is the unit coordinate vector, and i=1,2,…,2n, use PLSM to minimize Px,xk∗ and find a local minimum x′ of Px,xk∗, let xk+1=x,k=k+1, and then rotate by 2 degrees.(5)
(Stop criterion) ifk < C and it satisfies fxk∗−fxk−1∗≤ε , or k > C, the algorithm terminates; at this point, x∗=xk∗ is the global minimum of fx, and fx∗ is the global minimum; otherwise, make r≔10⋅r, increment: and turn 3°.Note: (1) In this paper,fx and Px,xk∗ are minimized. The selected local minimization method is “FMINCON” in MATLAB. (2) In 5°, when r is sufficiently large in theory, a solution that is better than the current local optimal solution can be found.


## 2.2.2. Fill Function Method

The filling function method is a transformation function method, that is, using auxiliary functions from a certain point of the objective function to find another local minimum point better than the current point (the new local minimum point has a smaller objective function value). Therefore, the filling function method is actually a combination of local optimization algorithm and constructor method to achieve the purpose of global optimization. The realization process is mainly divided into three steps: firstly, the local optimization algorithm is used to find a local minimumX1 of the objective function; then, an auxiliary function C is constructed at point X1, and local optimization algorithm is used to conduct local optimization on function C with X1 as the initial point, and a local minimum point X2 of C is obtained. Finally, the local optimization algorithm is used to optimize the objective function with X2 as the initial point, and a new local minimum point X3, where fX3<fX1, is obtained. Repeat the above three steps until you find the global minimum of the objective function fx, as follows:(20)CX,a=−fX−Fx1⋅X−X12+12a,where fx is the objective function, X1 is a local minimum point of the objective function, which is a positive number, and C(X, a) is the filling function constructed at the point X1. Figure 3 reflects the characteristics of the constructed filling function. The objective function fx=xsinx,x1=4.9132 in the figure is constructed by taking a = 0.3 and a = 0.5, respectively. It is clear from Figure 3 that, at a = 0.3, the function value increases. At a = 0.5, the function value increases to the maximum and the curve gets almost flattened.Figure 3
Characteristic diagram of filling function C(x).


## 2.3. Improved Optimization Algorithm of Intelligent Control System

The filling function optimization method starts from a local minimum point of the objective function and gradually converges to the global minimum point of the objective function. In each step, the corresponding local optimization algorithm should be called twice (once for the objective function and once for the filling function). This may lead to a longer optimization time, especially for the multimodal objective function. The specific idea of the improved intelligent control system optimization algorithm is as follows: (1) Use the optimization algorithm of the intelligent control system to optimize the objective function, stop after reaching the set number of iterations, and write down the best advantage obtained by searching; (2) the optimal advantage obtained by the optimization algorithm of the intelligent control system is used as the starting point and the filling function method is used to optimize, to obtain the global minimum value of the objective function. Combining the optimization algorithm of intelligent control system with the filling function method can not only reduce the difficulty of parameter setting of the optimization algorithm of intelligent control system, but also shorten the optimization time compared with the filling function method alone.The concrete implementation steps of the improved optimization algorithm of the intelligent control system using the filling function method are as follows:Step 1.
Latin square is used to set the initial position ofX (5 systems are used in parallel in the experimental part), set the initial value of ω,kp,ki,kd, and determinate the number of iteration steps and the initial values of fx, error Ε,∂fx/∂x, f_best, and f_allbest.Step 2.
Use equation (5) to update the position of X and judge whether X has crossed the feasible region. If it has crossed the feasible region, the optimal position of the current five systems will be assigned to the current system X (that is, X = X_AllBest). This method has achieved very good results in the experiment.Step 3.
Update the objective functionfx, error Ε,∂fx/∂x, and f_best and determine the best of the five f_best values (that is, updates f_allbest).Step 4.
Adjust the value ofω,kp,ki,kd with equations (9)∼(12).Step 5.
Judge whether the number of iteration steps has been reached. If the number of iteration steps has been reached, the value of F_ALLBEST will be recorded as the optimization result of this part and transferred to Step6; otherwise, the value of F_ALLBEST will be transferred to Step 2.Step 6.
Set the initial value ofX0,a,ξ,Mt,fmin, where X0=x_allbest, A is a positive number, ξ is a number greater than 1, and Mt is the upper limit of M (fmin<f_allbest).Step 7.
M sets zero to enter the first stage of the filling function method. Local optimization algorithm was used (FMINUNC function in MATLAB optimization toolbox was used in the experiment part) to optimize the objective functionfx; the starting point is X0. Obtain the local minimum point X1; if fX1≤fmin, then assign fX1 to fmin; otherwise, the a=ξ⋅a.Step 8.
Enter the second stage of the fill function method. UseX1 to construct the filling function C.Step 9.
Generate an appropriate random numbere1,e2,…,en of the same dimension as X1, and use X1=x11,x12,…,x1n to construct X=x11+e1β,x12+e2β,…,x1n+enβ, where β is the linear search step size in the second stage. The search direction S is defined as X−X1/X−X1. Let M = M+1; take X1 as the starting point and S as the search direction to search for the optimal linear search of C function.Step 10.
Continue the linear search until you reach pointX.(A)
IfX crosses the feasible region, jump to Step 11.(B)
IffX<fX1, set X0=X ; then jump to Step 6; otherwise go to Step 9.(C)
IfX is a local minimum of function c, let X0=X jump to Step 6; otherwise go to Step 9.Step 11.
IfM<Mt, skip to Step 7; otherwise, the optimal point found is taken as the global minimum point of the objective function, and the algorithm stops. Steps 1∼5 are the implementation of the optimization algorithm of intelligent control system (MAPID), and Steps 6∼11 are the implementation of the filling function (FF) method.


## 3. Result Analysis

To show that the performance of the improved optimization algorithm of intelligent control system by using the filling function method has been improved, a comparative experiment was conducted with the filling function method and the optimization algorithm of intelligent control system. This part adopts a total of 7 standard test functions from F1 to F7. The hardware environment of the experiment is IntelPentiumM 1.70 GHz processor and 512 MB memory, and the software environment is Windows operating system and Matlab7.0 version, respectively.The seven standard test functions are as follows:(21)F1=∑i=1nxi2−10cos2πxi+10,F2=−20exp1n∑i=1nxi2−0.2−exp1n∑i=1ncos2πxi+20+exp1,F3=14000∑i=1nxi2−Πi=1ncosxii+1,F4=πn10sin2πy1+yn−12+∑i=1n−1yi−12⋅1+10sin2πyi+1,yi=1+14xi+1,F5=1n∑i=1nxi4−16xi2+5xi,F6=∑i=1n−1100xi2−xi+12+xi−12,F7=∑i=111ai−x1bi2+bix2bi2+bix3+x42,a=0⋅1957,0⋅1947,0⋅1735,0⋅1600,0⋅0844,0⋅0627,0⋅0456,0⋅0342,0⋅0323,0⋅0235,0⋅0246,b=4,2,1,12,14,16,18,110,112,114,116.Table2 is the basic information of the 7 standard test functions, including the dimension and feasible region of independent variables, as well as the standard global optimal value of the objective function. Table 3 is the average results of 10 experiments of 7 standard test functions, in which the optimization results and CPU time of the intelligent control system optimization algorithm (MAPID), the filling function (FF) method, and the improved intelligent control system optimization algorithm (MAPID_FF) are listed, respectively. As can be seen from Table 3, MAPID_FF algorithm is clearly superior to MAPID algorithm in optimization accuracy and superior to FF method in CPU consumption. When MAPID algorithm is only used for optimization, the convergence of test functions F6 and F7 is not good. After MAPID_FF algorithm is used, the convergence and optimization precision of these two test functions are guaranteed, and the optimization time is shortened compared with the filling function method. Therefore, MAPID_ FF algorithm proves out to be the better suit for electromechanical system to handle low accuracy and untimely tendency of conventional systems which are used in various practical applications. It imparts higher accuracy and faster speed and thus its global optimal value is more accurate.Table 2
Providing basic information about the test functions.
Test functionsThe feasible regionGlobal optimal valueDimensionF1−5⋅12,5⋅12n030F2−32,32n030F3−600,600n030F4−50,50n030F5−5,5n−78.3323630F6−5,10n030F7−5,5n3.0749E − 44Table 3
Experimental results.
Test functionsMAPIDFFMAPID-FFThe minimum valueCPU time (s)The minimum valueCPU time(s)The minimum valueCPU time(s)6F19.18E − 110.25003.07800.270F26.79E − 103.1405.16E − 945.8901.25EE − 104.470F37.08E − 130.23500.45300.265F41.92E − 226.6871.42E − 161.5005.28E − 181.375F5−78.33231.390−78.33233.703−78.33232.578F628.00320.2036.7658E − 2410.5934E − 279.500F70.02150.2193.0749E − 40.5623.0749E − 40.204


## 4. Discussion

Because the model’s practical applicability involves several dimensions, due to computational restrictions, only a few learning algorithms have been created that explicitly optimize mean average precision (MAP). The work given here is for an electromechanical system that will manage the low precision and unreliability of traditional systems utilized in a variety of practical applications. The convergence of test functions F6 and F7 is poor when the MAPID technique is utilized only for optimization. The convergence and optimization precision of these two test functions are ensured when the MAPID-FF technique is applied, and the optimization time is reduced when compared to the filling function approach, turned proven to be superior. As a result, the MAPID_ FF algorithm appears to be a better fit for electromechanical systems to manage the low precision and untimely inclination of conventional systems employed in a variety of real applications. It provides more precision and faster speed, resulting in a more accurate global ideal value.


## 5. Conclusions

Global optimization problems have multiple local optimal solutions and make the algorithm easy to fall into local optimum in the iterative process of defects, the intelligent control system of the optimization algorithm were studied, according to the basic ideas of fusion algorithm, by designing a class of filled function algorithms and blending intelligent control system optimization algorithm, then a global optimization algorithm for intelligent electromechanical control system with improved filling function is proposed. Compared with the optimization algorithm of intelligent control system, the algorithm has an obvious improvement in the precision of searching, and the initial point closer to the local optimal solution can be selected in the process of the algorithm, so that the convergence speed of the algorithm to the global optimal solution can be improved. The work given here is for an electromechanical system that will manage the low precision and unreliability of traditional systems utilized in a variety of practical applications. Due to computational restrictions, only a few learning algorithms have been created that explicitly optimize mean average precision (MAP).



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