# Echo State Network for Extended State Observer and Sliding Mode Control of Vehicle Drive Motor with Unknown Hysteresis Nonlinearity

Authors: Xuehui Gao; Bo Sun; Xinyan Hu; Kun Zhu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

An echo state network (ESN) for extended state observer (ESO) and sliding mode control (SMC) of permanent magnet synchronous motor (PMSM) in an electric vehicle system is investigated in this paper. For the PMSM model, most researches neglect the hysteresis loss and other nonlinear factors, which reduces the accuracy of the PMSM model. We present a modified PMSM model considering the hysteresis loss and then transform the new PMSM model to a canonical form to simplify the controller design. In order to deal with the hysteresis loss, an ESN is utilized to estimate the nonlinearity. Considering that some states cannot be directly obtained, an ESO with ESN is proposed to estimate unknown system states of the electric vehicle PMSM system. Afterwards, an SMC is adopted to control the closed-loop system based on the ESO with ESN, and a double hyperbolic function instead of the sign function is used to suppress the chattering of the SMC. The stabilities of the observer and the controller are all guaranteed by Lyapunov functions. Finally, simulations are presented to verify the validity of the echo state network for extended state observer and the neural network sliding mode control.


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## Body

## 1. Introduction

Electric vehicles are the most important new energy resource vehicles that attract much attention in these years. Many researchers and factories have investigated the electric vehicles and got some well results. Different electric vehicles have been produced, and some have been accepted in the market. Even though the electric vehicles already appeared in the market, they also have a lot of problems that need to be investigated. For example, an accurate electric vehicle motor model is one of the important issues that needs to be investigated more such that it can improve the precision of the control system and the automation for electric vehicles.Most of the electric vehicles adopt permanent magnet synchronous motor (PMSM), and their models have been investigated for many years. But the electric vehicles not only include the PMSM but also have many other subsystems such as torque production dynamics and crankshaft dynamics. Na et al. [1] discussed the different models of vehicle and designed an input observer and adaptive estimations to approximate the unknown parameters. In order to precisely control the tracking errors, Na et al. [2] utilized the prescribed performance function to control the vehicles suspensions. Wang et al. [3, 4] designed an adaptive controller to precisely control the model of PMSM with funnel motion control that is similar to that in literature [2]. Huang et al. [5] proposed an approximation-free control strategy which does not need function approximators (e.g., neural networks and fuzzy logic systems) for the vehicles suspension [2]. Furthermore, Na et al. [6] investigated the nonlinear active suspension systems of vehicles with adaptive finite time fuzzy control strategy considering the input delay since the input delay commonly appears in the suspension systems of vehicle. And in literature [7], the autonomous vehicles were researched with extended Kalman filter designing minimum model error tracking control that considers the input saturation in real vehicle systems that always contains the issue of the input saturation. Compared with many other literatures that utilized linear tyre model, Li et al. [8] had investigated the nonlinear tyre model with vehicle MP algorithm. Sun et al. [9, 10] investigated the variable stiffness and damping model of magnetorheological (MR) vehicle suspension system. The MR damper is described by two Bouc–Wen hysteretic models, and TS fuzzy approach was used to model the quarter car system.All the above studies investigated the vehicles’ model or part of vehicles’ model (submodel), but these vehicle systems are not just pure electric vehicle systems, which also include hybrid electric vehicle systems. In pure electric vehicle systems, the motor model is the most important model such that it draws much attention. Many kinds of motor can be utilized for electric vehicles, and the permanent magnet synchronous motors (PMSMs) are the common motors in the vehicle systems. The PMSM model has been investigated for many years, and different approaches were applied for different conditions. Luo et al. [11] proposed a field-circuit-coupled parameter adaptive modeling method for PMSM which combined the merits of a mathematical model and a magnetic field model. Since the permanent magnet materials in the electric vehicles have hysteresis losses in practice, it has been investigated in different applications and many results were acquired. Egorov et al. [12] investigated the hysteresis loss of the permanent magnet using static history-dependent hysteresis model (HDHM) for PMSM with ferrite magnets of a rotor surface magnet. But in literature [13], a zero-sequence current hysteresis controller with space vector pulse-width modulation was presented to solve the system loss of the open-end winding permanent magnet synchronous motor, where the loss was caused by the limited maximum output power. Notwithstanding the hysteresis of the PMSM is widely researched [14, 15], and it also has much issues to be further researched such as using mathematical hysteresis models or intelligent models to describe the hysteresis or using an observer to estimate the unable direct measurement hysteresis parameters. Considering the U model theory [16–19], the structure of the model and the observer is presented under the inspiration of the model-independent framework for the U model. Therefore, the observer and the controller will be investigated based on the U model model-independent framework.Many real electric vehicle systems cannot obtain all the model states directly, and then the observer will be utilized to estimate the unknown states or parameters. In general, the observer is utilized to estimate two categories of unknowns in controlled systems: disturbances and states [4, 20–24]. Many different observers have been applied to approximate the states or disturbances for many years. In these observers, the extended state observer (ESO) is one of the excellent observers and has achieved well results. ESO is firstly proposed by Han [25], and it has been further researched by many excellent scholars. Xue et al. [26] proposed an ESO-based active disturbance rejection control (ADRC) to deal with the uncertainties of the system, where the ESO gain could automatically timely be adjusted for reducing the estimation errors. Xue et al. [27] proved that a certain ESO could serve as estimation through the augmented gain. Chen et al. [28] proposed an ESO to estimate both the high-order nonlinear system unknown states and uncertainties in order to facilitate the controller design. Sun et al. [29] focused on the multimotor servomechanism systems, using the ESO to approximate the unmeasured velocity of servo motors, and Wang et al. [30] also investigated the motor servomechanism systems, but it is different to [29]. The ESO was utilized to estimate the unknown dynamics in the control system (e.g., friction and disturbances) for its easy design on account of the system bandwidth. On the whole, most ESOs are utilized to estimate the “total disturbance,” which is regarded as the disturbance observer for the control systems. But in this paper, we will adjust the structure of the ESO to approximate the system states. The adjusted structure of ESO has designed high gain parameters that can fastly estimate the unknown system states and higher accuracy. In order to simplify the ESO design and guarantee the ESO convergence, the hysteresis nonlinearity of the PMSM model and other unknown sections as “general disturbance” will be handled by a proposed echo state network (ESN).ESN has been utilized for many nonlinear systems for its fewer nodes and computational requirements. Chen et al. [31] designed an adaptive ESN control for a class of constrained pure-feedback nonlinear systems, but Sun et al. [32] investigated a modified ESN dynamic surface controller for multi-input and multioutput nonlinear systems. These ESNs all have acquired well results. In additions, Wang et al. [33, 34] investigated the prescribed performance tracking control of nonlinear servo mechanisms with ESN, one chose robust adaptive control and the other selected dynamic surface control and verified by experiments. Different from the above literatures, Wang et al. [35] researched the two-inertia servo mechanisms using ESN and designed a prescribed performance function dynamic surface control. Wang et al. [34] considered the influence of the backlash for the servo mechanisms. Different from [34] that utilized ESN to dispose the backlash nonlinearities, we will adopt ESN to deal with the unknown hysteresis nonlinear sections using the ESO simplifying the ESO design in this paper.Sliding mode control (SMC) is one of the excellent control strategies which can be frequently used in practice. We have utilized a composite control which consists of a discrete inverse model-based control and a discrete adaptive sliding mode control to deal with a Hammerstein system in literature [36]. The Hammerstein system was composed of a linear dynamics connecting a hysteresis nonlinearity, where the order of linear dynamics was unknown and the hysteresis modeled by Preisach operator. Then, the effectiveness of the sliding mode control was verified by servo motor system experiments. In order to restrain the chattering of the sliding mode control, Gao et al. and Tao et al. [36, 37] adopted double hyperbolic function tanh⋅ instead of the sign⋅, which can obtain favourable reduced chattering. The sliding mode control is applied for many real systems. Zhao et al. [38] applied the SMC to the multimotor driving servo systems, and Chen et al. [39] utilized the SMC for an uncertain spacecraft system. In the literature [7], the sliding mode control was used for the autonomous vehicles. In this paper, the SMC will be designed to precisely control the proposed PMSM model system where the unknown states are estimated by the adjusted ESO with ESN.In this paper, we firstly consider the influence of the hysteresis nonlinearity for the PMSM model that mostly is neglected in the former models. Therefore, a new modified PMSM model including the hysteresis nonlinearities is established. Secondly, the modified PMSM model will be transformed to a canonical form to simplify the controller design. Then, the ESN is applied, and a new ESO will be proposed to estimate all the states whether that can be directly measured or not. A Lyapunov function guarantees the effectiveness of the observer results. Finally, an SMC with the ESN and the ESO is designed for this new modified PMSM model, and another Lyapunov function guarantees all the closed-loop signals bounded. In order to restrain the chattering of the SMC, we chose the continuous functiontanh⋅ replacing the discontinuous function sign⋅ and the simulations will verify the effectiveness of the proposed approaches. The contributions are listed as follows:(i)
The PMSM model is improved for suiting the electric vehicle system which considers the hysteretic losses. Besides, the modified PMSM model is transformed to a canonical form that can simplify the ESO and controller design.(ii)
Different from most ESOs of the disturbance observer, the structure of an ESO is adjusted to estimate the states of the modified PMSM model as well as the ESN is adopted to approximate the unknown hysteretic losses. That not only guarantees the convergence of ESO fastly but also simplifies the observer design.(iii)
An SMC is designed with the designed ESO for this transformed PMSM model in electric vehicle systems and utilizes the continuous functiontanh⋅ replacing the discontinuous function sign⋅ to restrain the SMC chatting.The rest of the paper is organized as follows: Section2 discusses the problem formulations that give the new PMSM model and the transformation of the canonical form. The introduction of ESN is provided in Section 3, and Section 4 gives the ESO design, which includes the ESN, and the observer effectiveness is guaranteed. The neural network sliding mode control design is discussed in Section 5, and Section 6 shows simulations. Section 7 concludes this paper.


## 2. Problem Formulations

The electric vehicle model includes different sections, which is composed of vehicle derive, engine, motor, battery, rear and front wheels, and other electrical and mechanical accessory models. In this paper, we discuss the motor model of the vehicle with hysteresis nonlinearity. According to reference [40], the permanent magnet synchronous motor (PMSM) d−q model can be described as follows:(1)diqdt=−RLqiq+ωLdLqid−KlLqω+1Lqvq,diddt=−RLdid−ωLqLdiq+1Ldvd,T=Ktiq+KlLd−Lqidiq,where id,iq,vd,andvq represent the motor d−q-axes currents and voltages, respectively; RandLdandLq mean the motor resistance and d−q-axes inductances, respectively; KlandKt are the motor-induced voltage constant and motor torque constant, respectively; T is the transmit torque; and ω represents the motor speed.From (1), it is obviously shown that the motor model parameters are viewed as constants. But it neglects hysteresis loss, cross coupling, eddy current loss, etc. In addition, according to the discussion of reference [40], since the number of rotor teeth is sufficiently high, one can assume that Ld≈Lq. Then, considering the effect of the neglected factors in former models, especially the hysteresis loss, the model (1) is modified as(2)diqdt=−RLqiq+ωLdLqid−KlLqω+1Lqvq+f1iq,id,diddt=−RLdid−ωLqLdiq+1Ldvd+f2iq,id,T≈Ktiq,where f1iq,id,f2iq,id,andg1iq,id are the unknown nonlinear smooth functions which represent the neglected factors such as hysteresis loss in (1). We rewrite the model (2) to the state-space form as follows:(3)ddtiqid︸x=−RLqωLdLq−ωLqLd−RLd︸Aiqid+1Lq1Ld︸Bvqvd+f1iq,idf2iq,id︸fx,TKt︸y=iq,where(4)x=x1x2=iqid,A=a11a12a21a22=−RLqωLdLq−ωLqLd−RLd,B=b1b2=1Lq1Ldfx=f1xf2x=f1iq,idf2iq,id.Many observers and controllers require that the system model should be in canonical form. To facilitate the observer and control design in this paper, a set of new state variables will be employed to transform the system model (2) into a canonical form. Define the new system states as(5)z1=x1,z2=z˙1.Then, according to the equationz2=z˙1=x˙1=−R/Lqiq+ωLd/Lqid−Kl/Lqω+1/Lqvq+f1iq,id, we have(6)z˙2=z¨1=−RLqz2+ωLdLq−RLdid−ωLqLdiq+1Ldvd+f2x+f˙1x=−ω2︸a¯1z1−RLq︸a¯2z2+ωLq︸b˘vd︸u−ωRLqid+ωLdLqf2x+f˙1x︸F¯x.In this paper, we adopt the Bouc–Wen hysteresis model to describe the hysteresis loss of the vehicle PMSMd−q model. The Bouc–Wen model is given as follows:(7)fx=ς1u+ς2χ,χ˙=ϖ0u˙−ϖ1u˙χh−1χ−ϖ2u˙χh,where ϖ0>0,ϖ1>ϖ2,h>1,χ0=0, and χ=χ1,χ2T∈ℝ2.Remark 1.
At three-phase motor system, we havevd=3/2Um,vq=0, where Um represents the root-mean-square (RMS) value of the input voltage by coordinate transform. Then, without loss of generality, we assume vq=0 during the control process.
Then, equation (6) can be deduced as(8)z˙2=a¯1z1+a¯2z2+b⌣u−ωRLqid+ωLdLqς12u+ς22χ2+ς11u˙+ς21χ˙1=a¯1z1+a¯2z2+b⌣+ωLdς12Lq︸b¯u−ωRLqid+ωLdLqς22χ2+ς11u˙+ς21χ˙1︸Fx,where χ1andχ2 are defined in (7).Remark 2.
The Bouc–Wen model is a mathematical model for hysteresis nonlinearity, where the parametersς1andς2 decide the direction of the hysteresis nonlinearity. If ϖ0=2,ϖ1=4,ϖ2=0.5,h=3,ς1=3,andς2=5, the input signal u=2.5sin1.5π; then, the curve of the Bouc–Wen model is illustrated in Figure 1. If the parameters ς1=−3andς2=−5 and other parameters are not changed, the curve is illustrated in Figure 2. It is obviously shown that the parameters ς1andς2 decide the direction of the Bouc–Wen model.
Then, the system model is transformed into the canonical form as(9)z˙1=z2,z˙2=a¯1z1+a¯2z2+b¯u+Fx,y=z1.Figure 1
The curve of the Bouc–Wen model withς1=3andς2=5.Figure 2
The curve of the Bouc–Wen model withς1=−3andς2=−5.


## 3. Echo State Network

The echo state network (ESN) is relatively a new recurrent neural network (RNN) structure for complex dynamical systems. The structure of ESN is composed of a dynamical recursive hidden layer and a memoryless output layer. Because of the feedback recursive hidden layers like the echoes, the structure of the neural network is named ESN.ESN has easily supervised training. Compared with some other NNs, the ESN can fastly, simply, and constructively realize the supervised learning. In addition, without changing all the weights between the input layer and hidden layer, only by changing the weights from the reservoir to the output layer, ESN can fastly obtain higher precision.The continuous-time leaky integrator ESN can be defined as(10)Ω˙=α−γΩ+ψWinν+WitΩ+Wbaξ,where Ω represents the n-dimensional dynamic state; α>0 is a time constant; γ is the leaking decay rate of the neuron’s set Ω; ψ represents a given real continuous function, in this paper, we chose the Gaussian function; and Win∈ℝn×m,Wit∈ℝn×n, and Wba∈ℝn×m are the input, internal, and feedback weight matrices, respectively.As suggested in [29], by choosing α=1andγ=1, equation (10) is rewritten as(11)Ω˙=−Ω+ψWinν+WitΩ+Wbaξ.Then, the kernel function is chosen as the Gaussian function andΩ is defined as Ωx=s1x,s2x,…,slx with l the neuron node number of the ESN output layer. The function is(12)six=exp−x−ϱiTx−ϱiρ2,where ϱi=ϱi1,ϱi2,…,ϱiqT,i=1,2,…,l; q means the input number; and ρ represents the width of the Gaussian function.Defining the output activation functionΘ is chosen as, the output of ESN is defined as(13)Γ=ΘWΩ.According to the literature [29], the output activation function Θ units had no memory so that their values at time i+1 depended only fractionally and indirectly on their previous values. Thus, those networks are best suited for modeling intrinsically discrete-time systems with a computational and jumpy flavor. It is difficult to use networks with continuous dynamics. But we can define the ESNs with continuous function so that the dynamics can be approximated by this continuous function.For any given real continuous functionG⋅:ℝn⟶ℝ on a sufficiently large compact set Ξ and arbitrary εM, the ESN system Γ satisfies(14)supΩ∈ΞG−Γ≤εM,where G is a given real continuous function on a sufficiently large compact set Ξ⊂ℝ with the supremum εM>0.Then, the functionG can be expressed as(15)G=W∗Ωx+ε∗,where ε∗≤εM is the ESN error.


## 4. Extended State Observer Design

Since the nonlinearities including hysteresis loss are considered, the controller cannot be directly designed, and the system states need an observer. Considering the normalized form (9), we will design an extended state observer (ESO) to approximate the system states. Before designing the ESO, an extended state z3 will be defined and the extended state system model is expressed as follows:(16)z˙1=z2,z˙2=z3+b¯u,z˙3=a¯1z1+a¯2z2+Fx.Then, the ESO is designed as(17)z^˙1=z^2+β1y−z^1,z^˙2=z^3+b¯u+β2y−z^1,z^˙3=a¯1z^1+a¯2z^2+F^x+β3y−z^1,where z^1,z^2,z^3,andF^x are the estimations of z1,z2,z3,andFx, respectively, and β1,β2,andβ3 are the designed high gain parameters, respectively.Define the errors(18)z˜i=zi−z^,i=1,2,3,y˜=y−y^,F˜x=Fx−F^x,where Fx is described by the ESN. Therefore, it has Fx=WΩx+εandF^x=W^Ωx. Then, we have F˜x=W˜Ωx+ε=WΩx+ε−W^Ωx, and define the update law of W˜ as(19)W˜˙=W^˙=Ue^Ωx−εe^W^,where U=UT is a constant matrix, ϵ is a designed positive constant, and e^ is defined in (36).Lemma 1.
[41] The ESN weights W^ in equation (19) are bounded by W^≤ϖM/ϵ, where ϖM is the bound of the ESN basis function vector, i.e., Ω≤ϖM.Proof.
From the previous discussion, the ESN basis function vectorΩ are Gaussian functions so that it is bounded obviously, i.e., Ω≤ϖM. We choose the Lyapunov function candidate as(20)VN=12UW^TW^,and then, considering equation (19), the time derivative of VN can be deduced as(21)V˙N=1UW^TW^˙=W^Te^Ωx−εe^W^≤−W^e^εW^−ϖM.
According to reference [41], W^ is bounded by W^≤ϖM/ϵ and thus W˜=W∗−W^ is also bounded as W˜≤ωM, where ωM=WN+ϖM/ϵ.
Then, considering (16) and (17) yields(22)z˜˙1=−β1z˜1+z˜2,z˜˙2=−β2z˜1+z˜3,z˜˙3=a¯1−β2z˜1+a¯2z˜2+F˜x.
Equation (22) can be rewritten as(23)z˜˙=A¯z˜+ζF˜x,where(24)A¯=−β110−β201a¯1−β3a¯20,ζ=001.
Considering that the high gain parametersβ1,β2,andβ3 are designed, the characteristic polynomial of A¯ is Hurwitz. Then, given a positive definite symmetric matrix P=PT>0, the existing positive definite symmetric matrix Q=QT>0 satisfies(25)A¯TP+PA¯≤−Q.Remark 3.
In most cases, ESO has been utilized to estimate the generalized disturbance. But in this paper, the structure of the ESO has been adjusted to approximate the system states. If appropriate high gainsβ1,β2,andβ3 are chosen, the ESO can fastly and precisely estimate the states. Besides, adjusting ESO can easily guarantee the convergence of the observer.
Therefore, the following theorem holds.Theorem 1.
Considering the vehicle drive PMSM model (2), which is transformed into canonical form (9), the unknown nonlinearity Fx is approximated by an ESN (15), the normalized system (9) is extended to (16), and then, all the signals are bounded and the states can be observed by an ESO (17).Proof.
Considering the ESO (17) and equation (22), the Lyapunov function candidate Vo for the state observer errors can be designed as(26)Vo=z˜TPz˜.
Considering (15), the derivative of Vo is deduced as(27)V˙o=z˜TA¯TP+PA¯z˜+2F˜xζTPz˜≤−z˜TQz˜+2W˜ΩxPz˜+2ε,where W˜=W−W^ and W^ is the estimation of the ESN weight, and the unknown nonlinearity Fx is approximated by ESN Fx=WΩx+ε; therefore, the estimation F^x is F^x=W^Ωx.
According to Lemma1 and reference [31], the error weight of ESN satisfies W˜≤wM, and Ωx is bounded, it has Ωx≤ϖM. Considering ε≤εM, equation (27) is expressed as(28)V˙o≤−z˜TQz˜+2wMφMλmaxPz˜+2εM≤−z˜TQz˜+2wMφMλmaxPz˜+2εM,where λmaxP represents the biggest norm of the matrix P. From Young’s inequality ab≤a2+b2/2, it has 2wMφMλmaxPz˜≤wM2+φ2+λmax2P+z˜2. Then, we have(29)V˙o≤−z˜TQz˜+wM2+φ2+λmax2P+z˜2+2εM.
We rewrite (29) as(30)V˙o≤−a⌣1Vo+a⌣2,where a⌣ can be deduced by A¯TP+PA¯≤−Q; a⌣2=wM2+φ2+λmax2P+z˜2+2εM. By integrating both sides of (30), the following equation can be obtained:(31)z˜≤Vo0e−a⌣1t+a⌣2/a⌣1λmaxP,and that indicates all the signals are bounded and the observer error z˜ is converged into a compact set around the zero. Therefore, according to the Lyapunov theory, the proposed ESO can estimate the system states.


## 5. Sliding Mode Control Design

This section will design a sliding mode control for the new vehicle drive PMSM model that the hysteresis loss is described by ESN. The control structure of the sliding mode control with the ESO and ESN is illustrated in Figure3. We firstly define a sliding mode manifold and then design the sliding mode controller. The stability of the control strategy finally is demonstrated.Figure 3
The structure of the controller.To design the sliding mode control, the tracking error is defined as(32)e=y−yd=z1−yd,and the sliding manifold is adopted as(33)s=ηe+e˙,where the designed parameters η>0 and the reference signal yd has continual derivative.From (32), the derivative of s can be deduced as(34)s˙=ηe˙+e¨.Because the tracking error is defined in (32), we have(35)e˙=z˙1−y˙d=z2−y˙d,e¨=z˙2−y¨d.We define the functions as(36)e^=y^−yd=z^1−yd,s^=ηe^+e^˙,s˜=s−s^,e˜=e−e^.The sliding mode controller is designed as(37)u=1b¯−ηts^−z^3−ηe^˙+y¨d−τsigns,where the designed parameters ηt>0 and s^ande^ represent the estimation of sande, respectively. In order to decrease the influence of the chattering, we choose 0<τ<1.Then, the following theorem can be obtained:Theorem 2.
For the vehicle drive PMSM model considering the hysteresis loss (9), which extends the state to (16), the states are observed by ESO (17); defining the sliding mode manifold in (33) and designing the controller in (37), then, all signals in the closed-loop are uniformly ultimately bounded (UUB).Proof.
To verify the stability of the closed loop, the Lyapunov function is selected as(38)V=12s2.
Considering the sliding manifold (33)–(35), the derivative of V is(39)V˙=ss˙=sηe˙+e¨=sηe˙+z˙2−y¨d.
Substitutingz˙2=z3+b¯u in (16) into (39) yields(40)V˙=sηe˙+z3+b¯u−y¨d.
By the substitution of sliding mode controller (37) into (40), the derivative of V can be deduced as(41)V˙=sηe˙+z3−ηts^−z^3−ηe^˙+y¨d−τsigns−y¨d.
Considering (36), the derivative of V can be deduced as(42)V˙=sηe˜˙+z˜3−ηts^−τsigns=sηe˜˙+z˜3−ηts−s˜−τsigns=−ηts2+sηe˜˙+z˜3+ηts˜−τsigns.
Substituting the sliding manifold (33) and s^=ηe^+e^˙ in (36) into s˜ yields(43)s˜=ηe˜+e˜˙.
By the substitution of (43) into (42), we get(44)V˙=−ηts2+sηe˜˙+z˜3+ηηte˜+ηte˜˙−τsigns=−ηts2+sη+ηte˜˙+z˜3+ηηte˜−τsigns.
According toe=z1−yd in (32), we have(45)e˜=z˜1−yd,e˜˙=z˜2−y˙d.
Substituting (22) and (45) into (44) yields(46)V˙=−ηts2+sη+ηtz˜2−η+ηty˙d+a¯1z˜1+a¯2z˜2+F˜x+ηηtz˜1−ηηtyd−τsigns=−ηts2+sa¯1+ηηtz˜1+a¯2+η+ηtz˜2−ηηtyd−η+ηty˙d+F˜x−τsigns.
Considering Theorem 1, the unknown bounded functionF˜x can be approximated by the ESN F˜x=W˜Ωx≤ωMφM and z˜1,z˜2,yd,andy˙d are all bounded, respectively. Then, defining a¯1+ηηtz˜1+a¯2+η+ηtz˜2−ηηtyd−η+ηty˙d+W˜Ωx≤Δmax, where Δmax≥0, equation (46) satisfies(47)V˙≤−ηts2+sΔmax−τssigns≤−ηts2+sΔmax−τs.
Ifτs>1, equation (47) can be rewritten as(48)V˙≤−ηts2+sΔmax=−μ+sΔmax,where μ=ηt>0.
If0≤τs≤1, it has(49)V˙≤−ηts2+sΔmax−τs2=−ηt+τs2+sΔmax=−μs2+sΔmax,where μ=ηt+τ>0.
According to Young’s inequality,sΔmax has sΔmax≤s2/2+Δmax2/2; then, equation (47) can be rewritten as(50)V˙≤−μs2+s22+Δmax22=−μ+12s2+Δmax22.
Whenμ>0.5, according to the definition of the Lyapunov function V in (38), equation (50) can be solved(51)s≤2μ−1V0+Δmax22μ−1Δmax.
Considering the definition ofs in (33) and the bounded (51), the margin of error e is obtained:(52)e≤s0+η/ϑϑ,where ϑ=2μ−1V0+Δmax2/2μ−1Δmax.
Therefore, according to the Lyapunov theory, all the signals of the closed loop are uniformly ultimately bounded.Remark 4.
In this controller design, we chose the robust functionsign⋅ to design the sliding mode control. But the function sign⋅ is a discontinuous function which can deduce the strong chattering for the sliding mode manifold. In order to restrain the chattering, many approaches are utilized. In this paper, we select the continuous function tanh⋅ instead of the discontinuous function sign⋅, which can substantially restrain the chattering of the SMC. The simulations are demonstrated for this approach.


## 6. Simulations

We design the simulations with Simulink of Matlab to verify the effectiveness of the proposed ESO and ESNSMC for the vehicle drive motor with hysteresis nonlinearity. The PMSM, the ESO, and the ESNSMC parameters are selected as follows:d−axis inductance Ld and q−axis inductance Lq are 0.47mH; torque T is 96.5N⋅m; angular speed ω is 3000rpm; resistance R is 0.033Ω; motor-induced voltage constant Kl is 0.147; and motor torque constant Kt is 0.72. The high gain parameters β1,β2,andβ3 are selected as β1=360,β2=5000,andβ3=28500. η is chosen as 5. τ can be selected as 0.3. In order to verify the proposed approaches, we restrain the errors in −0.5,0.5 in the simulations.In the section, we design two series of simulations with different values ofτsigns. One is τsigns=0.3signs, and the other is selected as τsigns=0.3tanhs. The reason of the design is to contrast the results of the ESNSMC to show the effect of the chattering in the sliding mode control.


### 6.1. Case 1

In this section, we selectτsigns=0.3signs in the controller of (37) to verify the proposed ESNSMC and ESO of this paper. The output tracking results of y,y˙, and y¨ are illustrated in Figure 4, and the tracking errors are shown in Figure 5. The observer results of z1,z2, and z3 of the proposed ESO are shown in Figure 6, and the observer errors are illustrated in Figure 7. Figure 8 is the proposed ESNSMC controller trajectory.Figure 4
The output resultsy,dy, and ddy of the ESNSMC with ESO τsigns=0.3signs.Figure 5
The errors ofy,dy, and ddy of the ESNSMC with ESO τsigns=0.3signs.Figure 6
The observer results ofz1,z2, and z3 with ESO τsigns=0.3signs.Figure 7
The observer errors ofz1,z2, and z3 with ESO τsigns=0.3signs.Figure 8
The trajectory of the ESNSMC controllerτsigns=0.3signs.From Figures4 and 6, it is obviously shown that both the proposed ESO and the ESNSMC are well tracking the reference signals. Based on the structure of the control strategy, the adjusting of the observer is in the closed loop and that follow the adjustment of the controller. Then, if the gains of ESO β1,β2,andβ3 are chosen enough higher, it can quickly approximate the states z1,z2,andz3. From Figure 6, one can clearly find that the proposed ESO worked well, and from Figure 4, one can also find that the output quickly tracks the reference signal. But from Figures 5 and 7, due to the existence of the chattering in the sliding mode control, it has been obviously shaking when the PMSM is veered. If we survey Figure 8, it can be seen that the trajectory of the proposed ESMSMC controller also has the same characteristics as well as the errors of ESNSMC control results and the ESO observer.From Figures4–8, one can easily find the influence of the chattering in the proposed ESNSMC, even though the ESN compensates the hysteresis nonlinearity. The chattering of the sliding mode control really degrades the precision of the control strategy. In order to alleviate the influence of the chattering of the ESNSMC, we chose continuous tanh⋅ function to take the place of the discontinuous sign⋅ function in the next case.


### 6.2. Case 2

In this section, the robustτsigns is chosen as τsigns=0.3tanhs. Then, Figures 9 and 10 illustrate the output results y,y˙,andy¨ of the proposed ESNSMC with ESO for τsigns=0.3tanhs, and the ESO estimates the states z1,z2,andz3 are shown in Figure 11. Figure 12 is the estimate errors of z1,z2,andz3 with the ESO observer. The controller input is illustrated in Figure 13 for the robust τsigns=0.3tanhs.Figure 9
The output resultsy,dy, and ddy of the ESNSMC with ESO τsigns is replaced by 0.3tanhs.Figure 10
The errors ofy,dy, and ddy of the ESNSMC with ESO τsigns is replaced by 0.3tanhs.Figure 11
The observer results ofz1,z2, and z3 with ESO τsigns is replaced by 0.3tanhs.Figure 12
The observer errors ofz1,z2, and z3 with ESO τsigns is replaced by 0.3tanhs.Figure 13
The trajectory of the ESNSMC controllerτsigns is replaced by 0.3tanhs.Compared with the control tracking results Figures4 and 9 and the controller errors Figures 5 and 10, one can obtained that the continuous function tanh⋅ can substantially restrain the chattering of the sliding mode control. Especially in contrast with the tracking errors Figures 5 and 10, the influence of the chattering at the reversing processes almost is not catched in Figure 10 with the continuous robust function τsigns=0.3tanhs. The same conclusions can also be deduced by the observer results and errors in Figures 11 and 12. Especially in contrast with Figures 7 and 12, one can find that the maximum of the error of z3 is about 0.08 at the reversing processes in Figure 7 and the maximum of the error of z3 is about 0.02 at the reversing processes in Figure 12. One can also obtain that, in Figure 12, the maximum of errors of z1,z2,andz3 appears at the dynamic processes with the continuous robust function τsigns=0.3tanhs, which are caused by the ESN, but in Figure 8, the maximum of errors of z1,z2,andz3 appears at the reversing processes with discontinuous robust function τsigns=0.3signs, which are caused by the chattering of the SMC.Table1 illustrates the tracking errors and the observer errors with the discontinuous robust function τsigns=0.3signs, and the tracking errors and the observer errors with the continuous robust function τsigns=0.3tanhs are shown in Table 2. From Tables 1 and 2, the MAEs of z˜1,z˜2,z˜3T are 2.702e−6,9.5797e−4,0.0187T and 2.289e−6,8.3078e−4,0.0116, respectively. It is illustrated that the continuous robust function tanh⋅ can substantially restrain the chattering of the SMC. But considering the MAE tracking errors y˜,y˜˙,y˜¨T are 0.0276,0.0269,0.0317T and 0.0775,0.0281,0.0312T, respectively, the MIN tracking errors are 0.0014,1.8734e−6,1.3852e−5 and 0.047,7.9464e−6,1.473e−5, respectively. That means the continuous robust function tanh⋅ can substantially restrain the chattering but degrade the tracking precision.Table 1
The tracking errors and the observer errors with the robustτsigns=0.3signs
Variables
MAE
Max
Min
y
˜
0.0276
0.5
0.0014
y
˜
˙
0.0269
0.5
1.8734e−6
y
˜
¨
0.0317
0.5
1.3852e−6
z
˜
1
2.702e−6
9.9556e−6
0
z
˜
2
9.5797e−4
0.0035
0
z
˜
3
0.0187
0.0763
0Table 2
The tracking errors and the observer errors with the robustτsigns=0.3tanhs.
Variables
MAE
Max
Min
y
˜
0.0775
0.5
0.047
y
˜
˙
0.0281
0.5
7.9464e−6
y
˜
¨
0.0312
0.5
1.473e−5
z
˜
1
2.2891e−6
8.3371e−6
0
z
˜
2
8.3078e−4
0.0032
0
z
˜
3
0.0116
0.0698
0Compared with Figures8 and 13, one can obviously find that the dynamic processes is influenced by the ESN in Figure 13 with the continuous robust function tanh⋅ which can well restrain the chattering of the SMC, but the discontinuous robust function sign⋅ will result in strong chattering and difficult to achieve stable process from Figure 8.


## 6.1. Case 1

In this section, we selectτsigns=0.3signs in the controller of (37) to verify the proposed ESNSMC and ESO of this paper. The output tracking results of y,y˙, and y¨ are illustrated in Figure 4, and the tracking errors are shown in Figure 5. The observer results of z1,z2, and z3 of the proposed ESO are shown in Figure 6, and the observer errors are illustrated in Figure 7. Figure 8 is the proposed ESNSMC controller trajectory.Figure 4
The output resultsy,dy, and ddy of the ESNSMC with ESO τsigns=0.3signs.Figure 5
The errors ofy,dy, and ddy of the ESNSMC with ESO τsigns=0.3signs.Figure 6
The observer results ofz1,z2, and z3 with ESO τsigns=0.3signs.Figure 7
The observer errors ofz1,z2, and z3 with ESO τsigns=0.3signs.Figure 8
The trajectory of the ESNSMC controllerτsigns=0.3signs.From Figures4 and 6, it is obviously shown that both the proposed ESO and the ESNSMC are well tracking the reference signals. Based on the structure of the control strategy, the adjusting of the observer is in the closed loop and that follow the adjustment of the controller. Then, if the gains of ESO β1,β2,andβ3 are chosen enough higher, it can quickly approximate the states z1,z2,andz3. From Figure 6, one can clearly find that the proposed ESO worked well, and from Figure 4, one can also find that the output quickly tracks the reference signal. But from Figures 5 and 7, due to the existence of the chattering in the sliding mode control, it has been obviously shaking when the PMSM is veered. If we survey Figure 8, it can be seen that the trajectory of the proposed ESMSMC controller also has the same characteristics as well as the errors of ESNSMC control results and the ESO observer.From Figures4–8, one can easily find the influence of the chattering in the proposed ESNSMC, even though the ESN compensates the hysteresis nonlinearity. The chattering of the sliding mode control really degrades the precision of the control strategy. In order to alleviate the influence of the chattering of the ESNSMC, we chose continuous tanh⋅ function to take the place of the discontinuous sign⋅ function in the next case.


## 6.2. Case 2

In this section, the robustτsigns is chosen as τsigns=0.3tanhs. Then, Figures 9 and 10 illustrate the output results y,y˙,andy¨ of the proposed ESNSMC with ESO for τsigns=0.3tanhs, and the ESO estimates the states z1,z2,andz3 are shown in Figure 11. Figure 12 is the estimate errors of z1,z2,andz3 with the ESO observer. The controller input is illustrated in Figure 13 for the robust τsigns=0.3tanhs.Figure 9
The output resultsy,dy, and ddy of the ESNSMC with ESO τsigns is replaced by 0.3tanhs.Figure 10
The errors ofy,dy, and ddy of the ESNSMC with ESO τsigns is replaced by 0.3tanhs.Figure 11
The observer results ofz1,z2, and z3 with ESO τsigns is replaced by 0.3tanhs.Figure 12
The observer errors ofz1,z2, and z3 with ESO τsigns is replaced by 0.3tanhs.Figure 13
The trajectory of the ESNSMC controllerτsigns is replaced by 0.3tanhs.Compared with the control tracking results Figures4 and 9 and the controller errors Figures 5 and 10, one can obtained that the continuous function tanh⋅ can substantially restrain the chattering of the sliding mode control. Especially in contrast with the tracking errors Figures 5 and 10, the influence of the chattering at the reversing processes almost is not catched in Figure 10 with the continuous robust function τsigns=0.3tanhs. The same conclusions can also be deduced by the observer results and errors in Figures 11 and 12. Especially in contrast with Figures 7 and 12, one can find that the maximum of the error of z3 is about 0.08 at the reversing processes in Figure 7 and the maximum of the error of z3 is about 0.02 at the reversing processes in Figure 12. One can also obtain that, in Figure 12, the maximum of errors of z1,z2,andz3 appears at the dynamic processes with the continuous robust function τsigns=0.3tanhs, which are caused by the ESN, but in Figure 8, the maximum of errors of z1,z2,andz3 appears at the reversing processes with discontinuous robust function τsigns=0.3signs, which are caused by the chattering of the SMC.Table1 illustrates the tracking errors and the observer errors with the discontinuous robust function τsigns=0.3signs, and the tracking errors and the observer errors with the continuous robust function τsigns=0.3tanhs are shown in Table 2. From Tables 1 and 2, the MAEs of z˜1,z˜2,z˜3T are 2.702e−6,9.5797e−4,0.0187T and 2.289e−6,8.3078e−4,0.0116, respectively. It is illustrated that the continuous robust function tanh⋅ can substantially restrain the chattering of the SMC. But considering the MAE tracking errors y˜,y˜˙,y˜¨T are 0.0276,0.0269,0.0317T and 0.0775,0.0281,0.0312T, respectively, the MIN tracking errors are 0.0014,1.8734e−6,1.3852e−5 and 0.047,7.9464e−6,1.473e−5, respectively. That means the continuous robust function tanh⋅ can substantially restrain the chattering but degrade the tracking precision.Table 1
The tracking errors and the observer errors with the robustτsigns=0.3signs
Variables
MAE
Max
Min
y
˜
0.0276
0.5
0.0014
y
˜
˙
0.0269
0.5
1.8734e−6
y
˜
¨
0.0317
0.5
1.3852e−6
z
˜
1
2.702e−6
9.9556e−6
0
z
˜
2
9.5797e−4
0.0035
0
z
˜
3
0.0187
0.0763
0Table 2
The tracking errors and the observer errors with the robustτsigns=0.3tanhs.
Variables
MAE
Max
Min
y
˜
0.0775
0.5
0.047
y
˜
˙
0.0281
0.5
7.9464e−6
y
˜
¨
0.0312
0.5
1.473e−5
z
˜
1
2.2891e−6
8.3371e−6
0
z
˜
2
8.3078e−4
0.0032
0
z
˜
3
0.0116
0.0698
0Compared with Figures8 and 13, one can obviously find that the dynamic processes is influenced by the ESN in Figure 13 with the continuous robust function tanh⋅ which can well restrain the chattering of the SMC, but the discontinuous robust function sign⋅ will result in strong chattering and difficult to achieve stable process from Figure 8.


## 7. Conclusion

A new neural network sliding mode control strategy was designed for PMSM of electric vehicles in this research. Firstly, a modified PMSM model considering hysteresis losses was proposed. Since the existence of the hysteresis nonlinearity complicated the PMSM model, a transformed canonical form was deduced to simplify the controller. Because most electric vehicle system states could not be measured directly, an ESO was adopted to estimate all the states including the indirectly obtained states, and the hysteresis nonlinearity was approximated by an ESN. Finally, we designed a sliding mode control based on the ESO results with the ESN in the closed loop to precisely control the electric vehicle PMSM systems, and the chattering of the SMC was restrained bytanh⋅. Two different Lyapunov functions guaranteed the observer validity and the controller effectiveness. The simulations demonstrated that the proposed observer and controller can achieve a good performance and all the signals were uniformly ultimately bounded.



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*Source: 2534038-2020-01-04.xml*