Research on Control Strategy of a Magnetorheological Fluid Brake Based on an Enhanced Gray Wolf Optimization Algorithm

Abstract

In order to improve the response characteristics of magnetorheological fluid brake (MRB) and reduce the braking fluctuation rate, an improved grey wolf optimization algorithm was proposed to adjust the parameters of the proportion integration differentiation (PID) controller. Firstly, an MRB system was designed and constructed, and its transfer function was determined. Moreover, by adopting the iterative method of logistic curve, an enhanced grey wolf optimization algorithm (EGWOA) was presented. Using the EGWOA, the parameters of the PID controller were optimized to improve the control performance of the system. Finally, the simulation and experiment were carried out. The results showed that EGWOA has a faster response output and overall better performance without overshoot compared with the conventional PID and grey wolf optimization algorithm (GWOA) PID controller.

1. Introduction

Magnetorheological fluid (MRF) with rheological properties is regarded as one of the most promising intelligent materials. When exposed to magnetic field, MRF can change from a free-fluid state to a solid-like state within a few milliseconds, resulting in shear yield stress, which is applied to many fields [1,2,3]. The magnetorheological fluid brake (MRB), as a new type of semi-active control device, has attracted more and more of researchers’ attention because of its quick response, easy control, strong controllability, simple structure, and low power consumption [4,5]. However, MRB has a high degree of nonlinearity, time delay and uncertainty in its working process, and it is difficult to control it accurately because of the interaction between current, magnetic field, magnetorheological fluid and structure [6,7]. The braking force of MRB depends on the shear stress in the magnetorheological fluid between the brake discs, which continuously changes by controlling the intensity of the external magnetic field. Therefore, by controlling the external magnetic fields and related temperature devices in real time, torque transferred can be easily and continually adjusted. Due to its rapid and reliable response, simple control, low energy consumption and strong anti-interference ability, MRB has attracted extensive research attention [8].

In order to realize the optimal control of MRB, many researchers improved the PID controller according to the characteristics of the equipment. Nguyen et al. proposed a strategy of DC motor speed control using an MR clutch, and designed a PID controller to adjust the output speed [9]. In order to analyze the braking performance, Wang et al. developed a magnetorheological brake implemented by a PID controller, and it showed that the designed MR brake system had smaller overshoot than the traditional hydraulic brake [10]. To improve the security and adaptability of this device, Okui et al. proposed a welding teaching device with an MRB, and a PID control strategy was adopted in a feedback system [11]. To eliminate the viscous torque of MRB, Shamieh et al. designed a PID controller in combination with a novel magnetorheological brake, and the closed-loop braking performance of the innovative MRB was demonstrated, which provided good braking torque capacity and had no residual viscous torque [12]. To investigate the motion characteristics of the prosthetic limbs used by MRF dampers, Nordin et al. proposed a fuzzy-PID controller to realize an energy-efficient MRF damper [13]. Hou et al. used a fuzzy PID control method to simulate the dynamic performance of mount, and the results showed that this method was effective in reducing the acceleration of the vehicle body [14]. Seung et al. proposed a haptic master activated by an MR clutch and MR brake, and the designed fuzzy plus PID controller had an excellent track control performances [15]. Wang et al. proposed a type of antilock braking system (ABS) using MRB, and it was proved that the ABS with the fuzzy logic sliding mode controller had a great braking control effect [16]. Shiao et al. designed a hybrid ABS with a multipole MRB, showing that the hybrid ABS is feasible in practical applications, which can effectively improve the braking performance and ensure driving stability [17]. Russo et al. proposed an adaptive approach for MRB feedback control and confirmed the effectiveness of the proposed adaptive controller [18].

In order to obtain higher control precision, optimizing intelligent algorithms is the main method of some researchers. Due to the nonlinear dynamic and hysteresis characteristics of MRFA, Diep et al. proposed a self-turning fuzzy controller used the fuzzy logic algorithm, and the experiment results showed that the designed controller had better control performance than PID controller and the open-loop control method [19]. Sohn et al. designed a magnetorheological brake with a fuzzy-PID control algorithm to investigate control performance [20]. Choi et al. designed a haptic cue vehicle accelerator pedal device by MRB, and a tactile cue algorithm to obtain the best gear shift was proposed [21]. In addition, different control strategies focus on improving the braking performance. Liu et al. proposed an IFOA, and the simulation and experiment results showed that IFOA is superior to the conventional PID and FOA PID controller [22]. Basing on integration of FOA and PSOA, Hua et al. proposed a combined braking approach of MRB [23].

Although many promising studies have been carried out on MRB, many problems are far from being solved. Firstly, the MRB still has some challenges in braking with large torque. Secondly, due to the influence of nonlinear factors, such as magnetic powder wear, hysteresis characteristics and system friction, the control of the braking system is not accurate.

In this work, a novel MRB was proposed with EGWOA-based PID controller. Initially, a disc MRB was designed, and the structure of magnetic field was simulated, which proved that the designed structure could meet the requirements of transmission. Additionally, the dynamic model and transfer function of the MRB system were derived. Moreover, an enhanced grey wolf optimization algorithm (EGWOA) was proposed to adjust PID parameters. Compared with the original algorithm, the algorithm could find the optimal solution faster and with higher accuracy. Finally, the system was simulated by using the unit step signal as the excitation source in MATLAB. The experimental results indicated that the proposed controller has the merits of small overshoot and a quick response compared with other controllers.

The remainder of this paper is organized as follows. In Section 2, the MRB is designed, and the transfer function of the system is obtained. In Section 3, the EGWOA is introduced and its feasibility is verified. In Section 4, the simulation example and experiments are carried out to verify efficiency of proposed controller. Conclusions and future works are summarized in Section 5.

2. Design and Transfer Function Acquisition of MRB

2.1. Structure of MRB

Some transmission devices need to meet the requirements of having a compact structure, long service life and low maintenance cost. MRB should have excellent sealing performance to ensure no leakage of MRF. In addition, for reducing the magnetic leakage phenomenon, the direction of magnetic induction line should be perpendicular to the effective working area of the MRF as far as possible, to ensure that the magnetic field meets the normal working requirements of the transmission device.

As shown in Figure 1, a kind of disk-shaped MRB was designed. In order to make the magnetic induction lines more distributed in the working area of the MRF and reduce the leakage of the magnetic induction lines and the magnetic resistance of the magnetic circuit, the coil was designed to be arranged inside the shell. When the coil is not energized, the MRF between the drive disks is in a Newton fluid state, and the driving and driven disks are in a separate state. When the coil is energized, magnetic particles in MRF are magnetized, and attract each other to form columnar structures parallel to the applied field. The structures restrict the flow of the fluid, which makes the driving disc and the brake disc of the MRB integrated. The braking torque of MRB is mainly produced by the field-induced yield stress of MRF, which can be adjusted by the exciting coil current.

The torque calculation principle of the MRB is shown in Figure 2; the effective working area is the area enclosed by R₁ and R₂. Taking the micro ring of width dr, the shear stress in the magnetized MRF in this micro ring region can be calculated by the following formula.

$$\mathit{dT} = \mathit{dF}·\mathit{r} = \mathit{dA}·\tau ·\mathit{r} = 2\pi \mathit{r}·\mathit{dr}·\tau ·{\mathit{r} = 2\pi \mathit{r}}^2·\tau ·\mathit{dr}$$

(1)

where τ is the shear stress in MRF.

The formula for calculating the shear strain rate γ of MRF can be expressed as follows.

$$\gamma =\frac{{r(\omega }_1{-\omega }_2)}{L}$$

(2)

where ω₁ is the input disc speed; ω₂ is the brake disc speed; and L is the distance between the input disc and the brake disc.

The constitutive relation of MRF is expressed by the Bingham model [10].

$$\begin{array}{l}{\tau =\tau }_0\left(H\right)\mathit{sgn}\left(\gamma \right)+\eta \gamma \left|\tau \right| > \left|{\tau }_0\right|\hfill \\ \tau =0 \left|\tau \right| < \left|{\tau }_0\right|\hfill \end{array}$$

(3)

where η is the viscosity coefficient of MRF.

In terms of Equations (2) and (4), the shear stress in MRF under the action of magnetic field can be obtained by substituting Equation (4) into Equation (5).

$${\tau }_M{=\tau }_0+\eta \frac{{r(\omega }_1{-\omega }_2)}{L}$$

(4)

The micro-transfer torque of MRF on micro torus can be obtained by substituting Equation (6) into Equation (3).

$${\mathit{dT}=2\pi \mathit{r}}^2{[\tau }_0+\eta \frac{{\mathit{r}(\omega }_1{-\omega }_2)}{\mathit{L}}]\mathit{dr}$$

(5)

Therefore, the braking torque of MRB can be calculated as follows:

$$\begin{array}{l}\mathit{T}={{\displaystyle \int }}_{{\mathit{R}}_1}^{{\mathit{R}}_2}{2\pi \mathit{r}}^2{[\tau }_0+\eta \frac{{\mathit{r}(\omega }_1{-\omega }_2)}{\mathit{L}}]\mathit{dr}\hfill \\ =\frac{2}{3}{\pi \tau }_0{(\mathit{R}}_2^3{ - \mathit{R}}_1^3)+\frac{\pi }{2\mathit{L}}{\eta \mathit{r}(\omega }_1{ - \omega }_2{)(\mathit{R}}_2^4{ - \mathit{R}}_1^4)\hfill \\ {=\mathit{T}}_{\mathit{B}}{+\mathit{T}}_{\eta }\hfill \\ {\mathit{T}}_{\mathit{B}}=\frac{2}{3}{\pi \tau }_0{(\mathit{R}}_2^3{ - \mathit{R}}_1^3)\hfill \\ {\mathit{T}}_{\eta }=\frac{\pi }{2\mathit{L}}{\eta (\omega }_1{ - \omega }_2{)(\mathit{R}}_2^4{ - \mathit{R}}_1^4)\hfill \end{array}$$

(6)

According to Equation (6), the braking torque of MRB is mainly composed of T_B produced by shear yield stress caused by the magnetization of magnetic particles, and ${\mathit{T}}_{\eta }$ produced by the viscosity of the base load fluid. The torque component ${\mathit{T}}_{\eta }$ is smaller than ${\mathit{T}}_{\mathit{B}}$, usually less than 2%.

2.2. Magnetic Circuit Simulation of MRB

In order to reduce the amount of calculation in simulation, the factors that have little influence on calculation such as bolts, sealing rings and chamfering in the transmission device were ignored. The simplified structure is shown in Figure 3, where different colors represent different areas, and loop reluctance can be analogous to Ohm’s law.

The closed loop of the magnetic circuit consists of a left shell, input disk, MRF, brake disk and right shell. However, some of the magnetic flux will be lost in the components and magnetic resistance. Loop reluctance can be determined analogous to Ohm’s Law:

$${\mathit{R}}_{\mathit{m}}=\frac{{\mathit{F}}_{\mathit{m}}}{\mathit{\Phi }}=\frac{{\mathit{N}}_{\mathit{c}}\mathit{I}}{\mathit{\Phi }}=\frac{{\mathit{L}}_{\mathit{c}}}{\mu \mathit{A}}$$

(7)

where R_m is the magnetic resistance, F_m is the magnetic potential, Φ is the magnetic flux, N_c is turns, I is the electric current, L_c is the length of the magnetic path, μ is the magnetic conductivity, and A is the Magnetic flux area.

The function of the excitation coil is to provide the magnetic field needed for the normal operation of the MRB, and considering coil heating, the resistance of the excitation coil should be as low as possible.

When the coil of excitation has poor cooling performance, the current density J allowed to pass over a long period of time shall not exceed 3~5 A/mm². When the transmission device has good heating dissipation structure, the value should not exceed 5~7 A/mm² [24]. The calculation formula is as follows:

$$\mathit{J}=\frac{4 \mathit{I}}{{\pi \mathit{d}}^2}$$

(8)

where d is the diameter of the excitation coil.

According to the abovementioned requirements, the design coil diameter was 0.96 mm and turns were 1000. When the maximum working current was 2 A, the current density could be calculated according to Formula (8) to be 2.77 A/mm², which met the working requirements.

Ansoft Maxwell (16.0) was adopted to simulate and analyze the electromagnetic field of MRB, where the number of turns of the excitation coil was 1300. As shown in Figure 4a, when the single-turn current of the coil was 2 A, the highest magnetic induction of about 2.1 T appeared at the junction of the left shell and the right shell with the outer shell, close to the magnetic saturation strength of No. 20 steel [25]. The magnetic flux density in the gap between the non-magnetic material and the air was the smallest, equal to about zero, indicating that it plays a good role in limiting the envelope of the magnetic field lines and can meet the experimental requirements. From Figure 4b, the magnetic field in the working area was basically perpendicular to the direction of the working surface, and the effective magnetic field lines accounted for more than 90% of the total magnetic field, which met the design requirements of MRB [26].

2.3. MRB System

In the MRB system, 40# steel was chosen for the magnetic circuit components, while 304 stainless steel was used for the non-magnetic parts. The MRF used in this work was MRF-350, the rheological performance parameters of which are shown in

Table 1. Rheological performance parameters of MRF-350.

Magnetic Particles	Density (g/mL)	Temperature (°C)	Zero Field Viscosity/mPas (1000 S−1, 0A)	Shear Stress/kPa (10 S−1, 4A)	Yield Stress/ kPa (4A)
Carbonyl iron powder	2.9~3.1	40	350~450	≥70	≥45

, and the B-H curve is shown in Figure 5.

The MRB system consists of a control system, data acquisition system and mechanical transmission system, as shown in Figure 6. The control system is comprised of an upper computer, programmable power supply, 2013 and a frequency converter. The mechanical transmission system consists of a three-phase asynchronous motor, torque speed sensor and MRB. Finally, the data acquisition system is comprised of a digital sensor indicator and data acquisition card, which is mainly used to collect torque data.

As can be seen from Figure 7, the designed MRB test-bed consists of a three-phase asynchronous motor, frequency converter, torque speed sensor, self-designed MRB, programmable power supply and computer. The parameters of the main devices are described in

Table 2. The main technical parameters of the self-designed MRB.

Radial Dimension	Axial Dimension	Average Maximum Magnetic Field	Current Range	Brake Torque	Weight	Inductance of Coil	Wire Diameter
300 mm	110 mm	0.6 T	0–2 A	4~30 N·m	24 kg	1.5 H	0.96 mm

,

Table 3. The main parameters of the PCI data acquisition card.

Type	Input Channel	Accuracy	Input Range	Sampling Rate	Manufacturer
PCI8735	Double 16	0.1%	±5 V	≤500 KHz	Beijing Art Science and Technology Development Co., Ltd.

and

Table 4. The main parameters of the torque speed sensor.

Type	Torque Signal	Torque Range	Speed Signal	Speed Range	Accuracy	Manufacturer
HCNJ-101	5~15 KHz	±500 N·m	60 pulse/roll	0~3000 r/min	<±0.5%	Beijing haibohua Technology Co., Ltd.

. The function of the three-phase asynchronous motor is to provide the required speed and power of the system; the frequency converter can obtain the rotating speed of the system; the upper computer completes the calculation and analysis of the data uploaded by the data acquisition card, and sends signals to the programmable power to realize the system speed regulation. The digital display instrument converts the digital quantity output by the sensor into analog quantity, and the data acquisition card receives the voltage signal from the digital display instrument, which is solved and stored by the upper computer.

The input signal current I₁ was adjusted to 0.5 A, 1.5 A and 2.5 A, respectively. The speed of the motor was increased from 100 r/min to 800 r/min by changing the frequency of frequency converter. The variation of braking torque at different slip speeds was tested, as shown in Figure 8. Under the same excitation current, the braking torque increased slightly with the increase of the slip speed. When the excitation coil current was 0.5 A, 1.5 A and 2.5 A, the corresponding braking torque increased from 61 N·m, 102.3 N·m and 110.1 N·m to 64.6 N·m, 111.2 N·m and 120.6 N·m, respectively. The increase range was 5.9%, 8.7% and 9.5%, respectively. This phenomenon indicated that MRB still has stable mechanical transmission performance under the influence of slip speed.

2.4. Transfer Function of MRB System

It is difficult to establish an accurate model for MRB systems with strong nonlinearity [27]. In this work, the MRB was controlled by an excitation current, which can be expressed as the first-order pure delay link: ${\mathit{G}}_{\mathit{S}}=\frac{\mathit{K}}{{(1+\mathit{T}}_1\mathit{s})}{\mathrm{e}}^{\mathit{ps}}$. The rheological effect of MRF was generated in milliseconds, where p is very small, and the eps could be replaced by 1/(1 + T₂s). Therefore, the transfer function of MRB can be expressed: ${\mathit{G}}_{\mathit{S}}=\frac{\mathit{K}}{{(1+\mathit{T}}_1{\mathit{s}) (1+\mathit{T}}_2\mathit{s})}$, where K is the gain of MRF brake, T₁ and T₂ are the time constants. The current-braking torque data were obtained by the data acquisition system, and the mathematical model of MRB was identified by using the system identification toolbox in MATLAB. A group of excitation current and output torque data was used as identification data, while another group of excitation current and output torque data was used as inspection data.

By selecting the model class and algorithm provided by the system software, the model parameters were calculated and the optimal value was selected to complete the mathematical model identification of the MRB system. The identification steps are as follows: Firstly, the identification data samples were written into the MATLAB command window and the start time and sampling time were set. Meanwhile, to ensure the reliability of the identification data in the real environment, there was no need for pre-processing and interference signal filtering. Secondly, we selected the model structure by clicking Estimate/Process Models, and selected Model Output to view the results. Finally, the best fit was 92.45%, which is a good fitting degree [22]. The transfer function of the MRB system can be expressed as follows:

$$\mathit{G}\left(\mathit{s}\right)=\frac{5.327}{{0.0667\mathit{s}}^2+0.5261\mathit{s}+5.327}$$

3. Torque Control Strategy for MRB

3.1. Gray Wolf Optimization Algorithm

The Gray Wolf Algorithm (GWO), with the features of simple structure and few parameters, is widely used to solve many problems. As an intelligent algorithm based on simulating the hunting behavior of wolves, the core idea of GWO is to complete the hunting task in the unique social hierarchy of wolves [28]. In the algorithm, the α wolf is dominant in the gray wolf population, the β wolf has a lower rank than the α wolf, and the δ wolf has a lower rank than the β wolf. The GWO consists of three steps: hunting, chasing, and attacking. The distance and position of wolves can be expressed as follows:

$$\mathit{D}=|\mathit{C}·{\mathit{X}}_{\mathit{p}}\left(\mathit{t}\right) - \mathit{X}\left(\mathit{t}\right)|$$

(9)

$${\mathit{X}(\mathit{t}+1)=\mathit{X}}_{\mathit{p}}\left(\mathit{t}\right) - \mathit{A}·\mathit{D}$$

(10)

where X_p(t) is the position vector of prey, X(t) is the position vector of the gray wolf, and t is the number of iterations. The coefficients C and A are defined as follows:

C = 2r₁

(11)

A = 2a₁*r₂−a₁

(12)

where r₁ and r₂ are random vectors in the interval [0, 1], and a₁ decreases linearly from 2 to 0 as the number of iterations increases.

After completing the process of surrounding the prey, the wolves enter the chasing process. During this process, ω wolves judge the approximate position of the prey according to the position of the three abovementioned gray wolves, as shown in Figure 9. At this point, the action of ω wolves is calculated shown in Formulas (13)–(15).

$$\begin{array}{c}{\mathit{D}}_{\alpha }{=|\mathit{C}}_1·{\mathit{X}}_{\alpha }\left(\mathit{t}\right)-\mathit{X}\left(\mathit{t}\right)|\\ {\mathit{D}}_{\beta }{=|\mathit{C}}_2·{\mathit{X}}_{\beta }\left(\mathit{t}\right)-\mathit{X}\left(\mathit{t}\right)|\\ {\mathit{D}}_{\delta }{=|\mathit{C}}_3·{\mathit{X}}_{\delta }\left(\mathit{t}\right)-\mathit{X}\left(\mathit{t}\right)|\end{array}$$

(13)

$$\begin{array}{c}{\mathit{X}}_1{=|\mathit{X}}_{\alpha }{\left(\mathit{t}\right)-\mathit{A}}_1·{(\mathit{D}}_{\alpha }\left)\right|\\ {\mathit{X}}_2{=|\mathit{X}}_{\beta }{\left(\mathit{t}\right)-\mathit{A}}_2·{(\mathit{D}}_{\beta }\left)\right|\\ {\mathit{X}}_3{=|\mathit{X}}_{\delta }{\left(\mathit{t}\right)-\mathit{A}}_3·{(\mathit{D}}_{\delta }\left)\right|\end{array}$$

(14)

$$\mathit{X}(\mathit{t}+1)=\frac{{\mathit{X}}_1{+\mathit{X}}_2{+\mathit{X}}_3}{3}$$

(15)

where D_α, D_β and D_δ represent the distance between ω wolves and corresponding individuals; C₁, C₂ and C₃ are random vectors; X_α(t), X_β(t) and X_δ(t) are the current positions of α, β and δ; and X(t) is the current position of ω.

3.2. Enhanced Gray Wolf Optimization Algorithm (EGWOA)

In the GWOA, ω wolves around the leadership gray wolves hunt prey, completing the determination of the target location. The position information of the three leadership wolves has an important role to the optimization results of ω wolves, where the central position of the three leadership wolves is more likely to be closer to the target position.

Therefore, in order to improve the optimization speed of GWOA, the central position information between the three leadership wolves is added, which is combined with the location information of the three leadership wolves to guide ω wolves to complete the search operation. The central position of three leadership wolves can be expressed as follows:

$$\begin{array}{c}{\mathit{X}}_{\alpha \beta }\left(\mathit{t}\right)=\frac{1}{2}{|\mathit{X}}_{\alpha }{\left(\mathit{t}\right)-\mathit{X}}_{\beta }\left(\mathit{t}\right)|\\ {\mathit{X}}_{\alpha \delta }\left(\mathit{t}\right)=\frac{1}{2}{|\mathit{X}}_{\alpha }{\left(\mathit{t}\right)-\mathit{X}}_{\delta }\left(\mathit{t}\right)|\\ {\mathit{X}}_{\beta \delta }\left(\mathit{t}\right)=\frac{1}{2}{|\mathit{X}}_{\beta }{\left(\mathit{t}\right)-\mathit{X}}_{\delta }\left(\mathit{t}\right)|\end{array}$$

(16)

$$\begin{array}{c}{\mathit{D}}_{\alpha \beta }{=|\mathit{C}}_4·{\mathit{X}}_{\alpha \beta }\left(\mathit{t}\right)-\mathit{X}\left(\mathit{t}\right)|\\ {\mathit{D}}_{\alpha \delta }{=|\mathit{C}}_5·{\mathit{X}}_{\alpha \delta }\left(\mathit{t}\right)-\mathit{X}\left(\mathit{t}\right)|\\ {\mathit{D}}_{\beta \delta }{=|\mathit{C}}_6·{\mathit{X}}_{\beta \delta }\left(\mathit{t}\right)-\mathit{X}\left(\mathit{t}\right)|\end{array}$$

(17)

$$\begin{array}{c}{\mathit{X}}_4{=|\mathit{X}}_{\alpha \beta }{\left(\mathit{t}\right)-\mathit{A}}_4{\mathit{D}}_{\alpha \beta }|\\ {\mathit{X}}_5{=|\mathit{X}}_{\alpha \delta }{\left(\mathit{t}\right)-\mathit{A}}_5{\mathit{D}}_{\alpha \delta }|\\ {\mathit{X}}_6{=|\mathit{X}}_{\beta \delta }{\left(\mathit{t}\right)-\mathit{A}}_6{\mathit{D}}_{\beta \delta }|\end{array}$$

(18)

$$\mathit{X}(\mathit{t}+1)=\frac{{\mathit{X}}_1{+\mathit{X}}_2{+\mathit{X}}_3{+\mathit{X}}_4{+\mathit{X}}_5{+\mathit{X}}_6}{6}$$

(19)

where D_αβ, D_αδ and D_βδ are the distance vectors between α and β wolves, α and δ wolves and β and δ wolves; X₄, X₅ and X₆ are the position vectors of the center points of the three distances, respectively.

In addition, to achieve fast convergence, the value range of a₁ is gradually linearly reduced in GWOA, and the surrounding space of the gray wolf is gradually restricted according to the increase of iteration times; however, in the initial stage of the algorithm, the gray wolf should be given a larger search range, which can help the gray wolf to jump out of the local optimal and complete the global optimal search process. Therefore, it is considered to slow down the decline speed of a₁ in the initial stage, and the end stage of the algorithm to improve the global search ability of the gray wolf and increase the decline speed of a₁ in the middle stage to improve the search speed of the gray wolf.

The modified curve of a₁ is defined as a₂ shown in Figure 10; the logistic curve was used to transform the decline curve of a₂ shown in Formula (20).

$${\mathit{a}}_2{=2\mathit{e}}^{-{0.0005\mathit{N}}^2}$$

(20)

where N is the independent variable, which represents the number of iterations in this work.

The EGWOA can significantly reduce the errors caused by the continuous search for the local optimal in the early stage of solving, reduce the convergence time of the algorithm and improve the accuracy of the optimal solution. It can search the global optimal solution in a short number of iterations, and has good global search ability and local convergence performance. The flow of EGWOA is shown in Figure 11.

The pseudo-code of EGWOA can be summarized as follows (Algorithm 1):

Algorithm 1 Pseudo-code of EGWOA

Initialize the population of grey wolves Find the initial negative value of parameters Calculate the fitness value of grey wolves Assign the value to xα; xβ; xδ While (t < Max Maximum number of iterations) For each search wolf Calculate the value of D1; D2; D3; Dαβ; Dαδ; Dβδ Update the positions of search wolves End for Update parameters Calculate the fitness value of each wolf Update xα; xβ; xδ t = t + 1 End while Return xα

The solving ability of the EGWOA that can be evaluated by test functions include the Beale function, Booth function, Dixon–Price function, Levy function, Rosen Brock function and Shubert function, with an accuracy of 0.001. During the solving process, the same initial conditions, population number and maximum number of iterations were set in different test functions. The solved fitness values and convergence curves of the test functions are shown in

Table 5. The fitness values of the test function.

Fitness Value	Beale	Booth	Dixon–Price	Levy	Rosen Brock	Shubert
Theoretical value	0	0	0	0	0	−186.7309
GWO	2.596 × 10−6	7.459 × 10−6	1.636 × 10−6	3.644 × 10−6	0.0839	−186.7291
IGWO	4.308 × 10−8	6.112 × 10−7	2.701 × 10−7	1.213 × 10−7	1.109 × 10−4	−186.7299

and Figure 12, respectively.

3.3. Tuning the PID Parameters Based on EGWOA

The principle of PID controller based on EGWOA is shown in Figure 13. The output u(t) can be expressed as follows:

$${\mathit{u}\left(\mathit{t}\right)=\mathit{K}}_{\mathit{P}}{\mathit{e}\left(\mathit{t}\right)+\mathit{K}}_{\mathit{I}}{{\displaystyle \int }}_0^{\mathit{t}}{\mathit{e}\left(\mathit{t}\right)\mathit{dt}+\mathit{K}}_{\mathit{D}}\frac{\mathit{de}\left(\mathit{t}\right)}{\mathit{dt}}$$

(21)

where K_P, K_I and K_D represent the coefficient of the proportional link, integral link and differential link of the system, respectively.

To optimize the PID controller’s parameters, the evaluation function system of absolute error time integral was established. In the evaluation function, u(t)² was used to reduce the possible overshoot of the control system, and the objective function was calculated as:

$$\mathrm{J}=\left\{\begin{array}{c}{{\displaystyle \int }}_0^{\infty }{(\omega }_1{\left|\mathit{e}\right(\mathit{t}\left)\right|+\omega }_2{\mathit{u}}^2{\left(\mathit{t}\right))\mathit{dt}+\omega }_3{\mathit{t}}_{\mathit{r}} e\left(\mathit{t}\right) \ge 0\\ {{\displaystyle \int }}_0^{\infty }{(\omega }_1{\left|\mathit{e}\right(\mathit{t}\left)\right|+\omega }_2{\mathit{u}}^2{\left(\mathit{t}\right)+\omega }_4{\left|\mathit{e}\right(\mathit{t}\left)\right|)\mathit{dt}+\omega }_3{\mathit{t}}_{\mathit{r}} e\left(\mathit{t}\right) < 0\end{array}\right.$$

(22)

To evaluate the comparison results of the system among the algorithms in more detail, the parameters overshoot σ, rise time t_r and adjusting time t_s are shown in

Table 6. The performance evaluation indexes of the three controllers.

Controller	σ (%)	tr (s)	ts (s)
Self-tuning PID	3.9	0.026	0.081
GWO PID	0	0.062	0.123
EGWO PID	0	0.031	0.057

, and the unit step response curve is presented in Figure 14. The overshoot σ of the self-tuning PID controller was 3.9%, and the rising time and adjusting time were 0.026 s and 0.081 s, respectively. As in Figure 14, the overshoot of the PID controller optimized by the GWOA decreased to 0, and the rising time and adjusting time decreased to 0.062 s and 0.123 s, respectively. The overshoot of the PID controller optimized by the EGWOA also decreased to 0, and the rise time and adjustment time decreased to 0.031 s and 0.057 s, respectively.

By comparing the performance of the self-tuning PID controller, GWOA PID and EGWOA PID under unit step response, the controller performance could be effectively enhanced by optimization of the PID controller parameters by an improved algorithm. Furthermore, the results of the EGWOA PID controller parameters were better than those of the GWOA PID, which met the control requirements of MRB.

4. Experimental Results and Analysis

The MRB test-bed shown in Figure 7 was used for the stability control experiment, where the data were collected by the torque speed sensor and data acquisition card, and the upper computer realized the automatic control of braking torque. Firstly, we adjusted the frequency of the frequency inverter to keep the rotating speed of the driving shaft at 400 r/min, and then, the target torque values were set in the interface of the upper computer platform and the output torques of the system under different target torques were recorded. Figure 15a–c show the braking torque of MRB when the rotation speed n was from 0 to 300, 400, and 500 r/min, and the target torque T was 60, 100 and 110 N·m, respectively.

The fast and stable output response of MRB is an important foundation for its application in the braking field. Stability control experiments of braking torque prove the stability of the MRB, that is, it can quickly reach the set output torque and keep it stable. The sooner the output torque reaches the target value, the better the torque response will be. Otherwise, the torque response performance will be worse. Moreover, the greater the fluctuation of the output torque value near the set torque value, the more stable the torque output is; on the contrary, the torque output becomes more unstable. In general, the output performance of the MRB is evaluated by static potential difference rate, fluctuation rate and response time [23].

Using the static potential difference rate to evaluate the stability of the system, the calculation formula can be expressed as follows:

$${\delta }_1=\frac{∆T}{T_m}\times 100\%$$

(23)

where ΔT is the difference between the maximum and the minimum output value of braking torque, and T_m is the average value of the output value.

The fluctuation rate is mainly used to evaluate the stability of the system after running for a period of time; the calculation formula is as follows:

$${\delta }_2=\frac{∆T}{T_0}\times 100\%$$

(24)

where T₀ is the target output value of the system.

Table 7. Comparison of controller performance indexes.

Type	Static Potential Difference Rate (%)			Fluctuation Rate (%)			Response Time (s)
	T1	T2	T3	T1	T2	T3	0 ~ T1	0 ~ T2	0 ~ T3
PID	8.33	6.34	9.85	8.42	6.4	9.91	2.2	2.1	2.2
GWO PID	4.47	4.08	2.85	4.49	4.18	2.89	1	1.2	1.3
EGWO PID	2.2	1.17	1.07	2.2	1.17	1.07	0.5	0.6	0.5

shows the MRB evaluation indexes of three different PID controllers under the target braking torque T1 (60 N·m), T2 (100 N·m) and T3 (110 N·m). It can be seen from the table that the maximum static potential difference rates of PID, GWO PID and EGWO PID controller were 9.85 %, 4.47% and 2.2%, respectively. Additionally, the fluctuation rate of the three modes was less than 9.91%, 4.49% and 2.20 %, respectively. When the target braking torque was adjusted from 0 to 200 N.m, the response time of EGWO PID controller was shorter than that of the GWO PID and PID controllers, ranging from 0.5 to 0.6 s. Therefore, compared with the conventional PID controller and GWO PID, the EGWO PID controller has a better effect of adjusting and controlling the output torque. Meanwhile, the output torque was more stable, the adjustment time was shorter, and the response speed was quicker.

As shown in Figure 16, the performance of MRB torque tracking control with EGWO PID was evaluated. The target torque was set to 60, 100 and 110 N·m, and it was kept for 10 s in each stage. It was observed that the braking torque of the black line matched the expected braking torque of the red line well, and the system had good torque tracking control performance.

The experimental results showed that the torque transmitted by the brake had good stability when the braking torque was constant. After adjusting the brake torque, the torque fluctuations of the PID controller optimized by the algorithm took less time than the conventional PID controller to restore to the stable state. The EGWOA PID controller not only takes less time, but also has more accurate adjustment results.

5. Conclusions and Future Works

In this paper, the braking system of magnetorheological fluid was designed and tested. Firstly, the relationship model between the excitation current and the shear stress in MRF was established, and the torque transfer function of the system was identified. On this basis, an EGWOA algorithm based on the GWO algorithm was proposed by using the logistic curve iteration method, and the effectiveness of the improvement was verified by testing functions. The performances of the self-tuning PID controller, GWOA-optimized PID controller and EGWOA-optimized PID controller were compared by unit step response. Finally, an experimental platform was set up for the torque control experiment, and the experimental results showed that the controller had a better control effect after using EGWOA to optimize the controller parameters.

In the future, other control strategies and optimization algorithms of MRB will be studied to obtain better response characteristics. Furthermore, the braking performance of MRB will also deteriorate with the extension of working time. Therefore, the rheological properties of MRF and the structure of MaRB will be optimized to improve the braking performance; in the next stage, some experiments related to different test signals will be conducted.