Variable-rate collaborative reaching law nonsingular fast integral terminal sliding mode control method

Abstract

The invention provides a variable-rate collaborative reaching law nonsingular fast integral terminal sliding mode control method. The method comprises the following steps: designing a disturbance observer, a nominal controller and a compensation controller; the interference observer constructs an observer through the intermediate state vector and the adjustable gain matrix in combination with a smooth gain function to obtain a lumped interference observation value; an improved command filter technology is introduced into the nominal controller, and signal smoothing processing is achieved by dynamically adjusting a weight factor; then, a filtering error compensation mechanism is designed to counteract a filtering error; in the compensation controller, a non-singular fast integral terminal sliding mode surface is constructed through a sliding mode surface function, a fusion tracking error, an error integral term and a smoothing function tanh; designing adaptive estimation; based on adaptive estimation, an adaptive variable rate cooperative reaching law is updated, so that the observation error of the observer is disturbed, and the approximation error of the saturation model is compensated. According to the method, the dynamic response speed and the steady-state precision of trajectory tracking control can be remarkably improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of trajectory tracking control, and specifically provides a variable rate cooperative approaching law nonsingular fast integral terminal sliding mode control method. BACKGROUND

[0002] In the trajectory tracking control of PMSM servo system, convergence and robustness are important performance indicators that must be considered. The sliding mode control method has significant advantages, such as robustness to external disturbances and parameter uncertainties, fast dynamic response, and simple algorithm implementation. However, the discontinuous switching term in its structure, although it can improve the robustness of the system, also causes chattering to the system. Therefore, some scholars use improved algorithms based on SMC to reduce system chattering while maintaining the robustness of traditional SMC methods, such as integral SMC (ISMC), terminal SMC (TSMC), and nonsingular fast SMC (NSFTSMC). The idea of ISMC is to add the integral term of the state variable to the linear sliding surface to reduce chattering and steady-state error. The TSMC method introduces a nonlinear function into the sliding surface to ensure that the tracking error converges to zero in a finite time. Although the nonlinear sliding surface has certain advantages, there are still some challenges in its application, for example, most algorithms can only achieve a second-order sliding mode, and there is a singularity problem. The NSFTSMC method is used to solve the singularity problem and achieve fast dynamic response. Non-smooth switching functions usually require a large switching gain to ensure the robustness of the system, resulting in significant chattering of the system. In addition, some SMC methods based on disturbance compensation reduce chattering by selecting a small switching gain value, sacrificing the dynamic response performance.

[0003] Therefore, there is an urgent need for a variable rate cooperative approaching law nonsingular fast integral terminal sliding mode control method to solve the above problems. SUMMARY

[0004] In order to overcome the above defects, the present application is proposed to provide a solution or partial solution to the above problems.

[0005] The application provides a variable rate cooperative approaching law non-singular fast integral terminal sliding mode control method, which comprises the following steps: designing an interference observer, a nominal controller and a compensation controller; the interference observer is used for estimating the lumped interference in the system in real time, an observer is constructed by combining a smooth gain function, an intermediate state vector and an adjustable gain matrix, and an observed value of the lumped interference is obtained; the nominal controller introduces an improved command filter technology, the improved command filter technology realizes signal smoothing processing by dynamically adjusting a weight factor; then a filtering error compensation mechanism is designed to offset the filtering error; in the compensation controller, a non-singular fast integral terminal sliding surface is constructed by combining a tracking error, an error integral term and a smooth function tanh through a sliding surface function; an adaptive estimation is designed; based on the adaptive estimation, an adaptive variable rate cooperative approaching law is updated, and then the observation error of the interference observer and the saturation model approximation error are compensated.

[0006] In one of the technical solutions of the variable rate cooperative approaching law non-singular fast integral terminal sliding mode control method, the interference observer is expressed as:

[0007]

[0008] In the formula, ψ i is an intermediate state vector of the lumped interference observer, is an observation vector of the lumped interference D i , η i =diag{η i1 , …, η im} is an adjustable gain matrix designed, η ij > 0, Q i (x i ) = η i x i is a smooth gain function, u is an actual control input signal, i = 1, …, n, j = 1, …, m, f1(x) is a known nonlinear function, x n is a system state vector.

[0009] In one of the technical solutions of the variable rate cooperative approaching law non-singular fast integral terminal sliding mode control method, the improved command filter is:

[0010]

[0011] In the formula, α1 is an input of the filter, is an output of the filter, χ α is a weight factor, and χ α > 0, initial conditions satisfy θ1(t) = α1(0) and θ2(0) = 0, is an adjustable gain matrix.

[0012] In one of the technical solutions of the variable rate cooperative approaching law non-singular fast integral terminal sliding mode control method, the filter error compensation mechanism is:

[0013]

[0014] In the formula, is the auxiliary vector of the designed filter error compensation mechanism, c1=diag{c 11 , …, c 1m}, c2=diag{c 21 , c 2m}, c 1j and c 2j are design parameters, and c 1j >0, c 2j >0, the initial condition satisfies δ p (0)=0, p=1, 2.

[0015] In one of the technical solutions of the variable rate cooperative approaching law non-singular fast integral terminal sliding mode control method, the constructed non-singular fast integral terminal sliding mode surface is:

[0016]

[0017] In the formula, λ1=diag{λ 11 , …, λ 1m}, λ2=diag{λ 21 , …, λ 2m}, λ3=diag{λ 31 , …, λ 3m}, 0<γ1<1, γ2 is a design adjustable parameter, τ is a differential operator, f n (x) is a nonlinear function, λ 2m , λ 3m , λ 0n-2 , λ 01 are elements in a diagonal matrix, is the n-1th order derivative, h v v0 is the nominal control input after input saturation processing, is the disturbance observation value, is the derivative of the disturbance observation value, is a gain vector.

[0018] In one of the technical solutions of the variable rate cooperative approaching law non-singular fast integral terminal sliding mode control method, the adaptive variable rate cooperative approaching law is designed as:

[0019]

[0020] In one of the technical solutions of the variable rate cooperative reaching law non-singular fast integral terminal sliding mode control method, based on adaptive estimation, the adaptive variable rate cooperative reaching law is updated, and the process of compensating the observation error of the disturbance observer, the saturation model approximation error includes:

[0021] The first order derivative of the non-singular fast integral terminal sliding surface time is calculated;

[0022] The compensated tracking error signal is defined as Ξ1=z1-δ1, and the first order derivative of time is calculated as:

[0023]

[0024] In one of the technical solutions of the variable rate cooperative reaching law non-singular fast integral terminal sliding mode control method, the first order derivative of the non-singular fast integral terminal sliding surface time is:

[0025]

[0026] Where λ1=diag{λ 11 , …, λ 1m}, is an unknown constant, u is the actual control input signal, is the mth element of the error vector, χ e is a weight factor, S(v) is a designed smooth input saturation model approximation error, is the disturbance estimation error;

[0027] The Lyapunov function V1 is selected:

[0028] Based on the expression of the error signal and the improved command filter, the first order derivative of V1 with respect to time is arranged as:

[0029]

[0030] The compensated tracking error signal is defined as Ξ2=z2-δ2, and the first order derivative of time is calculated as:

[0031]

[0032] The Lyapunov function V2 is constructed as follows:

[0033] The first order derivative of V2 with respect to time is:

[0034]

[0035] The method further comprises designing the nominal controller input as:

[0036]

[0037] where k2 = diag{k 21 , …, k 2m}, k2x is a design parameter, and k 2j > 0.

[0038] Substitute v0 into , and obtain:

[0039] where X δ is a weight factor.

[0040] In one of the technical solutions of the variable rate cooperative approaching law non-singular fast integral terminal sliding mode control method, based on adaptive estimation, the adaptive variable rate cooperative approaching law is updated, and the process of compensating for the observation error of the disturbance observer and the saturation model approximation error includes:

[0041] The expression of introducing adaptive estimation is:

[0042]

[0043] where β0, β1, …, β n > 1 are gain parameters, 1≤j≤n, is the derivative of the n-1 order state vector of the system;

[0044] The method further includes: the compensation controller input is designed as:

[0045]

[0046] where l1 = diag{l 11 , …, l 1m}, l2 = diag{l 21 , …, l 2m}, l3 = diag{l 31 , …, l 3m}, and l 1j , l 2j , and l 3j are adjustment parameters, and 1≤μ1<2, 0<μ2<1, l0 is a smooth cooperative function, χ σ is a weight factor, χ σ > 0, and I is a unit matrix.

[0047] The beneficial effects of the variable-rate cooperative reaching law non-singular fast integral terminal sliding mode control method provided by this invention are as follows: This method can significantly improve the dynamic response speed and steady-state accuracy of trajectory tracking control. Specifically, by adopting an improved command filtering technique in the nominal controller, the problem of virtual signal differential explosion in the backstepping method is effectively solved. Furthermore, the introduction of an exponential power function and the use of a continuous function to replace the discontinuous function further improves the trajectory tracking control accuracy and smoothness. In addition, the adaptive variable-rate cooperative reaching law designed with a compensation controller can achieve rapid convergence in different sliding stages of the system, quickly converge the system state to the sliding mode surface in the global space, reduce chattering, and dynamically compensate for lumped disturbance observation errors and saturated model approximation errors. Based on the Lyapunov criterion analysis, the stability of the entire system can be guaranteed. The effectiveness of this control strategy has been fully demonstrated by the experimental platform of the PMSM servo system, which is widely used in complex engineering. Attached Figure Description

[0048] The disclosure of this invention will become more readily understood with reference to the accompanying drawings. It will be readily understood by those skilled in the art that these drawings are for illustrative purposes only and are not intended to limit the scope of protection of this invention. Furthermore, similar numbers in the drawings are used to denote similar components, wherein:

[0049] Figure 1 This is a closed-loop control circuit diagram of an uncertain nonlinear system according to an embodiment of the present invention;

[0050] Figure 2 The graphs of different functions in Table 1.1 are based on an embodiment of the present invention.

[0051] Figure 3 According to an embodiment of the present invention, different weight values ​​χ are selected. e The smoothing function tanh(e 1j / χ e (Line graph)

[0052] Figure 4 The curves l of different functions in Table 1.2 according to an embodiment of the present invention 0j picture;

[0053] Figure 5 The curves (1-1) of different functions in Table 1.2 according to an embodiment of the present invention are 0j )picture;

[0054] Figure 6 It is a smoothing function l according to an embodiment of the present invention. 0j Selecting different weight values ​​χ σ A curve graph;

[0055] Figure 7is a smoothing function (1-l 0j ) according to an embodiment of the present application; σ is a graph of selecting different weight values χ

[0056] Figure 8 is a LINKS-RT based PMSM servo system structure diagram according to an embodiment of the present application;

[0057] Figure 9 is a NFITSMC strategy closed loop control diagram of PMSM servo system using adaptive variable rate cooperative approach law according to an embodiment of the present application;

[0058] Figure 10 is a speed diagram of PMSM servo system at 200 rpm according to an embodiment of the present application;

[0059] Figure 11 is a current diagram of PMSM servo system at 200 rpm according to an embodiment of the present application;

[0060] Figure 12 is a voltage diagram of PMSM servo system at 200 rpm according to an embodiment of the present application;

[0061] Figure 13 is a speed diagram of PMSM servo system at 600 rpm according to an embodiment of the present application;

[0062] Figure 14 is a current diagram of PMSM servo system at 600 rpm according to an embodiment of the present application;

[0063] Figure 15 is a voltage diagram of PMSM servo system at 600 rpm according to an embodiment of the present application;

[0064] Figure 16 is a speed diagram of PMSM servo system at 1000 rpm according to an embodiment of the present application;

[0065] Figure 17 is a current diagram of PMSM servo system at 1000 rpm according to an embodiment of the present application;

[0066] Figure 18 is a voltage diagram of PMSM servo system at 1000 rpm according to an embodiment of the present application. DETAILED DESCRIPTION

[0067] Some embodiments of the present application will be described below with reference to the accompanying drawings. It should be understood by those skilled in the art that these embodiments are only used to explain the technical principles of the present application, and are not intended to limit the protection scope of the present application.

[0068] In practical engineering, due to the limitations of the actuator's own physical characteristics leading to control input saturation, and the negative impact of model uncertainty and external disturbances on the system, the following PMSM servo system with input saturation uncertainty is considered:

[0069]

[0070] In the formula, Let u be the system state vector, j = 1, ..., n, j = 1, ..., m. u = Sat(v) = [Sat1(v1), ..., Sat...] m (v m )] T Let v be the actual control input vector containing actuator input saturation, and v be the designed control input vector. Given a nonlinear function, The gain vector is known. D i It is lumped interference, and D i =Δf i (x)+d ei , Δf i (x) represents f in the system i The uncertainty term of (x), d ei This indicates an unknown external disturbance in the system.

[0071] Consider the actuator input saturation phenomenon that occurs in actual systems, and describe it simply as follows:

[0072]

[0073] Among them, u max and u min These are the actuator control input signals v j The upper and lower bounds of the constraint.

[0074] When the control input signal v j Reaching v j >u max or v j min At the time, under the limit value u max and u min Frequent switching at the point of contact may cause actuator chatter, increase wear on the actuator, and thus disrupt the stable operation of the system.

[0075] The actuator input saturation (1-2) is approximated by the following smoothing model:

[0076]

[0077] In the formula, u M =(u​max +u min ) / 2+((u max -u min ) / 2)sign(vj), tanh(·)∈(-1,1). According to the Mean Value Theorem, if there exists a constant The actual control input u is then expressed as:

[0078] u=h(v)+S(v=h v v+S(v) (1-4)

[0079]

[0080] In the formula, 0 <h vj ≤1, j=1,…,m. and

[0081] Therefore, this invention discloses a variable-rate cooperative reaching law non-singular fast integral terminal sliding mode control method, such as... Figures 1-18 As shown, the specific steps include:

[0082] The design of a control system comprised of a disturbance observer, a nominal controller, and a compensation controller aims to ensure good tracking control performance even when faced with actuator input saturation, parameter uncertainty, and unknown external disturbances. The disturbance observer enhances the system's ability to suppress unknown external disturbances by estimating disturbances in the system in real time and providing feedback. The nominal controller is responsible for achieving the system's basic control objectives, ensuring performance under ideal conditions, such as stability and high-precision control characteristics. However, since real-world systems typically exhibit uncertainties, the compensation controller adjusts the nominal controller to adapt to changes in system parameters and rapid dynamic adjustments. Through the coordinated work of these three components, the control system effectively improves robustness, ensuring trajectory tracking performance even under external disturbances and changes in model parameters. Furthermore, the system can respond quickly and remain stable, meeting stringent time response requirements and avoiding performance degradation caused by actuator input saturation or disturbances.

[0083] To mitigate the impact of lumped interference in the system, the lumped interference observer is designed as follows:

[0084]

[0085] In the formula, ψ i This represents the intermediate state vector of the lumped disturbance observer. For lumped interference D i The observation vector, η i =diag{η i1 , ..., ηim} represents the adjustable gain matrix of the design, η ij >0, Q i (x i )=η i x i Let f1(x) be a smooth gain function, i = 1, ..., n, j = 1, ..., m, and f1(x) be a known nonlinear function. n This is the system state vector.

[0086] The observation error of lumped disturbance is defined as: The dynamics of its observation error are described as follows:

[0087]

[0088] The control input v is described as follows:

[0089] v = v0 + v σ (1-8)

[0090] In the formula, v0 and v σ These represent the nominal control input and the compensated control input, respectively. The closed-loop control scheme under the proposed strategy is as follows: Figure 1 As shown.

[0091] definition Let x0 be the tracking error of system (1-1), and x0 be the desired trajectory signal, i = 1, ..., n. To overcome the singularity and chattering problems of terminal sliding mode control, the sliding mode function is designed as follows:

[0092]

[0093] In the formula, χ e It is a weighting factor. λ0=diag{λ 01 ,…,λ 0n-2}, λ1=diag{λ 11 ,·…,λ 1m}, λ2=diag{λ 21 ,…,λ 2m}, λ3=diag{λ 31 ,…,λ 3m}, λ 0r , λ 1j , λ 2j and λ 3j To adjust the parameters, r = 1, ..., n-2.

[0094] in, It is a diagonal matrix. For γ p >0 indicates a smooth and monotonically increasing trend. tanh(·)=[tanh1(·),…,tanh m (·)] T Continuous function tanh j (·)∈(-1,1), p=1,2.

[0095] Taking the first time derivative of equation (1-9), we get:

[0096]

[0097] Table 1.1 Expressions of different functions

[0098]

[0099] To reduce the high-frequency chattering problem in sliding mode, the discontinuous sign function "sign(·)" of traditional sliding mode is replaced with the smoothing function tanh(·). Compared with the SF1(·) and SF2(·) functions in Table 1.1, the SF3(·) based on the tanh(·) function converges faster, as shown below. Figure 2 As shown. When the selected weighting factor χ e As the value approaches zero, the tanh(·) function can be gradually fitted to the discontinuous function sign(·), such as... Figure 3 As shown. Furthermore, on the sliding surface, use The function can realize different stages of error e 1j The goal is to achieve fast convergence. That is, when the system error moves far from the origin, the error term... It plays a dominant control role in equation (1-9); when the system error approaches the origin, the value in equation (1-9) becomes more significant. The system is dominated by a single factor. Therefore, the system error can converge to zero quickly in the global space.

[0100] To improve the accuracy of trajectory tracking control, the novel non-singular fast integral terminal sliding surface is designed as follows:

[0101]

[0102] Taking the first derivative of the nonsingular fast integral terminal sliding surface (1-11) in time, we get:

[0103]

[0104] in, It is an unknown constant.

[0105] Introducing improved command filter technology into the nominal controller design avoids the differential explosion problem of virtual signals in backstepping technology, thereby improving system stability and trajectory tracking control accuracy.

[0106] Based on the above analysis, the nominal form of equation (1-1) can be described as follows:

[0107]

[0108] The error signal of system (1-13) is defined as follows:

[0109]

[0110] The virtual control input α1 is designed as follows:

[0111]

[0112] Where, k1=diag{k 11 ,…,k 1m}, k 1j These are design parameters, and k 1j >0.

[0113] The improved command filter is designed as follows:

[0114]

[0115] In the formula, α1 is the input of the filter, and z 2f =θ1 is the output of the filter, χ α χ is a weighting factor, and α >0. The initial conditions satisfy θ1(0)=α1(0) and θ2(0)=0. It is an adjustable gain matrix.

[0116] To eliminate the impact of filtering errors on the system's tracking control accuracy, a filtering error compensation mechanism is designed as follows:

[0117]

[0118] In the formula, This is the auxiliary vector for the designed filtering error compensation mechanism. c1 = diag{c 11 ,…,c 1m}, c2=diag{c 21 , ..., 2 cm}, c 1j and c 2j For design parameters, and c 1j >0, c 2j >0. The initial conditions satisfy δ. p (0)=0, p=1,2.

[0119] Specifically, the compensated tracking error signal is defined as Ξ1 = z1 - δ1, and its first derivative in time is calculated as follows:

[0120]

[0121] Where, λ1=diag{λ 11 ,…,λ 1m}, Let u be an unknown constant, and let u be the actual control input signal. χ is the m-th element of the error vector. e S(v) is the weighting factor, and S(v) is the approximation error of the designed smooth input saturation model. This is to account for the estimation error due to interference.

[0122] Choose the Lyapunov function V1:

[0123]

[0124] Considering equations (1-14) and (1-16), the first derivative of V1 with respect to time is rearranged as follows:

[0125]

[0126] Define the compensation tracking error signal as Ξ2=z2-δ2, and calculate its first derivative with respect to time:

[0127]

[0128] Construct the following Lyapunov function:

[0129]

[0130] Taking the first derivative of equation (1-22) with respect to time, we get:

[0131]

[0132] The nominal control input is designed as follows:

[0133]

[0134] In the formula, k2=diag{k 21 , ..., k 2m}, k 2j These are design parameters, and k 2j >0.

[0135] Substituting equation (1-24) into equation (1-23), we get:

[0136]

[0137] Input compensation control v σ Designed as follows:

[0138]

[0139] In the formula, l1=diag{l 11 ,…,l 1m}, l2=diag{l 21 , ..., l 2m}, l3=diag{l 31 ,…,l 3m}, and l 1j l 2j and l 3j The parameters are set to be adjusted, and 1 ≤ μ1 < 2, 0 < μ2 < 1. L0 is a smoothing cooperative function, χ σ It is a weighting factor, χ σ >0, where I is the identity matrix. The expressions and curves for different smoothing functions are shown in Table 1.2 and... Figures 4-7 The information is provided in the text.

[0140] In the sliding mode reaching law, using The function can improve the convergence speed of sliding mode and reduce the potential chattering effect caused by the discontinuity of the sign function in the traditional reaching law. To further reduce the impact of high-frequency chattering that may occur during the sliding mode transition and improve convergence efficiency, a smoothing cooperative function l0 and (I-l0) are designed using the tanh(·) function to achieve a smooth transition between the two stages. Under the same weighting factor, compared with other functions in Table 1.2, the tanh(·)-based function l0... H0 Faster convergence speed, such as Figures 4-5 As shown. Furthermore, the convergence speed increases with the weighting factor χ. σ The decrease is accelerated, such as Figures 6-7 As shown. When the system's sliding mode is far from or far from the sliding surface, i.e., |σ j When |>1, in equation (1-26) The term plays a dominant controlling role. When the system's sliding mode approaches the sliding surface, i.e., |σ j When |≤1, in equation (1-26) This method plays a dominant and controlling role. Overall, this approach not only achieves rapid convergence at different sliding stages, but also smoothly and quickly converges the system state to the sliding surface in global space.

[0141] Experimental Verification—PMSM Servo System

[0142] The proposed method was validated using a surface-mount PMSM servo system. The PMSM servo system consists of a servo inverter, industrial computer, control box, motion control card transmission box, torque sensor, and a 130MB150A surface-mount PMSM. The torque sensor is model YH502, with a range of 0±50Nm, a power supply of DC24V, and an output of 10±5kHz. The control algorithm was implemented using Matlab modeling software, and the algorithm design was based on Simulink software. Code was generated through compilation, and the control algorithm was validated using the LINKS-RT real-time simulation software package. The structure of the PMSM servo system based on LINKS-RT is shown below. Figure 8 As shown. The experimental platform parameters are u. min =-50, the parameters of the drive motor and the load motor are the same. L d =L q =L=6.65mH, J m =0.0027 kg·m 2 n p =4, R s =1.84Ω, Φ=0.32Wb, u max =200,

[0143] The dynamics of the PMSM servo systems (1-30) and (1-31) can be described in the form of system (1-1) as follows:

[0144]

[0145] In the formula, x = [x1 x2] T =[J m ω m Li q ] T , u = u q x 10 =J m ω m0 ,ω m0 Let ΔR be the desired mechanical angular velocity. s =R st -R s ΔJ m =J mt -J m ΔL=L t -L and ΔΦ=Φ t -Φ represents the system model parameters R. s J m The uncertainty of Φ and L, with the subscript "t" indicating the real-time value.

[0146] Combining formulas (1-24) and (1-26), the nominal controller and compensation controller of the PMSM servo system are designed separately:

[0147]

[0148] The PMSM servo system employs a closed-loop control scheme based on the adaptive variable-rate cooperative reaching law NFITSMC strategy designed in this invention, as follows: Figure 9 As shown.

[0149] Experimental Results and Analysis

[0150] To fully demonstrate the superiority of the proposed strategy, it was compared with PI, ITSMC (1-26) without cooperative reaching law, and STSMC methods. The parameters of the proposed cooperative reaching law ITSMC were designed as follows: λ0 = 1, λ1 = λ2 = λ3 = 10, l1 = l2 = l3 = 10, γ1 = 0.001, μ1 = 7 / 5, μ2 = 0.001, β0 = β1 = β2 = 25, k1 = 25, k2 = 100, c1 = 20, c2 = 20, χ... e =χ δ =χ σ =0.001 and η1=η2=50. The parameters of STSMC are selected as α=20, λ=0.4, K=10. The PI controller parameters are k P =9,k I =100.

[0151] The experimental section verified the robustness of the PMSM servo system using six control methods under varying load torque disturbances. The variation in load torque disturbances is described as follows:

[0152]

[0153] To effectively and qualitatively evaluate the performance of the proposed control strategy, several different performance metrics were used for verification, including overshoot at several different speeds, speed reduction / increase due to disturbances, and recovery time due to disturbances.

[0154] Table 1.3 Performance Comparison Results of Different Methods

[0155]

[0156] like Figures 10-18As shown in the figure, the speed, q-axis current, and q-axis voltage response curves of the PMSM servo system using PI, ITSMC-SS, STSMC-SS, and the proposed cooperative reaching law NSFITSM-SS method are displayed when the speed tracking is 200 / 600 / 1000 rpm. It can be seen from the figure that when different speed control methods are implemented, the speed curves using the PI and STSMC-SS methods not only exhibit significant overshoot at 25 / 50 / 80 rpm and 0 / 33 / 100 rpm, but also show obvious chattering and a large steady-state error. When using ITSMC-SS to achieve different desired speed control, no overshoot occurs in the dynamic stage, but obvious chattering is present. When load disturbances change, the speed curves using the PI, ITSMC-SS, and STSMC-SS methods change significantly, resulting in speed reductions of 45 / 50 / 55 rpm, 20 / 25 / 25 rpm, and 42 / 55 / 58 rpm, and speed increases of 48 / 48 / 58 rpm, 25 / 30 / 25 rpm, and 43 / 53 / 56 rpm, respectively. The proposed cooperative reaching law NSFITMC-SS strategy not only achieves better speed control for different needs but also avoids overshoot. Furthermore, when load disturbances change, the proposed strategy ensures smaller steady-state errors and smaller speed reduction / increase effects, with speed reduction / increases of 16 / 18 / 16 rpm and 18 / 16 / 18 rpm, respectively, and recovery times of 0.2 / 0.2 / 0.2 s and 0.2 / 0.2 / 0.2 s, respectively. In summary, the proposed cooperative reaching law NSFITMC-SS strategy outperforms the other three methods discussed and other methods in the literature. Table 1.3 shows a detailed comparison of the speed regulation performance indicators of different methods.

[0157] In summary, compared with the other three methods, the proposed adaptive variable-rate cooperative reaching law NSFITSM-SS strategy significantly improves the speed of trajectory tracking adjustment, the accuracy of steady-state control, and the performance of strong disturbance suppression. Furthermore, it features lower current, more stable voltage, and lower energy consumption. In other words, this strategy achieves the goal of energy-saving optimization.

[0158] The technical solution of the present invention has been described above with reference to the preferred embodiments shown in the accompanying drawings. However, it will be readily understood by those skilled in the art that the scope of protection of the present invention is obviously not limited to these specific embodiments. Without departing from the principles of the present invention, those skilled in the art can make equivalent changes or substitutions to the original technical features, and the technical solutions resulting from these changes or substitutions will all fall within the scope of protection of the present invention.

Claims

1. A variable-rate cooperative reaching law nonsingular fast integral terminal sliding mode control method, characterized in that, include: Design an interference observer, a nominal controller, and a compensation controller; The interference observer is used to estimate the lumped interference in the system in real time. The observer is constructed by combining the intermediate state vector and the adjustable gain matrix with the smooth gain function to obtain the observed value of the lumped interference. The nominal controller introduces an improved command filter technique, which achieves signal smoothing by dynamically adjusting weighting factors; then, a filtering error compensation mechanism is designed to offset the filtering error. The compensation controller constructs a non-singular fast integral terminal sliding surface by fusing tracking error, error integral term and smoothing function tanh through a sliding surface function. Design adaptive estimation; Based on adaptive estimation, the adaptive variable rate cooperative approach law is updated, thereby compensating for the observation error of the interference observer and the approximation error of the saturated model.

2. The method according to claim 1, characterized in that, The interference observer is represented as: In the formula, ψ i This represents the intermediate state vector of the lumped disturbance observer. For lumped interference D i The observation vector, η i =diag{η i1 , ..., η im } represents the adjustable gain matrix of the design, η ij >0, Q i (x i )=η i x i Let f1(x) be the smoothing gain function, u be the actual control input signal, i = 1, ..., n, j = 1, ..., m, and f1(x) be a known nonlinear function. n This is the system state vector.

3. The method according to claim 1, characterized in that, The improved command filter is: In the formula, α1 is the input of the filter, and z 2f =θ1 is the output of the filter, χ α χ is a weighting factor, and α >0, the initial conditions satisfy θ1(0)=α1(0) and θ2(0)=0, It is an adjustable gain matrix.

4. The method according to claim 1, characterized in that, The filtering error compensation mechanism is as follows: In the formula, c1 = diag{c 11 c 1m }, c2=diag{c 21 c 2m }, c 1j and c 2j For design parameters, and c 1j >0, c 2j >0, initial conditions satisfy δ p (0)=0, p=1,2.

5. The method according to claim 2, characterized in that, The constructed non-singular fast integral terminal sliding surface is: In the formula, λ1=diag{λ 11 ,…λ 1m }, λ2=diag{λ 21 , …, λ 2m }, λ3=diag{λ 31 , …, λ 3m },0<γ1<1, γ2 is an adjustable parameter in the design, τ is the differential operator, and f n (x) is a nonlinear function, λ 2m , λ 3m , λ 0n-2 , λ 01 For elements in a diagonal matrix, h is the (n-1)th first derivative. v v0 is the nominal control input after input saturation processing. To interfere with the observed values, For the derivative of the interference observation, This is the gain vector.

6. The method according to claim 5, characterized in that, The adaptive variable-rate cooperative reaching law is designed as follows:

7. The method according to claim 6, characterized in that, The process of updating the adaptive variable-rate cooperative reaching law based on adaptive estimation, thereby compensating for the observation errors of the interference observer and the approximation errors of the saturated model, includes: Find the first derivative of the terminal sliding surface time of the nonsingular fast integral; The compensated tracking error signal is defined as Ξ1=z1-δ1, and its first time derivative is calculated as follows:

8. The method according to claim 7, characterized in that, The first derivative of the terminal sliding surface time of the nonsingular fast integral is: Where, λ1=diag{λ 11 , …, λ 1m }, Let u be an unknown constant, and let u be the actual control input signal. χ is the m-th element of the error vector. e S(v) is the weighting factor, and S(v) is the approximation error of the designed smooth input saturation model. This is to account for the estimation error due to interference. Choose the Lyapunov function V1: Based on the expression for the error signal and the improved command filter, the first-order time derivative of V1 is rearranged as follows: Define the compensation tracking error signal as Ξ2=z2-δ2, and calculate its first derivative with respect to time: Construct the following Lyapunov function V2: Taking the first derivative of V2 in time, we get:

9. The method according to claim 8, characterized in that, The method further includes: designing the nominal controller input as follows: In the formula, k2=diag{k 21 , ..., k 2m }, k 2j These are design parameters, and k 2j >0; Substitute v0 into In the middle, we get: In the formula, χ δ This is the weighting factor.

10. The method according to claim 7, characterized in that, The process of updating the adaptive variable-rate cooperative reaching law based on adaptive estimation, thereby compensating for the observation errors of the interference observer and the approximation errors of the saturated model, includes: Introducing adaptive estimation The expression is: In the formula, β0, β1, ..., β n >1 is the gain parameter, 1≤j≤n, The derivative of the (n-1)th order state vector of the system; The method further includes: designing the input of the compensation controller as follows: l0=diag{tanh1((σ1 / x σ ) 2 ),…,tanh m ((s m / h σ ) 2 )} In the formula, l1=diag{l 11 , ..., l 1m },l2=diag{l 21 , ..., l 2m },l3=diag{l 31 , ..., l 3m }, and l 1j l 2j and l 3j To adjust the parameters, and 1≤μ1<2, 0<μ2<1, l0 is a smoothing cooperative function, χ σ It is a weighting factor, χ σ >0, where I is the identity matrix.

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