Trajectory Tracking Control for Electro-Optical Tracking System Using ESO Based Fractional- Order Sliding Mode Control

In this study, to achieve high trajectory tracking performance in electro-optical tracking systems under strong nonlinear disturbances and uncertainties, we develop a nonlinear extended state observer (ESO) based fractional-order sliding mode control. When compared with previous work, the new compound control strategy is attractive in terms of the following three points. First, a novel controller is developed that integrates the advantages of a nonlinear ESO, a fractional-order nonsingular terminal sliding mode (FONTSM) manifold, and a super-twisting algorithm. Second, the nonlinear ESO is employed to estimate the disturbances and uncertainties without explicit knowledge of the system model. Third, a FONTSM manifold-based super-twisting algorithm is integrated into the controller to enhance the system robustness. The FONTSM manifold has a faster dynamical response, more flexible sliding manifold structure, and better control results than its integer-order NTSM counterpart. The finite-time convergences of the ESO and controller are both proved by the Lyapunov method. Finally, the comparative experimental results demonstrate the effectiveness and superiority of the developed control strategy with respect to existing approaches.


I. INTRODUCTION
An electro-optical tracking system (EOTS) is a complex, high-accuracy device that is integrated with optical, mechanical, and electronic devices, and it has attracted increasing attention in recent decades [1], [2]. It has been widely used to expand the capabilities of humans in observation, surveillance, search and rescue, navigation, mapping, and optoelectronic countermeasure applications, to name a few, from civil to military domains [3]. To exploit the full potential of EOTSs, better control performance, clearer observation results, and longer monitoring distances are urgently required [4]. To achieve these goals, many trajectory tracking control methods [5]- [8], have been developed. The Line-ofsight stabilization control methods [9]- [11] have also been proposed.
The associate editor coordinating the review of this manuscript and approving it for publication was Zheng Chen . However, although the abovementioned techniques are capable of achieving a certain performance, each of them has its limitations. Satisfactory tracking control is still hard to obtain for EOTS due to the strong nonlinear factors such as friction, uncertainties, disturbances, and model variation [3]. Thus, designing a novel control strategy for EOTS to tackle nonlinear factors is a challenging task that motivates us to do further research.
To tackle strong nonlinearities, many control methods were proposed such as SMC, adaptive backstepping method [12], model predictive control, fractional-order (FO) control and so on. SMC is a powerful technique for handling bounded disturbances and parameter uncertainties owing to its strong robustness and suitability for practical applications [13]- [15]. Nevertheless, chattering is a problem in SMC that cannot be neglected. Several methods have been proposed to reduce chattering, such as the nonsingular terminal sliding mode (NTSM) [16], high-order sliding mode VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ (HOSM) [17], observer compensation method [18], and reaching law method [19]. The NTSM method not only guarantees that the system state reaches its origin within a finite time, it also avoids the nonsingularity problem.
In [20], to tackle unknown parameters, disturbances, and uncertainties, a robust adaptive NTSM method was developed for an automatic train operation system, and NTSM control with neural networks has been designed for MEMS gyroscopes [21]. HOSM methods such as the twisting algorithm, suboptimal algorithm, and super-twisting algorithm (STA) can reduce chattering and ensure finite-time convergence [22]. In particular, the STA, which handles a system state with a relative degree of one, is the most effective. It only requires the sliding manifold information, whereas other STAs require the sliding manifold derivative information as well. Because of the discontinuous function under the integral term, the chattering of the STA is greatly attenuated. In [23], a finite-time super-twisting SMC was proposed for Mars entry vehicles. A super-twisting SMC with adaptive gains and time delay estimation was proposed for maritime autonomous surface ships [24]. Moreover, an adaptive super-twisting sliding mode controller was designed for a wing-sweep morphing aircraft [25]. FO control is an effective control method that has become a new branch of automatic control. It can improve the flexibility of parameter adjustment and controller design. It has been broadly shown to be more effective than integer-order (IO) control and is always combined with various techniques such as SMC and PID control. In [26], a FO sliding mode controller with an FO disturbance observer was designed for UAVs and maglev suspension systems. In [27], to handle nonlinearities, a continuous FO nonsingular terminal sliding mode controller with dynamic SM manifold was designed for a class of second-order systems. A FO-based NTSM surface and a fast terminal sliding mode-type reaching law with time-delay estimation was proposed for a cable-driven manipulator [28]. To handle complex lumped uncertainties, a continuous FONTSM controller with time delay estimation was designed for cable-driven manipulators [29].
Moreover, since the time-varying lumped uncertainties existing in SMC are difficult to obtain, the observer technique is an effective approach to estimating them for a system and suppressing the chattering in SMC. Many proposed observers include the extended state observer (ESO) [30], disturbance observer [31], and finite-time disturbance compensator [32], [33] have achieved good effect. The ESO extends the system to a new state to estimate the lumped uncertainties and needs only the knowledge of inaccurate mathematical models for a strongly nonlinear system. It has been utilized in various applications, such as permanent magnet synchronous motor (PMSM) servo systems [34], hydraulic systems [35], rigid spacecraft systems [36], and mobile robots [37].
To the best of our knowledge, due to the complex uncertainties and disturbances in EOTS, it is difficult to implement high-performance tracking control using any of the existing control methods such as SMC, ESO-based methods, model-based methods, and PID. Therefore, a nonlinear ESO-based FO nonsingular terminal super-twisting SMC strategy is developed in this study. The motivation for developing compound control strategy is to reduce the impact of complex uncertainties and disturbances, thus achieving a satisfactory tracking control for EOTS. The novel compound controller is developed that integrates the advantages of a nonlinear ESO, a fractional-order nonsingular terminal sliding mode (FONTSM) manifold, and a super-twisting algorithm. The ESO is used to estimate lumped disturbances in real time without the need for accurate system models and disturbance models. The sliding mode dynamic, which utilizes a FONTSM manifold, has a more flexible control structure than the IO controller, and responds rapidly in both the sliding mode and reaching phases. The super-twisting technique is integrated into the FO sliding mode controller to reduce chattering and enhance robustness. The finite-time convergence of the designed ESO is demonstrated. The finite-time stability of the controller was also analyzed by the Lyapunov theorem. In addition to the developed novel ESO based SMC strategy, we performed experiments on the EOTS platform to evaluate its effectiveness. The contributions of this study are as follows: 1 An improved nonlinear function ϕ(e 1 (t)) is employed in the ESO. The nonlinear ESO can estimate the lumped disturbance without requiring explicit information about the nonlinear factors and an accurate system model. 2 FO calculus is integrated with the NTSM manifold.
The FONTSM manifold guarantees a fast dynamical response, avoids the nonsingularity problem, increases the sliding manifold flexibility, and achieves better control than the ordinary IO-based NTSM method. 3 A FONTSM manifold based super-twisting technique is integrated into the controller to improve the system robustness. 4 The finite-time stability of both the observer and the controller are analyzed via the Lyapunov theorem. Comparative experiments were carried out on the EOTS platform to demonstrate that the developed control strategy is effective and outperforms the existing NTSM [38], observer-based ISM [39], and ESO based super-twisting SMC [40] methods. The remainder parts of this paper are organized as follows. In Section 2, the system model are introduced. Section 3 is devoted to describe the control design. Stability analysis of the developed control is given in Section 4. Experimental studies are given in Section 5. The conclusions are presented in Section 6.

II. PROBLEM STATEMENT AND SYSTEM MODEL
The system model of the EOTS is described as follows: where θ t and ω t denote angle and angular velocity. J represents the moment of inertia. T f indicates friction force. T d indicates disturbances including external disturbance, uncertainties, and model variation. T i is the electromagnetic force.
The system model (1) can be divided into two parts, the nominal part and the bias part, i.e., J = J * + J , where * is the nominal value and is the difference between the real model and the nominal model. It worth noting that an EOTS always operates in a harsh environment. Wind disturbance and changes in aircraft attitude could lead to external disturbances and uncertainties T * d + T d . Temperature variation and changes in the center of gravity could contribute to variation in the system model J and T f . Under such conditions, the trajectory tracking control performance will deteriorate significantly and the system may even become unstable.
Then the equation (1) can be reformulated as follows where h denotes the lumped disturbance in EOTS, It worth mention that EOTS is driven by permanent magnet synchronous motor (PMSM) directly. By using the field-oriented control (FOC) approach for PMSM, the electromagnetic force T i could be described as follows: where p denotes the number of pole pairs, ψ d represents the permanent magnet flux linkages, K t represents the torque coefficient and i d is the driving current input. Friction is one of the most complex nonlinear factors existing in EOTS. In this study, we adopt a modified friction model [41] with continuously differentiable characteristic to describe the friction behavior in EOTS. Owing to the using of a hyperbolic tangent and a differentiated Gaussian function, the modified friction model T f is continuously differentiable with practical engineering value.
where F s denotes the Stribeck peak force, F c represents the Coulomb friction force, µ s indicates the viscous coefficient, w s and w t are the Stribeck and the transition velocity, respectively. Let The system model of EOTS in scalar form can be re-expressed as follows: Thus, the objective we try to implement in this study can be described as: We design an effective controller u for the EOTS, to guarantee trajectory x 1 (t) to accurately and quickly track the predetermined reference trajectory r(t) in the presence of complex uncertainties and disturbances, i.e., the tracking error lim t→∞ ε 1 (t) = lim t→∞ (x 1 (t) − r(t)) = 0.
Assumption 1: T f , T d , and J are bounded. We assume that the lumped disturbance h and its derivativeḣ exist and are bounded, satisfying |h| ≤h ∈ R + , whereh is the unknown upper bound.

III. CONTROL DESIGN
In this section, the control design including ESO, FONTSM dynamic and super-twisting technique is described. A lemma is introduced to make the discussion more clearly.
Lemma 1 [42]: For dynamic systemẋ c = f (x c (t)) with f (0) = 0 and x c (t) ∈ R n , assume that the following relationship between a positive define function V c (x c (t)) and its differential term are satisfieḋ where τ 1 > 0, τ 2 > 0 and 0 < θ < 1. The dynamic system is stable. Besides, the convergence time T 1 is obtained: In

A. NONLINEAR ESO
A general linear ESO [40] for a second-order system can be presented as follows: where β 1 , β 2 and β 3 are the adjustable observer gains.
Consider the second-order system (6), an improved nonlinear ESO is designed as follows In contrast to general linear ESOs, the designed nonlinear ESO includes a nonlinear function ϕ(e 1 (t)). It combines the advantages of traditional linear and nonlinear ESOs. A detailed analysis is given in Remark 1.
The derivative of the function ϕ(e 1 (t)) is given as follows: Define an intermediate variable So the Eq.(13) can be rewritten aṡ The observer error is defined as Consider (6), (10) and (15), the observation error system is gained as follows

INTRODUCTION TO FRACTIONAL ORDER CALCULUS AND FRACTIONAL ORDER CONTROL
The theoretical research into FO calculus dates back to about 300 years ago. However, its application has only gradually attracted increasing attention in recent years. The FO differentiation and integration operators, which can be thought of as generalizations of their IO versions, are defined as follows: where R represents the fractional calculus; α represents the fractional order value; t 0 and t indicate the limits for the operator; and (α) means the real part of α.
As the development of FO calculus theory, the FO definition is diverse. The three most commonly used fractional order calculus definitions in engineering application field are Riemann-Liouville (RL), Grunwald-Letnikov (GL) and Caputo. It worth noting that the proposed controller in this paper is developed according to Caputo definition. The definition for Caputo FO calculus is expressed as follows in which m represents an integer, and (·) means the Gamma In term of the amplitude-frequency characteristic, the Laplace transform is adopted to describe the FO calculus. The Laplace transform for Caputo FO calculus is expressed as below: where s = jω means the Laplace transform variable. Besides, it is worth mentioning that a fractional-order controller is able to adjust the slope of the Bode diagram shape arbitrarily, whereas the IO controller can only adjust its slope by integer multiples of 20 dB/decade. To illustrate the different results of IO and FO calculus, two typical functions f 1 (t) = 1 and f 2 (t) = t are introduced as follows: The tracking error is defined as below: where ε 1 (t) and ε 2 (t) are angle tracking error and angle rate tracking error, respectively. r(t) is the reference trajectory and it is twice differentiable, continuous and bounded. By employing the derivative on (20), the tracking error system is obtained as 45894 VOLUME 9, 2021 Consider the following fractional-order nonsingular terminal sliding manifold s ∈ R n : where parameters α 1 , α 2 , χ 1 , χ 2 , γ 1 , γ 2 are positive. Differentiating on both sides of the sliding manifold (22), the following equation is gotten: When neglecting the lumped disturbance h(t), by settinġ s = 0, the equivalent controller is obtained as follows A generalized super-twisting algorithm [44] which could provide more robustness and faster convergence rate than standard super-twisting algorithm is adopted.
where φ 1 (s) and φ 2 (s) denote nonlinear functions with extra linear correction terms based on [44] Consider the equivalent controller, the generalized super-twisting algorithm and the ESO, the control input u is designed containing three components u eq , u st and u ob By substituting (27), (28) into (23), the sliding manifold dynamic system can be expressed aṡ where e 3 = z 3 − h(t). Define z s1 and z s2 as the new state Then the closed-loop system dynamic in scalar form can be written as According to Theorem 1, the observer error system is finite-time bounded. So supposed |ρ| ≤ with ∈ R + . Afterwards, the developed control strategy (27)(28) is presented in Fig.2.
The proof of stability of the controller based on super-twisting algorithm is given in section IV.

IV. STABILITY ANALYSIS A. CONVERGENCE OF ESO
Theorem 1: Consider the designed ESO (10) with Assumption 1, there exist suitable parameters β i (i = 1,2,3) and k 1 , k 2 , r such that is satisfied. The observation error system (16) converges to zero in a finite time.

Substituting (40) into (42), and yieldṡ
According to Lemma 1, the observation error (15) is able to converge into the region {||η|| ≥ 4HM λ min {Q} } within finite time T 1 . Besides, the upper reaching time is presented as follows where V e (0) is the initial value of V e (e(t)). This completes the proof.

B. CONVERGENCE OF THE CLOSED-LOOP SYSTEM
Theorem 2: Assume that there exist a positive and symmetric definite matrix ϒ = ϒ T > 0 such that the matrix inequality (45) or Algebraic Riccati Inequality (46) is satisfied. Then trajectories of the system state (30) converge to origin in a finite time less than T 2 for all bounded perturbations satisfying |ρ| ≤ .
Remark 1: Nonlinear function ϕ(e 1 (t)) (11) consists of two parts k 1 e 1 (t) and k 2 e 1 (t) (r+1)/2 . When k 1 = 1 and k 2 = 0, the designed nonlinear ESO reduces to the general linear ESO (9). Note that the second term k 2 e 1 (t) (r+1)/2 in (11) is based on an continuous finite-time convergent differentiator technique and exhibits a better chattering suppression performance and a better observation accuracy than traditional sliding mode observation [43]. For instance, if k 1 e 1 (t) > k 2 e 1 (t) (r+1)/2 when the error system is far away from zero, the term k 1 e 1 (t) in (11) dominates. In contrast, if k 1 e 1 (t) < k 2 e 1 (t) (r+1)/2 when the error system is very close to zero, the second term in (11) dominates. Thus, the appropriate selection of parameters r, k 1 , and k 2 according to the application will improve the results.
Remark 2: The FO differentiation and integration operators can be regarded as generalizations of their IO versions. The designed FO sliding manifold is better than the IO ones since that it is able to increase the degrees of freedom of the sliding mode controller and the parameter adjustment is more flexible, thus yielding better robustness and tracking accuracy. It can be seen that when χ 1 = 1 and Because that the additional parameters χ 1 and χ 2 can be set arbitrarily, the designed FO controller outperforms the IO controller in term of control accuracy and robustness. Moreover, the denominator λ min {Q} relies on parameters β 1 , β 2 , β 3 and r. Thus, when the parameter β 3 is set correctly, the numerator can be treated as a constant. Subsequently, we can adjust β 1 , β 2 and r to satisfy the condition (32) and make the value of the denominator as large as possible. Note that the upper region of the observation error ||η|| can be decreased to a small and negligible value, hence, the observed accuracy of the nonlinear ESO is sufficient.

V. EXPERIMENTAL STUDIES A. EXPERIMENTAL SETUP
The experimental setup is depicted in Fig.3. It contains mainly four parts: personal computer (PC), power supply (24V), digital signal processor (DSP) controller and EOTS. The schematic diagram of EOTS is presented in Fig.4, which includes infrared camera, visible light television camera, PMSM, driver magnetic Encoder (RENISHAW AksIM with 0.01 • measurement accuracy).
The reference trajectory is given by a personal computer (PC). The DSP controller receives the reference signals, implements the proposed algorithm (Period: 1ms), and sends the algorithm result to the motor driver. Next, the motor driver amplifies the algorithm's output and drives the PMSM in  the azimuth axis to track the desired trajectory. Additionally, the FO terms in controller is implemented by the oustaloup filter approximation method. The order of oustaloup filter is selected as five order and the frequency range is choose as 0.001-100 rad/s.

1) CONTROLLER PARAMETERS
The method of selecting the parameters consists of the following three steps. First, we determined the parameters of the ESO. The selection of β 1−3 , k 1−2 , and r for estimating the lumped disturbance can substantially affect the control performance. A large β 1−3 indicates a large bandwidth, but if it is too large, this leads to serious chattering in the control results, whereas a value of β 1−3 that is too small cannot ensure stability. Hence, we selected the set of β 1−3 and r shown in Table 1 to ensure the accuracy of the observed value without much chattering. We also adjusted k 1−2 to reduce the observed error slightly. Second, we determined the parameters of the fractional-order nonsingular terminal sliding manifold. The fractional-integral term R χ 2 −1 can be regarded as a low-pass filter. The value of χ 2 is selected so that the steady-state errors are reduced. However, a value of χ 2 that is too large will cause long-term integration, which could result in a steady-state oscillations. Suitable values of α 2 and γ 2 , as listed in Table 1, will further reduce the steady-state errors. For the fractional-differential term R χ 1 , the values of the fractional-differential parameter χ 1 together with those of α 2 and γ 2 listed in Table 1 are able to ensure  rapid response without overshoot. Third, the parameters of u st were determined. An appropriate value of K 1−2 can enhance the robustness. However, a value that is too large may lead to chattering. When the load changes, we can re-adjust the value of K 1−2 to handle it. Finally, based on the above discussion, we selected a set of parameters to ensure high tracking accuracy, good robustness, and rapid response. The values are listed in Table 1.

2) EXPERIMENTS
The following four control methods are taken for experimental comparison: Our developed control strategy;  The NTSM control method [38]; The ESO based integral sliding mode (ESO+ISM) method [39]; The ESO based super-twisting sliding mode (ESO+STA) method [40].
The following four experiments were designed to evaluate the effectiveness of the developed method. In the first experiment, a step reference trajectory with two angles was designed to evaluate the response speed. In the second experiment, a sinusoidal reference trajectory was tested, and in the third experiment, a triangular reference trajectory with two angular velocities was employed. A higher angular velocity indicates increased friction, enabling the friction compensation to be evaluated. In the fourth experiment, a 2 kg load was added to the EOTS with the same triangular reference trajectory used in the third experiment to check the robustness of the system against uncertainties.
Furthermore, to make the compared results more convincing, the following calculations are introduced: the absolute average error AAE = 1 N N i=1 |ε 1 (i)| and the root-meansquare error RMS = N i=1

1) STEP REFERENCE TRAJECTORY
In the first experiment, a step signal with two amplitudes, 5 • and 20 • , is used as the predetermined reference trajectory. The corresponding experimental results are shown in Figs.5(a)-(c). Fig.5(a) shows the tracking results, and Fig.5(b) presents the tracking errors. All four control methods are able to track the step reference trajectory accurately without any overshoot, while still converging quickly.
In particular, because of the adopted FONTSM manifold, our proposed control strategy is faster than the other three control methods for both the 5 • and 20 • signals.  Table 2.

2) SINUSOIDAL REFERENCE TRAJECTORY
In the second experiment, a sinusoidal angle with a peak amplitude of 30 • and frequency of 0.1 Hz was used as the predetermined reference trajectory. The corresponding experimental results are shown in Figs. 6(a)-(e). In Fig.6(a), the four control approaches are all capable of tracking the predetermined sinusoidal reference trajectory with very little error. Fig.6(b) shows the tracking errors of the four approaches. The maximal error peak occurs when the system reaches the peak angle amplitude and changes direction. This is caused by changes in the system from static friction to dynamic friction. Because of the STA with the FONTSM manifold, the error peaks converge to zero more rapidly than in the other three methods. Owing to the designed ESO and the super-twisting controller with the FONTSM manifold, the proposed control strategy yields the smallest tracking error over the entire motion procedure. Specifically, for AAE, the one yielded by the developed control scheme is 42.8%, 24.0% and 6.3% of the ones by the ESO+STA, ESO+ISM, and NTSM methods, respectively. For RMS, the result is 43.6%, 25.1%, and 7.2% of results for the ESO+STA, ESO+ISM, and NTSM method, respectively. Fig.6(c) shows that the observed result of z 1 is easily able to capture the real angle. Fig.6(d) shows the lumped disturbance z 3 , and Fig. 6(e) shows the control efforts. The AAE and RMS values of the comparative results of the four methods are shown in Fig.7.

3) TRIANGULAR REFERENCE TRAJECTORY
In the third experiment, a triangular signal with two angular velocities of 10 • /s and 15 • /s is used as the predetermined reference trajectory. The corresponding experimental results are shown in Figs.8(a)-(e). The results in Fig.8(a) show that the four control methods are able to track the predetermined triangular reference trajectory with small errors. Fig.8(b) presents the four tracking errors. The proposed nonlinear ESO enables the developed control strategy to achieve the smallest tracking error when the system is in the uniform motion period. The maximal error peak occurs when the system is changing direction. Again, this is caused by changes in the system from static friction to dynamic friction. Owing to the proposed STA with the FONTSM manifold, the error peaks converge to zero more quickly than for the other three methods. Obviously, as the angular velocity increases, the four tracking performances all degrade, which is caused by the increased friction. However, the increase in tracking error of the proposed control strategy is the smallest, and this strategy performs the best over the entire tracking process, regardless of whether the system is moving uniformly or changing direction. Specifically, for AAE, the one yielded by the developed control strategy is 53.6%, 32.1%, and 4.88% of the ones by the ESO+STA, ESO+ISM, and NTSM methods, respectively. For RMS, the result is 66.2%, 44.5%, and 13.7% of the results of the ESO+STA, ESO+ISM, and NTSM methods. Fig.8(c) shows that the observed result of z 1 is well able to capture the real angle. In Fig.8(d), the lumped disturbance z 3 is presented. Fig.8(e) shows the control inputs. The AAE and RMS values of the four methods are provided in Fig.9.

4) TRIANGULAR REFERENCE TRAJECTORY WITH LOAD
In the fourth experiment, a 2 kg load was added to the inner frame of the EOTS to check the robustness against uncertainties of the developed compound control strategy. The reference trajectory is the same as that used in the third experiment. In Fig.10(a), the trajectories with and without load track the predetermined reference trajectory with little error. Fig.10(b) shows the two tracking error curves. It can be seen that the tracking error with the load is slightly larger than that without the load. However, good control performance can still be obtained under an extra 2 kg load. Specifically, the withload AAE increases by 0.2%. For RMS, the with-load result shows an increase of 0.2%. Fig.10(c) shows system state z 3 , which estimates the lumped disturbance. Owing to the extra load, the lumped disturbance increases such that with-load z 3 is slightly larger than without-load z 3 . Fig.10(d) presents the two control inputs. To ensure control accuracy with an extra load, the with-load control input requires a slightly larger control effort to handle the additional uncertainties. However, the two control inputs consume the same level of control torque. Fig.11 compares the AAE and RMS results with and without the load.

VI. CONCLUSION
This study proposed a nonlinear ESO based FO nonsingular terminal super-twisting SMC strategy designed for an EOTS with disturbances and unknown uncertainties. By applying the newly developed nonlinear ESO, the lumped disturbances of the system could be estimated without requiring explicit knowledge of the system model. The use of the NTSM guarantees the fast dynamical response of the control system. The use of FO calculus gives the controller a more flexible structure and a superior control and more robustness than its IO counterpart. In addition, the super-twisting sliding mode algorithm is integrated into the controller to increase the tracking accuracy and robustness. The finite-time convergence performance of both the developed nonlinear ESO and the system tracking error were analyzed using Lyapunov stability theory. Comparative experiments were conducted to evaluate the effectiveness of the developed control strategy, which was shown to be superior to the existing NTSM method in [38], the ESO+ISM method in [39], and the ESO+STA method in [40]. The results illustrate that our developed controller is capable of achieving faster response, higher tracking accuracy, and better robustness than the existing methods.
In future studies, we will consider integrating some adaptive laws into the developed controller to reduce the chattering, raise the tracking accuracy, and further improve the control performance. His research interests include nonlinear control, robust control, electro-optical targeting systems, signal processing, and manufacture and integration of inertial sensors.