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

Gao, Xuehui; Sun, Bo; Hu, Xinyan; Zhu, Kun

doi:https://doi.org/10.1155/2020/2534038

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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From Complexity to Simplicity in U-Model Enhanced Control System Design

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Research Article | Open Access

Volume 2020 | Article ID 2534038 | https://doi.org/10.1155/2020/2534038

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

Xuehui Gao,¹Bo Sun,¹Xinyan Hu,¹and Kun Zhu²

Guest Editor: Weicun Zhang

Received27 Nov 2019

Accepted12 Dec 2019

Published04 Jan 2020

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.

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 (, 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 instead of the , 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 function replacing the discontinuous function 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 function replacing the discontinuous function to restrain the SMC chatting.

The rest of the paper is organized as follows: Section 2 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) model can be described as follows:where represent the motor -axes currents and voltages, respectively; mean the motor resistance and -axes inductances, respectively; 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 . Then, considering the effect of the neglected factors in former models, especially the hysteresis loss, the model (1) is modified aswhere 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:where

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

Then, according to the equation , we have

In this paper, we adopt the Bouc–Wen hysteresis model to describe the hysteresis loss of the vehicle PMSM model. The Bouc–Wen model is given as follows:where , and .

Remark 1. At three-phase motor system, we have , where represents the root-mean-square (RMS) value of the input voltage by coordinate transform. Then, without loss of generality, we assume during the control process.
Then, equation (6) can be deduced aswhere are defined in (7).

Remark 2. The Bouc–Wen model is a mathematical model for hysteresis nonlinearity, where the parameters decide the direction of the hysteresis nonlinearity. If the input signal ; then, the curve of the Bouc–Wen model is illustrated in Figure 1. If the parameters and other parameters are not changed, the curve is illustrated in Figure 2. It is obviously shown that the parameters decide the direction of the Bouc–Wen model.
Then, the system model is transformed into the canonical form as

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 aswhere represents the n-dimensional dynamic state; is a time constant; γ is the leaking decay rate of the neuron’s set ; ψ represents a given real continuous function, in this paper, we chose the Gaussian function; and , and are the input, internal, and feedback weight matrices, respectively.

As suggested in [29], by choosing , equation (10) is rewritten as

Then, the kernel function is chosen as the Gaussian function and is defined as with l the neuron node number of the ESN output layer. The function iswhere ; q means the input number; and ρ represents the width of the Gaussian function.

Defining the output activation function is chosen as, the output of ESN is defined as

According to the literature [29], the output activation function units had no memory so that their values at time 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 function on a sufficiently large compact set and arbitrary , the ESN system satisfieswhere is a given real continuous function on a sufficiently large compact set with the supremum .

Then, the function can be expressed aswhere 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 will be defined and the extended state system model is expressed as follows:

Then, the ESO is designed aswhere are the estimations of , respectively, and are the designed high gain parameters, respectively.

Define the errorswhere is described by the ESN. Therefore, it has . Then, we have , and define the update law of aswhere is a constant matrix, ϵ is a designed positive constant, and is defined in (36).

Lemma 1. [41] The ESN weights in equation (19) are bounded by , where is the bound of the ESN basis function vector, i.e., .

Proof. From the previous discussion, the ESN basis function vector are Gaussian functions so that it is bounded obviously, i.e., . We choose the Lyapunov function candidate asand then, considering equation (19), the time derivative of can be deduced asAccording to reference [41], is bounded by and thus is also bounded as , where .
Then, considering (16) and (17) yieldsEquation (22) can be rewritten aswhereConsidering that the high gain parameters are designed, the characteristic polynomial of is Hurwitz. Then, given a positive definite symmetric matrix , the existing positive definite symmetric matrix satisfies

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 gains 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 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 for the state observer errors can be designed asConsidering (15), the derivative of is deduced aswhere and is the estimation of the ESN weight, and the unknown nonlinearity is approximated by ESN ; therefore, the estimation is .
According to Lemma 1 and reference [31], the error weight of ESN satisfies , and is bounded, it has . Considering , equation (27) is expressed aswhere represents the biggest norm of the matrix P. From Young’s inequality , it has . Then, we haveWe rewrite (29) aswhere can be deduced by ; . By integrating both sides of (30), the following equation can be obtained:and that indicates all the signals are bounded and the observer error 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 Figure 3. We firstly define a sliding mode manifold and then design the sliding mode controller. The stability of the control strategy finally is demonstrated.

To design the sliding mode control, the tracking error is defined asand the sliding manifold is adopted aswhere the designed parameters and the reference signal has continual derivative.

From (32), the derivative of s can be deduced as

Because the tracking error is defined in (32), we have

We define the functions as

The sliding mode controller is designed aswhere the designed parameters and represent the estimation of , respectively. In order to decrease the influence of the chattering, we choose .

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 asConsidering the sliding manifold (33)–(35), the derivative of isSubstituting in (16) into (39) yieldsBy the substitution of sliding mode controller (37) into (40), the derivative of can be deduced asConsidering (36), the derivative of can be deduced asSubstituting the sliding manifold (33) and in (36) into yieldsBy the substitution of (43) into (42), we getAccording to in (32), we haveSubstituting (22) and (45) into (44) yieldsConsidering Theorem 1, the unknown bounded function can be approximated by the ESN and are all bounded, respectively. Then, defining , where , equation (46) satisfiesIf , equation (47) can be rewritten aswhere .
If , it haswhere .
According to Young’s inequality, has ; then, equation (47) can be rewritten asWhen , according to the definition of the Lyapunov function in (38), equation (50) can be solvedConsidering the definition of s in (33) and the bounded (51), the margin of error e is obtained:where .
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 function to design the sliding mode control. But the function 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 instead of the discontinuous function , 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: inductance and inductance are ; torque T is ; angular speed ω is ; resistance R is ; motor-induced voltage constant is 0.147; and motor torque constant is 0.72. The high gain parameters are selected as . η is chosen as . τ can be selected as . In order to verify the proposed approaches, we restrain the errors in in the simulations.

In the section, we design two series of simulations with different values of . One is , and the other is selected as . 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 select in the controller of (37) to verify the proposed ESNSMC and ESO of this paper. The output tracking results of , and are illustrated in Figure 4, and the tracking errors are shown in Figure 5. The observer results of , and 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.

From Figures 4 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 are chosen enough higher, it can quickly approximate the states . 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 Figures 4–8, 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 function to take the place of the discontinuous function in the next case.

6.2. Case 2

In this section, the robust is chosen as . Then, Figures 9 and 10 illustrate the output results of the proposed ESNSMC with ESO for , and the ESO estimates the states are shown in Figure 11. Figure 12 is the estimate errors of with the ESO observer. The controller input is illustrated in Figure 13 for the robust .

Compared with the control tracking results Figures 4 and 9 and the controller errors Figures 5 and 10, one can obtained that the continuous function 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 . 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 is about 0.08 at the reversing processes in Figure 7 and the maximum of the error of is about 0.02 at the reversing processes in Figure 12. One can also obtain that, in Figure 12, the maximum of errors of appears at the dynamic processes with the continuous robust function , which are caused by the ESN, but in Figure 8, the maximum of errors of appears at the reversing processes with discontinuous robust function , which are caused by the chattering of the SMC.

Table 1 illustrates the tracking errors and the observer errors with the discontinuous robust function , and the tracking errors and the observer errors with the continuous robust function are shown in Table 2. From Tables 1 and 2, the MAEs of are and , respectively. It is illustrated that the continuous robust function can substantially restrain the chattering of the SMC. But considering the MAE tracking errors are and , respectively, the MIN tracking errors are and , respectively. That means the continuous robust function can substantially restrain the chattering but degrade the tracking precision.

Compared with Figures 8 and 13, one can obviously find that the dynamic processes is influenced by the ESN in Figure 13 with the continuous robust function which can well restrain the chattering of the SMC, but the discontinuous robust function 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 by . 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.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Shandong Provincial Natural Science Foundation of China (ZR2017MF048), Shandong Provincial Key Research and Development Project (2019GGX101005 and 2016GGX105013), National Natural Science Foundation of China (61803216), and Shandong University Science and Technology Plan (J17KA214).

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Copyright © 2020 Xuehui Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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