Robust decomposed system control for an electro‐mechanical linear actuator mechanism under input constraints

This paper aims to develop a robust decomposed system control (RDSC) strategy under input constraints for an electro‐mechanical linear actuator (EMLA) facing model uncertainty and external disturbances. At first, a state‐space model of a complex multi‐stage gearbox EMLA system, driven by a permanent magnet synchronous motor (PMSM), is developed, and the non‐ideal characteristics of the ball screw are presented through the model. The result is a four‐order nonlinear strict‐feedback form (NSFF) system decomposed into three subsystems. As the paper's main result, a novel RDSC strategy with uniform exponential stability for controlling subsystem states is presented. This developed controller avoids the "explosion of complexity" problem associated with backstepping by treating the time derivative of the virtual control input as an uncertain system term. The proposed method, despite assuming load disturbances and input constraints with arbitrary bounds, offers a straightforward control approach for a broader range of applications. Further, the controller's performance is evaluated by simulating two distinct duty cycles, each representing different levels of demand on the actuator facing load disturbances near the rated motor performance.


INTRODUCTION
The robotics industry is projected to dominate the upcoming decade, as well as to experience considerable market growth.As such, advanced robotic systems are receiving significant attention from both academia and industry, especially concerning their essential role in the development of autonomous technology and battery electric vehicles (BEVs) to enhance the efficiency, safety, and sustainability of mobility. 1-3Mobile manipulators, which are robotic arms mounted on wheeled platforms for industrial use, can benefit from the development of BEVs, as they are capable of performing a wide range of tasks in various environments, from manufacturing and logistics to search and rescue operations and beyond. 4The on-board robotic arms of mobile manipulators can be equipped with hydraulic actuators, to enable their movement and manipulation capabilities, as they are commonly used in such scenarios due to their high power-to-weight ratio.Despite representing a mainstay in numerous industries for decades, the use of hydraulic actuators in mobile manipulators presents several challenges and limitations, the primary of which is that they are energy-inefficient. 5In addition, they are susceptible to leakage, which can lead to decreased pressure and diminished performance. 6 Electrified actuators have emerged as a popular alternative to hydraulic actuators due to their high potential performance, the primary advantage of which over hydraulic actuators lies in their superior efficiency and lesser maintenance needs as well. 7Furthermore, they offer more accurate control, as they integrate with sensors and advanced electronic control systems to enable greater precision when performing such tasks as lifting and moving heavy objects. 8,9The ongoing trend towards electrification in the field of mobile manipulators has led to the emergence of electro-mechanical linear actuators (EMLAs) as an alternative to hydraulic actuators, as they comprise a class of devices designed to transform electrical energy into linear motion, enabling the execution of various functions 10  and typically consisting of several components, including a motor, lead screw or ball screw, nut, and load-bearing component. 11,12 The motor is responsible for providing the required rotational force, and the lead or ball screw is utilized to convert the rotational motion of the motor into linear motion.Permanent magnet synchronous motors (PMSMs) are a common motor type utilized in EMLAs due to their numerous benefits, including high power and torque density, allowing them to generate substantial force and power relative to their size. 13,14In addition, PMSMs are highly efficient at converting electrical power into mechanical power, making them suitable for energy-efficient applications, such as mobile manipulators, whose batteries can store only limited energy. 15,16 EMLAs are considered a multidisciplinary challenge, whose interactions spark intricate dynamics that are difficult to model for control design. 17Accurately modeling the multi-stage components of an EMLA is crucial because of its complex behavior, which can vary under different operating conditions, such as variable loads and parameter uncertainty.In particular, the ball screw exhibits nonlinear and non-ideal characteristics that must be considered in modeling and control design. 17In this way, to simplify the structure of EMLAs, a nonlinear strict-feedback form (NSFF) is a proper option, as it can mathematically express a cascade of functions, each of which takes its input as the output from the previous function.The system input can be directly controlled, but the intermediate variables and system output are determined by the system dynamics.In general, NSFF systems are typically controlled using feedback control, such as adaptive control, nonlinear control, or backstepping control, among others, all of which enable the easy modification of parts of the entire system and a separate investigation of the performance of each subsystem for high-quality results. 18Hence, some recent research has utilized the NSFF control structure for the design of PMSM control to offer effective solutions to the challenges of achieving stability, tracking performance, and ensuring robustness in the presence of disturbances, uncertainties, and input constraints in PMSM control applications. 19-21 In recent years, backstepping control, which is one method compatible with NSFF systems, has gained attention as a research area to address robust control in the electric machinery field, like PMSMs, due to its autonomy and ability to handle complex nonlinear systems, providing a straightforward method for designing robust control systems.This recursive control method is based on the backstepping technique, which involves designing a sequence of controllers, each of which stabilizes one subsystem of the overall system. 22Furthermore, numerous studies in the literature have focused on enhancing the robustness of PMSMs using different types of adaptive backstepping control techniques, which have been demonstrated to possess significant potential for modeling complex systems and as adaptive controllers for nonlinear systems, as evidenced in previous works. 23-25In the field of PMSM control, most research and studies have been conducted with a focus on velocity control.However, the current work is being discussed in a context focused primarily on position servo control. 26,27Conversely, one drawback of backstepping algorithms that can engender complexity is called an "explosion of complexity" due to the repeated time derivative of the virtual control input, which should be solved in complex applications. 28Hence, Lu and Wang 29 proposed the use of a fractional-order command filter for PMSMs to address this issue, and a command filtering-based neural network control scheme was developed to handle input saturation in PMSMs.They, in another paper, 30 studied observer-based command-filtered adaptive neural network (NN) tracking control for a fractional-order chaotic PMSM subject to parameter uncertainties and external load disturbances.Likewise, to resolve the complexity of backstepping, they employed a command-filtering approach based on the first-order Levant differentiator.As demonstrated, most prior research tackled the issue by utilizing various filtering techniques, but the present work deviates from this approach and instead treats the time derivative of the virtual control input as an uncertain term in the system, without relying on any explicit filtering.
Precise measurements of electrical and mechanical parameters, including stator resistance and the friction coefficient, for PMSM, are often challenging.Moreover, commonly, the external load torque value is unknown and cannot be determined accurately.These factors highlight the critical need for the design of robust control methods, as the absence of which can jeopardize a system's performance and safety, as well as result in costly downtime or even catastrophic failure.Specifically, load disturbances can significantly impact the performance and stability of PMSMs, resulting in reduced efficiency, increased wear and tear, and potential damage to both the motor and drive system.Hence, a nonlinear disturbance observer was designed to estimate load disturbances by Liu et al., 31 to regulate PMSM speed when driving using backstepping control, and it combines the speed and current controllers.PMSM systems with unknown load torques were studied by Sun et al. 19 and Ali et al. 32 who proposed new adaptive backstepping control methods to address the position/speed tracking problem, ensuring the asymptotic stability of the overall system.Although recent studies on PMSM control have emphasized addressing external disturbances and uncertainties, they have not adequately addressed the issue of input constraints, which is a common problem in PMSMs.Input constraints are crucial considerations in the control of PMSMs, owing to the inherent physical limitations of the input voltage and current of these motors.The application of control signals exceeding these limits can result in system instability, motor overheating, and permanent damage.Furthermore, the presence of load disturbances and model uncertainties can also cause the control signals to surpass the physical limitations of the motor.In such situations, it is imperative to design control systems that ensure stable and dependable motor operation while adhering to these constraints, thus enabling the operation of the motor within a safe and stable setting and safeguarding it from damage.To address general disturbances and input saturation in PMSMs, Tan et al. 33 proposed a sensorless robust optimal control scheme utilizing two NN-based observers.The first was designed to estimate the back-electromotive force (EMF), while the second was designed to estimate the tracking errors of the rotor position and speed in uniform boundedness.
As extensively discussed earlier, in recent years, robust backstepping control has generated significant interest in PMSMs in NSFF structures.However, it has been observed that this control method has not been widely adopted in current studies concerning EMLAs actuated by PMSMs within NSFF structures.In addition, few studies have comprehensively addressed control problems in EMLAs, such as external disturbances, uncertainties, and input constraints.However, an adaptive backstepping tuning-function sliding-mode controller was proposed by Coban 34 for a class of NSFF systems to overcome quickly the varying parametric and unstructured uncertainties and avoid complexity.The effectiveness of the proposed controller was demonstrated by studying its implementation in an electro-mechanical system.Although Lin and Lee 35 discussed linear induction motors (LIMs), they may not be directly relevant to our application.In addition, they have also not addressed input constraints in the systems.
The present study makes several significant contributions to the field of EMLA control, as follows: (1) A complex EMLA system with a multi-stage gearbox, driven by a PMSM to carry loads, is modeled.(2) A robust control for the EMLA system under input constraints is designed by transforming the model into a four-order NSFF decomposed into three subsystems with uniformly exponential stability.(3) The "explosion of complexity" problem is solved without the need for filters.(4) Finally, a comprehensive analysis of the RDSC's performance with an EMLA across various duty cycles confirms its significance.
The remainder of the paper is organized as follows: 1. Section 2 presents the various components of the EMLA servomechanism, along with the development of related analytical models to examine the energy conversions occurring within the system.The state-space model of the EMLA is then obtained by considering the aforementioned sub-models, and it is presented as a function of linear motion on the load side of the servomechanism to facilitate position control purposes.Finally, the specifications of the EMLA components, including the PMSM, gearbox, and screw, are listed at the end of this section.2. In Section 3, we investigate the control problems of the EMLA system, including input constraints, external disturbances, and uncertainties.To do so, we transform the EMLA system into a four-order NSFF system, consisting of three subsystems, the first of which is associated with the linear motion dynamic of the EMLA, while the second and third are associated with the Park frame related to the PMSM.3. In Section 4, we define the RDSC and present implementable equations to meet the requirements of the EMLA system.4. Section 5 includes an extensive analysis of the uniformly exponential stability of the EMLA system. 5.The performance validity of the RDSC is investigated in Section 6 by considering two types of cycle duties, depending on the load type and the moment of loading and unloading, and the numerical performance of the RDSC for the EMLA is summarized in Tables 4-7.

Mathematical model of EMLA
This paper presents a study of an EMLA that utilizes a surface-mounted PMSM to convert electric power to mechanical power.The mathematical model for this EMLA is developed in the current section, and the system configuration is illustrated in Figure 1.The electric motor of the servomechanism is connected to a reduction gear, which in turn is coupled to a screw-nut mechanism to convert rotary motion into linear motion and to allow the load to move mechanically.
To simplify the EMLA modeling process, the paper considers three sub-models, namely, the PMSM, the gears, and the screw-nut mechanisms, as follows:

Permanent magnet synchronous motor
The rotary part of the PMSM is equipped with PMs, and their flux linking can be written in three ABC phases, as in (1): 36,37 PM cos ) where  PM is the PM flux linkage and  e is the electrical angle of the rotor.The total flux in each stator phase, which comprises the PM and three windings, can be calculated as a function of the self-inductance of the stator windings (L aa , L bb , and L cc ), the mutual inductance between stator windings (L ab , L bc , L ca , etc.), and the PM fluxes in each phase ( a ,  b , and  c ).This relationship is expressed as in (2): Lastly, the voltage across the stator windings (u a , u b , and u c ) can be determined based on the magnetic flux change rate for each stator winding and the stator current (i a , i b , and i c ), as given in (3): where the resistance of the stator winding is denoted as R s .To eliminate the dependence of the PMSM's three-phase voltage on the rotor angle, the Park transformation can be utilized, as it converts the voltage from the ABC frame to the dq0 frame, as illustrated in (4).
where L d , L q , and L 0 are inductances in the d-axis, q-axis, and zero sequence, respectively.Furthermore, the two currents of the PMSM in the d-axis and q-axis are represented as i d and i q .In addition,  is the mechanical rotational speed, and P is the number of rotor PM pole pairs.Subsequently, the electromagnetic torque produced by the PMSM ( m ) can be expressed as in (5): As the rotor of the studied surface-mounted PMSM has a non-salient shape, the inductances L d and L q are identical.Therefore, the electromagnetic torque equation can be simplified and expressed as a function of i q , as derived in (6): 38 The dynamic equation of EMLA motion on the motor side can be explained as in (7): where J m is the motor inertia;  f is the mechanical loss of the motor, including friction and windage;  g1 is the input torque of the gearbox; and θ is the angular acceleration of the rotor of the electric motor.

Reduction gear
The relationship between the input and output torques of the gearbox is as follows (8): 39 where n is gear ratio,  g2 is the output torque of the gearbox, and  GB is the efficiency of the gearbox.It is worth mentioning that  GB is a function of the applied torque and angular velocity of the gearbox, and the gear ratio can be defined as in (9): where n 1 and n 2 are the number of input and output gear teeth.The output torque of the gears is coupled to the transmission system of the EMLA to deliver power from the gears to the screw nut, and the motion equation can be considered as in (10): where J t and b t are the inertia and viscosity of the transmission system, respectively.In addition, θt and θt are the angular acceleration and angular velocity of the transmission system, respectively, while  BS is the torque applied to the screw mechanism, and it will be calculated in Section 2.1.3.

Screw-nut mechanism
The screw-nut mechanism of an EMLA is responsible for converting the rotary motion into linear motion, and the torque-force relationship motion equation of a screw mechanism can be written as in (11): 40,41    BS =  2 BS ( BS ,  BS ) F BS (11) where the direct, reverse, and general efficiencies of the screw mechanism, denoted as  d ,  i , and  BS , are obtained using ( 12): 42 In the mentioned equation,  is the friction coefficient of the screw system and  is the lead angle of the screw.For simplification, the coefficient of converting rotary movement to linear movement, called c RL , is defined as in (13): where  is the screw pitch of the screw mechanism.Meanwhile, the output force of the screw mechanism, called F BS , can be computed as in (14): where F L , m BS , b BS , and k BS are the load, the mass, damping coefficient, and stiffness of the screw mechanism.
As well, x L , ̇xL , and ẍL are the linear position, velocity, and acceleration of the screw mechanism at the load side.
A mathematical model of the EMLA can be derived by developing motion equations between its different components.To form the two-mass model of the system, the inertia of the EMLA can be expressed at a single point, as in (15): Meanwhile, the speed of different components can be written concerning the motor side, as in ( 16): Similarly, the equivalent stiffness at the motor side can be expressed as a function of motor twist, as in (17): By with considering Δx = F k and Δ =  k  , the stiffness of the system can be obtained using (18).
Finally, the equivalent viscosity of the entire system and the force coefficient can be written as in (19) and (20), respectively: As a result, the electromagnetic torque of the electric motor can be written as in (21), as the function of linear motion characteristics at the load side of EMLA (ẍ L , ̇xL , x L ): m = I eq ẍL + B eq ̇xL + K eq x L + f eq F L (21) where I eq , B eq , K eq , and f eq are the equivalent inertia, damping, spring effect, and load coefficient of the EMLA system, respectively.

State-space model of EMLA
To meet the control objectives, the state-space model equations are formulated in terms of the linear motion of the EMLA at the load side, and the equations for the derivatives of the motor current can be obtained using (22).
In this study, the objective of control is to maintain a load position while regulating the d-axis current, i d , to follow the reference current i * d = 0.By setting i d to zero, the magnetic field along the d-axis is eliminated, and the torque generated by the electric machine becomes independent of the rotor's position.As a result, the load position can be controlled without affecting the magnetic field along the d-axis.The q-axis current, i q , can be used to generate the torque necessary for position control.In addition to the time derivatives of the currents, the linear velocity and acceleration of EMLA at the load side can be found using (23): The state vectors of EMLA including the position at the load side, linear velocity, and currents in the dq-axis for the studied system can be defined using (24): Input vectors (control variables) of the system consist of input voltages of a PMSM along the dq-axis, which are given in (25): In this paper, the PMSM adopted for the study case is the 400 V 3.99 kW Nidec PMSM servo motor, as well as the ball screw BSG-ZUAZO model HLF "63×16-13-8-R".The characteristics of the EMLA components, including the motor, gearbox, and ball screw, are listed in Table 1.

PROBLEM FORMULATION AND PRELIMINARIES
In this section, the entire EMLA system, expressed in Section 2, is transformed into a fourth-order NSFF, decomposed into three subsystems, that consider the constraints on the control inputs S(⋅), which are introduced later in (30).Table 1, regarding the PMSM's specifications, expresses the input constraints, which are important for the motor.This means the control inputs i q , u d , and u q cannot be applied directly to each subsystem.For (k = 1, … , 4), x k represents the states of the subsystems, g k known coefficients, F k uncertainties, and d k external disturbances, including load effects, as shown in Figure 2. Here, F k and d k are assumed nonlinear and unknown.To clarify, Table 2 presents the definitions of each parameter used in NSFF within the EMLA system.The first subsystem has two sub-subsystems based on the dynamics of the EMLA, representing the linear position and velocity in (23), and a current-q control S(i q ) = i * q is proposed to manipulate the position and velocity.The second subsystem exploits the voltage-d control (S(u d )) for (22), after receiving the control value i * q from the previous subsystem, to force the states i d to track i * d (= 0), according to Park's concept.Finally, the last subsystem determines the voltage-q control (S(u q )) for (22) to force the real i q to track i * q , received from the first subsystem as the reference value.Because the RDSC uses the model-based EMLA, it must be robust enough to counter nonlinear load effects and modeling errors.During this process, the RDSC, which requires only reference trajectories and actual state values, can sufficiently produce the voltages and currents to tolerate a high load and uncertainties.For more details, the interconnection among the subsystems of the RDSC structure as it relates to the EMLA has been illustrated in Figure 3. Before conducting each subsystem analysis separately, we must provide some definitions, as follows: Definition 1. Assuming the function a k−1 values (for k = 1, … , 4) are virtual controls and differentiable, we define the following function to reduce backstepping complexity and uncertainties, as follows: F I G U R E 2 Decomposing the EMLA system into three subsystems in NSFF.

TA B L E 2
Actual physical meaning of NSFF parameters.

NSFF parameters EMLA parameters
x 1 (t) Linear position (x L ) x 2 (t) Linear velocity ( ̇xL ) x d2 (t) Linear velocity reference ( ̇xd ) where a 0 = a 2 = a 3 = 0.Meanwhile, θk , an adaptive function that can compensate for unknown effects imposed both outside and inside of the system, is introduced later in (35).By assuming that there is a positive constant Λ k ; and a smooth function r k , which is assumed always within the upper bound of uncertainties; d max(k) , as the bound of external disturbances; and a positive parameter Ω k , which all could be unknown, we can define: where x dk can be the reference value of the state x k .
To consider a robust adaptive law, we define the functions of subsystems to compensate for estimation errors, as follows: We can define 0 ≤  min = inf( k ), where  k is the constraint on control input and is used later in (32).Further,  * k is a finite function to adjust the adaptive estimation, and it is also unknown.All other parameters are assumed positive constants and will be defined later, and they can be unknown except  k for k = 1, … , 4, which will be used as a control design parameter.
Definition 2. Uniform exponential stability is a property of dynamic systems that ensures system solutions decay exponentially at a uniform rate.More formally, a dynamical system is considered uniformly and exponentially stable if there exist positive constants  and c, such that for any initial condition x(0), and positive μ, the distance between their trajectories decays exponentially at a uniform rate: where || ⋅ || denotes a norm on the state-space of the system, and t ≥ 0 is the time variable.Uniform exponential stability is an important concept in the field of control theory, referring to the property of a dynamical system where for any initial condition, the system's response converges exponentially quickly to an equilibrium point, and the rate of convergence is independent of the initial condition.

Definition 3.
As mentioned earlier, it is mandatory to consider control input constraints in the EMLA application.In this way, S(u(t)), as a function of the constraints on the control input u(t), is defined as follows: where u 1 and u 2 are the high and low bound values of u(t).To a greater extent: where and As we can see, Assumption 1.It is assumed that all system states are known and differentiable.
Assumption 2. It is assumed that external disturbances and uncertainties are unknown and bounded.Except for the upper and lower constraints of the control input (u 1 and u 2 ) based on the EMLA specification, we assume that no further knowledge about input saturation is required.
In the following, the RDSC for the EMLA system, expressed in Section 2, will be defined, and a stability analysis will be conducted.

EMLA CONTROL BASED ON RDSC
As expressed in Section 3, there are three subsystems comprised of four sub-subsystems.Now, by knowing that g 1 (= 1), g 2 , g 3 , and g 4 , as the factors of the control inputs, are non-zero, the virtual (a k ) and actual control (i q , u d , and u q ) for the three subsystems can be proposed as follows: F I G U R E 4 Schematic of the proposed controller for the EMLA.
where for k = 1, … , 4,  k follows (28). k is a positive constant, and P k = x k − a k−1 , where x k , as the tracking error, will be introduced later.As can be observed, virtual control is absent from the second and third subsystems due to their respective single subsystem order, each maintaining actual control.Meanwhile, θk , an adaptive function that can compensate for unknown effects imposed both outside and inside of the system, is introduced for all four sub-subsystems as follows: where  k and  k are assumed positive parameters.As mentioned earlier, the RDSC has a straightforward form and only requires simple mathematical equations, which are shown in (34) and (35).
By defining the adaptation error θk = θk −  * k and inserting it into (35), we have: where  * k follows (28) to tune the adaptation law and to compensate for the load effect, and uncertainties are unknown.The implementation of the proposed control approach, illustrated in Figure 3, in the EMLA system, enables a schematic representation of the RDSC for the EMLA, as observed in Figure 4.After obtaining the desired trajectories from the motion profile of the EMLA and retrieving the immediate states of the EMLA from the state detection section, the RDSC generates the necessary command voltages for the PMSM, similar to fig. 1 illustrated by Chai et al. 44   Remark 1.We defined many parameters, although most are considered for the stability analysis and can be unknown.Thus, only a few parameters will be used in the RDSC, including , , , and  as the control design parameters, which offer a straightforward control form.(31), we claim that all states provided in all three subsystems by following the RDSC satisfy the tracking process of the model, and they are uniformly exponentially bounded, as expressed in Definition 2. Now, we aim to investigate each subsystem via stability analysis and Theorem 1 proof separately, although the methods for each are highly similar.

CONTROL STABILITY ANALYSIS
Following the previous section, we can define the four-order NSFF system of the EMLA, according to Figure 2, as follows: In this way, we define the error of tracking (for k = 1, … , 4) as follows: x k = x k − x dk (38) where x dk is the reference trajectory defined in Table 2, such that x d1 = x d , x d2 = ̇xd , x d3 = i * d = 0, and x d4 = S(i q ) = i * q .We alter the form of the system to the tracking form, as shown: 45 where a 1 is a virtual control for the first sub-subsystem (x 1 ) introduced in (37) and a 0 = a 2 = a 3 = 0. To avoid the "explosion of the complexity", derived from Wang et al., 28 the time derivative of the virtual control input ( ȧ1 ) is treated as an uncertain term in the system, according to (26), in which F k = F k − ȧk−1 .In this way, by knowing g 1 = 1, differentiating (39) and inserting ( 37) and ( 38) therein, we establish a new form of the system, as follows: Next, a Lyapunov function is suggested as follows: After differentiating V 1 and inserting (40), we obtain: After conducting simple mathematical work, we obtain: By considering (27), we have: By using the mathematical concept (P 1 − P 2 ) 2 ≥ 0, we obtain: Because Λ 1 r 1 and d max (1) are meant to be finite, we can find positive values for  1 , and  1 , as follows: Then, by considering the definition of  * 1 in (28), we achieve: Now, by inserting a 1 and θ1 , we obtain: 1 , and considering (41), we obtain: where Then, from (50), we obtain: Because − 1 2  min  1 θ2 1 ≤ 0, we can eliminate it from the right side to reach: Likewise, the Lyapunov function V 2 is suggested as follows: where V 1 is the Lyapunov function introduced in (41).By differentiating V 2 and inserting (40), we obtain: Likewise, we continue by considering (53), inserted into (50) as follows: By inserting the control input constraint S(.) defined in ( 31) into (56), we obtain: Then, by inserting the current-q control i q from (34) into (57): By considering θ2 ≥ 0 according to Assumption 3, and because 0 ≤  min ≤  2 , we can argue: Similar to (46), by knowing (27) and the definition of  max in (33), we can assume: where: Then, by considering  * 2 in (28): In addition, by inserting θ2 from ( 36) into (62): where: By dividing − min  2 θ2 2 , and because − 1 2  min  2 θ2 2 ≤ 0, we can eliminate this part from the right side and reach: where: A third Lyapunov function is thus suggested as follows: All proof steps are the same as the previous one (V 2 ), except we consider the reference of state x d3 = i * d instead of x d2 = ̇xd .In this way, we obtain: where | ̇i * d | ≤ Ω 3 , according to (27).Therefore, as with the previous equations, we generate: where: Likewise, we can define the fourth Lyapunov function, as follows: All proof steps are the same as the previous ones (V 2 and V 3 ), except that we consider the reference of state x d4 = S(i q ) = i * q .In this way, we obtain: where | ̇i * q | ≤ Ω 4 according to (27).Similar to how (66) was obtained, we can argue: Then: For brevity, we introduce: Considering ( 41), ( 54), (67), and (71), we know that: Thus, considering V = V 4 , we can argue: where: Thus, according to (74) and (75), we obtain: Taken from the general solution for state-space representation, we can establish: Thus, we can solve (79), as follows: Considering (77), we can interpret (81) as follows: Because  k is a positive constant, we can express: Hence, a continuous function can be assumed as follows: Notice that the initial amount Z(0) in ( 84) equals (83).Now, it is obvious to say that there is a positive amount  ∈ , such that: By multiplying e (t−t 0 by (82), we reach: Because 0 ≤  <  4 , we can eliminate the decreasing element e −( 4 −)(t−t 0 ) from the right side of the inequality in (86): In this way, we can express the continuous and non-decreasing functions E 0 and E k : Then, by considering (87) and (88) and by performing some simple mathematical works, as well as eliminating the negative section, we achieve: Because E k is not decreasing, the left side of (89) will not decrease.Therefore, regarding the definition of E 0 in (88), we can establish: By defining: We can obtain: such that 0 < E 0 ≤ E and both E 0 and E are not decreasing, enabling * Z, as follows: Thus, (93) makes sense because  k can be an option to make Z small enough.Adding (93) into (92), we reach: Afterward, we Concerning the definition (88), we obtain: It is significant that: sup Thus, based on Definition 2, it is obvious from (96) that ||P|| is uniformly exponentially bounded by using the RDSC approach in (34), in consideration of the control input constraint defined in (31).Therefore, ||P|| will exhibit uniform exponential stability inside to a specific ball  (  0 ) , where:

NUMERICAL VALIDITY
To validate the effectiveness of the RDSC, we conducted a comprehensive performance analysis using three subsystems, as described in Figure 2, to simulate the behavior of the EMLA, modeled in Section 2. The analysis was achieved using two duties with different loads, where the EMLA tracked the same reference trajectories.Furthermore, a data-driven robust adaptive sliding mode control (DRBC), as described in Reference 46, and a NN adaptive backstepping control (RNABC), as outlined in Reference 47, were chosen as the standard benchmarks for conducting a comparative analysis of the linear position and velocity tracking in the same model of the EMLA.Contrary to the RDSC approach, both control algorithms provided in References 46 and 47 could not achieve uniformly exponential stability.The 22-step algorithm outlines the procedure for deploying the RDSC control.It offers both a concise overview and detailed, step-by-step instructions for the EMLA system, according to the specifications mentioned in Tables 1 and 2 (Algorithm 1).During this analysis, uncertainties, which are assumed unknown, were considered in the EMLA subsystem, according to the specifications provided in Table 2, as follows: eq (b eq x 2 + k eq x 1 ) + f 2 ⇒ f 1 = 0, f 2 = 2 (5 + 0.1 cos(0.5t))Arctan(900t) Algorithm 1. RDSC approach to the EMLA Input: system states x 1 = x L , x 2 = ̇xL , x 3 = i d , and x 4 = i q , and the linear position reference x d Output: d-voltage S(u d ) and q-voltage S(u q ) 1 Decompose X into x 1 , x 2 , x 3 , x 4 .Provide S(u d ) and S(u q ) to the system ( 22) and ( 23).

22
Repeat the steps from step 3 onward.
In addition, we considered the external disturbances affected by integrating loads into all three subsystems, as follows: ) The load force, denoted by F L , refers to the force required by the actuator to move the load in the desired direction, considering the load's weight and friction.The weight and distribution of the load are critical to determining the load force required, where a heavier load with an uneven distribution will require a higher load force compared to a lighter, more evenly distributed load.To evaluate the effectiveness of our RDSC design for EMLAs, we conducted two different duties involving loading and unloading procedures with various nonlinear load forces, which are supposed to be unknown.These duties included forward and backward motions to represent different movements, which tested the RDSC's capacity to control the EMLA.Our analysis focused on assessing the performance of the RDSC under different load conditions, motion types, and undesired forces, the results of which will provide valuable insights into the EMLA's effectiveness for practical applications.Figure 5 illustrates the goal trajectories of the EMLA modeled during both duty phases.The actuator is designed to move linearly forward from its initial position, as indicated by the blue line, until it reaches a point 100 cm away from the starting point in the first step.In the second step, the actuator returns halfway to its previous position, stopping at a point 50 cm from the starting point, as demonstrated by the green line.The actuator then moves forward again in the third step, following the red line, until it reaches a point 100 cm from the starting point.In the final and fourth step, the actuator returns to its original starting position, following the orange line.
Further, Table 3 presents information on the parameters used in the control design and on the saturation levels applied to both the current and voltage control inputs, which were kept constant across both duties.As Table 1 shows, the upper

RDSC parameters Value
1 ,   and lower limits for the motor current, i q max and i q min , are set to 7.9 A and −7.9 A, respectively.In addition, based on the motor's specifications, the acceptable maximum and minimum voltage limits are set to 380 V for u q max and u d max and to −380 V for u q min and u d min .It should be emphasized that the design control parameters for the RDSC of the EMLA are limited to only four, namely, , , , and .These few parameters contribute to the simplicity of the control process for the entire EMLA system, and the equations are essential to the design of the RSDC situated within boxes for convenient reference, including (31), (34), and (35).During duty cycles 1 and 2, we introduced a variety of loads to evaluate the performance of the RDSC in the presence of heavy and nonlinear loads, as well as to assess its robustness against unknown forces acting as uncertainties and disturbances, as defined in (99) to (100).

Duty cycle 1
In this duty, we considered two distinct load types (load 11 and load 12), as illustrated in Figures 6 and 7A In this duty, we considered significantly heavy and constant loads (80 and 40 kN), accompanied by a nonlinear component represented by a sine function to enable an uneven load distribution.Notably, the amplitude of Load 11 is twice that of Load 12. Based on the information provided, it can be observed that the loading and unloading times for duty 1, as well as the time spent tracking the trajectory, are as follows: at 145 s into the duty, Load 11 is applied to the EMLA at a position of 100 cm.The load is then unloaded at 320 s, when the EMLA has moved to a position of 50 cm (green line step).Then, at 420 s, Load 12 is retrieved by the EMLA at a position of 100 cm, and the load is then transported to its origin, arriving at 600 s, where it is subsequently unloaded (orange line step).

Duty cycle 2
In this duty, we introduced a wider range of loads, while maintaining the same tracking process, control parameters, and unknown uncertainties as in the previous duty.As illustrated in (102) and Figure 7B, it is apparent that the amplitude of the constant component and frequency of the sine function of the loads are lower than on the previous duty, while the number of loading and unloading operations is higher.In addition, the constant load component for the present duty is comparatively lighter than that of the previous duty, with magnitudes of 72, 24, and 36 kN as the constant parts: The loading and unloading positions can be observed in Figures 7B and 8.In contrast to duty 1, where the EMLA only carried loads during its backward movement, this duty includes additional loading tasks for transporting Load 21 and Load 23 forward.Moreover, the EMLA carries Load 22 and Load 24 during its return movement.It is noteworthy that Load 21 and Load 24 are the heaviest among those considered in this duty, while Load 22 has a third of their amplitude and Load 23 has half of their amplitude.Therefore, in this duty and according to Figures 7B and 8 After unloading Load 21, the EMLA carries Load 22 to the same position and disposes of it at a distance of 50 cm from its origin, at 310 s into the duty.Subsequently, at 350 s into the duty, the EMLA carries Load 23 to the same position and disposes of it 100 cm from the initial point using a forward motion approach at 450 s into the duty.Finally, the EMLA carries Load 24 at 510 s into the duty and disposes of it at the original point using a feedback motion approach.It is noteworthy that the duration of rest during duty 2 is slightly longer than that during duty 1 at the 50 cm position, due to the increased loading and unloading responsibilities.Similarly, the final step of loading and unloading in duty 2 takes longer than in duty 1, also due to the increased responsibilities.Therefore, it can be observed that for duty 2, the maximum load considered is 10% lighter than for duty 1, even though the number of loading and unloading duties doubled.
Furthermore, the loading and unloading process is faster than in the previous duty, as depicted in Figure 7.All these variations have been examined to assess the performance of RDSC for the EMLA under different conditions.

RDSC results for both duties
Figure 9 illustrates the performance of the RDSC in tracking the reference position of the EMLA in duty 1, with the results signifying its superiority to the other algorithms in close scale, in the presence of load effects and uncertainties.Specifically, the true position state x 1 performs accurate and fast tracking of the reference value x d1 = x d , and the figure indicates that the tracking error accuracy is approximately 2 × 10 −3 cm when the EMLA is unloaded, while the error increases to 4 × 10 −3 cm when the EMLA is loaded.Notably, when the EMLA is not loaded and there is a consistency in position, the tracking error is 2 × 10 −8 .Thus, the error of 2 × 10 −3 occurs only at the moment of stopping in a new position.Similar to duty 1, Figure 10 demonstrates the tracking performance of the RDSC for EMLA in duty 2. It is evident from the results that the tracking performance is also exceptional in duty 2, even in the presence of more variable load effects and the same uncertainties.The figure indicates that the tracking error accuracy of the RDSC is approximately 2 × 10 −3 cm when the EMLA is not loaded, while it increases to 3 × 10 −3 cm when the EMLA is loaded, outperforming the others.
Notably, when the EMLA is not loaded and there is no change in position, before 45 s and after 625 s in Figure 10, the tracking error is similar, at 2 × 10 −8 , even in the presence of uncertainties.Likewise, the error of 2 × 10 −3 occurs only during the stopping moment.As shown in Figures 9 and 10, the accuracy of position tracking using all three control algorithms with loads is superior in duty 2 to that in duty 1, a sensible result considering that in duty 2, although the number of loads is greater, their magnitudes were lower than those in duty 1.
Figure 11 displays the performance of linear velocity tracking using the RDSC in duty 1.The figure illustrates pulses in the real velocity when the load status changes as well as that most of the pulses correspond to the changing statuses of load tasks.Using the RDSC approach, the error in the reference velocity tracking is approximately 3 × 10 −5 m/s, and the tracking speed in this duty is about 0.1 s.Similar to duty 1, Figure 12 demonstrates the performance of linear velocity tracking using the RDSC in duty 2 as well as that the pulses remain in real velocity when the load status changes.The figure illustrates that most of the pulses correspond to changes in the statuses of load tasks, as in the previous duty.Using the RDSC approach, the error in reference velocity tracking is approximately 2.5 × 10 −5 m/s, and the tracking speed in this duty is about 0.07 s.In both duties, the results of velocity tracking using the RDSC approach are superior to those of the other algorithms.
It is worth noting that the number of velocity pulses in duty 2 is greater than that in duty 1, which can be attributed to the higher number of loading tasks in duty 2. In Figure 13, which pertains to duty 1, the real and saturated values of i q and u q are designated by red and blue lines, respectively, confirming the capacity of the RDSC under input constraints.Furthermore, according to the second subsystem, i d is expected to track zero, which is achieved with an accuracy of 0.45 A. The highest pulse in i d is observed when loading of Load 11 at 145 s.In addition, the value of u d is maintained below 30 V to force i d to track zero.Similar to duty 1, Figure 14, pertaining to duty 2, depicts the actual and saturated values for i q and u q respectively, represented by red and blue lines, which serve to confirm the RDSC's capability under input constraints.
Moreover, as per the second subsystem's requirement, i d is expected to track zero, which is achieved with an accuracy of 0.2 A. The highest pulse in i d is attributed to Load 24 at the time of loading, which occurs at 510 s.In addition, the value of u d is maintained below 20 V to force i d to track zero.Due to the higher frequency of load changes in duty 2, there are more variations and spikes in currents and voltages.Figure 15 presents a comparison of the torques produced in both duties.According to (6), because the model-based EMLA's produced torque is dependent on the i q value and because i q is constrained to a range of −7.9A to 7.9 A, the maximum torque value for both duties is consistent and identical, that is, 12.7 N.m.However, due to the longer duration required by the EMLA to carry the loads in duty 2, the torque associated with this duty is greater than that in duty 1.Furthermore, the start and finish times of the produced torque in both are identical because of the same moving tasks.
In summary, Tables 4,5,6, and 7 provide an overview of the accuracy and speed control performance of the RSDC approach compared with References 46 and 47 in controlling the modeled EMLA to follow the desired trajectories in both duties.They demonstrate that the performance of the control system proposed in this paper, including the errors in position and velocity tracking, as well as the convergence speeds in both duty cycles, is superior.This observation is substantiated by the exponential rate of stability exhibited by the RDSC.As previously mentioned, the control position accuracy in duty 2 outperforms that in duty 1 due to the lighter loads involved.Furthermore, the speed of position convergence in both duties is relatively similar and fast.This also holds true for the error convergence when tracking reference velocity using the RDSC, although the control velocity in duty 2 exhibits a slightly better performance when compared to the first duty.In essence, these findings indicate that the RSDC system effectively controls the EMLA in both duties with good accuracy and speed control performance.

F I G U R E 5
EMLA trajectories for both duty cycles (blue line: step 1, green line: step 2, red line: step 3, orange line: step 4).TA B L E 3 Control design parameters.

F I G U R E 13
Duty 1: Currents and voltages.F I G U R E 14 Duty 2: Currents and voltages.