Improving the Tracking Performance under Nonlinear Time-Varying Constraints in Motion Control Applications: From Theoretical Servo Model to Experimental Validation

In the high-accuracy control of an AC machine, the knowledge of pure system parameters, with no deviation in drive coeﬃcients and no disturbance or other nonlinear components, is a diﬃcult issue for operators, even though it is occasionally nonviable. To overcome these troubles, this paper introduces a robust adaptation strategy based on pseudo fuzzy logic and sliding mode control (PFSMC) for an AC servo drive subject to uncertainties and/or external disturbance. Owing to the robustness of the SMC technique, the reduced sensitivity to uncertainties, and the enhanced resistance to disturbances from the pseudo fuzzy mechanism, this control algorithm can guarantee not only system stability but also the improvement of tracking errors in the steady state. To validate the design eﬃciency of PFSMC, both simulation and laboratory tests of the proposed scheme and a conventional PID scheme were performed to compare them as follows. In a computer environment, test cases with and without certainties were implemented with two controllers to visualize the comparative responses. Then, the two control methods were integrated into a real-world hardware platform to acquire practical outcomes. From these results, it can be noted that our successful approach showed a feasible, eﬀective, and robust performance in AC drive.


Introduction
At present, the demands in high-precision machines have been sharply increased to correlate with numerous fields (manufacturing tools, aerospace, innovative robotics system, etc.). Servo mechanisms as well as servo actuators are indispensable components in advanced machines. e enlargement of servo systems would inspire the growth of other industries. Hence, many developers have focused on them, from a vibration-assisted milling machine [1], a hydraulic actuator coupled with a nonlinear physical system [2], and mechanical parameters in a two-mass servo drive system [3] to a duo-servo vehicle brake [4] or 3D pose tracking for underwater vehicles [5]. An integrated size and a lightweight motor are desired for high-performance applications. Consequently, the volume and dimensions may be important factors in medical robots. Depending on each industrial field, the technical specifications, electrical features, and dynamic characteristics may differ. A servo pneumatic system has a widespread range of applications in industrial automation, such as packaging, supervising, assembly, or clamping. is technique is significantly helpful for physical system manipulation, the fast movement of linear components, safe and clean technology, low cost and low maintenance, high mechanical efficiency and long working life [6,7]. Nonlinear characteristics generally have a negative effect on the precision of the overall system. e nonlinear frictional forces, the thermodynamics of the air flow in the cylinder, the varying load, the uncertain features of valve actuators, and the unprofitable energy are all limitations of servo pneumatic actuator systems. e control problems that most research focuses on are the response of exactness and velocity of individual actuators. e step change in reference input could lead to tests of position modification or point to point. e considerable developments in the current era have made an impact on various technologies of robot and automation systems close to each other. In the general industrial sector [8], automation relies on electric, mechanic, and information technology components to handle all processes. Specifically, if robotics is the main element of an automated system, then motion control techniques are the core of the robot system. Based on the servoing motion knowledge, an industrial robot could be manipulated for multifunctional missions, and its movement could be reprogrammed to carry materials, parts, tools, or devices through continuously varying motion. Currently, it is considered to be an implementing component of industries due to its performance in enhancing precision, repeatability, reliability, and flexible drive. As key factors in economic progress, industrial manipulators are replacing humans in harsh environments to perform tedious tasks.
Servo drives are commonly utilized in high-tech motion systems; AC servo motors in particular have brought many great benefits, such as high speed, better efficiency, small steady-state error, and low noise. Along with significant success in the semiconductor industry, many efforts have been expended to research the AC drive in order to yield commercial products in the global market. In the new trend, it is compulsory to combine the driving motor with an inverter and its controller. is kind of motor offers a speed near nominal values with the power supply frequency. Conversely, the ultra-high-velocity drive technique of a large-capacity motor diverges with high-velocity drive in the rotor surface, magnetic bearing, material, and assembly routine [9].

Background Work
From the above-mentioned applications, successful developments in robotics systems are obviously related to proper control laws in servo actuators. e primary objective of most of these implementations is to measure the corrective control signals to compensate for the many variations from different sources. e appropriate design helps to lessen the effects of external disturbances and causes the steady-state error to tend to zero whenever the system tracks precisely and nearly close to the target position [9]. e most popular scheme is the Proportional-Integral-Derivative (PID) control, significantly characterizing both industry and research. e challenge in this controller is to tune right gains either manually or automatically [11]. ese methods consist of the open/closed-loop Ziegler-Nichols method, the integral of time-weighted absolute error method, and the Cohen-Coon method [12]. e advanced tuning procedures are used to integrate with fuzzy, neural network, adaptive control or others to adapt with fluctuating in the system. Other types of PID scheme include fractional-order and event-based structures, which are typical for a generalization of standard integer-order and wireless transmitters and actuators, respectively. Together with the required industrial applications, different modern control strategies exist, such as computed torque control [13,14], sliding mode control [15], observer-based control [16], time-delayed control [17], and LQR [18]. Of these, the sliding mode control design reveals superior advantages as it shows robustness and proved stability in robotics systems. Modeling of servo actuators is often carried out to clarify the driving torque in motors. Although there are uncertainties in modeled parameters or external disturbances, the system can switch from one phase to another depending upon the current system's state.
In a high-precision AC machine, the complex integrations of numerous components involve the information of system parameters. Despite testing and validation efforts, knowledge of machinery data can be hard to identify exactly. As it cannot be determined occasionally, the identification procedure is a challenge in automation control [19]. As a result, machinery data estimation is considered a notable contribution in the development of AC drives. ese features enable the electric drives to govern the machine parameters and tune the control loop. Based on the working principles, many methods are clustered in several categories of numerical methods. ey rely on the modeling of finiteelement machines [20] and are only available for motor developers. Regular users who lack design data cannot successfully build an exact model of the machine. In the dynamo method [21], the dedicated instrumentation is set up as a speed-controlled follower and a driver. ey are connected together as twin motors, and a host computer plays the role of data logger. e driving mechanism with different input signals is activated and the observed response is obtained through the measurement of electrical variables (voltage, current, and power). With time, the recorded data are obtained along with environmental variables (temperature and magnetic and electromagnetic fields). With this information, a synthesizing step is performed to form a flux mapping. e parameter identification process is accurately predictable to enhance the positioning control. In the other method, standstill determination, which is partly sectioned into frequency domain [22] and time domain [23], is simpler and needs rotor locking. In this way, the testing machine is delivered in a single-phase grid with various signal patterns to approximate the system's response. e setting facilities that are proper for offline techniques include a signal generator, high-resolution measuring instrument, and data recorder. However, without good instrumentation, some fundamental hardware is also able to accomplish accurate characterization of an AC drive.
e self-commissioning technique [24] is a key classification system in modern drive systems. e characteristics of this alternative current drive are that it is built on a low-cost microcontroller, employs powerful computation using accessible sensors, does not need rotor locking and sources of driving torque, and requires the least operator intervention. e parameters of motorizing circuits can be simulated from steady-state measurements and standstill frequency responses by dissimilar excitation signals through a power inverter. Most modern commercial drives, in the initial routine, generate algorithms to identify the motor's parameters. Depending on the drive manufacturer, the identification process via the inverter may vary separately, but the main idea still depends on hardware resources and the control strategy. Consequently, the overview of research topics on servo systems is summarized in Figure 1, in which these approaches are grouped into various categories based on their operating principles.

Components of a Servo System.
e main components of a servo system include the driving actuator, sensing device, control mechanism, and data acquisition system. Some industrial motor drives are commonly utilized in different applications, such as Permanent Magnet Synchronous Motor (PMSM) and Brushless DC (BLDC) [25], which have many similarities but fundamental differences in Electromotive Force (EMF). Recently, a piezoelectric motor [26] driven by the change in shape of the piezoelectric material when an electric field is applied appeared in a highly precise control aspect. Furthermore, several articles on the advanced functions [27] or more complex structure [28] of piezo motors to optimize energy harvesting and to guarantee stable motion have been proposed.
Sensors play an important role in most industrial experiments because, if faulty, they may lead to failures in the drive system. Regularly, a linear incremental position encoder [29] is deployed for the accurate measurement of the displacement of a driving motor. Additionally, a linear potentiometer [30] coupled with the rod end of a lead-screw mechanism has also been utilized as a position transducer to evaluate the displacement. e accuracy, resolution, and data acquisition rate have significant impacts on the dynamics of a closed-loop system.

Modeling of a Servo
System. Compared to regular motors, a servo system is specifically launched with consideration for the moment of inertia of the rotor and the electrical responses in such a way that it can react to sudden alterations of voltage and current from the servo amplifier. Initially, the mathematical model of a servo drive can be illustrated by fourth-order equations [31]. In some cases, advanced control algorithms [32,33] are studied in order to achieve high performance. If the friction force is considered, then a compensator [34,35] should be added to reject unexpected disturbances.

Tracking Control of a Servo System.
In the field of control, one of the main research objects is the whole servo system, which usually uses both linear and nonlinear schemes. Each control method has its own advantages-for example, feedback linearization [36] and LQR [37] reveal a superior performance in optimal parameters. For the class of nonlinear control, several studies have been carried out-for instance, tracking control using asymptotic prescribed performance [38] and indirect adaptive control [39]. In recent years, intelligent algorithms including fuzzy [40] or neural networks [41] have been introduced into complex servo systems to boost their performance. Furthermore, to improve the tracking control in finite time, the nonsingular fast terminal sliding mode strategy has been used to deliver a rapid response and control precision via an extended state observer [42].

Motion Profile of a Servo
System. For a large number of applications, including medical automation, scientific instrumentation, and many types of general automation, point-to-point movement is employed more frequently than other profiles. One of the most popular motion profiles for a point-to-point approach is the S-curve or fourth-order symmetric polynomial [43]. For a large number of industrial applications, exponential trajectory planning [44] and trigonometric-function-based generators [45] are options to drive the servo system to track its desired path.

Vibrationless Control of a Servo System.
e real-world applications of the above profiles aim to achieve fast movement with low vibration [46,47] and nanopositioning measurement [48]. As for other aspects, smoothness and the time-optimal effect are also valuable for automated robots and machines [49]. From the viewpoint of filtering performance, some studies of applications in FIR (finite impulse response) [50] or exponential filter [51] have been conducted. Another research study focused on jerk-continuous profile in order to generate a smooth trajectory for robotic manipulators [52].

Identification Parameters of a Servo System.
Numerical approaches rely on finite element modeling of electrical machines [53]. e disadvantage of this method is that it is only accessible for motor designers, since others do not have enough data to establish a numerical or finite element model of the machine. In another approach, dynamo [21] requires a speed-controlled prime mover with dedicated instrumentation. It maps the flux of a machine and utilizes phasor interpretation to obtain equivalent circuit parameters. Furthermore, the standstill identification approach [54] is one of the simpler applications and usually needs rotor locking. It could be classified into a frequency response and time response based on the principle of parameter estimation utilized. In both cases, the automated machine has a single phase during the test with various signal patterns to evaluate the system parameters according to the response. Last but not least, the parameter identification techniques that require the rotor shaft to be free to rotate are categorized individually under the heading of free shaft [55].

Mathematical Model of a Servo System
e dynamics equation of an AC servo system is described in where θ is angular movement of motor; J is inertial moment of motor; B is viscous coefficient; τis driven torque; τ d is Mathematical Problems in Engineering external disturbance torque from unknown load on rotational motor shaft.
Equation (1) could be rewritten in state space form: where Assumption 2. ere exist unknown positive bounds e boundary conditions are satisfied as follows: e uncertainties Δa i and Δb i , i � 1, 2 denote variations in the system parameters and impact from bounded external perturbations. e upper bounds δ i and ε i ensure that the varying effect does not cause disturbances such that it would reverse the control action.

Proposed Approach
Sliding control is the constrained motion of states along trajectories on the sliding surface s that is chosen as follows: e coefficient c 1 is symbolized by λ|λ + c 1 � 0 from the sliding surface.
By selecting the Lyapunov candidate function the condition to stabilize in a servo system is To satisfy the above condition, the components of motion controller should be taken as where u Eq is the equivalent state of the controller; u Sl is the sliding state of the controller; and u Fz is the fuzzy state of the controller. e components of the controller can be found as follows.

Equivalent State.
e essential situation for the state path to maintain on the sliding surface s is _ s � 0. In this perfect case, the nominal system containing Δa i � 0, Δb i � 0 stays on the sliding surface s if For a closed-loop system, u Eq is assigned from the desired dynamics of the linear feedback system  Figure 1: Summary of related studies on servo systems.

Switching State.
If the servo works in this case, the system state x may not be located on the sliding surface s at the initial stage, or it leaves from surface s due to the effect of varying parameters or external disturbances. e motion controller must perform driving action of system state x back to sliding mode s � 0.

Theorem 1. Assume that dynamics system (1) with nonlinear characteristics satisfying Assumptions 1 and 2. If the sliding mode control is formed as
with equivalent control gain K 0 Eq given in (12) and sliding control u Sl and the sliding control signal is en, the system state x asymptotically converges to zero.
Proof. Using the positive definite Lyapunov function (7), we have From (10), _ V can be obtained: where I is an identity matrix 2 × 2 and f T j is the jth row of the matrix D − EK E . en, there are two possible cases for φ j (x). In the case that φ j (x) > 0, we have erefore, the control signal should be designed as Otherwise, φ j (x) < 0: To meet the condition, the output of the controller is From (20) and (22), the sliding control can be identified as follows: Consequently, the Lyapunov candidate function is ensured to be less than or equal to zero.

Fuzzy Control.
A novel fuzzy scheme is proposed to implement into the controller to enhance the overall performance. We consider that the fuzzy control signal u Fz,i is expressed as where c i is the output of the fuzzy scheme. e fuzzy signal u Fz,i must comply with fuzzy rules, which depend on variations of s i and Δs i . In sliding mode control, there are oscillations in the system's state when the state trajectory would tend toward a sliding surface rapidly. If the system state is far from the surface, then |s i | becomes larger. As a result, the sliding coefficients increase correspondingly. If the state deviates from plane (s i Δs i > 0) and |Δs i | is greater, the coefficients of the system must drive the state back. On the other hand, if the state trajectory moves toward the sliding plane closely (s i Δs i < 0) and |Δs i | is bigger, the sliding parameters would decline to suppress oscillations. For the above purposes, the fuzzing processes |s i | and |Δs i | in two fuzzy linguistic variables large and smallare defined as follows: where η s i and η Δs i are tuning coefficients positively. e fuzzy rules are described briefly in Table 1. Since it is necessary to maintain the real-time performance of the servo system, singleton membership functions of fuzzy output are selected instead of triangular or trapezoidal shapes.
where c m i is any proportional nonnegative constant. To defuzzify, we could form the output by using the maxmin method. Figure 2 is formed as

Theorem 2. Assume that dynamics system (1) with nonlinear characteristics satisfies Assumptions 1 and 2. e output controller in
where the equivalent control signal u Eq is given in (11), the sliding switching control signal u Sl is defined in (16), and the fuzzy control signal u Fz is identified in (24). en, the system state x(t) asymptotically converges to zero.
Proof. Using the Lyapunov function candidate in (7) and the controller signal in (28), the 1st derivative of V(t) is presently demonstrated as with φ ≠ 0; condition (8) could be satisfied if, and only if, If the fuzzy control signal is determined as (24), then It is known that φ[f T x + (1 + E)u Sl + ]] < 0 (from eorem 1), (1 + E) > 0 and c > 0. Hence, the sliding mode controller with the added fuzzy component can be expressed as Mathematical Problems in Engineering It can be seen that proposed controller (27) suppresses the approximation tracking error and uncertainties, permitting the positioning error to converge to zero asymptotically due to the Lyapunov principle.

Results of Numerical Simulation
By presenting the pseudo-fuzzy law with the robust sliding mode control, the effectiveness of proposed scheme was firstly validated by numerical simulation. e target AC motor and driver were from Higen Motor [56], which have an industrial performance and are certified by international accreditation.
e servo driver was operated in position control mode, which was measured by a 20-bit rotary encoder.
e driving specifications were chosen as J � 1, 6 × 10 − 4 kgm 2 , B � 10 − 3 (Nm/rad/s), ε � 0.1, and c 1 � 0.001. In the pseudo-fuzzy structure, the selected parameters were considered as η S i � η ΔS i � 100 and c m i � 25. As demonstrated previously, the nonlinear servo system model was chosen to verify its applicability in practical industry. For that reason, the simulation tests were classified into two types: with and without uncertainties. Currently, the most common controller from servo manufacturers is the PID scheme, attracting many practitioners due to its advantages. e coefficients of the PID controller are given by K P � 0.02; K I � 0.005; K D � 0.0015. e comparative simulation results from a square pulse as the reference signal are provided to visualize the superior performance of proposed control strategy. In Figure 3, the position, angular velocity, and torque response of a conventional PID scheme without nonlinear servo characteristics are shown, as simulated on a computer. e blue line represents the reference input signal while the red line symbolizes the output response of the servo system. As expected, the PID tracking performance seems to struggle to follow the reference data when the input changes suddenly. It requires the operator's experience to tune the acceptable control parameters. e overshoot in tracking position geometrically seems to be large in the transient period.
e PFSMC controller was implemented in the same configuration, and its results are demonstrated in Figure 4. Whenever the input signal changed, a nonzero tracking error appeared in the system state. e overall output of the control scheme adjusted the driving status to react in accordance with the fluctuation in the reference signal. e effectiveness of the proposed controller is revealed in the small overshoot and fast rising time.
In reality, nonlinear components or external disturbances will always exist in the driving model of an AC servo motor. ey induce more difficulties in the motor control method. To imitate these situations, elements of uncertainties, which were simulated in a numerical computer, were added in the dynamic equation. e computer-generated performance with uncertainties in the PID strategy and in the proposed strategy is shown in Figures 5 and 6, respectively. Based on the rapid change of the input value, the output control responds to alter the feedback value. However, the Proportional-Integral-Derivative structure seemed to hardly adapt with uncertainties. As it cannot predict the nonlinear influences in the servo model, the response of the PID control oscillates, even though the system stays in a steady state. Unlike the conventional PID controller, the PFSMC could accommodate the internal variations in the AC servo motor. It is thought that the cause of nonlinear characteristics is due to the alternatives of PFSMC and setting of the modifications rule again. e simulation results in the uncertain model confirmed that vibration in the control signal still occurs, but with a lesser amplitude, and the PFSMC attempts to drive the angular error to converge to zero immediately. e comparison of the simulation results between PID and PFSMC is listed in Table 2. To deliver a general view, we compared their performances in three categories: overshoot, rising time, and tracking error. In each comparative item, we obtained results in a perfect case (without certain elements) and an imperfect case (with certain elements). It can be seen clearly that the proposed PFSMC shows a better performance, with a fast response, low tracking error, and adaptive characteristics.

Results of Experimental Verification
e experimental hardware shown in Figure 7 comprises a servo motion system (embedded motion controller, servo driver, servo motor, ball-screw mechanism, load and linear scale ruler).
e system software includes a monitoring program (Windows-based operating system) and a real-time firmware (C-based programming language). In this subsection, the robust properties of controllers whose control algorithms are implemented in firmware are investigated. e host computer plays a principal role in the interaction with operators, while the embedded motion manages data flow at the board level. To prevent data loss, on each side, a circular buffer was programmed to store and obtain data simultaneously. As a result, data transmission was ensured at high speed and with high reliability. e servo driver was set up in position control mode and the velocity was found by backward difference of the position signal. e parameters of the controllers and the boundary conditions were the same as in the simulation tests.
A micro-processor-based controller was developed to verify the proposed control approach. In order to mimic practical scenarios, all design processes comprising electronic components, schematics, and PCB routing followed industrial standards. Figure 8 illustrates the physical connection from the host side to the servo side. In the Windows environment, a C-based thread in motion library, which helps to maintain continuous data transmission, executed read and write commands continuously. User access on the API functioned to monitor all system states in the graph of the control screen. e external link between the computer server and motion controller was by USB-to-serial wiring, while the internal communication in the embedded board used SPI and I2C protocol. Depending on the functional purpose, the design of the board is divided into three parts: CPU module, base module, and GPIO module. e central processor unit ARM Cortex M4 [57] was placed as a CPU module. It handles data to the host computer through a USB bridge IC and to other segments with the aid of a communication bus. is ARM chip is a highly efficient embedded processor, up to 80 MHz, with low power, a large memory with 256 kB Flash, 32 kB SRAM and 2 kB EEPROM; is of low cost; and has floating-point support. As the base module, the design was integrated with a PCL-6045 [58] chip from NIPPON PULSE, which is a high-performance motion IC with advanced control motion features, supported by four axes, with a linear/circular/continuous interpolation and trapezoidal/S-curve profile. Furthermore, some isolated circuits were used to protect the controller from backwardcurrent. Two differential signals went from pulse port and dir port to the servo interface. Synchronous in/out and manual pulse control were employed to match the input and output, respectively, and to generate the desired number of pulses. In a general purpose input/output module, a series of LEDs as well as connector pins designate the level of signals.
At the firmware level, the programming environment was IAR workbench, which uses a C/C++ compiler, assembler, and linker. e architecture of the embedded software is described in Figure 9. e controller constantly runs in the main control loop to manage data exchange, read information from sensors, check the servo status, and hold in/out. In one servo cycle, the motion profile generates a point-to-point trajectory plan from the polynomial function. e sampling interval is 0.25 ms, allowing enough time execution for the planner to accomplish a step of motion.
en, an interrupt signal is acknowledged by the host if the movement is complete or if any error takes place. To apply it in more areas, more functions and utilities can be realized in the controller. Henceforth, a manager subprogram administers motion commands, sensing commands, and multitasking commands. e servo platform would unavoidably be subject to parameter deviations, nonlinear forces, or troubles in highfrequency dynamics, so a theoretical analysis alone is not convincing. erefore, real-world tests were performed with the apparatus in Figure 7. In these cases, a point-to-point reference technique that is well known in numerous positioning motion systems was engaged to prove the control performance of the controllers under various circumstances. Figures 10 and 11 present the output results of the real investigations with the PID controller and the proposed controller with uncertainties, respectively. It can be seen that an oscillation exists in the measured signal with nonlinear uncertainties in the PID scheme. An enhanced outcome with a better performance was achieved with the proposed scheme. When the robust control law based on the PFSMC strategy is added to the servo system, it can attenuate the effect of uncertainties. e timing motion generation in the proposed control approach is shorter because it does not rely on vibrationless movement. Consequently, the superior specifications in terms of rapid reaction, high positioning precision, less vibration, and increased robustness were validated in the laboratory test.
In terms of practical application, there are several relations between fuzzy control and tuning coefficients. e effect of the fuzzy control component is larger when there is   an increase in the sliding function magnitude |Δs i |. Nevertheless, their signs are contradictory. When tuning, the fuzzy control has the same effect on the overall control signals. It is rather sensitive to both the sliding function s i and sliding vector s in the case that μ Δs i , μ s i , or μ φ i are subject to the magnitude of the sliding functions, whilst c m i or u Fz depend on the saturation condition of the control input signal.

Conclusions
In this paper, a novel control application for AC servo system was investigated. e contributions can be fully described as follows: (i) With the SMC approach, robustness and asymptotical stability based on the Lyapunov theory could be achieved. Combined with fuzzy principle, the tracking performance is expected to attenuate disturbances and decrease the sensitivity to uncertainties with a boundary threshold. e fuzzy-like architecture not only promises simple programming but also ensures real-time output. (ii) An AC servo model was built on a computer to simulate the system response. From mathematical equations, the dynamic characteristics and theoretical assumptions were perfectly performed to overcome the control problems. (iii) An industrial motion hardware framework was established to approve the proposed scheme. e printed circuit board of the motion controller was designed to embed the control algorithm using C code. e peripheral instruments, including a servo driver, servo motor, and mechanical actuators, were obtained from popular industry partners. (iv) e extensive comparisons of simulations and experiments clearly demonstrated the effectiveness of the proposed controller and validated that an improved robustness and a superior disturbance rejection result can be obtained by applying the SMC approach into the pseudo fuzzy scheme.
Using the proposed controller, the servo system could explicitly perform chattering-reduced effects with high insensitivity to load variations and unknown system parameters. e tracking error performance and the end-point path planning validated the robustness of the system employing our controller against external disturbances and parameter uncertainty. It is well acknowledged that a tradeoff occurs between robustness and chattering. e application of our technique to the field of industrial control requires further study. A powerful microprocessor should also be considered because of computation burdens. For multiaxis motion control, the effects on tracking performance and synchronization need to be verified. Last but not least, several optimal algorithms could be innovated to choose the proper gains of the controller.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.