# Magnetorheological Elastomer Precision Platform Control Using OFFO-PID Algorithm

Authors: Ying-Qing Guo; Jie Zhang; Dong-Qing He; Jin-Bao Li
Journal: Advances in Materials Science and Engineering
 (2020)

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## Abstract

The magnetorheological elastomer (MRE) is a kind of smart material, which is often processed as vibration isolation and mitigation devices to realize the vibration control of the controlled system. The key to the effective isolation of vibration and shock absorption is how to accurately and in real time determine the magnitude of the applied magnetic field according to the motion state of the controlled system. In this paper, an optimal fuzzy fractional-order PID (OFFO-PID) algorithm is proposed to realize the vibration isolation and mitigation control of the precision platform with MRE devices. In the algorithm, the particle swarm optimization algorithm is used to optimize initial values of the fractional-order PID controller, and the fuzzy algorithm is used to update parameters of the fractional-order PID controller in real time, and the fractional-order PID controller is used to produce the control currents of the MRE devices. Numerical analysis for a platform with the MRE device is carried out to validate the effectiveness of the algorithm. Results show that the OFFO-PID algorithm can effectively reduce the dynamic responses of the precision platform system. Also, compared with the fuzzy fractional-order PID algorithm and the traditional PID algorithm, the OFFO-PID algorithm is better.


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## Body

## 1. Introduction

The precision platform is the carrier of precision instrument work. The external excitation such as environmental vibration affects the performance of the precision instrument to a great extent, even affects its service life and the normal work of operators. Many researchers adopted the passive control method to mitigate the vibration of the platform due to its economic efficiency and convenience [1]. In this paper, the semiactive control method of the magnetorheological elastomer (MRE) device is used to realize the vibration control of the precision platform.The MRE is made of polymer and micron-sized soft magnetic materials. After solidification, the mechanical, electrical, and magnetic properties of the MRE will change with the change of the external magnetic field. Therefore, it belongs to a branch of MR materials. The MRE has the advantages of both MR material and elastomer, and it avoids the problem of easy settlement of MR fluid. That is to say, it retains the controllable properties of the MR material’s stiffness and damping, that is, its elasticity, shear storage modulus, and loss factor are controllable, and it overcomes the shortcomings of the MR fluid such as poor sedimentation and stability [2–5]. MREs can be used to make various shock absorber devices or vibration isolation and mitigation devices, such as shock absorber, suspension, engine frame, axle lining, shock vibration absorber, shock absorber, and base isolator [6–10].The system equipped with MR devices has strong nonlinearity and is time varying. If the MR devices are expected to have an effective vibration isolation and mitigation effect for the system, the key is to have appropriate control current (or control voltage) for MR devices. Many control algorithms on systems incorporated with MR devices have been proposed. In 1996, Dyke et al. proposed a clipped-optimal control strategy based on acceleration feedback for controlling MR dampers to reduce structural responses due to seismic loads. And a numerical example was given to illustrate the effectiveness of the control strategy [11]. In 2001, Schurter and Roschke used ANFIS (adaptive neuro-fuzzy inference system) to create a fuzzy controller based on acceleration of the building feedback to achieve the vibration mitigation [12]. In 2003, Xu et al. applied neural network technology to achieve semiactive control of structures with MR dampers [13]. In 2005, Wang and Liao presented an inverse neural network model for MR dampers to generate the control voltage when the MR damper is operating in a semiactive mode [14]. In 2008, Guo et al. proposed a fuzzy control strategy based on a neural network forecasting model of the building structure with MR dampers, in which a neural network forecasting model was developed to predict dynamic responses of the system with MR dampers, and a fuzzy controller is then designed to determine control currents of MR dampers [15]. In 2010, Wilson and Abdullah proposed a self-tuning fuzzy controller to regulate MR dampers’ properties and reduce structural responses of single degree-of-freedom seismically excited structures [16]. In 2015, Yu et al. presented a hybrid support vector regression-based model to predict the inherent hysteresis behaviors of the MRE-based isolator and also proposed a hybrid modeling method to characterize the nonlinear dynamics of MRE isolators [17, 18]. In 2016, different control algorithms, such as the fruit fly optimization algorithm (FFOA), are employed for model parameter identification using testing data of shear force, displacement, and velocity obtained from different loading conditions [19]. In 2017, Gu et al. developed an inverse model for MRE base isolator based on the optimal general regression neural network (GRNN) and used the LQR controller and the GRNN inverse model to carry out the numerical and experimental validation of a real-time semiactive controlled MRE [20]. In 2018, Yu et al. developed a novel nonparametric model based on the artificial neural network to describe the nonlinear characteristic of the MRE base isolator. Also, in this work, a novel binary-coded cat swarm optimization algorithm (BCDCSO) was proposed to optimize the input subsets of the ELM (extreme learning machine) [21]. Leng et al. proposed an artificial neutral network approach optimized by the fuzzy algorithm (ANNOFA) system for approximately capturing the nonlinear functional relationship between inputs (displacement, frequency, and current) and output (force) of the MRE isolator [22]. In 2019, to overcome the inherent nonlinearity and hysteresis of the MRE isolator, Gu et al. developed a radial basis function neural network-based fuzzy logic control algorithm due to its inherent robustness and capability in coping with uncertainties [23].In this paper, an optimization fuzzy fractional-order PID control algorithm (OFFO-PID) is proposed to control the isolation and vibration reduction system of the precision platform based on MRE, and the algorithm is composed of the fuzzy control algorithm and the fractional PID algorithm based on particle swarm optimization (PSO), which significantly improve the control accuracy of the system. In addition, using SIMULINK simulation platform, the mathematical model of the precise platform structure and the OFFO-PID control algorithm were simulated and analyzed. The simulation results of OFFO-PID are compared with the simulation results of unoptimized fuzzy fractional PID, no control condition, and traditional PID. The analysis results show that the OFFO-PID can effectively suppress the vibration response of the precision platform under certain vibration disturbance, and the simulation results show that it is obviously superior to the no control condition, unoptimized condition, and traditional PID control.


## 2. Mathematical Model of the Controlled System

### 2.1. Mathematical Model of the Precision Platform Structure

The precision platform discussed in this paper is the carrier of high-precision equipment operation, which is used to isolate or reduce the impact of external vibration on equipment operation. In order to isolate or reduce the influence of external vibration more effectively and make the high-precision equipment run better, MRE devices are installed between the installation base and the legs of the precision platform, as shown in Figure1. The MRE in the device is silicone rubber-based and anisotropic. Its ferromagnetic particle volume fraction is 0.38. Considering that the vertical movement of the platform cannot be too large, the device is provided with a limit device, and the limit displacement is not more than 1 mm. The maximum output of the MRE isolation layer is 5708 N. The precision platform is assumed to be square and axisymmetric. For simplifying the calculation, the entire precision platform structure is simplified to a single degree-of-freedom structure, and its mathematical model is shown in Figure 2. The vertical acceleration signals are as the external vibration interference excitation. The dynamic equation of the single degree-of-freedom model for the precision platform with the MRE devices is(1)mz¨+4c0z˙+4k0+kmz=−mz¨g,where m is the effective load mass of the precision platform; k0 is the inherent stiffness of the MRE device; km is the adjustable stiffness of the MRE device; c0 is the inherent damping of the MRE device; z, z˙, and z¨ are the vertical displacement, the vertical velocity, and the vertical acceleration of the precision platform countertop, respectively; and z¨g is the vertical acceleration excitation.Figure 1
The precision platform with MRE devices.Figure 2
The single degree-of-freedom model of the precision platform with MRE devices.


### 2.2. Adjustable Stiffness Model of the MRE Device

According to equation (1), it can be seen that the responses of the precision platform are affected by the adjustable stiffness of the MRE device. That means responses of the precision platform can be controlled by adjusting the adjustable stiffness of the MRE device. The adjustable stiffness km is [24](2)km=ΔGm⋅Atk,where A is the shear area, tk is the thickness of the vibration isolation and mitigation layer of MRE, and ΔGm is the magnetic-induced shear stiffness, which can be obtained through the dipole model [3, 4]:(3)ΔGm=36ϕμfμ0β2H02Rd3ξ,where ϕ is the volume fraction of ferromagnetic particles; μf is the relative magnetic permeability of the matrix material of the MRE device; μ0=4π×10−7N/A2 is the magnetic permeability in vacuum; H0 is the strength of the external magnetic field; R is the average radius of ferromagnetic particles; d is the average spacing between particles; and ξ=∑n=1∞1/n3≈1.202, β≈1.From equation (3), it can be seen that ΔGm can be adjusted by changing the intensity of the external magnetic field. And the intensity of the external magnetic field can be adjusted by changing the control currents. That means ΔGm is the function of the control current. In this paper, the MRE devices used in the precision platform were developed by Xu of the same research group [4]. According to the MRE device and the analysis of its magnetic circuit [25, 26], the N/Le ratio is 27900 (N is the number of turns of the coil in the device, and Le is the effective magnetic circuit length). According to the formula of magnetic field strength, H0=IcN/Le. So, the functional relationship between ΔGm and the control current Ic can be obtained as(4)ΔGm=2.78ϕμfμ0β2Ic2Rd3ξ×1010.


## 2.1. Mathematical Model of the Precision Platform Structure

The precision platform discussed in this paper is the carrier of high-precision equipment operation, which is used to isolate or reduce the impact of external vibration on equipment operation. In order to isolate or reduce the influence of external vibration more effectively and make the high-precision equipment run better, MRE devices are installed between the installation base and the legs of the precision platform, as shown in Figure1. The MRE in the device is silicone rubber-based and anisotropic. Its ferromagnetic particle volume fraction is 0.38. Considering that the vertical movement of the platform cannot be too large, the device is provided with a limit device, and the limit displacement is not more than 1 mm. The maximum output of the MRE isolation layer is 5708 N. The precision platform is assumed to be square and axisymmetric. For simplifying the calculation, the entire precision platform structure is simplified to a single degree-of-freedom structure, and its mathematical model is shown in Figure 2. The vertical acceleration signals are as the external vibration interference excitation. The dynamic equation of the single degree-of-freedom model for the precision platform with the MRE devices is(1)mz¨+4c0z˙+4k0+kmz=−mz¨g,where m is the effective load mass of the precision platform; k0 is the inherent stiffness of the MRE device; km is the adjustable stiffness of the MRE device; c0 is the inherent damping of the MRE device; z, z˙, and z¨ are the vertical displacement, the vertical velocity, and the vertical acceleration of the precision platform countertop, respectively; and z¨g is the vertical acceleration excitation.Figure 1
The precision platform with MRE devices.Figure 2
The single degree-of-freedom model of the precision platform with MRE devices.


## 2.2. Adjustable Stiffness Model of the MRE Device

According to equation (1), it can be seen that the responses of the precision platform are affected by the adjustable stiffness of the MRE device. That means responses of the precision platform can be controlled by adjusting the adjustable stiffness of the MRE device. The adjustable stiffness km is [24](2)km=ΔGm⋅Atk,where A is the shear area, tk is the thickness of the vibration isolation and mitigation layer of MRE, and ΔGm is the magnetic-induced shear stiffness, which can be obtained through the dipole model [3, 4]:(3)ΔGm=36ϕμfμ0β2H02Rd3ξ,where ϕ is the volume fraction of ferromagnetic particles; μf is the relative magnetic permeability of the matrix material of the MRE device; μ0=4π×10−7N/A2 is the magnetic permeability in vacuum; H0 is the strength of the external magnetic field; R is the average radius of ferromagnetic particles; d is the average spacing between particles; and ξ=∑n=1∞1/n3≈1.202, β≈1.From equation (3), it can be seen that ΔGm can be adjusted by changing the intensity of the external magnetic field. And the intensity of the external magnetic field can be adjusted by changing the control currents. That means ΔGm is the function of the control current. In this paper, the MRE devices used in the precision platform were developed by Xu of the same research group [4]. According to the MRE device and the analysis of its magnetic circuit [25, 26], the N/Le ratio is 27900 (N is the number of turns of the coil in the device, and Le is the effective magnetic circuit length). According to the formula of magnetic field strength, H0=IcN/Le. So, the functional relationship between ΔGm and the control current Ic can be obtained as(4)ΔGm=2.78ϕμfμ0β2Ic2Rd3ξ×1010.


## 3. Optimal Fuzzy Fractional-Order PID Control Algorithm

### 3.1. The Fractional-Order PID Controller

The theory of the fractional-order PID controller was originally proposed by Podlubny [27], and its general expression is PIλDμ. Compared with the traditional integer-order PID, it includes an integral order λ and a derivative order μ. Its transfer function is(5)Gcs=kp+kis−λ+kdsμ0<λ,μ<1,where kp, ki, and kd represent the proportional gain, integral gain, and differential gain, respectively. The fractional differential operators in equation (5) can be obtained by the improved Oustaloup algorithm [28]. The fractional differential operator is(6)sα≈dωhbds2+dωhsd1−αs2+bωhs+dα∏k=−NNs+ωk′s+ωk,where 0<α<1, b>0, and d>0, ωk=bωh/dα+2k/2N+1, and ωk′=dωb/bα−2k/2N−1.


### 3.2. Optimization of Fractional-Order PID Parameters

According to equation (5), there are five parameters (λ, μ, kp, ki, and kd ) in the fractional-order PID controller that need to be determined. In this paper, the particle swarm optimization (PSO) algorithm is used to determine the optimal initial values of these five parameters. In the paper, a linearly decreasing inertial weight (LPSO) is considered instead of nonlinear ones (NPSO) mainly because the NPSO has fast convergence speed and high accuracy when optimizing low-dimensional functions, while its accuracy is far less than the LPSO when optimizing high-dimensional functions. The system function in this paper is not a low-dimensional function. In order to improve the control accuracy of the system, LPSO is selected. The evolution equation of the PSO algorithm can be described as [29](7)ω=ωmax−iter/MaxIter∗ωmax−ωmin,vt+1=ωvt+c1r1Pt−xt+c2r2Gt−xt,xt+1=xt+vt+1,where ω is the inertia factor, ωmax and ωmin are the maximum and minimum values of the inertia factor; x is the position of the particle; v is the velocity of the particle; iter is the number of iterations; MaxIter is the maximum number of iterations; c1 and c2 are the acceleration constants; r1 and r2 are the random numbers in the interval [0, 1]; Pt is the optimal position that the particle has searched so far; and Gt is the best position that the whole particle swarm has searched so far.In the parameter optimization process, the fitness function (also called the objective function) uses the ITAE performance indicator function [30]:(8)J=∫0∞tetdt.


### 3.3. Design of the Fuzzy Controller

The vibration isolation and mitigation control system of the precision platform with the MRE device is a dynamic system with real-time changes in parameters, but the PSO algorithm is only used to optimize the initial values of the five parameters of the fractional-order PID controller, and it cannot adjust the five parameters in real time. The fuzzy control algorithm is a control method based on heuristic knowledge and language decision rules. It is beneficial to simulate the process and method of manual control and improve the adaptability of the control system. At the same time, the fuzzy control system has strong robustness, and the influence of disturbance and parameter variation on the control effect is greatly weakened, especially suitable for the control of nonlinear systems. Therefore, in the paper, the fuzzy controller is designed to adjust the five parameters of the fractional-order PID controller in real time.The fuzzy controller has two input variables and five output variables. These two input variables are the vertical acceleration responses of the platform with MRE devices and its rate of change under the external disturbance signal. These five output variables are the correction values (Δλ, Δμ, Δkp, Δki, and Δkd) of the five parameters in the fractional-order PID controller. The input and output variables all are discretized with seven grades: {NB (negative big), NM (negative medium), NS (negative small), ZO (zero), PS (positive small), PM (positive medium), and PB (positive big)}. Their membership function curves are plotted in Figure 3. According to the fuzzy rules that should be followed when selecting the function, the characteristics of various membership functions, and the experience of experts, the membership functions of A (the difference between the vertical vibration acceleration and the set value) and AC (the rate of change) are selected as the combination of Gaussian function and linear triangle function. The membership functions of the five key parameters of fractional PID are selected as linear triangle functions. The five fuzzy rule tables corresponding to the five output variables (Δλ, Δμ, Δkp, Δki, and Δkd) are Tables 1–5.Figure 3
Membership function curves of the input and output variables of the fuzzy controller. (a)A and AC. (b) Δλ and Δμ. (c) Δkp, Δki, and Δkd.
(a)
(b)
(c)Table 1
Fuzzy rules ofΔλ.
A
AC
NB
NM
NS
ZO
PS
PM
PB
NB
PB
PB
PM
MM
PS
PS
ZO
NM
PB
PB
PM
PS
PS
ZO
NS
NS
PM
PM
PM
PS
PS
ZO
NS
ZO
PM
PM
PM
NM
ZO
NS
NS
PS
PS
PS
ZO
NS
NS
NM
NM
PM
PS
ZO
NS
NM
NM
NM
NB
PB
ZO
ZO
NM
NM
NM
NB
NBTable 2
Fuzzy rules ofΔμ.
A
AC
NB
NM
NS
ZO
PS
PM
PB
NB
NB
NB
NM
PM
NS
ZO
ZO
NM
NB
NB
NM
NS
ZO
ZO
NS
NS
NM
NS
NM
NS
ZO
PS
PS
ZO
NM
NM
NS
ZO
PS
PM
PM
PS
NM
NS
ZO
PS
PS
PM
PB
PM
ZO
ZO
PS
PS
PM
PB
PB
PB
ZO
ZO
PS
PM
PM
PB
PBTable 3
Fuzzy rules ofΔkp.
A
AC
NB
NM
NS
ZO
PS
PM
PB
NB
PB
PB
PM
PM
PS
ZO
ZO
NM
PB
PB
PM
PS
PS
ZO
NS
NS
PM
PM
PM
PS
PS
ZO
NS
ZO
PM
PM
PM
PS
ZO
NS
NS
PS
PS
PS
ZO
NS
NS
NM
NM
PM
PS
ZO
NS
NM
NM
NM
NB
PB
ZO
ZO
NB
NS
NM
NB
NBTable 4
Fuzzy rules ofΔki.
A
AC
NB
NM
NS
ZO
PS
PM
PB
NB
NB
NB
NM
PM
NS
ZO
ZO
NM
NB
NB
NM
NS
ZO
ZO
NS
NS
NM
NS
NS
NS
ZO
PS
PS
ZO
NM
NM
NS
ZO
PS
PM
PM
PS
NM
NS
ZO
PS
PS
PM
PB
PM
ZO
ZO
PS
PS
PM
PB
PB
PB
ZO
ZO
PS
PM
PM
PB
PBTable 5
Fuzzy rules ofΔkd.
A
AC
NB
NM
NS
ZO
PS
PM
PB
NB
PS
NS
NB
NB
NB
NM
PS
NM
PS
NS
NB
NM
NM
NS
ZO
NS
ZO
NS
NM
NM
NS
NS
ZO
ZO
ZO
NS
NM
NM
NS
NS
ZO
PS
ZO
ZO
ZO
ZO
ZO
ZO
ZO
PM
PB
NS
PS
PS
PS
PS
PB
PB
PB
PM
PM
PM
PS
PS
PB


### 3.4. Optimal Fuzzy Fractional-Order PID Control Algorithm

In this paper, the optimal fuzzy fractional-order PID (OFFO-PID) control algorithm is proposed to realize the vibration isolation and mitigation control of the precision platform with MRE devices. In the algorithm, the particle swarm optimization algorithm is used to get optimal initial values of five parameters of the fractional-order PID controller; according to the vertical acceleration responses of the platform with MRE devices and its rate of change under the external disturbance signal, the fuzzy controller is used to update five parameters of the fractional-order PID controller in real time; and then the fractional-order PID controller is used to produce the control force of the precision platform with MRE devices. The control schematic diagram of the precision platform with the MRE device is shown in Figure4. It can be seen from the control schematic diagram that the correction values of the five control parameters of the fractional-order PID controller are obtained through calculation by the fuzzy controller, and the five variables of the fractional-order PID are tuned in combination with the particle swarm optimization algorithm. Finally, the fractional-order PID controller outputs the control current to the MRE devices. Among them, real-time adjustment of some parameters in the fractional-order PID controller is realized; thereby, real-time control of the system is realized.Figure 4
The control schematic diagram of the precision platform with the MRE device.


## 3.1. The Fractional-Order PID Controller

The theory of the fractional-order PID controller was originally proposed by Podlubny [27], and its general expression is PIλDμ. Compared with the traditional integer-order PID, it includes an integral order λ and a derivative order μ. Its transfer function is(5)Gcs=kp+kis−λ+kdsμ0<λ,μ<1,where kp, ki, and kd represent the proportional gain, integral gain, and differential gain, respectively. The fractional differential operators in equation (5) can be obtained by the improved Oustaloup algorithm [28]. The fractional differential operator is(6)sα≈dωhbds2+dωhsd1−αs2+bωhs+dα∏k=−NNs+ωk′s+ωk,where 0<α<1, b>0, and d>0, ωk=bωh/dα+2k/2N+1, and ωk′=dωb/bα−2k/2N−1.


## 3.2. Optimization of Fractional-Order PID Parameters

According to equation (5), there are five parameters (λ, μ, kp, ki, and kd ) in the fractional-order PID controller that need to be determined. In this paper, the particle swarm optimization (PSO) algorithm is used to determine the optimal initial values of these five parameters. In the paper, a linearly decreasing inertial weight (LPSO) is considered instead of nonlinear ones (NPSO) mainly because the NPSO has fast convergence speed and high accuracy when optimizing low-dimensional functions, while its accuracy is far less than the LPSO when optimizing high-dimensional functions. The system function in this paper is not a low-dimensional function. In order to improve the control accuracy of the system, LPSO is selected. The evolution equation of the PSO algorithm can be described as [29](7)ω=ωmax−iter/MaxIter∗ωmax−ωmin,vt+1=ωvt+c1r1Pt−xt+c2r2Gt−xt,xt+1=xt+vt+1,where ω is the inertia factor, ωmax and ωmin are the maximum and minimum values of the inertia factor; x is the position of the particle; v is the velocity of the particle; iter is the number of iterations; MaxIter is the maximum number of iterations; c1 and c2 are the acceleration constants; r1 and r2 are the random numbers in the interval [0, 1]; Pt is the optimal position that the particle has searched so far; and Gt is the best position that the whole particle swarm has searched so far.In the parameter optimization process, the fitness function (also called the objective function) uses the ITAE performance indicator function [30]:(8)J=∫0∞tetdt.


## 3.3. Design of the Fuzzy Controller

The vibration isolation and mitigation control system of the precision platform with the MRE device is a dynamic system with real-time changes in parameters, but the PSO algorithm is only used to optimize the initial values of the five parameters of the fractional-order PID controller, and it cannot adjust the five parameters in real time. The fuzzy control algorithm is a control method based on heuristic knowledge and language decision rules. It is beneficial to simulate the process and method of manual control and improve the adaptability of the control system. At the same time, the fuzzy control system has strong robustness, and the influence of disturbance and parameter variation on the control effect is greatly weakened, especially suitable for the control of nonlinear systems. Therefore, in the paper, the fuzzy controller is designed to adjust the five parameters of the fractional-order PID controller in real time.The fuzzy controller has two input variables and five output variables. These two input variables are the vertical acceleration responses of the platform with MRE devices and its rate of change under the external disturbance signal. These five output variables are the correction values (Δλ, Δμ, Δkp, Δki, and Δkd) of the five parameters in the fractional-order PID controller. The input and output variables all are discretized with seven grades: {NB (negative big), NM (negative medium), NS (negative small), ZO (zero), PS (positive small), PM (positive medium), and PB (positive big)}. Their membership function curves are plotted in Figure 3. According to the fuzzy rules that should be followed when selecting the function, the characteristics of various membership functions, and the experience of experts, the membership functions of A (the difference between the vertical vibration acceleration and the set value) and AC (the rate of change) are selected as the combination of Gaussian function and linear triangle function. The membership functions of the five key parameters of fractional PID are selected as linear triangle functions. The five fuzzy rule tables corresponding to the five output variables (Δλ, Δμ, Δkp, Δki, and Δkd) are Tables 1–5.Figure 3
Membership function curves of the input and output variables of the fuzzy controller. (a)A and AC. (b) Δλ and Δμ. (c) Δkp, Δki, and Δkd.
(a)
(b)
(c)Table 1
Fuzzy rules ofΔλ.
A
AC
NB
NM
NS
ZO
PS
PM
PB
NB
PB
PB
PM
MM
PS
PS
ZO
NM
PB
PB
PM
PS
PS
ZO
NS
NS
PM
PM
PM
PS
PS
ZO
NS
ZO
PM
PM
PM
NM
ZO
NS
NS
PS
PS
PS
ZO
NS
NS
NM
NM
PM
PS
ZO
NS
NM
NM
NM
NB
PB
ZO
ZO
NM
NM
NM
NB
NBTable 2
Fuzzy rules ofΔμ.
A
AC
NB
NM
NS
ZO
PS
PM
PB
NB
NB
NB
NM
PM
NS
ZO
ZO
NM
NB
NB
NM
NS
ZO
ZO
NS
NS
NM
NS
NM
NS
ZO
PS
PS
ZO
NM
NM
NS
ZO
PS
PM
PM
PS
NM
NS
ZO
PS
PS
PM
PB
PM
ZO
ZO
PS
PS
PM
PB
PB
PB
ZO
ZO
PS
PM
PM
PB
PBTable 3
Fuzzy rules ofΔkp.
A
AC
NB
NM
NS
ZO
PS
PM
PB
NB
PB
PB
PM
PM
PS
ZO
ZO
NM
PB
PB
PM
PS
PS
ZO
NS
NS
PM
PM
PM
PS
PS
ZO
NS
ZO
PM
PM
PM
PS
ZO
NS
NS
PS
PS
PS
ZO
NS
NS
NM
NM
PM
PS
ZO
NS
NM
NM
NM
NB
PB
ZO
ZO
NB
NS
NM
NB
NBTable 4
Fuzzy rules ofΔki.
A
AC
NB
NM
NS
ZO
PS
PM
PB
NB
NB
NB
NM
PM
NS
ZO
ZO
NM
NB
NB
NM
NS
ZO
ZO
NS
NS
NM
NS
NS
NS
ZO
PS
PS
ZO
NM
NM
NS
ZO
PS
PM
PM
PS
NM
NS
ZO
PS
PS
PM
PB
PM
ZO
ZO
PS
PS
PM
PB
PB
PB
ZO
ZO
PS
PM
PM
PB
PBTable 5
Fuzzy rules ofΔkd.
A
AC
NB
NM
NS
ZO
PS
PM
PB
NB
PS
NS
NB
NB
NB
NM
PS
NM
PS
NS
NB
NM
NM
NS
ZO
NS
ZO
NS
NM
NM
NS
NS
ZO
ZO
ZO
NS
NM
NM
NS
NS
ZO
PS
ZO
ZO
ZO
ZO
ZO
ZO
ZO
PM
PB
NS
PS
PS
PS
PS
PB
PB
PB
PM
PM
PM
PS
PS
PB


## 3.4. Optimal Fuzzy Fractional-Order PID Control Algorithm

In this paper, the optimal fuzzy fractional-order PID (OFFO-PID) control algorithm is proposed to realize the vibration isolation and mitigation control of the precision platform with MRE devices. In the algorithm, the particle swarm optimization algorithm is used to get optimal initial values of five parameters of the fractional-order PID controller; according to the vertical acceleration responses of the platform with MRE devices and its rate of change under the external disturbance signal, the fuzzy controller is used to update five parameters of the fractional-order PID controller in real time; and then the fractional-order PID controller is used to produce the control force of the precision platform with MRE devices. The control schematic diagram of the precision platform with the MRE device is shown in Figure4. It can be seen from the control schematic diagram that the correction values of the five control parameters of the fractional-order PID controller are obtained through calculation by the fuzzy controller, and the five variables of the fractional-order PID are tuned in combination with the particle swarm optimization algorithm. Finally, the fractional-order PID controller outputs the control current to the MRE devices. Among them, real-time adjustment of some parameters in the fractional-order PID controller is realized; thereby, real-time control of the system is realized.Figure 4
The control schematic diagram of the precision platform with the MRE device.


## 4. Dynamic Simulation Analysis

In order to verify the effectiveness of the OFFO-PID control algorithm, the dynamic simulation of the entire control system of the precision platform is carried out. The parameters of the single degree-of-freedom model (in Figure2) for the precision platform with the MRE devices are as follows: m=600kg, k0=4122kN/m, c0=42.64N⋅s/m, A=0.0515m2, and tk=10mm. Parameters of the designed PSO algorithm are the population size N=100, the maximum number of iterations T=100, the acceleration constants c1=c2=2, and the inertia factor ω decreases from 0.9 to 0.3 with the increase of iteration times. According to the traditional method, the parameters of the traditional PID controller are kp=40.34, ki=30.65, and kd=13.34; the parameters of the fractional PID controller based on the expert experience but not optimized by PSO algorithm are kp=75.52, ki=23.36, kd=50.57, λ=0.34, and μ=0.13; and the parameters of the fractional PID controller optimized by PSO are kp=43.78, ki=12.37, kd=37.27, λ=0.57, and μ=0.31. The external disturbance acceleration signal is shown in Figure 5, and the time interval is 0.01 s.Figure 5
The external disturbance acceleration signal.The dynamic simulation analysis of the entire precision platform system under different control methods is carried out by using MATLAB. Simulation results of the OFFO-PID control system are compared with those of the fuzzy fractional-order PID (FFO-PID, the parameters of the fractional PID controller are obtained from the expert experience rather than optimized using the PSO algorithm.) control system, the traditional PID control system, and the uncontrolled system. Figure6 shows the displacement response and acceleration response comparison between the OFFO-PID control system and the uncontrolled system. Figure 7 shows the displacement response and acceleration response comparison between the OFFO-PID control system and the traditional PID control system. Figure 8 shows the displacement response and acceleration response comparison between the OFFO-PID control system and the FFO-PID control system. Table 6 lists the maximum displacement and acceleration responses of the precision platform system under different control methods. Also, Table 7 lists reduction rate of the OFFO-PID algorithm compared with other control methods.Figure 6
Vibration response comparison between the OFFO-PID control system and the uncontrolled system. (a) The displacement responses. (b) The acceleration responses.
(a)
(b)Figure 7
Vibration response comparison between the OFFO-PID control system and the traditional PID control system. (a) The displacement responses. (b) The acceleration responses.
(a)
(b)Figure 8
Vibration response comparison between the OFFO-PID control system and the FFO-PID control system. (a) The displacement responses. (b) The acceleration responses.
(a)
(b)Table 6
The maximum response comparison under different control methods.
Uncontrolled
Traditional PID
FFO-PID
OFFO-PID
Displacement (mm)
1.366
0.6558
0.3728
0.2488
Acceleration (m/s2)
8.607
3.367
2.664
1.455Table 7
Reduction rate of the OFFO-PID algorithm compared with other control methods (%).
Uncontrolled
Traditional PID
FFO-PID
Displacement
81.78%
62.06%
33.26%
Acceleration
83.09%
56.78%
45.38%It can be seen from Figures6–8 and Tables 6 and 7 that both displacement responses and acceleration responses of the OFFO-PID control system are effectively reduced compared with those of the uncontrolled system, the traditional PID control system, and the FFO-PID control system. The maximum displacement response of the OFFO-PID control system is 0.249 mm. Compared with the maximum displacement response of the uncontrolled system, 1.366 mm, it is reduced by 81.77%. Compared with the maximum displacement response of the traditional PID control system, 0.656 mm, it is reduced by 62.06%. And compared with the maximum displacement response of the FFO-PID control system, 0.373 mm, it is reduced by 33.26%. The maximum acceleration response of the OFFO-PID control system is 1.455 m/s2. Compared with the maximum acceleration response of the uncontrolled system, 8.607 m/s2, it is reduced by 83.09%. Compared with that of the traditional PID control system, 3.367 m/s2, it is reduced by 56.78%. And compared with the maximum acceleration response of the FFO-PID control system, 2.664 m/s2, it is reduced by 45.38%.The comparison analysis shows that the fuzzy fractional-order PID controller based on particle swarm optimization can effectively suppress the vibration response of the precision platform under certain vibration disturbance, and the simulation results show that it is obviously superior to the uncontrolled, nonoptimal, and traditional PID control.


## 5. Conclusions

In this paper, an optimal fuzzy fractional-order PID (OFFO-PID) control algorithm is proposed to realize the vibration isolation and mitigation control of the precision platform with MRE devices. In the algorithm, the particle swarm optimization algorithm is used to get optimal initial values of five parameters of the fractional-order PID controller, the fuzzy control algorithm is used to update five parameters of the fractional-order PID controller in real time, and then the fractional-order PID controller is used to generate the control currents of MRE devices. In order to validate the effectiveness of the OFFO-PID control algorithm, dynamic simulation analysis for a platform with the MRE devices is carried out. Simulation results of the OFFO-PID control system are compared with those of the fuzzy fractional-order PID control system, the traditional PID control system, and the uncontrolled system. Analysis results show that the OFFO-PID control algorithm can effectively reduce the dynamic responses of the precision platform system, and the simulation results show that it is obviously superior to three other control methods.



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*Source: 3025863-2020-02-26.xml*