A reso-based filter backstepping sliding mode control method and system for electro-hydraulic servo system

Abstract

The application relates to the field of electro-hydraulic servo system control, and discloses a RESO-based filtering backstepping sliding mode control method and system for an electro-hydraulic servo system, wherein the method comprises the following steps: S1. establishing a system state equation of the electro-hydraulic servo system, and processing unmodeled friction, parameter time variation and unknown disturbance into matched disturbance and unmatched disturbance; S2. based on system state variables of the system state equation, designing RESO to estimate the matched disturbance and the unmatched disturbance in real time; S3. based on the RESO, designing a virtual control law by using a filtering backstepping control framework; S4. according to the virtual control law, designing an actual control input by combining a sliding mode control method; and S5. applying the actual control input to the electro-hydraulic servo system, and feeding back the matched disturbance and the unmatched disturbance estimated by the RESO to the controller for real-time compensation.

Description

Technical Field

[0001] This invention relates to the field of electro-hydraulic servo system control, and specifically to a method and system for filtering backstepping sliding mode control of an electro-hydraulic servo system based on RESO. Background Technology

[0002] Electro-hydraulic servo systems, with their advantages of high power-to-weight ratio, fast response speed, and high control precision, are widely used in industrial fields such as excavators, ship transportation, and water well drilling rigs. In recent years, with the continuous advancement of industrial transformation and upgrading, electro-hydraulic servo systems are steadily moving towards higher response and higher precision. To achieve this goal, a feasible path is to select more technologically advanced system mechanical components and higher-performance electro-hydraulic servo valves. However, this approach involves multiple professional fields and has a relatively long implementation cycle. In contrast, improving the overall control performance of the system by selecting and optimizing control strategies is more direct and efficient. Electro-hydraulic servo systems are essentially typical time-varying nonlinear systems, and in practical applications, they are also affected by many unknown disturbances. These nonlinear factors undoubtedly increase the complexity of control strategy design. Therefore, designing advanced control algorithms for electro-hydraulic servo systems to achieve high-precision tracking control still faces many challenges.

[0003] Over the past few decades, significant progress has been made in the control technology of electro-hydraulic servo systems. Existing control methods can be categorized into three types: linear control, nonlinear control, and observer-based control. Traditional PID technology and feedback linearization control algorithms are easier to implement in engineering. However, they only perform well under certain specific conditions and fail to achieve satisfactory performance under the influence of the aforementioned unknown disturbances. To improve the control performance of electro-hydraulic servo systems, researchers have conducted extensive studies and proposed many advanced nonlinear control methods, such as backstepping control and sliding mode control.

[0004] Backstepping control is one of the most commonly used methods in nonlinear control. It designs the control law in a step-by-step recursive manner by constructing a Lyapunov function. The unique feature of this method is its clear and explicit design steps, and the ability to guarantee the convergence of the closed-loop system using Lyapunov stability analysis. However, the control law designed using this method relies excessively on the exact model of the system and suffers from a "computational explosion" problem caused by repeatedly differentiating the virtual control law.

[0005] Observer-based control is an effective method for addressing uncertainties. The main idea is to design an observer to estimate uncertainties and compensate for their effects in the control loop. The most typical observer is the Extended State Observer (ESO), which integrates internal and external disturbances into an extended system state for estimation and compensation. However, traditional ESOs are only applicable to integral chain systems and cannot handle mismatched disturbances. While techniques exist to convert mismatched disturbances into matched disturbances, these require complex coordinate transformations. Furthermore, the order of the ESOs constructed using these techniques is often greater than the system order. For high-order systems like electro-hydraulic servo systems, this can easily amplify noise, causing peaking and reducing tracking control accuracy. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention proposes a filtered backstepping sliding mode control method and system for electro-hydraulic servo systems based on RESO (Resistant Optimizer), enabling the system to maintain high-precision and rapid tracking control under the influence of matched and unmatched disturbances, while avoiding the "computational explosion" problem. Specifically, firstly, the unmodeled friction, time-varying parameters, and unknown disturbances in the system are treated as matched and unmatched disturbances. A RESO is designed to observe and estimate the matched and unmatched disturbances in real time, improving the tracking accuracy and robustness of the system. Then, within the framework of backstepping control, a second-order command filter is introduced, and a sliding mode controller is designed, simplifying the calculations and further improving the robustness of the system.

[0007] To achieve the above objectives, this invention provides a method for filtering backstepping sliding mode control of an electro-hydraulic servo system based on RESO, comprising the following steps:

[0008] S1. Establish the system state equation of the electro-hydraulic servo system, and treat the unmodeled friction force, time-varying parameters and unknown disturbances as matched disturbances and unmatched disturbances;

[0009] S2. Based on the system state variables of the system state equation, design a RESO to estimate matched and unmatched disturbances in real time;

[0010] S3. Based on RESO, a virtual control law is designed using a filtered backstepping control framework;

[0011] S4. Based on the virtual control law, and in conjunction with the sliding mode control method, design the actual control input;

[0012] S5. Apply the actual control input to the electro-hydraulic servo system, and feed back the matched and unmatched disturbances estimated by RESO to the controller for real-time compensation.

[0013] Preferably, step S2 includes:

[0014] Based on the known nominal part of the system state equation and the system state variables, construct the auxiliary state equation of RESO;

[0015] The auxiliary state equations output estimates of matched and unmatched perturbations.

[0016] Preferably, step S3 includes:

[0017] Based on the desired trajectory and output deviation of the electro-hydraulic servo system, the first tracking error is defined;

[0018] Based on the first tracking error, a first Lyapunov function is constructed and a first virtual control law is designed in conjunction with a second-order instruction filter;

[0019] Based on the deviation between the first virtual control law and the state of the electro-hydraulic servo system, a second tracking error is defined;

[0020] Based on the estimated values ​​of the second tracking error and unmatched disturbance, a second Lyapunov function is constructed and a second virtual control law is designed in conjunction with a second-order instruction filter.

[0021] Preferably, step S4 includes:

[0022] Define the sliding mode function based on the second tracking error and its derivative;

[0023] Based on the sliding mode function and the estimated matching disturbance, a third Lyapunov function is constructed, and the actual control input is generated.

[0024] Preferably, step S5 includes: feeding back the estimated matched disturbance and the unmatched disturbance to the design step of the backstepping controller respectively; performing feedforward compensation on the matched disturbance when generating the actual control input, and performing feedforward compensation on the unmatched disturbance when generating the virtual control law.

[0025] The present invention also provides a RESO-based electro-hydraulic servo system filtering backstepping sliding mode control system, the system being used to implement the above method, comprising: a construction module, an estimation module, a first design module, a second design module, and a compensation module;

[0026] The building module is used to establish the system state equation of the electro-hydraulic servo system, and to process unmodeled friction, time-varying parameters and unknown disturbances into matched disturbances and unmatched disturbances.

[0027] The estimation module is used to design RESO to estimate matched and unmatched disturbances in real time based on the system state variables of the system state equation.

[0028] The first design module is used to design virtual control laws based on RESO and utilizing a filtered backstepping control framework;

[0029] The second design module is used to design the actual control input based on the virtual control law and the sliding mode control method;

[0030] The compensation module is used to apply the actual control input to the electro-hydraulic servo system and feed back the matched and unmatched disturbances estimated by RESO to the controller for real-time compensation.

[0031] Preferably, the workflow of the estimation module includes:

[0032] Based on the known nominal part of the system state equation and the system state variables, construct the auxiliary state equation of RESO;

[0033] The auxiliary state equations output estimates of matched and unmatched perturbations.

[0034] Preferably, the workflow of the first design module includes:

[0035] Based on the desired trajectory and output deviation of the electro-hydraulic servo system, the first tracking error is defined;

[0036] Based on the first tracking error, a first Lyapunov function is constructed and a first virtual control law is designed in conjunction with a second-order instruction filter;

[0037] Based on the deviation between the first virtual control law and the state of the electro-hydraulic servo system, a second tracking error is defined;

[0038] Based on the second tracking error and the estimated mismatch disturbance value, a second Lyapunov function is constructed and a second virtual control law is designed in conjunction with a second-order instruction filter.

[0039] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0040] This invention designs RESO to accurately estimate matched and unmatched disturbances in an electro-hydraulic servo system in real time, combines a command filter to process the virtual control law in backstep control to avoid peaking and computational explosion problems in the control, and integrates sliding mode control to enhance robustness. Under complex working conditions with unmodeled friction, time-varying parameters, and unknown disturbances, it significantly improves the system's tracking accuracy and response speed. Attached Figure Description

[0041] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0042] Figure 1 This is a schematic diagram of the electro-hydraulic servo system according to an embodiment of the present invention;

[0043] Figure 2 This is a block diagram of a RESO-based filtered backstepping sliding mode control system according to an embodiment of the present invention;

[0044] Figure 3 This figure shows a comparison of the tracking results of exponential step trajectories by RESO backstepping control (RESO-TB) and traditional ESO backstepping control (ESO-TB) according to an embodiment of the present invention. Figure 3 (a) represents the position tracking response curve; Figure 3 (b) represents the tracking error response curve;

[0045] Figure 4 This figure shows a comparison of observational results of the RESO backstepping control (RESO-TB) and the conventional ESO backstepping control (ESO-TB) for uncertain disturbances according to an embodiment of the present invention. Figure 4 (a) represents the dynamic observation response curves of the two observers to the mismatched disturbance; Figure 4 (b) represents the dynamic observation response curves of the two observers to the matched disturbance; Figure 4 (c) represents the response curves of the two observers to the observation error of the mismatched disturbance; Figure 4 (d) represents the response curves of the two observers to the observation error of the matched perturbation;

[0046] Figure 5 This image shows a comparison of the tracking results of a sinusoidal trajectory between RESO-Filtered Backstepping Sliding Mode Control (RESO-FBSMC) and traditional ESO-Backstepping Sliding Mode Control (ESO-BSMC) according to an embodiment of the present invention. Figure 5 (a) represents the position tracking response curve; Figure 5 (b) represents the tracking error response curve;

[0047] Figure 6 The comparison curves of the control input voltages of the RESO filtered backstepping sliding mode control (RESO-FBSMC) and the traditional ESO backstepping sliding mode control (ESO-BSMC) in this embodiment of the invention are shown.

[0048] Figure 7This figure shows a comparison of observational results of the RESO filtered backstepping sliding mode control (RESO-FBSMC) and the traditional ESO backstepping sliding mode control (ESO-BSMC) for uncertain disturbances according to an embodiment of the present invention; wherein, Figure 7 (a) represents the dynamic observation response curves of the two observers to the mismatched disturbance; Figure 7 (b) represents the dynamic observation response curves of the two observers to the matched disturbance. Detailed Implementation

[0049] 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. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0050] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0051] First, let's explain the characters. In the text, the superscript "." in the formulas indicates the first derivative and ".." indicates the second derivative.

[0052] RESO, or Reduced-Order Extended State Observer, is an advanced perturbation observation technique. Similar in core idea to the traditional Extended State Observer (ESO), it treats unmodeled internal dynamics and unknown external disturbances as a lumped "perturbation" and estimates it as a new state in real time.

[0053] However, the key improvement of RESO lies in "order reduction". Unlike traditional ESO, it does not need to reconstruct and estimate all system states. Instead, it uses some known system state information to directly build a lower-order observer specifically to estimate disturbances. This makes RESO simpler in structure, with fewer parameters and faster response speed. It can also effectively avoid the "peaking" phenomenon that may be caused by the high order of traditional ESO, thus exhibiting higher estimation accuracy and stronger noise resistance in high-order systems (such as electro-hydraulic servo systems).

[0054] Example 1:

[0055] This embodiment provides a filtered backstepping sliding mode control method for an electro-hydraulic servo system based on RESO, including the following steps:

[0056] S1. Establish the system state equation of the electro-hydraulic servo system, and treat the unmodeled friction force, time-varying parameters and unknown disturbances as matched disturbances and unmatched disturbances.

[0057] like Figure 1 The diagram shown illustrates the working principle of an electro-hydraulic servo system (EHSS), which consists of an electro-hydraulic servo valve and a hydraulic cylinder. The load is controlled by the electro-hydraulic servo valve, which converts the received electrical signal into a hydraulic signal, which then drives the hydraulic cylinder. Figure 1 The electro-hydraulic servo system under consideration For the mass of the load, The area of ​​the slider in the cavity. For piston displacement, For the displacement of the servo valve spool, For unmodeled friction and unknown disturbances in the system, The pressure inside the left chamber of the hydraulic cylinder. This refers to the pressure inside the right chamber of the hydraulic cylinder. To supply airflow to both rooms, To supply the return flow to both rooms, To alleviate supply pressure.

[0058] According to Newton's second law, the dynamic equation of a hydraulic cylinder is:

[0059]

[0060] in, The load pressure in the hydraulic actuator. ; The coefficient of viscous friction is given; considering the effect of internal leakage, the dynamic load pressure can be defined as:

[0061]

[0062] in Effective oil bulk modulus; For the time-varying uncertain flow rate of the hydraulic cylinder, For load flow rate, ; V is the total internal leakage coefficient of the hydraulic cylinder; t This indicates the total effective oil volume of the hydraulic cylinder.

[0063] The electro-hydraulic servo system employs a high-performance servo valve, which operates at a frequency much higher than the desired motion trajectory. Therefore, the relationship between valve spool displacement and control input is approximately: , where k i It is a positive constant, and u is the backstep control law. The load flow rate can be obtained as follows:

[0064]

[0065] in It's flow gain. , For flow coefficient, For the area gradient of the servo valve, This refers to the density of the hydraulic oil.

[0066] definition If the variables are state variables, then the state-space equation of the EHSS can be obtained from equations (1)-(3) as follows:

[0067]

[0068] in,

[0069]

[0070] Notice and These are system parameters.

[0071] In practical applications, it is often difficult to obtain accurate system parameters, therefore... and There is an uncertain part, which can be denoted as and . This refers to the mismatched disturbance term caused by unmodeled friction and external interference in the system. This refers to the matching disturbance term caused by uncertain system traffic. In this embodiment, all the above uncertainties are treated as lumped disturbances. and , can be represented as:

[0072]

[0073]

[0074] Therefore, equation (4) is rewritten as:

[0075]

[0076] in and For the known nominal part of the EHSS model, and For unknown lumped disturbances, this embodiment designs a RESO to estimate them in real time. and The estimated value is fed back to the controller to compensate for its impact on the control system, thus improving the output. Capable of quickly and accurately tracking the desired reference trajectory .

[0077] Considering the physical limitations of EHSS in practical applications, the following assumptions are made.

[0078] Assumption 1: Expected reference trajectory and its first-order time derivative Available.

[0079] Assumption 2: Lumped disturbance and And all of their derivatives are bounded, meaning there exist positive constants. , , and satisfy , , and , .

[0080] Therefore, the control objective of this embodiment is to design a suitable backstepping control law that combines RESO and backstepping control techniques to address the presence of matched and unmatched lumped disturbances in the EHSS. This causes the system output to be... High-precision and fast tracking of the desired trajectory .

[0081] S2. Based on the system state variables of the system state equation, design RESO to estimate matched and unmatched disturbances in real time.

[0082] To achieve the control objective of S1, this embodiment designs a RESO (Reduced-Order Extended State Observer) to estimate matched and unmatched disturbances.

[0083] Compared to traditional ESO, RESO has a faster response speed and can effectively avoid the "peaking phenomenon" present in traditional ESO under complex perturbations. The specific form of traditional RESO is:

[0084]

[0085]

[0086] in, Represents the lumped disturbance term and The estimated value of the sum, Indicates the auxiliary state of the observer. This indicates the observer bandwidth.

[0087] In the system state and Given the internal dynamics term and It can provide direct feedback to the observer and controller. Therefore, to reduce the burden on the observer, model-assisted RESO is designed as follows:

[0088]

[0089]

[0090] in, and Representing lumped disturbances respectively and The estimated value.

[0091] Define the lumped disturbance estimation error as Then, combining equations (8), (11), and (12), its first derivative form is as follows:

[0092]

[0093]

[0094] The estimation error of RESO is defined as follows: Then equation (14) can be further expressed as:

[0095]

[0096] in , , and when hour, It is a Hurwitz matrix, in which case, for any given positive definite matrix... Both have a positive definite matrix. satisfy:

[0097]

[0098] Compared to traditional ESOs, the RESO involved in this embodiment has the advantages of simple structure and small gain parameter. Specifically, the ESO structure designed for the system using the pole placement method is as follows:

[0099]

[0100]

[0101] in, , . yes The observed values, yes The observed values, and It is a state of system expansion. , . and This is the observer bandwidth of ESO, indicating that both types of observers share the same bandwidth. and .

[0102] By comparison, it is clear that RESO has a lower order than ESO; therefore, the design of RESO is much simpler than that of ESO. Furthermore, both observers have the same bandwidth. It is an adjustable parameter, therefore it can be used at the same bandwidth. Under these conditions, the value of the ESO gain parameter is much larger than that of RESO, which may amplify the impact of high-frequency noise on the system and lead to system instability.

[0103] S3. Based on RESO, design virtual control laws using a backstepping control framework.

[0104] This step involves designing a filtered backstepping sliding mode controller based on a reduced-order linear state observer of S2 to achieve the trajectory tracking control objective.

[0105] The tracking error is defined as follows:

[0106]

[0107] in, These are the virtual state variables after SCF (Self-Processing Function); then, the backstepping control law is derived using instruction filtering, recursive backstepping, and sliding mode control. e1 represents the first tracking error; e2 represents the second tracking error; and e3 represents the third tracking error.

[0108] Define the first Lyapunov function:

[0109]

[0110] According to equations (8) and (19), we can obtain The derivative is:

[0111]

[0112] Then for Taking the derivative, we get:

[0113]

[0114] In order to Negative, first virtual control law Designed as:

[0115]

[0116] in, These are positive design parameters. To avoid affecting the first virtual control law in subsequent steps... Repeated differentiation, Its estimated value is obtained after passing through a second-order filter SCF. and its derivative The filtering error is defined as Substituting equation (23) into equation (22) yields:

[0117]

[0118] Define the second Lyapunov function:

[0119]

[0120] According to equations (8) and (19), we can obtain The derivative is:

[0121]

[0122] Then for Taking the derivative, we get:

[0123]

[0124] In order to Negative, second virtual control law Designed as:

[0125]

[0126] in, Positive design parameters For obtained through RESO The estimated value. Similarly, to avoid affecting the second virtual control law in subsequent steps. Repeated differentiation, Its estimated value is obtained after passing through a second-order filter SCF. and its derivative Define the filtering error Substituting equation (28) into equation (27) yields:

[0127]

[0128] in, .

[0129] S4. Design the actual control input based on the virtual control law and the sliding mode control method.

[0130] First, define the sliding mode function:

[0131]

[0132] in, , It is a positive constant.

[0133] According to equations (8) and (19), we can obtain The derivative is:

[0134]

[0135] Differentiating equation (30) yields:

[0136]

[0137] Define the third Lyapunov function:

[0138]

[0139] Then for Taking the derivative, we get:

[0140]

[0141] In order to A negative value indicates the backstep control law of the actual control input. Designed as:

[0142]

[0143] in, Positive design parameters The result obtained through equation (12) The estimated value. Substituting equation (35) into equation (34) yields:

[0144]

[0145] in, .

[0146] Based on the above design steps, the RESO-based backstepping controller structure designed in this embodiment is as follows:

[0147]

[0148] The block diagram of the filter backstepping sliding mode control system based on model-assisted RESO proposed in this embodiment is as follows: Figure 2 As shown.

[0149] S5. Apply the actual control input to the electro-hydraulic servo system and feed back the matched and unmatched disturbances estimated by RESO to the controller for real-time compensation.

[0150] This embodiment will utilize Lyapunov stability theory to analyze the convergence of the system error signal and the stability of the control system under the proposed control method. The stability of the control system is given by the following theorem:

[0151] Theorem 1: For the RESO proposed for EHSS containing matching and non-matching uncertainties, under the conditions that Assumptions 1 and 2 are satisfied, the estimated error of the designed RESO is eventually bounded and its norm satisfies:

[0152] .

[0153] The proof is as follows: Define the Lyapunov function of RESO as:

[0154] .

[0155] right Taking the derivative and substituting equation (16) into the equation, we get:

[0156] ,

[0157] in, express The smallest eigenvalue of . Therefore, within a finite time interval, the norm of the estimation error is bounded as follows:

[0158] ,

[0159] By selecting an appropriate observer gain This allows the estimation error of RESO to be arbitrarily small.

[0160] Theorem 2: Considering an EHSS containing matched and unmatched uncertainties, under the conditions that Assumption 1 and Assumption 2 are satisfied, by applying the RESO of Equations (11)-(12) and the backstepping controller of Equation (37), by selecting appropriate design parameters, it is ensured that all signals in the control system are consistent and eventually bounded, and the tracking error converges to a small neighborhood near zero.

[0161] The proof is as follows:

[0162] Based on the design in step S3, the final Lyapunov for backstepping control is designed as follows:

[0163]

[0164] According to equations (22), (29) and (36). The derivative can be expressed as:

[0165]

[0166] According to Young's inequality, the latter terms of equation (43) can be transformed as follows:

[0167] ,

[0168] Substituting equation (44) into equation (43), we get:

[0169] .

[0170] Equation (45) can be rearranged into a compact inequality as follows:

[0171] ,

[0172] in:

[0173] .

[0174] Selecting design parameters To ensure Then, solving inequality (46) yields:

[0175] ,

[0176] when , It will converge to the upper boundary. Therefore, the compensated error Bounded. According to Assumption 2, we know... Since the control system is bounded, all error signals are bounded. Furthermore, selecting appropriate control parameters... It can make the region of convergence arbitrarily small.

[0177] Example 2:

[0178] This embodiment also provides a RESO-based filtered backstepping sliding mode control system for an electro-hydraulic servo system, comprising: a construction module, an estimation module, a first design module, a second design module, and a compensation module. The construction module is used to establish the system state equation of the electro-hydraulic servo system, and to process unmodeled friction, time-varying parameters, and unknown disturbances into matched disturbances and unmatched disturbances. The estimation module is used to design RESO to estimate matched and unmatched disturbances in real time based on the system state variables of the system state equation. The first design module is used to design a virtual control law based on RESO using the wave-wave backstepping control framework. The second design module is used to design the actual control input based on the virtual control law and combined with the sliding mode control method. The compensation module is used to apply the actual control input to the electro-hydraulic servo system and to feed back the matched and unmatched disturbances estimated by RESO to the controller for real-time compensation.

[0179] The estimation module's workflow includes: constructing auxiliary state equations for RESO based on the known nominal parts of the system state equations and the system state variables; and outputting estimates of matched and unmatched disturbances through the auxiliary state equations.

[0180] The workflow of the first design module includes: defining a first tracking error based on the desired trajectory of the electro-hydraulic servo system and the deviation of its output; constructing a first Lyapunov function based on the first tracking error and designing a first virtual control law in combination with a second-order instruction filter; defining a second tracking error based on the first virtual control law and the deviation of the electro-hydraulic servo system state; and constructing a second Lyapunov function based on the second tracking error and the estimated mismatch disturbance value and designing a second virtual control law in combination with a second-order instruction filter.

[0181] Example 3:

[0182] To verify the effectiveness and superiority of the control method of this invention, a simulation experiment will be conducted in the MATLAB / Simulink environment using EHSS as the object.

[0183] The physical parameters of the EHSS system are selected as shown in Table 1:

[0184] Table 1

[0185] .

[0186] The controller parameters are selected as follows: Set the system's time-varying, unpredictable flow rate as follows: N, at this point, the matching perturbation N.

[0187] To verify the advantages of RESO compared to ESO mentioned in Table 1, this embodiment conducted a performance comparison test on the two observers. To better demonstrate the observer's performance, the controllers in this embodiment all use a traditional backstepping (TBC) design. The controller structure is as follows:

[0188] ,

[0189] in, and It is a virtual control law. , , .

[0190] The desired trajectory is to have an initial state of zero and a steady state of zero. The exponential signal of m, i.e. m, and apply a time-varying sinusoidal external perturbation to the EHSS. N, at this point, the system's unmatched disturbance is N, the observer parameters are selected as follows: , .

[0191] Performance comparison test of the two observers, as follows Figure 3 As shown. Among them, Figure 3 (a) is the step function position tracking curve. Figure 3 (b) is the step function position tracking error curve. Figure 4 (a) and Figure 4 (b) shows the observation curves of the two observers for unmatched perturbations and the observation curves of the two observers for matched perturbations, respectively. (c) and (d) represents the observation error result curves for unmatched perturbations and matched perturbations, respectively.

[0192] from As can be seen in (a), both observers can stably and quickly track the desired step function. Compared to the ESO observer, the RESO observer performs better, exhibiting stronger tracking capabilities. As shown in (b), under the same disturbance conditions, compared with the two observers using the backstep control strategy, the tracking error of RESO is significantly lower than that of ESO, which indicates that RESO exhibits higher performance in estimation accuracy.

[0193] from (a) and In (b), it can be clearly observed that the RESO observer can accurately and quickly estimate the unmatched and matched disturbances of the system. In contrast, the ESO observer exhibits phase lag and has certain shortcomings in estimation accuracy. Through comparison... (c) and As shown in (d), the RESO observer exhibits a smaller observation error compared to the ESO observer. This advantage directly reflects the higher observation accuracy and faster observation speed of the RESO, thus proving that its performance is significantly improved compared to the traditional ESO.

[0194] To further verify the superiority of the proposed control scheme, the following two controllers are compared.

[0195] RESO-FBSMC: This is the control scheme proposed in this invention, namely a RESO-based filtered backstepping sliding mode controller. The backstepping controller parameters are set as follows: , , The parameters of the second-order command filter are set as follows: , The parameters of the sliding mode controller are set as follows: , At this time, the bandwidth of RESO is set as follows: , .

[0196] ESO-BSMC: This is a backstepping sliding mode controller based on ESO. The parameters of the backstepping sliding mode controller are the same as those of RESO-FBSMC, namely: , , , , The observer uses the same bandwidth, that is: , The structure of the ESO-based backstepping sliding mode controller is as follows:

[0197]

[0198] The controller tracks a smooth sinusoidal motion trajectory, that is... m, and apply external perturbation to EHSS N, at this point, the unmatched disturbance N.

[0199] Simulation results of the electro-hydraulic servo system under two control schemes are as follows: - As shown. Among them, of (a), (b) are the sinusoidal function position tracking curve and the sinusoidal function position tracking error curve, respectively. It is the system control input curve. Figure 7 (a)- Figure 7 (b) shows the observation curves for unmatched and matched perturbations, respectively.

[0200] from As can be seen from (a), both control schemes can effectively track the sine function. However, further comparison... As can be clearly seen in (b), the RESO-FBSMC scheme performs better in error control, and its accuracy is significantly higher than that of the other scheme.

[0201] from As can be seen, under the same disturbance conditions, the control input curves of both control schemes maintain boundedness, continuity, and stability.

[0202] according to As shown in (a), the proposed control scheme demonstrates a certain advantage in observing unmatched disturbances, while the ESO-BSMC scheme exhibits phase lag in comparison. Further analysis from... As can be clearly seen in (b), the RESO-FBSMC scheme is both accurate and rapid in observing matched disturbances. In contrast, the ESO-based control scheme suffers from significant phase lag and its estimation accuracy is also insufficient.

[0203] To further quantitatively analyze the observer's performance, mean absolute error was used to compare the two control schemes. Root mean square error absolute error in integration time Three performance indicators were analyzed. For example... As shown.

[0204] Table 2

[0205] .

[0206] The performance index comparison results under different control schemes are as follows: As shown, it is evident that the proposed RESO-FBSMC has the smallest indices of all two control schemes. Specifically, compared to ESO-BSMC, the proposed RESO-FBSMC has the smallest mean absolute error. Improved by 82%, root mean square error This represents an 83.3% improvement. Experimental results show that this control scheme has the best tracking accuracy. Furthermore, the absolute error in integral time... This represents a 56.8% improvement, meaning that this control scheme has better anti-interference capabilities and stronger robustness against time-varying disturbances compared to ESO-BSMC.

[0207] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A filtering backstepping sliding mode control method for an electro-hydraulic servo system based on RESO, characterized in that, Includes the following steps: S1. Establish the system state equation of the electro-hydraulic servo system, and treat the unmodeled friction force, time-varying parameters and unknown disturbances as matched disturbances and unmatched disturbances; S2. Based on the system state variables of the system state equation, design a RESO to estimate matched and unmatched disturbances in real time; S3. Based on RESO, a virtual control law is designed using a filtered backstepping control framework; The steps in S3 include: Based on the desired trajectory and output deviation of the electro-hydraulic servo system, the first tracking error is defined; Based on the first tracking error, a first Lyapunov function is constructed and a first virtual control law is designed in conjunction with a second-order instruction filter; Based on the deviation between the first virtual control law and the state of the electro-hydraulic servo system, a second tracking error is defined; Based on the estimated values ​​of the second tracking error and unmatched disturbance, a second Lyapunov function is constructed and a second virtual control law is designed in conjunction with a second-order instruction filter; S4. Based on the virtual control law, and in conjunction with the sliding mode control method, design the actual control input; the steps include: First, define the sliding mode function: ； Where c1 and c2 are positive constants; e1 is the first tracking error; e2 is the second tracking error; and e3 is the third tracking error. The derivative of e3 is: ； In this formula, the superscript "." indicates the first derivative; z 2d The derivative of the estimated value z2 obtained by passing the second virtual law through the second-order filter SCF; b is the system parameter; g is the control gain function; u represents the backstepping control law; f k2 For the known nominal part of the EHSS model; f u2 For lumped disturbances; Differentiating the sliding mode function S, we get: ； z 1d This represents the derivative of the estimated value z1 obtained by passing the first virtual rate through the second-order filter SCF; x1, x2, and x3 are system state variables; f k1 For the known nominal part of the EHSS model; x d Represents the desired trajectory; Define the third Lyapunov function: ； V2 represents the second Lyapunov function; Taking the derivative of V3, we can obtain: ； k1 and k2 represent positive design parameters; and Representing the lumped disturbance f respectively u1 and f u2 The estimated value; In order to A negative value indicates the backstep control law of the actual control input. Designed as: ； Where k3 is a positive design parameter; α2 represents the second virtual rate; By substituting into the formula, we can obtain: ； in, ;e s1 e s2 Indicates the filtering error; Based on the above design steps, the designed RESO-based backstepping controller structure is as follows: ； S5. Apply the actual control input to the electro-hydraulic servo system, and feed back the matched and unmatched disturbances estimated by RESO to the controller for real-time compensation.

2. The method for filtering, backstepping, and sliding mode control of an electro-hydraulic servo system based on RESO according to claim 1, characterized in that, The steps in S2 include: Based on the known nominal part of the system state equation and the system state variables, construct the auxiliary state equation of RESO; The auxiliary state equations output estimates of matched and unmatched perturbations.

3. The method for filtering, backstepping, and sliding mode control of an electro-hydraulic servo system based on RESO according to claim 1, characterized in that, The steps in S4 include: Define the sliding mode function based on the second tracking error and its derivative; Based on the sliding mode function and the estimated matching disturbance, a third Lyapunov function is constructed, and the actual control input is generated.

4. The method for filtering, backstepping, and sliding mode control of an electro-hydraulic servo system based on RESO according to claim 3, characterized in that, The steps of S5 include: feeding back the estimated matched disturbance and the unmatched disturbance to the design steps of the backstepping controller respectively; performing feedforward compensation on the matched disturbance when generating the actual control input; and performing feedforward compensation on the unmatched disturbance when generating the virtual control law.

5. A RESO-based electro-hydraulic servo system filtering backstepping sliding mode control system, said system being used to implement the method described in any one of claims 1-4, characterized in that, include: The module includes a construction module, an estimation module, a first design module, a second design module, and a compensation module. The building module is used to establish the system state equation of the electro-hydraulic servo system, and to process unmodeled friction, time-varying parameters and unknown disturbances into matched disturbances and unmatched disturbances. The estimation module is used to design RESO to estimate matched and unmatched disturbances in real time based on the system state variables of the system state equation. The first design module is used to design virtual control laws based on RESO and utilizing a filtered backstepping control framework; The workflow of the first design module includes: Based on the desired trajectory and output deviation of the electro-hydraulic servo system, the first tracking error is defined; Based on the first tracking error, a first Lyapunov function is constructed and a first virtual control law is designed in conjunction with a second-order instruction filter; Based on the deviation between the first virtual control law and the state of the electro-hydraulic servo system, a second tracking error is defined; Based on the second tracking error and the estimated unmatched disturbance value, a second Lyapunov function is constructed and a second virtual control law is designed in conjunction with a second-order instruction filter; The second design module is used to design the actual control input based on the virtual control law and the sliding mode control method; The compensation module is used to apply the actual control input to the electro-hydraulic servo system and feed back the matched and unmatched disturbances estimated by RESO to the controller for real-time compensation.

6. The RESO-based electro-hydraulic servo system filtering backstepping sliding mode control system according to claim 5, characterized in that, The workflow of the estimation module includes: Based on the known nominal part of the system state equation and the system state variables, construct the auxiliary state equation of RESO; The auxiliary state equations output estimates of matched and unmatched perturbations.

Images