A two-axis electromechanical coupling system asymptotic preset output feedback control method

Abstract

This invention discloses an asymptotic preset output feedback control method for a two-axis electromechanical coupling system. A two-axis state-space model containing lumped unknown dynamics is constructed for the system. With only angular position measured, an improved high-gain observer with an error correction term is designed to reconstruct states such as angular velocity within a finite time. A tracking error surface is constructed based on the tracking error, and a preset performance function is introduced to pre-define the transient and steady-state boundaries of the error surface. Through error transformation, the performance constraints are converted into a bounded problem with unconstrained variables. An output feedback control law containing a constant gain term and a time-varying gain adaptive PI compensation term is designed, and a saturation element is introduced on the observer output side to suppress the "peak" phenomenon of the high-gain observer. This invention ensures that the tracking error evolves strictly within the predefined performance boundaries and converges asymptotically under continuous bounded control input, while improving stability accuracy and response speed under conditions of large inertia and strong disturbances.

Description

Technical Field

[0001] This invention relates to the field of two-axis electromechanical coupling system control technology, and specifically to an asymptotic preset output feedback control method for a two-axis electromechanical coupling system. Background Technology

[0002] The two-axis electromechanical coupling system is the core end effector of a modern high-performance stabilization platform, and its performance directly determines the load's ability to achieve high-precision pointing and holding during maneuvering. Modern mission scenarios emphasize "rapid acquisition and stable tracking," placing stringent performance requirements on the system for "high-precision stabilization under dynamic conditions." In this context, the stabilization system must quickly and accurately drive and stabilize the load axis toward the predetermined target when the carrier is subjected to complex external excitations resulting in multi-degree-of-freedom strong vibration disturbances. Therefore, this system is a crucial component for achieving high-precision positioning and stable tracking in dynamic environments.

[0003] However, two-axis electromechanical coupling systems face severe control challenges in actual operation due to large inertia and strong disturbance coupling. On the one hand, to improve functional density and load-bearing capacity, the integration of load platforms is increasing, leading to a significant increase in their rotational inertia. Large inertia reduces the system's natural frequency, exacerbates mechanical resonance, limits the bandwidth of the control system, and results in sluggish dynamic response. On the other hand, when the carrier moves in a complex disturbance environment, strong random vibration disturbances caused by random external excitations are transmitted to the platform and load axis through the foundation, forming continuous, wide-bandwidth external interference. These strong external disturbances are intertwined with the inherent complex nonlinear factors within the system, such as frictional nonlinearity, transmission backlash, structural flexibility, and the dynamics of electromechanical actuators, making the entire system highly nonlinear, strongly coupled, and exhibiting parameter uncertainty. This poses an extreme test to the controller's dynamic response speed, steady-state accuracy, and robustness.

[0004] To address these challenges, domestic and international research has primarily focused on the application of advanced nonlinear control strategies in electromechanical stability systems. While traditional PID control methods are simple in structure and mature in engineering applications, their linear nature makes them difficult to effectively compensate for the strong nonlinearity and uncertainty of the system. They often require compromises between dynamic performance and steady-state accuracy, failing to meet high-performance requirements. Therefore, various modern nonlinear control methods have been introduced into this field: adaptive control and robust control enhance the system's adaptability to uncertainty through online parameter adjustment or worst-case design; sliding mode control is known for its strong invariance to matched disturbances, but its inherent high-frequency chattering problem may damage the actuator and excite unmodeled dynamics; active disturbance rejection control (ADRC) estimates and compensates for the "total disturbance" by extending the state observer, reducing the dependence on accurate models and demonstrating good potential for engineering applications; intelligent control methods based on neural networks and fuzzy systems utilize their powerful nonlinear approximation capabilities to compensate for complex unmodeled dynamics. These methods have improved the system's tracking performance and robustness to a certain extent.

[0005] However, existing control strategies still have significant limitations when facing the extreme combined condition of "large inertia and strong disturbances". Most methods either rely too heavily on precise model parameters or require full-state feedback (such as angular acceleration signals), resulting in complex engineering implementations; or, while they can guarantee that the tracking error will eventually be uniformly bounded, they are difficult to achieve asymptotic convergence in theory and practice. More importantly, these methods mostly focus on improving steady-state accuracy, while lacking direct, pre-set constraint mechanisms for the transient performance of the system response (such as overshoot and convergence rate). Therefore, developing a high-dynamic control technology with a relatively simple structure, requiring only output feedback, capable of simultaneously and strictly guaranteeing preset transient and steady-state performance, and achieving asymptotically stable tracking is of vital importance for breaking through the performance bottleneck of the next generation of two-axis electromechanical coupling systems. Summary of the Invention

[0006] The purpose of this invention is to provide an asymptotic preset output feedback control method for a two-axis electromechanical coupling system. This strategy integrates an improved finite-time convergent high-gain observer (HGO) for state estimation and a preset performance function (PPF) to constrain the transient and steady-state performance of the tracking error. Detailed controller design and closed-loop system stability analysis were completed. It was proven that under the proposed control law, all system signals are bounded and the tracking error asymptotically converges to zero, ensuring the system's transient and steady-state performance, achieving the specified tracking performance, and realizing the system's asymptotic stability.

[0007] The technical solution to achieve the purpose of this invention is: an asymptotic preset output feedback control method for a two-axis electromechanical coupling system, comprising the following steps:

[0008] Step 1: Establish the mathematical model of the two-axis electromechanical coupling system, then proceed to Step 2.

[0009] Step 2: Based on the mathematical model of the two-axis electromechanical coupling system, design an asymptotic preset output feedback controller based on a finite-time convergent high-gain observer, and proceed to Step 3.

[0010] Step 3: Using Lyapunov stability theory, prove the stability of the asymptotically preset output feedback controller based on a finite-time convergent high-gain observer, and obtain the result that the system tracking error is limited to the specified constraints and is asymptotically stable.

[0011] Compared with the prior art, the significant advantages of this invention are: (1) It proposes an output feedback control architecture that integrates preset performance control and finite-time state observation. By designing a novel tracking error transformation and a finite-time high-gain observer, it achieves fast and accurate estimation and compensation of the unmeasured state and lumped disturbance of the system under the condition that only angle feedback is required; (2) By combining adaptive robust integral and preset performance function, a simplified asymptotic preset output feedback controller is constructed. It can not only pre-set and guarantee the transient and steady-state performance boundaries of the tracking error, but also theoretically prove the asymptotic stability of the closed-loop system. It achieves the control effect of asymptotic convergence of the tracking error to zero under continuous strong disturbance. The simulation results verify its superior comprehensive performance. Attached Figure Description

[0012] Figure 1 This is a schematic diagram illustrating the principle of the asymptotic preset output feedback control method for the two-axis electromechanical coupling system of the present invention.

[0013] Figure 2 This is a structural diagram of a two-axis electromechanical coupling system.

[0014] Figure 3 This is a schematic diagram of the electric cylinder mounting structure.

[0015] Figure 4 This is a load platform position tracking diagram under the action of the asymptotic preset output feedback controller designed in this invention.

[0016] Figure 5 This is a load shaft position tracking diagram under the action of the asymptotic preset output feedback controller designed in this invention.

[0017] Figure 6 This is the position tracking error diagram under the action of the asymptotic preset output feedback controller designed in this invention.

[0018] Figure 7 This is a high-low tracking error diagram under the action of the asymptotic preset output feedback controller designed in this invention.

[0019] Figure 8This is a comparison of the tracking error of the load platform under the action of the asymptotic preset output feedback controller designed in this invention and under the action of the traditional PID controller.

[0020] Figure 9 This is a comparison of the tracking error of the load axis under the action of the asymptotic preset output feedback controller designed in this invention and under the action of the traditional PID controller.

[0021] Figure 10 A comparison of the directional performance indicators of the asymptotic preset output feedback controller designed in this invention and the traditional PID controller.

[0022] Figure 11 A comparison chart of high and low performance indicators under the action of the asymptotic preset output feedback controller designed in this invention and under the action of the traditional PID controller. Detailed Implementation

[0023] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0024] Combination Figure 1 , Figure 2 and Figure 3 The present invention provides an asymptotic preset output feedback control method for a two-axis electromechanical coupling system, comprising the following steps:

[0025] Step 1: Establish a mathematical model of the two-axis electromechanical coupling system.

[0026] Step 1-1: For a typical two-axis electromechanical coupling system, build a mathematical model and a controller model for the two-axis electromechanical coupling system.

[0027] To characterize the overall motion characteristics of the two-axis electromechanical coupling system, it is equivalent to a system consisting of a carrier, a load platform, and a load axis; the mass of the load platform is denoted as . The mass of the load shaft is denoted as . The rotation angle of the load platform relative to the carrier in the azimuth direction is denoted as . The rotation angle of the load axis relative to the load platform in the vertical direction is denoted as . The distance between the rotation center of the load platform and the rotation center of the load support structure is denoted as . The distance from the rotation center to the end of the load shaft load support structure is denoted as . .

[0028] When modeling, the load platform plane is selected as the zero potential energy surface. Therefore, the potential energy at the center of gravity of the load platform is set to zero, and the potential energy of the load axis relative to this plane is expressed by its mass and the height of its center of gravity. Based on this, the potential energies of the load platform and the load axis are written in their respective functional forms, denoted as: , Kinetic energy is expressed by the rotational kinetic energy of each rigid body and denoted as... , :

[0029] ,

[0030] ,

[0031] ,

[0032] ,

[0033] In the formula, Let be the angular velocity of the load platform's rotation relative to the carrier's orientation. The angular velocity of the load shaft load support structure relative to the load platform in the vertical direction is denoted as . is the gravitational constant.

[0034] According to the definition of the Lagrange system:

[0035] ,

[0036] This represents the system's Lagrange quantity.

[0037] Let the driving torques of the load platform and the load shaft be respectively , Then, from the Euler–Lagrange equation, we get:

[0038] ,

[0039] Among them, subscript , Indicates the load platform. Indicates the load axis.

[0040] Will , , and Substituting into equation (6), the mechanical dynamics model of the two-axis electromechanical coupling system is:

[0041] ,

[0042] ,

[0043] In the formula, Let be the angular acceleration of the load platform's rotation relative to the carrier's orientation. The angular acceleration of the load shaft load support structure relative to the load platform in the vertical direction; For the unmodeled disturbance terms of the load platform, This represents the unmodeled disturbance term for the load axis, which includes the effects of unmodeled friction, clearance, flexibility, launch impact, and road vibration uncertainties.

[0044] Based on the existing mechanical dynamics model of the two-axis electromechanical coupling system, it is also necessary to model the motor system and couple it with the mechanical model to construct a complete mechatronic dynamics model. The load platform uses a high-precision servo motor and a multi-stage gear reduction mechanism as the actuator. The servo motor stator is fixed inside the load platform, and the lower gear ring is fixed to the chassis of the motion platform by bolts. During operation, the interaction between the motor armature current and the excitation flux generates electromagnetic torque, which drives the load platform to rotate around the carrier after being reduced in speed by a multi-stage gear, thereby achieving azimuth control.

[0045] When modeling the motor, its simplified servo motor dynamics equations can be described as follows:

[0046] ,

[0047] ,

[0048] In the formula, This refers to the rotational inertia of the servo motor. For the rotation angle of the servo motor, For the electromagnetic torque of the servo motor, The viscous damping coefficient of the servo motor shaft. For gear input torque, This is the ratio of the servo motor torque amplification factor to the circuit resistance. For the control input of the servo motor, This is the ratio of the electromotive force coefficient of the servo motor to the circuit resistance. The angular velocity representing the rotation angle of the servo motor. This refers to the angular acceleration that represents the rotation angle of the servo motor.

[0049] Since the load platform requires a mechanical reduction transmission device composed of multi-stage gear mechanisms to reduce speed to meet the needs of the load platform and load shaft, a continuous backlash model is introduced to characterize the backlash effect in multi-stage gear transmissions. This model considers the input torque of the gear mechanism as... With output torque The relationship is represented as:

[0050] ,

[0051] In the formula, The transmission ratio of a multi-stage gear mechanism; This refers to the unmodeled error in the transmission process of a multi-stage gear mechanism.

[0052] As can be seen from the gear reduction,

[0053] ,

[0054] Among them, order , Represents the zeroth derivative. Denotes the first derivative. Represents the second derivative;

[0055] Substituting equations (10) and (11) into equation (9), we get:

[0056] ,

[0057] Therefore, the input torque of the load platform can be expressed as:

[0058] ,

[0059] Will Substituting into equation (7), the electromechanical integrated analytical dynamic model of the load platform of the two-axis electromechanical coupling system is obtained as follows:

[0060] .

[0061] The load shaft uses an electric cylinder as the actuator. The electric cylinder consists of a servo motor, reduction gears, and a ball screw, and features a compact structure, high transmission accuracy, and convenient installation and maintenance, making it suitable for servo stabilization systems operating under high-mobility conditions. The tail end of the electric cylinder is fixed to a support ring on the top of the load platform, and the other end of the telescopic rod is hinged to one side of the rocker arm. To describe the above process, starting from the servo motor shaft, its rotational balance equation and voltage balance equation are derived, resulting in a dynamic equation containing electromagnetic torque, damping, moment of inertia, control voltage, and back electromotive force terms. Its dynamic model is described as follows:

[0062] ,

[0063] In the formula, The electromagnetic torque of the motor. This indicates the output torque of the motor. It is the equivalent moment of inertia of the motor and its connected components. The rotation angle of the motor. The angular velocity is the rotation angle of the motor. The angular acceleration is the angle of rotation of the motor. It is the viscous damping coefficient at the motor shaft end. It is the ratio of the motor's torque coefficient to the total circuit resistance. This is the control input voltage for the servo motor. It is the torque output by the electric cylinder to the load. This refers to the equivalent rotational inertia of the deceleration and lead screw drive components. This is the viscous damping coefficient of the transmission link.

[0064] Subsequently, the relationship between the motor output torque and the screw shaft output thrust is introduced, and expressed as follows:

[0065] ,

[0066] ,

[0067] In the formula, The lead of a ball screw is a key geometric parameter that converts rotary motion into linear motion. The thrust output by the telescopic rod of the electric cylinder. It represents the overall efficiency of the electric cylinder transmission system and reflects the energy loss in actual transmission. This refers to the transmission ratio of the electric cylinder, describing the speed-torque conversion relationship from the motor to the lead screw. This is an electric cylinder.

[0068] From a geometric constraint perspective, the upper and lower fulcrums of the electric cylinder, the trunnion, and the axis of the load support structure of the load shaft satisfy certain spatial geometric relationships. For example... Figure 3 As shown, where This is the distance between the upper pivot point of the electric cylinder and the center of the trunnion within the load platform; This is the distance between the lower fulcrum of the electric cylinder and the center of the trunnion. This is the initial length of the electric cylinder; This is the current vertex. This is the initial angle.

[0069] in Then the displacement of the electric cylinder telescopic rod It can be represented as:

[0070] ,

[0071] Substituting equation (19) into equation (18), we get

[0072] ,

[0073] From equation (17), we can obtain

[0074] ,

[0075] The displacement of the telescopic rod can be determined by geometric constraints. Expressed as elevation angle The function is used to derive the thrust of the telescopic rod. Contribution to high and low directional torque. Equivalent driving torque of the electric cylinder on the load shaft. It can be written as:

[0076] ,

[0077] In the formula, angle Let be the angle between the thrust of the electric cylinder and the orientation axis of the load shaft support structure. This angle is introduced due to the nonlinearity of the mechanism introduced during the conversion of the linear motion of the electric cylinder's telescopic rod into the rotational motion of the load shaft support structure. It can be obtained from the trigonometric relationships in [the equation / reference].

[0078] ,

[0079] Therefore, according to equations (17) and (21), the input torque of the load shaft can be obtained as:

[0080] ,

[0081] In the formula, This represents the modeling error of the load shaft actuator. Substituting into equation (8), we can obtain the analytical dynamic model of the load axis electromechanical integration of the all-electric two-axis electromechanical coupling system as follows:

[0082] ,

[0083] In steps 1-2, to facilitate controller design, state variables are defined, and the mathematical model of the two-axis electromechanical coupling system is transformed into state-space equations, as follows:

[0084] The system dynamics model considers frictional nonlinearity and other uncertainties. To facilitate subsequent controller design, state variables are defined. , The mathematical model of the entire two-axis electromechanical coupling system can then be represented by the following state-space equations:

[0085] ,

[0086] In the formula, the inertia matrix Control gain matrix Control Matrix Coriolis force Gravity term vector Friction term vector External disturbance vector .

[0087] The specific expressions for the elements in each matrix in equation (26) are as follows:

[0088] ,

[0089] ,

[0090] ,

[0091] ,

[0092] ,

[0093] ,

[0094] in, The azimuth Coulomb friction coefficient is... For the azimuth friction shape parameters, The high and low Coulomb friction coefficients are, These are the high and low friction shape parameters. This is the azimuth linear damping coefficient. The linear damping coefficients are for both high and low directions. For the gravity eccentricity of the load platform, The eccentricity of the load platform due to gravity.

[0095] The following section will design a nonlinear controller for equation (26). The control objective is to design a bounded control input. This enables the load shaft to stabilize the system output. Track the reference motion angle signal of the load axis stabilization system as accurately as possible. And ensure that the tracking error is as small as possible.

[0096] To facilitate subsequent control design and closed-loop system stability analysis, the following lemma is introduced:

[0097] Lemma 1: For real numbers and constant ,in For the summation index, there exists .

[0098] Lemma 2: Consider a given initial value positive definite function ,if The time derivative satisfies the following inequality: , where the coefficient ,coefficient real numbers The range is ,So It can reach zero, and The upper bound is calculated using the following formula: .

[0099] Lemma 3: Consider the function and variables We obtain the inequality: .

[0100] Lemma 4: Consider variables , where positive numbers ,if If it is bounded, then it is determined. and It is also bounded; furthermore, if Then the condition is met. and .

[0101] To facilitate controller design, the following assumptions are made:

[0102] Assumption 1: Desired motion trajectory Second-order continuous differentiable bounded.

[0103] Proceed to step 2.

[0104] Step 2: Based on the mathematical model of the two-axis electromechanical coupling system, design an asymptotically preset output feedback controller based on a finite-time convergent high-gain observer. The specific steps are as follows:

[0105] Step 2-1: Design a finite-time convergent high-gain observer (HGO) for state estimation, as detailed below;

[0106] The two-axis electromechanical coupling system follows the desired motion trajectory given by the servo stabilization system. In continuous bounded control input This ensures that the motion platform meets firing requirements while moving. The tracking error and its derivative with respect to time are defined as follows:

[0107] ,

[0108] Among them, reference signal and its derivative It is bounded.

[0109] Based on equation (33), system (26) can be formulated as:

[0110] ,

[0111] In the formula, To aggregate unknown dynamics,

[0112] ;

[0113] in Reference signal The second derivative is derived as follows:

[0114] ,

[0115] in , , For positive constants, the maximum value is... scalar function Note the joint angular acceleration. Typically, this is achieved by analyzing joint position signals. Differentiation reveals that this may amplify noise in the joint position signal. To overcome this problem, an improved high-gain observer (HGO) with finite-time convergence will be designed to reconstruct the system state.

[0116] Although the HGO has been successfully applied in many industrial fields, due to its simple structure and direct parameter tuning process, the observer error can only reach an eventual bound. To address this issue, the HGO is modified to estimate the velocity error. :

[0117] ,

[0118] In the formula, and These respectively represent displacement errors and speed error The estimated value, These are positive numbers chosen by the designer; axis indexes. , Represents the orientation towards the load platform. Representing the high and low load axes, the following Both represent this meaning.

[0119] The gain matrix in equation (36) and gain matrix Described as:

[0120] ,

[0121] In the formula, the gain coefficient Defined as , It is a positive number.

[0122] Lemma 5: Consider the designed observer (36), which claims to estimate in finite time They converge to their truth values, that is, for time... , and .

[0123] Proof: We first define the estimation error as... and ,therefore and Relative to time The derivative can be calculated as:

[0124] ,

[0125] in Representing vectors The element in the i-th row, The row element represents the direction of the load platform. The time represents the row element corresponding to the high and low load axes.

[0126] Then, the Lyapunov function is chosen as:

[0127] ,

[0128] in Represents the integral variable, upper limit of integration. Defined as Lower limit of points Defined as Given It is an increasing and continuous bijective function, such that the inverse function... , Existence and It is continuous, and It is homogeneous in the zero limit, and its homogeneous approximation function is... .

[0129] Therefore, according to the definition of homogeneity in the zero limit, it can be derived that when superscript hour, and Furthermore, according to the derivation rules of inverse functions, we obtain...

[0130] ,

[0131] superscript Based on the above discussion, we know It is a continuously differentiable function. Then, according to (40), exist and In all cases, they are positive definite differentiable functions, and if ,but According to Leibniz's integral rule, time derivative Together with equation (38), the result is:

[0132] ,

[0133] in,

[0134] ,

[0135] ,

[0136] Note the auxiliary function and They are homogeneous in the two limits. In fact, In weight , and degree , The two limits are homogeneous. Furthermore, we can derive... In weight , and degree The two limits are homogeneous. In weight and degree It is also homogeneous in the double limit.

[0137] Therefore, according to the lemma, there exists a positive constant. This makes it possible for all parameters , The upper bound can be

[0138] ,

[0139] in It is a positive number.

[0140] Considering exist The time series is homogeneous and has positive constants. We have:

[0141] ,

[0142] Therefore, combining equations (44) and (45), equation (41) is further derived as follows:

[0143] ,

[0144] in It is a positive constant, such that , Given the maximum magnitude of the model's uncertainty, it can be deduced that there exists a normal constant. , making Then, if by appropriately selecting positive constants... Conditions can be guaranteed Then equation (46) holds true. Note that... and Is with Irrelevant parameters. Based on the above analysis, the second derivative can be derived. It must be bounded to ensure that the proposed HGO converges in finite time.

[0145] Therefore, based on Lemma 2, we can conclude that It is bounded and can reach zero in finite time, such that the estimates converge to their true values ​​in finite time, and the convergence time is limited by the following:

[0146] ,

[0147] Among the larger ones It helps improve the convergence speed of the observer error, but excessively large errors can lead to problems. This may lead to potential chattering; therefore, the trade-off between convergence speed and the smooth response of estimation error should be carefully considered.

[0148] In step 2-2, a preset performance function PPF is designed to constrain the transient and steady-state performance of the tracking error, as detailed below:

[0149] Generally, most available control methods based on a predefined performance function (PPF) aim to keep the transient and steady-state convergence performance of the tracking error within predefined boundaries. However, within this framework, controller design follows a recursive design process similar to inversion methods. Furthermore, complex pre-calculations of the time derivatives of these specific signals are necessary, leading to the well-known "complexity explosion" problem. To overcome this problem, tracking error surfaces... Designed as

[0150] ,

[0151] Among them, the constant term matrix ,constant ,constant , It is a two-dimensional real number space.

[0152] In addition, a specified performance set is defined. To characterize the tracking error surface Boundary:

[0153] ,

[0154] Among the positive numbers Monotonically decreasing function PPF , The set of all 2×2 real matrices.

[0155] ,

[0156] in It is the initial value. It is the allowable final steady-state error boundary. ; positive numbers , control The rate of decrease; based on equations (49)-(50), it was found that the specified performance set The size is determined by the parameter , , and Decide.

[0157] Lemma 6: Consider the tracking error surface in equation (48) PPF in formula (50) The following conditions must be met:

[0158] 1) If the conditions are met and Then guarantee , It is bounded, and Belongs to set :

[0159] ,

[0160] Wherein the initial upper bound coefficient of position error , Represents the set of real numbers;

[0161] 2) If ,but ;

[0162] Proof: The tracking error surface equation (48) can be rewritten as:

[0163] ,

[0164] in For displacement error The derivative of .

[0165] By solving the differential equation (52), based on and The facts lead to the following inequality:

[0166] ,

[0167] ,

[0168] According to equation (54), it can be concluded that the tracking error converges to The boundaries of the constraints. In other words, The performance envelope can be directly converted The performance envelope. Furthermore, according to... It can be further obtained ,in .

[0169] Based on equation (53) and Using L'Hôpital's rule and the squeeze theorem, and For a statement to be true, the condition is... It can be guaranteed.

[0170] Note that equation (49) is used to predefine the tracking error surface. The boundary, which can be further formulated as equation (51), is based on the designed PPF equation (50) and Equation (48) is used to describe the tracking error. The convergence boundary. Simultaneously, it can guarantee the tracking error... The performance constraints imposed.

[0171] In steps 2-3, an asymptotically preset output feedback controller is designed to ensure that the tracking error is within the predefined transient and steady-state convergence boundaries, and to achieve the asymptotic stability of the closed-loop system, as detailed below:

[0172] Following the above discussion, an asymptotically preset output feedback controller is designed. This controller not only ensures that the tracking error remains within the predefined transient and steady-state convergence boundaries, but also achieves asymptotic stability of the closed-loop system. This is because the tracking error surface in equation (48)... Includes unmeasurable variables We first use the estimates derived from the modified HGO(36) to replace This makes the available tracking error surface Defined as:

[0173] ,

[0174] in , It is a positive number. It is a saturation function, given as follows:

[0175] ,

[0176] Note that the introduced saturation function helps to correct the potential peaking phenomenon caused by HGO(36). To achieve the specified performance requirements described in the previous section, the transformation error estimate implemented in the output feedback controller is... Described as

[0177] ,

[0178] Among them, the estimated value of auxiliary variables , .

[0179] Regarding transformation error It should be noted that most existing results directly use the transformation error to design the controller, resulting in rather conservative system stability results. To address this issue and improve overall tracking accuracy, an adaptive PI compensation term estimate is designed by combining the transformation error equation (57). for

[0180] ,

[0181] Where the system estimation parameters , The error estimate is a constant gain. , It is a diagonal positive definite matrix. The time-varying gain is represented as follows:

[0182] ,

[0183] in,

[0184] ,

[0185] in It is an adaptive estimate of the unknown parameters of the system. for The derivative, constant term matrix System state estimation matrix ,function , and These are positive integers; note the function. Meet the conditions and , For positive constants, the function It is an integrally bounded function. To further ensure the boundedness of the estimated parameters, this paper adopts an exponentially decreasing function, i.e. ,in , It is a positive number.

[0186] Based on the observer (36), the transformation error estimate (57), and the compensation term estimate (58), the asymptotically specified output feedback controller can be designed as follows:

[0187] ,

[0188] Among them, the derivative matrix of the preset performance function The proposed controller (61) mainly consists of a constant gain. Time-varying gain The adaptive PI compensation term estimate and feedback items Control. Estimated value of adaptive PI compensation term. This is used to compensate for unknown uncertainties and nonlinearities in the system, while achieving asymptotic output feedback control with specified performance. Specifically, the constant gain involved... This is used to maintain the boundedness of the transformation error, ensuring the specified performance of the tracking error. Furthermore, the time-varying gain is driven by an adaptive law. With the estimated value and bounded functions of integration Adaptive adjustment not only handles unknown uncertainties, but also helps to ensure that the tracking error asymptotically approaches zero with the help of variables.

[0189] Proceed to step 3.

[0190] Step 3: Using Lyapunov stability theory, prove the stability of the asymptotically preset output feedback controller based on a finite-time convergent high-gain observer. The results show that the system tracking error is limited to the specified constraints and is asymptotically stable, as detailed below:

[0191] We will analyze the asymptotic stability of the closed-loop system with the proposed finite-time convergent HGO (36) and asymptotically preset output feedback controller (61).

[0192] Theorem 1: Consider system (26), using the proposed observer form (36) and controller form (61), if the chosen parameters and Then there exists a positive constant. This makes it possible for all The closed-loop system is stable, all signals are bounded, and the tracking error is... Strictly in sets It evolves internally and converges asymptotically to zero.

[0193] Proof: The proof consists of two parts. The first part proves that when the auxiliary variable... and its estimated value All in the collection Within the first part, the estimation error is bounded and converges to zero; the second part proves the asymptotic stability of the closed-loop system.

[0194] Part 1: First, satisfy the inequality variables Furthermore, the saturation level should meet the following requirements. This makes it possible to obtain Furthermore, due to the specified performance function It is continuous, therefore it can be claimed that there exists a relationship with... Unrelated finite time constant , making ,for Furthermore, during the time interval At that point, one can follow the tracking error surface. (48) and controller (61) Calculate variables The time derivative is:

[0195] ,

[0196] Note that, by using saturation operations, we obtained the following: Auxiliary variable estimates , Representing a bounded set means that for Both are bounded. Furthermore, based on equation (62) and for... ,condition Its derivative with respect to time can be solved. And for We obtain the following inequality:

[0197] ,

[0198] in It is a positive number. Therefore, as long as The following inequality holds.

[0199] ,

[0200] It can be seen that equations (63) and (64) describe the origin of the set Any auxiliary variables The maximum growth rate. Therefore, based on and The continuity of this can lead to the derivation of a finite time constant. (Notice ,and (Irrelevant), making the condition and right Established.

[0201] Now, choose the Lyapunov function. ,in It is given in equation (39). Recalling the proof of lemma 5, we know that there exists a positive constant. , making For all This holds true. Based on Lemma 1, we obtain...

[0202] ,

[0203] The minimum gain parameter of the controller The minimum parameter that dominates convergence The other minimum parameter that governs convergence Therefore, by Lemma 2, we obtain that in finite time... Bounded and capable of reaching zero, convergence time Bounded to , For an explicit upper bound function of convergence time, the convergence time can be found. along with The value decreases as the value increases. Therefore, we can conclude that positive constants exist. And for all We have

[0204] ,

[0205] For simplicity, we use symbols to replace As can be seen from the above analysis, the existence time Used to describe The first time from Therefore, we can say that, regarding the situation of escaping, ,have ,for ,have .

[0206] Part Two: In this part, we will continue with the time intervals. The proof above. Due to the finite-time convergence of the observer equation (36), for We have , The adaptive algorithm equations (60) and (62) can be reformulated as follows:

[0207] ,

[0208] ,

[0209] in, They represent the estimated values ​​respectively. The true value of . Note that the detailed mathematical derivation of the above symbols can be found in equations (57)-(60).

[0210] We select Lyapunov functions as candidates.

[0211] ,

[0212] in, For system parameters, express The estimation error.

[0213] Combining equations (67) and (68), the time derivative of V can be calculated as follows:

[0214] ,

[0215] Using Lemma 3, equation (35), and Young's inequality, we can obtain the following inequality:

[0216] ,

[0217] Then, It can be rewritten along equation (71) as follows:

[0218] ,

[0219] in The smallest eigenvalue of a positive definite matrix is ​​. It is a positive constant.

[0220] Through Integrating both sides of equation (72) above, where You can get

[0221] ,

[0222] in It is a positive number. Furthermore, through... Definition and for , The facts, among which It is a finite time constant, from which we can derive for It is bounded. Then, It is also bounded.

[0223] Therefore, based on equation (73), we can obtain for It is bounded, which means that, , , for Both are bounded. According to Lemma 4, for It is also bounded. Furthermore, when When it approaches infinity, only if tending to We can start from the current hour and The boundedness derivation and Never from the set He escaped from the middle, so we can claim It is true. Furthermore, from... The definition can be derived when hour, As shown in Lemma 6, the position error can be guaranteed. and speed error The boundedness of tracking means that the tracking error will evolve strictly within a predetermined defined boundary.

[0224] Finally, we will demonstrate the tracking error. In It converges to zero as it approaches positive infinity. According to equation (73), we know that... right It is bounded. Furthermore, when At that time, and ,from In the definition, Guaranteed It is bounded. Therefore, using Barbalat's lemma, we can obtain... Then, we know the conditions. Only when It can only be maintained in time, therefore it can be deduced that if it can be guaranteed ,but It is true.

[0225] Therefore, we can assert the result. Yes, that's true, and that completes the proof.

[0226] Therefore, we can conclude that by adjusting the control gain, the asymptotically preset output feedback controller designed for a two-axis electromechanical coupling system can enable the tracking error of the two-axis electromechanical coupling system to asymptotically converge to 0 according to the specified constraints. A schematic diagram of the asymptotically preset output feedback controller for a two-axis electromechanical coupling system is shown below. Figure 1 As shown.

[0227] Example

[0228] To evaluate the performance of the designed controller, the physical parameters of the two-axis electromechanical coupling system in the simulation are shown in Table 1:

[0229] Table 1 System Physical Parameters

[0230] Given the desired instructions of the system rad, rad.

[0231] The following controller is used for comparison in the simulation:

[0232] Asymptotic preset output feedback controller for two-axis electromechanical coupling system: gain selection , , , , , Take adjustable parameters , , , , , , , , , , , , , , .

[0233] PID controller: The steps for selecting PID controller parameters are as follows: First, ignoring the nonlinear dynamics of the two-axis electromechanical coupling system, obtain a set of controller parameters through the PID parameter self-tuning function in Matlab. Then, after adding the nonlinear dynamics of the system, fine-tune the obtained self-tuning parameters to achieve the best tracking performance of the system.

[0234] The following parameters are considered in the two-axis electromechanical coupling system: load platform position tracking, load axis position tracking, APOFC azimuth tracking error, APOFC elevation tracking error, load platform comparison tracking error, load axis comparison tracking error, and a comparison of the azimuth performance indicators of the two controllers. Figure 4 , Figure 5 , Figure 6 , Figure 7 , Figure 8 , Figure 9 , Figure 10 and Figure 11 As shown. Figure 4 and Figure 5 It demonstrates well that stable tracking can be achieved under APOFC control. Figure 6 and Figure 7 It demonstrates that, with the help of PPF, the APOFC controller can ensure that the tracking error is within a predefined boundary, can converge the tracking error to zero in steady state, and the APOFC controller produces very little fluctuation in the steady state of the tracking error. Figure 8 and Figure 9 The comparison of tracking errors between the APOFC controller and the PID controller is shown for a load platform and load axis with a sinusoidal angle signal. Figure 10 and Figure 11Comparing the azimuth and elevation performance indicators of these two controllers, the APOFC controller outperforms the PID controller in terms of standard deviation, peak value, and mean value. The APOFC controller ensures the load axis stabilization system achieves optimal tracking performance. Even when performing servo tracking angle commands under high-maneuverability conditions, the load platform's stability accuracy remains at 0.1071 mrad, and the load axis's stability accuracy reaches 0.0105 mrad. It can be seen that the tracking error of the asymptotic preset output feedback controller for the two-axis electromechanical coupling system proposed in this invention is significantly smaller than that of the PID controller. Furthermore, the adaptive asymptotic preset performance tracking controller ensures that the error remains within the preset performance boundary, resulting in superior tracking performance.

Claims

1. A method for asymptotic preset output feedback control of a two-axis electromechanical coupling system, characterized in that, Includes the following steps: Step 1: Establish the mathematical model of the two-axis electromechanical coupling system, then proceed to Step 2; Step 2: Based on the mathematical model of the two-axis electromechanical coupling system, design an asymptotic preset output feedback controller based on a finite-time convergent high-gain observer, and proceed to Step 3; Step 3: Using Lyapunov stability theory, prove the stability of the asymptotically preset output feedback controller based on a finite-time convergent high-gain observer, and obtain the result that the system tracking error is limited to the specified constraints and is asymptotically stable.

2. The asymptotic preset output feedback control method for a two-axis electromechanical coupling system according to claim 1, characterized in that, In step 1, a mathematical model of the two-axis electromechanical coupling system is established, as follows: Step 1-1: For a typical two-axis electromechanical coupling system, build a mathematical model and a controller model for the two-axis electromechanical coupling system; Steps 1-2: To facilitate controller design, define state variables and convert the mathematical model of the two-axis electromechanical coupling system into state-space equations.

3. The asymptotic preset output feedback control method for a two-axis electromechanical coupling system according to claim 2, characterized in that, In step 1-1, for a typical two-axis electromechanical coupling system, an analytical dynamic model of the load platform and the load axis electromechanical integration of the two-axis electromechanical coupling system is built, resulting in the mathematical model of the two-axis electromechanical coupling system, as follows: To characterize the overall motion characteristics of the two-axis electromechanical coupling system, the system is equivalent to a system consisting of a carrier, a load platform, and a load axis. The mass of the load platform is denoted as . The mass of the load shaft is denoted as The rotation angle of the load platform relative to the carrier in the azimuth direction is denoted as... The rotation angle of the load axis relative to the load platform in the vertical direction is denoted as . The distance between the rotation center of the load platform and the rotation center of the load support structure is denoted as . The distance from the rotation center of the load shaft load support structure to its end is denoted as . ; When modeling, the load platform plane is selected as the zero potential energy surface. Therefore, the potential energy at the center of gravity of the load platform is set to zero, and the potential energy of the load axis relative to this plane is expressed by its mass and the height of its center of gravity. Based on this, the potential energies of the load platform and the load axis are written in their respective functional forms, denoted as: , Kinetic energy is expressed by the rotational kinetic energy of each rigid body and denoted as... , : ， ， ， ， In the formula, Let be the angular velocity of the load platform's rotation relative to the carrier's orientation. The angular velocity of the load shaft load support structure relative to the load platform in the vertical direction; It is the gravitational constant; According to the definition of the Lagrange system: ， Represents the Lagrangian of the system; Let the driving torques of the load platform and the load shaft be respectively , Then, from the Euler–Lagrange equation, we get: ， Among them, subscript , Indicates the load platform. Indicates the load axis; Will , , and Substituting into equation (6), the mechanical dynamics model of the two-axis electromechanical coupling system is: ， ， In the formula, Let be the angular acceleration of the load platform's rotation relative to the carrier's orientation. The angular acceleration of the load shaft load support structure relative to the load platform in the vertical direction; For the unmodeled disturbance terms of the load platform, This represents the unmodeled disturbance term for the load axis, which includes the effects of unmodeled friction, clearance, flexibility, launch impact, and road vibration uncertainties. Based on the mechanical dynamics model of the two-axis electromechanical coupling system, it is also necessary to model the motor system and couple it with the mechanical model to construct a complete mechatronic dynamics model. When modeling the motor, its simplified servo motor dynamics equations are described as follows: ， ， In the formula, This refers to the rotational inertia of the servo motor. For the rotation angle of the servo motor, For the electromagnetic torque of the servo motor, The viscous damping coefficient of the servo motor shaft. The input torque of the gear mechanism is... This is the ratio of the servo motor torque amplification factor to the circuit resistance. For the control input of the servo motor, This is the ratio of the electromotive force coefficient of the servo motor to the circuit resistance. The angular velocity representing the rotation angle of the servo motor. Angular acceleration, representing the rotation angle of a servo motor; Since the load platform requires a mechanical reduction transmission device composed of multi-stage gear mechanisms to reduce speed to meet the needs of the load platform and load shaft, a continuous backlash model is introduced to characterize the backlash effect in multi-stage gear transmissions. This model considers the input torque of the gear mechanism as... With output torque The relationship is represented as: ， In the formula, The transmission ratio of a multi-stage gear mechanism; This refers to the unmodeled error in the transmission process of a multi-stage gear mechanism. As can be seen from the gear reduction, ， Among them, order , Represents the zeroth derivative. Denotes the first derivative. Represents the second derivative; Substituting equations (10) and (11) into equation (9), we get: ， Therefore, the input torque of the load platform can be expressed as: ， Will Substituting into equation (7), the electromechanical integrated analytical dynamic model of the load platform of the two-axis electromechanical coupling system is obtained as follows: ， The load shaft uses an electric cylinder as the actuator; to describe the above process, starting from the servo motor shaft, its rotational balance equation and voltage balance equation are derived, resulting in a dynamic equation containing electromagnetic torque, damping, moment of inertia, control voltage, and back EMF terms. Its dynamic model is described as follows: ， In the formula, The electromagnetic torque of the motor. This indicates the output torque of the motor. It is the equivalent moment of inertia of the motor and its connected components. The rotation angle of the motor. The angular velocity is the rotation angle of the motor. The angular acceleration is the angle of rotation of the motor. It is the viscous damping coefficient at the motor shaft end. It is the ratio of the motor's torque coefficient to the total circuit resistance. This is the control input voltage for the servo motor. It is the torque output by the electric cylinder to the load. This refers to the equivalent rotational inertia of the deceleration and lead screw drive components. This is the viscous damping coefficient of the transmission link; Subsequently, the relationship between the motor output torque and the screw shaft output thrust is introduced, and expressed as follows: ， ， In the formula, The lead of a ball screw is a key geometric parameter that converts rotary motion into linear motion. The thrust output by the telescopic rod of the electric cylinder. It represents the overall efficiency of the electric cylinder transmission system and reflects the energy loss in actual transmission. This refers to the transmission ratio of the electric cylinder, describing the speed-torque conversion relationship from the motor to the lead screw. It is an electric cylinder; From a geometric constraint perspective, the upper and lower fulcrums of the electric cylinder, the trunnion, and the axis of the load support structure of the load shaft satisfy certain spatial geometric relationships; among which... This is the distance between the upper pivot point of the electric cylinder and the center of the trunnion within the load platform; This is the distance between the lower fulcrum of the electric cylinder and the center of the trunnion. This is the initial length of the electric cylinder; This is the current vertex. This is the initial angle; in Then the displacement of the electric cylinder telescopic rod Represented as: ， Substituting equation (19) into equation (18), we get ， From equation (17), we can obtain ， Displacement of the telescopic rod by geometric constraints Expressed as elevation angle The function is used to derive the thrust of the telescopic rod. Contribution to high and low directional torque; equivalent driving torque of the electric cylinder on the load shaft. Written as: ， In the formula, angle Let be the angle between the thrust of the electric cylinder and the orientation axis of the load shaft support structure. This angle is introduced due to the nonlinearity of the mechanism introduced during the conversion of the linear motion of the electric cylinder's telescopic rod into the rotational motion of the load shaft support structure. Based on the trigonometric relationships in [the equation / reference], we can obtain: ， Therefore, the input torque of the load shaft is obtained according to equations (17) and (21). for: ， In the formula, This indicates the modeling error of the load shaft actuator; Substituting into equation (8), the analytical dynamic model of the load axis electromechanical integration of the all-electric two-axis electromechanical coupling system is obtained as follows: 。 4. The asymptotic preset output feedback control method for a two-axis electromechanical coupling system according to claim 3, characterized in that, In steps 1-2, to facilitate controller design, state variables are defined, and the mathematical model of the two-axis electromechanical coupling system is transformed into state-space equations, as follows: The system dynamics model considers frictional nonlinearity and other uncertainties; to facilitate subsequent controller design, state variables are defined. State variables ; For transpose; The mathematical model of the entire two-axis electromechanical coupling system is represented by the following state-space equations: In the formula, the inertia matrix Control gain matrix Control Matrix Coriolis force Gravity term vector Friction term vector External disturbance vector ; The specific expressions for the elements in each matrix in equation (26) are as follows: ， ， ， ， ， ， in, The azimuth Coulomb friction coefficient is... For the azimuth friction shape parameters, The high and low Coulomb friction coefficients are, These are the high and low friction shape parameters. This is the azimuth linear damping coefficient. The linear damping coefficients are for both high and low directions. For the gravity eccentricity of the load platform, The eccentricity of the load platform due to gravity; The following section will design a nonlinear controller for equation (26). The control objective is to design a bounded control input. This enables the load shaft to stabilize the system output. Track the reference motion angle signal of the load axis stabilization system as accurately as possible. And ensure that the tracking error is as small as possible; To facilitate subsequent control design and closed-loop system stability analysis, the following lemma is introduced: Lemma 1: For real numbers and constant ,in For the summation index, there exists ; Lemma 2: Consider a given initial value positive definite function ,if The time derivative satisfies the following inequality: , where the coefficient ,coefficient real numbers The range is ,So It can reach zero, and The upper bound is calculated using the following formula: ; Lemma 3: Consider the function and variables We obtain the inequality: ; Lemma 4: Consider variables , where positive numbers ,if If it is bounded, then it is determined. and It is also bounded; furthermore, if Then the condition is met. and .

5. The asymptotic preset output feedback control method for a two-axis electromechanical coupling system according to claim 4, characterized in that, In step 1, for the convenience of controller design, the following assumptions are made: Assumption 1: Desired motion trajectory Second-order continuous differentiable bounded; Proceed to step 2.

6. The asymptotic preset output feedback control method for a two-axis electromechanical coupling system according to claim 5, characterized in that, Step 2: Based on the mathematical model of the two-axis electromechanical coupling system, design an asymptotic preset output feedback controller. The specific steps are as follows: Step 2-1: Design a finite-time convergent high-gain observer (HGO) for state estimation; Step 2-2: Design a preset performance function PPF to constrain the transient and steady-state performance of the tracking error; Steps 2-3: Design an asymptotic preset output feedback controller to ensure that the tracking error is within the predefined transient and steady-state convergence boundaries, and to achieve the asymptotic stability of the closed-loop system.

7. The asymptotic preset output feedback control method for a two-axis electromechanical coupling system according to claim 6, characterized in that, In step 2-1, a finite-time convergent high-gain observer (HGO) is designed for state estimation, as follows: The two-axis electromechanical coupling system follows the desired motion trajectory given by the servo stabilization system. In continuous bounded control input The following ensures that the motion platform meets firing requirements during movement; define the tracking error and its derivative with respect to time. for: ， Among them, reference signal and its derivative It is bounded; Based on equation (33), system (26) is formulated as follows: ， In the formula, To aggregate unknown dynamics, ； in Reference signal The second derivative is derived as follows: ， in , , For positive constants, the maximum value is... scalar function Note the joint angular acceleration. Typically, this is achieved by analyzing joint position signals. Differentiation reveals that this may amplify noise in the joint position signal. To overcome this problem, an improved high-gain observer (HGO) with finite-time convergence will be designed to reconstruct the system state. Although the HGO has been successfully applied in many industrial fields, due to its simple structure and direct parameter tuning process, the observer error can only reach an eventual bound. To solve this problem, the HGO is modified to estimate the velocity error. : ， In the formula, and These respectively represent displacement errors and speed error The estimated value; These are positive numbers chosen by the designer; axis indexes. ; The gain matrix in equation (36) and gain matrix Described as ， In the formula, the gain coefficient Defined as , It is a positive number; Lemma 5: Consider the designed observer (36), which claims to estimate in finite time They converge to their truth values, that is, for time... , and .

8. The asymptotic preset output feedback control method for a two-axis electromechanical coupling system according to claim 6, characterized in that, In step 2-2, a preset performance function PPF is designed to constrain the transient and steady-state performance of the tracking error, as detailed below: Let the tracking error surface be... Designed as: (48)， Among them, the constant term matrix ,constant ,constant , It is a two-dimensional real number space; In addition, a specified performance set is defined. To characterize the tracking error surface Boundary: (49)， Among the positive numbers Monotonically decreasing function PPF , The set of all 2×2 real matrices; (50)， in It is the initial value. It is the allowable final steady-state error boundary. ; positive numbers , control The rate of decrease; based on equations (49)-(50), it is found that the specified performance set The size is determined by the parameter , , and Decide; Lemma 6: Consider the tracking error surface in equation (48) PPF in formula (50) The following conditions must be met: 1) If the conditions are met and Then guarantee , It is bounded, and Belongs to set : (51)， Wherein the initial upper bound coefficient of position error , Represents the set of real numbers; 2) If ,but .

9. The asymptotic preset output feedback control method for a two-axis electromechanical coupling system according to claim 6, characterized in that, In steps 2-3, an asymptotically preset output feedback controller is designed to ensure that the tracking error is within the predefined transient and steady-state convergence boundaries, and to achieve the asymptotic stability of the closed-loop system, as detailed below: Following the above discussion, an asymptotically preset output feedback controller is designed. This controller not only ensures that the tracking error is within the predefined transient and steady-state convergence boundaries, but also achieves the asymptotic stability of the closed-loop system. Due to the tracking error surface in equation (48) Includes unmeasurable variables We first use the unmeasurable variables derived from equation (36) to estimate to replace This makes the available tracking error surface Defined as: (55)， in , It is a positive number. It is a saturation function, given as follows: (56)， Note that the introduced saturation function helps to correct the potential peaking phenomenon caused by HGO(36); in order to achieve the specified performance requirements described in the previous section, the transformation error estimate implemented in the output feedback controller is... Described as: (57)， Among them, the estimated value of auxiliary variables , ; Regarding transformation error It should be noted that most existing results directly use the transformation error to design the controller, resulting in rather conservative system stability results. To solve this problem and improve the overall tracking accuracy, the adaptive PI compensation term estimate is designed by combining the transformation error equation (57). for: (58)， Where the system estimation parameters , The error estimate is a constant gain. diagonal positive definite matrix Time-varying gain As shown below: (59)， in, (60)， in, It is an adaptive estimate of the unknown parameters of the system. for The derivative, constant term matrix System state estimation matrix ,function , and These are positive integers; note the function. Meet the conditions and , For positive constants, the function It is an integral bounded function. To further ensure the boundedness of the estimated parameters, an exponentially decreasing function is used, i.e. ,in , It is a positive number; Based on the observer (36), the transformation error estimate (57), and the compensation term estimate (58), the asymptotically specified output feedback controller can be designed as follows: (61)， Among them, the derivative matrix of the preset performance function The proposed controller (61) consists of a constant gain. Time-varying gain The adaptive PI compensation term estimate and feedback items Control; Estimation of adaptive PI compensation term This is used to compensate for unknown uncertainties and nonlinearities in the system, while achieving asymptotic output feedback control with specified performance; specifically, the constant gain involved... To maintain the boundedness of the transformation error, ensuring the specified performance of the tracking error; furthermore, the time-varying gain is driven by an adaptive law. With the estimated value and bounded functions of integration Adaptive adjustment not only handles unknown uncertainties, but also helps to ensure that the tracking error asymptotically approaches zero with the help of variables; Proceed to step 3.

10. The asymptotic preset output feedback control method for a two-axis electromechanical coupling system according to claim 9, characterized in that, Step 3 uses Lyapunov stability theory to prove the stability of the asymptotically preset output feedback controller based on a finite-time convergent high-gain observer. The results show that the system tracking error is limited to the specified constraints and is asymptotically stable, as detailed below: We will analyze the asymptotic stability of the closed-loop system with finite-time convergence HGO (36) and asymptotically preset output feedback controller (61); Theorem 1: Consider system (26), using the proposed observer form (36) and controller form (61), if the chosen parameters and Then there exists a positive constant. This makes it possible for all The closed-loop system is stable, all signals are bounded, and the tracking error is... Strictly in sets It evolves internally and converges asymptotically to zero; Proof: The proof consists of two parts. The first part proves that when the auxiliary variable... and its estimated value All in the collection The estimation error is bounded and converges to zero within the first part; the second part proves the asymptotic stability of the closed-loop system. Part 1: First, satisfy the inequality variables In addition, the saturation level should meet the following requirements. This makes it possible to obtain Furthermore, due to the specified performance function It is continuous, therefore it can be claimed that there exists a relationship with... Unrelated finite time constant , making ,for ; In addition, at the time interval At that point, one can follow the tracking error surface. (48) and controller (61) Calculate variables time derivative for: (62)， Choose the Lyapunov function , (39)， in For integration variables, The lower limit of integration, This is the maximum score. It can be proven that the estimation error is bounded and converges to zero; Part Two: In this part, we will continue with the time intervals. The proof above, in which This is the lower limit of the effective time. This is the upper limit of the effective time; due to the finite-time convergence of the observer equation (36), for We have , The adaptive algorithm is re-represented; The Lyapunov function candidates are: (69)， in, For system parameters, express The estimation error; The stability was proven using Lyapunov stability theory, and the system tracking error was found to be constrained under specified constraints and asymptotically stable.

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