Fuzzy adaptive backstepping control method for finite-time electromechanical servo system

Abstract

This invention relates to a fuzzy adaptive backstepping control method for electromechanical servo systems based on finite-time conditions. Based on the structure of the electromechanical servo system, a model is created and a problem description is provided. Combining the filtering backstepping method and fuzzy control, a fuzzy logic system is used to approximate the unknown nonlinear dynamics while simultaneously constructing an adaptive controller. Stability analysis based on finite-time theory proves that the designed controller can guarantee the convergence of the system's tracking error within a finite time. To address the unknown external disturbances and unmodeled dynamics in the system, the fuzzy logic system is used to approximate the unknown nonlinear dynamics while simultaneously constructing an adaptive controller, combining the filtering backstepping method and fuzzy control theory. Considering the computational explosion problem caused by differential calculations, a finite-time command filter is constructed to reduce the system's computational complexity. Furthermore, a filter error compensation mechanism is designed to compensate for the filter error term, ensuring the approximation capability of the filtered signal and improving the system's tracking control performance.

Description

Technical Field

[0001] This invention relates to a fuzzy adaptive backstepping control method for electromechanical servo systems based on finite time. Background Technology

[0002] Electromechanical servo systems are servo systems that use electric motors as the power drive element. They are widely used in industrial fields, and control accuracy is one of the important indicators in their system design. To ensure high-precision manufacturing processes, industry demands increasingly higher performance from electromechanical servo systems. However, due to the existence of unmodeled dynamics in actual systems and the unavoidable influence of unknown disturbances, servo performance is severely affected. Currently, to address these issues, nonlinear control methods are commonly used to achieve high-precision control of electromechanical servo systems.

[0003] To address external disturbances in a system, robust control is typically employed; however, the tracking accuracy of this method is often insufficient. Particularly concerning specific position signal noise disturbances, the system becomes exceptionally sensitive to noise when high-order derivatives exist in the control feedback signal, severely impacting control performance. To address internal disturbances, a disturbance observer is usually designed to estimate and compensate for disturbances in real time, achieving good tracking control performance.

[0004] In addition to the factors mentioned above, electromechanical servo systems are also affected by unmodeled dynamics such as fluctuations in unknown system parameters and nonlinear factors during operation. Adaptive methods are often used to estimate unknown parameters; however, these methods heavily rely on the system's dynamic model, and the parameter estimates are often suboptimal when external disturbances are present, sometimes even leading to system instability. Fuzzy adaptive methods can effectively approximate unmodeled dynamics, but no relevant research has been found to date.

[0005] The aforementioned control methods can guarantee the global asymptotic stability of the system, but the system cannot stabilize in a short time. In recent years, many finite-time methods have been proposed, which not only enable the system to converge quickly within a finite time interval, but also allow controllers designed based on finite-time methods to have higher tracking accuracy and anti-interference capabilities. Meanwhile, traditional backstepping control methods require multiple derivatives of the virtual signal when designing controllers, leading to the computational explosion problem.

[0006] Currently, in the angular displacement tracking control of inertial loads in electromechanical servo systems, there is no research combining fuzzy adaptive methods and finite-time command filtering methods to address unknown disturbances, unmodeled dynamics, and computational explosion problems in the system. Summary of the Invention

[0007] The purpose of this invention is to overcome the shortcomings of the existing technology and provide a fuzzy adaptive backstepping control method for electromechanical servo systems based on finite time.

[0008] The objective of this invention is achieved through the following technical solution:

[0009] A fuzzy adaptive backstepping control method for electromechanical servo systems based on finite-time conditions is characterized by the following steps:

[0010] Step 1: Based on the structure of the electromechanical servo system, model the electromechanical servo system and provide a problem description;

[0011] Step 2: Combining the filtering backstepping method and fuzzy control, a fuzzy logic system is used to approximate the unknown nonlinear dynamics while constructing an adaptive controller;

[0012] Step 3: Based on the finite-time theory, perform stability analysis to prove that the designed controller can guarantee that the tracking error of the system converges within a finite time.

[0013] Furthermore, the above-mentioned fuzzy adaptive backstepping control method for electromechanical servo systems based on finite time combines the filtering backstepping method and fuzzy control theory, and uses a fuzzy logic system to approximate the unknown nonlinear dynamics while constructing an adaptive controller.

[0014] Step 1: The electromechanical servo system directly drives the inertial load via a motor, resulting in the following dynamic model:

[0015]

[0016] In equation (1): x1 represents the angular displacement of the inertial load; x2 represents the angular velocity of the inertial load; τ represents the measurement noise; C t Φ(x1,x2) represents the control coefficients; Φ(x1,x2) represents the unmodeled dynamics; χ represents the unknown external disturbances; the controller design relies on the known control coefficients C. t And the measurable inertial load angular displacement x1 and angular velocity x2;

[0017] Assume the reference signal y of the electromechanical servo system is 1. d and Both exist and are bounded;

[0018] Assume that the system noise τ and the unknown external disturbance χ are both bounded;

[0019] Lemma 1: Given a set φ, and f(w) is a continuous function, there exists a fuzzy logic system that satisfies the following inequalities:

[0020]

[0021] Wherein, the weight vector λ=[w1,w2,...,w N ] T ∈R N, basic function vector Gaussian function W i (w) = exp[-(w - t i ) T (w - t i ) / a i 2 ; in the fuzzy logic system, t i = [t i,1 ,..., t i,n ) T is the center vector, and a i is the width;

[0022] Lemma 2 For any real number υ i , i = 1,..., n, and 0 < δ < 1, the following inequality holds:

[0023]

[0024] Lemma 3 For any real numbers τ, υ, and any real variables a, b, and c, the following inequality holds:

[0025]

[0026] Definition 1 Consider the following nonlinear system:

[0027]

[0028] When x ∈ R n represents a state variable, u ∈ R m represents the system input, and f: D → R n is continuous on an open neighborhood D near the origin; if for each x(t0) = x0, there exists υ > 0 and a time 0 < T(x0, υ) < ∞ such that ||x(t)|| < υ for t > t0 + T, the system is semi-globally finite-time stable;

[0029] Lemma 4 Consider a nonlinear system. Suppose there exists a C 1 function V(x) on D → R n near the origin, and scalars A > 0, γ ∈ (0, 1), 0 < B < ∞. If V(x) has V(0) = 0 and is positive on D and its derivative satisfies:

[0030]

[0031] The trajectories of the system are semi-globally finite-time stable, and V(x) satisfies:

[0032]

[0033] where 0 < ε < 1, and the reaching time Tr yes:

[0034]

[0035] Lemma 5 instruction filter form is as follows:

[0036]

[0037] Where △1 and △2 represent positive design parameters; α i and This indicates the input and output of the instruction filter.

[0038] Furthermore, in the aforementioned fuzzy adaptive backstepping control method for electromechanical servo systems based on finite time, step 2 involves constructing a controller using a finite-time command filtering backstepping method. The controller design process is as follows:

[0039] First, to facilitate formula derivation, some symbols in the model are replaced; let f1 = n, f2 = f(x1,x2) + d, and the following model representation is obtained:

[0040]

[0041] In controller design, state variables are transformed in the following way:

[0042]

[0043] Among them, κ i ν i y d These represent the tracking error, the command filter output, and the desired signal, respectively.

[0044] Since the use of command filters can introduce errors that affect the tracking of the desired signal, an error compensation signal is introduced, defined as follows:

[0045]

[0046] Where, ρ i (i = 2, ..., n-1) represents the error compensation signal, ρ i (0)=0,μ i >0;

[0047] The compensation tracking error takes the following form:

[0048]

[0049] The following finite-time control laws are constructed using the backstepping method:

[0050]

[0051] Among them, h i >0; Wi (i = 1, 2) represents the basis function vector; γ (0 < γ < 1) represents a positive constant; For the Θ estimate, Θ = max{||λ i || 2 ;i=1,2,...,n;

[0052] The backstepping method for constructing a controller involves decomposing the electromechanical servo system into second-order subsystems and designing the virtual control variables and Lyapunov functions for each system.

[0053] Design a first-order Lyapunov function and a virtual control variable:

[0054] Step 1: To Taking the derivative, we get:

[0055]

[0056] This paper addresses the computational explosion problem by employing a finite-time instruction filtering method. The control signal α is processed using a finite-time instruction filter. i Quickly obtain the control quantity y d The derivative value is obtained, which effectively reduces the computational complexity;

[0057] Define the Lyapunov function V1 as follows:

[0058]

[0059] Differentiating equation (16) and substituting equation (15) into it, we get:

[0060]

[0061] The unknown function f1 is determined by the fuzzy logic system. Approximation, according to Lemma 1, for any given ε>0, we have Furthermore, the approximation error δ1 satisfies ||δ1||≤ε1; according to Young's inequality and W1 T If W1≤1, then:

[0062]

[0063] Substituting equations (12), (14), and (18) into equation (17), we obtain the following form:

[0064]

[0065] This yields an inequality for the derivative of the first-order Lyapunov function, which will later be used to prove that the system is stable in finite time.

[0066] Design a second-order Lyapunov function and a virtual control variable:

[0067] Step 2: Take The derivative:

[0068]

[0069] Define the Lyapunov function V2 as follows:

[0070]

[0071] Differentiating equation (21), we get:

[0072]

[0073] The unknown function f2 is generated by the fuzzy logic system. Approximation, according to Lemma 1, for any given ε>0, we have Furthermore, the approximate error of δ2 satisfies ||δ2||≤ε2; according to Young's inequality and W2 T If W2≤1, then...

[0074]

[0075] Substituting equations (12), (14), and (23) into equation (22), we get:

[0076]

[0077] The inequality for the derivative of the second-order Lyapunov function is obtained and used to prove that the system is stable in finite time.

[0078] Define a Lyapunov function V and provide an adaptive rate for real-time online estimation of uncertain parameters in electromechanical servo systems;

[0079] Define the Lyapunov function V as:

[0080]

[0081] Where σ is a positive parameter, and

[0082] Differentiating both sides of the equation and combining with (24) and (25), we get:

[0083]

[0084] The adaptive law is constructed in the following form:

[0085]

[0086] Combining equations (26) and (27), we can obtain:

[0087]

[0088] Based on Lemma 2, we can obtain:

[0089]

[0090] Where Ξ = 2 γ min{μ m}, m=1,2.

[0091] According to the perfect square inequality, the following inequality holds:

[0092]

[0093] Substituting equations (29) and (30) into (28), we get:

[0094]

[0095] Where A = {minβ} γ ,Ξ};

[0096] The inequalities of the derivatives of Lyapunov functions of each order are obtained, and stability analysis is performed to prove that the designed controller can make the error of the inertial load angular displacement of the electromechanical servo system converge to the region near the far point in a finite time, and all variables of the system are semi-global finite time stable.

[0097] According to Lemma 3, let a = 1 - γ, b = γ, τ = 1, The following inequalities exist:

[0098]

[0099] Based on equations (31) and (32), we can obtain:

[0100]

[0101] in

[0102] Define arrival time T r for:

[0103]

[0104] Where V(x(0)) represents the initial value of V(x); according to Lemma 4, This indicates that all closed-loop variables are semi-global finite-time stable;

[0105] At the same time, according to the definition of V, for The following inequalities hold:

[0106]

[0107] This indicates that in a finite time T r Within this area, the tracking error enters a small region around the origin;

[0108] The steps and conclusions of the controller design described above are summarized as follows:

[0109] Theorem 1 considers an electromechanical servo system model under assumptions 1 and 2; if the finite-time filter is as shown in equation (9), the virtual control signal and control law are as shown in α1 in equation (14), and the error compensation mechanism is as shown in equation (12), then the control law can be selected as shown in u in equation (14) to make the tracking error κ i The system converges to a region near the far point within a finite time, and all variables of the system are semi-global finite-time stable.

[0110] Furthermore, the aforementioned fuzzy adaptive backstepping control method for electromechanical servo systems based on finite-time theory, through stability analysis, proves that the designed controller can guarantee that the system's tracking error converges within a finite time.

[0111] according to i = 1, 2, to consider κ i The boundedness of ρ needs to be considered. i Boundedness;

[0112] For the input signal α i ,satisfy:

[0113] ||C i (ν i+1 -α i )||≤η i ψ i ,i=1,...,n-1 (36)

[0114] Among them, C i η represents the control coefficient. i , ψ i Representing the upper limit of the instruction filter error and C respectively. i According to equation (12), the Lyapunov function is defined as follows:

[0115]

[0116] Using equation (12) and the perfect square formula, the derivative of Vρ is as follows:

[0117]

[0118] in

[0119] According to Lemma 3, let a = 1 - γ, b = γ, τ = 1, Then the following inequality holds:

[0120]

[0121] Combining equations (38) and (39), we can obtain:

[0122]

[0123] in,

[0124] Similar to (34),

[0125] It has been proven that ρ i (i=1,2) is semi-global finite-time stable; at the same time, according to equations (11) and (13), κ can be obtained. i It is bounded; by selecting appropriate controller design parameters, it can be ensured that all closed-loop variables in the electromechanical servo system are bounded, and the angular displacement of the system's inertial load reaches the desired value and stabilizes within a finite time.

[0126] Compared with the prior art, the present invention has significant advantages and beneficial effects, specifically reflected in the following aspects:

[0127] ① To address the unknown external disturbances and unmodeled dynamics in the system, a fuzzy logic system is used to approximate the unknown nonlinear dynamics while constructing an adaptive controller, combining the filtering backstepping method and fuzzy control theory. At the same time, considering the computational explosion problem caused by differential calculation, a finite-time instruction filter is constructed to reduce the computational complexity of the system. Furthermore, a filtering error compensation mechanism is designed to compensate for the filtering error term, ensuring the approximation capability of the filtered signal and improving the tracking control performance of the system.

[0128] ②Using a fuzzy logic system to approximate unmodeled dynamics and external disturbances gives the system higher robustness and anti-interference ability;

[0129] ③ Construct a finite-time instruction filter and design a filtering error compensation mechanism to compensate for the filtering error term, ensuring the approximation capability of the filtered signal and improving the transient and steady-state performance of the system;

[0130] ④ Through stability analysis, it was proven that the designed controller can ensure that the tracking error of the system converges within a finite time.

[0131] Other features and advantages of the invention will be set forth in the following description, and will be apparent in part from the description, or may be learned by practicing specific embodiments of the invention. The objects and other advantages of the invention may be realized and obtained by means of the structures particularly pointed out in the written description and the accompanying drawings. Attached Figure Description

[0132] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0133] Figure 1 This is a schematic diagram of the system structure of the present invention;

[0134] Figure 2 This is a design flowchart of the present invention;

[0135] Figure 3 This is a diagram illustrating the tracking effect of the controller in an embodiment of the present invention.

[0136] Figure 4 This is a comparison chart of the tracking errors of four controllers in this embodiment of the invention;

[0137] Figure 5 Adaptive rate in embodiments of the present invention

[0138] Figure 6 The error compensation signal ρ in this embodiment of the invention i ;

[0139] Figure 7 To compensate for the tracking error ω in the embodiments of the present invention i ;

[0140] Figure 8 The control input u is used in this embodiment of the invention. Detailed Implementation

[0141] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0142] It should be noted that similar reference numerals and letters in the following figures indicate similar items; therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures. Furthermore, in the description of this invention, directional and ordinal terms are used only to distinguish descriptions and should not be construed as indicating or implying relative importance.

[0143] like Figure 1 As shown, the electromechanical servo system includes a controller 101, a current loop PID controller 103, an amplification and processing circuit 104, and a DC motor 105. Based on the desired signal y from the controller 101... d The input signal u is obtained by the controller 101 and, together with the current loop PID controller 103 and the amplification and processing circuit 104, forms a driver 102 to drive the DC motor 105 to control the angular displacement of the inertial load 106.

[0144] This invention addresses the problem of angular displacement tracking control of inertial loads in electromechanical servo systems under unknown disturbances. It considers the impact of unmodeled dynamics on the system and provides a finite-time fuzzy adaptive command filtering backstepping control method, which effectively compensates for unknown disturbances and unmodeled dynamics in the system and has good angular displacement tracking control performance for inertial loads.

[0145] This invention relates to a fuzzy adaptive backstepping control method for electromechanical servo systems based on finite-time conditions, comprising the following steps:

[0146] Step 1: Based on the structure of the electromechanical servo system, model the electromechanical servo system and provide a problem description;

[0147] Step 2: Combining the filtering backstepping method and fuzzy control, a fuzzy logic system is used to approximate the unknown nonlinear dynamics while constructing an adaptive controller;

[0148] Step 3: Based on the finite-time theory, perform stability analysis to prove that the designed controller can guarantee that the tracking error of the system converges within a finite time.

[0149] Step 1: The electromechanical servo system directly drives the inertial load via a motor, resulting in the following dynamic model:

[0150]

[0151] In equation (1): x1 represents the angular displacement of the inertial load; x2 represents the angular velocity of the inertial load; τ represents the measurement noise; C t Φ(x1,x2) represents the control coefficients; Φ(x1,x2) represents the unmodeled dynamics; χ represents the unknown external disturbances; the controller design relies on the known control coefficients C. t And the measurable inertial load angular displacement x1 and angular velocity x2;

[0152] Assume the reference signal y of the electromechanical servo system is 1. d and Both exist and are bounded;

[0153] Assume that the system noise τ and the unknown external disturbance χ are both bounded;

[0154] Lemma 1: Given a set φ, and f(w) is a continuous function, there exists a fuzzy logic system that satisfies the following inequalities:

[0155]

[0156] Wherein, the weight vector λ=[w1,w2,...,w N ] T ∈R N Basic function vectors Gaussian function W i (w)=exp[-(wt i ) T (wt i ) / a i 2 In fuzzy logic systems, t i =[t i,1 ,...,t i,n ] T It is the center vector, a i It is the width;

[0157] Lemma 2 For any real number υ i For any i = 1, ..., n, and 0 < δ < 1, the following inequalities hold:

[0158]

[0159] Lemma 3 For any real numbers τ and υ, and any real variables a, b, and c, the following inequality holds:

[0160]

[0161] Definition 1 Consider the following nonlinear system:

[0162]

[0163] When \(x\in\mathbb{R}\) n represents a state variable, \(u\in\mathbb{R}\) m represents the system input, \(f:D\rightarrow\mathbb{R}\) n is continuous on an open neighborhood \(D\) near the origin; if for each \(x(t_0)=x_0\), there exists \(\upsilon>0\) and an interval \(0 < T(x_0,\upsilon)<\infty\) such that \(\|x(t)\|<\upsilon\) for \(t > t_0+T\), the system is semi - globally finite - time stable;

[0164] Lemma 4 Consider a nonlinear system. Suppose there exists a \(C\) 1 function \(V(x)\) on \(D\rightarrow\mathbb{R}\) n near the origin, and scalars \(A>0\), \(\gamma\in(0,1)\), \(0 < B<\infty\). If \(V(x)\) has \(V(0)=0\) is positive on \(D\) and its derivative satisfies:

[0165]

[0166] The trajectories of the system are semi - globally finite - time stable, and \(V(x)\) satisfies:

[0167]

[0168] where \(0 < \varepsilon<1\), and the reaching time \(T\) r is:

[0169]

[0170] Lemma 5 The command filter is of the following form:

[0171]

[0172] where \(\triangle_1\) and \(\triangle_2\) represent positive design parameters; \(\alpha\) i and represent the input and output of the command filter.

[0173] Step 2, construct a controller through finite - time command - filtering backstepping. The controller design process:

[0174] First, for the convenience of formula derivation, replace some symbols in the model; let \(f_1 = n\), \(f_2 = f(x_1,x_2)+d\), and obtain the following model representation:

[0175]

[0176] The coordinate transformation of the state variables is converted into the following form:

[0177]

[0178] Among them, κ i ν i y d These represent the tracking error, the command filter output, and the desired signal, respectively. The use of the command filter introduces errors, which affect the tracking performance of the desired signal. Therefore, an error compensation signal is introduced, defined as follows:

[0179]

[0180] Where, ρ i (i = 2, ..., n-1) represents the error compensation signal, ρ i (0)=0,μ i >0;

[0181] The compensation tracking error takes the following form:

[0182]

[0183] The following finite-time control laws are constructed using the backstepping method:

[0184]

[0185] Among them, h i >0;W i (i = 1, 2) represents the basis function vector; γ (0 < γ < 1) represents a positive constant; For the Θ estimate, Θ = max{||λ i || 2 ;i=1,2,...,n;

[0186] The above-mentioned backstepping method for constructing a controller involves decomposing the electromechanical servo system into second-order subsystems, then designing the virtual control variables and Lyapunov functions for each system, thereby completing the controller design and ultimately achieving tracking control of the system to meet the desired performance requirements.

[0187] The following section describes the design of the first-order Lyapunov function and the virtual control variable:

[0188] Step 1: To Taking the derivative, we get:

[0189]

[0190] To address the computational explosion problem, a finite-time instruction filtering method is employed. This method involves introducing a finite-time instruction filter to process the control signal α. i Quickly obtain the control quantity y d The derivative value is obtained, which effectively reduces the computational complexity;

[0191] Define the Lyapunov function V1 as follows:

[0192]

[0193] Differentiating equation (16) and substituting equation (15) into it, we get:

[0194]

[0195] The unknown function f1 is determined by the fuzzy logic system. Approximation, according to Lemma 1, for any given ε>0, we have Furthermore, the approximation error δ1 satisfies ||δ1||≤ε1; according to Young's inequality and W1 T If W1≤1, then:

[0196]

[0197] Substituting equations (12), (14), and (18) into equation (17), we obtain the following form:

[0198]

[0199] The design of the second-order Lyapunov function and virtual control variable is as follows:

[0200] Step 2: Take The derivative:

[0201]

[0202] Define the Lyapunov function V2 as follows:

[0203]

[0204] Differentiating equation (21), we get:

[0205]

[0206] The unknown function f2 is generated by the fuzzy logic system. Approximation, according to Lemma 1, for any given ε>0, we have Furthermore, the approximate error of δ2 satisfies ||δ2||≤ε2; according to Young's inequality and W2 T If W2≤1, then...

[0207]

[0208] Substituting equations (12), (14), and (23) into equation (22), we get:

[0209]

[0210] An inequality for the derivative of the second-order Lyapunov function was obtained, which was used to prove that the system is stable in finite time.

[0211] The Lyapunov function V will be defined below, and the adaptive rate will be given to perform real-time online estimation of uncertain parameters in electromechanical servo systems.

[0212] Define the Lyapunov function V as:

[0213]

[0214] Where σ is a positive parameter, and

[0215] Differentiating both sides of the equation and combining with (24) and (25), we get:

[0216]

[0217] The adaptive law is constructed in the following form:

[0218]

[0219] Combining equations (26) and (27), we can obtain:

[0220]

[0221] Based on Lemma 2, we can obtain:

[0222]

[0223] Where Ξ = 2 γ min{μ m}, m=1,2.

[0224] According to the perfect square inequality, the following inequality holds:

[0225]

[0226] Substituting equations (29) and (30) into (28), we get:

[0227]

[0228] Where A = {minβ} γ ,Ξ};

[0229] Through the above steps, the inequalities of the derivatives of Lyapunov functions of each order are obtained. Stability analysis will be performed to prove that the designed controller can make the error of the inertial load angular displacement of the electromechanical servo system converge to the region near the far point in a finite time, and that all variables of the system are semi-global finite time stable.

[0230] According to Lemma 3, let a = 1 - γ, b = γ, τ = 1, The following inequalities exist:

[0231]

[0232] Based on equations (31) and (32), we can obtain:

[0233]

[0234] in

[0235] Define arrival time T r for:

[0236]

[0237] Where V(x(0)) represents the initial value of V(x); according to Lemma 4, This indicates that all closed-loop variables are semi-global finite-time stable;

[0238] At the same time, according to the definition of V, for The following inequalities hold:

[0239]

[0240] This indicates that in a finite time T r Within this area, the tracking error enters a small region around the origin;

[0241] The basic conclusions of the above controller design steps are summarized as follows:

[0242] Theorem 1 considers an electromechanical servo system model under assumptions 1 and 2; if the finite-time filter is as shown in equation (9), the virtual control signal and control law are as shown in α1 in equation (14), and the error compensation mechanism is as shown in equation (12), then the control law can be selected as shown in u in equation (14) to make the tracking error κ i The system converges to a region near the far point within a finite time, and all variables of the system are semi-global finite-time stable.

[0243] Stability analysis based on finite-time theory proves that the designed controller can guarantee that the tracking error of the system converges in finite time:

[0244] according to i = 1, 2, to consider κ i The boundedness of ρ needs to be considered. i Boundedness;

[0245] For the input signal α i ,satisfy:

[0246] ||C i (ν i+1 -α i )||≤η i ψ i ,i=1,...,n-1 (36)

[0247] Among them, C i η represents the control coefficient. i , ψ i Representing the upper limit of the instruction filter error and C respectively. i According to equation (12), the Lyapunov function is defined as follows:

[0248]

[0249] Using equation (12) and the perfect square formula, V ρ The derivative is as follows:

[0250]

[0251] in

[0252] According to Lemma 3, let a = 1 - γ, b = γ, τ = 1, Then the following inequality holds:

[0253]

[0254] Combining equations (38) and (39), we can obtain:

[0255]

[0256] in,

[0257] Similar to (34),

[0258] It has been proven that ρ i (i=1,2) is semi-global finite-time stable; at the same time, according to equations (11) and (13), κ can be obtained. i It is bounded; by selecting appropriate controller design parameters, it can be ensured that all closed-loop variables in the electromechanical servo system are bounded, and the angular displacement of the system's inertial load reaches the desired value and stabilizes within a finite time.

[0259] Simulations were performed in Matlab / Simulink, and the iterative operation flow of the controller was as follows: Figure 2 As shown. To verify the effectiveness of the fuzzy adaptive command filtering backstepping control method for finite-time electromechanical servo systems, comparative experiments were conducted with various controllers.

[0260] The parameters for the electromechanical servo system are selected as follows: C t =0.8 N·m / V, the model uncertainty of the system is Φ(x1,x2)=x1+x2, the uncertainty of the system such as unknown disturbances is χ=sint, and the measurement noise amplitude of the position signal is 2.5×10 -6 dB, with a sampling period of 0.2ms.

[0261] The parameters for the finite-time fuzzy adaptive filtering backstepping control method (FA-FT-CFB) for electromechanical servo systems are selected as follows: h1 = 10, h2 = 10, μ1 = 30, μ2 = 30, γ = 0.65, β = 1, σ = 0.1.

[0262] Compare the results with the following three controllers:

[0263] (1) Backstepping controller (IT-TB), whose control law is as follows:

[0264]

[0265] (2) Robust backstepping controller (IT-RB), whose control law is as follows:

[0266]

[0267] (3) Finite Instruction Filtered Backstepping Controller (FT-CFB), whose control law is as follows:

[0268]

[0269] The system's input position signal is: y d = 2sin(πt)rad, the maximum angular velocity is 2πrad / s; Figure 3 The controller tracking results show that the actual position accurately tracks the set desired motion trajectory.

[0270] Figure 4 The tracking error of the four controllers is compared. As can be seen from the figure, the four controllers perform under the same control gain (k1=10, k2=30).

[0271] Traditional backstepping controllers failed to converge. The IT-RB, without an error compensation mechanism, had a tracking error range of 0–0.4152 rad. Although this method achieved convergence, its handling of model uncertainties and external disturbances was poor, potentially failing to meet control accuracy requirements in practical applications. In contrast, the FT-CFB and FA-FT-CFB had tracking errors of 0–0.0871 rad and 0–0.0714 rad, respectively, exhibiting smaller tracking errors and better tracking performance.

[0272] To compare the transient response performance of the three control methods—FA-FT-CFB, IT-RB, and FT-CFB—the IAE criterion is used to evaluate the controllers. The objective function is: Where |e(t)| is the tracking error, i.e., |x1(t)-y d (t)|, where time t is selected between 0s and 1s, and the sampling period is 2ms. The statistical results are shown in Table 1 below:

[0273] Table 1 Comparison of the three controllers

[0274]

[0275] Because the control effects of FA-FT-CFB and FT-CFB are not easily obtained from Figure 4 This can be seen from the text. Therefore, Figure 4 The partial plots in the image separately show the tracking control errors of the two methods between 0s and 1s.

[0276] Among them, FA-FT-CFB reaches a steady state around 0.28s, while FT-CFB reaches a steady state around 0.33s. As can be seen from the figure, compared with FT-CFB, FA-FT-CFB can better approximate unmodeled dynamics and external disturbances. Therefore, FA-FT-CFB has a faster transient response and stronger anti-interference ability.

[0277] Based on the performance of the four controllers mentioned above, it can be concluded that FA-FT-CFB has fast transient response and high control accuracy at steady-state levels.

[0278] Figure 5 For adaptive rate We can see that the adaptive rate gradually converges and stabilizes at 0 to 2 × 10⁻⁶ after 3 seconds. 5 between; Figure 6 The system's error compensation signal ρ i The error compensation signal ρ can be seen. i It converges to the region near 0 after 0.4s; Figure 7 To compensate for tracking errors We can see the compensation tracking error The curve converges and is relatively smooth, with no obvious jitter; from Figure 6 , Figure 7 The designed error compensation mechanism can effectively compensate for filtering errors and improve the approximation capability of the filtered signal. Figure 8 As can be seen from the control input u, the controller input curve is relatively smooth with no obvious jitter and converges between 0V and 26V.

[0279] The simulation results of the above implementation cases show that the fuzzy adaptive backstepping control method for electromechanical servo systems based on finite time can effectively compensate for unmodeled dynamics and external disturbances, and reduce the influence of measurement noise on the system position signal. At the same time, the effectiveness and excellence of the control method are verified by comparing it with three controllers through simulation experiments.

[0280] This invention employs a fuzzy logic system to approximate unmodeled dynamics and external disturbances, giving the system high robustness and anti-interference capability. A finite-time instruction filter is constructed, and a filter error compensation mechanism is designed to compensate for the filter error term, ensuring the approximation capability of the filtered signal and improving the transient and steady-state performance of the system.

[0281] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the invention should be included within the scope of protection of the invention. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0282] The above are merely specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A fuzzy adaptive backstepping control method for electromechanical servo systems based on finite-time conditions, characterized in that: Includes the following steps: Step 1: Based on the structure of the electromechanical servo system, model the electromechanical servo system and provide a problem description; The electromechanical servo system directly drives the inertial load by a motor, and its dynamic model is as follows: (1) In formula (1): Angular displacement representing inertial load; Angular velocity representing inertial load; Represents measurement noise; Represents the control coefficient; Represents unmodeled dynamics; Represents unknown external disturbances; controller design relies on known control coefficients. and measurable inertial load angular displacement angular velocity ; Assumption 1: Reference signal of the electromechanical servo system and Both exist and are bounded; Assumption 2 System noise and unknown external interference Both are bounded; Lemma 1 Given a set , If is a continuous function, then there exists a fuzzy logic system that satisfies the following inequalities: (2) Among them, the weight vector Basic function vectors Gaussian function In fuzzy logic systems, It is the center vector. It is the width; Lemma 2 For any real number , ,and The following inequalities hold: (3) Lemma 3 For any real number , and any real variable , and The following inequalities hold: (4) Definition 1 Consider the following nonlinear system: (5) when Represents a state variable. Indicates system input, It is continuous in an open neighborhood D near the origin; if for each ,exist and ,but right The system is found to be semi-globally finite-time stable. Lemma 4 Consider a nonlinear system, and assume there exists a function exist The value is near the origin, and the scalar... , , ,if have It is positive on D and its derivative satisfies: (6) The system trajectory is semi-global finite-time stable, and satisfy: (7) in and arrival time yes: (8) Lemma 5 The instruction filter takes the following form: (9) in, and Indicates positive design parameters; and Indicates the input and output of the command filter; Step 2: Combining the filtering backstepping method and fuzzy control, a fuzzy logic system is used to approximate the unknown nonlinear dynamics while constructing an adaptive controller; Controller design process: First, to facilitate formula derivation, some symbols in the model are replaced; let , The model can be represented as follows: (10) The coordinates of the state variables are transformed into the following form: (11) in, , , These represent the tracking error, command filter output, and desired signal, respectively. Since the use of the command filter can introduce errors, an error compensation signal is introduced, defined as follows: (12) in, This represents the error compensation signal. , , ; The compensation tracking error takes the following form: (13) The following finite-time control laws are constructed using the backstepping method: (14) in, ; Represents a basis function vector. ; To represent a positive integer, ; for estimate, ; The backstepping method for constructing a controller involves decomposing the electromechanical servo system into second-order subsystems and designing the virtual control variables and Lyapunov functions for each system. Design a first-order Lyapunov function and a virtual control variable: Step 1: To Taking the derivative, we get: (15) To address the computational explosion problem, a finite-time instruction filtering method is employed, which involves introducing a finite-time instruction filter to process the control signal. To obtain the control quantity The derivative value; Define Lyapunov functions for: (16) Differentiating equation (16) and substituting equation (15) into it, we get: (17) Unknown function Fuzzy logic system Approximation, according to Lemma 1, for any given ,have And approximation error satisfy According to Young's inequality and ,have: (18) Substituting equations (12), (14), and (18) into equation (17), we obtain the following form: (19) Design a second-order Lyapunov function and a virtual control variable: Step 2: Take The derivative: (20) Define Lyapunov functions for: (21) Differentiating equation (21), we get: (22) Unknown function Fuzzy logic system Approximation, according to Lemma 1, for any given ,have ,and Approximation error satisfies According to Young's inequality and ,have (23) Substituting equations (12), (14), and (23) into equation (22), we get: (24) The inequality for the derivative of the second-order Lyapunov function is obtained and used to prove that the system is stable in finite time. Define Lyapunov functions It also provides an adaptive rate for real-time online estimation of uncertain parameters in electromechanical servo systems; Define Lyapunov functions for: (25) in, It is a positive parameter, and ; Differentiating both sides of the equation and combining with (24) and (25), we get: (26) The adaptive law is constructed in the following form: (27) Combining equations (26) and (27), we can obtain: (28) Based on Lemma 2, we can obtain: (29) in , According to the perfect square inequality, the following inequality holds: (30) Substituting equations (29) and (30) into (28), we get: (31) in ; The inequalities of the derivatives of Lyapunov functions of each order are obtained, and stability analysis is performed to prove that the designed controller can make the error of the inertial load angular displacement of the electromechanical servo system converge to the region near the far point in a finite time, and all variables of the system are semi-global finite time stable. According to Lemma 3, let , , , , The following inequalities exist: (32) Based on equations (31) and (32), we can obtain: (33) in Define arrival time for: (34) in, express The initial value; according to Lemma 4, This indicates that all closed-loop variables are semi-global finite-time stable. At the same time, according to The definition, for The following inequalities hold: (35) This indicates that within a finite time Within this area, the tracking error enters a small region around the origin; Step 3: Perform stability analysis based on finite-time theory to prove that the designed controller can guarantee that the tracking error of the system converges within a finite time.

2. The fuzzy adaptive backstepping control method for electromechanical servo systems based on finite time as described in claim 1, characterized in that: The controller design steps can be summarized as follows: Theorem 1 Consider the electromechanical servo system model under assumptions 1 and 2; if the finite-time filter is as shown in equation (9), and the virtual control signal and control law are as shown in equation (14). As shown, the error compensation mechanism is as shown in equation (12), and the control law that can be selected is as shown in equation (14). As shown, the tracking error is reduced. The system converges to a region near the far point within a finite time, and all variables of the system are semi-global finite-time stable.

3. The fuzzy adaptive backstepping control method for electromechanical servo systems based on finite time as described in claim 1, characterized in that: Stability analysis based on finite-time theory proves that the designed controller can guarantee that the tracking error of the system converges in finite time: according to , For consideration The boundedness of needs to be considered. Boundedness; For the input signal ,satisfy: (36) in, This represents the control coefficient. , These represent the upper limit of the instruction filter error and, respectively. According to equation (12), the Lyapunov function is defined as follows: (37) Using equation (12) and the perfect square formula, The derivative is as follows: (38) in , , ; According to Lemma 3, let , , , , Then the following inequality holds: (39) Combining equations (38) and (39), we can obtain: (40) in, ; Similar to (34), ; prove It is semi-global finite-time stable; and according to equations (11) and (13), we can obtain that... It is bounded; by selecting appropriate controller design parameters, it can be ensured that all closed-loop variables in the electromechanical servo system are bounded, and the angular displacement of the system's inertial load reaches the desired value and stabilizes within a finite time.

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