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Abstract

The unmatched perturbation of load torque is the essential problem in the Electrical Vehicle (EV) based DC motor which requires deep control concern for solution. In this study, the proposed solution to such a control problem is to design three observer-backstepping control schemes for speed trajectory tracking of EVs. The backstepping control algorithm has been developed based on three observers: Adaptive Disturbance Estimator (ADE), Nonlinear Disturbance Observer (NDO), and Quasi-Sliding Mode disturbance Observer (QSMO). Based on Lyapunov stability analysis, the ultimate boundedness of proposed controllers has been detailedly analyzed, assessed, and evaluated in the presence of unmatched perturbation. In the case of backstepping control based on ADE, the stability analysis has been conducted in the presence of stationary and time-varying disturbances to take into account the real load at the EV system. Moreover, a novel mathematical analysis has been conducted to determine the ultimate bound of disturbance estimation error in the case of backstepping based on QSMO. For the NDO-based backstepping controller, the design has been developed by proposing a new structure of nonlinear disturbance observer. The evaluation of proposed controllers has been verified with numerical simulations.

Keywords

DC motor drive backstepping control unmatched disturbance adaptive disturbance estimator quasi-sliding mode disturbance observer

Introduction

In order to ensure the reduction of fossil fuels and to account clean climate, engines of internal combustion are replaced by electric drive engines in most life requirements. The technology of Electric Vehicles (EV) is a suitable and efficient solution due to its reliability and sustainability. In general, to actuate electric vehicles, different types of electric motors can be used like DC motors, synchronous motors, induction motors, and reluctance motors.¹ Due to its simplicity, ease of manufacturing, and maintenance, the DC motor is a suitable choice for most EV applications.² The weight of passengers and the disturbances due to bumpy roads are the potential uncertainties encountering the speed control of EVs. In spite that the model of the DC motor is easy to develop, the presence of unmatched perturbation represents a challenging problem when evaluating the performance of robust controllers in terms of load torque.¹ To tackle the uncertainty problem, one may use the disturbance estimation together with the proposed controller to compensate for the mismatched perturbation as a hard solution. The other solution is to design a control law that can only reduce the effect of perturbation. However, the assessment of the proposed control solution depends on how it can reduce the ultimate bound of convergence error.² Accordingly, many control strategies have been devoted to solve this control problem and mitigating the effects of mismatched disturbances such as high-order sliding mode controllers and approximated sliding mode controllers such as by using disturbance observers, and adaptive estimators.³

The Backstepping controller (BS) is a nonlinear control tool that belongs to the control theory. The BS algorithm adopts the strict feedback class of systems to deal with. Thus it is employed in various situations of complex nonlinear dynamic systems. The main feature of the BS algorithm is to subdivide the system into sub-systems and to convert the whole control objective into a series of simpler, interconnected control tasks by stepping back through these channels. Using such an iterative design of virtual controllers for each channel, made the BS algorithm be suitable approach for stabilizing nonlinear strict-feedback form systems and applied in a wide range of applications.⁴ However, the robustness characteristics of the BS algorithm against uncertainty or perturbations are still premature and a point of attraction for many control researchers.

In the nonlinear systems analysis, perturbations represent an inherent phenomenon of physical systems. It can be divided into sets and groups according to their impact and effects on the nonlinear systems. Vanishing perturbations are the perturbations that vanish in the equilibrium set while the non-vanishing perturbations affect the equilibrium.⁵ Also, the matched perturbations are characterized by their impact on controller column space, opposite to the unmatched (mismatched) perturbations where their impact lies in the controller null space. For the systems subjected to perturbation, the hardest situation is the unmatched non-vanishing perturbation, where the best result can be got is the uniform ultimate boundedness. In this sense, the competition of controllers is based on how well one can give the least ultimate bound.⁴

Disturbance observers represent an advanced level of control theory. Extending the overall system dynamics gives the ability to the disturbance observer to estimate the perturbation (disturbance).⁶ According to the adopted technique, convergence of the estimated perturbation to the real one will judge the reliability and rigidness of the observer. Adaptive disturbance estimation has been proposed firstly as an extension of the model reference adaptive control which consists of the reference model as the observer dynamics and the adaptive law as an auxiliary dynamics.^7,8 The nonlinear disturbance observer uses the Luenberger observer estimation term for the observer to estimate the perturbation.⁹ Another variable structure control disturbance observer based on is Sliding Mode disturbance Observer (SMO), considered a brilliant concept to estimate the system variables or perturbations robustly. The main feature of such observer is the sliding variable that possesses a unity relative degree with respect to the observer injection term. The observation mechanism consists of two phases: the reaching and the sliding phases. The observer forces the sliding variable to converge to zero from any initial condition in the reaching phase. While, for the sliding phase, the disturbance estimation error converges to zero asymptotically.¹⁰ Due to injection term discontinuity, the disturbance estimation will be corrupted with chattering. One of the treatments to rid of this chattering is to use a low pass filter.¹¹ Furthermore, the chattering in the injection term which will represent the disturbance estimation prevents employing the injection term in the backstepping virtual control due to the impossibility of tracking high-frequency components of the disturbance estimation.¹² However, for backstepping-based SMO a chattering free of disturbance estimation is required. Thus, the hyperbolic (tanh) function is used instead of the signum function.^13,14

Backstepping enhancements to tackle the mismatched perturbations become a priority in the nonlinear control theory community.¹⁵ Achievements in that field are represented by fusing estimating techniques to the virtual controller of the backstepping like backstepping-based approximated SMC in Mattei and Monaco¹⁶ and extended observer-based backstepping in Rashad et al.¹⁷ other contributions consist of a higher-order SMC direct compensation like mixed $H_{\infty}$ ISMC control in Castanos and Fridman,¹⁸ hierarchal HOSM quasi-continuous controller in Estrada and Fridman,¹⁹ and adaptive higher-order SMC disturbance estimation in Guo et al.²⁰ In Li et al.,^21–23 the researcher applied a model predictive controller hybridized with other robust controllers and observers to control the EV actuated by a Permanent Magnet Synchronous Motor (PMSM) under matched disturbances.

The following recent works, relevant to the scope of the paper, are briefly highlighted here. A nonlinear integral controller is proposed for DC motors affected by an external load, the integral term is considered as a saturation error function to gain robust properties against the external torque in Al-Samarraie et al.²⁴ A backstepping based on the nonlinear PI controller is used as for the DC motor in AL-Samarraie and Abbas²⁵ DC motor speed control with load torque is considered in Humaidi et al.²⁶ with robust MRAC is designed and tested. An optimized PID controller is considered for the DC motor in Ekinci et al.²⁷ where a heuristic optimization algorithm named Harris Hawk’s Optimization (HHO) algorithm is adopted. Speed control of a Brushed DC motor is implemented with the manually tuned PID control algorithm in Hammoodi et al.²⁸ Fixed point induction and zero average dynamics and controllers driving a Buck converter to DC motor speed control are designed in Hoyos et al.²⁹ Fuzzy controllers-based genetic algorithm optimization was considered for the DC motors in Lotfy et al.³⁰ Robust tracking performance and regulatory effective control for a brushed DC motor are designed and assessed based on active disturbance reduction control based on linear extended observer.³¹ Trajectory tracking of rotor angular velocity for converter-fed system DC motor with load torque is designed using a continuous non-singular terminal sliding mode control.³² Adaptive law-based backstepping cooperated with integral sliding mode control to the angular speed control of DC motor with external load are assessed in Afifa et al.³³ A comparative control analysis of P, PI, and PID is conducted in Barkas et al.³⁴ Velocity trajectory tracking of DC motor based on hierarchical flatness control has been addressed in Roldán-Caballero et al.³⁵ Fractional order PID tuning and design for Electric vehicle drive based on DC motor has been designed and tuned in Patil et al.³⁶

It is worthy to mention that the DC motor actuator is the core of the EV and the load changes due to physical conditions are all lumped into the unmatched uncertainties within its dynamic model. The motivation for this paper is how to propose a control scheme that can solve the speed control presence of unmatched uncertainties due to real conditions of EV motion.³⁷ Enriching this study with complex, nonlinear, adaptive, and variable structure control theories could significantly enhance the performance of EVs under different environmental conditions.^38–40

In this work, the problem of speed trajectory tracking control for EV DC-drive systems subjected to unmatched load torque will be addressed and assessed based on disturbance-based backstepping control methodologies. Three observer-based backstepping control schemes have been proposed, represented by a Backstepping controller based on Adaptive Disturbance Estimator (BS-ADE), Backstepping controller based Nonlinear Disturbance Observer-based (BS-NDO), and Backstepping controller based Quasi-Sliding Mode disturbance Observer (BS-QSMO). The design and stability analysis has been developed to ensure the ultimate boundedness of tracking and estimation convergence errors. In addition to the conventional backstepping control algorithm, the proposed observer-based controllers have been compared in terms of tracking and estimation convergence errors. The main challenging task of these controllers is how to manage the unmatched disturbance in the framework of backstepping control. Figure 1 presents the schematic configuration of the proposed backstepping controller based on the three disturbance observers for the EV based on the DC drive control system.

Figure 1.

The proposed observer-based backstepping control schemes.

The main contribution of this paper is to design a backstepping control algorithm based on three various disturbance observers to achieve speed trajectory tracking control for EVs based on a DC motor drive system in the presence of unmatched load torque. Therefore, one can elaborate the main contribution into the following control tasks:

Development of a Backstepping control algorithm based on an Adaptive Disturbance Estimator (BS-ADE) using Lyapunov-based stability analysis in case of slow-varying (stationary) and time-varying disturbances. The latter case has been firstly analyzed, discussed, and verified and it is an important contributing point of this study.

Development of backstepping based on quasi-sliding mode disturbance observer (BS-QSMO) using Lyapunov stability. The contribution key of this part is conducting a novel mathematical analysis to determine the ultimate bound of disturbance estimation error.

Development of a backstepping control algorithm based on the proposed nonlinear disturbance observer (BS-NDO) using Lyapunov stability analysis. The proposed structure of NDO and the determination of the region of attraction and ultimate bound is another contributing point in this study,

Conducting a comparison study of the performance among the three proposed observer-based controllers.

The rest of the paper is organized into Section “Control problem formulation,” the control problem formulation and assumptions, Section “Control design,” the Lyapunov stability analysis-based controllers design, the simulation results are discussed in Section “Simulation Results” and finally Section “Conclusions,” which ends with the conclusions.

Control problem formulation

The DC motor is the main part of EV which works to convert the electric energy to mechanical torque and speed. In this paper, a separately-excited armature control DC motor has been applied for vehicle actuation as shown in Figure 2.

Figure 2.

DC motor circuit.

According to Newton’s second law and Kirchhoff’s Voltage Law (KVL), the following model can be established¹

{\overset{\cdot}{x}}_{1} = - a_{1} x_{1} + a_{2} x_{2} + d

(1)

{\overset{\cdot}{x}}_{2} = - a_{3} x_{1} + a_{4} x_{2} + \frac{1}{L_{a}} u

(2)

where, $x_{1} = ω$ and $x_{2} = i_{a}$ , and $i_{a}$ is the armature current, $ω$ represents the armature angular speed, $L_{a}$ denotes the winding inductance of armature. The disturbance (load torque) is represented by $d = \frac{1}{J} T_{l}$ such that $| d | = \frac{1}{J} | T_{l} | \leq d_{1}$ , where $d_{1}$ is the upper bound of load torque. The parameters $a_{i}$ ( $i = 1, \dots, 4$ ) can be defined by $a_{1} = b / J$ , $a_{2} = K_{b} / J$ , $a_{3} = K_{b} / L_{a}$ , $a_{4} = K_{b} / R_{a}$ , where $R_{a}$ represents the resistance of the armature windings. The signal $u$ represents the control action to actuate the DC motor system.

It is clear from equations (1) and (2) that the model includes unmatched perturbation (load torque), which is considered a challenging control problem to be addressed in this study. Therefore, the backstepping controller-based disturbance observer is the control objective for the above model to make the angular speed $x_{1}$ tracks $x_{1 d}$ which represents the desired trajectory of the accelerator pedal taking into account the unmatched perturbation $d$ which it assumed to be bounded with known bound. Also, all the states are available for feedback.

The system of equations (1) and (2) represents a strict-feedback system and by letting the speed error $e_{1} = x_{1} - x_{1 d}$ , then the speed error dynamics can be written as

{\overset{\cdot}{e}}_{1} = - a_{1} x_{1} - {\overset{\cdot}{x}}_{1 d} + d + a_{2} x_{2}

(3)

Remark 1: error ultimate bound represents the control objective such that $| e_{1} | \leq ρ (d_{1})$ , where $ρ$ is a K-class function which arbitrarily minimized via either adjusting the design parameters to force the error to be kept in the origin neighbor or the upper bound $d_{1}$ and its first derivative.

Control design

There are many control extensions of the backstepping algorithm that deal with unmatched disturbance. Fusing the observer techniques with the backstepping control could effectively tackle the unmatched problem, which motivated a rapid rise of observer-based backstepping control theory. In what follows, the backstepping control design based on three versions of observers will be analyzed in details for the DC drive system under unmatched disturbance (load torque).

Backstepping

A virtual controller for the unperturbed system of equation (3) can be designed as:

x_{2}^{*} = - a_{1} x_{1} + {\overset{\cdot}{x}}_{1 d} - c e_{1}

(4)

Using a Lyapunov candidate for the unperturbed individual channel of equation (3), and $x_{2}^{*}$ of equation (4) replacing $x_{2}$ to have

V_{e_{1}} = \frac{1}{2} e_{1}^{2}

(5)

or,

{\overset{\cdot}{V}}_{e_{1}} = - {ce}_{1}^{2} < 0 \forall e_{1} \neq 0

(6)

The error $z_{2}$ which represents the second system of the EV assumed as $z_{2} = x_{2} - x_{2}^{*}$ , then

{\overset{\cdot}{e}}_{1} = - c e_{1} + d + z_{2}

(7)

The $z_{2}$ dynamics can be expressed as

\begin{matrix} {\overset{\cdot}{z}}_{2} = - a_{2} a_{3} x_{1} - a_{2} a_{4} x_{2} - c^{2} e_{1} + c z_{2} - a_{1}^{2} x_{1} + a_{1} a_{2} x_{2} \\ - {\overset{\cdot\cdot}{x}}_{1 d} + \frac{a_{2}}{L_{a}} u + cd + a_{1} d \end{matrix}

(8)

By defining a total Lyapunov function of,

V = \frac{1}{2} (e_{1}^{2} + z_{2}^{2})

(9)

Taking the time derivative of $V (e_{1}, z_{2})$ to get

\begin{matrix} \dot{V} = - c e_{1}^{2} + e_{1} d + z_{2} (- a_{2} a_{3} x_{1} - a_{2} a_{4} x_{2} - c^{2} e_{1} + c z_{2} \\ - a_{1}^{2} x_{1} + a_{1} a_{2} x_{2} - {\overset{\cdot\cdot}{x}}_{1 d} + \frac{a_{2}}{L_{a}} u + (c + a_{1}) d) \end{matrix}

(10)

Let the lumped perturbations represented as $ζ_{1} = (c + a_{1}) d$ where $| ζ_{1} | \leq μ_{1} > 0 \in R$ , then the control action can be chosen as

\begin{matrix} u = \frac{L_{a}}{a_{2}} (a_{2} a_{3} x_{1} + a_{2} a_{4} x_{2} + a_{1}^{2} x_{1} - a_{1} a_{2} x_{2} \\ - (1 - c^{2}) e_{1} - 2 c z_{2} + {\overset{\cdot\cdot}{x}}_{1 d}) \end{matrix}

(11)

Using equation (11), equation (10) becomes

\begin{matrix} \overset{\cdot}{V} = - c (e_{1}^{2} + z_{2}^{2}) + e_{1} d + z_{2} ζ_{1} \leq - c ‖ (e, z) ‖^{2} \\ + \sqrt{{| d |}^{2} + {| ζ_{1} |}^{2}} ‖ (e, z) ‖ \end{matrix}

(12)

Let $‖ D_{1} ‖ = \sqrt{d_{1}^{2} + μ_{1}^{2}}$ , then as in Khalil,⁴ Let $0 < ϑ < 1$ then,

\overset{\cdot}{V} \leq - c (1 - ϑ) ‖ (e, z) ‖^{2} + ‖ D_{1} ‖ ‖ (e, z) ‖ - c ϑ ‖ (e, z) ‖^{2}

(13)

\overset{\cdot}{V} \leq - c (1 - ϑ) ‖ (e, z) ‖^{2} \forall ‖ (e, z) ‖ > \frac{1}{c ϑ} ‖ D_{1} ‖

(14)

Referring to Khalil⁴ (Theorem 4.18 and Lemma 9.2), the ultimate bound is

‖ (e, z) ‖ \leq ρ (‖ D_{1} ‖)

(15)

where $ρ (‖ D_{1} ‖) = \frac{1}{c} ‖ D_{1} ‖$ then,

‖ (e, z) ‖ \leq \frac{1}{c} ‖ D_{1} ‖

(16)

Equation (16) represents the challenge of this paper. Despite that the backstepping controller could stabilize the system of equation (3) separately, certain terms are required to do this properly. Hence, the ultimate bound that the backstepping controller (mathematically) can provide depends on the upper bound of $‖ D_{1} ‖$ and the design parameter $c$ .

Backstepping control based on adaptive disturbance estimator (BS-ADE)

Including the unmatched perturbation estimation in the virtual controller of the backstepping is the key feature for robustness enhancement of the backstepping controller. Nevertheless, the perturbation upper bound is not a prerequisite for the adaptive estimator. To proceed, two cases of the perturbation will be considered:

Case I: Slow varying perturbation

An important assumption has to be strictly followed

The assumption of $\overset{\cdot}{d} = 0$ imposed on the first derivative of disturbance indicates that the disturbance has slow-varying behavior or it has stationary characteristics.

A virtual controller is assigned to the upper sub-system of equation (3)

x_{2}^{*} = - a_{1} x_{1} + {\overset{\cdot}{x}}_{1 d} - c e_{1} - \hat{d}

(17)

where estimation of the disturbance is represented by $\hat{d}$ , and $c > 0 \in R$ , represents the design parameter of error convergence rate, the error dynamics of equation (3) can be managed by pursuing the same analysis of backstepping with the use of a virtual controller of equation (17) instead of equation (4) as

{\overset{\cdot}{e}}_{1} = - c e_{1} + z_{2} + \bar{d}

(18)

Where the disturbance estimation error is represented by $\bar{d} = d - \hat{d}$ . Using equation (17) and the fact of $\overset{\cdot}{\bar{d}} = - \overset{\cdot}{\hat{d}}$ then equation (8) can be rewritten as

\begin{matrix} {\overset{\cdot}{z}}_{2} = - a_{3} a_{2} x_{1} - a_{2} a_{4} x_{2} - c^{2} e_{1} + c z_{2} - a_{1}^{2} x_{1} + a_{1} a_{2} x_{2} \\ - {\overset{\cdot\cdot}{x}}_{1 d} + \overset{\cdot}{\hat{d}} + \frac{a_{2}}{L_{a}} u + c \bar{d} + a_{1} d \end{matrix}

(19)

The suggested control law can be given by

u = \frac{L_{a}}{a_{2}} (u_{1} + u_{2})

(20)

where,

\begin{matrix} u_{1} = a_{2} a_{3} x_{1} + a_{2} a_{4} x_{2} - (1 - c^{2}) e_{1} - 2 c z_{2} + a_{1}^{2} x_{1} \\ - a_{1} a_{2} x_{2} + {\overset{\cdot\cdot}{x}}_{1 d} - \overset{\cdot}{\hat{d}} \end{matrix}

(21)

Thereafter,

{\overset{\cdot}{z}}_{2} = - c z_{2} - e_{1} + u_{2} + a_{1} d + c \bar{d}

(22)

A Lyapunov function consisting estimation error term can be chosen as

V_{2} = 0.5 (e_{1}^{2} + z_{2}^{2} + γ^{- 1} {\bar{d}}^{2})

(23)

where $γ > 0 \in R$ stands for the learning rate of adaptive law. The time derivative of the equation (23) can be given as

{\overset{\cdot}{V}}_{2} = - c (e_{1}^{2} + z_{2}^{2}) + z_{2} (u_{2} - a_{1} d) + \bar{d} (e_{1} + c z_{2} - γ^{- 1} \overset{\cdot}{\hat{d}})

(24)

Using $u_{2} = a_{1} \hat{d}$ , then equation (24) becomes

{\overset{\cdot}{V}}_{2} = - c (e_{1}^{2} + z_{2}^{2}) + \bar{d} ((c - a_{1}) z_{2} + e_{1} - γ^{- 1} \overset{\cdot}{\hat{d}})

(25)

To ensure ${\overset{\cdot}{V}}_{2} \leq 0$ , the following adaptive law can be deduced

\overset{\cdot}{\hat{d}} = γ (e_{1} + (c - a_{1}) z_{2})

(26)

This leads to

{\overset{\cdot}{V}}_{2} = - c (e_{1}^{2} + z_{2}^{2})

(27)

Using Barbalat’s Lemma,⁴¹ the asymptotic convergence of $V (e_{1}, z_{2}, \bar{d}) \to 0$ is guaranteed.

Case II: Time-varying perturbation

In the case of time-varying perturbation $\overset{\cdot}{d} \neq 0$ , one may assume the boundedness of $| \overset{\cdot}{d} | = \frac{1}{J} | {\overset{\cdot}{T}}_{l} | \leq d_{2}$ has to be satisfied. Then, equation (24) becomes

{\dot{V}}_{2} = e_{1} {\dot{e}}_{1} + z_{2} {\dot{z}}_{2} + γ^{- 1} \dot{d} \bar{d} - γ^{- 1} \bar{d} \dot{\hat{d}}

(28)

Again, using the adaptive law of equation (26) and the control law of equations (20), (21), and $u_{2} = a_{1} \hat{d}$ results in

{\overset{\cdot}{V}}_{2} = - c (e_{1}^{2} + z_{2}^{2}) + γ^{- 1} \overset{\cdot}{d} \bar{d}

(29)

Then, a uniform ultimate boundedness of ${\overset{\cdot}{V}}_{2}$ can be guaranteed; that is,

{\overset{\cdot}{V}}_{2} \leq - c ‖ (x, z, \bar{d}) ‖^{2} + γ^{- 1} ‖ (x, z, \bar{d}) ‖ | \overset{\cdot}{d} |

(30)

Let $0 < ϑ < 1$ , then

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - c ‖ (x, z, \bar{d}) ‖^{2} + γ^{- 1} ‖ (x, z, \bar{d}) ‖ | \overset{\cdot}{d} | + c ϑ ‖ (x, z, \bar{d}) ‖^{2} \\ - c ϑ ‖ (x, z, \bar{d}) ‖^{2} \end{matrix}

or,

{\overset{\cdot}{V}}_{2} \leq - c (1 - ϑ) ‖ (x, z, \bar{d}) ‖^{2} + γ^{- 1} ‖ (x, z, \bar{d}) ‖ | \overset{\cdot}{d} | - c ϑ ‖ (x, z, \bar{d}) ‖^{2}

This yields the following results

{\overset{\cdot}{V}}_{2} \leq - c (1 - ϑ) ‖ (x, z, \bar{d}) ‖^{2} \forall ‖ (x, z, \bar{d}) ‖ > \frac{γ^{- 1} | \overset{\cdot}{d} |}{c ϑ}

(31)

Referring to Akhtar and Patil² (Theorem 4.18 and Lemma 9.2), the ultimate bound is

‖ (x, z, \bar{d}) ‖ \leq \frac{γ^{- 1} d_{2}}{c}

(32)

Equation (32) represents the ultimate bound of the tracking and disturbance estimation errors which yields a brilliant achievement, in addition to the design parameter $c$ , the ultimate bound depends on the learning rate reciprocal $γ^{- 1}$ where a faster adaptation due to increasing the learning rate $γ$ the less ultimate bound acquired.

Backstepping controller-based nonlinear disturbance observer (BS-NDO)

Regarding the separation principle in the observer, a design that gives the ability to separate the backstepping controller and the disturbance observer can be achieved. Considering the estimation of mismatched disturbance, the estimates of the designed observer will be used to synthesize the virtual controller of the backstepping control algorithm and this is the improvement key in the backstepping control based on nonlinear disturbance observer (BS-NDO). In this part of the observer-based controller, the assumption of Case I in the previous section will be pursued.

The following observer is proposed

{\dot{\hat{x}}}_{1} = a_{2} x_{2} - a_{1} x_{1} + \hat{d}

(33)

Defining the estimation error as ${\bar{x}}_{1} = x_{1} - {\hat{x}}_{1}$ , the estimation error dynamics are given by

{\dot{\bar{x}}}_{1} = \bar{d} = d - \hat{d}

(34)

Choosing $\hat{d} = k_{n 1} (x_{1} - {\hat{x}}_{1})$ , where $k_{n 1} > 0 \in R$ is design parameter, and taking the assumption $\overset{\cdot}{d} = 0$ into account, then the derivative of equation (34) becomes

\dot{\bar{d}} = - \dot{\hat{d}} = - k_{n 1} {\dot{\bar{x}}}_{1} = - k_{n 1} \bar{d}

(35)

Following the previous analysis (equations (17)–(19)), the error dynamics for $e_{1}$ and $z_{2}$ becomes

{\overset{\cdot}{e}}_{1} = - c e_{1} + \bar{d} + z_{2}

(36)

\begin{matrix} {\overset{\cdot}{z}}_{2} = - a_{2} a_{3} x_{1} - a_{2} a_{4} x_{2} + \frac{a_{2}}{L_{a}} u - c^{2} e_{1} + c z_{2} - a_{1}^{2} x_{1} \\ + a_{1} a_{2} x_{2} - {\overset{\cdot\cdot}{x}}_{1 d} + \overset{\cdot}{\hat{d}} + a_{1} d + c \bar{d} \end{matrix}

(37)

The control law can be chosen to be expressed in terms of its components as,

u = \frac{L_{a}}{a_{2}} (u_{1} + u_{2})

(38)

where, $u_{1}$ is defined as

\begin{matrix} u_{1} = a_{2} a_{3} x_{1} + a_{2} a_{4} x_{2} - (1 - c^{2}) e_{1} - 2 c z_{2} + a_{1}^{2} x_{1} \\ - a_{1} a_{2} x_{2} + {\overset{\cdot\cdot}{x}}_{1 d} + c \hat{d} \end{matrix}

(39)

Equation (37) can be rewritten as

{\overset{\cdot}{z}}_{2} = - c z_{2} - e_{1} + u_{2} + \overset{\cdot}{\hat{d}} + (c - a_{1}) d

(40)

Let $ζ_{2} = \overset{\cdot}{\hat{d}} + cd - a_{1} d$ , assuming ${\overset{\cdot}{ζ}}_{2} = 0$ , so

{\overset{\cdot}{z}}_{2} = - c z_{2} - e_{1} + u_{2} + ζ_{2}

(41)

The following observer for $z_{2}$ is proposed as,

{\dot{\hat{z}}}_{2} = - c z_{2} - e_{1} + {\hat{ζ}}_{2} + u_{2}

(42)

where ${\hat{ζ}}_{2} = k_{n 2} (z_{2} - {\hat{z}}_{2}) = k_{n 2} {\bar{z}}_{2}$ , $k_{n 2} > 0 \in R$ is a design parameter, ${\bar{z}}_{2}$ represents the estimation error. Accordingly, the observer error dynamics is simplified to

{\dot{\bar{z}}}_{2} = ζ_{2} - {\hat{ζ}}_{2} = {\bar{ζ}}_{2}

(43)

Since ${\overset{\cdot}{ζ}}_{2} = 0$ , then

\dot{{\bar{ζ}}_{2}} = - \dot{{\hat{ζ}}_{2}} = - k_{n 2} {\dot{\bar{z}}}_{2} = - k_{n 2} {\bar{ζ}}_{2}

(44)

This indicates the asymptotic stability of dynamics. The control part $u_{2}$ can be chosen as $u_{2} = - {\hat{ζ}}_{2}$ , then equation (41) becomes

{\overset{\cdot}{z}}_{2} = - c z_{2} - e_{1} + {\bar{ζ}}_{2}

(45)

Taking the time derivative of Lyapunov candidate $V (e_{1}, z_{2})$ to have

\overset{\cdot}{V} = - c (e_{1}^{2} + z_{2}^{2}) + e_{1} \bar{d} + z_{2} {\bar{ζ}}_{2}

(46)

or,

\overset{\cdot}{V} \leq - c ‖ (x, z) ‖^{2} + ‖ (x, z) ‖ ‖ {\bar{D}}_{2} ‖

(47)

where $‖ {\bar{D}}_{2} ‖ = \sqrt{{| \bar{d} |}^{2} + {| {\bar{ζ}}_{2} |}^{2}}$ .

Letting $0 < ϑ < 1$ , then

\overset{\cdot}{V} \leq - c (1 - ϑ) ‖ (x, z) ‖^{2} + ‖ (x, z) ‖ ‖ {\bar{D}}_{2} ‖ - c ϑ ‖ (x, z) ‖^{2}

or,

\overset{\cdot}{V} \leq - c (1 - ϑ) ‖ (x, z) ‖^{2} \forall ‖ (x, z) ‖ > \frac{‖ {\bar{D}}_{2} ‖}{c ϑ}

(48)

Referring to Akhtar and Patil² Lemma (9.2), Theorem (4.18) then the bound is

‖ (x, z) ‖_{2} \leq \leq \frac{‖ {\bar{D}}_{2} ‖}{c}

(49)

The ultimate bound dependency of equation (49) is related to the disturbance estimation error representing a considerable achievement.

Backstepping control based on quasi-sliding mode disturbance observer

In this part, two sub-observers are designed: one for the system channel of equation (1) to account for unmatched disturbance, while the design of the other observer is devoted to cope with the projection of lumped disturbance to the second channel. Firstly, the observer of the first channel is designed based on equation (1)

{\dot{\hat{x}}}_{1} = - a_{1} x_{1} + a_{2} x_{2} + \hat{d}

(50)

The estimation error is defined by ${\bar{x}}_{1} = x_{1} - {\hat{x}}_{1}$ and its dynamics can be described

{\bar{x}}_{1} = d - \hat{d} = \bar{d}

(51)

Selecting $\hat{d} = k_{1} \tanh ({\bar{x}}_{1} / η)$ , where $1 >> η > 0 \in R$ , and $k_{1} > 0 \in R$ are design parameters. The stability analysis based on the sliding mode stability analysis gives:

{\bar{x}}_{1} {\bar{x}}_{1} \leq - | {\bar{x}}_{1} | (k_{1} \tanh ({\bar{x}}_{1} / η) - | d |)

(52)

where $k_{1} > d_{1}$ is a necessary condition to satisfy equation (52). Using the same procedure followed by equations (17)–(19), to get

{\overset{\cdot}{e}}_{1} = - c e_{1} + z_{2} + \bar{d}

(53)

u = \frac{L_{a} (u_{1} + u_{2})}{a_{2}}

(54)

where,

\begin{matrix} u_{1} = a_{2} a_{3} x_{1} + a_{2} a_{4} x_{2} + a_{1}^{2} x_{1} - a_{1} a_{2} x_{2} + {\overset{\cdot\cdot}{x}}_{1 d} \\ + c \hat{d} - (1 - c^{2}) e_{1} - 2 c z_{2} \end{matrix}

(55)

Then,

{\dot{z}}_{2} = - c z_{2} - e_{1} + u_{2} + c d - a_{1} d + \dot{\hat{d}}

(56)

letting $ζ_{2} = (c - a_{1}) d + \dot{\hat{d}}$ , where $| ζ_{2} | \leq μ_{2}$ , one can obtain

{\overset{\cdot}{z}}_{2} = - c z_{2} - e_{1} + ζ_{2} + u_{2}

(57)

The proposed structure for the second sub-observer can be proposed as

{\dot{\hat{z}}}_{2} = - c z_{2} - e_{1} + {\hat{ζ}}_{2} + u_{2}

(58)

Let ${\hat{ζ}}_{2} = k_{2} \tanh ({\bar{z}}_{2} / η)$ , where $k_{2} > 0 \in R$ is a design parameter, then the estimation error dynamics can be written as:

{\dot{\bar{z}}}_{2} = ζ_{2} - {\hat{ζ}}_{2} = {\bar{ζ}}_{2}

(59)

One can establish the reaching condition as follows:

{\bar{z}}_{2} {\dot{\bar{z}}}_{2} \leq - | {\bar{z}}_{2} | (k_{2} \tanh ({\bar{z}}_{2} / η) - ζ_{2})

(60)

where $k_{2} > μ_{2}$ is required. If the control part is assigned as $u_{2} = - {\hat{ζ}}_{2}$ , then

{\overset{\cdot}{z}}_{2} = - c z_{2} - e_{1} + {\bar{ζ}}_{2}

(61)

Using equation (9), then $\overset{\cdot}{V}$ becomes

\overset{\cdot}{V} = - c (e_{1}^{2} + z_{2}^{2}) + e_{1} \bar{d} + z_{2} {\bar{ζ}}_{2}

(62)

The estimation error ultimate bound of the ${\bar{x}}_{1}$ and ${\bar{z}}_{2}$ can be determined by using the following Lyapunov function

V_{{\bar{x}}_{1}} = \frac{{\overset{\cdot}{\bar{x}}}_{1}^{2}}{2}

(63)

{\dot{V}}_{{\bar{x}}_{1}} = {\bar{x}}_{1} \dot{d} - {\bar{x}}_{1} \dot{\hat{d}} \leq - | {\bar{x}}_{1} | (| \dot{\hat{d}} | - d_{2})

(64)

Since $\hat{d} = k_{1} \tanh ({\bar{x}}_{1} / η)$ , then then its time derivative is given by $\overset{\cdot}{\hat{d}} = k_{1} \frac{{\overset{\cdot}{\bar{x}}}_{1}}{η} (1 - \tanh^{2} (\frac{{\bar{x}}_{1}}{η}))$ . Accordingly, $\dot{\hat{d}}$ ultimate bound can be found to be $| \overset{\cdot}{\hat{d}} | \leq \frac{k_{1} | {\bar{x}}_{1} |}{η}$ at the worst case where $| {\bar{x}}_{1} | = 0$ . Then,

{\overset{\cdot}{V}}_{{\bar{x}}_{1}} \leq - | {\overset{\cdot}{\bar{x}}}_{1} | (\frac{k_{1} | {\overset{\cdot}{\bar{x}}}_{1} |}{η} - d_{2})

(65)

To ensure ${\overset{\cdot}{V}}_{{\bar{x}}_{1}} \leq 0$ , the following has to be satisfied:

| {\overset{\cdot}{\bar{x}}}_{1} | = | \bar{d} | \leq \frac{η d_{2}}{k_{1}}

(66)

Similarly, for the ${\bar{ζ}}_{2}$

| {\overset{\cdot}{\bar{z}}}_{2} | = | {\bar{ζ}}_{2} | \leq \frac{η | \overset{\cdot}{ζ} |}{k_{2}}

(67)

Accordingly, the uniform ultimate boundedness can be guaranteed for $\overset{\cdot}{V}$

\overset{\cdot}{V} \leq - c {(x, z)}^{2} + \frac{η}{k} {\bar{D}}_{3} (x, z)

(68)

where ${\bar{D}}_{3} = \sqrt{{(\frac{k}{k_{1}} d_{2})}^{2} + {(\frac{k}{k_{2}} | \overset{\cdot}{ζ} |)}^{2}}$ . Letting $0 < ϑ < 1$ , one can deduce

\begin{matrix} \begin{matrix} \overset{\cdot}{V} \leq - c ‖ (x, z) ‖^{2} + \frac{η}{k} ‖ (x, z) ‖ ‖ {\bar{D}}_{3} ‖ + c ϑ ‖ (x, z) ‖^{2} - c ϑ ‖ (x, z) ‖^{2} \end{matrix} \\ \overset{\cdot}{V} \leq - c (1 - ϑ) ‖ (x, z) ‖^{2} + \frac{η}{k} ‖ (x, z) ‖ ‖ {\bar{D}}_{3} ‖ - c ϑ ‖ (x, z) ‖^{2} \\ \overset{\cdot}{V} \leq - c (1 - ϑ) ‖ (x, z) ‖^{2} \forall ‖ (x, z) ‖_{2} > \frac{η ‖ {\bar{D}}_{3} ‖}{ck ϑ} \end{matrix}

(69)

Then, according to Lemma (9.2) and Theorem (4.18) of Khalil,⁴ the ultimate bound can be finally established

‖ (x, z) ‖_{2} \leq \leq \frac{η ‖ {\bar{D}}_{3} ‖}{ck}

(70)

The ultimate bound of equation (70) represents the new contribution achieved in this study, the ultimate bound is related to the disturbance estimation error, which in turn relies on the upper bound of differentiable disturbance. That bound is minimized by decreasing the parameters $η$ and increasing the gain $k$ . On the basis of stability analysis, using equations (51) and (59), the reaching phase can be avoided or crossed over by selecting ( ${\bar{x}}_{1} (t_{0}) = 0, {\bar{z}}_{2} (t_{0}) = 0$ ), which means that the disturbance estimation will take place from the beginning $t \geq t_{0}$ .

Simulation results

The verification of proposed disturbance-based controllers for the DC motor system has been made based on numerical simulation using MATLAB according to the operating region of EV system as in Lin et al.³⁷ The setting of DC motor parameters used in the simulation results are listed in Table 1.^42,43

Table 1.

DC motor system parameters.

Parameters	Values	Units
$J$	7.95 × 10⁻⁵	kg m²/rad
$L_{a}$	105 × 10⁻⁶	H
$R_{a}$	0.7	$Ω$
$b$	0	N m s/rad
$K_{b}$	59 × 10⁻³	N m/A or V s/rad

The desired speed trajectory which represents a commanded speed by pressing the accelerator pedal to multi-speed levels which can be described as follows

x_{1 d} = {\begin{matrix} 104.72 0 \leq t \leq 3 \\ 136.13 3 \leq t \leq 7 \\ 115.2 7 \leq t \end{matrix} rad / s

The load torque represents the traction load either due to changes in passengers’ weight, road terrains, road bumps, changes of the center of gravity (COG) for EV, or the friction changes caused by slippery roads.³⁷ Considering a shifted sinusoidal signal as: $T_{l} = 0.05 + 0.03 \sin (1.2 π t) + 0.02 \sin (4 π t)$ (N m). For all controllers, choosing the design parameters as: $γ = 0.152$ , $c = 200$ , $k_{n 2} = 4000$ , $k_{n 1} = 2200$ , $k_{2} = 755000$ , $k_{1} = 1300$ , $η_{1} = 0.05$ , and $η_{2} = 5$ . Furthermore, the parameters are primarily selected based on the performance requirements. However, the parameter $c$ represents the convergence rate of the tracking error and can be chosen freely. When it comes to the adaptive disturbance estimator, the learning rate $γ$ is determined through trial and error and should be adjusted according to the expected frequency of the disturbance. The gains of the nonlinear disturbance observer, $k_{n 1}$ , and $k_{n 2}$ , should be chosen in relation to the expected disturbance frequency, with the observer dynamics needing to be faster than the disturbance. As for the quasi-sliding mode observer, the approximation rate $η$ is determined through trial and error, and the observer’s gains, $k_{1}$ and $k_{2}$ , must be greater than the upper bound of the disturbance. The initial conditions of $x_{1} (t_{0}) = 0$ and $x_{2} (t_{0}) = 0$ ; the initial conditions of the observers are chosen as ${\hat{x}}_{1} (t_{0}) = x_{1} (t_{0})$ , ${\hat{e}}_{2} (t_{0}) = e_{2} (t_{0})$ . The objective of the controller is to ensure a minimum robust ultimate bound of error convergence the error $e_{1}$ .

Figure 3 shows the state $x_{1}$ time response for the three controllers BS-ADE, BS-QSMO, and BS-NDO. In terms of transient characteristics, the BS-ADE shows an overshoot as compared to other controllers. On the other hand, the BS-QSMO and BS-NDO have no over-shoot due to the exponential convergence of the tracking error. However, the response of BS-NDO is faster than that based on BSQSMO. Figure 4 shows the behavior of tracking error $e_{1}$ based on the three observer-based controllers and it is clear that the least steady-state error is given by BS-QSMO, while BS-ADE gives the highest stated state error. Table 2 reports the RMS values of tracking error $e_{1}$ for all controllers where quantitatively, the lowest speed tracking error are achieved by BS-QSMO then BS-NDO and then BS-ADE.

Figure 3.

The behavior of $x_{1}$ for the suggested controllers.

Figure 4.

The behavior of $e_{1}$ for the suggested controllers.

Table 2.

Performance and robustness characteristics.

Controller	Performance factor in RMS
	$e_{1}$	$u$	$T_{l} - {\hat{T}}_{l}$
BS-ADE	3.2933	6.5661	0.0172
BS-NDO	2.8332	6.5555	1.3993 × 10⁻⁴
BS-QSMO	2.7844	6.5550	5.5548 × 10⁻⁶

Figure 5 shows the behaviors of control actions due to proposed observer-based controllers. It is evident that the controlled system based on BS-ADE has the highest initial control effort as compared to other controllers due to that adaptive law initial condition while in NDO and QSMO the observers guaranteed the disturbance estimation from $t \geq t_{0}$ due to the choice of estimation error nullity. On the other hand, the initial efforts based on all the controllers have low and reasonable values and the physical actuators could easily provide these levels of control signals in the real environment. The evaluation of control efforts based on proposed controllers in RMS values is reported in Table 2 where quantitatively, the lowest control effort are required from BS-QSMO then BS-NDO and then BS-ADE.

Figure 5.

Control action $u$ for the proposed observer-based controllers.

Figure 6 shows the tracking error $e_{2}$ of the lower sub-system. One can notice that the unmatched perturbations have a direct effect on errors for all controllers. However, the least effect of perturbation can be detected in the case of BS-QSMO, while there is a noticeable effect of the disturbance on $e_{2}$ based on controllers BS-NDO and BS-ADE. In the latter case, one can see that the state $x_{2}$ can hardly follow the virtual controller due to the effect of the perturbation.

Figure 6.

Armature current error $e_{2}$ for the suggested controllers.

Figure 7 shows the responses of armature currents, represented by state variable $x_{2}$ , due to proposed controllers. It is worth mentioning that the state $x_{2}$ tries to strictly follow the virtual controller, which in turn enables the state variable $x_{1}$ to track the desired speed. According to the figure, one can observe that the highest initial current can be produced by BS-ADO controller, while the other controllers give reasonable initial values of armature current.

Figure 7.

The behavior of $x_{2}$ for the proposed controllers.

Figure 8 shows the errors of the state estimation of both BS-QSMO and BS-NDO where it is obvious that the BS-QSMO controller has less estimation error than BS-NDO. Figure 9 shows the actual load torque and its estimates resulting from observers. One can see that all observers are capable of estimating the load torque with considerably small errors. Due to the adaptive law inherited in the mechanism of controller BS-ADE, more time is required for this controller to give acceptable estimation performance. Figure 10 represents the estimation error between the actual load torque $T_{l}$ and its estimate ${\hat{T}}_{l}$ . As previously proved in the stability analysis, it is clear that the ultimate bound can be reached by all proposed controllers. However, the ultimate bound produced by BS-QSMO is the minimum as compared to that based on NDO and ADE. Disturbance estimation error quantitative evaluation based on RMS values has been reported in Table 2 where the lowest ultimate bound has been satisfied by the BS-QSMO.

Figure 8.

State estimation $x_{1}, {\hat{x}}_{1}$ and $e_{2}, {\hat{e}}_{2}$ for BS-NDO and BS-QSMO controllers.

Figure 9.

The behaviors of actual torque $T_{l}$ and estimated torque ${\hat{T}}_{l}$ .

Figure 10.

The estimation errors ( $T_{l}$ − ${\hat{T}}_{l}$ ).

Conclusions

This study has addressed the problem of ultimate boundedness for DC-actuated electrical vehicle system in the presence of unmatched disturbance (torque). The backstepping control based on three different disturbance observers has been proposed. A rigorous stability analysis and design have been presented and developed based on the Lyapunov theorem to establish robust observer-based backstepping controllers under unmatched load disturbance. In terms of transient characteristics, the BS-ADE showed the highest peak overshoot with the fastest response as compared to other controllers. However, the lowest convergence tracking error is given by BS-QSMO, while BS-ADE gives the highest ultimate bound of convergence error. In terms of control effort, the controlled system based on BS-ADE has the highest initial control effort as compared to other controllers. On the other hand, the initial efforts based on all controllers have low and reasonable values to be applied in the real control environment. The numerical simulation showed that the state variable $x_{2}$ , armature current, tries to strictly follow the virtual controller, which in turn enables the state variable $x_{1}$ (actual speed) to track the desired speed.

Footnotes

Handling Editor: Xiaodong Sun

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Amjad J Humaidi

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Ultimate bounded observer-based control of electrical vehicle driven by DC motor system with unmatched load torque

Abstract

Keywords

Introduction

Control problem formulation

Control design

Backstepping

Backstepping control based on adaptive disturbance estimator (BS-ADE)

Case I: Slow varying perturbation

Case II: Time-varying perturbation

Backstepping controller-based nonlinear disturbance observer (BS-NDO)

Backstepping control based on quasi-sliding mode disturbance observer

Simulation results

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References