A Review on Model and Control of Electromagnetic Active Engine Mounts

State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130022, China School of Automotive Engineering, Shandong Jiaotong University, Jinan 250023, China Zhejiang Provincial Key Laboratory of Modern Textile Machinery, Zhejiang Sci-tech University, Hangzhou 310018, China School of Aviation Operations and Services, Aviation University of Air Force, Changchun 130022, China

One of the main objectives of the present work is to summarize and show general information about the model of AEMs. Models can be categorized as theoretical models, finite-element models, and identification [1], which can provide deep comprehension of the dynamic behaviour of AEMs and improve the control performance in designing AEM controllers. A second aim of this present work is a quantitative comparison of the various control strategies of AEMs regarding their weighting functions, the order of controllers, the sampling frequency, and the step size filter length. Based on this comparison, the merits and weaknesses are discussed. e goal is to identify the focus of the literature on the dynamic modelling and control of AEMs.
is article is organized as follows: General Considerations of Electromagnetic AEMs relates the main considerations and parameters needed for a deep comprehension of AEMs. Model describes a review of the different models of AEMs: Section 3.1 reviews the theoretical model, Section 3.2 reviews the finite-element model, and Section 3.3 reviews identification (or the experimental model). In Control Strategies, classical control problems of AEMs are discussed, and adaptive control, such as LMS, filtered-x least mean squares, minimal controller synthesis (MCS), robust control, and 2DOF control, are reviewed. Finally, concluding remarks are presented in Conclusions.

General Considerations of Electromagnetic AEMs
2.1. Actuator. Electromagnetic actuators, such as solenoids or voice coil (moving coil) actuators, are shown in Figure 2 and have the characteristics of compact structure, low energy consumption, sensitive response, work densities, easily control, and good force.

Electromagnetic AEMs with a Voice Coil
Actuator. e passive HEM and the voice coil actuator are designed as the passive components and the active components of the electromagnetic AEM, respectively. e voice coil actuator produces the dynamic force and drives the upper chamber or the lower chamber of the AEM. As shown in Figure 3, Fursdon et al. [51] developed an electromagnetic AEM, which is installed on the Audi S8 [4]. e dynamic force of the actuator drives the main fluid chamber of AEM. Vahdati and Heidari [69] proposed an engine mounting design. e dynamic force of the actuator drives the compensation chamber of the AEM.

Electromagnetic AEM with a Solenoid Actuator.
e coil of the solenoid actuator generates a magnetic field, which attracts the iron component to the coil [50]. Similar to the structure of the electromagnetic AEM with a voice coil actuator, the passive HEM and the solenoid actuator are designed as the passive components and active components of the electromagnetic AEM with a solenoid actuator, respectively. As shown in Figure 4, the excitation plate is actuated by the solenoid actuator, which changes the liquid pressure of the chamber. Mansour et al. [71][72][73] proposed an electromagnetic AEM with a solenoid actuator. e solenoid actuator is placed on the inertia track plate of the HEM, and the passive structure of the HEM is retained. Continental [54,55] developed an electromagnetic AEM with a solenoid actuator for testing on vehicles. e electromagnetic AEM with a solenoid actuator shown in Figure 4 was applied on the Honda INSPIRE [61,70,74]. Kitayama et al. [75][76][77][78][79][80][81][82][83] studied the linear electromagnetic actuator for AEM. e permanent magnet is bonded on a diaphragm. A magnetic field is produced by the electric current in the solenoid coil, which can produce mechanical force and alter the pressure of fluid in the chamber [84].
From the above discussion, the passive HEM and actuator are designed as the passive components and active components of the electromagnetic AEM, respectively. e voice coil actuator subsystem consists of a permanent magnet and a coil, as shown in Figure 5; the electrical differential equation of voltage applied on the voice coil actuator can be expressed as where u(t), R, L, and x a (t) are the input voltage applied on the voice coil, the electrical resistance of the voice coil, the   Shock and Vibration inductance of the voice coil, and the position of the moving diaphragm, respectively. e magnetic flux generated by the permanent magnet interacts with the current in the coil, and the actuator produces a Lorentz force, which can be expressed as where B, l w , and i denote the field density, wire length, and current flowing in the wire, respectively. e decoupling membrane of the HEM is replaced by the voice coil actuator or the solenoid actuator. e actuator acting on the diaphragm is usually regarded as a mass-spring-damper system, as shown in Figure 6, which can be expressed by where f a is the actuator force, p u is the pressure inside the upper fluid chamber, m a is the actuator mass, d a is the damping coefficient of the actuator, and k a is the spring constant of actuator. When the actuator is turned on, the displacement of the chassis is small. If it is supposed that the chassis is fixed, then x c (t) is zero.

Fluid.
e fluid in an AEM is assumed to be incompressible. e continuity equation of the upper chamber is expressed as where A a , A i , and A m denote the actuator diaphragm area, the cross-sectional area of the inertia track, and the equivalent piston area of the main rubber spring, respectively. x a , x i , x e , and x c denote the displacement of the actuator, the fluid in the inertia track, the AEM at the engine side, and the AEM at the chassis side, respectively. e volumetric compliance element is expressed as  Shock and Vibration 3 Frequency-dependent stiffness and damping of the fluid are generated in the inertia track of the HEM [86]. e static pressure of the AEM upper chamber is relieved only in the inertia track [65]. A column of fluid is created in the inertia track. e friction generated by the fluid flow in the track is not directly transmitted to the chassis. e inertia track is assumed to be connected to the absolute reference frame [66]. e fluid in the inertial track is forced to flow by the pressure in the upper chamber, which can be represented by where m i and d i are the equivalent mass and damping coefficient of the fluid in the inertia track, respectively. e stiffness of the lower chamber is smaller than that of the upper chamber [87][88][89], and the stiffness of the lower chamber k i can be neglected.

Elastomeric.
e AEM can be designed by incorporating a conventional passive HEM [88][89][90][91][92] with the actuator. e dynamic stiffness of a passive engine mount depends on the frequency, temperature, amplitude, and types of external excitation [93,94]. Meanwhile, the main rubber spring or elastomer rubber element of the passive engine mount has amplitude-and frequency-related behaviours [95]. e damping, stiffness, and bulking properties of the main rubber spring or elastomer rubber element of the AEM contribute substantially to the dynamic characteristics of the AEM.
e frequency-dependent dynamics model of the elastic coupling unit can be traced back to the 18 th century; Maxwell et al. studied the behaviour of viscoelastic materials [96]. Some viscoelastic material models have been widely used in simulations, such as the Maxwell [97] and Kelvin-Voigt models [98]. In the AEM mathematical model, the main rubber springs or elastomeric rubber elements are typically modelled using linear spring elements [99,100] or a spring in parallel to a damper (the Kelvin-Voigt model shown in Figure 6(a)) [101][102][103], which overestimates both stiffness and damping at higher frequencies.
To settle the problem of the Kelvin-Voigt model, as shown in Figure 6(b), the main rubber spring and its bulking properties are modelled as one spring in parallel to two dampers and another spring. Lambertz et al. [104][105][106][107][108] applied this approach in studying conventional HEMs, and the transfer function of the elastomer element in the Laplace domain is expressed as where K dyn (s) is the dynamic properties of the main rubber spring or elastomer rubber. e damping and stiffness of the high frequency simulation are in good agreement with those of the test [85]. e Kelvin-Voigt model is consistent with the measured loss angle at only one design frequency [109].

Force
Transmitted to the Engine and the Chassis. e force transmitted to the engine, chassis, and body through the AEM, F e (t), and F c (t), respectively, can be represented by Similar equations [66,85,110] are used to deduce the transfer functions to study the dynamic characteristics of electromagnetic AEMs.

Model
As discussed in Section 2, the dynamic behaviour of AEMs is nonlinear. Accurate AEM models can improve the controller performance in designing model-based controllers and facilitate the accurate study of the dynamic behaviour of AEMs [111][112][113][114]. ere are three general models, namely, theoretical models, finite-element models, and identification [1]. e theoretical model is obtained by applying methods from calculus to equations derived from physics. e finite-element models make use of a virtual development environment, such as ADAMS. Identification, which is also named experimental models, is a mathematical model derived from measurements.

eoretical
Model. e theoretical model describes the dynamic characteristics of the AEM, which is expressed by the transmission performance of force between the chassis and the engine, and the secondary path transfer function of the AEM between the actuator input signal and the output force (displacement or acceleration) on the chassis (or engine) side. It is generally assumed that (1) the forces transmitted to the engine and the chassis are always equal to each other, (2) the displacement or acceleration of the chassis is zero, (3) the stiffness or damping of the elastomeric rubber is independent of the frequency or preload, and (4) the dynamics of the chassis or the mass of the engine are ignored, and the characteristics of the AEM are derived therefrom [88,115].
Assuming that the chassis displacement is zero and the dynamics of the engine, Lee and Lee [65] proposed a theoretical model to describe the dynamic characteristics of AEMs, which is expressed by the secondary path transfer function between the control voltage and the chassis, and the transmitted force in terms of the engine excitation force and the actuator motion. e theoretical model is verified by experiments, and the analytical results based on the proposed theoretical model agree well with the experimental results and confirm that the proposed theoretical model accurately describes the dynamic behaviour of an AEM. Considering the dynamics of actuator, the dynamics of the fluid in the inertia track, the structural parameter of the AEM, and the chassis displacements, Lee and Lee [66] proposed a modified linear AEM model that is expressed by the transmitted force in terms of the engine excitation force and the chassis force and the secondary path transfer function between the active force of the actuator and the engine excitation force or chassis force. e modified linear AEM model is verified by an experimental apparatus for measurement of the dynamic characteristics of an ACM, which shows that the simulation results based on the proposed modified linear AEM model and the experimental test are in agreement with fair accuracy. Considering the dynamics of the actuator, the dynamics of the fluid in the inertia track, the structural parameter of the AEM, the chassis displacements, the frequency-dependent characteristics of the volumetric stiffness of the main liquid chamber, and the complex stiffness of the main rubber spring, Hausberg [103] proposed a theoretical model expressed in the Laplace domain to describe the dynamic characteristics of an AEM. e proposed theoretical model is expressed by the cross point and driving point dynamic stiffness of the AEM at the engine side and the secondary path transfer function between the control voltage and the chassis (engine). e simulated curves of theoretical model proposed by Hausberg are in good agreement with the experimental results.

Finite-Element Model.
Olsson [116] studied a threepoint suspended 5-cylinder combustion diesel engine, as shown in Figure 7, which is attached to the vehicle body on the LHS (left-hand side) and RHS (right-hand side) via two rubber engine mounts and connected to the subframe via a TR (torque rod) with rubber bushings at both ends. en, a finite-element model was proposed by ADAMS. e engine and torque rod are modelled with 12 kinematical degrees of freedom (DOFs) using rigid body representation. e LHS engine mount, RHS engine mount, and rubber bushings are modelled using 6 DOFs. Finally, the body and subframe attachment points are regarded as rigid in all directions. e finite-element model was not compared with the theoretical model or experimental model by Olsson in this article.

Identification (Experimental Model).
e identification of the AEM shown in Table 1 is significant in the verification of the theoretical model, the controller design, and the dynamic characteristics of the AEM. e primary path is the transfer path between the error sensor and the disturbance source. e transfer path between the error sensor and the controller output is named the secondary path. e identification of the primary path or the secondary path is required for model-based control. Identification techniques include harmonic identification [57,66], LMS-or FBLMSbased finite impulse response (FIR) identification [69],   [117] 1995 Identification FXLMS/AC -V Nakaji et al. [57] 1999 FRT identification SFX/AC -R/V Aoki et al. [9] 1999 -SFX/AC -V Lee et al. [ [120], and neural network identification [138], all of which have been implemented in the identification of AEMs.

Frequency Response Measurements.
Scholars have estimated AEM models by harmonic identification [129,139]. For the identification of the actuator dynamics, the harmonic identification tests shown in Figure 8 were applied using a commercial rubber testing machine by Lee et al. [100]. Fakhari et al. [135] carried out the identification test shown in Figure 9(a), which identified the transfer function of the passive parts between the control force of the electromagnetic actuator and the output displacement of the AEM on the engine side. As shown in Figure 9(b), the transfer function of the active parts between the control force and the input current of the electromagnetic actuator is identified. en, the secondary path transfer function of the AEM is proposed through the transfer function of the passive part times the transfer function of the active part.
Yang et al. [118] carried out a test to analyse the transfer properties of the AEM between the input current and the output force. en, the approximate transfer function of the AEM was proposed by solving a nonlinear curve fitting problem.

LMS and FBLMS Identification.
A finite impulse response (FIR) filter can be modelled as the estimate of the secondary path. Modelling errors between the secondary path and its estimate may lead to instability or serious performance degradation.
e control system may be unstable when the phase error between the secondary path and its estimate is not less than 90° [140]. Broadband white noise can be used as an input signal in dynamical models, and the transfer function of the secondary path can be identified by the LMS identification algorithm. As shown in Figure 10, an LMS filter can identify the secondary path.
However, the LMS identification algorithm cannot cancel interference from the input engine vibration, which can be resolved by a fast-block LMS (FBLMS) in the frequency domain. FBLMS identification is robust to large system parameter variations, unknown or unmodelled dynamics, and nonlinear effects. Shynk [142] explained the general structure of FBLMS. As shown in Figure 11, the FBLMS identification algorithm is applied to a MIMO AEM system. e MIMO FBLMS filter updates the equation of the K th length L data block [127], which is expressed as where B 1 (z k ) and B 2 (z k ) can be expressed by where where F represents the fast Fourier transform (FFT).

Other
Identification. Subspace identification [143,144] was carried out to achieve the frequency responses from the input to the output of the system, as shown in Figure 12(a). e transfer functions G dyd represent the frequency response between the disturbance and the response. In general, G io represents the transfer function from the input i to the output o. e transfer function between the secondary source (the input u) and the error signal (the output y u ) and the transfer function between the disturbance signal and the error signal were identified by Seba et al. [120]. e neural network identification shown in Figure 12(b) can be defined mathematically for the NARMA-L2 neural network controller as the following equation: where u(k) is the system input, y(k) is the system output, and d is the delay of the parameters. e NARMA-L2 neural network controller identifies the inverse of the control plant.

Control Strategies
To improve the vibration control performance of AEM, many investigators have implemented various control algorithms, such as PID [50], adaptive control [121], robust control [132], and 2DOF control. Table 1 shows a summary of the control strategies applied during the last 2 decades, which are discussed in the following section.

Classical
Control. PID control applied in AEMs continuously calculates an error value between a desired set point and the measured variable [124,125,128,145]. e PID controller is mainly applied in the control of singleinput-single-output (SISO) systems. It is difficult to control MIMO systems with the PID controller based on the transfer function. e linear quadratic regulator (LQR) can overcome the aforementioned drawbacks of PID [126,128,146]. e weights of Q and R are adjusted, and the value of the gain K is calculated. e optimal results are obtained when R and Q are determined by various iterations. With Q � diag (10 9 , 10 9 , 10 9 , 10 9 ), the LQR controller R was set to be unity by Mahil et al. [128] to obtain a better AEM performance. Classical control has been assessed but found to be deficient.

Adaptive Control.
To tackle the high uncertainty of complex and variable environments, modern adaptive control is increasingly being applied. e AEM system is a time-varying system, which requires adaptive control. is control method can attenuate the vibration by adjusting the frequency and amplitude of the AEM actuator.

LMS.
In 1960, a highly simplified recursive algorithm was developed by Widrow and Hoff [147] to calculate the optimal filter, named the LMS algorithm. To simplify the control strategies, as shown in Figure 13, the LMS algorithm can be verified by setting G to 1 and by setting error signal as the input signal for controlling the AEM.  Vehicle test results show that the transmitted force, seat rail vibration, and interior noise can be simultaneously attenuated by the modified LMS controller over a wide frequency band. e LMS controller only can reduce the amplitude at a certain frequency corresponding to a 8 Shock and Vibration reference signal and cannot simultaneously attenuate the amplitude over a wide frequency band.

Filtered-X Least Mean Squares.
Conover et al. [148,149] proposed the filtered-x least mean squares (FXLMS) algorithm, which has been described in detail by other scholars [111][112][113][114]. e FXLMS control algorithm has become the primary tool to reduce active vibration or noise. As shown in Table 1, the FXLMS algorithm is the most common control strategy for controlling AEMs. Figure 14 shows the block diagram of the normalized FXLMS algorithm for AEM control, which uses the engine RPM as the reference signal and the transmitted force from the engine to the chassis or body as the error signal for the generation of the actuator signal. e main design parameters of the FXLMS algorithm are the step size or the convergence coefficient, filter length, and leaky factor. e convergence speed of the FXLMS algorithm is influenced by the step size. When the filter converges to a steady state, the step size has a slight influence on the performance. e overall system stability is influenced by the leaky factor. e sampling rate and available computer specifications are influenced by the filter length. Table 2 shows that the sampling frequency, step size, filter length, and leaky factor vary in different studies. e FXLMS algorithm requires an  Shock and Vibration estimate of the secondary path of the AEM because model errors can cause performance degradation or instability of the FXLMS algorithm. To reduce the errors of model, an online system identification scheme is employed by Bao et al. [133]. In addition, the robustness of algorithm [150][151][152] must be improved to reduce the impact of errors on the control algorithm.
To control more than one AEM, Hillis [127] applied a MIMO narrow-band FXLMS algorithm to a system of two AEMs. e control equation is defined as e reference signal is defined by where ω e is the engine crankshaft rotation frequency, i is the engine order, and Δ is the sampling interval. e weight vectors {w 1 (k) ∈ R q×1 , w 2 (k) ∈ R q×1 } are updated according to S 21 (z) Figure 11: MIMO online system identification [127].  Figure 12: Block diagram of identification: (a) subspace identification [120]; (b) neural network identification [124,125,138].  10 Shock and Vibration , where μ and ] are the step size and leakage factor, respectively. To avoid the accumulation of numerical rounding errors, the leakage factor is applied to the tap weights.
x 1 ′ (k), x 1 ′ (k) ∈ R q×p are the filtered matrices of the reference signals x 1 , x 2 , which can be expressed by where S T (k) ∈ R p×q is the matrix of the estimation of the secondary path transfer function, which is described by e proposed MIMO FXLMS algorithm was applied to a two-mount/two-sensor system fitted to a saloon car equipped with a four-cylinder two-litre turbo-diesel engine. Vehicle tests indicate that the controller typically reduces chassis vibration by 50 percent to 90 percent under normal driving conditions. Synchronized filtered-x least mean square (SFX) is a modified form of the FXLMS algorithm with a limited application to cyclic phenomena [56,57]. Under this application environment, SFX has a computational advantage compared with the FXLMS algorithm. SFX has the ability to control higher order components and straight convergence. e control output of SFX is expressed as e filter weights and reference signal are, respectively, expressed by Vehicle tests show that reductions in the vehicle idle vibration, the boom noise during driving, and the higher order boom noise during idling are achieved by the proposed SFX controller [56].
Lee et al. [3,134,153] proposed the Newton FXLMS algorithm, the filter weight of which is expressed by Simulations show that the Newton FXLMS control has an equal convergence rate at different engine speeds, while the FXLMS control has different convergence rate at different engine speeds. To tackle the limited convergence time and the tracking speed, as shown in Figure 15, online adapted look-up tables were incorporated as parametermaps or parallel-maps, respectively, into the proposed Newton FXLMS algorithm. e look-up tables store the initial conditions of the Newton FXLMS, which is the filter weight vector w(0) that is passed from the look-up tables to F e (t) Correlated signal (tachometer) Engine AEM Figure 14: Diagram of normalized FXLMS [65]. the Newton FXLMS before activating the Newton FXLMS with the parameter-map. e look-up table is applied in parallel to the Newton FXLMS, which is the Newton FXLMS with the parallel-map. e output w LU of the parallel-map can be expressed as e vector of the table data is adapted online with the algorithm, which is expressed by e step size μ LU is different from the step size μ of the FXLMS. μ is set to 0.001 and μ LU <0.0005, which will suppress short-term variations of the training signal.
Vehicle tests show that the proposed control strategy applied on the AEM can reduce the convergence time. When μ is set to 0.001, the Newton FXLMS with the parameter-map or the parallel-map completely attenuates the vibration with almost no convergence time.
e tracking behaviour is improved by the Newton FXLMS algorithm with the parallel-map control applied on the AEM, while the tracking performance remains unchanged for the Newton FXLMS algorithm with the parameter-map.
As shown in Figure 16, body input point inertance (IPI) and extended FXLMS were proposed by Guo et al. [137]. e transmitted force to the chassis is converted to an acceleration, which is set to the error signal. e IPI is expressed as where € X, F, and ω are the acceleration, transmitted force to the chassis, and angular frequency of the engine crankshaft angle, respectively. e weight coefficient can be expressed as where α is the leakage coefficient, μ is the step size, and R ′ (n) is the "filtered" reference signal vector. Compared with the uncontrolled AEM, the simulation results show that the acceleration on the AEM has decreased by 80 percent for most of the time domain after applying the extended FXLMS controller.

Minimal Controller Synthesis (MCS).
e minimal control synthesis (MCS) requires no knowledge of the parameters and achieves stability and robustness [154]. e Er-MCSI controller shown in Figure 17 is a derivation of the MCS controller. e control law of the Er-MCSI is expressed as where x e (n) is the state error and x I is the scalar discretetime integral of the output error signal. e update law of the controller gains can be expressed as K e (t) � K e (n − 1) + βq e (n) − σq e (n − 1), where q e � y e x T e , q I � y e X I , where α and β are weights and y e and Δ are the output error and the sampling interval, respectively. e proposed Er-MCS controller was applied on an AEM, which is fitted to a car equipped with a four-cylinder engine. Test results show that the Er-MCS algorithm performed similarly to the FXLMS algorithm in terms of the cancellation level and convergence speed, and the Er-MCS algorithm has a significant computational advantage compared with the FXLMS algorithm. When the Er-MCSI is applied to control multiple AEMs and multiple sensor systems, the computational advantage will become more significant.
To tackle narrow-band error signals or the narrowband component of broadband signals, the narrow-band MCS (NBMCS) controller shown in Figure 18 was developed based on the Er-MCSI by Hillis et al. [119]. For the MCS algorithm, the plant parameters are unknown or where x s (t) and x c (t) are described by x c (t) � cos(ωt).
e proposed NBMCS algorithm was compared with the Er-MCSI algorithm using simulations and implementation with an AEM fitted to a diesel engine saloon car [119]. When the frequency of the disturbance is known or measured, the simulations and vehicle tests show that the NBMCS algorithm outperforms the broadband Er-MCSI algorithm in narrow-band applications and overcomes the gain windup problem in the Er-MCSI.

Robust Control.
Accurate models of the AEM system are difficult in model-based control, and controllers may not achieve the desired performance due to model uncertainties, disturbances, and noise. Robust controllers have robust performance in the presence of perturbations, such as disturbance, noise, parametric uncertainties, and unmodelled dynamics [155,156].
To attenuate transmitted forces over a large frequency band from the engine to the chassis, the gain scheduled H 2 controllers applied on an AEM by Olsson [116] can work well with system nonlinearities and engine vibration, but they do not work well at extremely high ramping speeds and nominal engine torque.
To isolate the engine vibration and prevent actuator saturation in the case of perturbations, Fakhari et al. [110,126,130] applied the H 2 and H ∞ controllers shown in Figure 19. IPI estimation X r (n) f 1 (n) Figure 16: Extended FXLMS algorithm [137].

Reference model
Integrator C
x c (t) x E (t) Controller Figure 18: Block diagram of the NBMCS controller [119].
input current to the actuator, respectively. μ-analysis was used to evaluate the robust stability of the controllers. Using the Hankel-norm approximation method, the order of the H 2 and H ∞ controllers can be set to 16 and 14, respectively, for low cost, easy commissioning, high reliability, and the maintenance of the controller. Simulations were carried out by Fakhari and Ohadi [130] to evaluate the effectiveness of an AEM in vibration suppression of a four-cylinder engine using the closed-loop system with H 2 and H ∞ . Compared with the H ∞ control, the simulation shown in Figure 20 indicates that the H 2 control requires a greater control effect to achieve the control performances. Meanwhile, robust performance and stability of the closed-loop system are achieved with H 2 and H ∞ .

2DOF Control.
To solve the disadvantage of nonadaptive control and adaptive control, researchers have studied 2DOF controllers, such as integrating open-and closed-loop control methods for AEMs [157]. In the case of perturbations, such as disturbance, noise, and unmodelled dynamics, the conventional adaptive control system may become unstable, while the robust approaches are still robust [155,156]. When there are nonparametric uncertainties, the robust adaptive strategy can conquer the behaviour of the adaptive controller. e robust adaptive controller can overcome the conservative behaviour of the robust controller.
By selecting a proper reference model, the robust model reference adaptive control (MRAC) method based on the modified gradient method was proposed by Ioannou et al. [135] and can be expressed as where θ → is the controller parameters and ω → � [ ω →T 1 ω →T 2 y p r] is the state variables of the controller. e adaptive law can be expressed as where c > 0.
[Γ] and c denote the adaptive gains. sgn and ρ are the sign function and the estimate of ρ * , respectively. ρ * can be expressed as ρ * � k p /k m . Figure 19: e closed-loop system [110]. An experimental setup was carried out by Ioannou et al. [135] to evaluate the vibration performance of an AEM with the proposed robust MRAC. Tests show that the proposed robust MRAC provides a better control performance than the P control with a phase shift in the case of large uncertainties. However, in the case of no uncertainties, the P control with a phase shift provides a better control performance than the proposed robust MRAC in a certain high frequency range.
To control the vibrations of the engine-chassis system, Yang [118] proposed a 2DOF control strategy applied on an AEM, which is formed of an H ∞ robust feedback controller and a FXLMS feedforward controller. e weight is represented as W(n + 1) � W(n) + μE(n)X c (n), (35) where X c (n) is the filtered input and E(n) is the error. e robust feedback controller is designed by μ-synthesis. To achieve control through a personal computer, the order of the robust controller was reduced from 23 to 7, which attenuated the vibration from the engine to the chassis, and robustness was achieved. As shown in Figure 21, open-loop control is designed based on the manifold absolute pressure and crank speed. Based on the single-tone adaptive feedforward control, closed-loop control is designed. e update algorithm in discrete-time implementation is expressed as u c,new � u c,old − cG * (jpω)Qr c,old .
Simulations show that fast response times and robustness to vehicle variations can be achieved at the same time by integrating open-loop and closed-loop controls.

Conclusions
Based on the above review, the difficulties and trends in the research of modelling and controlling AEMs are summarized as follows: (1) Most theoretical models are based on assumptions that are not always consistent with the working conditions of AEMs. e relative displacements (actuator, engine attachment point, and chassis attachment point), the frequency-dependent and amplitude-dependent characteristics of the elastomeric stiffness, and the damping should be considered. Similarly, theoretical models are oversimplified and do not consider complex interactions between the preload and the mass of the chassis, which can yield a vibration solution for one AEM fitted to different vehicles equipped with different engines.
(2) e method of AEM identification should be proposed to verify the above complex theoretical models and finite-element models of AEMs. Meanwhile, the effects of the AEM parameters (structural parameters and performance parameters) on the AEM dynamics should be carried out to design, manufacture, and control AEMs. (3) e proposed closed-loop control system of AEM must be stable or robust in the case of high uncertainty in complex and variable environments, such as disturbance, noise, and unmodelled dynamics. e implementation of some controllers is sometimes costly and time-consuming. us, the demand of AEM control is the development and application of more advanced control approaches based on the principles of stability, robustness, intelligence, and optimality. (4) e goal of the next generation of electromagnetic AEMs will be a compromise among a small volume, light weight, large actuator force, low cost, intelligence, and effectiveness under all working conditions.

Conflicts of Interest
e authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Shock and Vibration 15