Preset time tracking control method for electro-hydraulic servo system with uncertainty

Abstract

The application discloses a preset time tracking control method for an electro-hydraulic servo system with uncertainty, which is based on a traditional backstepping design control method, combines an improved finite time function to scale a control error, and designs a preset time controller to ensure that output tracking error converges to zero within fixed time. The control method is proposed for the following problem: the existing nonlinear control method for the electro-hydraulic servo system has poor error transient convergence performance, and can only ensure that the tracking error approaches zero or a small bounded range as time tends to infinity, while in actual application, it is always desired that the tracking error can converge to zero as soon as possible. The disclosed control method can achieve very ideal convergence within the preset time, effectively solves the problem of poor convergence performance of the traditional nonlinear control method, and can explicitly determine the convergence time and make it independent of the initial state condition, thereby obtaining better tracking performance.

Description

TECHNICAL FIELD

[0001] The present application relates to electro-hydraulic servo control technology, and in particular to a preset time tracking control method for an electro-hydraulic servo system with uncertainty. BACKGROUND

[0002] The electro-hydraulic servo system is widely used in aerospace, weapons, ships and other high-end equipment, and intelligent robots, engineering machinery, metallurgical industry and other fields due to its large power density, fast dynamic response, strong anti-load rigidity, large output force / torque and other outstanding advantages, and often plays a key role in control and power transmission. However, the electro-hydraulic servo system is a typical highly nonlinear system, which has many nonlinear characteristics and model uncertainties, and is a bottleneck factor restricting the improvement of its control performance. The control performance of the electro-hydraulic servo system directly determines the core performance of the high-end equipment, and the advanced servo control algorithm is the key to realizing high-performance servo control.

[0003] Many methods have been proposed for the nonlinear control problem of the electro-hydraulic servo system, such as sliding mode control, adaptive robust control (ARC), error sign robust integral (RISE) control, etc. However, the sliding mode control often leads to the problem of discontinuity and chattering of the control input in the physical system; the adaptive robust control can only guarantee that the tracking error is bounded in theory when both parameter uncertainty and unmodeled disturbance exist; and the RISE controller ensures asymptotic tracking performance due to the high robustness of the integral robust control law to any smooth time-varying disturbance, but the value of the nonlinear robust gain in the controller designed by this control method needs to satisfy certain conditions, which are closely related to the upper bounds of the first and second derivatives of the modeling uncertainty with respect to time. It is worth noting that most of the existing nonlinear control methods for the electro-hydraulic servo system can only guarantee that the system tracking error approaches zero or becomes arbitrarily small as time goes to infinity. For practical engineering applications, we always hope that the tracking error converges to zero as soon as possible. Therefore, the fast convergence performance of the above control methods needs to be further improved. In order to solve this problem, finite time control has received extensive attention in the control community in the past few decades. The existing finite time control methods are generally divided into three categories, including homogeneous method, terminal sliding mode control and fractional power method based on state feedback. However, the final stable time of the system under the action of the existing finite time control method depends on the initial state or the design parameters, and the convergence time cannot be predetermined. SUMMARY

[0004] The present application relates to electro-hydraulic servo control technology, and in particular to a preset time tracking control method for an electro-hydraulic servo system with uncertainty.

[0005] The technical solution for achieving the purpose of the present application is as follows: a preset time tracking control method for an electro-hydraulic servo system with uncertainty, comprising the following steps:

[0006] Step 1, a mathematical model of the electro-hydraulic servo system is established, and step 2 is entered.

[0007] Step 2, a preset time tracking controller is designed according to the mathematical model of the electro-hydraulic servo system, and step 3 is entered.

[0008] Step 3, the Lyapunov stability theory is used to prove the stability of the electro-hydraulic servo system under the preset time tracking controller, and the result that the tracking error of the system converges to zero within the preset time is obtained.

[0009] Compared with the prior art, the present application has the following advantages: the problem of poor transient tracking error convergence performance of the traditional nonlinear method is effectively solved, the system stable time is not dependent on the initial state condition, and better tracking performance is obtained. The simulation results verify the effectiveness. BRIEF DESCRIPTION OF DRAWINGS

[0010] Figure 1 is a schematic diagram of the electro-hydraulic servo system of the present application.

[0011] Figure 2 is a schematic diagram of the preset time tracking control method of the electro-hydraulic servo system.

[0012] Figure 3 is a tracking error curve of the system under the preset time tracking controller (PTC), the feedback linearization controller (FLC) and the traditional PID controller designed by the present application when the expected command of the system is x 1d = 10·sin(π·t) mm.

[0013] Figure 4 is a tracking process curve of the system output under the preset time tracking controller (PTC) designed by the present application when the expected command of the system is x 1d = 10·sin(π·t) mm.

[0014] Figure 5 is a tracking error curve of the system under the preset time tracking controller (PTC) designed by the present application when the expected command of the system is x 1d = 10·sin(π·t) mm.

[0015] Figure 6 is a control input curve of the system under the preset time tracking controller (PTC) designed by the present application when the expected command of the system is x 1d = 10·sin(π·t) mm.

[0016] Figure 7 is a control input curve of the system under the preset time tracking controller (PTC) designed by the present application when the expected command of the system is x 1dThe tracking error comparison curve of the system under the action of the preset time tracking controller (PTC), the feedback linearization FLC controller and the traditional PID controller designed by the application when the displacement is 10mm.

[0017] Figure 8 is the desired command of the system 1d The curve of the system control input changing with time under the action of the preset time tracking controller (PTC) designed by the application when the displacement is 10mm. DETAILED DESCRIPTION

[0018] The preset time tracking control method for the electro-hydraulic servo system with uncertainty is provided, the improved finite time function is introduced to scale the virtual error, the preset time controller is designed, the problem of poor transient tracking error convergence performance of the traditional nonlinear method is effectively solved, the stable time of the system is not dependent on the initial state condition, and better tracking performance is obtained.

[0019] The application will be further described in detail below in combination with the drawings and specific embodiments.

[0020] In combination Figures 1-2 The preset time tracking control method for the electro-hydraulic servo system with uncertainty, comprising the following steps:

[0021] Step 1, establishing a mathematical model of the electro-hydraulic servo system;

[0022] S1.1, assuming that the electro-hydraulic servo system is directly driven by a valve-controlled double-rod hydraulic cylinder to drive an inertial load. The control target is to make the inertial load track any smooth motion trajectory as accurately as possible.

[0023] Therefore, according to Newton's second law, the motion equation of the electro-hydraulic servo system is:

[0024]

[0025] In formula (1), m is the mass of the inertial load, y is the displacement of the inertial load, P L is the pressure difference of the hydraulic cylinder load, A is the effective action area of the piston, B v represents the effective viscous damping coefficient, f(t) is the uncertainty of the system that is not matched, is the velocity of the inertial load, is the acceleration of the inertial load, and t represents the running time.

[0026] The pressure dynamic equation of the electro-hydraulic servo system is:

[0027]

[0028] In formula (2), V tThe total volume of the system control cavity. P represents L The derivative of β e C is the effective volumetric elastic modulus of the hydraulic cylinder. t Q is the total leakage coefficient. L Let q(t) represent the load flow rate, and let q(t) represent the model uncertainty caused by complex internal leakage, parameter uncertainty, and unmodeled dynamic effects.

[0029] Ignoring valve spool dynamics, the load flow equation for the servo valve is:

[0030]

[0031] In equation (3), P s The constant oil supply pressure of the fluid, u is the system control input, i.e., the preset time tracking controller, and the flow gain. C d Here, is the flow coefficient, w is the valve area gradient, ρ is the oil density, and the sign function sign(u) is defined as:

[0032]

[0033] S1.2, Define the state variable x: Then the motion equation (1) is transformed into the state equation, which is the mathematical model of the electro-hydraulic servo system:

[0034]

[0035] In equation (5), the coefficients g and g are... coefficient coefficient All are nominal values ​​and are known.

[0036]

[0037] The system's unmatched disturbance term d1 = f(t) / m, and the system's matched disturbance term d2 = 4β e Aq(t) / (mV t x1 represents the displacement of the inertial load, x2 represents the velocity of the inertial load, and x3 represents the acceleration of the inertial load. Denotes the derivative of x1. Denotes the derivative of x². Let x3 be the derivative.

[0038] For ease of controller design, the following assumptions are made:

[0039] Assumption 1: The motion trajectory tracked by the system is third-order continuous, differentiable, and bounded;

[0040] Assumption 2: The electro-hydraulic position servo system works in normal condition, the pressure P1 and P2 of the left and right chambers of the hydraulic cylinder must satisfy the following conditions: 0 < P1 < P2 < P r s r s ; P r represents the system back pressure;

[0041] Assumption 3: The uncertainty of system matching and mismatching is bounded, that is: |d1|≤δ1, |d2|≤δ2, where δ1, δ2 are unknown normal numbers.

[0042] Go to Step 2.

[0043] Step 2, according to the mathematical model of electro-hydraulic servo system, design the preset time tracking controller, the steps are as follows:

[0044] S2.1, define the improved finite time function μ(t) ∈ [0, t f ):

[0045]

[0046] In formula (6), t f is the preset convergence time, n is the order of electro-hydraulic servo system, n1 is a positive integer term, a is a normal number term, formula (6) satisfies:

[0047]

[0048] L ∞ [0, t f ) represents a bounded function set on [0, t f ), it is easy to prove that μ(0) = 1 + a, μ(t f ) = +∞.

[0049] S2.2, define the tracking error e = x1-x 1d , x 1d is the position command that the system expects to track and the command is three-order continuous and differentiable; use the improved finite time function μ(t) to scale the virtual error, in order to facilitate the controller design later, define two variables ω i and η i , ω i represents the virtual error, η i represents the scaling value of the virtual error, the independent variable i = 1, 2, 3:

[0050]

[0051] In formula (7), α1, α2 represent the virtual control of the system.

[0052] ​​​S2.3. Based on the mathematical model of the electro-hydraulic servo system, design a preset time tracking controller u.

[0053] S2.3.1, According to the first equation in equation (5) and ω in equation (7) i By definition, α1 is chosen as the virtual control, so that equation Tend to a stable state:

[0054]

[0055] For virtual error ω i Scaling is performed to obtain the scaling value η of the virtual error. i

[0056] η1=μω1=μe=μ(x1-x 1d (9)

[0057] In the above formula, μ represents the abbreviation of μ(t); ω1 represents the first virtual error, and η1 represents the scaling value of the first virtual error.

[0058] Differentiating both sides of equation (9) and applying equations (7) and (8), we get:

[0059]

[0060] In the above formula, ω2 represents the first derivative of variable *; ω2 represents the second virtual error, and η2 represents the scaling value of the second virtual error.

[0061] Define Lyapunov functions Differentiating both sides, we get:

[0062]

[0063] Design virtual control α1:

[0064]

[0065] In equation (11), the adjustable gain k1 > 0. If the term is static damping, then:

[0066]

[0067] S2.3.2, According to the second equation in equation (5) and ω in equation (7) i By definition, α2 is chosen as the virtual control to make the equations tend to a stable state:

[0068]

[0069] Scaling the virtual error and taking the derivative of both sides of the equation gives

[0070]

[0071] In the above equation, ω3represents the third virtual error, and η3represents the scaling value of the third virtual error. The Lyapunov function is defined as Taking the derivative of both sides gives

[0072]

[0073] Based on the assumption in hypothesis 3 that |d1|≤δ1φ1, the constant term φ1=1, and using Young's inequality:

[0074]

[0075]

[0076] In equation (17), the constant term Δ1=max{1,δ1} is defined, and the intermediate function The constant θ1>0, (j) represents the jth derivative of *.

[0077] Using equations (16) and (18), the virtual control α2is designed as

[0078]

[0079] In equation (19), the adjustable gain k2>0, then:

[0080]

[0081] S2.3.3, according to the third equation in equation (5) and the definition of ω i in equation (7), we have:

[0082]

[0083] Scaling the virtual error and taking the derivative of both sides of the equation gives

[0084]

[0085] Taking the derivative of both sides of equation (19) gives

[0086]

[0087] The Lyapunov function is defined as Taking the derivative of both sides gives

[0088]

[0089] The constant term Δ2 in formula (24) is defined as Δ2 = max{1, δ1, δ2}, and the intermediate function The constant θ2 > 0.

[0090] According to formula (24), the preset time tracking controller u based on the mathematical model of the electro-hydraulic servo system is as follows:

[0091]

[0092] Substituting formula (25) into formula (24) gives:

[0093]

[0094] Go to step 3.

[0095] Step 3, the Lyapunov stability theory is used to prove the stability of the electro-hydraulic servo system under the preset time tracking controller, and the result that the tracking error of the system converges to zero within the preset time is obtained, which is as follows:

[0096] The Lyapunov function is defined as follows:

[0097]

[0098] Using formula (13), (20) and (26) gives:

[0099]

[0100] In formula (28), the intermediate function k = min{k1, k2, k3}, and the constant term The constant term θ ≤ min{θ1, θ2}; solving the inequality gives:

[0101]

[0102] Where τ and s represent the independent variables of the integral function, and V(0) represents the initial value of the function V.

[0103] The right side term of formula (29) is calculated as:

[0104]

[0105] Using formula (29) and (30) gives:

[0106]

[0107] In formula (31) is a monotonically decreasing function and is always positive. Therefore, it is obtained that V is bounded, and ω i and η i are bounded, and since η1∈L ∞ [0, tf ) and η1= μ(t)ω1, according to the definition of μ(t), it can be proved that the tracking of the system converges to zero in the preset time under the action of the adjustment gains k1, k2, k3, i.e.

[0108] To prove that all internal signals of the controller are bounded on [0, t f ), first of all, since ω i and η i belong to L ∞ [0, t f ), L ∞ [0, t f ) represents a set of functions bounded on [0, t f ), and according to assumptions 1 and 2, it is obvious that ω1= μ -1 (t)η1、 and are bounded, it can be determined that α1∈ L ∞ [0, t f ), according to formula (7), it can be obtained that x2 is bounded. To prove that α2 is bounded, the boundedness of is demonstrated. According to formulas (11) and (13), it can be obtained that then:

[0109]

[0110] In formula (32), the constant term Since η1∈ L ∞ [0, t f ), there exist constants c1 and c2 such that:

[0111]

[0112] According to formula (34), L'Hospital's rule is repeatedly used to obtain:

[0113]

[0114] According to formula (35), when t→ t f , the speed at which η1 approaches zero is not less than the (n1+n-1)th power of (t f -t), and L'Hospital's rule is continuously used to derive formula (35) to obtain:

[0115]

[0116] According to formula (36), when t→ t f , the speed at which approaches zero is not less than the (n1+n-2)th power of (t f -t), then:

[0117]

[0118] In formula (37), denotes the second derivative of the variable.

[0119] Therefore, the boundedness of a2can be guaranteed. According to formula (7), it can be obtained that x3is also bounded. Following this inference rule and according to the L'Hopital rule, it can be proved that [0, t f ) a2, and u(t) are bounded.

[0120] Therefore, it is concluded that the preset time tracking controller designed for the electro-hydraulic servo system (such as formula (5)) can make the system obtain a preset time stable result, and the adjustment gain k1, k2, k3can make the tracking error of the system converge to zero within the preset time. The schematic diagram of the preset time tracking control principle of the electro-hydraulic servo system is shown in Figure 2 .

[0121] Embodiment

[0122] Fundamentally, the preset time tracker is realized by controlling the gain, and in the case of ensuring that the state and control input remain continuous and bounded, the control gain reaching ∞ as t→t f ∞ makes the system stable within the preset time. However, in practice, the solution adopted is to add a saturation value in the time-varying high gain function before the specified time. This method promotes the actual system implementation by avoiding infinite gain, slightly sacrificing the convergence accuracy, and this method will be used in subsequent simulation.

[0123] In order to examine the performance of the designed controller, the following parameters are taken in the simulation to model the electro-hydraulic servo system:

[0124] The inertial load parameter m = 30 kg, the viscous friction coefficient B v = 4000 N·s / m; the torque amplification coefficient k i = 5 N·m / V, the piston pressure area A = 904.78 mm 2 , the constant oil supply pressure P s = 10 MPa, the total volume of the system control cavity V t = 7.96 × 10 - 5 m 3 , the effective volume liquid elastic modulus β e = 700 MPa, the flow gain k u = 1.1969 × 10 -8 m 3 / s / V / Pa 1 / 2 , the total leakage coefficient C t = 1 × 10 -13m 3 / s / Pa;

[0125] According to two different system conditions, the simulation process is divided into two parts:

[0126] 1. The desired command of the system is given as x 1d = 10 · sin(π · t) mm

[0127] The following controller is taken for comparison:

[0128] The preset time tracking controller (PTC): the controller parameters are taken as t f = 1.5 s, k1 = 6 × 10 -6 , k2 = 4 × 10 -7 , k3 = 7 × 10 -6 , θ1 = 10 -26 , θ2 = 5 × 10 -30 , and n = 3, n1 = 1

[0129] The feedback linearization controller (FLC): the controller is designed as follows:

[0130]

[0131]

[0132] The controller parameters are taken as k1 = 1100, k2 = 70, and k3 = 1300;

[0133] The PID controller: the proportional-integral-derivative controller, and the controller parameters are taken as k P = 150, k I = 100, and k D = 0, which respectively represent the P gain, the I gain, and the D gain.

[0134] The comparison curves of the system tracking error under the action of the PTC, the FLC, and the PID controller, the tracking curve of the system output to the desired command under the action of the PTC, and the tracking error curve of the system under the action of the PTC are respectively shown in FIGS. 1 Figure 3 , Figure 4 , and Figure 5 . It can be seen from FIGS. 1 Figure 3 that, compared with the PID, the FLC has a good ability to compensate for modeling uncertainty, and the tracking error obtained is smaller. By introducing the finite time function, the PTC achieves the result that the tracking error converges to zero after a specified time, and the tracking performance is greatly improved. Therefore, compared with the FLC and the PID, the PTC achieves the best tracking performance.

[0135] Figure 6 The desired command of the system is given as x 1dThe graph shows the change of control input of the electro-hydraulic system under PTC action over time when the input is 10·sin(π·t) mm. As can be seen from the graph, the obtained control input is a continuous signal, which is more conducive to execution in practical applications.

[0136] ② Given the desired instruction of the system as x 1d When the thickness is 10mm, the following controller is used for comparison:

[0137] Preset Time Tracking Controller (PTC): Takes controller parameter t f =1.5s, k1 = 6 × 10 -6 k2 = 4 × 10 -7 k3 = 7 × 10 -6 θ1=10 -26 θ2=5×10 -30 , and n = 3, n1 = 1.

[0138] Feedback linearization controller (FLC): Take controller parameters k1 = 350, k2 = 25, k3 = 410.

[0139] PID controller: Proportional-Integral-Derivative controller, controller parameter is k P =70,k I =30,k D =0.

[0140] This operating condition compares the step response performance of different controllers. Figure 7 The figure shows a comparison of tracking errors under three different controllers. The tracking error of the system under the proposed PTC still converges to zero within a specified time. The overall control performance is better than PID and FLC in both transient and steady-state tracking performance. Figure 8 Given the desired instruction of the system as x 1d When the diameter is 10mm, the control input of the electro-hydraulic system is under the action of PTC.

Claims

1. A preset time tracking control method for an electro-hydraulic servo system with uncertainty, characterized in that, Includes the following steps: Step 1: Establish the mathematical model of the electro-hydraulic servo system, then proceed to Step 2; Step 2: Based on the mathematical model of the electro-hydraulic servo system, design a preset time tracking controller, as follows: S2.

1. Define the improved finite-time function μ(t)∈[0,t] f ); S2.2, Define the system tracking error e = x1 - x 1d x1 represents the displacement of the inertial load, x 1d It is the position command that the system expects to track, and this command is three times continuously differentiable; S2.

3. Based on the mathematical model of the electro-hydraulic servo system, the preset time tracking controller u is designed as follows: Among them, coefficient coefficient All coefficients g are nominal values ​​and are known. x2 represents the velocity of the inertial load, x3 represents the acceleration of the inertial load, ω3 represents the third virtual error, η3 represents the scaling value of the third virtual error, Φ2 represents the intermediate function, k3 represents the adjustable gain, and θ1 is a constant. Proceed to step 3; Step 3: The stability of the electro-hydraulic servo system under the preset time tracking controller is proved by using Lyapunov stability theory, and the result is that the tracking error of the system converges to zero within the preset time.

2. The preset time tracking control method for an electro-hydraulic servo system with uncertainty according to claim 1, characterized in that, Step 1 establishes the mathematical model of the electro-hydraulic servo system, as follows: S1.1 Assume the electro-hydraulic servo system directly drives the inertial load through a valve-controlled dual-bar hydraulic cylinder, and the control objective is to make the inertial load track any smooth motion trajectory as accurately as possible: Therefore, according to Newton's second law, the equation of motion for the electro-hydraulic servo system is: In equation (1), m is the mass of the inertial load, y is the displacement of the inertial load, and P L Let A be the pressure difference under the load of the hydraulic cylinder, and B be the effective working area of ​​the piston. v Let f(t) represent the effective viscous damping coefficient, and f(t) represent the uncertainty of system mismatch. For the speed of inertial load, Let t represent the acceleration of the inertial load and t represent the running time. The pressure dynamic equation of the electro-hydraulic servo system is: In equation (2), V t The total volume of the system control cavity. P represents L The derivative of β e C is the effective volumetric elastic modulus of the hydraulic cylinder. t Q is the total leakage coefficient. L Let q(t) represent the load flow rate, and q(t) represent the model uncertainty caused by complex internal leakage, parameter uncertainty, and unmodeled dynamic effects. Ignoring valve spool dynamics, the load flow equation for the servo valve is: In equation (3), P s The constant oil supply pressure of the fluid, u is the system control input, i.e., the preset time tracking controller, and the flow gain. C d Here, is the flow coefficient, w is the valve area gradient, ρ is the oil density, and the sign function sign(u) is defined as: S1.2, Define the state variable x: Then the motion equation (1) is transformed into the state equation, which is the mathematical model of the electro-hydraulic servo system: In equation (5), the coefficients g and g are... coefficient coefficient All are nominal values ​​and are known; The system's unmatched disturbance term d1 = f(t) / m, and the system's matched disturbance term d2 = 4β e Aq(t) / (mV t x1 represents the displacement of the inertial load, x2 represents the velocity of the inertial load, and x3 represents the acceleration of the inertial load. Denotes the derivative of x1. Denotes the derivative of x². Let x3 be the derivative; For ease of controller design, the following assumptions are made: Assumption 1: The motion trajectory tracked by the system is third-order continuous, differentiable, and bounded; Assumption 2: Under normal operating conditions, the pressures P1 and P2 in the left and right chambers of the hydraulic cylinder must meet the following conditions: 0 <P r <P1<P s 0 <P r <P2<P s ;P r Indicates the system return oil pressure; Assumption 3: The uncertainty of both matching and mismatch in the system is bounded, i.e., |d1|≤δ1, |d2|≤δ2, where δ1 and δ2 are unknown positive constants.

3. The preset time tracking control method for an electro-hydraulic servo system with uncertainty according to claim 2, characterized in that: The uncertainty f(t) of system mismatch includes unmodeled nonlinearity and external disturbances.

4. The preset time control method for the electro-hydraulic servo system according to claim 1, characterized in that, In step 2, based on the mathematical model of the electro-hydraulic servo system, a preset time tracking controller u is designed. The specific steps are as follows: S2.

1. Define the improved finite-time function μ(t)∈[0,t] f ): In equation (6), t f Let n be the order of the electro-hydraulic servo system, n1 be a positive integer term, and a be a positive constant term. Equation (6) satisfies: L ∞ [0,t f ) represents [0,t f Given a bounded set of functions on ), it is easy to prove that μ(0) = 1 + a, μ(t) = 1 + a. f ) = +∞; S2.2, Define the system tracking error e = x1 - x 1d x 1d The system expects to track the position command, and this command is third-order continuously differentiable. A modified finite-time function μ(t) is used to scale the virtual error. To facilitate subsequent controller design, two variables ω are defined. i and η i ω i Represents the virtual error, η i The scaling value represents the virtual error, with independent variables i = 1, 2, 3: In equation (7), α1 and α2 both represent the virtual control of the system; S2.

3. Based on the mathematical model of the electro-hydraulic servo system, design a preset time tracking controller u; According to the first equation in equation (5) and ω in equation (7) i By definition, α1 is chosen as the virtual control, so that equation Tend to a stable state: For virtual error ω i Scaling is performed to obtain the scaling value η of the virtual error. i η1=μω1=μ(x1-x 1d ) (9), In the above formula, μ represents the abbreviation of μ(t); ω1 represents the first virtual error, and η1 represents the scaling value of the first virtual error; Differentiating both sides of equation (9) and applying equations (7) and (8), we get: In the above formula, ω2 represents the first derivative of variable *; ω2 represents the second virtual error, and η2 represents the scaling factor of the second virtual error; Define Lyapunov functions Differentiating both sides, we get: Design virtual control α1: In equation (11), the adjustable gain k1 > 0. If the term is static damping, then: According to the second equation in equation (5) and ω in equation (7) i By definition, α2 is chosen as the virtual control to make the equations tend to a stable state: Scaling the virtual error and differentiating both sides of the equation, we get: In the above formula, ω3 represents the third virtual error, and η3 represents the scaling value of the third virtual error; the Lyapunov function is defined. Differentiating both sides, we get: Based on assumption 3, |d1|≤δ1φ1, and the constant term φ1=1, using Young's inequality: In equation (17), the constant term Δ1 is defined as max{1,δ1}, and the intermediate function is... The constant θ1>0,* (j) This represents the j-th derivative of *. Using equations (16) and (18), design a virtual control: In equation (19), the adjustable gain k2 > 0, then: According to the third equation in equation (5) and ω in equation (7) i From the definition, we get: Scaling the virtual error and differentiating both sides of the equation, we get: Differentiating the virtual control α2 in equation (19) yields: Define Lyapunov functions Differentiating both sides, we get: In equation (24), the constant term Δ2 = max{1,δ1,δ2} is defined, and the intermediate function... The constant θ2 > 0; According to equation (24), based on the mathematical model of the electro-hydraulic servo system, the preset time tracking controller u is as follows: Substituting equation (25) into equation (24), we get:

5. The preset time tracking control method for an electro-hydraulic servo system with uncertainty according to claim 1, characterized in that, In step 3, the stability of the electro-hydraulic servo system under the preset time tracking controller is proven using Lyapunov stability theory, and the result is obtained that the tracking error of the system converges to zero within the preset time, as detailed below: The Lyapunov function is defined as follows: Using equations (13), (20), and (26), we get: In equation (28), the intermediate function k = min{k1,k2,k3}, and the constant term... The constant term θ ≤ min{θ1, θ2}; solving the inequality yields: Where τ and s both represent the independent variables of the integral function, and V(0) represents the initial value of the function V; The right-hand side of formula (29) is calculated as follows: Using equations (29) and (30), we get: In formula (31) It is a monotonically decreasing function and is always positive; therefore, V is bounded, and ω i and η i Both are bounded, since η1∈L ∞ [0,t f And η1=μ(t)ω1. According to the definition of μ(t), it can be proved that the tracking of the system converges to zero within a preset time under the action of adjusting the gain k1, k2, k3.

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