Hybrid Model Predictive Control of Semiactive Suspension in Electric Vehicle with Hub-Motor

applied sciences

Article Hybrid Model Predictive Control of Semiactive Suspension in Electric Vehicle with Hub-Motor

Hong Jiang 1,*, Chengchong Wang 1, Zhongxing Li 2 and Chenlai Liu 2

1 School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China; [email protected] 2 School of Automotive and Trafﬁc Engineering, Jiangsu University, Zhenjiang 212013, China; [email protected] (Z.L.); [email protected] (C.L.) * Correspondence: [email protected] or [email protected]

Abstract: In hub-motor electric vehicles (HM-EVs), the unbalanced electromagnetic force generated by the HM will further deteriorate the dynamic performance of the electric vehicle. In this paper, a semiactive suspension control method is proposed for HM-EVs. A quarter HM-EV model with an electromechanical coupling effect is established.The model consists of three parts: a motor model, road excitation model and vehicle model. A hybrid model predictive controller (HMPC) is designed based on the developed model, taking into account the nonlinear constraints of damping force. The focus is on improving the vertical performance of the HM-EV. Then, a Kalman ﬁlter is designed to provide the required state variables for the controller. The proposed control algorithm and constrained optimal control (COC) algorithm are simulation compared under random road excitation and bump road excitation, and the results show that the proposed control algorithm can improve ride comfort, reduce motor vibration, and improve handling stability more substantially.

Keywords: semiactive suspension; hub motor; electric vehicle; hybrid model predictive control

Citation: Jiang, H.; Wang, C.; Li, Z.; 1. Introduction Liu, C. Hybrid Model Predictive Due to increased pressure for environmental protection and energy shortages, elec- Control of Semiactive Suspension in tric vehicles’ development is being highly valued by many research institutions [1]. Elec- Electric Vehicle with Hub-Motor. tric vehicles are environmentally friendly and excel in terms of sporting performance Appl. Sci. 2021, 11, 382. https://doi. due to the fast and precise torque of the electromotor [2,3]. Hub-motor electric vehicles org/10.3390/app11010382 (HM-EVs) have the advantages mentioned above and eliminate the need for a powertrain,

Received: 7 December 2020 improving transmission efficiency and space utilization, making power control easier to Accepted: 29 December 2020 achieve [4,5]. Thanks to its unique technical advantages, HM-EV has received unprece- Published: 2 January 2021 dented attention and development. It has become one of the focuses and hotspots of electric vehicle technology research and has become a unique direction for electric vehicles’ devel- Publisher’s Note: MDPI stays neu- opment. The industry also refers to the HMs as the final drive form of electric vehicles [6]. tral with regard to jurisdictional clai- However, because the HM-EV highly integrates the motor, reduction mechanism, ms in published maps and institutio- and brake in the wheel, it increases the unsprung mass of the vehicle and deteriorates the nal affiliations. vertical performance of the vehicle [7]. Additionally, tire bounce, support shaft bending, bearing wear caused by the excitation of the road will cause the stator and rotor to be eccentric and cause uneven distribution of the motor air gap along the circumference [8,9]. The unbalanced electromagnetic force generated by the uneven air gap will be transmitted Copyright: © 2021 by the authors. Li- to the wheels through the reduction mechanism or directly, which is equivalent to adding censee MDPI, Basel, Switzerland. a certain external load to the wheels, thus affecting the dynamic characteristics of the This article is an open access article suspension [10]. To solve these problems, many scholars have suppressed various adverse distributed under the terms and conditions of the Creative Commons At- effects of HMs on vehicles from both structural and control perspectives. Nagaya et al. [7] tribution (CC BY) license (https:// used special springs and dampers to connect the motor to the unsprung mass, to turn creativecommons.org/licenses/by/ the motor into a dynamic vibration absorber (DVA) for the unsprung mass, reducing the 4.0/). vibration of the sprung mass and improving the smoothness of the vehicle ride, as well

Appl. Sci. 2021, 11, 382. https://doi.org/10.3390/app11010382 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 382 2 of 23

as reducing the input to the motor vibration. Luo et al. [11] proposed a new type of HM structure with rubber bushing between the stator and rotor. Through the comparison and simulation of two HM schemes with bushing and without bushing, it was concluded that the vertical performance of the motor with bushing was significantly improved. Qin et al. [12] proposed a new method of vibration damping based on the DVA structure, and the simulation results show that the ride comfort and handling stability were effectively improved after adopting the new DVA structure. Finally, a multibody simulation was carried out using muti-body dynamic software (LMS motion) to verify the feasibility of the proposed DVA structure. There are many scholars already working on suppressing the adverse effects caused by HM from an active suspension perspective. Liu et al. [13] proposed an optimal control strategy for electric vehicle wheels, which optimizes the parameters of the DVA structure according to the typical working conditions, and then adopts the fuzzy control method and linear quadratic regulator (LQR) method to control the actuator force of the DVA structure and suspension force, respectively, to reduce the impact on the HM and improve the ride comfort. Shao et al. [14] present an HM-EV active suspension fault-tolerant fuzzy H∞ controller that takes into account sprung mass variation, actuator faults, and control input constraints, and verifies the effectiveness of the controller by simulation. He et al. [15] proposed a fuzzy optimal sliding mode control method using normalization and hierarchical analysis to select weighting coefficients, and verified the superiority of using this control method for electric vehicles by simulation. Li et al. [16] proposed a multiobjective optimization control method for active suspension, using a particle swarm algorithm to generate the optimal parameters of the active suspension and the simulation results show that the optimized active suspension can effectively reduce the unbalanced electromagnetic force while simultaneously turning the motor eccentricity so it is kept within a reasonable range. Wu et al. [17] proposed an H2 active suspension control scheme with good robustness, and the simulation results show that the suspension system has a satisfactory control effect under complex electromagnetic force excitation and dynamic perturbation. Shao et al. [18] proposed a hybrid controller consisting of an H∞ suspension controller and a switched reluctance motor controller, which consists of current chopping control and pulse width modulation control, and simulation results show that the proposed control method can effectively reduce the unbalanced electromagnetic force and air gap eccentricity. Liu et al. [19] proposed a two-stage optimization method based on the mathematical model of a quarter vehicle with active suspension equipped with a dynamic vibration absorber. Firstly, an LQR controller was designed to suppress the vibration of the wheel motor. Then, a controller with finite frequency H∞ was designed to improve the ride comfort of the vehicle. Although the active suspension exhibits an excellent performance, its structure is complex and expensive, limiting the promotion of its application. The emergence and mature development of controllable dampers has led to a greater focus on semiactive suspension. Xu et al. [20] proposed a hybrid controller based on the hybrid acceleration drive damping algorithm and used the multiobjective optimization method to determine the parameters of the controller, and finally the simulation was carried out under the random road excitation and bump excitation simulation; the results show that the proposed control method can effectively reduce the vibration of the sprung mass and motor. Mauricio et al. [21] used four different semiactive suspension controllers in an HM-EV and evaluated and compared the whole vehicle to show that it is feasible to use four different semiactive suspension controllers for the vehicles. From the available research, it can be seen that there are few studies on semiactive suspension in HM-EVs. This research proposes a damping control method for the HM-EV semiactive suspension to improve the ride comfort, handling stability and reduce the eccentricity between the stator and rotor. The primary contributions of this study are as follows: (1) developed a hub motor direct drive-air suspension cooperative system; (2) designed a hybrid model predictive controller to suppress the vibration of the entire system; (3) designed a Kalman filter to provide state variables that cannot be measured for the hybrid model predictive controller. Appl. Sci. 2021, 11, 382 3 of 23

The rest of this article is organized as follows: In the second section, the establishment of the 1/4 cooperative system model is introduced. Then, the control problem is proposed in the third section. To address this problem, a hybrid model predictive controller and a Kalman ﬁlter are designed. In the ﬁfth section, a numerical simulation is performed to verify the effectiveness of the proposed method and summarize the conclusions.

2. Electromechanical Coupling Model HM-EVs are complex electromechanical coupling systems, which includes HM, suspension, and wheels. In this research, an integrated model of a quarter HM-EV is established, which contains three parts: the permanent magnet brushless direct current (PMBLDC) motor model, which provides an unbalanced electromagnetic force to the wheels and suspension; vehicle model, which emphasizes the vehicle dynamic response; road excitation model, which provides external excitation to the system.

2.1. BLDC Motor Model In recent years, the PMBLDC motor has received more interest in automotive applications due to the higher reliability, lower maintenance and quieter operation the BLDC motor has compared to its brushed DC counterpart [22]. This paper uses an outer rotor PMBLDC motor as the hub drive motor. The current equation of the outer rotor PMBLDC is as follows:            ua R 0 0 ia L − M 0 0 ia ea  ub  =  0 R 0  ib  +  0 L − M 0 pop ib  +  eb  (1) uc 0 0 R ic 0 0 L − M ic ec

where ua, ub, and uc represent the three-phase winding phase voltage, R is the stator phase resistance, L is the winding inductance, M is the mutual inductance, pop is the differential operator, ia, ib, and ic represent the three-phase winding phase current, and ea, eb, and ec represent the three-phase winding back electromotive force. According to Equation (1), the electromagnetic torque equation and mechanical motion equation of motor are as follows:

1 Te = (eaia+ebib+ecic) (2) ωn dω J n +Bω = (T −T ) (3) n dt n e L

where Te is the electromagnetic torque of the motor, TL is the load torque of the motor, Jn is the moment of inertia of the motor, B is the friction coefficient, and ωn is the angular velocity of the rotor. Due to the complex installation conditions and mechanical environment of the hub motor on an electric vehicle, the motor is under vertical load during operation, which causes the motor stator-rotor to be eccentric and thus generates unbalanced electromagnetic force. The motor noneccentricity and eccentricity are shown in Figure1. The air gap magnetic field of a PMBLDC motor includes the permanent magnet magnetic field and the armature reaction magnetic field, which can be added together by the linear superposition method to obtain the radial air gap magnetic field and the tangential air gap magnetic field of a PMBLDC motor—the equations are as follows [23]:

Ber(r, α, t) = [Bmr(r, α, t)+Bar(r, α, t)]εδ (4)

Bet(r, α, t) = [Bmt(r, α, t)+Bat(r, α, t)]εδ (5)

where r is the radius at the intended position, α is the stator angle, t is time, Ber is the radial air gap magnetic field in the eccentric state, Bmr is the radial magnetic field of the permanent magnet, Bar is the radial magnetic field of armature reaction, Bet is the tangential air gap magnetic field in the eccentric state, Bmt is the tangential magnet field of the permanent Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 25

Bet(r,α,t)=[Bmt(r,α,t)+Bat(r,α,t)]εδ (5)

where r is the radius at the intended position, α is the stator angle, t is time, Ber is the radial air gap magnetic field in the eccentric state, Bmr is the radial magnetic field of the permanent magnet, Bar is the radial magnetic field of armature reaction, Bet is the tangential air Appl. Sci. 2021, 11, 382 4 of 23 gap magnetic field in the eccentric state, Bmt is the tangential magnet field of the permanent magnet, Bat is the tangential magnetic field of armature reaction, and εδ is the correction coefficient of magnetic conductivity in the eccentric state. magnet, Bat is the tangential magnetic ﬁeld of armature reaction, and εδ is the correction coefﬁcient of magnetic conductivity in the eccentric state.

Z Stator Z Stator

O’ e O X O X w w

Rotor Rotor

(a) (b)

FigureFigure 1. 1. EccentricityEccentricity diagram diagram of statorof stator and rotor.and rotor. (a) Noneccentricity. (a) Noneccentricity. (b) Eccentricity. (b) Eccentricity.

Finally, the vertical and longitudinal unbalanced electromagnetic forces caused by the eccentricityFinally, betweenthe vertical the stator and and longitudinal rotor of the motor unba arelanced calculated electromagnetic as follows: forces caused by the eccentricity between the stator and rotor of the motor are calculated as follows:  Lr R 2πnh 2 2i o  Fez = 2µ 0 Ber(r, α, t) −Bet(r, α, t) sinα + 2[Ber(r, α, t) · Bet(r, α, t)]cosα dα Lr 0 2π (6) Lr R 2πnh 22 22i o ⎧F F=ex = B Ber(r(,rα, α,,tt)) -−BBet((rr,αα,,t) cossinα −α2+2[Ber[B(r, α(,rt),α· ,Btet)⋅(Br, α,(tr)],αsin,tα)]dcosα αdα ez 2µ0 0 er et er et ⎪ 2µ 0 0 where L is the axial2π length of the motor and µ0 is vacuum permeability. (6) ⎨ Lr 2 2 ⎪ Fex= Ber(r,α,t) -Bet(r,α,t) cos α -2[Ber(r,α,t)⋅Bet(r,α,t)] sin αdα 2.2. Quarter2 Vehicleµ Model ⎩ 0 0 Despite its simplicity, the quarter vehicle model is widely used in vertical dynamics whereanalysis L is and the controller axial length design of [24 the]. Figure motor2 depicts and µ the0 is physical vacuum structure permeability. and equivalent model of the quarter vehicle model for HM-EV. The dynamics equations can be described 2.2.according Quarter to Vehicle Newton’s Model law of motion as follows: Despite its simplicity, the quarter · · vehicle model is widely used in vertical dynamics .. ( − ) cs zs−zus  z = ks zs zus − − udam analysis and controllers m designs [24].m Figures 2m depictss the physical structure and equivalent  · · model of the quarter.. vehicle( − ) modelcs z sfor−zus HM-EV.( The− )dynamicsF equations can(7) be described z = ks zs zus + − kb zus zur + ump + udam according to Newton’sus mlawus of motionmus as follows:mus mus ms   .. kb(zus−zur) kt(zur−q) Fump  zur = − − ksm(urzs-zus) cs(mzsur-zus) umdamur ⎧ zs=- - - ⎪ .. ms ms ms · where ms is sprung⎪ mass, zs is vertical acceleration of sprung mass, zs is vertical velocity k (z -z ) c (z -z ) k (z -z ) Fump u of sprung mass, zs is verticals s us displacements s us of sprungb us ur mass, mus is statordam and residual ..zus= + - + + · (7) unsprung mass,⎨zus is verticalmus accelerationmus of statormus and residualmus unsprungms mass, zus is vertical velocity⎪ of statork and(z residual-z ) unsprungk z -q mass,F zus is vertical displacement of stator ⎪ b us ur t ur ump .. and residual unsprungzur= mass, mur is rotor,- rim and - tire mass, z ur is vertical acceleration of ⎩ · mur mur mur rotor, rim and tire mass, zur is vertical velocity of rotor, rim and tire mass, zur is vertical displacement rotor, rim and tire mass, cs is damping coefﬁcient, ks is spring stiffness, kb is where ms is sprung mass, zs is vertical acceleration of sprung mass, zs is vertical velocity spring stiffness, kt is tire stiffness, q is road excitation and udam is damping force. of sprung mass, zs is vertical displacement of sprung mass, mus is stator and residual unsprung mass, zus is vertical acceleration of stator and residual unsprung mass, zus is vertical velocity of stator and residual unsprung mass, zus is vertical displacement of stator and residual unsprung mass, mur is rotor, rim and tire mass, zur is vertical acceleration of rotor, rim and tire mass, zur is vertical velocity of rotor, rim and tire mass, zur is vertical displacement rotor, rim and tire mass, cs is damping coefficient, ks is spring stiffness, kb is spring stiffness, kt is tire stiffness, q is road excitation and udam is damping force. Appl. Sci. 2021, 11, 382 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 23 5 of 25

Suspension z ms s Tire

Wheel rim ks cs Permanent magnet Electrically zus controlled valve Electronic control mus Bearing ball kb Fump z m ur Brake disc Stator ur k Rotor t q

(a) (b)

FigureFigure 2. Hub-motor2. Hub-motor electric electric vehiclevehicle (HM-EV) (HM-EV) system: system: (a)( physicala) physical structure; structure; (b) equivalent (b) equivalent model. model.

The air suspensionThe air has suspension a low frequency has a low of frequency vibration, of canvibration, change can the change height the of height the of the vehicle and canvehicle improve and control can improve stability control and stability ride comfort and ride [25 comfort]. Therefore, [25]. Therefore, in this paper, in this paper, air springs have been chosen as the elastic element of the suspension system. Hengjia air springs have been chosen as the elastic element of the suspension system. Hengjia states states that the differential equation for the change in air pressure in an air spring with time that the differentialcan be equation expressed for as the [26]: change in air pressure in an air spring with time can be expressed as [26]: κp· 𝑞m κp Vas · as - as (8) κpasqm pκasp=asVas pas = − mas Vas (8) mas Vas Assuming that the air pressure, volume and air mass changes inside the air spring Assuming thatare much the air less pressure, than its volumeinitial state, and the air above mass equation changes can inside be linearized the air spring as: are much less than its initial state, the above equation can be linearized as: κp 𝑞m κp Vas as0 - as0 (9) pas= · mas0 Vas0 · κpas0qm κpas0Vas pas = − (9) Xinbo indicates that themas air0 springV modelas0 can be ultimately represented as [27]: p A (z +z -z )=κRT𝑞 -κp A (z -z ) Xinbo indicates that the air spring modelas as canas0 s beus ultimatelym representedas as s us as [27]: (10)

·Considering that the rate of change in air pressure · · and suspension dynamic deflec- tionp hasas A asa (largez as0 +influencezs−zus) on = κtheRTq dynamicm−κpas characteristicsAas zs − zus of the system and(10) therefore is retained, and the current air pressure and current suspension dynamic deflection are re- Consideringplaced that the by the rate initial of change air pressure in air pressureand initial and suspension suspension dynamic dynamic deflection, deflection the linearized has a large influenceair spring on the model dynamic can be characteristics represented as: of the system and therefore is retained, and the current air pressure and current suspension dynamic𝑞 - deflection - are replaced by pasAaszas0=κRT m κpas0Aas(zs zus) (11) the initial air pressure and initial suspension dynamic deflection, the linearized air spring model can be representedwhere pas is as: the air pressure of the air spring, Vas is the volume of the air spring, Vas0 is the initial volume of the air spring, mas is the air mass in the air spring, mas0 is the initial air mass of the· air spring, κ is the air polytropic · exponent,· R is the perfect gas constant, T is p Aaszas0= κRTqm−κp Aas zs − zus (11) the air temperature,as zas0 is the initialas 0air spring displacement, Aas is the air spring cross- section area (assuming it remains constant) and qm is mass flow of air flowing into the air where pas is the airspring. pressure of the air spring, Vas is the volume of the air spring, Vas0 is the initial volume of theThe air air spring, springm isas notis theinflated air mass or deflated in the in air this spring, paper,m soas0 theis air the spring initial model air can be mass of the air spring,representedκ is the as: air polytropic exponent, R is the perfect gas constant, T is the air temperature, zas0 is the initial air spring displacement, - Aas is the - air spring cross-section pasAaszas0= κpas0Aas(zs zus) (12) area (assuming it remains constant) and qm is mass flow of air flowing into the air spring. The air spring is not inflated or deflated in this paper, so the air spring model can be represented as: · · · pas Aaszas0= −κpas0 Aas zs − zus (12) Appl. Sci. 2021, 11, 382 6 of 23

From the above equation, it follows that

2 pas Aas= −κpas0 Aas (z s−zus)/Vas0 (13)

The gas spring stiffness can therefore be expressed as:

2 ks= κpas0 Aas /Vas0 (14)

As the main components of the suspension system, dampers have a significant effect on the smoothness and handling stability of the vehicle. The damping coefficient of traditional dampers is generally fixed, making it difficult to meet the performance requirements of vehicles under different working conditions. In this paper, a controllable dampener was used, which can achieve a constantly adjustable damping coefficient by adjusting the opening of the damping valve. The controllable damper force Fcs can be expressed as a function of the control current i and the relative velocity of the suspension. The formulas are described as follows [28]: · · Fcs= f (i, xs − xus (15)

Many methods can be used to describe such a function and this paper adopts a nonparametric model [29], in which the controllable damper force is expressed as:

 · · = ( ) −  Fcs A i Sb xs xus   k  n A(i) = ∑ ani (16) n=0  · ·  −b |xs−xus|  · · · · ( 0 )  V0  Sb xs − xus = sgn xs − xus [1 − e ]

where A(i) is the maximum damping force at a given control current, an is a polynomial parameter, k is the largest order of the polynomial, Sb is a shape function, b0 and V0 are the shape function parameter. The parameters are shown in Table1[ 28] with the force-velocity characteristics depicted in Figure3.

Table 1. Parameters for damping force equation.

. . . . Tension (xs−xus≥0) Compressed (xs−xus<0) + − a0 4002.72 a0 −2002.45 a+ −1567.91 a− 801.58 1+ 1− b0 3.41 b0 9.48 + − V0 1.31 V0 3.38

Think of the damping coefﬁcient of a system as a combination of uncontrollable cmin and controllable parts [0, cmax-cmin]. According to JASOC602:2001 [30], the damping force when the piston speed is 0.31 m/s represents the equivalent damping force, and the corresponding equivalent damping coefﬁcient can be obtained accordingly. For a controllable damper, the nonadjustable part of the damper is always at work, so it is assumed that cs=cmin. Parameters of the quarter HM-EV model are shown in Table2. Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 25 Appl. Sci. 2021, 11, 382 7 of 23

4000 0.2A 3000 0.4A 0.6A 0.8A 2000 1A 1.2A

1000

-1000

-2000 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 velocity(m/s) FigureFigure 3. The force-velocityforce-velocity characteristics characteristics of controllable of controllable damper. damper.

ThinkTable of 2.theParameters damping of quarter coefficient HM-EV of model. a system as a combination of uncontrollable cmin and controllable parts [0, cmax-cmin]. According to JASOC602:2001 [30], the damping Parameters Value Parameters Value force when the piston speed is 0.31 m/s represents the equivalent damping force, and the m m Sprung mass, s (kg)corresponding equivalent 335 damping Stator and coefficient residual unsprung can be mass,obtainedus (kg) accordingly. 57.5 For a control- Rotor, rim and tire mass, mur (kg) 66.5 Tire rolling radius, Rt (m) 0.256 6 5 Bearing stiffness, kb (N/m)lable damper,5 the× 10 nonadjustable part Tireof the stiffness, damperkt (N/m) is always at work,2.5 so× it10 is assumed 2 Initial air spring sectional area,thatA0 (m cs=)cmin. Parameters0.009 of theMaximum quarter HM-EV damping coefficient,model arecmax shown(Ns/m) in Table 2 9000 5 Initial relative air spring pressure, P0 (pa) 4.6478 × 10 Minimum Damping coefficient, cmin (Ns/m) 1000 5 Atmospheric pressure, Pa (pa) Table1 2.× Parameters10 of quarterAir HM-EV polytropic model. exponent, κ (-) 1.4 3 Initial air spring volume, V0 (m ) 0.0024 Parameters Value Parameters Value 2.3. Road Excitation Stator and residual unsprung mass, mus Sprung mass, ms (kg) 335 57.5 The road excitation is the main excitation of the(kg) suspension system, so a suitable road Rotor, rim and tire mass,excitation mur (kg) model is needed 66.5 for control effect Tire rolling verification. radius, The R changet (m) in the height 0.256 of the Bearing stiffness, kbpavement (N/m) relative to 5 × the 10 reference6 surfaceTire along stiffness, the length kt (N/m) of the roadway is commonly 2.5 × 105 referred to as the pavement unevenness function. In this paper, two types of road excitation Initial air spring sectional area, A0 Maximum damping coefficient, cmax were used: random0.009 road excitation and bump road excitation. 9000 (m2) (Ns/m) Random road surface unevenness can be represented by the power spectral density Initial relative air spring pressure, Minimum Damping coefficient (PSD) function [314.6478]. The × PSD 105 of velocity in the spatial frequency domain of the road1000 surface P0 (pa) is shown in the following equation cmin (Ns/m) Atmospheric pressure, Pa (pa) 1 × 105 Air polytropic exponent, κ (-) 1.4 2 Gv(n)= 4π n0Gq(n )v (17) Initial air spring volume, V0 (m3) 0.0024 0

where n is the spatial frequency, n0 is the reference spatial frequency, Gq(n0) is the power 2.3.spectral Road density Excitation at the reference spatial frequency and v is the vehicle velocity. The road speedexcitation signal is is the white main noise excitation with a finite of the bandwidth, suspension so system, the white so noise a suitable can road excitationbe passed throughmodel is a filterneeded to generate for control a time-domain effect verification. signal of the The road change elevation. in the The height filter of the is shown in the following equation pavement relative to the reference surface along the length of the roadway is commonly referred to as the pavement unevenness functionq . In this paper, two types of road excita- 2π Gq(n 0)v tion were used: random road exciG(station) = and bump road excitation. (18) s + f Random road surface unevenness can be2π re0 presented by the power spectral density (PSD) function [31]. The PSD of velocity in the spatial frequency domain of the road surface is shown in the following equation

2 Gv(n)=4π n0Gq(n0)v (17) Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 25

where n is the spatial frequency, n0 is the reference spatial frequency, Gq(n0) is the power spectral density at the reference spatial frequency and v is the vehicle velocity. The road speed signal is white noise with a finite bandwidth, so the white noise can be passed through a filter to generate a time-domain signal of the road elevation. The filter is shown in the following equation

2πGq(n0)v 𝐺(s) = (18) s+2πf0

where 𝑓 is the cutting frequency and 𝑓 = 0.01 HZ. Road excitation at 20 m/s on a Class C road and on a Class B road is shown in Figure 4. Then, the road time-domain signal can be expressed as follows:

( ) ( ) q t =-2πf0q t +2πGq(n0)vw(t) (19)

In order to study the dynamic response characteristics of the HM-EV in this paper introduces bump road excitation as shown in the following equation.

Appl. Sci. 2021, 11, 382 a 2πv l 8 of 23 1-cos t ,0≤t≤ z z(t)= 2 L v (20) l 0 ,t> v where f0 is the cutting frequency and f0 = 0.01 HZ. Road excitation at 20 m/s on a Class where a representsC roadthe height and on aof Class the Bbump, road is l shownrepresents in Figure the4 .length Then, theof roadthe bump time-domain and v signal can represents the vehiclebe expressed velocity. as In follows: this paper, two different heights of bump road were selected as excitations: a = 0.1 m and a = 0.02· m. For other parameters,q v = 1.5 m/s, l = 0.85 ( )= − ( )+ ( ) ( ) m was chosen. Bump road excitation is shownq t in 2Figureπ f 0q t 5. 2π Gq n 0 vw t (19)

0.015

0.01

0.005

-0.005

-0.01 012345678910 t(s) (a) (b)

FigureFigure 4. Road 4. Roadexcitation: excitation: (a) C-class (a) C-class 20 m/s 20 m/s road road excitation; excitation; (b ()b B-class) B-class 20 20 m/s m/s road road excitation.

In order to study the dynamic response characteristics of the HM-EV in this paper introduces bump road excitation as shown in the following equation. ( a 1 − cos 2πv t, 0 ≤ t ≤ l z(t) = 2 Lz v l (20) 0 , t > v l v Appl. Sci. 2021, 11, x FOR PEER REVIEW where a represents the height of the bump, represents the length of the9 bump of 26 and represents the vehicle velocity. In this paper, two different heights of bump road were selected as excitations: a = 0.1 m and a = 0.02 m. For other parameters, v = 1.5 m/s, l = 0.85 m was chosen. Bump road excitation is shown in Figure5.

0.03

0.02

0.01 q (m)

-0.01 012345 t (s) (a) (b)

Figure 5. Bump excitation:Figure (a 5.) aBump = 0.1 excitation:m; (b) a = (0.02a) a =m 0.1. m; (b) a = 0.02 m.

3. Hybrid Model Predictive Controller Design Suspension performance directly affects passenger comfort and vehicle driving safety. Conventional suspensions have difficulty adapting to changes in the road surface, speed and load due to their stiffness and damping, making it difficult to meet the high demands of vehicle performance in changing conditions. This paper proposes a hybrid model predictive controller for the nonlinear constraint of damping forces to improve the overall system vertical performance deterioration due to the HM. For traditional control methods, the optimization problem with state constraints has been a research challenge [32]. Model predictive control (MPC) can handle optimization problems with constraints in a finite time domain. At each sampling time interval, MPC transforms the system control problem into an open-loop optimal control problem solving a finite time domain with a step length of N according to the amount of state of the system x(k) at the current moment, obtains the optimal control variables {u(k), u(k + 1), ..., u(k + N - 1)} within N steps by an online calculation method, applies the first step control amount u(k) to the system and corrects the future output at the next moment according to the error between the system output and the predicted output to realize the closed-loop of the overall rolling optimization. The control procedure is shown in the Figure 6.

Cost function Constraints Disturbance

y (k) ref u(k) Controlled y(k) Optimizer object

Predictive x(k) model MPC controller

Figure 6. Model predictive control (MPC) control procedure.

For linear MPC, the optimization problem can be described by the following equations: Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 25

0.03

0.02

0.01 q (m) q (m)

-0.01 01234 t (s) (a) (b) Appl. Sci. 2021, 11, 382 9 of 23 Figure 5. Bump excitation: (a) a = 0.1 m; (b) a = 0.02 m.

3. Hybrid Model Predictive Controller Design 3. Hybrid Model Predictive Controller Design Suspension performance directly affects passenger comfort and vehicle driving safety.Suspension Conventional performance suspensions directly have affectsdifficulty passenger adapting comfort to changes and vehicle in the road driving surface, safety. speedConventional and load suspensions due to their have stiffness difficulty and adaptingdamping, to making changes it in difficult the road to surface, meet speedthe high and load due to their stiffness and damping, making it difficult to meet the high demands of demands of vehicle performance in changing conditions. This paper proposes a hybrid vehicle performance in changing conditions. This paper proposes a hybrid model predic- model predictive controller for the nonlinear constraint of damping forces to improve the tive controller for the nonlinear constraint of damping forces to improve the overall system overall system vertical performance deterioration due to the HM. vertical performance deterioration due to the HM. For traditional control methods, the optimization problem with state constraints has For traditional control methods, the optimization problem with state constraints has been a research challenge [32]. Model predictive control (MPC) can handle optimization been a research challenge [32]. Model predictive control (MPC) can handle optimiza- problems with constraints in a finite time domain. At each sampling time interval, MPC tion problems with constraints in a finite time domain. At each sampling time interval, transforms the system control problem into an open-loop optimal control problem solving MPC transforms the system control problem into an open-loop optimal control prob- a finite time domain with a step length of N according to the amount of state of the system lem solving a finite time domain with a step length of N according to the amount of x(k) at the current moment, obtains the optimal control variables {u(k), u(k + 1), ..., u(k + N state of the system x(k) at the current moment, obtains the optimal control variables - 1)} within N steps by an online calculation method, applies the first step control amount {u(k), u(k + 1), ..., u(k + N - 1)} within N steps by an online calculation method, applies the u(k)first to step the system control and amount corrects u(k) the to future the system output and at the corrects next moment the future according output to at the the error next betweenmoment the according system output to the and error the between predicted the ou systemtput to output realize andthe closed-loop the predicted of the output over- to allrealize rolling the optimization. closed-loop ofThe the control overall procedure rolling optimization. is shown in Thethe Figure control 6. procedure is shown in the Figure6.

Cost function Constraints Disturbance

y (k) ref u(k) Controlled y(k) Optimizer object

Predictive x(k) model MPC controller

Figure 6. Figure 6. ModelModel predictive predictive control control (MPC) (MPC) control control procedure. procedure. For linear MPC, the optimization problem can be described by the following equations: For linear MPC, the optimization problem can be described by the following equations: N N min J = ∑ y0(k)Qy(k) + ∑ u0(k)Ru(k) k=1 k=1  ( + )= ( )+ ( )  x k 1 Ax k Buu k (21)  y(k)= Cx(k)+D u(k) Subject to u  umin ≤ u(k) ≤ umax  xmin ≤ x(k) ≤ xmax

where Q and R are the corresponding weighting matrixes. The optimization problem described above can be translated into a standard form of linear quadratic programming.

1 T T min 2 x Gx + f x A x = b (22) Subject to e e Ajx ≤ bj

Equation (7) can be expressed in the form of a state-space:

. x(t) = Ax(t) + B u(t) + B ω(t) u ω (23) y(t) = Cx(t) + Duu(t) Appl. Sci. 2021, 11, 382 10 of 23

· · · T with the state variable x = [zs zus zur zs−zus zus−zur zur −q ] , u = [u ], unmeasured dis- dam h · .. T turbance ω = qF and output y = [zs zus−zur kt(z −q)] . ump ur Since MPC solvs the problem in the discrete time domain, it is necessary to discretize the existing continuous time state equations. The equation of state after discretization can be written as follows:

x(k + 1) =A x(k) + B u(k) + B ω(k) d du dω (24) y(k) =Cdx(k) + Dduu(k)

ATs R Ts Aτ T where Ts is the sampling time, Ad= e , Bd = τ e dτ [B u Bw] , Cd= C, Dd= D. For the quarter vehicle model for HM-EVs, the evaluation indicators are usually the .. acceleration of vertical vibration of the sprung mass zs, the eccentricity of the motor zus−zur and tire dynamic load kt(z us −q). The acceleration of the vertical vibration of the sprung mass represents the comfort of the vehicle; the eccentricity of the motor represents the safety of the motor—too large an eccentricity will damage the motor, but also increase the electromagnetic torque fluctuation and deteriorate the comfort and handling stability of the vehicle; the dynamic load of the tire represents the grounding of the tire, and its value is smaller, indicating that the vehicle handling stability is better. Excessive suspension dynamic deflection can lead to an increase in the probability of hitting a limit during its movement, so it is necessary to constrain the suspension dynamic deflection. Suspension limit deflection is typically ±60 mm, and the probability of the suspension hitting the limit is 0.3% when the suspension limit deflection is three times larger than the suspension dynamic travel. Therefore, it is necessary to limit the suspension dynamic deflection to ±20 mm. For the damping force, the direction of the damping force must be the same as the direction of the relative speed of the suspension and must also meet the maximum damping force constraint. The constraints are shown in the following equation:

−20 mm ≤ x4(k) ≤ 20 mm 2 (25) 0 ≤ u(k)(x 1(k) − x2(k)) ≤ (c max−cmin)(x 1(k) − x2(k))

Since both pavement excitation and unbalanced electromagnetic forces are unmeasur- able, the chosen predictive equations are as follows:

x(k + 1) =A x(k) + B u(k) d du (26) y(k) =Cdx(k) + Dduu(k)

The optimization problem can be expressed as the following equation:

N N min J = ∑ y0(k)Qy(k) + ∑ u0(k)Ru(k) k=1 k=1  x(k + 1)= Ax(k)+Buu(k) (27)   y(k)= Cx(k)+Duu(k) Subject to  u(k)(x1(k)−x2(k)) ≥ 0  2 0 ≤ u(k)(x1(k) − x2(k)) ≤ (c max−cmin)(x1 (k)−x2(k))

In the above constraint, the computational equation used for the constraint is related to the logical relationship between state variables, so not only the magnitude of the constraint has to be satisﬁed, but the correct constraint also has to be switched according to the logical relationship between the state variables. It is clear that traditional linear MPC does not handle these types of constraints. In order to better deal with the above constraints, a hybrid model in the damping adjustment process needs to be established. Appl. Sci. 2021, 11, 382 11 of 23

Due to the complex engineering background, there is no unified way to construct hybrid system models for different research fields. The main models currently used to describe hybrid systems are: stepwise structural hybrid system models, hybrid automata models, piecewise affine models, mixed logical dynamical (MLD) models and linear com- plementary models [33]. The MLD model is a modeling framework for hybrid systems proposed by Bemporad and Morari in 1999, which can effectively solve the problem of modeling continuous dynamic processes coupled with discrete events in hybrid systems, and at the same time, with certain theoretical derivations, the MLD model can also be transformed into other model structures of hybrid systems, thus it has a strong generality. The MLD-based model can effectively handle the integrated problem of optimizing control laws for hybrid systems. Therefore, in this paper, the MLD modeling approach was used to model the mixing dynamics in the damping regulation process. The standard form of the MLD model is as follows [34]:

 + = + + +  x(k 1) Ax(k) B1u(k) B2δ(k) B3z(k) y(k)= Cx(k)+D1u(k)+D2δ(k)+D3z(k) (28)  E2δ(k)+E3z(k) ≤ E1u(k)+E3x(k)+E5

where x is the state variable, y is the output variable, u is the system input and δ is the auxiliary variable. x, y, u and δ can be continuous and logical variables. Two logical and continuous auxiliary variables need to be introduced in order to accurately describe the nonlinear constraints.

h · · i δv= 1] ↔ [zs − zus ≥ 0 [δ = 1] ↔ [u ≥ 0] u (29) [δv= 1] → [δu= 1]

[δv= 1] → [δu= 0]

where δv, δu are auxiliary continuous variables.

 · ·  u − (c max−cmin)(x1−x2) (zs − zus ≤ 0 F = · · (30)  −u + (c max−cmin)(x1−x2) (zs − zus > 0

The hybrid system description language (HYSDEL) can be used to translate the above problems into a standard MLD model. The optimization problem can then be transformed into the following form

N N min J = ∑ y0(k)Qy(k) + ∑ u0(k)Ru(k) k=1 k=1  x(k + 1)= Ax(k)+B u(k)+B δ(k)+B z(k) (31)  1 2 3 Subject to y(k)= Cx(k)+D1u(k)+D2δ(k)+D3z(k)  E2δ(k)+E3z(k) ≤ E1u(k)+E3x(k)+E5   q1 0 0 where Q = 0 q2 0 , R =[r] and N is the prediction step. The following vectors 0 0 q3 are deﬁned:

" x(k + 1|k) # " y(k + 1|k) # " u(k|k) # " δ(k|k) # " z(k) # . . . . . X(k + |k) = . Y(k + |k) = . U(k + |k) = . (k)= . zk . 1 . , 1 . , 1 . , ∆δ . , . . x(k + N|k) y(k + N|k) u(k + N − 1|k) δ(k + N − 1|k) z(k + N − 1) Appl. Sci. 2021, 11, 382 12 of 23

Then, the prediction equation of the system can be written as:

 ∼ ∼ ∼ ∼  X(k + 1|k) = Ax(k) + B1U(k) + B2∆δ(k) + B3z(k) ∼ ∼ ∼ ∼ (32)  Y(k + 1|k) = Cx(k) + D1U(k) + D2∆δ(k) + D3z(k)

where  B1 0 0 . . . 0   CB1 D1 0 . . . 0  ...... " A # " CA #  AB1 B1 0 .   CAB1 CB1 D1 .  . .  . .   . .  ...... Ae = ,Ce = Be1 =  . De1 =  .  . . . AB1 B1 0 . CAB1 CB1 0 N N     A CA ......  . . .   . . .  N−2 . . N−2 . . A B1 . 0 CA B1 . D1 N−1 N−2 N−1 N−2 A B1 A B1 ... AB1 B1 CA B1 CA B1 ... CAB1 CB1     B2 0 0 . . . 0 CB2 D2 0 . . . 0  .. .   .. .   AB2 B2 0 . .   CAB2 CB2 D2 . .     .  B =  . .. ,D =  . ..  e2  . AB2 B1 . 0  e2  . CAB2 CB2 . 0   .   .   N−2 . .. ..   N−2 . .. ..   A B2 . . . 0   CA B2 . . . D2  N−1 N−2 N−1 N−2 A B2 A B2 ... AB2 B2 CA B2 CA B2 ... CAB2 CB2     B3 0 0 . . . 0 CB3 D3 0 . . . 0  .. .   .. .   AB3 B3 0 . .   CAB3 CB3 D3 . .     .  B =  . .. , D =  . ..  e3  . AB3 B3 . 0  e3  . CAB3 CB3 . 0   .   .   N−2 . .. ..   N−2 . .. ..   A B3 . . . 0   CA B2 . . . D3  N−1 N−2 N−1 N−2 A B3 A B3 ... AB3 B3 CA B2 CA B3 ... CAB3 CB3 The performance metrics of the system can be further expressed as:

T U ΘU ∼T ∼ J = +xT(k)ΨU+xT(k)C QCx(k) (33) 2 where  ∼ ∼ ∼ ∼ ∼ ∼  D1QD1+R D2QD1 D3QD1 h i h T T T i ∼ ∼ ∼ ∼ ∼ ∼ = T( ) T( ) T( ) = ∼ ∼ ∼ ∼ ∼ ∼ =  U U k ∆δ k S k , Ψ C D C QD C QD , Θ  D1QD2 D2QD2 D3QD2  1 2 3  ∼ ∼ ∼ ∼ ∼ ∼  D1QD3 D2QD3 D3QD3

The constraints can be further translated as follows: ∼ ∼ ΛU ≤ E4x(k) + E5 (34)

where

 −E ...... E ...... E ......      1 2 3 E4 E5  ......  . . Λ =  . .. . . E . . E . , Ee =  . , Ee =  .   2 3  4  .  5  .  . . . E E .... −E1 .... E2 .... E3 4 5

By removing the constant term, the above problem can be transformed into a mixed integer quadratic programming problem, which is mathematically described as follows:

( T JU = min : U ΘU +xT(k)ΨU ∼ 2 ∼ (35) ΛU ≤ E4x(k) + E5 Appl. Sci. 2021, 11, 382 13 of 23

For the mixed-integer quadratic planning problem, the main methods are: the branch delimitation method, genetic algorithm, etc. In this paper, we use the branch delimitation method and use the Gurobi solver to improve the solution speed.

4. HM-EV Semiactive Suspension Kalman Filter Design In the hybrid model predictive controller design, we assumed that all state variables were measurable, but, in reality, not all state variables are measurable. Some noise and interference is inevitable in the input and output measurement data of the system. It is then necessary to reconstruct the system using a state observer and correct the state and output values of the reconstructed system by the error between the measured output and the observer output, so that the state and output values of the observer approximate the real state and output of the system. The equation of state for a multivariate linear system in a stochastic environment, as described by the Kalman ﬁlter, is as follows:

x(k + 1) = Ax(k) + Bu(k) + Gω(k) (36) y(k) = Cx(k) + Du(k) + Fω(k) + v(k)

ω, v are a white process noise and measurement noise, which are subjected to a multivariate Gaussian distribution and are independent of each other, as follows:

p(ω) ∼ N(0, Q) (37) p(v) ∼ N(0, R)

where Q is the process noise covariance matrix and R is the measurement noise covariance matrix. The principle of Kalman ﬁltering is to use the Kalman gain to correct the state prediction so that it approximates the actual value. The state variable can be estimated by the following equation:

xˆ(k + 1) = Axˆ(k)+Bu(k)+L(k)(y (k)−Cxˆ(k)−Du(k)) (38)

where the optimal observer gain L can be calculated by the discrete Riccati equation.

 T T −1  L(k) = (AP(k)C +N (CP(k)C +R  −1  M(k) = P(k)CT(CP(k)C T +R   T T  Z(k)= (I − M(k)C)P(k)(I − M(k)C) +M(k)RM (k)  −1 −1 −1 P(k + 1) = A − NR C )Z(k)(A− NR C )T+Q − NR NT (39)   = T  Q GQG  T T T  R = R + FN + N F +FQF  T  N = G(QF +N)

The Kalman filter minimizes the state estimation error covariance by solving the discrete algebraic Riccati equation to obtain the optimal gain matrix L(k). For an HM-EV semiactive suspension system, the measurable outputs are the acceleration of vertical vibration of the sprung mass and the suspension dynamic deflection. Thus, using these two measurable via the Kalman filter to estimate the state of the entire system. Under the disturbance of white noise, the observer results under C-class road excitation are shown in the figure below. As shown in Figures7–16, for each state variable, the difference between the estimated and actual values of the Kalman filter is very small under different noise disturbances and can meet the control requirements. Appl. Sci.Appl.2021Appl. Sci., Sci.11 2021, 2021 382, ,11 11, ,x x FOR FOR PEER PEER REVIEW REVIEW 1414 of of 25 1425 of 23

Relative error of observation Relative error of observation

(a(a) ) (b(b) )

Figure 7.FigureFigureObserved 7. 7. Observed Observed results results ofresults vertical of of vertical vertical velocity velocity velocity of the of sprungof the the sprung sprung mass. mass. (mass.a) Comparison ( a(a) )Comparison Comparison of observed of of observed observed and actualand and actual actual values. values. values. (b) Relative( b(b) ) RelativeRelative error error between between observed observed and and actual actual values. values. error between observed and actual values. Relative error of observation Relative error of observation Vertical velocity of stator and residual unsprung mass (m/s) unsprung residual and stator of velocity Vertical Vertical velocity of stator and residual unsprung mass (m/s) unsprung residual and stator of velocity Vertical

(a(a) ) (b(b) ) Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 26 Figure 8.FigureFigureObserved 8. 8. Observed Observed results results results of vertical of of vertical vertical velocity velocity velocity of stator of of stator stator and residualand and residual residual unsprung unsprung unsprung mass. mass. mass. (a) Comparison(a(a) )Comparison Comparison of of observedof observed observed and and and actual actualactual values. values. ( b(b) )Relative Relative error error between between ob observedserved and and actual actual values. values. values. (b) Relative error between observed and actual values. Relative error of observation Relative error Relative error of observation Relative error Relative error of observation Relative error Vertical velocity of rotor, rim and tire (m/s) tire and rim rotor, of velocity Vertical Vertical velocity of rotor, rim and tire (m/s) tire and rim rotor, of velocity Vertical Vertical velocity of rotor, rim and tire (m/s) tire and rim rotor, of velocity Vertical

(a) (b)

FigureFigure 9. Observed 9. Observed results results of verticalof vertical velocity velocity of of rotor,rotor, rim and and tire. tire. (a) (Comparisona) Comparison of observed of observed and actual and values. actual ( values.b) Relative error between observed and actual values. (b) Relative error between observed and actual values. Motor eccentricity (m) eccentricity Motor Relative error of observation

(a) (b) Figure 10. Observed results of motor stator eccentricity. (a) Comparison of observed and actual values. (b) Relative error between observed and actual values. Tire deflection (m) deflection Tire Relative error of observation

Appl.Appl. Sci. Sci. 2021 2021, ,11 11, ,x x FOR FOR PEER PEER REVIEW REVIEW 1515 of of 25 25

(a(a) ) (b(b) ) Appl. Sci. 2021, 11, 382 15 of 23 FigureFigure 9. 9. Observed Observed results results of of vertical vertical velo velocitycity of of rotor, rotor, rim rim and and tire. tire. ( a(a) )Comparison Comparison of of observed observed and and actual actual values. values. ( b(b) ) RelativeRelative error error between between observed observed and and actual actual values. values. Motor eccentricity (m) eccentricity Motor Motor eccentricity (m) eccentricity Motor Relative error of observation Relative error of observation

(a(a) ) (b(b) ) Figure 10. Observed results of motor stator eccentricity. (a) Comparison of observed and actual values. (b) Relative error FigureFigure 10. Observed 10. Observed results results of motor of moto statorr stator eccentricity. eccentricity. ( a(a)) Comparison Comparison of of observed observed and and actual actual values. values. (b) Relative (b) Relative error error betweenbetween observed observed and and actual actual values. values. between observed and actual values. Tire deflection (m) deflection Tire Tire deflection (m) deflection Tire Relative error of observation Relative error of observation

(a(a) ) (b(b) ) Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 25 Figure Figure 11.FigureObserved 11. 11. Observed Observed results results results of tire of of tire deﬂection.tire deflection. deflection. (a )( a( Comparisona) )Comparison Comparison ofof of observedobserved observed and and actual actual actual values. values. values. ( b(b) ()Relative bRelative) Relative error error error between between between observedobservedobserved and actual and and actual actual values. values. values. UnderUnder the the disturbance disturbance of of grey grey noise, noise, the the ob observerserver results results under under C-class C-class road road excita- excita- 0.25 4% observed value tiontion are are shown shown in in the the figure figure below. below. 0.2 actual value 2% 0.15

0.1 0% 0.05

0 -2%

-0.05 -4% -0.1

-0.15 -6% Vertical velocity of sprung mass velocity (m/s) Vertical -0.2

-0.25 -8% 012345 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t (s) t (s) (a) (b)

Figure 12.FigureObserved 12. Observed results results of vertical of vertical velocity velocity of of the the sprungsprung mass. mass. (a) ( aComparison) Comparison of observed of observed and actual and actualvalues. values.(b) Relative error between observed and actual values. (b) Relative error between observed and actual values.

0.5 observed value actual value

0.3

0.1

-0.1

-0.3 Relative error of observation of error Relative

-0.5 Vertical velocity of stator and residual unsprung mass (m/s) mass unsprung residual and stator of velocity Vertical 012345 t (s) (a) (b) Figure 13. Observed results of vertical velocity of stator and residual unsprung mass. (a) Comparison of observed and actual values. (b) Relative error between observed and actual values. Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 25

0.25 4% observed value 0.2 actual value 2% 0.15

0.1 0% 0.05

0 -2%

-0.05 -4% -0.1

-0.15 -6% Vertical velocity of sprung mass velocity (m/s) Vertical -0.2

-0.25 -8% 012345 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t (s) t (s) (a) (b) Appl. Sci. 2021Figure, 11, 382 12. Observed results of vertical velocity of the sprung mass. (a) Comparison of observed and actual values. (b) 16 of 23 Relative error between observed and actual values.

0.5 observed value actual value

0.3

0.1

-0.1

-0.3 Relative error of observation of error Relative

-0.5 Vertical velocity of stator and residual unsprung mass (m/s) mass unsprung residual and stator of velocity Vertical 012345 t (s) (a) (b) Appl.Appl. Sci. Sci. 2021 2021, 11, 11, x, xFOR FOR PEER PEER REVIEW REVIEW 1717 of of 25 25 Figure Figure 13. Observed 13. Observed results results of vertical of vertical velocity velocity of of stator statorand and residualresidual unsprung unsprung mass. mass. (a) (Comparisona) Comparison of observed of observed and and actual values. (b) Relative error between observed and actual values. actual values. (b) Relative error between observed and actual values.

0.60.6 observedobserved value value actualactual value value 0.40.4

0.20.2

0 0

-0.2-0.2 Relative error of observation of error Relative Relative error of observation of error Relative -0.4-0.4 Vertical velocity of rotor, rim and tire (m/s) rim and of rotor, velocity Vertical Vertical velocity of rotor, rim and tire (m/s) rim and of rotor, velocity Vertical -0.6-0.6 012345012345 t (ts ()s) (a(a) ) (b(b) ) FigureFigure 14.FigureObserved 14. 14. Observed Observed results results results of ofvertical of vertical vertical velocity ve velocitylocity of of rotor, rotor,rotor, rim rim and and and tire. tire. tire. ( a()a )Comparison( Comparisona) Comparison of of observed observed of observed and and actual actual and values. values. actual ( b( values.b) ) (b) RelativeRelativeRelative error error error between between between observed observed observed and and actual actualactual values. values.

10-4-4 10 15%15% observedobserved value value 88 actual value actual value 10%10% 66 5%5% 44 0 22 0

0 0 5%5%

-2-2 -10%-10% -4-4 -15% -6-6 -15%

-8-8 -20% 012345 -20% 012345 00.511.522.533.544.5500.511.522.533.544.55 t (s) t (s) t (s) t (s) (a(a) ) (b(b) ) Figure 15. Observed results of motor eccentricity. (a) Comparison of observed and actual values. (b) Relative error between FigureFigure 15. Observed 15. Observed results results of motor of motor eccentricity. eccentricity. (a ()a Comparison) Comparison of observed observed and and actual actual values. values. (b) Relative (b) Relative error errorbetween between observedobserved and and actual actual values. values. observed and actual values.

20%20% observedobserved value value 0.02 15% 0.02 actualactual value value 15%

10%10%

0.010.01 5%5%

0%0% 0 0 5%5%

-10% -0.01-0.01 -10%

-15%-15%

-0.02-0.02 012345 -20%-20% 012345 00 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3 3.5 3.5 4 4 4.5 4.5 5 5 t (ts ()s) t (s) t (s) Appl. Sci. 2021, 11, x FOR PEER REVIEW 18 of 26 Appl. Sci. 2021 , 11, 382 17 of 23

20% observed value 0.02 actual value 15%

10%

0.01 5%

0% 0 5% Tire deflection (m) Tire

-0.01 -10%

-15% Appl. Sci. 2021, 11, x FOR PEER REVIEW 18 of 25 -0.02 -20% 012345 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t (s) t (s) (a) (b) (a) (b) Figure 16.FigureObserved 16. Observed results results of tire of deflection. tire deflection. (a) Comparison(a) Comparison of of observed observed andand actualactual values. values. (b ()b Relative) Relative error error between between observed and actual values. Figureobserved 16. Observed and actual results values. of tire deflection. (a) Comparison of observed and actual values. (b) Relative error between observed and actual values. 5. Simulation Results Under the disturbance of grey noise, the observer results under C-class road excitation 5. Simulationare shownIn in orderResults the figureto verify below. the effectiveness of the proposed control strategy, a simulation analysis of the control strategy is required. The overall control framework is shown in 5.In Simulation Figureorder 17to Resultsasverify follows: the First,effectiveness the HM-EV of model the proposed inputs its controlmeasurable strategy, sprung amass simulation acceleration and suspension dynamic deflection into the Kalman filter, which estimates analysisIn of order the control to verify strategy the effectiveness is required of. the The proposed overall controlcontrol strategy, framework a simulation is shown in Figure 17the as system’s follows: state First, variables the HM-EVin real-time. model Then, inputs the MLD its modelmeasurable generates sprung the objective mass acceler- analysisfunction of the from control the values strategy of state is variables required. obtained The from overall the controlKalman filter framework in a predictive is shown ationin and Figureiterative suspension 17 processas follows: combineddynamic First, with deflection the a HM-EVreference into modeltrajectory, the Kalman inputs which its filter,is measurablethen whichtransformed estimates sprung into massa the system’sacceleration statestandard variables andform suspension of in a mixedreal-time. integer dynamic Then, quadratic deflection the programMLD into model (MIQP) the Kalman generates problem filter, to beth whichesolved. objective estimatesThen, function fromthe the system’sthe values damping stateof statereverse variables variables model in real-time.generates obtained a Then, controlfrom the thecurrent MLD Kalman to model the filter dampers generates in a basedpredictive the on objective the iterative processfunction optimalcombined from damping the with values forcea reference of stateoutput variables fromtrajectory, the obtained controller which from andis thethen the Kalman relativetransformed filter velocity in ainto predictiveof thea standard suspension. Finally, dampers generate the damping force provided to the model, thus formiterative of a mixed process integer combined quadratic with program a reference (MIQP) trajectory, problem which to is thenbe solved. transformed Then, the into damp- a standardclosing form the loop of a mixedon the entire integer control quadratic strategy. program (MIQP) problem to be solved. Then, ing reverse model generates a control current to the dampers based on the optimal damp- the damping reverse model generates a control current to the dampers based on the optimal ing dampingforce output force outputfrom the from controller the controller and and the the relative relative velocity velocityof of the the suspension. suspension. Finally, Finally, dampersdampers generate generate the the damping damping force force provided to to the the model, model,Road thus Disturbancethus closing closing the loopthe loop on the on the entireentire control control strategy. strategy. i Dampers Reverse IWM-EV damper Model Model u(k)

Reference Trajectory Optimization Road Disturbance MIQP Measurable Output Target Suspension relative velocities y(k) x(k) x1(k)-x2(k) Predictive i Kalman MLD Dampers Reverse IWM-EV damper Filter Iteration Model State variable Model u(k) x(k) HMPC Controller Reference Trajectory Optimization MIQP Measurable Output TargetFigure 17. Hybrid model predictive controller Suspension(HMPC) control relative scheme. velocities y(k) x1(k)-x2(k) For comparisonx(k) purposes, the clipped optimal control (COC) was selected as the Predictive Kalman comparator. InMLD active suspension systems, optimal control can be achieved by offline Iteration calculations or by updating the optimal feedbackState variablematrix online according toFilter driving x(k) conditions. For the semiactive suspension system, the equivalent mathematics is HMPC Controller

FigureFigure 17. 17.HybridHybrid model model predictive predictive controller controller (HMPC)(HMPC) control control scheme. scheme.

ForFor comparison comparison purposes, purposes, the the clipped optimaloptimal control control (COC) (COC) was was selected selected as the as the comparator.comparator. In active In active suspension suspension systems, systems, op optimaltimal control control can can be be achieved achieved by ofﬂineoffline calculations or by updating the optimal feedback matrix online according to driving conditions. For the semiactive suspension system, the equivalent mathematics is described as a complex nonlinear problem due to its controllable force energy dissipation constraint and state-dependent control force time-varying constraint [35]. Typical COC can be obtained as follows: 1 T J = 𝑙𝑖𝑚 x(t)TQx(t)+u(t)TRu(t)+2x(t)TNx(t) (40) → 𝑇 0 where Q is the semipositive matrix and R and N are the positive definite matrixes. The system dynamics model can be represented as

𝑥(t)=Ax(t)+Bu(t) (41)

u(t) can be expressed as linear state feedback.

u(t)=-Kx(t) (42)

where the state feedback gain matrix can be calculated by the following equation: Appl. Sci. 2021, 11, 382 18 of 23

calculations or by updating the optimal feedback matrix online according to driving conditions. For the semiactive suspension system, the equivalent mathematics is described as a complex nonlinear problem due to its controllable force energy dissipation constraint and state-dependent control force time-varying constraint [35]. Typical COC can be obtained as follows: 1 Z T J = lim x(t)TQx(t)+u(t)TRu(t)+2x(t)TNx(t) (40) T→∞ T 0 where Q is the semipositive matrix and R and N are the positive deﬁnite matrixes. The system dynamics model can be represented as

. x(t) = Ax(t) + Bu(t) (41)

u(t) can be expressed as linear state feedback.

u(t) = −Kx(t) (42)

where the state feedback gain matrix can be calculated by the following equation:

K = −R−1BTS · T (43) −S = A S + SA − SBR−1BTS + Q

Thus, the optimal control when unconstrained is as follows:

F = −R−1BTSx(t) (44)

There is a constraint on the damping force of the semiactive suspension, so the ﬁnal output control force after passing the constraint is expressed as

− F = sat(−R 1BTSx(t)) (45)

where sat denotes a damping force constraint. Assuming the vehicle is driving at 20 m/s on a C-class road surface and at 20 m/s on a B-class road surface, the damping coefficient of the passive suspension is 3000 N·s·m−1. In order to make the results comparable, for the selection of the weight coefficients of the controller, the trial-and-error method was used to make an appropriate selection to ensure that the performance of the system could be optimized under the control of the controller. For the weighting coefficients of the hybrid model predictive controller (HMPC) controller, this paper chose Q = diag ([1, 10 × 107, 10 × 104]) and R = 0.001. For the weighting coefficients of the COC controller, this paper chose Q = diag ([2.5 × 104, 2 × 1011, 5 × 109]) and R = 0.005. Figures 18–20 shows vertical acceleration of sprung mass, motor eccentricity and tire dynamic load with a time under 20 m/s of C-class road excitation. Table3 gives the root mean square (RMS) for the suspension performance index under B-class and C-class road. As shown in Figures 18–20 and Table3, both COC and HMPC effectively improve the system performance compared to passive suspension under, but the HMPC can improve the vertical performance of the vehicle to a greater extent in comparison. The improvement of the evaluation index by HMPC is approximately the same for both the C-class road with a larger amplitude and the B-class road with a smaller amplitude, which proves the effectiveness of HMPC. Appl. Sci. 2021, 11, x FOR PEER REVIEW 19 of 25

K=-R-1BTS (43) -S =ATS+SA-SBR-1BTS+Q Thus, the optimal control when unconstrained is as follows:

-1 T F=-R B Sx(t) (44)

There is a constraint on the damping force of the semiactive suspension, so the final output control force after passing the constraint is expressed as

-1 T F=sat(-R B Sx(t)) (45)

where 𝑠𝑎𝑡 denotes a damping force constraint. Assuming the vehicle is driving at 20 m/s on a C-class road surface and at 20 m/s on a B-class road surface, the damping coefficient of the passive suspension is 3000 N·s·m−1. In order to make the results comparable, for the selection of the weight coefficients of the controller, the trial-and-error method was used to make an appropriate selection to ensure that the performance of the system could be optimized under the control of the controller. For the weighting coefficients of the hybrid model predictive controller (HMPC) controller, this paper chose Q = diag ([1, 10 × 107, 10 × 104]) and R = 0.001. For the weighting coefficients of the COC controller, this paper chose Q = diag ([2.5 × 104, 2 × 1011, 5 × 109]) and R = 0.005. Figures 18–20 shows vertical acceleration of sprung mass, motor eccentricity and tire dynamic load with a time under 20 m/s of C-class road excitation. Table 3 gives Appl. Sci. 2021, 11, 382 the root mean square (RMS) for the suspension performance index under B-class and C- 19 of 23 class road. ) 2 Vertical acceleration of sprungVertical mass (m/s

Appl. Sci. 2021, 11, x FOR PEER REVIEW 20 of 25 Appl. Sci. 2021, 11, x FOR PEER REVIEW 20 of 25 FigureFigure 18. 18.Vertical Vertical acceleration acceleration of of sprung sprung mass. mass.

FigureFigure 19. 19. MotorMotor eccentricity. eccentricity. Figure 19. Motor eccentricity. 2500 COC 2500 passive COC HMPC 2000 passive 2000 HMPC 1500 1500 1000 1000 500 500 0 0 -500 -500 Tire dynamic load (N) load dynamic Tire -1000 Tire dynamic load (N) load dynamic Tire -1000 -1500 -1500 -2000 -2000 -2500 012345678910 -2500 012345678910t (s) t (s) Figure 20. Tire dynamic load. FigureFigure 20. 20. TireTire dynamic load. load. Table 3. The RMS values of HM-EV evaluation index. Table 3. The RMS values of HM-EV evaluation index. Road Excitation Performance Index Passive COC HMPC Road Excitation Performance Index Passive COC HMPC Vertical acceleration 1.2261 0.9927 Vertical acceleration 1.3325 1.2261 0.9927 of sprung mass 1.3325 (↑7.36%) (↑ 24.99%) 20 m/s C-class of sprung mass (↑7.36%) (↑ 24.99%) 20 m/s C-class 1.0376 × 10 8.6154 × 10 Motor eccentricity 1.1369 × 10 1.0376 × 10 8.6154 × 10 Motor eccentricity 1.1369 × 10 (↑ 8.73%) (↑ 24.22%) (↑ 8.73%) (↑ 24.22%) 630 607 Tire dynamic load 689 630 607 Tire dynamic load 689 (↑ 8.56%) (↑ 11.9%) (↑ 8.56%) (↑ 11.9%) Appl. Sci. 2021, 11, 382 20 of 23

Table 3. The RMS values of HM-EV evaluation index.

Road Excitation Performance Index Passive COC HMPC 1.2261 0.9927 Appl. VerticalSci. 2021, 11 acceleration, x FOR PEER REVIEW of sprung mass 1.3325 21 of 25 (↑7.36%) ( ↑ 24.99 %) 1.0376 × 10−4 8.6154 × 10−5 20 m/s C-class Motor eccentricity 1.1369 × 10−4 Vertical acceleration 0.6086( ↑ 8.73%) 0.4909 (↑ 24.22%) 0.6618 of sprung mass (↑ 8.03%) 630 (↑ 25.21%) 607 Tire dynamic load 689 5.2015 ×( 10↑8.56% 4.3713) × 10 ( ↑ 11.9%) 20 m/s B-class Motor eccentricity 5.6844 × 10 (↑ 8.49%)0.6086 (↑ 23.10%) 0.4909 Vertical acceleration of sprung mass 0.6618 315 (↑ 8.03%) 305 ( ↑ 25.21%) Tire dynamic load 344 (↑ 8.43%) −(↑5 11.34%) −5 −5 5.2015 ×10 4.3713 ×10 20 m/s B-class Motor eccentricityAs shown in Figures5.6844 18–20 and×10 Table 3, both COC and HMPC effectively improve ( ↑ 8.49%) ( ↑ 23.10%) the system performance compared to passive suspension under, but the HMPC can improve the vertical performance of the vehicle to a greater315 extent in comparison. The305 im- Tire dynamic load 344 provement of the evaluation index by HMPC is approximately( ↑ 8.43%) the same for (both↑ 11.34% the C-) class road with a larger amplitude and the B-class road with a smaller amplitude, which proves the effectiveness of HMPC. Figures 21–23 Figuresshow 21–23 the time-domain show the time-domain transient transien responset response of of passivepassive suspension, suspension, COC, COC, and HMPC forand HM-EV HMPC evaluationfor HM-EV evaluation indexes indexes under under bump bump road road excitation excitation (a (=a 0.1= 0.1m). m).

Appl. Sci. 2021, 11, x FOR PEER REVIEW 22 of 25

Figure 21. Vertical accelerationFigure 21. ofVertical sprung acceleration mass. of sprung mass. Motor eccentricity (m) Motor eccentricity

Figure 22. Motor eccentricity. Figure 22. Motor eccentricity. Tire dynamic load (N) Tire

Figure 23. Tire dynamic load.

In order to better compare the effect of different control methods, peak-to-peak (PTP) values were defined which represent the maximum values minus the minimum values. Table 4 shows the PTP values of each evaluation index under two types of bump road excitation.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 22 of 25

Motor eccentricity (m) Motor eccentricity

Appl. Sci. 2021, 11, 382 21 of 23 Figure 22. Motor eccentricity. Tire dynamic load (N) Tire

Figure 23. Tire dynamic load. Figure 23. Tire dynamic load.

In order to betterIn order compare to better the compare effect the of effect different of different control control methods, methods, peak-to-peak peak-to-peak (PTP) (PTP) values were defined which represent the maximum values minus the minimum values. values were definedTable which 4 shows represent the PTP values the maximum of each evalua valuestion minusindex under the minimum two types values.of bump Tableroad 4 shows the PTP valuesexcitation. of each evaluation index under two types of bump road excitation.

Table 4. The peak-to-peak (PTP) values of HM-EV evaluation indexes.

Road Excitation Performance Index Passive COC HMPC 10.7764 8.9044 Vertical acceleration of sprung mass 13.4729 (↑20.01%) ( ↑ 33.91% ) −4 −5 − 8.0169 × 10 6.1764 × 10 a = 0.1 m Motor eccentricity 10 × 10 4 (↑ 19.83%) (↑ 38.24%) 4685.1 3611.4 Tire dynamic load 6119.2 ( ↑ 23.44%) ( ↑ 40.98%) 2.1553 1.8280 Vertical acceleration of sprung mass 2.6946 (↑ 20.01%) ( ↑ 32.16%) −4 −4 − 1.6034 ×10 1.2476 ×10 a = 0.02 m Motor eccentricity 2.0913 ×10 4 ( ↑ 23.33%) ( ↑ 40.35%) 937 736 Tire dynamic load 1223 ( ↑ 23.39%) ( ↑ 39.82%)

It can be seen from Figures 21–23 and Table4 that HMPC can reduce the amplitude of the evaluation index to a greater extent under bump excitation compared to COC. In addition, the improvement of PTP values under different amplitudes bumps excitation of HMPC so it is basically remains the same, which reﬂects that HMPC can perform well under different amplitude of road excitation, further verifying the effectiveness of HMPC.

6. Conclusions and Discussion In this paper, we proposed a semiactive suspension control method that can consider the nonlinear constraint of damping force to suppress the vibration of the sprung mass and motor in order to address the deterioration of the vertical performance of HM-EV. A state observer was designed based on the Kalman ﬁlter algorithm for HM-EV to meet the control needs of the controller. The main conclusions are as follows: (1) The state observer designed based on the Kalman ﬁlter algorithm can effectively accomplish the observation task. Simulations were performed to verify the observation effect under white noise random roads and bump roads, and the results show that the errors between the observed and actual values are small under both road excitations. Appl. Sci. 2021, 11, 382 22 of 23

(2) The proposed control algorithm can effectively improve the vertical performance of the vehicle. Simulation analysis was carried out and compared with the COC control algorithm under different magnitudes of random road excitation and bump road excitation. The results show that the proposed control algorithm can effectively reduce the sprung mass and motor vibration regardless of the excitation magnitude and outperforms the COC control algorithm. The effectiveness of the proposed control algorithm was veriﬁed. The COC control method ignores the constraints in the optimization calculation and then feeds back to the system after limiting the calculated forces according to the current state, which destroys the optimality of the control. The HMPC control algorithm was able to consider the nonlinear constraints of the damping force directly in the controller, but this also greatly increased the computational effort of the controller and increased the computational time. The HMPC algorithm requires online optimization calculations which increases the actual workload and computation time of the control system and violates the real-time requirements of the control during the operation of the actual vehicle. In order to shorten the computation time, the online computation process of the control law can be transformed into a simple ofﬂine table look-up process using multiparameter planning techniques so that the proposed control algorithm can be applied to the real vehicle. This is one of the future research directions.

Author Contributions: Funding acquisition, Z.L.; Methodology, H.J.; Project administration, H.J.; Software, C.W.; Validation, H.J.; Writing—original draft, C.W.; Writing—review and editing, C.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of china, grant number 51975254. Institutional Review Board Statement: Not applicable for studies not involving humans or animals. Informed Consent Statement: Not applicable for studies not involving humans. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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