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Scientific Reports volume 14, Article number: 23889 (2024 ) Cite this article Micro Positioning Stage
To address the electromechanical coupling and multi-source disturbance problems of the seeker stabilized platform, this paper constructs an electromechanical coupling model of the seeker stabilized platform based on the Lagrange-Maxwell equation. To mitigate the influence of electromechanical coupling on the control performance of the seeker, a super-twisting controller based on a fractional-order terminal sliding mode surface (FOSTSMC) is proposed. Additionally, to handle various disturbances in the system, this paper introduces a method that combines the extended state observer (ESO) with the proposed controller to enhance the system’s stability and anti-disturbance performance. The Lyapunov function is designed to prove that the proposed controller can reach a convergence state within finite time. Finally, the proposed control method is compared with PID control, fuzzy PID control, linear sliding mode control, and super spiral control combined with a disturbance observer (DOB). Multiple simulation experiments demonstrate that, under the influence of electromechanical coupling and multi-source disturbance, the FOSTSMC-ESO significantly improves the stability and anti-disturbance performance of the seeker stabilization platform.
The seeker is a key component in the guidance system that can detect and track the target, thereby guiding weapons such as missiles, torpedoes, and drones to accurately hit their targets. The seeker stabilization platform plays a vital role in the seeker system. It is mainly used to maintain the stability and accuracy of the seeker, ensuring that it can continuously and accurately detect and track the target under different flight conditions. The stabilization platform senses the attitude changes of the carrier through gyroscopes and other devices and performs real-time compensation so that the seeker is not affected by these changes. During flight, the seeker is subject to various disturbances, such as engine vibration, airflow disturbance, friction, and mechanical structure vibration. These disturbances generate various disturbance torques on the frame of the seeker stabilization platform, which are transmitted to the motor shaft, causing the voltage and current inside the motor to fluctuate. This fluctuation, in turn, causes instability in motor torque and speed, and the fluctuation in motor speed leads to vibrations in the stabilization platform frame, forming a cyclic electromechanical coupling phenomenon with negative effects1,2,3. Therefore, it is of great significance to study the electromechanical coupling effect between the electrical system and mechanical structure of the seeker stabilization platform and to design an effective servo control method to improve the seeker’s performance.
Accurate modeling of the seeker stabilization platform plays a fundamental and critical role in controller design. It not only directly affects the performance and stability of the control system but also has a profound impact on the selection, development, and optimization of the controller. Currently, scholars have studied various methods to model the stabilization platform.
Arthur K. Rue4 used the PIOGRAM diagram method to analyze the kinematic and dynamic mechanisms of the optoelectronic stabilization mechanism. Sungpil Yoon5 employed the Lagrangian dynamic equation to analyze a two-axis stabilization platform. Peter J. Kennedy6 established the kinematic and dynamic equations of a two-axis stabilization platform based on the fixed-point Euler dynamic equation, Euler coordinate transformation, and Newton-Euler theory. Yin7 developed the dynamic model of a three-axis inertial stabilization platform using the Lagrangian dynamic equation, analyzing the coupling effects of axis friction and base vibration on its dynamics. Zhu8 applied the Local POE formula to the kinematic modeling of a roll and pitch seeker based on robot theory, decomposed the inverse kinematic problem of aligning the visual axis and the optical axis into Paden-Kahan subproblems, and obtained the incremental angle of the seeker. Zhao9 simplified the kinematic and dynamic modeling process using the mathematical methods of Lie group Lie algebra and spinor theory, providing a closed solution of the mechanism’s dynamic equation in a simple mathematical iterative form. Wang10 analyzed the disturbance torque of the seeker wire and optimized its design. Zhang11 studied system friction and implemented effective compensation measures. Yu12 and Yang13 discussed the mass imbalance torque of the two-axis stabilization platform frame. Zhang11 analyzed the system’s coupling torque and proposed suppression measures. However, these modeling methods are limited to the mechanical structure and do not consider the electromechanical coupling effect of the stabilization platform. Therefore, this paper constructs an electromechanical coupling model while building the mechanical model of the seeker stabilization platform and taking into account the impact of electrical factors on the system.
As a typical mechatronic system, the dynamic characteristics of the stabilization platform’s motion mechanism often interact with motor characteristics. Changes in the operating state of the motion mechanism impact the voltage, current, and speed of the motor, and vice versa, forming an electromechanical coupling effect1,2,3. This electromechanical coupling effect can prevent the stabilization platform from timely identifying and tracking the target, seriously affecting system performance. Therefore, the electromechanical coupling relationship between the servo drive and the seeker stabilization platform cannot be ignored. Currently, there is no relevant research on the electromechanical coupling modeling of the seeker. Therefore, this paper combines knowledge from mechanics and electrical engineering, utilizing the Lagrange-Maxwell equation to establish an electromechanical coupling dynamic model between the electrical system and the mechanical structure of the seeker stabilization platform, and analyzes the coupling characteristics between the electrical and mechanical systems.
Due to the electromechanical coupling effect in the stabilization platform, traditional control methods can no longer effectively suppress the resulting system oscillation. Therefore, it is urgent to design a high-performance controller to eliminate the impact of electromechanical coupling on the performance of the seeker. Considering the purpose and working environment of the seeker, the designed controller must have stronger anti-disturbance capabilities while addressing the electromechanical coupling problem.
The stabilization platform typically employs a PID controller and its various improved forms, but these have poor anti-disturbance capabilities against nonlinear disturbance factors14,15,16,17. When the system encounters significant external disturbances or changes in the controlled object, the tracking performance of the system decreases. In response to this, Tang18 designed an error-based feedforward controller that suppresses disturbances solely through the controller without a separate disturbance suppression compensation strategy, resulting in limited anti-disturbance capability. Zhou19 developed a non-singular terminal sliding mode controller based on an extended state observer, which mitigates disturbances through the non-singular terminal sliding surface of the velocity loop. However, this method of improving the sliding surface alone has limited disturbance suppression capability. Yang20 proposed an adaptive ANCESO, which relies on the accuracy of the system reference model. Reis M. F21. designed a super-twisting controller and utilized simulation methods to verify the effectiveness of the control strategy. However, this method requires the disturbance derivative to be bounded. Saied H22. integrated super-twisting with feedforward control to achieve more stable and smooth robot operation. Liu23 applied super-twisting to the trajectory tracking control of unmanned boats, enhancing the system’s resilience to time-varying disturbances. Reference24 combined super-twisting with a fuzzy control strategy to improve the robustness of PMSM control systems. Considering the structural complexity of the coupled system and the presence of various nonlinear factors, A. Tepljakov25 designed a Fractional-Order PID (FOPID) controller. The results demonstrated that the fractional-order controller significantly enhances the stability of nonlinear systems. Zhang26 reviewed the application and development of Fractional-Order controllers in optoelectronic stabilization platforms, providing important guidance for technological innovations in stabilization platforms. Li27 proposed a fractional-order anti-disturbance control strategy, with the main contribution being the design of a Fractional-Order Extended State Observer (FOESO), which improved the system’s ability to observe disturbances. R.S. Sharma28 applied a Fractional-Order Sliding Mode Control (FOSMC) method to the visual robust control of mobile robots, eliminating the need for actual depth data in image visual. Zhang29,30 proposed a composite control method that integrates a controller and an observer, specifically combining an extended state observer with a super-twisting sliding mode controller. The results demonstrated that this composite control method effectively suppresses nonlinear factors in the seeker stabilization platform, but the system still exhibits certain amplitude oscillation phenomena31. specifically analyzes and designs the control system for the dual-axis stabilized platform, which has important guiding significance for the control of the dual-axis stabilized platform32. introduces a neural network-based adaptive control approach to address the complexities of pure-feedback fractional-order systems. This study tackles challenges such as output constraints and actuator nonlinearities, offering valuable insights for mitigating the electromechanical coupling effects in seekers. Additionally, it serves as a crucial reference for the design of the fractional-order controller proposed in this paper. In33] and [34, scholars conducted in-depth research and detailed the nonlinearity of the controlled object, proposing corresponding control strategies. These strategies greatly improved the control performance of the nonlinear system and provided significant guidance for the controller design in this paper. In35, scholars propose a backstepping sliding mode control strategy based on a nonlinear disturbance observer for nonlinear systems. This method significantly suppresses nonlinear factors within the system and provides important insights for the stable control of electromechanical coupling systems and the mitigation of nonlinear effects. These contributions are of significant importance for guiding the suppression of nonlinear factors.
However, the aforementioned control strategies are designed without considering the electromechanical coupling effect. Since this paper is the first to study the electromechanical coupling effect in the seeker stabilization platform, the control performance of the previously mentioned methods may not be optimal under such conditions. Therefore, this paper proposes an improved control strategy based on the electromechanical coupling and multi-source disturbance status of the seeker stabilized platform and compares it with the existing control methods to verify its effectiveness.
By analyzing the advantages and limitations of the aforementioned control strategies, this paper integrates the fractional-order principle with the super-twisting algorithm. In the speed control loop, a super-twisting controller based on the fractional-order terminal sliding mode surface (FOSTSMC) is designed to mitigate the adverse effects of electromechanical coupling and multi-source disturbances on system performance. The super-twisting algorithm is employed to mitigate system oscillations, while the fractional-order algorithm is utilized to attenuate the electromechanical coupling effect within the system. Furthermore, to further enhance the system’s anti-disturbance capability, a composite control method combining Extended State Observer (ESO) with the proposed Fractional-Order Super-Twisting Sliding Mode Controller (FOSTSMC), denoted as FOSTSMC-ESO, is introduced. The novelty of this paper lies in the initial development of the electromechanical coupling model for the seeker stabilization platform, as well as the application of the fractional-order principle to the super-twisting controller.
The paper encompasses the following key components:
Introduction of the novel exploration into electromechanical coupling within stabilization platforms, presenting the construction of a dynamic model for electromechanical coupling within the seeker stabilization platform by amalgamating multidisciplinary knowledge.
Analysis of the electromechanical coupling impact between the mechanical and electrical systems of the seeker through simulation experiments.
Addressing the electromechanical coupling effects and multi-source disturbance challenges of the seeker by designing a robust super-twisting control strategy grounded on the fractional-order terminal sliding mode surface (FOSTSMC). The stability of the strategy is validated through the design of a Lyapunov function. Additionally, an extended state observer (ESO) is integrated into the speed loop to enhance the system’s resilience against multi-source disturbances.
Execution of numerous simulation experiments to compare the efficacy of FOSTSMC-ESO against traditional PID control, Fuzzy-PID control, linear sliding mode control, and FOSTSMC-DOB. This comparison aims to ascertain whether FOSTSMC-ESO exhibits superior performance in suppressing electromechanical coupling effects and various disturbances.
In simple terms, the research objectives of this paper are twofold: first, to construct an electromechanical coupling model of the seeker; second, to design a fractional-order super-twisting sliding mode control strategy and analyze its impact on system performance. The chapters are arranged as follows: Chap. 2 presents the dynamic modeling of the seeker stabilization platform. Chapter 3 designs and analyzes the stability of the fractional-order sliding mode controller. Chapter 4 verifies its control performance through Simulink simulations. Chapter 5 summarizes and evaluates the advantages and disadvantages of the proposed method and suggests directions for future improvements.
The structure of the seeker stabilization platform36 is shown in Fig. 1. The following sections will model and analyze the mechanical system, electrical system, and electromechanical coupling. Firstly, a mathematical model of the frame’s mechanical structure is constructed based on the principles of theoretical mechanics. Secondly, a torque motor model is developed, considering the influence of electromechanical coupling factors on electromagnetic torque, voltage, current, frame torque, frame rotation speed, and rotation angle within the system. Finally, the mechanical and electrical systems are integrated using the Lagrange-Maxwell equation, and the coupling relationships between the parameters in the system are examined through coupling simulation experiments.
A pitch-yaw two-axis, two-frame seeker structure.
The kinematic and dynamic analysis of the mechanical system of the seeker stabilization platform involves multiple coordinate systems, including the inertial coordinate system, the translational coordinate system of the inertial system, the projectile coordinate system, and the internal and external frame coordinate systems36. The coordinate system definitions are shown in Table 1.
The coordinate systems are shown in Fig. 2.
Seeker stabilization platform coordinate systems.
The transformation matrix6 from the missile body coordinate system to the external frame coordinate system is given below:
According to the coordinate relationship:
where \({\omega _o}\) represents the angular velocity in the external frame coordinate system, \({\omega _m}\) represents the angular velocity in the missile body coordinate system, and \({\dot {\lambda }_y}\) represents the azimuth angular velocity of the external frame.
The transformation matrix6 from the external frame coordinate system to the internal frame coordinate system is given below:
According to the coordinate relationship:
where \({\omega _i}\) represents the angular velocity in the internal frame coordinate system and \({\dot {\lambda }_z}\) represents the pitch angular velocity of the internal frame.
In theoretical mechanics36, the moment of momentum theorem can be expressed as:
where L represents the angular momentum of the frame, M represents the external torque acting on the rotation center, and \({a_m}\) represents the tangential acceleration of the rotation center.
Based on the spatial geometric relationships and the principles of compound motion, we can deduce that:
where \({a_x}\) ,\({a_y}\) and \({a_z}\) represent the acceleration of the missile in three directions, and \(\rho\) represents the distance between the center of rotation and the center of mass of the missile.
The rotational inertia of a rigid body can be expressed as37:
where \({J_{xy}}\) , \({J_{yz}}\) , and \({J_{xz}}\) represent the product of the moments of inertia of the rigid bodies.
From (5), the dynamic equation of rigid body rotation can be determined as follows:
The dynamics equation of the internal frame can be determined from (8).
The external torques acting on the pitch axis include motor drive torque, friction torque, and wire disturbance torque. Therefore, (9) can be expressed as:
The internal frame will exert a reactive torque on the external frame. This can be expressed as:
The transformation matrix6 from the internal frame coordinate system to the external frame coordinate system is given below.
According to the coordinate relationship:
From (8) and (13), the dynamic equation of the outer frame can be determined as follows:
The external torques acting on the yaw axis include motor drive torque, friction torque, and wire disturbance torque. Therefore, (14) can be expressed as:
Compared with other motors, brushless DC motors38 can provide greater initial torque at start, so they are widely used in applications where large torque is required for starting or acceleration. Additionally, torque motors offer advantages such as smooth operation, low noise, and long lifespan, which further contribute to their widespread use. The equivalent circuit of a brushless DC motor is shown in Fig. 3.
The equivalent circuit diagram of a brushless DC motor.
The three-phase windings of the brushless DC motor are perfectly symmetrical.
where \({u_a}\) , \({u_b}\) , and \({u_c}\) represent three-phase voltages, R represents resistance, \({i_a}\) , \({i_b}\) , and \({i_c}\) represent three-phase currents, \({e_a}\) , \({e_b}\) , and \({e_c}\) represent back electromotive force, L represents self-inductance, and M represents mutual inductance.
It is approximately assumed that only two phases are active in the motor during operation.
The equivalent circuit of the motor during operation is shown in Fig. 4.
Equivalent circuit diagram of the motor during operation.
here J represents the moment of inertia equivalent to the motor shaft.
\({T_e}\) represents the electromagnetic torque, which can be regarded as motor drive torque in (10) and (15).
Typically, (21) can be determined as:
Where \({n_p}\) represents the number of pole pairs, and \({k_e}\) denotes the back electromotive force coefficient.
\({T_d}\) represents the disturbance torque. From (10) and (15), the disturbance torque can be expressed as:
B represents the coefficient of viscous friction. The friction torque can be expressed as:
Building upon the established mechanical and electrical system models, the Lagrange-Maxwell equation was employed to derive the electromechanical coupling dynamic equation between the motor drive system of the seeker stabilization platform and the frame mechanical system. Utilizing this model, an analysis was conducted on the coupling characteristics between the voltage and current of the brushless DC motor and the frame torque.
In this system, the Lagrange-Maxwell operator is represented as39:
where T represents kinetic energy, and W represents magnetic field energy.
where \({E_r}\) represents the dissipated energy, and \({Q_i}\) represents the generalized force.
The system’s kinetic energy is as follows:
The system’s magnetic field energy is as follows:
where \({\varphi _f}\) represents the magnetic flux.
The energy dissipated in the system mainly consists of resistive losses and frictional losses.
where \({R_r}\) denotes the rotor resistance, and \({k_f}\) signifies the friction coefficient between the motor shaft and the stabilization platform frame.
Combining the models of the mechanical and electrical systems, the global electromechanical coupling dynamics equation of the seeker stabilization platform is obtained as:
To investigate the electromechanical coupling effect in the seeker stabilization platform, this paper develops mathematical models for both the mechanical structure and the electrical system of the seeker. Subsequently, using the Lagrange-Maxwell equation, the two systems are coupled to form a comprehensive electromechanical dynamics model of the seeker. Finally, the modeling and simulation are conducted in MATLAB/Simulink.
The electromechanical coupling simulation analysis of the seeker stabilization platform was conducted in MATLAB/Simulink. The coupling relationship between the motors in two different directions and the frame’s mechanical structure within the stabilization platform was examined. Additionally, the changes in voltage and current within the motor, influenced by the frame’s vibrations, were detected. The parameters of the seeker stabilization platform are presented in Table 2.
When the azimuth motor is inactive, only the pitch motor is powered, with its speed set to 100 deg/sec. Consequently, under the influence of the frame, the azimuth axis rotates synchronously with the pitch axis, as depicted in Fig. 5.
Speeds of the pitch and azimuth axes.
As depicted in Fig. 5(a), the rotation speed of the pitch axis eventually stabilizes at 100 deg/sec. Furthermore, as illustrated in Fig. 5(b), the azimuth axis, propelled by the frame, exhibits a swinging motion following the pitch axis, with an amplitude of 0.5 deg/sec. These simulation results comprehensively showcase the coupling effect of the mechanical structure on the motor.
To further investigate the influence of the mechanical structure on the voltage and current within the motor, we assume that both motors are unpowered, the missile is not subjected to overload, and the distance between the missile’s center of mass and the frame’s center of rotation is 1.5 m. However, there is an attitude disturbance with a frequency of 3 Hz and an amplitude of 1 deg in both the azimuth and pitch directions.
The three-phase voltage and current generated by the azimuth motor due to the frame’s attitude disturbance are illustrated in Fig. 6.
Three-phase voltage and current of the azimuth motor.
The three-phase voltage and current generated by the pitch motor due to the frame’s attitude disturbance are illustrated in Fig. 7.
Three-phase voltage and current of the pitch motor.
As shown in Fig. 6(a) and Fig. 7(a), under the influence of mechanical vibration, the azimuth motor and the pitch motor generate disturbance voltages with peak values of 0.028 V and 0.012 V, respectively. Similarly, as shown in Fig. 6(b) and Fig. 7(b), the azimuth motor and the pitch motor generate disturbance currents with peak values of 0.25 A and 0.1 A, respectively. Additionally, as illustrated in Fig. 7(a) and Fig. 7(b), the voltage and current induced by the frame vibration in the pitch motor are continuous and change continuously. However, this phenomenon is not observed in the azimuth motor. Therefore, it can be inferred that the pitch motor is more sensitive to mechanical vibration.
The simulation experiment indicates that the electromechanical coupling effect impacts the rotation speed of both the azimuth and pitch axes. Additionally, the vibration of the frame structure causes changes in the voltage and current within the motor, resulting in fluctuations in motor torque. These fluctuations, in turn, induce frame vibrations, creating a vicious cycle that severely degrades the performance of the stabilization platform. Therefore, it is essential to design a high-performance controller to mitigate the electromechanical coupling effect in the seeker stabilization platform.
During the operation of the stabilization platform, various disturbance factors arise, including internal disturbances caused by electromechanical coupling effects and external uncertain disturbance torques, such as wind resistance torque. These factors reduce the system’s stability and significantly impact the identification and tracking accuracy of the platform.
The seeker stabilization platform is a typical mechatronic servo system. The motion characteristics of the seeker stabilization platform include inertia and damping, and are affected by nonlinear factors such as saturation effect, friction, coupling effect, and dynamic load changes. These characteristics make it exhibit complex nonlinear behavior during control and stabilization. This results in the output not changing proportionally with changes in the system input. The system may also exhibit a variety of dynamic behaviors, such as chaos, periodic oscillations, and limit cycles. Common phenomena include saturation, dead zones, and hysteresis, all of which affect system stability. Additionally, stability analysis is relatively complex and requires the use of advanced tools, such as the Lyapunov method. Therefore, it is necessary to design an advanced controller to suppress this electromechanical coupling effect and nonlinear factors.
To address these factors, a super-twisting control method based on the fractional-order terminal sliding surface is proposed, combined with an ESO to further enhance control stability. Firstly, FOSTSMC is a method based on fractional-order sliding mode control theory, which can provide better dynamic performance and robustness in complex systems. Compared with traditional integer-order control methods, fractional-order controllers offer stronger adjustment flexibility and can more accurately modulate the system’s dynamic response. This feature is particularly important for the electromechanical coupling effect in the seeker stabilization platform, as this effect induces nonlinearity and uncertainty in the system, and fractional-order controllers can better manage these complexities. Secondly, Super-Twisting Sliding Mode Control (STSMC) is an improved sliding mode control method that achieves fast convergence without causing high-frequency chattering. This is crucial for managing multi-source disturbances in the seeker stabilization platform. STSMC mitigates the common chattering problem in traditional sliding mode control by introducing continuous control input, thereby improving the system’s control accuracy and stability. The introduction of the extended state observer (ESO) further enhances the system’s robustness and anti-disturbance capability. ESO can estimate unknown states and external disturbances in real time and effectively compensate for them, thereby improving the system’s dynamic performance and steady-state accuracy. In an environment characterized by electromechanical coupling and multi-source disturbances, ESO can effectively compensate for disturbances caused by electrical and mechanical factors, enhancing the system’s stability and response speed.
Compared with existing stabilization platform control methods, the FOSTSMC-ESO control strategy offers enhanced robustness, improved accuracy, reduced jitter, and strong adaptability. It is significantly more effective than current seeker stabilization platform control methods, particularly in addressing electromechanical coupling effects and multi-source disturbances. The advantages of the proposed method in terms of stability and anti-disturbance are verified through experimental comparisons with traditional PID control, fuzzy PID control, linear sliding mode control, and FOSTSMC-DOB. The principle of fractional-order super-twisting sliding mode control is illustrated in Fig. 8.
Schematic diagram of fractional-order super-twisting sliding mode control.
The complex structure and interactions within the system often lead to fractional-order dynamic characteristics40. In a system with fractional-order dynamic characteristics, the dynamic response may not follow the traditional exponential decay law but instead exhibit more complex behavior. Therefore, the introduction of fractional-order calculus theory can more accurately capture and analyze the dynamic characteristics of these systems, providing a richer set of tools and methods for system modeling, analysis, and control.
Fractional calculus theory is an extension of traditional calculus, allowing the order to be real or complex. In the context of this study, the order of fractional differentiation and integration is real. Its intuitive expression is41:
Based on the structure of the seeker stabilization platform and the BLDCM electromagnetic torque Eq. (18), the state equation of the system can be formulated as follows:
The fractional-order terminal sliding surface is designed as follows:
where \(si{g^\beta }({x_1})={\left| {{x_1}} \right|^\beta }sign({x_1})\) ,0 < β < 1;
\({k_1}\) and \({k_2}\) are sliding mode parameters.
Introducing the nonlinear term \(si{g^\beta }({x_1})\) aims to create a steeper sliding surface near the origin of the phase plane, allowing the system to reach convergence at the fastest possible speed.
Derivative of the fractional-order sliding surface:
\(\dot {s}=0\) , the equivalent controller:
where \(\hat {d}\) represents the disturbance estimate provided by the disturbance observer for the speed loop.
To mitigate the chattering problem in sliding mode control and ensure that the system state converges further to the sliding mode surface, the super-twisting sliding mode control rate is selected.
Finally, the control rate of the super-twisting sliding mode, based on the fractional-order terminal sliding mode surface, is designed as follows:
The term \(\hat {d}\) in (41) can be estimated using the following ESO.
where \(\omega\) represents the actual system speed, \(\hat {\omega }\) represents the system speed observed by ESO, and i represents the output signal of the controller.
When the system state reaches the sliding surface,\(s=\dot {s}=0\) , substitute (41) into (38).
where \(\bar {d}=d - \hat {d}\) represents the estimation error of the disturbance.
\(\left| {\bar {d}} \right| \leqslant a, a \in {R^+}\) .
\(\left| {\dot {\bar {d}}} \right| \leqslant b, b \in {R^+}\) .
Equation (42) can be further expressed as:
To streamline the proof process, the following methodology is implemented with the assistance of the approach outlined in43.
Choose the phasor \({\zeta ^T}=\left[ {{\zeta _1},{\zeta _2}} \right]=\left[ {{{\left| {{\mu _1}} \right|}^{0.5}}sign({\mu _1}),{\mu _2}} \right]\) , Eq. (43) can be expressed as:
where \(\left| {{\zeta _1}} \right|={\left| {{\mu _1}} \right|^{0.5}}\) ,\(\frac{1}{{\left| {{\zeta _1}} \right|}}={\left| {{\mu _1}} \right|^{ - 0.5}}\) ,\(A=\left[ {\begin{array}{cc} { - 0.5{k_3}}&{0.5} \\ { - {k_4} - \dot {\bar {d}} \cdot sign({\mu _1})}&0 \end{array}} \right]\) .
Construct the following Lyapunov function:
where \(P=\left[ {\begin{array}{cc} {4{k_4}+k_{3}^{2}}&{ - {k_3}} \\ { - {k_3}}&2 \end{array}} \right]\)
where \({\lambda _{\hbox{min} }}(P)\) is the smallest eigenvalue of matrix P, \({\lambda _{\hbox{max} }}(P)\) is the largest eigenvalue of matrix P, and \({\left\| \zeta \right\|_2}\) is the 2-norm of the vector \(\zeta\) .
Derivative of the Lyapunov function (45):
where \(A={A_1}+{A_2}\) , \({A_1}=\left[ {\begin{array}{cc} { - 0.5{k_3}}&{0.5} \\ { - {k_4}}&0 \end{array}} \right]\) , \({A_2}=\left[ {\begin{array}{cc} 0&0 \\ { - \dot {\bar {d}} \cdot sign({\mu _1})}&0 \end{array}} \right]\) ,
According to assumption 4, (47) can be further derived as:
where \({Q_1}=\left[ {\begin{array}{cc} {2{k_3}}&{ - 2} \\ { - 2}&0 \end{array}} \right]\dot {\bar {d}} \cdot sign({\mu _1})\) ,\({Q_2}=\left[ {\begin{array}{cc} {2{k_3}{k_4}+k_{3}^{3} - 2{k_3}b}&* \\ { - (k_{3}^{2}+2b)}&{{k_3}} \end{array}} \right]\) .
To ensure that \(\dot {V}(\mu ) \leqslant 0\) is true, only \({Q_2}\) needs to be a positive definite matrix. According to the sufficient and necessary conditions of the positive definite matrix, \({k_3}\) and \({k_4}\) satisfy the following relation.
Therefore, by appropriately designing the relevant parameters to satisfy the constraints of Eq. (53), the system can reach a convergent state within a finite time and maintain stability.
This chapter will conduct a simulation comparison between the proposed FOSTSMC-ESO and traditional PID control, fuzzy PID control, linear sliding mode control, and FOSTSMC-DOB. It will verify the superiority of the proposed control algorithm in the boresight control of the seeker stabilization platform from the following three aspects:
Analysis of speed signal tracking.
Analysis of displacement signal tracking.
During the operation of the seeker stabilization platform, the position or velocity tracking signal provided by the front-end optoelectronic device changes in real-time according to the target’s movement state. Therefore, in this part of the experiment, a step signal is used to simulate a fixed target state, while a sinusoidal signal is used to simulate a moving target state.
The seeker locks onto a fixed target. A step signal is used to simulate the fixed target state. The experiment is conducted as follows: a 10 deg/sec speed input signal is applied to the stable platform. The simulation results are depicted in Fig. 9. To facilitate comparison of the control effect, the section highlighted in red in Fig. 9(a) is enlarged as depicted in Fig. 9(b).
Response curve of seeker step speed tracking.
As depicted in Fig. 9(b), the speed response curves of traditional PID control, fuzzy PID control, and linear sliding mode control all exhibit significant fluctuations, revealing the influence of electromechanical coupling effects on the system. The steady-state error of the PID system is 0.018 deg/sec, while that of the Fuzzy-PID system is 0.015 deg/sec, and the SMC system is 0.017 deg/sec. In contrast, the speed response curve of the FOSTSMC-ESO proposed in this paper exhibits a smaller oscillation amplitude, with a steady-state error of only 0.001 deg/sec. This is 0.002 deg/sec smaller than the steady-state error of FOSTSMC-DOB. These results demonstrate that the FOSTSMC-ESO proposed in this paper can effectively mitigate the negative impact of electromechanical coupling effects on the system and reduce system oscillations. Detailed experimental data are presented in Table 3.
Here, the steady-state error is calculated by subtracting the set value from the peak value of oscillation.
The seeker locks onto a moving target. In this experiment, a sinusoidal signal is used to simulate the moving target state. Specifically, a sinusoidal input signal with an amplitude of 10 deg/sec and a frequency of 5 Hz is applied to the stable platform. The simulation results are depicted in Fig. 10, while the speed-tracking error is illustrated in Fig. 11.
Response curve of seeker sinusoidal speed tracking.
As shown in Fig. 11, for a sinusoidal input signal, the PID tracking error peak is 0.807 deg/sec; the Fuzzy-PID tracking error peak is 0.494 deg/sec; the SMC tracking error peak is 0.213 deg/sec. The FOSTSMC-DOB tracking error peak is 1/15 of the SMC error peak, only 0.014 deg/sec, while the FOSTSMC-ESO tracking error is even smaller at only 0.004 deg/sec. These results demonstrate that the FOSTSMC-ESO has superior tracking performance for sudden changes in the system input. The experimental data are presented in Table 4.
The results from the simulation experiments and data analysis indicate that irrespective of whether the target is stationary or mobile, the FOSTSMC-ESO proposed in this paper exhibits superior performance in terms of response speed, steady-state error, and suppression of system vibrations. However, it is important to acknowledge that while the proposed method demonstrates clear advantages, it is not without its limitations. For instance, despite the overall improved performance, the method still struggles to completely eliminate the effects of electromechanical coupling, resulting in slight oscillations within the seeker.
During operation, the stabilization platform can be affected by various disturbance signals, including those from the external environment, mechanical vibrations, system errors, and other sources. To address these disturbances, the following simulation experiments are conducted: instantaneous disturbance, mechanical vibration, wind disturbance, and sensor noise.
During system operation, sudden changes or transient responses often occur due to various factors. The resulting disturbances can be simulated using a step signal.
The experiment proceeds as follows: the stable platform operates at a speed of 10 degrees per second, and at the 0.3-second mark, there is a sudden increase in the load on the stable platform. At this point, a torque step signal of 0.3 Nm is applied to the stabilization platform to simulate the abrupt change in system load.
The step signal is depicted in Fig. 12.
The simulation results are illustrated in Fig. 13(a). To facilitate comparison of the control effect, the highlighted red section in Fig. 13(a) is enlarged and shown in Fig. 13(b).
Speed tracking curve of seeker under step disturbance.
As shown in Fig. 13(b), when the system load suddenly increases, the speed of the PID and Fuzzy-PID systems decreases by 0.169 deg/sec and 0.109 deg/sec, respectively, with a recovery time exceeding 0.4 s. The speed of the SMC system decreases by 0.121 deg/sec, with a recovery time of 0.02 s. The speed of the FOSTSMC-DOB system decreases by only 0.036 deg/sec, with a recovery time of 0.01 s. The motor speed of the FOSTSMC-ESO system is almost unaffected. These results indicate that the FOSTSMC-ESO exhibits superior stability in the presence of instantaneous disturbances. The detailed experimental data are presented in Table 5.
Here, the steady-state error is calculated by subtracting the set value from the peak value of oscillation.
The stabilization platform is subject to mechanical vibrations due to carrier motion, structural imperfections, ground vibrations, and other factors. These vibration signals are typically periodic and are thus simulated using a sinusoidal signal.
The experiment proceeds as follows: the stabilization platform operates steadily at a speed of 10 degrees per second. At 0.3 s, mechanical vibrations are introduced due to the internal structure of the seeker. Specifically, a sinusoidal disturbance signal with an amplitude of 0.3 Nm and a frequency of 14 Hz is applied to the stabilization platform at 0.3 s to simulate these mechanical vibrations.
The sinusoidal signal is depicted in Fig. 14.
The simulation results are illustrated in Fig. 15(a). To facilitate comparison of the control effect, the highlighted red section in Fig. 15(a) is enlarged and shown in Fig. 15(b).
Speed tracking curve of seeker under sinusoidal disturbance.
As illustrated in Fig. 15(b), when subjected to sinusoidal disturbance, the system’s speed response curve exhibits sinusoidal oscillation. The oscillation amplitudes of the PID, Fuzzy-PID, and SMC systems are 0.164 deg/sec, 0.102 deg/sec, and 0.108 deg/sec, respectively. The oscillation amplitude of the FOSTSMC-DOB system is only 1/18 of that of the SMC system, amounting to merely 0.006 deg/sec. Remarkably, the speed of the FOSTSMC-ESO system shows negligible oscillation under sinusoidal disturbance. These results underscore the robust capability of the FOSTSMC-ESO proposed in this study to suppress time-varying disturbances. Detailed experimental data are presented in Table 6.
Changes in wind direction and strength can introduce disturbances, particularly for systems such as aircraft or missiles. During operation, these disturbances are often random and difficult to predict. Consequently, random signals are used for the simulation.
The experiment proceeds as follows: the stable platform operates steadily at a speed of 10 degrees per second. At 0.3 s, the motor responsible for maintaining the visual axis experiences torque fluctuations due to wind disturbance. Specifically, a random disturbance torque signal is applied to the stabilization platform at this time to simulate the effects of wind disturbance.
The random signal is depicted in Fig. 16.
The simulation results are illustrated in Fig. 17(a). To facilitate comparison of the control effect, the highlighted red section in Fig. 17(a) is enlarged and shown in Fig. 17(b).
Speed tracking curve of seeker under random disturbance.
As shown in Fig. 17(a), under the influence of random disturbances, the speeds of the PID, Fuzzy-PID, and SMC systems exhibit significant irregular oscillations. The peak speed oscillation of the PID system reaches 0.39 deg/sec, while the peaks for the Fuzzy-PID and SMC systems are 0.136 deg/sec and 0.072 deg/sec, respectively. The FOSTSMC-DOB system shows a peak oscillation of 0.029 deg/sec, which is smaller than that of the SMC system. Notably, the speed response curve under the control of the FOSTSMC-ESO system has an oscillation peak of only 0.006 deg/sec. The experimental data demonstrate that, when subjected to random irregular disturbances, the FOSTSMC-ESO system proposed in this paper exhibits superior anti-disturbance capability. The specific experimental data are shown in Table 7.
In real-world control systems, sensors often exhibit detection errors due to material or structural factors. In this experiment, a sensor error value with an amplitude of 0.5 deg/sec and a frequency of 10 Hz was introduced into the feedback circuit.
The sensor error is depicted in Fig. 18.
The simulation results are illustrated in Fig. 19(a). To facilitate comparison of the control effect, the highlighted red section in Fig. 19(a) is enlarged and shown in Fig. 19(b).
Speed tracking curve of seeker with sensor error.
As depicted in Fig. 19(b), the speed curves corresponding to PID, Fuzzy-PID, and SMC control exhibit fluctuations due to sensor detection errors. Their fluctuation amplitudes are measured at 0.089 deg/sec, 0.061 deg/sec, and 0.053 deg/sec, respectively. Remarkably, the speed fluctuation amplitude of the FOSTSMC-DOB system is only 1/15 that of the PID system, amounting to merely 0.006 deg/sec. Furthermore, the fluctuation amplitude of the FOSTSMC-ESO system is reduced by 0.004 deg/sec compared to FOSTSMC-DOB, totaling only 0.002 deg/sec. This reduction significantly mitigates the influence of sensor errors, enabling stable speed tracking. The specific experimental data are presented in Table 8.
Through experiments involving instantaneous disturbance, mechanical vibration, wind disturbance, and sensor noise, the results consistently demonstrate that the speed response curve under FOSTSMC-ESO control exhibits reduced fluctuations, exceptional anti-disturbance capability, and maintains stable speed tracking. These findings demonstrate that the proposed FOSTSMC-ESO approach enables the seeker to maintain excellent stability under conditions of electromechanical coupling, mechanical vibration, wind disturbances, and enemy jamming. However, it is also critical to recognize that despite these promising results, the proposed method still has areas that require improvement. For instance, while the method significantly reduces fluctuations, it does not entirely eliminate them.
Compared to single closed-loop control, dual closed-loop control can further enhance system stability and anti-disturbance capabilities, thereby achieving superior control effectiveness. The dual closed-loop system for the speed displacement of the seeker stable platform is depicted in Fig. 20.
The experiment is conducted as follows: a displacement input signal of 5 degrees is applied to the seeker. At 0.3 s, a torque disturbance signal with an amplitude of 0.3 Nm and a frequency of 14 Hz is introduced to the stable platform. The sinusoidal signal is depicted in Fig. 14. The input signal for displacement tracking experiment is shown in the Fig. 21.
Input signal for displacement tracking experiment.
The simulation results are illustrated in Fig. 22(a). To facilitate comparison of the control effect, the highlighted red section in Fig. 22(a) is enlarged and shown in Fig. 22(b).
As depicted in Fig. 22(a), the response curve of the dual closed-loop system is smoother and more stable. Figure 22(b) illustrates that the error value of the dual closed-loop system is smaller than that of the single closed-loop system, with an order of magnitude 1/10th of the single closed-loop system. The amplitudes of the displacement tracking errors under PID, Fuzzy-PID, and SMC control are 0.0023 deg, 0.0015 deg, and 0.0008 deg, respectively. The error of FOSTSMC-DOB is 0.0002 smaller than that of SMC, which is 0.0006 deg. The displacement tracking error of the FOSTSMC-ESO system is only 0.0001 deg, with almost no oscillation in the displacement tracking curve. Specific experimental data are provided in Table 9.
Here, the steady-state error is calculated by subtracting the set value from the peak value of oscillation.
This chapter conducts performance analysis experiments in three aspects: speed signal tracking, disturbance response, and displacement signal tracking. Through multiple sets of experimental data, it is found that although FOSTSMC-ESO exhibits a slightly slower response speed, it demonstrates significant advantages in maintaining stable tracking of input signals, ensuring system stability, reducing electromechanical coupling effects, minimizing system oscillations, and enhancing the system’s ability to suppress disturbances. However, it is also important to note that the slightly slower response speed may limit the method’s applicability in scenarios requiring rapid adjustments. Additionally, while the method effectively reduces electromechanical coupling effects and system oscillations, it does not completely eliminate them.
This paper applies electromechanical coupling to the line-of-sight control of the seeker stabilization platform. Based on the theories of rigid body dynamics and motor science, as well as the Lagrange-Maxwell equations, a global electromechanical coupling model of the seeker stabilization platform is constructed. The coupling relationship between the electrical and mechanical systems in the stabilization platform is analyzed using this model. The interaction between various physical quantities in the seeker mechanical structure, such as frame mass, the distance between the frame’s rotation center and its center of mass, the frame’s moment of inertia, various torque disturbances in the frame, and the electromagnetic torque, voltage, current, and various device parameters in the motor, is demonstrated through simulation. To reduce or even eliminate the coupling phenomena within the seeker, this paper proposes a super-twisting control method based on a fractional-order sliding surface. Given the complex structure of the electromechanical coupling model established and the diverse disturbances in the system, the fractional-order super-twisting sliding mode controller is combined with ESO to further enhance the system’s disturbance rejection capability. To verify the effectiveness of the proposed method, it is compared with traditional PID control, fuzzy PID control, and first-order linear sliding mode control methods in the following three aspects: speed tracking control, dual-loop tracking control, and disturbance rejection analysis.
The conclusions drawn are as follows:
There is a significant electromechanical coupling effect within the seeker stabilization platform. Torque fluctuations in the platform frame directly affect motor parameters such as electromagnetic torque, voltage, current, and back electromotive force, which in turn cause fluctuations in motor speed. These speed fluctuations are further transmitted through the yaw and elevation axes, causing torque fluctuations in the frame, and forming a cyclical repetition process.
Due to the interconnectedness of the frame, the two motors located on the y-axis and z-axis also produce the same effects.
Compared to traditional control methods for the stabilization platform, the proposed fractional-order super-twisting control combined with the Extended State Observer can significantly mitigate the internal electromechanical coupling issues in the seeker. This conclusion can be drawn from the experimental results on speed fluctuation.
Under the same disturbances, the stability and disturbance rejection capabilities of the FOSTSMC-ESO system are far superior to those of traditional control systems.
In conclusion, this paper systematically investigates the electromechanical coupling effect and the fractional-order super-twisting control strategy in the seeker stabilization platform. The findings offer valuable theoretical guidance and practical references for researchers and engineers working in this domain. Furthermore, this research contributes to the advancement of seeker stabilization platform technology, potentially broadening its range of applications. However, it is important to recognize that the proposed methods are not without their limitations. While the fractional-order super-twisting control strategy shows promise, its slightly slower response speed may hinder its effectiveness in situations requiring rapid adjustments. Additionally, the control strategy, although it reduces electromechanical coupling effects and system oscillations, does not completely eliminate them. These residual effects indicate that there is still room for improvement in the control algorithm. Moreover, the robustness of the FOSTSMC-ESO approach under extreme or highly variable conditions has not been fully assessed, which raises questions about its reliability in all possible operational environments. Future work will focus on optimizing these control strategies to address these shortcomings and exploring their implementation in diverse operational environments. This will enhance the robustness and versatility of seeker systems, ensuring they can perform reliably under a wider range of conditions.
The prospects for modeling and controller design for seeker stabilization platforms are highly promising, driven by advancements in computational techniques such as machine learning and system identification algorithms. These developments will facilitate the creation of more accurate models that capture the complex dynamics of real-world systems, enabling the design of robust control strategies to address nonlinearities, uncertainties, and disturbances. The integration of multi-physics modeling will enhance the understanding of subsystem interactions, leading to optimized design and performance. Additionally, decoupling techniques will further improve control performance and stability by addressing subsystem coupling issues. The rise of distributed control architectures and AI-driven methodologies will enhance scalability, adaptability, and resilience, unlocking new deployment opportunities in dynamic and demanding settings. As technology continues to advance, the next generation of seeker stabilization platforms will set new standards for precision, reliability, and versatility across domains ranging from aerospace to robotics. Leveraging these advancements will open exciting opportunities for research, development, and practical implementation in the pursuit of ever-improving stabilization and control capabilities.
The datasets used and analysed during the current study available from the corresponding author on reasonable request.
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This work was supported by Natural Science Foundation of Shandong Province (ZR2021QF031), China Postdoctoral Science Foundation(2023M743757), and Tai Shan Scholar Foundation (tshw201502042).
College of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao, 266061, China
College of Sino-German Science and Technology, Qingdao University of Science and Technology, Qingdao, 266061, China
Qingdang Li & Mingyue Zhang
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Yanyu Song conducted the experiments and authored the manuscript, Mingyue Zhang supervised and guided the entire project, and Qingdang Li provided funding and additional project support.
The authors declare no competing interests.
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Song, Y., Li, Q. & Zhang, M. Electromechanical coupling modeling and fractional-order control of the seeker stabilization platform. Sci Rep 14, 23889 (2024). https://doi.org/10.1038/s41598-024-73478-6
DOI: https://doi.org/10.1038/s41598-024-73478-6
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