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Optimized PID Controller Based on Beetle Antennae Search Algorithm for Electro-Hydraulic Position Servo Control System

Optimized PID Controller Based on Beetle Antennae Search Algorithm for Electro-Hydraulic Position... sensors Article Optimized PID Controller Based on Beetle Antennae Search Algorithm for Electro-Hydraulic Position Servo Control System Yuqi Fan, Junpeng Shao * and Guitao Sun School of Mechanical and Power Engineering, Harbin University of Science and Technology, Harbin 150080, China; yuqifan77@163.com (Y.F.); sunguitao86@163.com (G.S.) * Correspondence: Sjp566@hrbust.edu.cn Received: 1 April 2019; Accepted: 14 June 2019; Published: 18 June 2019 Abstract: To improve the controllability of an electro-hydraulic position servo control system while simultaneously enhancing the anti-jamming ability of a PID controller, a compound PID controller that combines the beetle antennae search algorithm with PID strategy was proposed, and used to drive the position servo control system of the electro-hydraulic servo system. A BAS-PID controller was designed, and the beetle antennae search algorithm was used to tune PID parameters so that the disturbance signal of the system was e ectively restrained. Initially, the basic mathematical model of the electro-hydraulic position servo control system was established through theoretical analysis. The transfer function model was obtained by identifying system parameters. Then, the PID parameter-tuning problem was converted into a class of three-dimensional parameter optimization problem, and gains of PID controllers were adjusted using the beetle antennae search algorithm. Finally, by comparing the e ectiveness of di erent algorithms, simulation and experimental results revealed that the BAS-PID controller can greatly enhance the performance of the electro-hydraulic position servo control system and inhibit external disturbances when di erent interference signals are used to test the system’s robustness. Keywords: PID controller; electro-hydraulic servo system; beetle antennae search algorithm 1. Introduction Electro-hydraulic servo systems (EHSSs) have been widely used in the industry due to their advantages of high response, fast response, good sti ness, and high force [1]. With the development of digital technology, control theory, pattern recognition technology, and electronic technology, EHSSs have become one of the most popular topics in the fields of both scientific research and industrial engineering [2,3]. EHSSs including a position servo control system and a force servo control system are a typical closed-loop torque control system [4,5]. Position servo control systems play a major role in marine operating systems, industrial automation systems, and crane systems [6]. Due to the complexity of such control systems, it is considered urgent to develop control technology. Various control methods have been presented, such as the feedback control method [7], tracking control method [8], and adaptive control method [9]. Among these, the PID control system is the most popular control method. Due to its easy control structure, high robustness, and high accuracy, the conventional PID control system is the most widely used in industrial processes [10–12]. The chief function of PID controllers is to regulate feedback signals to be as similar to input signals as possible. The control performance of PID controllers is mainly determined by a proportional parameter, an integral parameter, and a di erential parameter [13,14]. It has been shown that rapid response and high accuracy of velocity can be obtained by selecting appropriate PID parameters in the Sensors 2019, 19, 2727; doi:10.3390/s19122727 www.mdpi.com/journal/sensors Sensors 2019, 19, 2727 2 of 21 control system. Therefore, the method of tuning PID parameters has become a popular research topic, and some significant achievements have been made in recent years [15,16]. Over the past few decades, several methods which can enhance the capacity of obtaining reasonable PID parameters have been announced. One of the most popular methods is the Z-N tuning method [17], proposed by Ziegler and Nichols in 1942. This method is widely used in industry to solve the traditional PID parameter tuning problem [18]. However, the mechanism of the controlled object is complex in actual industrial circumstances; many factors need to be considered, such as high-order duplication, time variation, and non-linearity. Consequently, it is dicult to achieve a perfect control result by using the Z-N tuning method. However, since the method’s publication, it has been extensively developed for tuning PID parameters. With the development of artificial intelligence science, artificial intelligence algorithms have been widely used in PID controllers to select reasonable PID parameters, for example the Genetic Algorithm (GA) [19,20], Particle Swarm Optimization Algorithm (PSO) [21,22], Firefly Algorithm (FA) [23,24], Cuckoo Search Optimization Algorithm (CS) [25], Grey Wolf Optimization Algorithm (GWO) [26], Ant Colony Algorithm (ACA) [27], and Bacterial Foraging Optimization Algorithm (BFO) [28]. The Beetle Antennae Search Algorithm (BAS) is an ecient and intelligent search algorithm proposed by Xiangyuan Jiang and Shuai Li in 2017 [29]. It is an evolutionary computation derived from the flocking and swarming behavior, foraging behavior and courtship behavior of beetles. The algorithm not only has the ability of individual recognition and environmental recognition, but also does not need to know the gradient information of the function. Furthermore, its program code is simple to implement, and it has an especially strong ability to obtain the best solution within a stable convergence property. BAS is used in neural network optimization [30], multi-objective energy management [31], investment portfolio problems [32], the prediction of the content of dissolved gases in power transformer oil [33], the determination of Young’s modulus in jet grouted coalcretes [34], prediction of permeability and unconfined compressive strength of pervious concrete [35], multi-objective optimization based BAS-WPT [36], and unmanned aerial vehicle (UAV) sensing and avoidance [37]. To improve the PID controllability of an electro-hydraulic position servo control system, this paper proposes a BAS-PID controller whose parameters are tuned by BAS. To verify the performance of the BAS-PID controller in the electro-hydraulic position servo control system, the controllability of the BAS-PID controller was compared to the controllability of PID controllers using other algorithms under di erent interference signals. Experimental results showed the superiority of the BAS-PID controller. 2. Related Works BAS is extremely ecient in solving di erent optimization problems, and the original BAS, as well as BAS variants, have been successfully applied in various scientific and engineering fields. Yuantian Sun et al. [30] proposed combination of the back-propagation neural network (BPNN) and BAS, termed the BPNN-BAS model. The BPNN-BAS model was shown to be more e ective than multiple regression and logistic regression. Additionally, in another study [34], support vector machine (SVM) was combined with BAS to produce the SVM-BAS model for the determination of Young’s modulus in jet grouted coalcretes. Zongyao Zhu et al. [31] applied BAS in microgrids to reduce environmental pollution and minimize pollutant treatment cost. Tingting Chen et al. [32] proposed the Beetle Swarm Optimization Algorithm (BSO), which combines BAS and PSO. Shengwei Fei et al. [33] developed the BAS based mixed kernel relevance vector regression (BAS-MkRVR) model to calculate the gas content of power systems. Junbo Sun et al. [35] introduced an ESVR-BAS model to predict the PC and UCS in pervious concrete. Xiangyuan Jiang and Shuai Li [36], who are designers of BAS, advanced the research on BAS by releasing tuning parameters to deal with multi-objective optimization. Qing Wu et al. [37] invented a new path algorithm called the Obstacle Avoidance Beetle Antennae Search Algorithm (OABAS), which is used for global path planning for UAVs. Meijin Lin et al. [38] proposed a hybrid PSO-BAS algorithm, while the same lead author together with Qinghao Li [39] used PSO-BAS to distribute economic loads in power systems. Qinhao Li [40] used BAS to solve Sensors 2019, 19, 2727 3 of 21 optimal scheduling problems of a wind–PV–hydro–storage power complementary system. Chen Wang et al. [41] employed BAS to enhance the precision of the evaluation of straightness error. Table 1 provides a brief comparison among BAS variants. Table 1. Variant ways and advantages of BAS variants. BAS Variants Variant Way Advantages BAS was used to train back propagation  High BPNN training speed BPNN-BAS [30] neural network (BPNN)  Make weights optimal Greater global search ability PSO-BAS [32,38,39]  BAS was combined with PSO  Better searching capacity than that of standard PSO BAS was used to select the appropriate kernel parameters and controlled  Stronger prediction capacity than BAS-MkRVR [33] parameters of the mixed kernel relevance RBFRVR and Sigmoid RVR. vector regression (MkRVR) The hyper-parameters of the support  Less time-consuming SVM-BAS [34] vector machine (SVM) were firstly tuned  Low-cost by BAS  Non-destructive The hyper-parameters of the evolved  More ecient than random ESVR-BAS [35] support vector regression was tuned by hyper-parameter selection BAS  High predictive capability Not require prior parameter tuning Normalization method and penalty BAS-WPT [36]  Simple to design, implement, and has function method were used to extend BAS less complexity Wider search range and the breakneck OABAS was designed on the basic of OABAS [37] search speed BAS to improve the path planning Shorter path length In [42] Yinyan Zhang, Shuai Li, and Bin Xu summarized and analyzed the convergence of BAS and its applications, which provided further theoretical proof of its extreme convergence ability. For a complex optimization problem with multiple parameters, the complexities of the solution space increase with increasing dimensionality, which may cause low convergence speed. For potential extensions and future work, in order to obtain a more accurate solution, the structure of BAS should be improved. Further research may consider an encoding scheme to introduce better diversity into the solution search space of BAS to raise the convergence of the algorithm. In this study, we will apply advanced BAS to solve more practical questions about electro-hydraulic systems. Table 2 shows the uniqueness of BAS among algorithms including PSO [32,37–39], GA [39], FA, BA [42], and the Artificial Bee Colony Algorithm (ABC) [37]. Table 2. Uniqueness of BAS among algorithms. Contrastive Relevant Recent Advantages of BAS Algorithms References Faster iteration speed PSO [32,37–39] Stronger ability to jump local optimal solution Does not need binary to represent decimal numbers GA [39] The BAS program runs faster Does not need more initial parameters FA This paper BAS program code simple BA Simple to implement, and has less complexity [42] Higher eciency ABC [37] Lower time complexity Sensors 2019, 19, x FOR PEER REVIEW 4 of 21 Sensors 2019, 19, 2727 4 of 21 Higher efficiency ABC [37] Lower time complexity From Table 2, it can be determined that BAS not only has abilities of individual recognition and From Table 2, it can be determined that BAS not only has abilities of individual recognition and environmental recognition but also allows the balance in exploratory stages, which shows that it is environmental recognition but also allows the balance in exploratory stages, which shows that it is necessary to use BAS to develop better sensing systems. necessary to use BAS to develop better sensing systems. 3. Electro-Hydraulic Position Servo Control System 3. Electro-Hydraulic Position Servo Control System 3.1. Working Principles of the System 3.1. Working Principles of the System The electro-hydraulic position servo system studied in this paper is chiefly composed of a hydraulic The electro-hydraulic position servo system studied in this paper is chiefly composed of a cylinder, an electro-hydraulic servo valve, a servo amplifier, a position sensor, and some other parts. hydraulic cylinder, an electro-hydraulic servo valve, a servo amplifier, a position sensor, and some A schematic diagram of the system is displayed in Figure 1. other parts. A schematic diagram of the system is displayed in Figure 1. Figure 1. Block diagram of the electro-hydraulic position servo control system. Figure 1. Block diagram of the electro-hydraulic position servo control system. The working principle of the position servo system is characterized as follows. Firstly, the voltage The working principle of the position servo system is characterized as follows. Firstly, the signal is transformed into an electric current signal by the servo amplifier and then input to the servo voltage signal is transformed into an electric current signal by the servo amplifier and then input to valve, and the slide valve of the main valve will be moved depending on the signal of the servo the servo valve, and the slide valve of the main valve will be moved depending on the signal of the valve. Then, the flow rate of the hydraulic cylinder is influenced by the movement of the slide valve. servo valve. Then, the flow rate of the hydraulic cylinder is influenced by the movement of the slide The piston rod is linked to the position sensor. Finally, depending on analogue–digital conversion, valve. The piston rod is linked to the position sensor. Finally, depending on analogue–digital the signal will be fed back to the controller of the control system. Overall, the error between the actual conversion, the signal will be fed back to the controller of the control system. Overall, the error signal and the target signal will be constantly decreased by the position servo control system. between the actual signal and the target signal will be constantly decreased by the position servo In practical engineering problems, due to complexities of the structure, manufacturing errors, large control system. delay time, inaccurate measurement, and other factors, it is dicult to accurately describe the whole In practical engineering problems, due to complexities of the structure, manufacturing errors, working process by building a precise mathematical model. There is not a determined mathematical large delay time, inaccurate measurement, and other factors, it is difficult to accurately describe the model, and therefore the chief aim of this paper is to make the electro-hydraulic servo system model as whole working process by building a precise mathematical model. There is not a determined similar to the really original model as possible. mathematical model, and therefore the chief aim of this paper is to make the electro-hydraulic servo 3.2. Model of the Electro-Hydraulic Position Servo Control System system model as similar to the really original model as possible. The electro-hydraulic position servo system studied in this paper is an electro-hydraulic flow 3.2. Model of the Electro-Hydraulic Position Servo Control System servo valve which controls cylinder structure. The electro-hydraulic servo valve is used as an electrThe electro-h o-hydraulic transfer ydraulic position servo sy apparatus, and is put stem studie into a lower d in this pape input signal r is an to obtain electro e ective -hydraulic flo hydraulic w pr servo v essure alve ener w gy hich cont and to achieve rols cylin the der aim struc oftcontr ure. Th ole e for lec hydraulic tro-hydraulic systems. servo valve is used as an electro- hydraulic transfer apparatus, and is put into a lower input signal to obtain effective hydraulic x (s) pressure energy and to achieve the aim of control for hydraulic systems. K = (1) I(s) xs () K = (1) where I is the output current, x is the spool displacement of the main valve, and K is the flow gain of v v v Is () the electro-hydraulic servo valve. The input voltage signal in the system is transformed into an electric current signal by the servo where I is the output current, xv is the spool displacement of the main valve, and Kv is the flow gain amplifier. Then the electric current signal is passed into the pilot valve. Therefore, the output current of the electro-hydraulic servo valve. of the servo amplifier can be seen as proportional to the input voltage, which can be regarded as a purely proportional stage and its mathematical model can be expressed as Sensors 2019, 19, 2727 5 of 21 I(s) K = (2) U(s) where I is the output current, U is the input voltage, and K is the amplification coecient of the amplifier. A position sensor which has the advantages of small size, low weight and fast response is used in the feedback stage. The sensor acquires the position signal by calculating and converting it into a feedback signal. The response frequency of the position sensor is much higher than the response frequency of the whole system. Hence, the process can be seen as a purely proportional stage, and the transfer function can be expressed as U = K x (3) m P where U is the output voltage, x is the piston displacement, and K is the magnification coecient of f P m the position sensor. The linearized load flow equation, hydraulic cylinder flow continuity equation, and the force-balance equation of the hydraulic cylinder need to be calculated to illustrate the model of the asymmetrical hydraulic cylinder. The linearized load flow equation from the servo valve to the hydraulic actuator can be written as q = K x K p (4) L q v c L 3 2 where q is the load flow of the servo valve (m /s), K is the flow gain coecient (m /s), x is the open L q v amount of valve spool (m), K is the flow pressure coecient (m /(Ns)), and p is the loading pressure of the cylinder (Pa). The flow continuity equation of the hydraulic cylinder can be expressed as dy dp q = A + C p + (5) L tc L dt 4 e dt 2 5 where A is the e ective area of cylinder piston (m ), C is the total leakage coecient (m /(Ns)), V is tp t the total volume of the cavity (m ), is the e ective bulk modulus (Pa). The force balance equation of the hydraulic cylinder without considering the friction and quality of the oil can be expressed as d y dy Ap = m + + Ky (6) L c dt dt where m is the colligation quality of the cylinder piston discreteness (kg), is the viscous damping coecient of the piston and load (N/(m/s)), K is the spring sti ness (N/mm), and y is the displacement of the piston (m). Equations (4) to (6) represent dynamic models of the hydraulic cylinder. Their Laplace transforms can be written as Q = K X K P (7) L q v c L Q = A Ys + C P + P s (8) L 1 tc L L 4 e AP = ms Y + B sY + KY (9) The transfer function of the position servo control system can be deduced by Equations (1) to (9) K (K +C ) q c tp X (1 + s)P v L A Av 4 (K +C ) e c tp G(s) = (10) m(K +C ) B (K +C ) K(K +C ) mV c tp V c c tp KV c tp t t t 3 2 s + ( + )s + (1 + + )s + 2 2 2 2 2 2 4 A A 4 A A 4 A A e e e Sensors 2019, 19, 2727 6 of 21 4. BAS-PID Control System 4.1. Beetle Antennae Search Algorithm BAS is a meta-heuristic intelligent optimization algorithm based on group optimization. The position of each beetle represents an achievable optimized solution. Beetles use antennae on two sides of their body to find food resources. When the antenna on one side is closer to food, the food odor received by the antenna is stronger, and the individual will move to that side. This new meta-heuristic algorithm BAS taking inspiration from detecting and searching behavior of longhorn beetles. We denote the strength of the food odor at position x to be the value at point of the optimized function. To explore the initial unknown environment, the initial beetle searching is supposed to move randomly in any direction. A vector with a random direction can be built to be representative and normalize in spaces of any dimension. A random searching direction can be normalized as rnd(dim, 1) b = (11) krnd(dim, 1)k where rnd(.) denotes a random function and dim represent the dimensions of the position. Beetles do not know the precise location of food when foraging. They use both antennae to detect the food odor and move in the direction of the odor. The positions of the right and left antenna can be obtained as t t x = x + d  b (12) t t x = x d  b (13) where t is the iteration number, x denotes the position of the right antenna, x denotes the position of t t the left-antenna, x is the position of the beetle, and d is the sensing length of the antennae. The beetle will move to the left if the left antenna receives a stronger scent than the right antenna; otherwise, it will move to the right. The beetle chooses its search behavior based on the direction of the detected odor. Thus, we can determine the next position of a beetle by judging the strength of the odor. The next position of the beetle can be determined by t t1 t x = x + c   b  sign( f(x ) f(x )) (14) r l where  is the step size of searching, sign(.) represents a sign function, f(.) is an optimized function, and c is the direction of beetle movement. If the aim is to find the maximum value, c = 1. If the problem is to find the minimum value, c = +1. t t The sensing length of the antennae d and the step size of searching  can be updated as t t1 d = 0.95d + 0.01 (15) t t1 = 0.95 (16) The iterative process of the beetle antennae search algorithm can be presented as follows: Step 1: Set the maximum number of iterations t . Randomly initialize N beetle positions max X (I = 1, 2, ::: , N). Define all beetle antennae search algorithm parameters, including the initial step size of searching  and the initial sensing length of the antennae d. Define the optimized function. Set direction of the beetle movement c according to the optimization purpose of the optimized function. Step 2: Generate random searching directions using Equation (11). Step 3: Update the positions of the right antennae of beetles using Equation (12). Update positions of the left antennae of beetles using Equation (13). Sensors 2019, 19, 2727 7 of 21 Step 4: Modify beetle positions according to Equation (14). Step 5: Calculate all feasible solutions and compare their fitness values to determine the optimal solution in the current generation. After comparing the current minimum fitness value with the previous minimum fitness value, update the global optimum solution if there is a better solution. Step 6: Update the sensing length of antennae using Equation (15). Update the step size of beetles using Equation (16). Step 7: Update the number of iterations t = t + 1 and return to Step 2. Repeat until t = t . max Step 8: Output the global optimum solution. In order to explain the BAS more clearly, we have detailed the steps of the BAS in Algorithm 1. Algorithm 1: BAS Input: Set the maximum number of iterations t . Define the evaluation function f(.). Randomly set N beetle max positions x (i = 1, 2, ::: , N). Set t = 0. Set the value of c according to the optimization purpose. Record the 0 0 initial sensing length of antennae d , the initial step size  Record the initial optimum solution, x , and the best initial optimum value, g best . Output: x , g . best best 1: while (t < T ) max 2: For i = 1: N 3: Update the searching direction b using Equation (11) 4: Update the right antenna position x using Equation (12) ri 5: Update the left antenna position x using Equation (13) li t+1 t 6: Update the next position x of the beetle x using Equation (14). i i 7: End for 8: For i = 1: N t+ 1 9: Calculate the function value f (x ) of ith beetle. t+ 1 10: If f (x )is better than g i best t+ 1 11: x = x best i t+ 1 12: g = f (x ) best i 13: End if 14: End for 15: Update the sensing length of the antennae d using Equation (15). 16: Update the step size of searching  using Equation (16). 17: t = t + 1 18: End while 4.2. PID Control System PID controllers which have high eciency and robust performances, are linear controllers. Thanks to their easy control structure, high robustness, and high accuracy, PID controllers are commonly used in engineering to enhance both transient and steady-state behaviors. It has been shown that rapid response speed and high controllability can be obtained by determining appropriate PID parameters. PID controllers are widely used in di erent control systems. There are three parameters in a PID controller, namely the proportional parameter K , the di erential parameter K , and the integral p d parameter K . These three parameters have dramatic e ects on the performance of control systems: K i p a ects the response speed of the system, K a ects dynamic performance, and K a ects the steady d i state error. The current PID controller can be divided into two main modes: the continuous form and the discrete form. The continuous form of the PID controller is described as de(t) u(t) = K e(t) + K e(t)dt+K (17) i d dt 0 Sensors 2019, 19, 2727 8 of 21 where K is the proportional parameter, K is the derivative parameter, K is the integral parameter, p d i u(t) is the output of the PID control signal, and e(t) is the system error signal. Let the sampling instant replace continuous time. Let the integral item and di erentiation item be discretized, and the discrete form of PID controller is written as u(k) = K e(k) + K T e(k) + [e(k) e(k 1)] (18) p i k=0 where T is the sampling period, k is the sampling number, and m is the total number of sampling times. 4.3. System Evaluation Function The value of the evaluation function is used to measure the speed of the response of the control system. It is necessary to select an evaluation function before the optimization. The evaluation function of the control system includes the integration of the absolute value of error (IAE), the integral of time multiplied by the absolute value of error (ITAE), the integral of the square value of error (ISE), the mean of the square of the error (MSE). For the PID control system, the objective function based on a single measurement value only evaluates a part of the system. The IAE, which give importance to the absolute error, only takes single factors into account. The IAE is often used in the digital simulation of systems, since it is somewhat dicult to obtain the absolute value of the error in analytic form. So, in order to comprehensively evaluate the performance of electro-hydraulic servo system, this paper do not select the IAE as the evaluation function. The ISE is the square of the error. Large errors are penalized more than smaller ones as the square of a large error is much bigger. Systems designated to minimize ISE tend to weaken large errors rapidly; however, they will have to tolerate small errors continuing for long periods. The gradual accumulation of small errors leads to low control accuracy in the later stage of ISE error calculation. The MSE improves the shortcomings of the ISE by calculating the mean of the ISE. However, the system needs to run for a long period of time in order to reduce the square of a large error calculated by the ISE. Therefore, the MSE is only applied in long-running systems. The ITAE, which is the absolute error multiplied by time, penalizes errors which exist after a long time, and is considered as a measure of system performance. The ITAE weights errors which exist after a long time and has an additional time multiplication in the error function, which emphasizes long duration errors and allows a faster response compared to the ISE and IAE. Thus, ITAE can solve problems more eciently than other evaluation functions [43–47]. It has been shown that ITAE is a better evaluation function in the PID control system. In this paper, double measurement value ITAE is selected as the evaluation function of a PID control system. The ITAE can be written as ITAE = t e(t) dt (19) Furthermore, the discrete ITAE is written as ITAE = kT e(k) T (20) k=o where T is the sampling period, k is the sampling number, and m is the total number of sampling times. 4.4. BAS-PID Control System The working principle of the BAS-PID controller is shown in Figure 2, where u(t) is the output signal of the BAS-PID controller, y(t) is the output signal from the control system, r(t) is the input Sensors 2019, 19, x FOR PEER REVIEW 9 of 21 ITAE = t e() t dt (19) Furthermore, the discrete ITAE is written as ITAE = kT e() k T (20) ko = where T is the sampling period, k is the sampling number, and m is the total number of sampling times. 4.4. BAS-PID Control System Sensors 2019, 19, 2727 9 of 21 The working principle of the BAS-PID controller is shown in Figure 2, where u(t) is the output signal of the BAS-PID controller, y(t) is the output signal from the control system, r(t) is the input signal to the control system, and e(t) is the error. PID parameters will be tuned automatically by BAS, signal to the control system, and e(t) is the error. PID parameters will be tuned automatically by BAS, after three PID parameter adjustment ranges are selected. BAS will tune three parameters according to after three PID parameter adjustment ranges are selected. BAS will tune three parameters according the ITAE value of the control system in real time. to the ITAE value of the control system in real time. TIME ITAE BAS ki kp kd e(t) r(t) u(t) y(t) PID Cont rolled object controller BAS -PID cont roller Figure 2. Block diagram of the BAS-PID controller. Figure 2. Block diagram of the BAS-PID controller. The control quality of the PID controller depends on three parameters. To obtain the optimal PID The control quality of the PID controller depends on three parameters. To obtain the optimal controller, in this paper, the PID parameter-tuning problem is converted into a class of three-dimensional PID controller, in this paper, the PID parameter-tuning problem is converted into a class of three- parameter optimization questions. The three parameters are seen as the beetle’s position in dimensional parameter optimization questions. The three parameters are seen as the beetle’s position three-dimensional space. The ITAE is regarded as the evaluation function. Beetle positions are in three-dimensional space. The ITAE is regarded as the evaluation function. Beetle positions are randomly generated, BAS is run, and then the beetle positions are input into the PID controller as three randomly generated, BAS is run, and then the beetle positions are input into the PID controller as parameters to calculate the evaluation function ITAE. The beetle position which minimizes the ITAE is three parameters to calculate the evaluation function ITAE. The beetle position which minimizes the considered to have the optimum PID parameters. The position of the beetle which minimizes the ITAE ITAE is considered to have the optimum PID parameters. The position of the beetle which minimizes is used to update the optimum PID parameters in the current iteration. If the controllability of the the ITAE is used to update the optimum PID parameters in the current iteration. If the controllability PID control system meets the requirements of the engineering application or the searching procedure of the PID control system meets the requirements of the engineering application or the searching reaches the maximum number of iterations, the optimal position of the beetle will be chosen as the procedure reaches the maximum number of iterations, the optimal position of the beetle will be final PID parameters. chosen as the final PID parameters. The parameter-tuning steps of the BAS-PID controller are as follows: The parameter-tuning steps of the BAS-PID controller are as follows: Step 1: Initialize all parameters and ranges. Step 1: Initialize all parameters and ranges. Randomly generate N beetle positions, X = [K , K , K ], where each parameter uses n (n = 1, 2, ::: , N) i d Randomly generate N beetle positions, X (n = 1, 2, …, N) = [Kp, Ki, Kd], where each parameter uses the the real number coding. The discrete ITAE is regarded as the evaluation function. Set the maximum real number coding. The discrete ITAE is regarded as the evaluation function. Set the m 0 aximum number of iterations, T . Set c =1 and T = 0. Set the initial sensing length, d , and the initial step MAX number of iterations, TMAX. Set c = –1 and T = 0. Set the initial sensing length, d , and the initial step size,  . size, δ . Step 2: Normalize searching directions. To expand the exploration environment, the searching directions of beetles can be normalized in random dimensional space using Equation (11). A random searching direction can be calculated as: rnd(3,1) b = krnd(3,1)k Step 3: Update the positions of the right and left antennae of one beetle. Beetles do not know the precise location of food when foraging. They use their antennae to determine their next direction. When the antenna on one side is closer to food, the food odor received by the antenna is stronger, and the beetle will move to that side. The positions of the beetle’s right and left antennae are determined using Equations (12) and (13). The position of the right antenna can be ! ! T T T T T T obtained as X = X + d  b , and the position of the left antenna can be obtained as X = X d  b . nr n n nl Step 4: Update the next position of one beetle. Operate the control system. The positions of the beetle’s right and left antennae, which can be seen as PID parameters, are carried over to the PID controller. Calculate the ITAE values of the evaluation Sensors 2019, 19, 2727 10 of 21 function of the position of the right antenna, f (X ). Calculate evaluation function ITAE values of ITAE nr left antenna position: f (X ). Equation (14) is used to determine the next position of a beetle to ITAE nl obtain a new set of beetle positions. Step 5: Calculate the ITAE of the evaluation function. Operate the control system. Then, new beetle positions are input into the PID controller as three parameters to calculate the ITAE of evaluation function. By comparing all fitness values, the beetle position which minimizes the ITAE is used to determine the current optimum PID parameters in the current generation. Record the current optimum PID parameters and current minimum ITAE and use in the next step. Step 6: Update the global optimum beetle position. After comparing the current minimum ITAE with the previous minimum ITAE, the global optimum beetle position is updated, and the global optimum beetle position is chosen to be the optimal PID parameters. Step 7: Update the sensing length of the antennae and the next step size. T T1 The sensing length of the antennae can be updated using Equation (15): d = 0.95d + 0.01. n n T T1 The step size of searching can be updated using Equation (16):  = 0.95 . By updating, the sensing n n length of the antennae and the step size of searching will be carried over to the next generation. Step 8: Judge iterations. Calculate and judge whether iterations achieve terminating conditions. Calculate the number of iterations: T = T + 1. Judge whether iterations achieve the terminating condition T = T . If T MAX meets the terminating condition, the global optimum beetle position is chosen to be the optimal PID parameters. If T does not meet the terminating condition, return to Step 2 and start the next loop iteration. Step 9: Stop. Output the global optimum position of the beetle as the final three PID parameters. The parameter-tuning flowchart of the BAS-PID controller is shown in Figure 3. Sensors 2019, 19, 2727 11 of 21 Sensors 2019, 19, x FOR PEER REVIEW 11 of 21 START Initialize all parameters and ranges. Randomly generate N beetle positions xn (n=1,2,…,N) Randomly normalize the searching direction of each beetle using Equation (11): The right-left antenna position can be obtained using Equations (12) and (13) Operate the control system. The right-left antenna position of beetle which can be seen PID parameters is carried over to PID controller. Calculate evaluation function ITAE T T values of right-left antenna positions: fITAE (X ) and fITAE(X ). The next position of the nr nl beetle can be generated using Equation (14). Output the new position of the beetle. K K p i K CONTROL SYSTEM Controlled Plant PID Controller Calculate the ITAE of evaluation function. Record current optimum PID parameters and current minimum ITAE. By comparing current minimum ITAE with the minimum ITAE of previous generation, the global optimum beetle position is updated. Update the sensing length of the antennae d using Equation (15). Update the size of searching δ using Equation (16). NO T=T+1 T≧TMAX YES STOP Figure 3. Parameters tuning flowchart of the BAS-PID controller. Figure 3. Parameters tuning flowchart of the BAS-PID controller. 5. Simulation and Analysis 5. Simulation and Analysis 5.1. Simulation Environment 5.1. Simulation Environment The electro-hydraulic servo valve is an FF102-30 (AVIC Nanjing Servo Control System Co., Ltd., Nanjing, China)., with a Ps of 21 MPa, a rated current of 50 mA, a  of 0.5, a K of 0.006 m /(sA), sv sv The electro-hydraulic servo valve is an FF102-30 (AVIC Nanjing Servo Control System Co., Ltd., 4 3 and a no-load flow of 2.315 10 (m /s). The saturation value of the servo amplifier contr 3 ol voltage Nanjing, China)., with a Ps of 21 MPa, a rated current of 50 mA, a ξsv of 0.5, a Ksv of 0.006 m /(sA), and is10V, the length of the piston −4 3displacement is 35 mm, the piston rod area is 0.001 m , the position a no-load flow of 2.315 × 10 (m /s). The saturation value of the servo amplifier control voltage is sensor gain is 50 V/m, the position sensor range is 7100 mm, the stroke of the cylinder 2 is 200 mm, ±10V, the length of the piston displacement is 35 mm, the piston rod area is 0.001 m , the position the rated flow of the cylinder is 30 L/min, and the oil supply pressure is 4.5 MPa. sensor gain is 50 V/m, the position sensor range is 7100 mm, the stroke of the cylinder is 200 mm, the All the model parameters of the electro-hydraulic position servo control system were obtained rated flow of the cylinder is 30 L/min, and the oil supply pressure is 4.5 MPa. by identifying system parameters. The identification techniques include the product look, numerical All the model parameters of the electro-hydraulic position servo control system were obtained derivation, experimental testing, and engineering experience [48–51]. The natural parameters of the by identifying system parameters. The identification techniques include the product look, numerical electro-hydraulic servo valve in the model are obtained from the servo valve product book. The natural derivation, experimental testing, and engineering experience [48–51]. The natural parameters of the electro-hydraulic servo valve in the model are obtained from the servo valve product book. The natural frequency, damping, and conversion factor of the servo valve were obtained from dynamic Sensors 2019, 19, 2727 12 of 21 frequency, damping, and conversion factor of the servo valve were obtained from dynamic characteristic test curves by numerical derivations. Natural parameters of the servo amplifier and the position sensor were obtained from the product book. Moreover, structural parameters, including the e ective piston area of the servo cylinder, the total piston stroke of the servo cylinder, and the volume of the oil pipe were obtained from the factory data of the cylinder. The working parameters—including system supply oil pressure, system return oil pressure, and sensor gain—were obtained from the experimental testing. Other parameters were selected based on engineering experience. The viscous damping coecient can be ignored. Therefore, the model of the controlled object obtained by identification is as 4.63 G(s) = 12 5 9 4 6 3 2 4.528 10 s + 4.1988 10 s + 5.1725 10 s + 0.002s + s To verify the performance of BAS, this paper also compared the result of the proposed BAS-PID controller with those of PID controllers based other popular artificial intelligence algorithms, including PSO, GA, and FA. For PSO, parameters were set as: learning factors c = c = 1, inertial weight w = 1. 1 2 For GA, roulette wheel selection with an elitism mechanism was used, and parameters were set as: crossover probability P = 0.7, mutation probability P = 0.02. cross m For FA, parameters were set as: initial attractiveness = 1. For BAS, parameters were fixed as: initial sensing length of the antennae d = 2, the initial step size  = 1. For all algorithms, the population size was set as 50 and the maximum number of evaluations was set as 200. All of the initial population positions of the di erent algorithms were generated from a uniform distribution. The lower and upper bounds of the search space of three parameters were given by (0, 100). Each optimization method was implemented over 10 independent runs in the MATLAB software (MathWorks, Natick, MA, USA). The input and disturbance signals were step signals. Through the iteration of the ITAE, the process of PID parameter-tuning can be seen as finding the solution which minimizes the ITAE value. The total sampling time was 100 and the sampling period was 0.01 s. 5.2. Simulation Results and Analysis Table 3 shows the results of the ITAE, overshoots M , the rise time t , the settling time t and their p r d corresponding PID controller parameters. The overshoot M reflects the stability of the system, the rise time t reflects the response capability of the system, and the settling time t reflects the adaptability r d of the system. As seen from Table 3, the BAS parameter-tuning method has the lowest overshoot, the fastest rise time and settling time, and the smallest value of ITAE of all algorithms. Table 3. PID parameters and performances of the control system. BAS FA GA PSO K 7.9927 9.5773 8.0156 8.2923 K 0.1412 6.9449 0 0.0225 K 0.0532 0.0714 0.0978 0 ITAE 0.0275 0.0384 0.0294 0.0358 M 0.0067 0.1439 0.1015 0.1503 t 0.022 0.051 0.056 0.029 t 0.023 0.026 0.029 0.036 Response curves derived from the step input are shown in Figure 4. From the figure, it can be seen that, compared with PIDs using other algorithms, BAS-PID shows the best performance. The step Sensors 2019, 19, 2727 13 of 21 response curve of the BAS -PID controller converges to the desired set value with the minimum overshoot and least steady time. Using the BAS-PID controller, the system can maintain high dynamic Sensors Sensors 2019 2019 , 19, , x FO 19, x FO R PE R PE ER R ER R EVIEW EVIEW 13 of 13 of 21 21 characteristics and stability precision, and the performance of the system is una ected by outside dynam dynam ic ch ic ch aract aract erie st ri ics stics anan d st d st abil ab itil y pr ity pr ecis ecis ion, ion, and and the perform the perform ana ce of the sy nce of the sy stem is un stem is un affec affec ted by ted by interference signals, showing perfect robustness. outside interference signals, showing perfect robustness. outside interference signals, showing perfect robustness. 1.2 1.2 BA BA S S FA FA GA GA 1 1 PSPS O O 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0 0 Time(s) Time(s) Figure 4. Response curves of the step signal. Figure 4. Response curves of the step signal. Figure 4. Response curves of the step signal. Figure 5 illustrate the average convergence curves of all algorithms disposing ITAE functions over Fig Fig ure u 5 illustrate the re 5 illustrate the aver aver age age converge converge nce curv nce curv es of al es of al l al lgori algori thm thm s di s di sposi sposi ng ITAE ng ITAE functi functi ons ons 10 independent runs. As can be observed in the Figure 5, BAS has the fastest convergence speed in all over over 10 10 indep indep endent runs. endent runs. As can As can be observed be observed in th in th e Fe F igur igur e 5, e 5, BAS has the BAS has the fastest convergence speed fastest convergence speed algorithms, showing better performance when optimizing PID controller parameters. Additionally, in ina lla lal l gorithms, showing better perf algorithms, showing better perf orma orma nce nce whwen optimizing PID hen optimizing PID controlle controlle r pr p aram aram eters. eters. BAS achieved the highest iteration speed of the four algorithms. The BAS convergence curve tends to Addition Addition ally ally , BA , BA S ach S ach ieve ieve d the h d the h ighe ighe st iteration st iteration spee spee d of the d of the four four alg alg orithms. T orithms. T he BA he BA S conve S conve rgence rgence accelerate as the number of iterations increases, and quickly converges towards the optimal value after curve tends to a curve tends to a ccel ccel eraete rate as the number of as the number of itera itera tions tions incre incre ases ases , an , an d q d q uicukly ickly con con verges t verges t ow oaw rd as t rds t heh e completing almost half of the iterations. BAS demonstrates outstanding convergence stability and opti opti mama l vl v alue a alue a fter compl fter compl etieng ting almost ha almost ha lf of lf of the iterat the iterat ions. ions. BAS d BAS d emonstrates outst emonstrates outst anding anding competence in jumping out from limited optimum. convergence stability and competence convergence stability and competence in jumping out from limited optimum. in jumping out from limited optimum. 0.2 0.2 BA BA S S 0.18 0.18 FA FA GA GA 0.16 0.16 PSPS O O 0.14 0.14 0.12 0.12 0.10 0.10 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0 50 50 100 100 150 150 200 200 Iterations Iterations Figure 5. Average convergence curves of ITAE. Figure 5. Figure 5. Aver Aver age convergence age convergence curves of curves of IT IT AE. AE. The box plot is shown in Figure 6 displays a set of scattered data. The stability of the control The box plot The box plot is shown is shown in F in F igur igur e 6 d e 6 d ispilays splays a se a se t of t of sca sca ttered da ttered da ta. The st ta. The st aba ilb itiy of lity of the control the control system can be shown by examining the position of the median, upper quartile, and lower quartile system can be shown by examining the position of the median, upper quartile, and lower quartile in system can be shown by examining the position of the median, upper quartile, and lower quartile in in the boxplot. The boxplot measures the dispersion of the ITAE value by graphical means. We note the boxplot. The boxplot measures the d the boxplot. The boxplot measures the d ispersion o ispersion o f th f th e ITAE value e ITAE value by graph by graph ical ical means. means. We n We n ote that ote that the value of ITAE calculated by BAS has fewer outliers and a lower degree of dispersion than other the value of ITAE calculated by BAS has fewer outliers and a lower degree of dispersion than other alg alg orit oh rit ms, hms, in i dicat ndicat ing t ing t hah t t ah t t e propose he propose d BA d BA S h S h as ex as ex cellent cellent stabi stabi lity. The lity. The medi medi ana , upper n, upper qu q ar utar ile, tile, an an d d lower quartile of the ITAE calculated by BAS are lower than for other algorithms, which proves that lower quartile of the ITAE calculated by BAS are lower than for other algorithms, which proves that BAS ha BAS ha s excell s excell ent opti ent opti miz miz atia on ca tion ca pap bi ali bi ty li. ty. ITAE Amplitude ITAE Amplitude Sensors 2019, 19, 2727 14 of 21 that the value of ITAE calculated by BAS has fewer outliers and a lower degree of dispersion than other algorithms, indicating that the proposed BAS has excellent stability. The median, upper quartile, and lower quartile of the ITAE calculated by BAS are lower than for other algorithms, which proves Sensors 2019, 19, x FOR PEER REVIEW 14 of 21 Sensors 2019, 19, x FOR PEER REVIEW 14 of 21 that BAS has excellent optimization capability. 1.00 1.00 0.09 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 BAS FA GA PSO BAS FA GA PSO Algorithms Algorithms Figure 6. Boxplot of ITAE. Figure 6. Boxplot of ITAE. Figure 6. Boxplot of ITAE. To further prove the reliability of the BAS-PID controller, response results of PID controllers To further prove the reliability of the BAS-PID controller, response results of PID controllers To further prove the reliability of the BAS-PID controller, response results of PID controllers based on di erent algorithms are presented in Figures 7 and 8, when disturbance signals are the based on different algorithms are presented in Figures 7 and 8, when disturbance signals are the based on different algorithms are presented in Figures 7 and 8, when disturbance signals are the triangular wave signal and the sawtooth signal, respectively. Figure 7a shows the response curves of triangular wave signal and the sawtooth signal, respectively. Figure 7a shows the response curves of triangular wave signal and the sawtooth signal, respectively. Figure 7a shows the response curves of the triangular wave signal, and Figure 7b shows the local amplification of the response curves. Under the triangular wave signal, and Figure 7b shows the local amplification of the response curves. Under the triangular wave signal, and Figure 7b shows the local amplification of the response curves. Under the control of the BAS-PID controller, the amplitude error is the smallest and the smoothing speed is the control of the BAS-PID controller, the amplitude error is the smallest and the smoothing speed is the control of the BAS-PID controller, the amplitude error is the smallest and the smoothing speed is the highest. Figure 8a shows the response curve of the sawtooth wave signal, and Figure 8b shows the the highest. Figure 8a shows the response curve of the sawtooth wave signal, and Figure 8b shows the highest. Figure 8a shows the response curve of the sawtooth wave signal, and Figure 8b shows local amplification of the response curve. Figure 7 shows that the BAS-PID controller can not only the local amplification of the response curve. Figures 7 shows that the BAS-PID controller can not the local amplification of the response curve. Figures 7 shows that the BAS-PID controller can not restrain the disturbance signal but also enhance the dynamic characteristics and the steadiness of only restrain the disturbance signal but also enhance the dynamic characteristics and the steadiness only restrain the disturbance signal but also enhance the dynamic characteristics and the steadiness the robustness. For the two di erent interference signals, the system has a remarkable capability to of the robustness. For the two different interference signals, the system has a remarkable capability of the robustness. For the two different interference signals, the system has a remarkable capability maintain the stability and reduce the shaking and concussion when the BAS-PID controller is selected, to maintain the stability and reduce the shaking and concussion when the BAS-PID controller is to maintain the stability and reduce the shaking and concussion when the BAS-PID controller is and the performance of the system is una ected by external interference. selected, and the performance of the system is unaffected by external interference. selected, and the performance of the system is unaffected by external interference. 0.6 0.6 BAS BAS BAS FA BAS FA FA -0.1 FA GA GA 0.4 -0.1 GA 0.4 GA PSO PSO PSO PSO -0.2 0.2 -0.2 0.2 -0.3 -0.3 -0.4 -0.2 -0.4 -0.2 -0.4 -0.5 -0.4 -0.5 -0.6 -0.6 -0.6 -0.6 0 0 0.05 0.10 0.15 0.20 0.25 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 0 0.05 0.10 0.15 0.20 0.25 Time(s) Time(s) Time(s) Time(s) (a) (b) (a) (b) Figure 7 Response curves of the triangular wave signal. (a) Whole response curves. (b) Local Figure 7. Response curves of the triangular wave signal. (a) Whole response curves. (b) Local Figure 7 Response curves of the triangular wave signal. (a) Whole response curves. (b) Local amplification curves. amplification curves. amplification curves. Amplitude Amplitude ITAE ITAE Amplitude Amplitude Sensors 2019, 19, 2727 15 of 21 Sensors 2019, 19, x FOR PEER REVIEW 15 of 21 Sensors 2019, 19, x FOR PEER REVIEW 15 of 21 0.4 1.0 BAS BAS 0.4 1.0 FA FA BAS GA 0.3 BAS GA 0.8 FA FA PSO PSO GA 0.3 GA 0.8 PSO PSO 0.2 0.6 0.2 0.6 0.1 0.4 0.1 0.4 0.2 0 0.2 0 -0.1 -0.1 -0.2 -0.2 -0.2 -0.2 0 0.5 1.0 1.5 2.0 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 0 0.5 1.0 1.5 2.0 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 Time(s) Time(s) Time(s) Time(s) (a) (b) (a) (b) Figure 8. Response curves of the sawtooth wave signal. (a) Whole response curves. (b) Local Figure 8. Response curves of the sawtooth wave signal. (a) Whole response curves. (b) Local Figure 8. Response curves of the sawtooth wave signal. (a) Whole response curves. (b) Local amplification curves. amplification curves. amplification curves. 6 6. . Ex Experimental perimental An Analysis alysis 6. Experimental Analysis To certify the e ectiveness of the proposed BAS-PID controller in an actual working environment, To certify the effectiveness of the proposed BAS-PID controller in an actual working To certify the effectiveness of the proposed BAS-PID controller in an actual working environment, the designed the environment, the designed designed controller was use controlle controlle d in a compr r w r wa as s ehensive use used d in in a c experiment a c oo mprehensiv mprehensiv of the e experiment e experiment operation of of the operat the of the operat electro-hydraulic ion of ion of position the electro-hydraulic p servo control system. osition serv The electro control o-hydraulicsystem. The semi-physical electro-hy experimental draulic platform semi-physic is illustrated al the electro-hydraulic position servo control system. The electro-hydraulic semi-physical experimental platform is illustrated in Figure 9. experimental pla in Figure 9. tform is illustrated in Figure 9. (b) (a) (a) (b) Figure 9. The electro-hydraulic semi-physical experiment platform. (a) Experiment platform. Figure 9. The electro-hydraulic semi-physical experiment platform. (a) Experiment platform. Figure 9. The electro-hydraulic semi-physical experiment platform. (a) Experiment platform. (b) Electro-hydraulic servo system. ( (b b) ) El Electr ectro-hydraul o-hydraulic ic se servo rvo system. system. Under the experimental conditions, the power of the hydraulic source was 5.5 k, the rated flow Under the experimental conditions, the power of the hydraulic source was 5.5 k, the rated flow Under the experimental conditions, the power of the hydraulic source was 5.5 k, the rated flow rate was 30 L/min, the rated pressure was 5 MPa, an ADVAN-TECH PCL1710HG multi-function rate was 30 L/min, the rated pressure was 5 MPa, an ADVAN-TECH PCL1710HG multi-function board rate was 30 L/min, the rated pressure was 5 MPa, an ADVAN-TECH PCL1710HG multi-function board was selected, the sampling time was 0.01 s, and the rated current was 40 mA. was selected, the sampling time was 0.01 s, and the rated current was 40 mA. board was selected, the sampling time was 0.01 s, and the rated current was 40 mA. A sinusoidal signal is usually chosen as the performance testing signal. The frequency response A sinusoidal signal is usually chosen as the performance testing signal. The frequency response char A sin acterist usoidal ic of the signal syis u stem under sually cho a s sen as the pe inusoidal sig rformance testing signal. The fre nal can be implemented to test the system quency response characteristic of the system under a sinusoidal signal can be implemented to test the system performance. performance. The frequency response, i.e., the response of the system under a sinusoidal signal, can characteristic of the system under a sinusoidal signal can be implemented to test the system The frequency response, i.e., the response of the system under a sinusoidal signal, can be used to be used to determine the resonance frequency, impedance, dynamic stiffness, and vibration stability performance. The frequency response, i.e., the response of the system under a sinusoidal signal, can determine of the system. The amplitude fre the resonance frequency q,uency impedance, characterist dynamic ic, Aωsti , which is ness, and defin vibration ed as the stability amplitude r of the atisystem. o of be used to determine the resonance frequency, impedance, dynamic stiffness, and vibration stability The the input amplitude signfr al t equency o the ide characteristic, al signal, is inA voked a , which s th is e defined index in t ashthe e frequ amplitude ency resratio ponse. If of the Aωinput is closer signal of the system. The amplitude frequency characteristic, Aω, which is defined as the amplitude ratio of to 1, the system is more stable. In the experiment, five sinusoidal signals with different amplitudes to the ideal signal, is invoked as the index in the frequency response. If A is closer to 1, the system the input signal to the ideal signal, is invoked as the index in the frequency response. If Aω is closer were used as disturbance signals. For the five sinusoidal signals, the angular velocity was 4π, the is more stable. In the experiment, five sinusoidal signals with di erent amplitudes were used as to 1, the system is more stable. In the experiment, five sinusoidal signals with different amplitudes initial phase was zero, and the amplitudes were 2, 4, 6, 8, and 10, respectively. The PID controllers disturbance signals. For the five sinusoidal signals, the angular velocity was 4, the initial phase was were used as disturbance signals. For the five sinusoidal signals, the angular velocity was 4π, the based on different algorithms, including FA, GA, and PSO, were selected for experiments in the same zero, and the amplitudes were 2, 4, 6, 8, and 10, respectively. The PID controllers based on di erent initial phase was zero, and the amplitudes were 2, 4, 6, 8, and 10, respectively. The PID controllers experimental environment. The ranges of PID parameters for all algorithms and iterations were the algorithms, including FA, GA, and PSO, were selected for experiments in the same experimental based on different algorithms, including FA, GA, and PSO, were selected for experiments in the same same as in Section 5. The experimental results were contrasted and analyzed. The response results environment. The ranges of PID parameters for all algorithms and iterations were the same as in experimental environment. The ranges of PID parameters for all algorithms and iterations were the are shown in Figures 10–14. Section 5. The experimental results were contrasted and analyzed. The response results are shown in same as in Section 5. The experimental results were contrasted and analyzed. The response results Figures 10–14. are shown in Figures 10–14. Amplitude Amplitude Amplitude Amplitude Sensors Sensors 2019 2019,, 19 19,, x FO 2727 R PEER REVIEW 16 16 of of 21 21 Sensors 2019, 19, x FOR PEER REVIEW 16 of 21 Sensors 2019, 19, x FOR PEER REVIEW 16 of 21 2.5 BAS ideal amplitude 2.5 FA 2.5 BAS BAS BAS 2 GA ideal amplitude ideal amplitude FA FA FA BAS PSO 2 BAS GA GA 2.0 GA FA PSO PS FA O GA PSO 2.0 GA 2.0 PSO PSO 1.5 0 1.5 1.5 1.0 -1 1.0 1.0 -1 -1 0.5 -2 0.5 0.5 -2 -2 -3 0.10 0.20 0.30 -300.2 0.4 0.6 0.8 1.0 -3 0.10 0.20 0.30 0 Time(s) 0 0.10 0.20 0.30 00.2 0.4 Time(s)0.6 0.8 1.0 00.2 0.4 0.6 0.8 1.0 Time(s) Time(s) Time(s) (a) (b) Time(s) (a) (b) (b) (a) Figure 10. Response curves of the sinusoidal signal whose amplitude is 2. (a) Whole response curves. Figure 10. Response curves of the sinusoidal signal whose amplitude is 2. (a) Whole response curves. (b) Local amplification curves. Figure Figure 10. 10. Response Response cu cur rves ves o off the s the sinusoidal inusoidal sig signal nal whose whose a amplitude mplitude is is 2. 2. ( (a a) ) Whole respo Whole response nse cu curves. rves. (b) Local amplification curves. ((b b)) Lo Local cal amplification amplification curves. curves. BAS 5 5.0 FA BAS ideal amplitude 5.0 GA BAS 4 5.0 3 FA BAS PS FAO ideal amplitude GA ide FA al amplitude 4.0 GA 3 BAS GA 3 PSO BAS PSO FA 4.0 PSO FA 4.0 2 GA 2 GA PSO PSO 3.0 3.0 3.0 -1 2.0 -1 -2 -1 2.0 2.0 -2 -2 -3 1.0 -3 -3 -4 1.0 1.0 -4 -5 -4 00.2 0.4 0.6 0.8 1.0 0.10 0.20 0.30 -5 0 -5 Time(s) 0 00.2 0.4 0.6 0.8 1.0 Time(s) 0 0.10 0.20 0.30 00.2 0.4 0.6 0.8 1.0 0 0.10 0.20 0.30 Time(s) Time(s) Time(s) Time(s) (a) (a) (b) (b) (a) (b) Figure 11. Response curves of the sinusoidal signal whose amplitude is 4. (a) Whole response curves. Figure 11. Response curves of the sinusoidal signal whose amplitude is 4. (a) Whole response curves. Figure 11. Response curves of the sinusoidal signal whose amplitude is 4. (a) Whole response curves. ( Figure 11. b) Local amplification Response cu curves. rves of the sinusoidal signal whose amplitude is 4. (a) Whole response curves. (b) Local amplification curves. (b) Local amplification curves. (b) Local amplification curves. 7.0 BAS ideal amplitude 6 7.0 FA 7.0 BAS BAS GA BAS 6.0 ideal amplitude 6 FA iFA deal amplitude 6 PS FAO BAS GA 6.0 GA BAS GA 6.0 FA PSO 5.0 PSO FA PSO 4 GA 4 GA PSO 5.0 5.0 PSO 4.0 4.0 4.0 0 3.0 -2 3.0 3.0 2.0 -2 -2 -4 2.0 2.0 -4 -4 1.0 -6 1.0 1.0 -6 -6 -8 0.10 0.20 0.30 00.2 0.4 0.6 0.8 1.0 -8 -8 Time(s) 0 0.10 0.20 0.30 00.2 0.4 Time(s)0.6 0.8 1.0 0.10 0.20 0.30 00.2 0.4 0.6 0.8 1.0 Time(s) Time(s) Time(s) Time(s) (a) (b) (a) (b) (a) (b) Figure 12. Response curves of the sinusoidal signal whose amplitude is 6. (a) Whole response curves. Figure 12. Response curves of the sinusoidal signal whose amplitude is 6. (a) Whole response curves. Figure 12. Response curves of the sinusoidal signal whose amplitude is 6. (a) Whole response curves. (b) Local amplification curves. Figure 12. Response curves of the sinusoidal signal whose amplitude is 6. (a) Whole response curves. (b) Local amplification curves. (b) Local amplification curves. (b) Local amplification curves. Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Amp Am litu pd li etude Amplitude Amp Am litud ple itude Amplitude Amplitude Amp Am litud plie tude Amplitude Amplitude Amplitude Sensors 2019, 19, 2727 17 of 21 Sensors 2019, 19, x FOR PEER REVIEW 17 of 21 Sensors 2019, 19, x FOR PEER REVIEW 17 of 21 9.0 BAS 9.0 ideal amplitude BA FA S 8.0 BAS ideal amplitude FA GA 8.0 BA FA S PSO GA GA 7.0 FA PSO PSO GA 4 7.0 PSO 4 6.0 6.0 5.0 5.0 4.0 -2 4.0 -2 3.0 -4 3.0 -4 2.0 -6 2.0 -6 1.0 -8 1.0 -8 -10 0.10 0.20 0.30 0 0 -1000.2 0.4 0.6 0.8 1.0 0.10 0.20 0.30 00.2 0.4 0.6 0.8 1.0 Time(s) Time(s) Time(s) Time(s) (a) (b) (a) (b) Figure 13. Response curves of the sinusoidal signal whose amplitude is 8. (a) Whole response curves. Figure 13. Figure 13.Re Response sponse cu cur rves ves o of f the s the sinusoidal inusoidal sig signal nal whose whose aamplitude mplitude is is 8. 8. (a (a ) Whole respo ) Whole response nse cu cu rves rves. . (b) Local amplification curves. ((b b) Lo ) Local cal amplification amplification curves. curves. 11.0 11.0 BAS ideal amplitude 10.0 BA FA S ide BA al a Smplitude 10.0 FA GA 8 BAS 9.0 FA PSO GA 8 GA 9.0 FA PSO GA PSO 8.0 PSO 8.0 7.0 2 7.0 6.0 6.0 5.0 -2 5.0 -2 -4 4.0 -4 4.0 -6 3.0 -6 3.0 -8 2.0 -8 2.0 -10 1.0 -10 -12 1.0 -1200.2 0.4 0.6 0.8 1.0 0 0 0.10 0.20 0.30 00.2 0.4 0.6 0.8 1.0 Time(s) 0.10 0.20 0.30 Time(s) Time(s) Time(s) (a) (b) (a) (b) Figure 14. Response curves of the sinusoidal signal whose amplitude is 10. (a) Whole response curves. Figure 14. Response curves of the sinusoidal signal whose amplitude is 10. (a) Whole response curves. Figure 14. Response curves of the sinusoidal signal whose amplitude is 10. (a) Whole response curves. (b) Local amplification curves. (b) Local amplification curves. (b) Local amplification curves. Figures 10b, 11b, 12b, 13b, and 14b show the local amplification curves for the five sinusoidal Figures 10b, 11b, 12b, 13b, and 14b show the local amplification curves for the five sinusoidal Figures 10b, 11b, 12b, 13b, and 14b show the local amplification curves for the five sinusoidal signals. It can be seen from Figures 10–14 that the response curves for the BAS-PID controller have the signals. It can be seen from Figures 10–14 that the response curves for the BAS-PID controller have sign smallest als. It can distance be s between een from the Figure ideal s amplitude 10–14 that the and re the sponse curves actual amplitude. for the Ther BA efor S-Pe, ID controlle we can infer r have that the smallest distance between the ideal amplitude and the actual amplitude. Therefore, we can infer the BAS-PID controller has outstanding performance including vibrational stability, strong dynamic the smallest distance between the ideal amplitude and the actual amplitude. Therefore, we can infer that the BAS-PID controller has outstanding performance including vibrational stability, strong sti ness, and high mechanical impedance. that the BAS-PID controller has outstanding performance including vibrational stability, strong dynamic stiffness, and high mechanical impedance. Frequency response indices for di erent PID controllers are listed in Table 4, where DA! = dynamic stiffness, and high mechanical impedance. |1 – A!|. From Table 4, it can be seen that the frequency response of the system controlled by the Frequency response indices for different PID controllers are listed in Table 4, where∆Aω BAS-PID controller still is clearly superior to the other PID controllers. The amplitude frequency Frequency response indices for different PID controllers are listed in Table 4, where ∆Aω characteristic = |1 – AωA |. From Table ! of the system 4, it can be controlled seen t by hat the fr the BAS-PID equenc contr y respons oller e of the is the closest system controlled to 1. This shows = |1 – Aω|. From Table 4, it can be seen that the frequency response of the system controlled by the BAS-PID controller still is clearly superior to the other PID controllers. The that the BAS-PID controller can asymptotically maintain system stability when the interference signal by the BAS-PID controller still is clearly superior to the other PID controllers. The is alter amplit ed dramatically ude frequenc . y characteristic Aω of the system controlled by the BAS-PID controller amplitude frequency characteristic Aω of the system controlled by the BAS-PID controller is the closest to 1. This shows that the BAS-PID controller can asymptotically maintain is the closest to 1. This shows that the BAS-PID controller can asymptotically maintain system stability when the interference signal is altered dramatically.Table 4. Frequency system stability when the interference signal is altered dramatically.Table 4. Frequency response characteristics response characteristics Amplitude Index BAS PSO GA FA Amplitude Index BAS PSO GA FA Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Sensors 2019, 19, 2727 18 of 21 Table 4. Frequency response characteristics. Sensors 2019, 19, x FOR PEER REVIEW 18 of 21 Sensors 2019, 19, x FOR PEER REVIEW 18 of 21 Amplitude Index BAS PSO GA FA Table 4. Frequency response characteristics Aω 0.9870 1.0270 0.9695 1.0165 A 0.9870 1.0270 0.9695 1.0165 2 ! ∆Aω 0.0130 0.0270 0.0305 0.0165 DA 0.0130 0.0270 0.0305 0.0165 Amplitude Index ! BAS PSO GA FA Aω 0.9878 1.0267 0.9693 1.0175 A 0.9878 1.0267 0.9693 1.0175 Aω 0.9870 1.0270 0.9695 1.0165 ∆Aω DA 0.0 0.0122122 0 0.0267.0267 0 0.0307.0307 0 0.0175.0175 ∆Aω 0.0130 0.0270 0.0305 0.0165 A 0.9878 1.0271 0.9186 1.0177 Aω ! 0.9878 1.0271 0.9186 1.0177 Aω 0.9878 1.0267 0.9693 1.0175 DA 0.0122 0.0271 0.0814 0.0177 4 ! ∆Aω 0.0122 0.0271 0.0814 0.0177 ∆Aω 0.0122 0.0267 0.0307 0.0175 A 0.9879 1.0267 0.9878 1.0174 Aω 0.9879 1.0267 0.9878 1.0174 Aω 0.9878 1.0271 0.9186 1.0177 8 DA 0.0121 0.0267 0.0122 0.0174 ∆Aω 0.0121 0.0267 0.0122 0.0174 ∆Aω A 0.0122 0.9878 0 1.0191.0271 0.96960.0814 1.01900.0177 Aω 0.9878 1.0191 0.9696 1.0190 DA 0.0122 0.0191 0.0304 0.0190 Aω ! 0.9879 1.0267 0.9878 1.0174 ∆Aω 0.0122 0.0191 0.0304 0.0190 ∆Aω 0.0121 0.0267 0.0122 0.0174 Aω 0.9878 1.0191 0.9696 1.0190 To further display the performance of the BAS-PID controller, the response results of the system To further display the performance of the BAS-PID controller, the response results of the system ∆Aω 0.0122 0.0191 0.0304 0.0190 under di erent PID controllers when interference signals were random signals are presented in under different PID controllers when interference signals were random signals are presented in Figures 15 and 16. Figures 15 and 16. To further display the performance of the BAS-PID controller, the response results of the system under different PID controllers when interference signals were random signals are presented in 1.4 1.5 BAS BAS Figures 15 and 16. FA FA 1.4 GA 1.2 GA 1.5 PSO BAS BAS PSO FA FA GA 1.2 GA PSO 1.0 PSO 1.0 1.0 0.8 1.0 0.8 0.6 0.5 0.6 0.4 0.5 0.4 0.2 0.2 00.1 0.2 0.3 0.4 0.5 02 48 6 10 Time(s) Time(s) 0.3 00.1 0.2 0.4 0.5 02 48 6 10 Time(s) (a) (b) Time(s) (a) Figure 15. Response curves of the random signal l. (a) Whole response curves. ( (b b) ) Local amplification Figure 15. Response curves of the random signal l. (a) Whole response curves. (b) Local Figure curv 1615 es. . Response curves of the random signal l. (a) Whole response curves. (b) Local amplification amplification curves. 1.2 1.2 curves. BAS BAS FA FA 1.2 1.2 GA 1.0 GA 1.0 BAS BAS PSO PSO FA FA GA 1.0 GA 1.0 PSO 0.8 PSO 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 01 24 3 5 0.3 00.1 0.2 0.4 0.5 0 Time(s) Time(s) 01 24 3 5 0.3 00.1 0.2 0.4 0.5 (a) (b) Time(s) Time(s) (b) Figure 16. Response cu rves (a) of the random signal 2. (a) Whole response curves. (b) Local amplification curves. Figure 16. Response curves of the random signal 2. (a) Whole response curves. (b) Local amplification Figure 16. Response curves of the random signal 2. (a) Whole response curves. (b) Local curves. amplification curves. Local enlarged drawings clearly show that the BAS-PID controller can not only rapidly suppress interference signals but also prevent excessive overshoot. For the unusual interference signals, the Amplitude Amplitude Amplitude Amplitude AmpAm litud pe litude Amplitude Amplitude Sensors 2019, 19, 2727 19 of 21 Local enlarged drawings clearly show that the BAS-PID controller can not only rapidly suppress interference signals but also prevent excessive overshoot. For the unusual interference signals, the BAS-PID controller has a remarkable capability to maintain the system stability and reduce the shaking, and the performance of the system is una ected by external interference. In other words, the BAS-PID controller has anti-interference ability and provides remarkable balance, which can enhance the anti-seismic properties of the electro-hydraulic position servo control system in an unknown environment. 7. Conclusions In order to enhance the control accuracy and ability of an electro-hydraulic position servo control system, the paper addressed the problem of determining three parameters of PID controllers. A PID parameter tuning method based on the beetle antennae search algorithm was applied to an electro-hydraulic position servo control system. A transfer function model was obtained by system parameter identification. A basic mathematical model of the electro-hydraulic position servo control system was established through theoretical analysis. The PID tuning problem was converted into a three-dimensional optimization question. The performance of the BAS tuning method was tested by ITAE and compared with that of the PSO, GA, and FA algorithms. An analysis of the performance of the BAS-PID controller with the electro-hydraulic position servo control system showed that the BAS algorithm can e ectively adjust three parameters of the PID controller. The BAS-PID controller can bring many advantages, such as restraining system interference and meeting the requirement that the control system can maintain the robustness when there are the di erent external signals, which can better maintain the control needs of the electro-hydraulic position servo control system. Author Contributions: Conceptualization, Y.F., J.S. and G.S; Investigation, Y.F., J.S. and G.S; Resources, J.S.; Project administration J.S.; Supervision, J.S. and G.S.; Writing—review & editing, Y.F., J.S. and G.S. Funding: This research was funded by the International Cooperation Project (grant no. 2012DFR70840). Conflicts of Interest: The authors declare no conflicts of interest. References 1. Yao, J.; Jiao, Z.; Ma, D.; Yan, L. High-Accuracy Tracking Control of Hydraulic Rotary Actuators with Modeling Uncertainties. IEEE ASME Trans. Mechatron. 2014, 19, 633–641. [CrossRef] 2. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Sensors (Basel, Switzerland) Pubmed Central

Optimized PID Controller Based on Beetle Antennae Search Algorithm for Electro-Hydraulic Position Servo Control System

Sensors (Basel, Switzerland) , Volume 19 (12) – Jun 18, 2019

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Abstract

sensors Article Optimized PID Controller Based on Beetle Antennae Search Algorithm for Electro-Hydraulic Position Servo Control System Yuqi Fan, Junpeng Shao * and Guitao Sun School of Mechanical and Power Engineering, Harbin University of Science and Technology, Harbin 150080, China; yuqifan77@163.com (Y.F.); sunguitao86@163.com (G.S.) * Correspondence: Sjp566@hrbust.edu.cn Received: 1 April 2019; Accepted: 14 June 2019; Published: 18 June 2019 Abstract: To improve the controllability of an electro-hydraulic position servo control system while simultaneously enhancing the anti-jamming ability of a PID controller, a compound PID controller that combines the beetle antennae search algorithm with PID strategy was proposed, and used to drive the position servo control system of the electro-hydraulic servo system. A BAS-PID controller was designed, and the beetle antennae search algorithm was used to tune PID parameters so that the disturbance signal of the system was e ectively restrained. Initially, the basic mathematical model of the electro-hydraulic position servo control system was established through theoretical analysis. The transfer function model was obtained by identifying system parameters. Then, the PID parameter-tuning problem was converted into a class of three-dimensional parameter optimization problem, and gains of PID controllers were adjusted using the beetle antennae search algorithm. Finally, by comparing the e ectiveness of di erent algorithms, simulation and experimental results revealed that the BAS-PID controller can greatly enhance the performance of the electro-hydraulic position servo control system and inhibit external disturbances when di erent interference signals are used to test the system’s robustness. Keywords: PID controller; electro-hydraulic servo system; beetle antennae search algorithm 1. Introduction Electro-hydraulic servo systems (EHSSs) have been widely used in the industry due to their advantages of high response, fast response, good sti ness, and high force [1]. With the development of digital technology, control theory, pattern recognition technology, and electronic technology, EHSSs have become one of the most popular topics in the fields of both scientific research and industrial engineering [2,3]. EHSSs including a position servo control system and a force servo control system are a typical closed-loop torque control system [4,5]. Position servo control systems play a major role in marine operating systems, industrial automation systems, and crane systems [6]. Due to the complexity of such control systems, it is considered urgent to develop control technology. Various control methods have been presented, such as the feedback control method [7], tracking control method [8], and adaptive control method [9]. Among these, the PID control system is the most popular control method. Due to its easy control structure, high robustness, and high accuracy, the conventional PID control system is the most widely used in industrial processes [10–12]. The chief function of PID controllers is to regulate feedback signals to be as similar to input signals as possible. The control performance of PID controllers is mainly determined by a proportional parameter, an integral parameter, and a di erential parameter [13,14]. It has been shown that rapid response and high accuracy of velocity can be obtained by selecting appropriate PID parameters in the Sensors 2019, 19, 2727; doi:10.3390/s19122727 www.mdpi.com/journal/sensors Sensors 2019, 19, 2727 2 of 21 control system. Therefore, the method of tuning PID parameters has become a popular research topic, and some significant achievements have been made in recent years [15,16]. Over the past few decades, several methods which can enhance the capacity of obtaining reasonable PID parameters have been announced. One of the most popular methods is the Z-N tuning method [17], proposed by Ziegler and Nichols in 1942. This method is widely used in industry to solve the traditional PID parameter tuning problem [18]. However, the mechanism of the controlled object is complex in actual industrial circumstances; many factors need to be considered, such as high-order duplication, time variation, and non-linearity. Consequently, it is dicult to achieve a perfect control result by using the Z-N tuning method. However, since the method’s publication, it has been extensively developed for tuning PID parameters. With the development of artificial intelligence science, artificial intelligence algorithms have been widely used in PID controllers to select reasonable PID parameters, for example the Genetic Algorithm (GA) [19,20], Particle Swarm Optimization Algorithm (PSO) [21,22], Firefly Algorithm (FA) [23,24], Cuckoo Search Optimization Algorithm (CS) [25], Grey Wolf Optimization Algorithm (GWO) [26], Ant Colony Algorithm (ACA) [27], and Bacterial Foraging Optimization Algorithm (BFO) [28]. The Beetle Antennae Search Algorithm (BAS) is an ecient and intelligent search algorithm proposed by Xiangyuan Jiang and Shuai Li in 2017 [29]. It is an evolutionary computation derived from the flocking and swarming behavior, foraging behavior and courtship behavior of beetles. The algorithm not only has the ability of individual recognition and environmental recognition, but also does not need to know the gradient information of the function. Furthermore, its program code is simple to implement, and it has an especially strong ability to obtain the best solution within a stable convergence property. BAS is used in neural network optimization [30], multi-objective energy management [31], investment portfolio problems [32], the prediction of the content of dissolved gases in power transformer oil [33], the determination of Young’s modulus in jet grouted coalcretes [34], prediction of permeability and unconfined compressive strength of pervious concrete [35], multi-objective optimization based BAS-WPT [36], and unmanned aerial vehicle (UAV) sensing and avoidance [37]. To improve the PID controllability of an electro-hydraulic position servo control system, this paper proposes a BAS-PID controller whose parameters are tuned by BAS. To verify the performance of the BAS-PID controller in the electro-hydraulic position servo control system, the controllability of the BAS-PID controller was compared to the controllability of PID controllers using other algorithms under di erent interference signals. Experimental results showed the superiority of the BAS-PID controller. 2. Related Works BAS is extremely ecient in solving di erent optimization problems, and the original BAS, as well as BAS variants, have been successfully applied in various scientific and engineering fields. Yuantian Sun et al. [30] proposed combination of the back-propagation neural network (BPNN) and BAS, termed the BPNN-BAS model. The BPNN-BAS model was shown to be more e ective than multiple regression and logistic regression. Additionally, in another study [34], support vector machine (SVM) was combined with BAS to produce the SVM-BAS model for the determination of Young’s modulus in jet grouted coalcretes. Zongyao Zhu et al. [31] applied BAS in microgrids to reduce environmental pollution and minimize pollutant treatment cost. Tingting Chen et al. [32] proposed the Beetle Swarm Optimization Algorithm (BSO), which combines BAS and PSO. Shengwei Fei et al. [33] developed the BAS based mixed kernel relevance vector regression (BAS-MkRVR) model to calculate the gas content of power systems. Junbo Sun et al. [35] introduced an ESVR-BAS model to predict the PC and UCS in pervious concrete. Xiangyuan Jiang and Shuai Li [36], who are designers of BAS, advanced the research on BAS by releasing tuning parameters to deal with multi-objective optimization. Qing Wu et al. [37] invented a new path algorithm called the Obstacle Avoidance Beetle Antennae Search Algorithm (OABAS), which is used for global path planning for UAVs. Meijin Lin et al. [38] proposed a hybrid PSO-BAS algorithm, while the same lead author together with Qinghao Li [39] used PSO-BAS to distribute economic loads in power systems. Qinhao Li [40] used BAS to solve Sensors 2019, 19, 2727 3 of 21 optimal scheduling problems of a wind–PV–hydro–storage power complementary system. Chen Wang et al. [41] employed BAS to enhance the precision of the evaluation of straightness error. Table 1 provides a brief comparison among BAS variants. Table 1. Variant ways and advantages of BAS variants. BAS Variants Variant Way Advantages BAS was used to train back propagation  High BPNN training speed BPNN-BAS [30] neural network (BPNN)  Make weights optimal Greater global search ability PSO-BAS [32,38,39]  BAS was combined with PSO  Better searching capacity than that of standard PSO BAS was used to select the appropriate kernel parameters and controlled  Stronger prediction capacity than BAS-MkRVR [33] parameters of the mixed kernel relevance RBFRVR and Sigmoid RVR. vector regression (MkRVR) The hyper-parameters of the support  Less time-consuming SVM-BAS [34] vector machine (SVM) were firstly tuned  Low-cost by BAS  Non-destructive The hyper-parameters of the evolved  More ecient than random ESVR-BAS [35] support vector regression was tuned by hyper-parameter selection BAS  High predictive capability Not require prior parameter tuning Normalization method and penalty BAS-WPT [36]  Simple to design, implement, and has function method were used to extend BAS less complexity Wider search range and the breakneck OABAS was designed on the basic of OABAS [37] search speed BAS to improve the path planning Shorter path length In [42] Yinyan Zhang, Shuai Li, and Bin Xu summarized and analyzed the convergence of BAS and its applications, which provided further theoretical proof of its extreme convergence ability. For a complex optimization problem with multiple parameters, the complexities of the solution space increase with increasing dimensionality, which may cause low convergence speed. For potential extensions and future work, in order to obtain a more accurate solution, the structure of BAS should be improved. Further research may consider an encoding scheme to introduce better diversity into the solution search space of BAS to raise the convergence of the algorithm. In this study, we will apply advanced BAS to solve more practical questions about electro-hydraulic systems. Table 2 shows the uniqueness of BAS among algorithms including PSO [32,37–39], GA [39], FA, BA [42], and the Artificial Bee Colony Algorithm (ABC) [37]. Table 2. Uniqueness of BAS among algorithms. Contrastive Relevant Recent Advantages of BAS Algorithms References Faster iteration speed PSO [32,37–39] Stronger ability to jump local optimal solution Does not need binary to represent decimal numbers GA [39] The BAS program runs faster Does not need more initial parameters FA This paper BAS program code simple BA Simple to implement, and has less complexity [42] Higher eciency ABC [37] Lower time complexity Sensors 2019, 19, x FOR PEER REVIEW 4 of 21 Sensors 2019, 19, 2727 4 of 21 Higher efficiency ABC [37] Lower time complexity From Table 2, it can be determined that BAS not only has abilities of individual recognition and From Table 2, it can be determined that BAS not only has abilities of individual recognition and environmental recognition but also allows the balance in exploratory stages, which shows that it is environmental recognition but also allows the balance in exploratory stages, which shows that it is necessary to use BAS to develop better sensing systems. necessary to use BAS to develop better sensing systems. 3. Electro-Hydraulic Position Servo Control System 3. Electro-Hydraulic Position Servo Control System 3.1. Working Principles of the System 3.1. Working Principles of the System The electro-hydraulic position servo system studied in this paper is chiefly composed of a hydraulic The electro-hydraulic position servo system studied in this paper is chiefly composed of a cylinder, an electro-hydraulic servo valve, a servo amplifier, a position sensor, and some other parts. hydraulic cylinder, an electro-hydraulic servo valve, a servo amplifier, a position sensor, and some A schematic diagram of the system is displayed in Figure 1. other parts. A schematic diagram of the system is displayed in Figure 1. Figure 1. Block diagram of the electro-hydraulic position servo control system. Figure 1. Block diagram of the electro-hydraulic position servo control system. The working principle of the position servo system is characterized as follows. Firstly, the voltage The working principle of the position servo system is characterized as follows. Firstly, the signal is transformed into an electric current signal by the servo amplifier and then input to the servo voltage signal is transformed into an electric current signal by the servo amplifier and then input to valve, and the slide valve of the main valve will be moved depending on the signal of the servo the servo valve, and the slide valve of the main valve will be moved depending on the signal of the valve. Then, the flow rate of the hydraulic cylinder is influenced by the movement of the slide valve. servo valve. Then, the flow rate of the hydraulic cylinder is influenced by the movement of the slide The piston rod is linked to the position sensor. Finally, depending on analogue–digital conversion, valve. The piston rod is linked to the position sensor. Finally, depending on analogue–digital the signal will be fed back to the controller of the control system. Overall, the error between the actual conversion, the signal will be fed back to the controller of the control system. Overall, the error signal and the target signal will be constantly decreased by the position servo control system. between the actual signal and the target signal will be constantly decreased by the position servo In practical engineering problems, due to complexities of the structure, manufacturing errors, large control system. delay time, inaccurate measurement, and other factors, it is dicult to accurately describe the whole In practical engineering problems, due to complexities of the structure, manufacturing errors, working process by building a precise mathematical model. There is not a determined mathematical large delay time, inaccurate measurement, and other factors, it is difficult to accurately describe the model, and therefore the chief aim of this paper is to make the electro-hydraulic servo system model as whole working process by building a precise mathematical model. There is not a determined similar to the really original model as possible. mathematical model, and therefore the chief aim of this paper is to make the electro-hydraulic servo 3.2. Model of the Electro-Hydraulic Position Servo Control System system model as similar to the really original model as possible. The electro-hydraulic position servo system studied in this paper is an electro-hydraulic flow 3.2. Model of the Electro-Hydraulic Position Servo Control System servo valve which controls cylinder structure. The electro-hydraulic servo valve is used as an electrThe electro-h o-hydraulic transfer ydraulic position servo sy apparatus, and is put stem studie into a lower d in this pape input signal r is an to obtain electro e ective -hydraulic flo hydraulic w pr servo v essure alve ener w gy hich cont and to achieve rols cylin the der aim struc oftcontr ure. Th ole e for lec hydraulic tro-hydraulic systems. servo valve is used as an electro- hydraulic transfer apparatus, and is put into a lower input signal to obtain effective hydraulic x (s) pressure energy and to achieve the aim of control for hydraulic systems. K = (1) I(s) xs () K = (1) where I is the output current, x is the spool displacement of the main valve, and K is the flow gain of v v v Is () the electro-hydraulic servo valve. The input voltage signal in the system is transformed into an electric current signal by the servo where I is the output current, xv is the spool displacement of the main valve, and Kv is the flow gain amplifier. Then the electric current signal is passed into the pilot valve. Therefore, the output current of the electro-hydraulic servo valve. of the servo amplifier can be seen as proportional to the input voltage, which can be regarded as a purely proportional stage and its mathematical model can be expressed as Sensors 2019, 19, 2727 5 of 21 I(s) K = (2) U(s) where I is the output current, U is the input voltage, and K is the amplification coecient of the amplifier. A position sensor which has the advantages of small size, low weight and fast response is used in the feedback stage. The sensor acquires the position signal by calculating and converting it into a feedback signal. The response frequency of the position sensor is much higher than the response frequency of the whole system. Hence, the process can be seen as a purely proportional stage, and the transfer function can be expressed as U = K x (3) m P where U is the output voltage, x is the piston displacement, and K is the magnification coecient of f P m the position sensor. The linearized load flow equation, hydraulic cylinder flow continuity equation, and the force-balance equation of the hydraulic cylinder need to be calculated to illustrate the model of the asymmetrical hydraulic cylinder. The linearized load flow equation from the servo valve to the hydraulic actuator can be written as q = K x K p (4) L q v c L 3 2 where q is the load flow of the servo valve (m /s), K is the flow gain coecient (m /s), x is the open L q v amount of valve spool (m), K is the flow pressure coecient (m /(Ns)), and p is the loading pressure of the cylinder (Pa). The flow continuity equation of the hydraulic cylinder can be expressed as dy dp q = A + C p + (5) L tc L dt 4 e dt 2 5 where A is the e ective area of cylinder piston (m ), C is the total leakage coecient (m /(Ns)), V is tp t the total volume of the cavity (m ), is the e ective bulk modulus (Pa). The force balance equation of the hydraulic cylinder without considering the friction and quality of the oil can be expressed as d y dy Ap = m + + Ky (6) L c dt dt where m is the colligation quality of the cylinder piston discreteness (kg), is the viscous damping coecient of the piston and load (N/(m/s)), K is the spring sti ness (N/mm), and y is the displacement of the piston (m). Equations (4) to (6) represent dynamic models of the hydraulic cylinder. Their Laplace transforms can be written as Q = K X K P (7) L q v c L Q = A Ys + C P + P s (8) L 1 tc L L 4 e AP = ms Y + B sY + KY (9) The transfer function of the position servo control system can be deduced by Equations (1) to (9) K (K +C ) q c tp X (1 + s)P v L A Av 4 (K +C ) e c tp G(s) = (10) m(K +C ) B (K +C ) K(K +C ) mV c tp V c c tp KV c tp t t t 3 2 s + ( + )s + (1 + + )s + 2 2 2 2 2 2 4 A A 4 A A 4 A A e e e Sensors 2019, 19, 2727 6 of 21 4. BAS-PID Control System 4.1. Beetle Antennae Search Algorithm BAS is a meta-heuristic intelligent optimization algorithm based on group optimization. The position of each beetle represents an achievable optimized solution. Beetles use antennae on two sides of their body to find food resources. When the antenna on one side is closer to food, the food odor received by the antenna is stronger, and the individual will move to that side. This new meta-heuristic algorithm BAS taking inspiration from detecting and searching behavior of longhorn beetles. We denote the strength of the food odor at position x to be the value at point of the optimized function. To explore the initial unknown environment, the initial beetle searching is supposed to move randomly in any direction. A vector with a random direction can be built to be representative and normalize in spaces of any dimension. A random searching direction can be normalized as rnd(dim, 1) b = (11) krnd(dim, 1)k where rnd(.) denotes a random function and dim represent the dimensions of the position. Beetles do not know the precise location of food when foraging. They use both antennae to detect the food odor and move in the direction of the odor. The positions of the right and left antenna can be obtained as t t x = x + d  b (12) t t x = x d  b (13) where t is the iteration number, x denotes the position of the right antenna, x denotes the position of t t the left-antenna, x is the position of the beetle, and d is the sensing length of the antennae. The beetle will move to the left if the left antenna receives a stronger scent than the right antenna; otherwise, it will move to the right. The beetle chooses its search behavior based on the direction of the detected odor. Thus, we can determine the next position of a beetle by judging the strength of the odor. The next position of the beetle can be determined by t t1 t x = x + c   b  sign( f(x ) f(x )) (14) r l where  is the step size of searching, sign(.) represents a sign function, f(.) is an optimized function, and c is the direction of beetle movement. If the aim is to find the maximum value, c = 1. If the problem is to find the minimum value, c = +1. t t The sensing length of the antennae d and the step size of searching  can be updated as t t1 d = 0.95d + 0.01 (15) t t1 = 0.95 (16) The iterative process of the beetle antennae search algorithm can be presented as follows: Step 1: Set the maximum number of iterations t . Randomly initialize N beetle positions max X (I = 1, 2, ::: , N). Define all beetle antennae search algorithm parameters, including the initial step size of searching  and the initial sensing length of the antennae d. Define the optimized function. Set direction of the beetle movement c according to the optimization purpose of the optimized function. Step 2: Generate random searching directions using Equation (11). Step 3: Update the positions of the right antennae of beetles using Equation (12). Update positions of the left antennae of beetles using Equation (13). Sensors 2019, 19, 2727 7 of 21 Step 4: Modify beetle positions according to Equation (14). Step 5: Calculate all feasible solutions and compare their fitness values to determine the optimal solution in the current generation. After comparing the current minimum fitness value with the previous minimum fitness value, update the global optimum solution if there is a better solution. Step 6: Update the sensing length of antennae using Equation (15). Update the step size of beetles using Equation (16). Step 7: Update the number of iterations t = t + 1 and return to Step 2. Repeat until t = t . max Step 8: Output the global optimum solution. In order to explain the BAS more clearly, we have detailed the steps of the BAS in Algorithm 1. Algorithm 1: BAS Input: Set the maximum number of iterations t . Define the evaluation function f(.). Randomly set N beetle max positions x (i = 1, 2, ::: , N). Set t = 0. Set the value of c according to the optimization purpose. Record the 0 0 initial sensing length of antennae d , the initial step size  Record the initial optimum solution, x , and the best initial optimum value, g best . Output: x , g . best best 1: while (t < T ) max 2: For i = 1: N 3: Update the searching direction b using Equation (11) 4: Update the right antenna position x using Equation (12) ri 5: Update the left antenna position x using Equation (13) li t+1 t 6: Update the next position x of the beetle x using Equation (14). i i 7: End for 8: For i = 1: N t+ 1 9: Calculate the function value f (x ) of ith beetle. t+ 1 10: If f (x )is better than g i best t+ 1 11: x = x best i t+ 1 12: g = f (x ) best i 13: End if 14: End for 15: Update the sensing length of the antennae d using Equation (15). 16: Update the step size of searching  using Equation (16). 17: t = t + 1 18: End while 4.2. PID Control System PID controllers which have high eciency and robust performances, are linear controllers. Thanks to their easy control structure, high robustness, and high accuracy, PID controllers are commonly used in engineering to enhance both transient and steady-state behaviors. It has been shown that rapid response speed and high controllability can be obtained by determining appropriate PID parameters. PID controllers are widely used in di erent control systems. There are three parameters in a PID controller, namely the proportional parameter K , the di erential parameter K , and the integral p d parameter K . These three parameters have dramatic e ects on the performance of control systems: K i p a ects the response speed of the system, K a ects dynamic performance, and K a ects the steady d i state error. The current PID controller can be divided into two main modes: the continuous form and the discrete form. The continuous form of the PID controller is described as de(t) u(t) = K e(t) + K e(t)dt+K (17) i d dt 0 Sensors 2019, 19, 2727 8 of 21 where K is the proportional parameter, K is the derivative parameter, K is the integral parameter, p d i u(t) is the output of the PID control signal, and e(t) is the system error signal. Let the sampling instant replace continuous time. Let the integral item and di erentiation item be discretized, and the discrete form of PID controller is written as u(k) = K e(k) + K T e(k) + [e(k) e(k 1)] (18) p i k=0 where T is the sampling period, k is the sampling number, and m is the total number of sampling times. 4.3. System Evaluation Function The value of the evaluation function is used to measure the speed of the response of the control system. It is necessary to select an evaluation function before the optimization. The evaluation function of the control system includes the integration of the absolute value of error (IAE), the integral of time multiplied by the absolute value of error (ITAE), the integral of the square value of error (ISE), the mean of the square of the error (MSE). For the PID control system, the objective function based on a single measurement value only evaluates a part of the system. The IAE, which give importance to the absolute error, only takes single factors into account. The IAE is often used in the digital simulation of systems, since it is somewhat dicult to obtain the absolute value of the error in analytic form. So, in order to comprehensively evaluate the performance of electro-hydraulic servo system, this paper do not select the IAE as the evaluation function. The ISE is the square of the error. Large errors are penalized more than smaller ones as the square of a large error is much bigger. Systems designated to minimize ISE tend to weaken large errors rapidly; however, they will have to tolerate small errors continuing for long periods. The gradual accumulation of small errors leads to low control accuracy in the later stage of ISE error calculation. The MSE improves the shortcomings of the ISE by calculating the mean of the ISE. However, the system needs to run for a long period of time in order to reduce the square of a large error calculated by the ISE. Therefore, the MSE is only applied in long-running systems. The ITAE, which is the absolute error multiplied by time, penalizes errors which exist after a long time, and is considered as a measure of system performance. The ITAE weights errors which exist after a long time and has an additional time multiplication in the error function, which emphasizes long duration errors and allows a faster response compared to the ISE and IAE. Thus, ITAE can solve problems more eciently than other evaluation functions [43–47]. It has been shown that ITAE is a better evaluation function in the PID control system. In this paper, double measurement value ITAE is selected as the evaluation function of a PID control system. The ITAE can be written as ITAE = t e(t) dt (19) Furthermore, the discrete ITAE is written as ITAE = kT e(k) T (20) k=o where T is the sampling period, k is the sampling number, and m is the total number of sampling times. 4.4. BAS-PID Control System The working principle of the BAS-PID controller is shown in Figure 2, where u(t) is the output signal of the BAS-PID controller, y(t) is the output signal from the control system, r(t) is the input Sensors 2019, 19, x FOR PEER REVIEW 9 of 21 ITAE = t e() t dt (19) Furthermore, the discrete ITAE is written as ITAE = kT e() k T (20) ko = where T is the sampling period, k is the sampling number, and m is the total number of sampling times. 4.4. BAS-PID Control System Sensors 2019, 19, 2727 9 of 21 The working principle of the BAS-PID controller is shown in Figure 2, where u(t) is the output signal of the BAS-PID controller, y(t) is the output signal from the control system, r(t) is the input signal to the control system, and e(t) is the error. PID parameters will be tuned automatically by BAS, signal to the control system, and e(t) is the error. PID parameters will be tuned automatically by BAS, after three PID parameter adjustment ranges are selected. BAS will tune three parameters according to after three PID parameter adjustment ranges are selected. BAS will tune three parameters according the ITAE value of the control system in real time. to the ITAE value of the control system in real time. TIME ITAE BAS ki kp kd e(t) r(t) u(t) y(t) PID Cont rolled object controller BAS -PID cont roller Figure 2. Block diagram of the BAS-PID controller. Figure 2. Block diagram of the BAS-PID controller. The control quality of the PID controller depends on three parameters. To obtain the optimal PID The control quality of the PID controller depends on three parameters. To obtain the optimal controller, in this paper, the PID parameter-tuning problem is converted into a class of three-dimensional PID controller, in this paper, the PID parameter-tuning problem is converted into a class of three- parameter optimization questions. The three parameters are seen as the beetle’s position in dimensional parameter optimization questions. The three parameters are seen as the beetle’s position three-dimensional space. The ITAE is regarded as the evaluation function. Beetle positions are in three-dimensional space. The ITAE is regarded as the evaluation function. Beetle positions are randomly generated, BAS is run, and then the beetle positions are input into the PID controller as three randomly generated, BAS is run, and then the beetle positions are input into the PID controller as parameters to calculate the evaluation function ITAE. The beetle position which minimizes the ITAE is three parameters to calculate the evaluation function ITAE. The beetle position which minimizes the considered to have the optimum PID parameters. The position of the beetle which minimizes the ITAE ITAE is considered to have the optimum PID parameters. The position of the beetle which minimizes is used to update the optimum PID parameters in the current iteration. If the controllability of the the ITAE is used to update the optimum PID parameters in the current iteration. If the controllability PID control system meets the requirements of the engineering application or the searching procedure of the PID control system meets the requirements of the engineering application or the searching reaches the maximum number of iterations, the optimal position of the beetle will be chosen as the procedure reaches the maximum number of iterations, the optimal position of the beetle will be final PID parameters. chosen as the final PID parameters. The parameter-tuning steps of the BAS-PID controller are as follows: The parameter-tuning steps of the BAS-PID controller are as follows: Step 1: Initialize all parameters and ranges. Step 1: Initialize all parameters and ranges. Randomly generate N beetle positions, X = [K , K , K ], where each parameter uses n (n = 1, 2, ::: , N) i d Randomly generate N beetle positions, X (n = 1, 2, …, N) = [Kp, Ki, Kd], where each parameter uses the the real number coding. The discrete ITAE is regarded as the evaluation function. Set the maximum real number coding. The discrete ITAE is regarded as the evaluation function. Set the m 0 aximum number of iterations, T . Set c =1 and T = 0. Set the initial sensing length, d , and the initial step MAX number of iterations, TMAX. Set c = –1 and T = 0. Set the initial sensing length, d , and the initial step size,  . size, δ . Step 2: Normalize searching directions. To expand the exploration environment, the searching directions of beetles can be normalized in random dimensional space using Equation (11). A random searching direction can be calculated as: rnd(3,1) b = krnd(3,1)k Step 3: Update the positions of the right and left antennae of one beetle. Beetles do not know the precise location of food when foraging. They use their antennae to determine their next direction. When the antenna on one side is closer to food, the food odor received by the antenna is stronger, and the beetle will move to that side. The positions of the beetle’s right and left antennae are determined using Equations (12) and (13). The position of the right antenna can be ! ! T T T T T T obtained as X = X + d  b , and the position of the left antenna can be obtained as X = X d  b . nr n n nl Step 4: Update the next position of one beetle. Operate the control system. The positions of the beetle’s right and left antennae, which can be seen as PID parameters, are carried over to the PID controller. Calculate the ITAE values of the evaluation Sensors 2019, 19, 2727 10 of 21 function of the position of the right antenna, f (X ). Calculate evaluation function ITAE values of ITAE nr left antenna position: f (X ). Equation (14) is used to determine the next position of a beetle to ITAE nl obtain a new set of beetle positions. Step 5: Calculate the ITAE of the evaluation function. Operate the control system. Then, new beetle positions are input into the PID controller as three parameters to calculate the ITAE of evaluation function. By comparing all fitness values, the beetle position which minimizes the ITAE is used to determine the current optimum PID parameters in the current generation. Record the current optimum PID parameters and current minimum ITAE and use in the next step. Step 6: Update the global optimum beetle position. After comparing the current minimum ITAE with the previous minimum ITAE, the global optimum beetle position is updated, and the global optimum beetle position is chosen to be the optimal PID parameters. Step 7: Update the sensing length of the antennae and the next step size. T T1 The sensing length of the antennae can be updated using Equation (15): d = 0.95d + 0.01. n n T T1 The step size of searching can be updated using Equation (16):  = 0.95 . By updating, the sensing n n length of the antennae and the step size of searching will be carried over to the next generation. Step 8: Judge iterations. Calculate and judge whether iterations achieve terminating conditions. Calculate the number of iterations: T = T + 1. Judge whether iterations achieve the terminating condition T = T . If T MAX meets the terminating condition, the global optimum beetle position is chosen to be the optimal PID parameters. If T does not meet the terminating condition, return to Step 2 and start the next loop iteration. Step 9: Stop. Output the global optimum position of the beetle as the final three PID parameters. The parameter-tuning flowchart of the BAS-PID controller is shown in Figure 3. Sensors 2019, 19, 2727 11 of 21 Sensors 2019, 19, x FOR PEER REVIEW 11 of 21 START Initialize all parameters and ranges. Randomly generate N beetle positions xn (n=1,2,…,N) Randomly normalize the searching direction of each beetle using Equation (11): The right-left antenna position can be obtained using Equations (12) and (13) Operate the control system. The right-left antenna position of beetle which can be seen PID parameters is carried over to PID controller. Calculate evaluation function ITAE T T values of right-left antenna positions: fITAE (X ) and fITAE(X ). The next position of the nr nl beetle can be generated using Equation (14). Output the new position of the beetle. K K p i K CONTROL SYSTEM Controlled Plant PID Controller Calculate the ITAE of evaluation function. Record current optimum PID parameters and current minimum ITAE. By comparing current minimum ITAE with the minimum ITAE of previous generation, the global optimum beetle position is updated. Update the sensing length of the antennae d using Equation (15). Update the size of searching δ using Equation (16). NO T=T+1 T≧TMAX YES STOP Figure 3. Parameters tuning flowchart of the BAS-PID controller. Figure 3. Parameters tuning flowchart of the BAS-PID controller. 5. Simulation and Analysis 5. Simulation and Analysis 5.1. Simulation Environment 5.1. Simulation Environment The electro-hydraulic servo valve is an FF102-30 (AVIC Nanjing Servo Control System Co., Ltd., Nanjing, China)., with a Ps of 21 MPa, a rated current of 50 mA, a  of 0.5, a K of 0.006 m /(sA), sv sv The electro-hydraulic servo valve is an FF102-30 (AVIC Nanjing Servo Control System Co., Ltd., 4 3 and a no-load flow of 2.315 10 (m /s). The saturation value of the servo amplifier contr 3 ol voltage Nanjing, China)., with a Ps of 21 MPa, a rated current of 50 mA, a ξsv of 0.5, a Ksv of 0.006 m /(sA), and is10V, the length of the piston −4 3displacement is 35 mm, the piston rod area is 0.001 m , the position a no-load flow of 2.315 × 10 (m /s). The saturation value of the servo amplifier control voltage is sensor gain is 50 V/m, the position sensor range is 7100 mm, the stroke of the cylinder 2 is 200 mm, ±10V, the length of the piston displacement is 35 mm, the piston rod area is 0.001 m , the position the rated flow of the cylinder is 30 L/min, and the oil supply pressure is 4.5 MPa. sensor gain is 50 V/m, the position sensor range is 7100 mm, the stroke of the cylinder is 200 mm, the All the model parameters of the electro-hydraulic position servo control system were obtained rated flow of the cylinder is 30 L/min, and the oil supply pressure is 4.5 MPa. by identifying system parameters. The identification techniques include the product look, numerical All the model parameters of the electro-hydraulic position servo control system were obtained derivation, experimental testing, and engineering experience [48–51]. The natural parameters of the by identifying system parameters. The identification techniques include the product look, numerical electro-hydraulic servo valve in the model are obtained from the servo valve product book. The natural derivation, experimental testing, and engineering experience [48–51]. The natural parameters of the electro-hydraulic servo valve in the model are obtained from the servo valve product book. The natural frequency, damping, and conversion factor of the servo valve were obtained from dynamic Sensors 2019, 19, 2727 12 of 21 frequency, damping, and conversion factor of the servo valve were obtained from dynamic characteristic test curves by numerical derivations. Natural parameters of the servo amplifier and the position sensor were obtained from the product book. Moreover, structural parameters, including the e ective piston area of the servo cylinder, the total piston stroke of the servo cylinder, and the volume of the oil pipe were obtained from the factory data of the cylinder. The working parameters—including system supply oil pressure, system return oil pressure, and sensor gain—were obtained from the experimental testing. Other parameters were selected based on engineering experience. The viscous damping coecient can be ignored. Therefore, the model of the controlled object obtained by identification is as 4.63 G(s) = 12 5 9 4 6 3 2 4.528 10 s + 4.1988 10 s + 5.1725 10 s + 0.002s + s To verify the performance of BAS, this paper also compared the result of the proposed BAS-PID controller with those of PID controllers based other popular artificial intelligence algorithms, including PSO, GA, and FA. For PSO, parameters were set as: learning factors c = c = 1, inertial weight w = 1. 1 2 For GA, roulette wheel selection with an elitism mechanism was used, and parameters were set as: crossover probability P = 0.7, mutation probability P = 0.02. cross m For FA, parameters were set as: initial attractiveness = 1. For BAS, parameters were fixed as: initial sensing length of the antennae d = 2, the initial step size  = 1. For all algorithms, the population size was set as 50 and the maximum number of evaluations was set as 200. All of the initial population positions of the di erent algorithms were generated from a uniform distribution. The lower and upper bounds of the search space of three parameters were given by (0, 100). Each optimization method was implemented over 10 independent runs in the MATLAB software (MathWorks, Natick, MA, USA). The input and disturbance signals were step signals. Through the iteration of the ITAE, the process of PID parameter-tuning can be seen as finding the solution which minimizes the ITAE value. The total sampling time was 100 and the sampling period was 0.01 s. 5.2. Simulation Results and Analysis Table 3 shows the results of the ITAE, overshoots M , the rise time t , the settling time t and their p r d corresponding PID controller parameters. The overshoot M reflects the stability of the system, the rise time t reflects the response capability of the system, and the settling time t reflects the adaptability r d of the system. As seen from Table 3, the BAS parameter-tuning method has the lowest overshoot, the fastest rise time and settling time, and the smallest value of ITAE of all algorithms. Table 3. PID parameters and performances of the control system. BAS FA GA PSO K 7.9927 9.5773 8.0156 8.2923 K 0.1412 6.9449 0 0.0225 K 0.0532 0.0714 0.0978 0 ITAE 0.0275 0.0384 0.0294 0.0358 M 0.0067 0.1439 0.1015 0.1503 t 0.022 0.051 0.056 0.029 t 0.023 0.026 0.029 0.036 Response curves derived from the step input are shown in Figure 4. From the figure, it can be seen that, compared with PIDs using other algorithms, BAS-PID shows the best performance. The step Sensors 2019, 19, 2727 13 of 21 response curve of the BAS -PID controller converges to the desired set value with the minimum overshoot and least steady time. Using the BAS-PID controller, the system can maintain high dynamic Sensors Sensors 2019 2019 , 19, , x FO 19, x FO R PE R PE ER R ER R EVIEW EVIEW 13 of 13 of 21 21 characteristics and stability precision, and the performance of the system is una ected by outside dynam dynam ic ch ic ch aract aract erie st ri ics stics anan d st d st abil ab itil y pr ity pr ecis ecis ion, ion, and and the perform the perform ana ce of the sy nce of the sy stem is un stem is un affec affec ted by ted by interference signals, showing perfect robustness. outside interference signals, showing perfect robustness. outside interference signals, showing perfect robustness. 1.2 1.2 BA BA S S FA FA GA GA 1 1 PSPS O O 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0 0 Time(s) Time(s) Figure 4. Response curves of the step signal. Figure 4. Response curves of the step signal. Figure 4. Response curves of the step signal. Figure 5 illustrate the average convergence curves of all algorithms disposing ITAE functions over Fig Fig ure u 5 illustrate the re 5 illustrate the aver aver age age converge converge nce curv nce curv es of al es of al l al lgori algori thm thm s di s di sposi sposi ng ITAE ng ITAE functi functi ons ons 10 independent runs. As can be observed in the Figure 5, BAS has the fastest convergence speed in all over over 10 10 indep indep endent runs. endent runs. As can As can be observed be observed in th in th e Fe F igur igur e 5, e 5, BAS has the BAS has the fastest convergence speed fastest convergence speed algorithms, showing better performance when optimizing PID controller parameters. Additionally, in ina lla lal l gorithms, showing better perf algorithms, showing better perf orma orma nce nce whwen optimizing PID hen optimizing PID controlle controlle r pr p aram aram eters. eters. BAS achieved the highest iteration speed of the four algorithms. The BAS convergence curve tends to Addition Addition ally ally , BA , BA S ach S ach ieve ieve d the h d the h ighe ighe st iteration st iteration spee spee d of the d of the four four alg alg orithms. T orithms. T he BA he BA S conve S conve rgence rgence accelerate as the number of iterations increases, and quickly converges towards the optimal value after curve tends to a curve tends to a ccel ccel eraete rate as the number of as the number of itera itera tions tions incre incre ases ases , an , an d q d q uicukly ickly con con verges t verges t ow oaw rd as t rds t heh e completing almost half of the iterations. BAS demonstrates outstanding convergence stability and opti opti mama l vl v alue a alue a fter compl fter compl etieng ting almost ha almost ha lf of lf of the iterat the iterat ions. ions. BAS d BAS d emonstrates outst emonstrates outst anding anding competence in jumping out from limited optimum. convergence stability and competence convergence stability and competence in jumping out from limited optimum. in jumping out from limited optimum. 0.2 0.2 BA BA S S 0.18 0.18 FA FA GA GA 0.16 0.16 PSPS O O 0.14 0.14 0.12 0.12 0.10 0.10 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0 50 50 100 100 150 150 200 200 Iterations Iterations Figure 5. Average convergence curves of ITAE. Figure 5. Figure 5. Aver Aver age convergence age convergence curves of curves of IT IT AE. AE. The box plot is shown in Figure 6 displays a set of scattered data. The stability of the control The box plot The box plot is shown is shown in F in F igur igur e 6 d e 6 d ispilays splays a se a se t of t of sca sca ttered da ttered da ta. The st ta. The st aba ilb itiy of lity of the control the control system can be shown by examining the position of the median, upper quartile, and lower quartile system can be shown by examining the position of the median, upper quartile, and lower quartile in system can be shown by examining the position of the median, upper quartile, and lower quartile in in the boxplot. The boxplot measures the dispersion of the ITAE value by graphical means. We note the boxplot. The boxplot measures the d the boxplot. The boxplot measures the d ispersion o ispersion o f th f th e ITAE value e ITAE value by graph by graph ical ical means. means. We n We n ote that ote that the value of ITAE calculated by BAS has fewer outliers and a lower degree of dispersion than other the value of ITAE calculated by BAS has fewer outliers and a lower degree of dispersion than other alg alg orit oh rit ms, hms, in i dicat ndicat ing t ing t hah t t ah t t e propose he propose d BA d BA S h S h as ex as ex cellent cellent stabi stabi lity. The lity. The medi medi ana , upper n, upper qu q ar utar ile, tile, an an d d lower quartile of the ITAE calculated by BAS are lower than for other algorithms, which proves that lower quartile of the ITAE calculated by BAS are lower than for other algorithms, which proves that BAS ha BAS ha s excell s excell ent opti ent opti miz miz atia on ca tion ca pap bi ali bi ty li. ty. ITAE Amplitude ITAE Amplitude Sensors 2019, 19, 2727 14 of 21 that the value of ITAE calculated by BAS has fewer outliers and a lower degree of dispersion than other algorithms, indicating that the proposed BAS has excellent stability. The median, upper quartile, and lower quartile of the ITAE calculated by BAS are lower than for other algorithms, which proves Sensors 2019, 19, x FOR PEER REVIEW 14 of 21 Sensors 2019, 19, x FOR PEER REVIEW 14 of 21 that BAS has excellent optimization capability. 1.00 1.00 0.09 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 BAS FA GA PSO BAS FA GA PSO Algorithms Algorithms Figure 6. Boxplot of ITAE. Figure 6. Boxplot of ITAE. Figure 6. Boxplot of ITAE. To further prove the reliability of the BAS-PID controller, response results of PID controllers To further prove the reliability of the BAS-PID controller, response results of PID controllers To further prove the reliability of the BAS-PID controller, response results of PID controllers based on di erent algorithms are presented in Figures 7 and 8, when disturbance signals are the based on different algorithms are presented in Figures 7 and 8, when disturbance signals are the based on different algorithms are presented in Figures 7 and 8, when disturbance signals are the triangular wave signal and the sawtooth signal, respectively. Figure 7a shows the response curves of triangular wave signal and the sawtooth signal, respectively. Figure 7a shows the response curves of triangular wave signal and the sawtooth signal, respectively. Figure 7a shows the response curves of the triangular wave signal, and Figure 7b shows the local amplification of the response curves. Under the triangular wave signal, and Figure 7b shows the local amplification of the response curves. Under the triangular wave signal, and Figure 7b shows the local amplification of the response curves. Under the control of the BAS-PID controller, the amplitude error is the smallest and the smoothing speed is the control of the BAS-PID controller, the amplitude error is the smallest and the smoothing speed is the control of the BAS-PID controller, the amplitude error is the smallest and the smoothing speed is the highest. Figure 8a shows the response curve of the sawtooth wave signal, and Figure 8b shows the the highest. Figure 8a shows the response curve of the sawtooth wave signal, and Figure 8b shows the highest. Figure 8a shows the response curve of the sawtooth wave signal, and Figure 8b shows local amplification of the response curve. Figure 7 shows that the BAS-PID controller can not only the local amplification of the response curve. Figures 7 shows that the BAS-PID controller can not the local amplification of the response curve. Figures 7 shows that the BAS-PID controller can not restrain the disturbance signal but also enhance the dynamic characteristics and the steadiness of only restrain the disturbance signal but also enhance the dynamic characteristics and the steadiness only restrain the disturbance signal but also enhance the dynamic characteristics and the steadiness the robustness. For the two di erent interference signals, the system has a remarkable capability to of the robustness. For the two different interference signals, the system has a remarkable capability of the robustness. For the two different interference signals, the system has a remarkable capability maintain the stability and reduce the shaking and concussion when the BAS-PID controller is selected, to maintain the stability and reduce the shaking and concussion when the BAS-PID controller is to maintain the stability and reduce the shaking and concussion when the BAS-PID controller is and the performance of the system is una ected by external interference. selected, and the performance of the system is unaffected by external interference. selected, and the performance of the system is unaffected by external interference. 0.6 0.6 BAS BAS BAS FA BAS FA FA -0.1 FA GA GA 0.4 -0.1 GA 0.4 GA PSO PSO PSO PSO -0.2 0.2 -0.2 0.2 -0.3 -0.3 -0.4 -0.2 -0.4 -0.2 -0.4 -0.5 -0.4 -0.5 -0.6 -0.6 -0.6 -0.6 0 0 0.05 0.10 0.15 0.20 0.25 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 0 0.05 0.10 0.15 0.20 0.25 Time(s) Time(s) Time(s) Time(s) (a) (b) (a) (b) Figure 7 Response curves of the triangular wave signal. (a) Whole response curves. (b) Local Figure 7. Response curves of the triangular wave signal. (a) Whole response curves. (b) Local Figure 7 Response curves of the triangular wave signal. (a) Whole response curves. (b) Local amplification curves. amplification curves. amplification curves. Amplitude Amplitude ITAE ITAE Amplitude Amplitude Sensors 2019, 19, 2727 15 of 21 Sensors 2019, 19, x FOR PEER REVIEW 15 of 21 Sensors 2019, 19, x FOR PEER REVIEW 15 of 21 0.4 1.0 BAS BAS 0.4 1.0 FA FA BAS GA 0.3 BAS GA 0.8 FA FA PSO PSO GA 0.3 GA 0.8 PSO PSO 0.2 0.6 0.2 0.6 0.1 0.4 0.1 0.4 0.2 0 0.2 0 -0.1 -0.1 -0.2 -0.2 -0.2 -0.2 0 0.5 1.0 1.5 2.0 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 0 0.5 1.0 1.5 2.0 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 Time(s) Time(s) Time(s) Time(s) (a) (b) (a) (b) Figure 8. Response curves of the sawtooth wave signal. (a) Whole response curves. (b) Local Figure 8. Response curves of the sawtooth wave signal. (a) Whole response curves. (b) Local Figure 8. Response curves of the sawtooth wave signal. (a) Whole response curves. (b) Local amplification curves. amplification curves. amplification curves. 6 6. . Ex Experimental perimental An Analysis alysis 6. Experimental Analysis To certify the e ectiveness of the proposed BAS-PID controller in an actual working environment, To certify the effectiveness of the proposed BAS-PID controller in an actual working To certify the effectiveness of the proposed BAS-PID controller in an actual working environment, the designed the environment, the designed designed controller was use controlle controlle d in a compr r w r wa as s ehensive use used d in in a c experiment a c oo mprehensiv mprehensiv of the e experiment e experiment operation of of the operat the of the operat electro-hydraulic ion of ion of position the electro-hydraulic p servo control system. osition serv The electro control o-hydraulicsystem. The semi-physical electro-hy experimental draulic platform semi-physic is illustrated al the electro-hydraulic position servo control system. The electro-hydraulic semi-physical experimental platform is illustrated in Figure 9. experimental pla in Figure 9. tform is illustrated in Figure 9. (b) (a) (a) (b) Figure 9. The electro-hydraulic semi-physical experiment platform. (a) Experiment platform. Figure 9. The electro-hydraulic semi-physical experiment platform. (a) Experiment platform. Figure 9. The electro-hydraulic semi-physical experiment platform. (a) Experiment platform. (b) Electro-hydraulic servo system. ( (b b) ) El Electr ectro-hydraul o-hydraulic ic se servo rvo system. system. Under the experimental conditions, the power of the hydraulic source was 5.5 k, the rated flow Under the experimental conditions, the power of the hydraulic source was 5.5 k, the rated flow Under the experimental conditions, the power of the hydraulic source was 5.5 k, the rated flow rate was 30 L/min, the rated pressure was 5 MPa, an ADVAN-TECH PCL1710HG multi-function rate was 30 L/min, the rated pressure was 5 MPa, an ADVAN-TECH PCL1710HG multi-function board rate was 30 L/min, the rated pressure was 5 MPa, an ADVAN-TECH PCL1710HG multi-function board was selected, the sampling time was 0.01 s, and the rated current was 40 mA. was selected, the sampling time was 0.01 s, and the rated current was 40 mA. board was selected, the sampling time was 0.01 s, and the rated current was 40 mA. A sinusoidal signal is usually chosen as the performance testing signal. The frequency response A sinusoidal signal is usually chosen as the performance testing signal. The frequency response char A sin acterist usoidal ic of the signal syis u stem under sually cho a s sen as the pe inusoidal sig rformance testing signal. The fre nal can be implemented to test the system quency response characteristic of the system under a sinusoidal signal can be implemented to test the system performance. performance. The frequency response, i.e., the response of the system under a sinusoidal signal, can characteristic of the system under a sinusoidal signal can be implemented to test the system The frequency response, i.e., the response of the system under a sinusoidal signal, can be used to be used to determine the resonance frequency, impedance, dynamic stiffness, and vibration stability performance. The frequency response, i.e., the response of the system under a sinusoidal signal, can determine of the system. The amplitude fre the resonance frequency q,uency impedance, characterist dynamic ic, Aωsti , which is ness, and defin vibration ed as the stability amplitude r of the atisystem. o of be used to determine the resonance frequency, impedance, dynamic stiffness, and vibration stability The the input amplitude signfr al t equency o the ide characteristic, al signal, is inA voked a , which s th is e defined index in t ashthe e frequ amplitude ency resratio ponse. If of the Aωinput is closer signal of the system. The amplitude frequency characteristic, Aω, which is defined as the amplitude ratio of to 1, the system is more stable. In the experiment, five sinusoidal signals with different amplitudes to the ideal signal, is invoked as the index in the frequency response. If A is closer to 1, the system the input signal to the ideal signal, is invoked as the index in the frequency response. If Aω is closer were used as disturbance signals. For the five sinusoidal signals, the angular velocity was 4π, the is more stable. In the experiment, five sinusoidal signals with di erent amplitudes were used as to 1, the system is more stable. In the experiment, five sinusoidal signals with different amplitudes initial phase was zero, and the amplitudes were 2, 4, 6, 8, and 10, respectively. The PID controllers disturbance signals. For the five sinusoidal signals, the angular velocity was 4, the initial phase was were used as disturbance signals. For the five sinusoidal signals, the angular velocity was 4π, the based on different algorithms, including FA, GA, and PSO, were selected for experiments in the same zero, and the amplitudes were 2, 4, 6, 8, and 10, respectively. The PID controllers based on di erent initial phase was zero, and the amplitudes were 2, 4, 6, 8, and 10, respectively. The PID controllers experimental environment. The ranges of PID parameters for all algorithms and iterations were the algorithms, including FA, GA, and PSO, were selected for experiments in the same experimental based on different algorithms, including FA, GA, and PSO, were selected for experiments in the same same as in Section 5. The experimental results were contrasted and analyzed. The response results environment. The ranges of PID parameters for all algorithms and iterations were the same as in experimental environment. The ranges of PID parameters for all algorithms and iterations were the are shown in Figures 10–14. Section 5. The experimental results were contrasted and analyzed. The response results are shown in same as in Section 5. The experimental results were contrasted and analyzed. The response results Figures 10–14. are shown in Figures 10–14. Amplitude Amplitude Amplitude Amplitude Sensors Sensors 2019 2019,, 19 19,, x FO 2727 R PEER REVIEW 16 16 of of 21 21 Sensors 2019, 19, x FOR PEER REVIEW 16 of 21 Sensors 2019, 19, x FOR PEER REVIEW 16 of 21 2.5 BAS ideal amplitude 2.5 FA 2.5 BAS BAS BAS 2 GA ideal amplitude ideal amplitude FA FA FA BAS PSO 2 BAS GA GA 2.0 GA FA PSO PS FA O GA PSO 2.0 GA 2.0 PSO PSO 1.5 0 1.5 1.5 1.0 -1 1.0 1.0 -1 -1 0.5 -2 0.5 0.5 -2 -2 -3 0.10 0.20 0.30 -300.2 0.4 0.6 0.8 1.0 -3 0.10 0.20 0.30 0 Time(s) 0 0.10 0.20 0.30 00.2 0.4 Time(s)0.6 0.8 1.0 00.2 0.4 0.6 0.8 1.0 Time(s) Time(s) Time(s) (a) (b) Time(s) (a) (b) (b) (a) Figure 10. Response curves of the sinusoidal signal whose amplitude is 2. (a) Whole response curves. Figure 10. Response curves of the sinusoidal signal whose amplitude is 2. (a) Whole response curves. (b) Local amplification curves. Figure Figure 10. 10. Response Response cu cur rves ves o off the s the sinusoidal inusoidal sig signal nal whose whose a amplitude mplitude is is 2. 2. ( (a a) ) Whole respo Whole response nse cu curves. rves. (b) Local amplification curves. ((b b)) Lo Local cal amplification amplification curves. curves. BAS 5 5.0 FA BAS ideal amplitude 5.0 GA BAS 4 5.0 3 FA BAS PS FAO ideal amplitude GA ide FA al amplitude 4.0 GA 3 BAS GA 3 PSO BAS PSO FA 4.0 PSO FA 4.0 2 GA 2 GA PSO PSO 3.0 3.0 3.0 -1 2.0 -1 -2 -1 2.0 2.0 -2 -2 -3 1.0 -3 -3 -4 1.0 1.0 -4 -5 -4 00.2 0.4 0.6 0.8 1.0 0.10 0.20 0.30 -5 0 -5 Time(s) 0 00.2 0.4 0.6 0.8 1.0 Time(s) 0 0.10 0.20 0.30 00.2 0.4 0.6 0.8 1.0 0 0.10 0.20 0.30 Time(s) Time(s) Time(s) Time(s) (a) (a) (b) (b) (a) (b) Figure 11. Response curves of the sinusoidal signal whose amplitude is 4. (a) Whole response curves. Figure 11. Response curves of the sinusoidal signal whose amplitude is 4. (a) Whole response curves. Figure 11. Response curves of the sinusoidal signal whose amplitude is 4. (a) Whole response curves. ( Figure 11. b) Local amplification Response cu curves. rves of the sinusoidal signal whose amplitude is 4. (a) Whole response curves. (b) Local amplification curves. (b) Local amplification curves. (b) Local amplification curves. 7.0 BAS ideal amplitude 6 7.0 FA 7.0 BAS BAS GA BAS 6.0 ideal amplitude 6 FA iFA deal amplitude 6 PS FAO BAS GA 6.0 GA BAS GA 6.0 FA PSO 5.0 PSO FA PSO 4 GA 4 GA PSO 5.0 5.0 PSO 4.0 4.0 4.0 0 3.0 -2 3.0 3.0 2.0 -2 -2 -4 2.0 2.0 -4 -4 1.0 -6 1.0 1.0 -6 -6 -8 0.10 0.20 0.30 00.2 0.4 0.6 0.8 1.0 -8 -8 Time(s) 0 0.10 0.20 0.30 00.2 0.4 Time(s)0.6 0.8 1.0 0.10 0.20 0.30 00.2 0.4 0.6 0.8 1.0 Time(s) Time(s) Time(s) Time(s) (a) (b) (a) (b) (a) (b) Figure 12. Response curves of the sinusoidal signal whose amplitude is 6. (a) Whole response curves. Figure 12. Response curves of the sinusoidal signal whose amplitude is 6. (a) Whole response curves. Figure 12. Response curves of the sinusoidal signal whose amplitude is 6. (a) Whole response curves. (b) Local amplification curves. Figure 12. Response curves of the sinusoidal signal whose amplitude is 6. (a) Whole response curves. (b) Local amplification curves. (b) Local amplification curves. (b) Local amplification curves. Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Amp Am litu pd li etude Amplitude Amp Am litud ple itude Amplitude Amplitude Amp Am litud plie tude Amplitude Amplitude Amplitude Sensors 2019, 19, 2727 17 of 21 Sensors 2019, 19, x FOR PEER REVIEW 17 of 21 Sensors 2019, 19, x FOR PEER REVIEW 17 of 21 9.0 BAS 9.0 ideal amplitude BA FA S 8.0 BAS ideal amplitude FA GA 8.0 BA FA S PSO GA GA 7.0 FA PSO PSO GA 4 7.0 PSO 4 6.0 6.0 5.0 5.0 4.0 -2 4.0 -2 3.0 -4 3.0 -4 2.0 -6 2.0 -6 1.0 -8 1.0 -8 -10 0.10 0.20 0.30 0 0 -1000.2 0.4 0.6 0.8 1.0 0.10 0.20 0.30 00.2 0.4 0.6 0.8 1.0 Time(s) Time(s) Time(s) Time(s) (a) (b) (a) (b) Figure 13. Response curves of the sinusoidal signal whose amplitude is 8. (a) Whole response curves. Figure 13. Figure 13.Re Response sponse cu cur rves ves o of f the s the sinusoidal inusoidal sig signal nal whose whose aamplitude mplitude is is 8. 8. (a (a ) Whole respo ) Whole response nse cu cu rves rves. . (b) Local amplification curves. ((b b) Lo ) Local cal amplification amplification curves. curves. 11.0 11.0 BAS ideal amplitude 10.0 BA FA S ide BA al a Smplitude 10.0 FA GA 8 BAS 9.0 FA PSO GA 8 GA 9.0 FA PSO GA PSO 8.0 PSO 8.0 7.0 2 7.0 6.0 6.0 5.0 -2 5.0 -2 -4 4.0 -4 4.0 -6 3.0 -6 3.0 -8 2.0 -8 2.0 -10 1.0 -10 -12 1.0 -1200.2 0.4 0.6 0.8 1.0 0 0 0.10 0.20 0.30 00.2 0.4 0.6 0.8 1.0 Time(s) 0.10 0.20 0.30 Time(s) Time(s) Time(s) (a) (b) (a) (b) Figure 14. Response curves of the sinusoidal signal whose amplitude is 10. (a) Whole response curves. Figure 14. Response curves of the sinusoidal signal whose amplitude is 10. (a) Whole response curves. Figure 14. Response curves of the sinusoidal signal whose amplitude is 10. (a) Whole response curves. (b) Local amplification curves. (b) Local amplification curves. (b) Local amplification curves. Figures 10b, 11b, 12b, 13b, and 14b show the local amplification curves for the five sinusoidal Figures 10b, 11b, 12b, 13b, and 14b show the local amplification curves for the five sinusoidal Figures 10b, 11b, 12b, 13b, and 14b show the local amplification curves for the five sinusoidal signals. It can be seen from Figures 10–14 that the response curves for the BAS-PID controller have the signals. It can be seen from Figures 10–14 that the response curves for the BAS-PID controller have sign smallest als. It can distance be s between een from the Figure ideal s amplitude 10–14 that the and re the sponse curves actual amplitude. for the Ther BA efor S-Pe, ID controlle we can infer r have that the smallest distance between the ideal amplitude and the actual amplitude. Therefore, we can infer the BAS-PID controller has outstanding performance including vibrational stability, strong dynamic the smallest distance between the ideal amplitude and the actual amplitude. Therefore, we can infer that the BAS-PID controller has outstanding performance including vibrational stability, strong sti ness, and high mechanical impedance. that the BAS-PID controller has outstanding performance including vibrational stability, strong dynamic stiffness, and high mechanical impedance. Frequency response indices for di erent PID controllers are listed in Table 4, where DA! = dynamic stiffness, and high mechanical impedance. |1 – A!|. From Table 4, it can be seen that the frequency response of the system controlled by the Frequency response indices for different PID controllers are listed in Table 4, where∆Aω BAS-PID controller still is clearly superior to the other PID controllers. The amplitude frequency Frequency response indices for different PID controllers are listed in Table 4, where ∆Aω characteristic = |1 – AωA |. From Table ! of the system 4, it can be controlled seen t by hat the fr the BAS-PID equenc contr y respons oller e of the is the closest system controlled to 1. This shows = |1 – Aω|. From Table 4, it can be seen that the frequency response of the system controlled by the BAS-PID controller still is clearly superior to the other PID controllers. The that the BAS-PID controller can asymptotically maintain system stability when the interference signal by the BAS-PID controller still is clearly superior to the other PID controllers. The is alter amplit ed dramatically ude frequenc . y characteristic Aω of the system controlled by the BAS-PID controller amplitude frequency characteristic Aω of the system controlled by the BAS-PID controller is the closest to 1. This shows that the BAS-PID controller can asymptotically maintain is the closest to 1. This shows that the BAS-PID controller can asymptotically maintain system stability when the interference signal is altered dramatically.Table 4. Frequency system stability when the interference signal is altered dramatically.Table 4. Frequency response characteristics response characteristics Amplitude Index BAS PSO GA FA Amplitude Index BAS PSO GA FA Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Sensors 2019, 19, 2727 18 of 21 Table 4. Frequency response characteristics. Sensors 2019, 19, x FOR PEER REVIEW 18 of 21 Sensors 2019, 19, x FOR PEER REVIEW 18 of 21 Amplitude Index BAS PSO GA FA Table 4. Frequency response characteristics Aω 0.9870 1.0270 0.9695 1.0165 A 0.9870 1.0270 0.9695 1.0165 2 ! ∆Aω 0.0130 0.0270 0.0305 0.0165 DA 0.0130 0.0270 0.0305 0.0165 Amplitude Index ! BAS PSO GA FA Aω 0.9878 1.0267 0.9693 1.0175 A 0.9878 1.0267 0.9693 1.0175 Aω 0.9870 1.0270 0.9695 1.0165 ∆Aω DA 0.0 0.0122122 0 0.0267.0267 0 0.0307.0307 0 0.0175.0175 ∆Aω 0.0130 0.0270 0.0305 0.0165 A 0.9878 1.0271 0.9186 1.0177 Aω ! 0.9878 1.0271 0.9186 1.0177 Aω 0.9878 1.0267 0.9693 1.0175 DA 0.0122 0.0271 0.0814 0.0177 4 ! ∆Aω 0.0122 0.0271 0.0814 0.0177 ∆Aω 0.0122 0.0267 0.0307 0.0175 A 0.9879 1.0267 0.9878 1.0174 Aω 0.9879 1.0267 0.9878 1.0174 Aω 0.9878 1.0271 0.9186 1.0177 8 DA 0.0121 0.0267 0.0122 0.0174 ∆Aω 0.0121 0.0267 0.0122 0.0174 ∆Aω A 0.0122 0.9878 0 1.0191.0271 0.96960.0814 1.01900.0177 Aω 0.9878 1.0191 0.9696 1.0190 DA 0.0122 0.0191 0.0304 0.0190 Aω ! 0.9879 1.0267 0.9878 1.0174 ∆Aω 0.0122 0.0191 0.0304 0.0190 ∆Aω 0.0121 0.0267 0.0122 0.0174 Aω 0.9878 1.0191 0.9696 1.0190 To further display the performance of the BAS-PID controller, the response results of the system To further display the performance of the BAS-PID controller, the response results of the system ∆Aω 0.0122 0.0191 0.0304 0.0190 under di erent PID controllers when interference signals were random signals are presented in under different PID controllers when interference signals were random signals are presented in Figures 15 and 16. Figures 15 and 16. To further display the performance of the BAS-PID controller, the response results of the system under different PID controllers when interference signals were random signals are presented in 1.4 1.5 BAS BAS Figures 15 and 16. FA FA 1.4 GA 1.2 GA 1.5 PSO BAS BAS PSO FA FA GA 1.2 GA PSO 1.0 PSO 1.0 1.0 0.8 1.0 0.8 0.6 0.5 0.6 0.4 0.5 0.4 0.2 0.2 00.1 0.2 0.3 0.4 0.5 02 48 6 10 Time(s) Time(s) 0.3 00.1 0.2 0.4 0.5 02 48 6 10 Time(s) (a) (b) Time(s) (a) Figure 15. Response curves of the random signal l. (a) Whole response curves. ( (b b) ) Local amplification Figure 15. Response curves of the random signal l. (a) Whole response curves. (b) Local Figure curv 1615 es. . Response curves of the random signal l. (a) Whole response curves. (b) Local amplification amplification curves. 1.2 1.2 curves. BAS BAS FA FA 1.2 1.2 GA 1.0 GA 1.0 BAS BAS PSO PSO FA FA GA 1.0 GA 1.0 PSO 0.8 PSO 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 01 24 3 5 0.3 00.1 0.2 0.4 0.5 0 Time(s) Time(s) 01 24 3 5 0.3 00.1 0.2 0.4 0.5 (a) (b) Time(s) Time(s) (b) Figure 16. Response cu rves (a) of the random signal 2. (a) Whole response curves. (b) Local amplification curves. Figure 16. Response curves of the random signal 2. (a) Whole response curves. (b) Local amplification Figure 16. Response curves of the random signal 2. (a) Whole response curves. (b) Local curves. amplification curves. Local enlarged drawings clearly show that the BAS-PID controller can not only rapidly suppress interference signals but also prevent excessive overshoot. For the unusual interference signals, the Amplitude Amplitude Amplitude Amplitude AmpAm litud pe litude Amplitude Amplitude Sensors 2019, 19, 2727 19 of 21 Local enlarged drawings clearly show that the BAS-PID controller can not only rapidly suppress interference signals but also prevent excessive overshoot. For the unusual interference signals, the BAS-PID controller has a remarkable capability to maintain the system stability and reduce the shaking, and the performance of the system is una ected by external interference. In other words, the BAS-PID controller has anti-interference ability and provides remarkable balance, which can enhance the anti-seismic properties of the electro-hydraulic position servo control system in an unknown environment. 7. Conclusions In order to enhance the control accuracy and ability of an electro-hydraulic position servo control system, the paper addressed the problem of determining three parameters of PID controllers. A PID parameter tuning method based on the beetle antennae search algorithm was applied to an electro-hydraulic position servo control system. A transfer function model was obtained by system parameter identification. A basic mathematical model of the electro-hydraulic position servo control system was established through theoretical analysis. The PID tuning problem was converted into a three-dimensional optimization question. The performance of the BAS tuning method was tested by ITAE and compared with that of the PSO, GA, and FA algorithms. An analysis of the performance of the BAS-PID controller with the electro-hydraulic position servo control system showed that the BAS algorithm can e ectively adjust three parameters of the PID controller. 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Published: Jun 18, 2019

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