A method for variable speed approach and disturbance compensation control of a pump-controlled electro-hydraulic steering system

Abstract

This invention proposes a variable speed approach and disturbance compensation control method for a pump-controlled electro-hydraulic steering system. The method uses a variable speed approach terminal sliding mode disturbance observer to accurately estimate the lumped disturbance of the pump-controlled electro-hydraulic servo steering system, and uses an adaptive incomplete compensation method for associated steering angle errors to feedforward compensate for the system's lumped disturbance. The coefficient of this disturbance compensation automatically adjusts according to the error changes, balancing steering accuracy and steering stability. The method includes the following steps: Step S1, establishing a mathematical model of the pump-controlled electro-hydraulic steering system; Step S2, designing a variable speed sliding mode approach law; Step S3, designing a variable speed approach terminal sliding mode disturbance observer; Step S4, designing an incomplete disturbance compensation variable speed approach sliding mode controller; Step S5, designing a Lyapunov stability analysis function for the closed-loop system. This invention achieves high-precision steering control without requiring an accurate model of the pump-controlled electro-hydraulic steering system or an upper bound on the disturbance.

Description

Technical Field

[0001] This invention relates to the field of electro-hydraulic servo steering technology for heavy vehicles, and in particular to a method for speed change approach and disturbance compensation control of a pump-controlled electro-hydraulic steering system. Background Technology

[0002] Heavy-duty vehicles are widely used in numerous fields such as wind power, firefighting, rescue, and construction, and are indispensable key technical equipment for national economic development. Heavy-duty vehicles are characterized by their long bodies, heavy loads, and numerous steering axles, typically employing electro-hydraulic steering systems to achieve high-load drive and flexible all-terrain steering. Traditional electro-hydraulic steering systems often use proportional valves or servo proportional valves as control elements, relying on significant energy dissipation through numerous valve ports to ensure high-precision dynamic steering of heavy-duty vehicles. However, with global warming and the increasingly severe energy crisis, there is an urgent need to develop highly efficient and energy-saving electro-hydraulic steering systems.

[0003] In recent years, with the rapid development of electronic control technology, pump-controlled electro-hydraulic steering systems have been widely studied in industry and academia. This system directly changes the output flow of the hydraulic pump to drive the power steering cylinder by varying the speed or displacement, essentially eliminating the overflow and throttling losses of traditional valve-controlled electro-hydraulic servo steering systems, thus significantly improving system efficiency. However, pump-controlled electro-hydraulic steering systems are only controlled by high pressure on one side of the circuit, resulting in low system stiffness, small open-loop gain, and insufficient dynamic performance. Furthermore, the reduced valve damping in the pump-controlled system further decreases the system damping ratio, making it prone to over-adjustment when using high-gain control methods. Simultaneously, the pump-controlled electro-hydraulic steering system uses a servo motor to drive a fixed-displacement pump to output hydraulic fluid to control a pair of asymmetrical cylinders driving a trapezoidal steering mechanism, resulting in a complex structure with severe nonlinearity, making it difficult to establish an accurate mathematical model. Additionally, the pump-controlled electro-hydraulic servo steering system needs to overcome the adverse effects of uncertainties and nonlinear factors such as parameter perturbations, unknown steering resistance torque, and unmodeled dynamics during steering. When an accurate model of the pump-controlled electro-hydraulic steering system and the upper bound of disturbances caused by uncertain nonlinear factors cannot be obtained, model-based nonlinear methods are insufficient to achieve high-performance dynamic steering control. Although ZL202210398704.3 improves the overshoot suppression capability of pump-controlled electro-hydraulic steering systems under large load disturbances through the invention barrier Lyapunov method, it still has some shortcomings and limitations, mainly manifested in:

[0004] 1. Pump-controlled electro-hydraulic steering systems are characterized by low stiffness, low damping, and strong nonlinearity, making it difficult to simultaneously achieve high-precision and high-stability steering control. Pump-controlled electro-hydraulic steering systems rely solely on high-pressure oil control, resulting in low system stiffness. When subjected to strong disturbances, control accuracy rapidly decreases. Furthermore, due to its low damping characteristics, simply using a high-gain controller to forcibly suppress disturbances can easily lead to over-adjustment, compromising system stability. In addition, the complex structure of pump-controlled electro-hydraulic steering systems, the inability to measure external load disturbances, and internal parameter perturbations make accurate modeling difficult, hindering the achievement of high-precision steering control through model compensation.

[0005] 2. Traditional disturbance observers cannot achieve finite-time convergence of disturbance observations, and directly applying them to the feedforward channel can easily lead to system instability. Traditional disturbance observers can only guarantee asymptotic convergence of the observer to system disturbances, making it difficult for the system to accurately compensate for rapidly changing strong disturbances. Furthermore, since constructing a disturbance observer requires the system's higher-order states, and this higher-order state information is usually fed back from the angle information measured by sensors, there is serious measurement noise and time delay. If the observed disturbance is directly applied to the feedforward channel for complete compensation, it will cause system instability, while using fixed-gain incomplete compensation will result in a loss of accuracy. Summary of the Invention

[0006] This invention proposes a variable speed approach and disturbance compensation control method for a pump-controlled electro-hydraulic steering system, which can achieve high-precision steering control without obtaining an accurate model of the pump-controlled electro-hydraulic steering system and the upper limit of the disturbance.

[0007] The present invention adopts the following technical solution.

[0008] A variable speed approach and disturbance compensation control method for a pump-controlled electro-hydraulic steering system is disclosed. The method uses a variable speed approach terminal sliding mode disturbance observer to accurately estimate the lumped disturbance of the pump-controlled electro-hydraulic servo steering system, and uses an adaptive incomplete compensation method for associated steering angle error to perform feedforward compensation for the lumped disturbance of the system. The coefficient of the disturbance compensation is automatically adjusted according to the error change, taking into account both steering accuracy and steering stability. The method includes the following steps.

[0009] Step S1: Combine the differential equations of the mechanical and hydraulic parts of the pump-controlled electro-hydraulic steering system, and use the input-output linearization method to establish the mathematical model of the pump-controlled electro-hydraulic steering system.

[0010] Step S2: Introduce a variable power function 1-ε1sig(s) to construct a variable speed sliding mode reaching law. Where s is the sliding surface, k1 is the coefficient of the exponential reaching term, k2 is the coefficient of the variable speed reaching term, ε1 is the power convergence coefficient for adjusting the variable speed sliding mode reaching law, 0 < ε1 < 1; sgn(s) is the sign function with respect to the sliding surface; sig(s) is the activation function for adjusting the rate of change of the variable speed sliding mode reaching law.

[0011] Step S3: Construct the sliding mode surface of the sliding mode disturbance observer, introduce the variable speed sliding mode approaching law associated with the sliding mode surface, and design the variable speed approaching Terminal sliding mode disturbance observer by combining the mathematical model of the pump-controlled electro-hydraulic steering system.

[0012] Step S4: Combining the model equivalent control law, variable speed sliding mode approach law and variable speed approach Terminal sliding mode disturbance observer, an adaptive disturbance compensation function with associated turning angle error is introduced to design an incomplete disturbance compensation variable speed approach sliding mode controller.

[0013] In step S1, the mathematical model corresponds to the heavy vehicle pump-controlled electro-hydraulic servo steering system, which includes a steering axle and an electro-hydraulic pump unit (EPU); the steering axle is composed of a left power assist cylinder (8), a right power assist cylinder (12), a left tire (9), a right tire (11) and a trapezoidal steering mechanism (10);

[0014] The electrostatic pump unit (EPU) consists of a servo motor (1), a hydraulic pump (4), an oil tank, a first check valve (3), a second check valve (15), a first hydraulically controlled check valve (6), a second hydraulically controlled check valve (14), a first relief valve (7), and a second relief valve (13). The servo motor drives the hydraulic pump to supply oil to the symmetrically installed left and right power cylinders, thereby driving the trapezoidal steering mechanism to steer the left and right tires.

[0015] In step S1, for the pump-controlled electro-hydraulic steering system of heavy vehicles, a dynamic model of the trapezoidal steering mechanism and a hydraulic model of the pump-controlled dual-power cylinder are built, and the control and guidance model of the pump-controlled electro-hydraulic steering system is derived; specifically, for the mechanical motion part of the trapezoidal steering mechanism of the pump-controlled electro-hydraulic steering system, the dynamic model is expressed by the following formula:

[0016] Formula 1;

[0017] In the formula: α and β are the turning angles of the left wheel and the right wheel, respectively; J is the partial derivative of the right wheel's steering angle with respect to the left wheel; L and J R These are the equivalent rotational inertia of the left and right wheels, respectively; C L and C R These are the equivalent damping coefficients for the left and right wheels, respectively; T L and T Rθ3' and θ3' are the steering assist torques of the left and right wheels, respectively; n is the distance between the steering cylinder actuation point and the kingpin; θ3' and θ3' are the angles between the forces of the left and right steering assist cylinders and the velocities of their actuation points, respectively; F L and F R These are the output forces of the left and right power booster cylinders, respectively.

[0018] For the hydraulic component of the pump-controlled electro-hydraulic steering system, considering that the response frequency of the servo motor is much higher than that of the hydraulic system, the relationship between the servo motor control voltage and the rotational speed is as follows:

[0019] n m =k m Formula 2;

[0020] Where: n m k is the motor speed. m U is the motor gain coefficient; u is the servo motor control voltage;

[0021] For the hydraulic component of the pump-controlled electro-hydraulic steering system, considering internal and external leakage of the bidirectional fixed displacement pump, its flow balance equation is established as follows:

[0022]

[0023] In the formula: q1 and q2 represent the inlet and outlet flow rates of the dual power steering cylinder, respectively; D p k is the displacement of the hydraulic pump. m C is the motor gain coefficient; i and C e These represent the internal and external leakage coefficients of the hydraulic pump, respectively; p1 and p2 represent the inlet and outlet pressures of the dual power steering cylinders, respectively.

[0024] For the hydraulic part of the pump-controlled electro-hydraulic steering system, considering the internal and external leakage of the dual power steering cylinders and the oil compression in the cavities on both sides, the flow balance equation for the dual power steering cylinders is established:

[0025]

[0026] In the formula: V t The total volume of the dual power steering cylinders; A and a represent the effective areas of the large and small chambers of the power steering cylinder, respectively; x L and x R The left and right piston displacements are x, respectively. L =an,x R =βn; and The piston speeds of the left and right booster cylinders are respectively... n is the distance between the steering cylinder actuation point and the kingpin; C ip and C ep These represent the internal and external leakage coefficients of the power steering cylinder, respectively; β e The effective elastic modulus of hydraulic oil;

[0027] For the hydraulic component of the pump-controlled electro-hydraulic steering system, the model needs to be simplified to facilitate the design of a nonlinear controller. The model is simplified and specifically defined as follows: V1 = V t / 2+Anα+anβ,V2=V t / 2-anα-Anβ,T T =T L α β +T R J T =J L α β 2 +J R , g1=ancosθ3+Ancosθ3′α β C2 = 2C ip +C i g2=Ancosθ3+ancosθ′α β C1 = 2C ip +2C ep +C e +C i ,

[0028] Using the right wheel steering angle β as the feedback signal and the servo motor control voltage u as the control input, define the state variables of the pump-controlled electro-hydraulic servo steering system. The state equations of the pump-controlled electro-hydraulic servo steering system are derived, and the mathematical model of the pump-controlled electro-hydraulic servo steering system is established:

[0029]

[0030] The variables in the formula are represented as follows:

[0031] g=g1β e D p k m n / (J T V1)+g2β e D p k m n / (J T V2) Formula 6;

[0032] f=-(Anα β +an)g1x2 / (J T V1)-(anα β +An)g2x2 / (J T V2) Formula 7

[0033]

[0034] d=Δgu+Δf+d uFormula 9;

[0035] In the formula, β, and These represent the right wheel steering angle, steering speed, and steering acceleration of the electro-hydraulic servo steering system, respectively; x represents the state variable of the pump-controlled electro-hydraulic steering system. and Let x1, x2, and x3 represent the derivatives, respectively; f and g are the dynamic and control gain coefficients for the pump-controlled electro-hydraulic steering system modeling, respectively; Δg and Δf are the control gain deviation and system dynamic deviation caused by parameter uncertainties, respectively; d u d represents the disturbance that includes unmodeled system dynamics and unknown steering resistance torque; d represents the lumped disturbance that includes system parameter perturbations, unmodeled dynamics, and unknown steering resistance torque.

[0036] Step S2 involves designing the variable-speed sliding mode reaching law.

[0037] First, define the angle error of the pump-controlled electro-hydraulic steering system:

[0038]

[0039] In the formula, e1, e2 and e3 are the steering angle error, steering speed error and steering acceleration error of the pump steering system, respectively;

[0040] Further obtain the state equation for the steering error:

[0041]

[0042] Subsequently, the sliding surface was designed based on the rotation angle error:

[0043]

[0044] In the formula, s is the sliding surface; λ is the bandwidth of the sliding surface;

[0045] By introducing the activation function of the associated sliding surface, a variable-speed sliding mode reaching law of the form:

[0046]

[0047] In the formula, s is the sliding surface, where s = e³ + 2λe² + λ 2 e1, e1 = x1 - x 1d , e1, e2, and e3 represent steering angle error, steering angular velocity error, and steering angular acceleration error, respectively; x 1d , and Let k1 be the steering angle, steering angular velocity, and steering angular acceleration of the desired steering signal, respectively; k2 be the coefficient of the exponential reaching term; k3 be the coefficient of the variable speed reaching term; ε1 be the power convergence coefficient for adjusting the variable speed sliding mode reaching law, 0 < ε1 < 1; sgn(s) be the sign function with respect to the sliding surface; sig(s) be the activation function for adjusting the rate of change of the variable speed sliding mode reaching law, expressed as:

[0048]

[0049] In the formula, τ1 is the shrinkage coefficient of the activation function;

[0050] The variable speed sliding mode approach law accelerates the approach speed when the state of the pump-controlled electro-hydraulic steering system is far from the sliding mode surface, and reduces the approach speed when the state of the system is close to the sliding mode surface, so as to reduce chattering and prevent system overshoot. The power term coefficient is adjusted by the activation function sig(s), and the larger the value of τ1, the faster the adjustment speed, so as to respond to rapid changes in disturbance more quickly.

[0051] The variable-speed sliding mode reaching law is robust to disturbance handling. The specific verification method is as follows:

[0052] Define the Lyapunov function V1 for the variable-speed approach to the Terminal sliding mode disturbance observer:

[0053] V1 = 0.5s 2 Formula 15;

[0054] Differentiate with respect to V1:

[0055]

[0056] Furthermore, the formula can be rewritten as:

[0057]

[0058]

[0059] The variable-speed sliding mode approach law asymptotically converges to the origin, then 1-ε1sig(1-|s|)=1-ε1′, since Therefore, the following inequality holds.

[0060] |s|≥D / k1 Formula 19;

[0061] at this time If this holds true, V1 will asymptotically converge to V1≤0.5(D / k1). 2 , making

[0062] |s|≤D / k1 (Formula 20)

[0063] Using the same method, the following inequalities are obtained;

[0064]

[0065] In the formula, ε1′ is the power term coefficient at the final moment of the variable-speed sliding mode reaching law, and ε1′ = ε1|sig(s)|;

[0066] It can be seen from the above formula that the corner error e1 is obtained by connecting s in series with two low-pass filters:

[0067]

[0068] In the formula, p is the Laplace operator, p = d / dt; λ is the control bandwidth of the sliding mode surface;

[0069] The convergence domain of the sliding mode surface and the corner error of the pump-controlled electro-hydraulic steering system is obtained by combining the above formula

[0070]

[0071]

[0072] In step S3, the specific steps for designing a variable-speed reaching Terminal sliding mode disturbance observer are as follows:

[0073] In step S3.1, introduce an auxiliary variable z and define the observer sliding mode surface s d :

[0074] s d = z - x3 Formula 25;

[0075] In step S3.2, the auxiliary variable of the variable-speed reaching Terminal sliding mode disturbance observer is designed as:

[0076]

[0077]

[0078] In the formula, k3, k4, and k5 are all design parameters of the disturbance observer; p and q are finite-time convergence factors, both positive odd numbers, and p < q; ε2 is the power convergence coefficient for adjusting the variable-speed reaching Terminal sliding mode disturbance observer, 0 < ε2 < 1; sgn(s d ) is the sign function with respect to the observer sliding mode surface; sig(s d ) is the activation function for adjusting the change rate of the variable-speed reaching Terminal sliding mode disturbance observer, which can be quickly adjusted when the system is subjected to rapidly changing disturbances to achieve accurate observation of the lumped disturbance of the system; τ2 is the contraction coefficient of the activation function;

[0079] The dynamic expression of the sliding mode surface of the variable-speed reaching Terminal sliding mode disturbance observer is:

[0080]

[0081] Step S3.3, design the variable speed approaching Terminal sliding mode interference observer as

[0082]

[0083] In the formula, For variable speed approaching Terminal sliding mode interference observer.

[0084] The aforementioned variable-speed approach terminal sliding mode disturbance observer can accurately observe the lumped disturbance of a pump-controlled electro-hydraulic steering system that changes rapidly and over a wide range, and can converge to the actual lumped disturbance of the system within a finite time. The specific proof method is as follows:

[0085] Define the observer Lyapunov function V2:

[0086]

[0087] Differentiating with respect to V2, we get:

[0088]

[0089] By choosing a sufficiently large k4, the following inequality always holds.

[0090]

[0091] The expression can then be rewritten as:

[0092]

[0093] Therefore, the sliding surface of the variable-speed approach terminal sliding mode disturbance observer can converge to the origin in a finite time, and the convergence time is:

[0094]

[0095] In the formula, t0 is the initial time, and T is the final approach time;

[0096] Combining the above formula, we get

[0097]

[0098] As can be seen from the equation, if the auxiliary variable s can converge to the equilibrium point within a certain time, then the approach error of the observer can also converge within the effective time. Due to unavoidable noise and time delays in actual measured signals, there may be deviations between the estimated and actual values ​​of lumped disturbances. Directly applying the disturbance observations for full compensation in the feedforward channel can lead to overcompensation and system instability. Therefore, this invention innovatively proposes an incomplete compensation method to improve the robustness and stability of the pump-controlled electro-hydraulic steering system.

[0099] The design of the incomplete disturbance compensation variable speed approach sliding mode controller in step S4 is as follows: The incomplete disturbance compensation variable speed approach sliding mode controller consists of three parts: an equivalent control law, a variable speed sliding mode approach law, and an incomplete disturbance compensator. The expression is:

[0100] u = u eq +u sw +u d Formula 36;

[0101] In the formula, u is the incomplete disturbance compensation variable speed approach sliding mode controller; u eq This is the equivalent control law; u sw For variable speed sliding mode reaching law; u d For incomplete disturbance compensators; their expressions are as follows:

[0102]

[0103]

[0104]

[0105] In the formula, κ is the disturbance compensation coefficient, 0≤κ≤1. Setting κ as a parameter will sacrifice some robustness. To fully utilize the robustness of disturbance compensation control, an activation function based on steering error is designed to adaptively adjust the disturbance compensation coefficient according to changes in steering angle error. Its specific expression is as follows:

[0106]

[0107] In the formula, κ0 is the initial disturbance compensation coefficient; τ3 is the contraction factor of the disturbance compensation coefficient;

[0108] The compensation coefficient κ of the incomplete disturbance compensation variable speed approach sliding mode controller gradually increases with the increase of steering error, thereby improving the robustness of the pump-controlled electro-hydraulic steering system; the larger the value of τ3, the faster the adjustment rate of the compensation coefficient κ, enabling the controller to respond to changes in disturbance in a timely manner.

[0109] The incomplete disturbance compensation variable speed approach sliding mode controller can control the system sliding surface and steering error within a smaller attraction domain. The proof method is as follows:

[0110] Substituting the expression into the equation, we get...

[0111]

[0112] The formula can be further rewritten as:

[0113]

[0114]

[0115] The variable-speed sliding mode approach law will asymptotically converge to the origin, therefore 1-ε1sig(1-|s|)=1-ε′1′, since

[0116] Therefore, the following inequality holds.

[0117] |s|≥(1-κ)D / k1 Formula 44;

[0118] at this time If true, V1 will asymptotically converge to V1≤0.5[(1-κ)D / k1] 2 , making

[0119] |s|≤(1-κ)D / k1 Formula 45;

[0120] Using the same method, we obtain the following inequality.

[0121]

[0122] In the formula, ε′1′ is the coefficient of the power term at the final moment of the variable speed sliding mode approach law, ε′1′=ε1|sig(s)|;

[0123] As can be seen from the equation, the rotation angle error e1 is obtained by connecting two low-pass filters in series with s:

[0124]

[0125] In the formula, p is the Laplace operator, p = d / dt; λ is the control bandwidth of the sliding surface;

[0126] The convergence region of the sliding surface and angular error of the pump-controlled electro-hydraulic steering system is obtained by combining the above formula.

[0127]

[0128]

[0129] The method to prove the stability of the entire closed-loop system is as follows:

[0130] Define the Lyapunov function for closed-loop control of the pump-controlled electro-hydraulic steering system:

[0131]

[0132] Differentiating the expression, we get:

[0133]

[0134]

[0135] In the formula, The formula can be further rewritten as: In the formula, Solving the differential equation yields

[0136]

[0137] In the formula, V(0) is the initial value of the Lyapunov function V.

[0138] The variable speed approach and disturbance compensation control method can improve the control accuracy of the input system and ensure system stability without requiring an accurate model and disturbance upper bound for the pump-controlled electro-hydraulic steering system.

[0139] The proof that the angular error of the pump-controlled electro-hydraulic steering system can converge within a finite time is as follows:

[0140]

[0141] As can be seen from the equation, by using an incomplete disturbance compensation variable speed approach sliding mode controller, the pump control steering error will be expressed as a function of speed. The exponential curve converges to the region of convergence in time T. r .

[0142] This invention discloses a variable speed approach and disturbance compensation control method for a pump-controlled electro-hydraulic steering system. The method designs a variable speed sliding mode approach law, which effectively improves the problems of insufficient dynamics, low accuracy, low damping, and easy overshoot in pump-controlled systems. Simultaneously, it innovates a novel variable speed approach terminal sliding mode disturbance observer to accurately estimate the lumped disturbance of the pump-controlled electro-hydraulic servo steering system, and proposes an adaptive incomplete compensation method for associated steering angle errors to perform feedforward compensation for the system's lumped disturbance. The disturbance compensation coefficient can automatically adjust according to error changes, simultaneously considering steering accuracy and steering stability. This control method achieves strong disturbance rejection and high-precision steering control without requiring an accurate model of the pump-controlled electro-hydraulic steering system or an upper bound on the disturbance, and has broad application prospects in heavy vehicles, engineering equipment, and other fields.

[0143] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0144] 1) This invention solves the problem of low stiffness, low damping, and strong nonlinearity in pump-controlled electro-hydraulic steering systems, making it difficult to simultaneously achieve high-precision and high-stability steering. A novel variable-speed sliding mode reaching law is designed, which converges rapidly when the system state is far from the sliding surface, improving system responsiveness; and slows down the convergence speed when the system state is close to the sliding surface, preventing over-adjustment and improving system stability. Furthermore, a novel reaching law switching function is constructed based on an activation function, which can smoothly switch between the two reaching rates of the variable-speed reaching law according to the system's motion state, further improving system stability. Finally, Lyapunov theory is used to prove the convergence domain and robustness to disturbances of the novel variable-speed sliding mode reaching law.

[0145] 2) This invention solves the problems of traditional observation methods failing to achieve accurate observation of rapidly changing disturbances and the instability caused by complete disturbance compensation. A novel Terminal sliding mode disturbance observer is constructed by introducing a variable-speed sliding mode reaching law. This observer can achieve finite-time convergence when the system is subjected to rapidly changing strong disturbances. Furthermore, a novel adaptive incomplete disturbance compensation control strategy is proposed, combining fixed disturbance compensation and adaptive disturbance compensation. This strategy adjusts the disturbance compensation gain according to the tracking error of the steering system, further compensating for the shortcomings of the fixed compensation gain, and ultimately achieving a balance between high-precision steering and high-stability steering. Attached Figure Description

[0146] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0147] Appendix Figure 1 This is a schematic diagram illustrating the principle of the variable speed approach and disturbance compensation control method for a pump-controlled electro-hydraulic steering system.

[0148] Appendix Figure 2 This is a schematic diagram of the pump-controlled electro-hydraulic steering system.

[0149] Appendix Figure 3 This is a schematic diagram of the simulation results under the control of four different controllers. Detailed Implementation

[0150] like Figure 1 As shown, a variable speed approach and disturbance compensation control method for a pump-controlled electro-hydraulic steering system is disclosed. The method uses a variable speed approach terminal sliding mode disturbance observer to accurately estimate the lumped disturbance of the pump-controlled electro-hydraulic servo steering system, and uses an adaptive incomplete compensation method for associated steering angle error to perform feedforward compensation for the lumped disturbance of the system. The coefficient of the disturbance compensation is automatically adjusted according to the error change, taking into account both steering accuracy and steering stability. The method includes the following steps.

[0151] Step S1: Combine the differential equations of the mechanical and hydraulic parts of the pump-controlled electro-hydraulic steering system, and use the input-output linearization method to establish the mathematical model of the pump-controlled electro-hydraulic steering system.

[0152] Step S2: Introduce a variable power function 1-ε1sig(s) to construct a variable speed sliding mode reaching law. Where s is the sliding surface, k1 is the coefficient of the exponential reaching term, k2 is the coefficient of the variable speed reaching term, ε1 is the power convergence coefficient for adjusting the variable speed sliding mode reaching law, 0 < ε1 < 1; sgn(s) is the sign function with respect to the sliding surface; sig(s) is the activation function for adjusting the rate of change of the variable speed sliding mode reaching law.

[0153] Step S3: Construct the sliding mode surface of the sliding mode disturbance observer, introduce the variable speed sliding mode approaching law associated with the sliding mode surface, and design the variable speed approaching Terminal sliding mode disturbance observer by combining the mathematical model of the pump-controlled electro-hydraulic steering system.

[0154] Step S4: Combining the model equivalent control law, variable speed sliding mode approach law and variable speed approach Terminal sliding mode disturbance observer, an adaptive disturbance compensation function with associated turning angle error is introduced to design an incomplete disturbance compensation variable speed approach sliding mode controller.

[0155] like Figure 2 As shown, in step S1, the mathematical model corresponds to the pump-controlled electro-hydraulic servo steering system of heavy vehicles, which includes a steering axle and an electro-hydraulic pump unit (EPU); the steering axle is composed of a left power steering cylinder 8, a right power steering cylinder 12, a left tire 9, a right tire 11, and a trapezoidal steering mechanism 10.

[0156] The electro-hydraulic pump unit (EPU) consists of a servo motor 1, a hydraulic pump 4, an oil tank, a first check valve 3, a second check valve 15, a first hydraulically controlled check valve 6, a second hydraulically controlled check valve 14, a first relief valve 7, and a second relief valve 13. The servo motor drives the hydraulic pump to supply oil to the symmetrically installed left and right power steering cylinders, which in turn drives the trapezoidal steering mechanism to steer the left and right tires. The oil tank components indicated by 2, 5, and 10 in the figure belong to the same oil tank.

[0157] Figure 2 The proposed solution eliminates the need for a steering control valve, directly controlling wheel steering by altering the output flow and direction of the hydraulic pump. This avoids overflow and throttling losses, thus achieving energy savings. However, this method of steering system control without relying on energy dissipation suffers from low stiffness and low damping, making high-precision steering control challenging. Furthermore, this pump-controlled electro-hydraulic steering system uses a single pump to drive a pair of series-mounted power steering cylinders, exhibiting strong nonlinearity, strong coupling, and uncertainty, significantly increasing control difficulty. Therefore, this invention proposes a variable speed approach and disturbance compensation control method for pump-controlled electro-hydraulic steering systems to address these issues.

[0158] In step S1, for the pump-controlled electro-hydraulic steering system of heavy vehicles, a dynamic model of the trapezoidal steering mechanism and a hydraulic model of the pump-controlled dual-power cylinder are built, and the control and guidance model of the pump-controlled electro-hydraulic steering system is derived; specifically, for the mechanical motion part of the trapezoidal steering mechanism of the pump-controlled electro-hydraulic steering system, the dynamic model is expressed by the following formula:

[0159]

[0160] In the formula: α and β are the turning angles of the left wheel and the right wheel, respectively; J is the partial derivative of the right wheel's steering angle with respect to the left wheel; L and J R These are the equivalent rotational inertia of the left and right wheels, respectively; C L and C R These are the equivalent damping coefficients for the left and right wheels, respectively; T L and T R θ3' and θ3' are the steering assist torques of the left and right wheels, respectively; n is the distance between the steering cylinder actuation point and the kingpin; θ3' and θ3' are the angles between the forces of the left and right steering assist cylinders and the velocities of their actuation points, respectively; F L and F R These are the output forces of the left and right power booster cylinders, respectively.

[0161] For the hydraulic component of the pump-controlled electro-hydraulic steering system, considering that the response frequency of the servo motor is much higher than that of the hydraulic system, the relationship between the servo motor control voltage and the rotational speed is as follows:

[0162] n m =k m Formula 2;

[0163] Where: n m k is the motor speed. m U is the motor gain coefficient; u is the servo motor control voltage;

[0164] For the hydraulic component of the pump-controlled electro-hydraulic steering system, considering internal and external leakage of the bidirectional fixed displacement pump, its flow balance equation is established as follows:

[0165]

[0166] In the formula: q1 and q2 represent the inlet and outlet flow rates of the dual power steering cylinder, respectively; D p k is the displacement of the hydraulic pump. m C is the motor gain coefficient; i and C e These represent the internal and external leakage coefficients of the hydraulic pump, respectively; p1 and p2 represent the inlet and outlet pressures of the dual power steering cylinders, respectively.

[0167] For the hydraulic part of the pump-controlled electro-hydraulic steering system, considering the internal and external leakage of the dual power steering cylinders and the oil compression in the cavities on both sides, the flow balance equation for the dual power steering cylinders is established:

[0168]

[0169] In the formula: V t The total volume of the dual power steering cylinders; A and a represent the effective areas of the large and small chambers of the power steering cylinder, respectively; x L and x R The left and right piston displacements are x, respectively. L =an,x R =βn; and The piston speeds of the left and right booster cylinders are respectively... n is the distance between the steering cylinder actuation point and the kingpin; C ip and C ep These represent the internal and external leakage coefficients of the power steering cylinder, respectively; β e The effective elastic modulus of hydraulic oil;

[0170] For the hydraulic component of the pump-controlled electro-hydraulic steering system, the model needs to be simplified to facilitate the design of a nonlinear controller. The model is simplified and specifically defined as follows: V1 = V t / 2+Anα+anβ,V2=V t / 2-anα-Anβ,T T =T L α β +T R J T =J L α β 2 +J R , g1=ancosθ3+Ancosθ3′α β C2 = 2C ip +C i g2=Ancosθ3+ancosθ′α β C1 = 2C ip +2C ep +C e +C i ,

[0171] Using the right wheel steering angle β as the feedback signal and the servo motor control voltage u as the control input, define the state variables of the pump-controlled electro-hydraulic servo steering system. The state equations of the pump-controlled electro-hydraulic servo steering system are derived, and the mathematical model of the pump-controlled electro-hydraulic servo steering system is established:

[0172]

[0173] The variables in the formula are represented as follows:

[0174] g=g1β e D p k m n / (J T V1)+g2β e D p k m n / (J T V2) Formula 6;

[0175] f=-(Anα β +an)g1x2 / (J T V1)-(anα β +An)g2x2 / (J T V2) Formula 7

[0176]

[0177] d=Δgu+Δf+d u Formula 9;

[0178] In the formula, β, and These represent the right wheel steering angle, steering speed, and steering acceleration of the electro-hydraulic servo steering system, respectively; x represents the state variable of the pump-controlled electro-hydraulic steering system. and Let x1, x2, and x3 represent the derivatives, respectively; f and g are the dynamic and control gain coefficients for the pump-controlled electro-hydraulic steering system modeling, respectively; Δg and Δf are the control gain deviation and system dynamic deviation caused by parameter uncertainties, respectively; d u d represents the disturbance that includes unmodeled system dynamics and unknown steering resistance torque; d represents the lumped disturbance that includes system parameter perturbations, unmodeled dynamics, and unknown steering resistance torque.

[0179] Step S2 involves designing the variable-speed sliding mode reaching law.

[0180] First, define the angle error of the pump-controlled electro-hydraulic steering system:

[0181]

[0182] In the formula, e1, e2 and e3 are the steering angle error, steering speed error and steering acceleration error of the pump steering system, respectively;

[0183] Further obtain the state equation for the steering error:

[0184]

[0185] Subsequently, the sliding surface was designed based on the rotation angle error:

[0186]

[0187] In the formula, s is the sliding surface; λ is the bandwidth of the sliding surface;

[0188] By introducing the activation function of the associated sliding surface, a variable-speed sliding mode reaching law of the form:

[0189]

[0190] In the formula, s is the sliding surface, where s = e³ + 2λe² + λ 2 e1, e1 = x1 - x 1d , e1, e2, and e3 represent steering angle error, steering angular velocity error, and steering angular acceleration error, respectively; x 1d , and Let k1 be the steering angle, steering angular velocity, and steering angular acceleration of the desired steering signal, respectively; k2 be the coefficient of the exponential reaching term; k3 be the coefficient of the variable speed reaching term; ε1 be the power convergence coefficient for adjusting the variable speed sliding mode reaching law, 0 < ε1 < 1; sgn(s) be the sign function with respect to the sliding surface; sig(s) be the activation function for adjusting the rate of change of the variable speed sliding mode reaching law, expressed as:

[0191]

[0192] In the formula, τ1 is the shrinkage coefficient of the activation function;

[0193] The variable speed sliding mode approach law accelerates the approach speed when the state of the pump-controlled electro-hydraulic steering system is far from the sliding mode surface, and reduces the approach speed when the state of the system is close to the sliding mode surface, so as to reduce chattering and prevent system overshoot. The power term coefficient is adjusted by the activation function sig(s), and the larger the value of τ1, the faster the adjustment speed, so as to respond to rapid changes in disturbance more quickly.

[0194] The variable-speed sliding mode reaching law is robust to disturbance handling. The specific verification method is as follows:

[0195] Define the Lyapunov function V1 for the variable-speed approach to the Terminal sliding mode disturbance observer:

[0196] V1 = 0.5s 2 Formula 15;

[0197] Differentiate with respect to V1:

[0198]

[0199] Furthermore, the formula can be rewritten as:

[0200]

[0201]

[0202] The variable-speed sliding mode approach law asymptotically converges to the origin, then 1-ε1sig(1-|s|)=1-ε′1′, since Therefore, the following inequality holds.

[0203] |s|≥D / k1 Formula 19;

[0204] at this time If this holds true, V1 will asymptotically converge to V1≤0.5(D / k1). 2 , making

[0205] |s|≤D / k1 (Formula 20)

[0206] Using the same method, the following inequalities are obtained;

[0207]

[0208] In the formula, ε′1′ is the coefficient of the power term at the final moment of the variable speed sliding mode approach law, ε′1′=ε1|sig(s)|;

[0209] From the above equation, we can see that the rotation angle error e1 is obtained by connecting two low-pass filters in series with s:

[0210]

[0211] In the formula, p is the Laplace operator, p = d / dt; λ is the control bandwidth of the sliding surface;

[0212] The convergence region of the sliding surface and angular error of the pump-controlled electro-hydraulic steering system is obtained by combining the above formula.

[0213]

[0214]

[0215] In step S3, the specific steps for designing the variable-speed approach terminal sliding mode disturbance observer are as follows:

[0216] Step S3.1: Introduce auxiliary variable z and define the observer sliding surface s. d :

[0217] s d =z-x3 Formula 25;

[0218] Step S3.2, design the auxiliary variables of the variable speed approaching Terminal sliding mode disturbance observer as follows:

[0219]

[0220]

[0221] where k3, k4, and k5 are all design parameters of the disturbance observer; p and q are finite-time convergence factors, both being positive odd numbers, and p < q; ε2 is the power convergence coefficient for adjusting the variable-speed reaching Terminal sliding-mode disturbance observer, 0 < ε2 < 1; sgn(s d ) is the sign function with respect to the observer sliding surface; sig(s d ) is the activation function for adjusting the change rate of the variable-speed reaching Terminal sliding-mode disturbance observer, which can be quickly adjusted when the system is subjected to rapidly changing disturbances to achieve accurate observation of the lumped disturbance of the system; τ2 is the contraction coefficient of the activation function;

[0222] The dynamic of the sliding surface of the variable-speed reaching Terminal sliding-mode disturbance observer is expressed as:

[0223]

[0224] Step S3.3, design the variable-speed reaching Terminal sliding-mode disturbance observer as

[0225]

[0226] where is the variable-speed reaching Terminal sliding-mode disturbance observer.

[0227] The variable-speed reaching Terminal sliding-mode disturbance observer can accurately observe the lumped disturbance of the pump-controlled electro-hydraulic steering system with rapid and large-range changes, and can converge to the actual lumped disturbance of the system within a finite time. The specific proof method is as follows:

[0228] Define the observer Lyapunov function V2:

[0229]

[0230] Take the derivative of V2 to get:

[0231]

[0232] By selecting a sufficiently large k4, the following inequality can always hold

[0233]

[0234] Then the equation can be rewritten as:

[0235]

[0236] Therefore, the sliding surface of the variable-speed approach terminal sliding mode disturbance observer can converge to the origin in a finite time, and the convergence time is:

[0237]

[0238] In the formula, t0 is the initial time, and T is the final approach time;

[0239] Combining the above formula, we get

[0240]

[0241] As can be seen from the equation, if the auxiliary variable s can converge to the equilibrium point within a certain time, then the approach error of the observer can also converge within the effective time. Due to unavoidable noise and time delays in actual measured signals, there may be deviations between the estimated and actual values ​​of lumped disturbances. Directly applying the disturbance observations for full compensation in the feedforward channel can lead to overcompensation and system instability. Therefore, this invention innovatively proposes an incomplete compensation method to improve the robustness and stability of the pump-controlled electro-hydraulic steering system.

[0242] The design of the incomplete disturbance compensation variable speed approach sliding mode controller in step S4 is as follows: The incomplete disturbance compensation variable speed approach sliding mode controller consists of three parts: an equivalent control law, a variable speed sliding mode approach law, and an incomplete disturbance compensator. The expression is:

[0243] u = u eq +u sw +u d Formula 36;

[0244] In the formula, u is the incomplete disturbance compensation variable speed approach sliding mode controller; u eq This is the equivalent control law; u sw For variable speed sliding mode reaching law; u d For incomplete disturbance compensators; their expressions are as follows:

[0245]

[0246]

[0247]

[0248] In the formula, κ is the disturbance compensation coefficient, 0≤κ≤1. Setting κ as a parameter will sacrifice some robustness. To fully utilize the robustness of disturbance compensation control, an activation function based on steering error is designed to adaptively adjust the disturbance compensation coefficient according to changes in steering angle error. Its specific expression is as follows:

[0249]

[0250] In the formula, κ0 is the initial disturbance compensation coefficient; τ3 is the contraction factor of the disturbance compensation coefficient;

[0251] The compensation coefficient κ of the incomplete disturbance compensation variable speed approach sliding mode controller gradually increases with the increase of steering error, thereby improving the robustness of the pump-controlled electro-hydraulic steering system; the larger the value of τ3, the faster the adjustment rate of the compensation coefficient κ, enabling the controller to respond to changes in disturbance in a timely manner.

[0252] The incomplete disturbance compensation variable speed approach sliding mode controller can control the system sliding surface and steering error within a smaller attraction domain. The proof method is as follows:

[0253] Substituting the expression into the equation, we get...

[0254]

[0255] The formula can be further rewritten as:

[0256]

[0257]

[0258] The variable-speed sliding mode approach law will asymptotically converge to the origin, therefore 1-ε1sig(1-|s|)=1-ε′1′, since Therefore, the following inequality holds.

[0259] |s|≥(1-κ)D / k1 Formula 44;

[0260] at this time If true, V1 will asymptotically converge to V1≤0.5[(1-κ)D / k1] 2 , making

[0261] |s|≤(1-κ)D / k1 Formula 45;

[0262] Using the same method, we obtain the following inequalities.

[0263]

[0264] In the formula, ε′1′ is the coefficient of the power term at the final moment of the variable speed sliding mode approach law, ε′1′=ε1|sig(s)|;

[0265] As can be seen from the equation, the rotation angle error e1 is obtained by connecting two low-pass filters in series with s:

[0266]

[0267] In the formula, p is the Laplace operator, p = d / dt; λ is the control bandwidth of the sliding surface;

[0268] The convergence region of the sliding surface and angular error of the pump-controlled electro-hydraulic steering system is obtained by combining the above formula.

[0269]

[0270]

[0271] The method to prove the stability of the entire closed-loop system is as follows:

[0272] In step S5, the specific method for designing the Lyapunov stability analysis function of the closed-loop system is to define the Lyapunov function for the closed-loop control of the pump-controlled electro-hydraulic steering system:

[0273]

[0274] Differentiating the expression, we get:

[0275]

[0276]

[0277] In the formula, The formula can be further rewritten as: In the formula, Solving the differential equation yields

[0278]

[0279] In the formula, V(0) is the initial value of the Lyapunov function V.

[0280] The variable speed approach and disturbance compensation control method can improve the control accuracy of the input system and ensure system stability without requiring an accurate model and disturbance upper bound for the pump-controlled electro-hydraulic steering system.

[0281] The proof that the angular error of the pump-controlled electro-hydraulic steering system can converge within a finite time is as follows:

[0282]

[0283] As can be seen from the equation, by using an incomplete disturbance compensation variable speed approach sliding mode controller, the pump control steering error will be expressed as a function of speed. The exponential curve converges to the region of convergence in time T. r .

[0284] In summary, the speed approach and disturbance compensation control method for a pump-controlled electro-hydraulic steering system of the present invention can significantly improve the control accuracy of the system and ensure system stability without requiring an accurate model and upper limit of disturbance for the pump-controlled electro-hydraulic steering system.

[0285] Example:

[0286] To evaluate the controller's performance, this example is based on Figure 2 The schematic diagram of the pump-controlled electro-hydraulic steering system is shown. A model of the heavy-duty vehicle pump-controlled electro-hydraulic servo steering system is built in MATLAB for simulation verification.

[0287] The detailed parameters of the pump-controlled electro-hydraulic steering system are shown in Table 1 below.

[0288] Table 1 Parameters of Pump-Controlled Electro-hydraulic Steering System

[0289]

[0290] The given steering target command for the pump-controlled electro-hydraulic steering system is: x 1d =20sin(0.3πt)(°), and the following four controllers are used for comparison:

[0291] C1: PI controller

[0292]

[0293] Take the control parameter as k p =300; k i =30.

[0294] C2: Fast power-law approaching sliding mode controller without disturbance compensation, the approaching law is:

[0295]

[0296] The control parameters are set as follows: λ = 35; k1 = 20, k2 = 50, ε1 = 0.5.

[0297] C3: Variable speed reaching law sliding mode controller without disturbance compensation, the reaching law is:

[0298]

[0299] The control parameters are set as follows: λ = 35; k1 = 20, k2 = 50, ε1 = 0.5, τ1 = 2.

[0300] C4: The proposed variable speed reaching law sliding mode controller with incomplete disturbance compensation is shown in the equation:

[0301] The control parameters are set as λ = 35; k1 = 20, k2 = 50, ε1 = 0.5, τ1 = 2; the observer parameters are k3 = 9, k4 = 0.5, k5 = 0.5, ε2 = 0.5, τ2 = 1, p = 5, q = 9; and the incomplete disturbance compensator parameters are κ0 = 0.5, τ2 = 1.

[0302] Under the control of four controllers, the tracking of steering target commands, steering error, and lumped disturbance estimation values ​​are as follows: Figure 3As shown in the figure, the tracking curve and steering error curve reveal that C1 has the largest error. This is mainly because C1, being a PI controller, can only eliminate errors through error correction and cannot compensate for disturbances in the pump-controlled electro-hydraulic steering system, making it difficult to achieve high-precision steering control under disturbances. Compared to C1, C2's steering error is reduced, primarily due to C2's ability to compensate for part of the modeling dynamics and to suppress unknown disturbances to a certain extent through a fast power-law approach, significantly reducing the steering angle error. Compared to C2, the C3 controller employs the variable-speed approach law proposed in this invention to further suppress disturbances, thereby achieving higher control accuracy. Finally, C4 achieves the best steering control accuracy, with the error consistently controlled within 0.5° even under high noise disturbances, demonstrating strong anti-disturbance capability and control performance.

[0303] Simulation results further demonstrate that the variable speed approach and disturbance compensation control method of the pump-controlled electro-hydraulic steering system of the present invention does not require an accurate model of the pump-controlled electro-hydraulic steering system and an upper limit of disturbance, thus achieving high-precision steering control.

[0304] The patent is not limited to the above-described preferred embodiments. Anyone can derive other forms of speed-approaching and disturbance compensation control methods for electro-hydraulic steering or pump-controlled electro-hydraulic systems based on the teachings of this patent. All changes and modifications made within the scope of the patent application of this invention shall fall within the scope of this patent.

Claims

1. A method for speed change approach and disturbance compensation control of a pump-controlled electro-hydraulic steering system, characterized in that: The method uses a variable speed approach terminal sliding mode disturbance observer to accurately estimate the lumped disturbance of the pump-controlled electro-hydraulic servo steering system, and uses an adaptive incomplete compensation method for associated steering angle error to feedforward compensate the lumped disturbance of the system. The coefficient of the disturbance compensation is automatically adjusted according to the error change, taking into account both steering accuracy and steering stability. The method includes the following steps. Step S1: Combine the differential equations of the mechanical and hydraulic parts of the pump-controlled electro-hydraulic steering system, and use the input-output linearization method to establish the mathematical model of the pump-controlled electro-hydraulic steering system. Step S2: Introduce a variable power function 1-ε1sig(s) to construct a variable speed sliding mode reaching law. Where s is the sliding surface, k1 is the coefficient of the exponential reaching term, k2 is the coefficient of the variable speed reaching term, ε1 is the power convergence coefficient for adjusting the variable speed sliding mode reaching law, 0 < ε1 < 1; sgn(s) is the sign function with respect to the sliding surface; sig(s) is the activation function for adjusting the rate of change of the variable speed sliding mode reaching law. Step S3: Construct the sliding mode surface of the sliding mode disturbance observer, introduce the variable speed sliding mode approaching law associated with the sliding mode surface, and design the variable speed approaching Terminal sliding mode disturbance observer by combining the mathematical model of the pump-controlled electro-hydraulic steering system. Step S4: Combining the model equivalent control law, variable speed sliding mode approach law and variable speed approach Terminal sliding mode disturbance observer, an adaptive disturbance compensation function with associated turning angle error is introduced to design an incomplete disturbance compensation variable speed approach sliding mode controller. In step S1, the mathematical model corresponds to the heavy vehicle pump-controlled electro-hydraulic servo steering system, which includes a steering axle and an electro-hydraulic pump unit (EPU); the steering axle is composed of a left power assist cylinder (8), a right power assist cylinder (12), a left tire (9), a right tire (11) and a trapezoidal steering mechanism (10); The electrostatic pump unit (EPU) consists of a servo motor (1), a hydraulic pump (4), an oil tank, a first check valve (3), a second check valve (15), a first hydraulically controlled check valve (6), a second hydraulically controlled check valve (14), a first relief valve (7), and a second relief valve (13). The servo motor drives the hydraulic pump to supply oil to the symmetrically installed left and right power cylinders, thereby driving the trapezoidal steering mechanism to steer the left and right tires. In step S1, for the pump-controlled electro-hydraulic steering system of heavy vehicles, a dynamic model of the trapezoidal steering mechanism and a hydraulic model of the pump-controlled dual-power cylinder are built, and the control and guidance model of the pump-controlled electro-hydraulic steering system is derived; specifically, for the mechanical motion part of the trapezoidal steering mechanism of the pump-controlled electro-hydraulic steering system, the dynamic model is expressed by the following formula: In the formula: α and β are the turning angles of the left wheel and the right wheel, respectively; J is the partial derivative of the right wheel's steering angle with respect to the left wheel; L and J R These are the equivalent rotational inertia of the left and right wheels, respectively; C L and C R These are the equivalent damping coefficients for the left and right wheels, respectively; T L and T R These are the steering assist torques for the left and right wheels, respectively. n is the distance between the steering cylinder's actuation point and the kingpin; θ3' and θ3 are the angles between the forces acting on the left and right power steering cylinders and the velocities at their points of action, respectively; F L and F R These are the output forces of the left and right power booster cylinders, respectively. For the hydraulic component of the pump-controlled electro-hydraulic steering system, considering that the response frequency of the servo motor is much higher than that of the hydraulic system, the relationship between the servo motor control voltage and the rotational speed is as follows: n m =k m Formula 2; Where: n m k is the motor speed. m U is the motor gain coefficient; u is the servo motor control voltage; For the hydraulic component of the pump-controlled electro-hydraulic steering system, considering internal and external leakage of the bidirectional fixed displacement pump, its flow balance equation is established as follows: In the formula: q1 and q2 represent the inlet and outlet flow rates of the dual power steering cylinder, respectively; D p k is the displacement of the hydraulic pump. m C is the motor gain coefficient; i and C e These represent the internal and external leakage coefficients of the hydraulic pump, respectively; p1 and p2 represent the inlet and outlet pressures of the dual power steering cylinders, respectively. For the hydraulic part of the pump-controlled electro-hydraulic steering system, considering the internal and external leakage of the dual power steering cylinders and the oil compression in the cavities on both sides, the flow balance equation for the dual power steering cylinders is established: In the formula: V t The total volume of the dual power steering cylinders; A and a represent the effective areas of the large and small chambers of the power steering cylinder, respectively; x L and x R The left and right piston displacements are x, respectively. L =an,x R =βn; and The piston speeds of the left and right booster cylinders are respectively... n is the distance between the steering cylinder actuation point and the kingpin; C ip and C ep These represent the internal and external leakage coefficients of the power steering cylinder, respectively; β e The effective elastic modulus of hydraulic oil; For the hydraulic component of the pump-controlled electro-hydraulic steering system, the model needs to be simplified to facilitate the design of a nonlinear controller. The model is simplified and specifically defined as follows: V1 = V t / 2+Anα+anβ,V2=V t / 2-anα-Anβ,T T =T L α β +T R J T =J L α β 2 +J R , g1=ancosθ3+Ancosθ′3α β C2 = 2C ip +C i , g2=An cosθ3+an cosθ′α β C1 = 2C ip +2C ep +C e +C i , Using the right wheel steering angle β as the feedback signal and the servo motor control voltage u as the control input, define the state variables of the pump-controlled electro-hydraulic servo steering system. The state equations of the pump-controlled electro-hydraulic servo steering system are derived, and the mathematical model of the pump-controlled electro-hydraulic servo steering system is established: The variables in the formula are represented as follows: g = g1β e D p k m n / (J T V1) + g2β e D p k m n / (J T V2); Formula 6 f = -(Anα β + an)g1x2 / (J T V1) - (anα β + An)g2x2 / (J T V2) Equation 7 d = Δgu + Δf + d u Formula 9; In the formula, β, and These represent the right wheel steering angle, steering speed, and steering acceleration of the electro-hydraulic servo steering system, respectively; x represents the state variable of the pump-controlled electro-hydraulic steering system. and Let x1, x2, and x3 represent the derivatives, respectively; f and g are the dynamic and control gain coefficients for the pump-controlled electro-hydraulic steering system modeling, respectively; Δg and Δf are the control gain deviation and system dynamic deviation caused by parameter uncertainties, respectively; d u d represents the disturbance that includes unmodeled system dynamics and unknown steering resistance torque; d represents the lumped disturbance that includes system parameter perturbations, unmodeled dynamics, and unknown steering resistance torque. Step S2 involves designing the variable-speed sliding mode reaching law. First, define the angle error of the pump-controlled electro-hydraulic steering system: In the formula, e1, e2 and e3 are the steering angle error, steering speed error and steering acceleration error of the pump steering system, respectively; Further obtain the state equation for the steering error: Subsequently, the sliding surface was designed based on the rotation angle error: In the formula, s is the sliding surface; λ is the bandwidth of the sliding surface; By introducing the activation function of the associated sliding surface, a variable-speed sliding mode reaching law of the form: In the formula, s is the sliding surface, where s = e³ + 2λe² + λ 2 e1, e1 = x1 - x 1d , e1, e2, and e3 represent steering angle error, steering angular velocity error, and steering angular acceleration error, respectively; x 1d , and Let k1 be the steering angle, steering angular velocity, and steering angular acceleration of the desired steering signal, respectively; k2 be the coefficient of the exponential reaching term; k3 be the coefficient of the variable speed reaching term; ε1 be the power convergence coefficient for adjusting the variable speed sliding mode reaching law, 0 < ε1 < 1; sgn(s) be the sign function with respect to the sliding surface; sig(s) be the activation function for adjusting the rate of change of the variable speed sliding mode reaching law, expressed as: In the formula, τ1 is the shrinkage coefficient of the activation function; The variable speed sliding mode approach law accelerates the approach speed when the state of the pump-controlled electro-hydraulic steering system is far from the sliding mode surface, and reduces the approach speed when the state of the system is close to the sliding mode surface, so as to reduce chattering and prevent system overshoot. The power term coefficient is adjusted by the activation function sig(s), and the larger the value of τ1, the faster the adjustment speed, so as to respond to rapid changes in disturbance more quickly.

2. The method for speed change approach and disturbance compensation control of a pump-controlled electro-hydraulic steering system according to claim 1, characterized in that: The variable-speed sliding mode reaching law is robust to disturbance handling. The specific verification method is as follows: Define the Lyapunov function V1 of the variable-speed reaching Terminal sliding mode disturbance observer: V1 = 0.5 s 2 Formula 15; Differentiate with respect to V1: Furthermore, the formula can be rewritten as: The variable-speed sliding mode approach law asymptotically converges to the origin, then 1-ε1sig(1-|s|)=1-ε1′, since Therefore, the following inequality holds. |s|≥D / k1 Formula 19; at this time If this holds true, V1 will asymptotically converge to V1≤0.5(D / k1). 2 , making |s|≤D / k1 (Formula 20) Using the same method, the following inequalities are obtained; In the formula, ε′1 is the coefficient of the power term at the final moment of the variable speed sliding mode approach law, and ε′1=ε1|sig(s)|; From the above equation, we can see that the rotation angle error e1 is obtained by connecting two low-pass filters in series with s: In the formula, p is the Laplace operator, p = d / dt; λ is the control bandwidth of the sliding surface; The convergence region of the sliding surface and angular error of the pump-controlled electro-hydraulic steering system is obtained by combining the above formula.

3. The method for speed change approach and disturbance compensation control of a pump-controlled electro-hydraulic steering system according to claim 1, characterized in that: In step S3, the specific steps for designing the variable-speed approach terminal sliding mode disturbance observer are as follows: Step S3.1: Introduce auxiliary variable z and define the observer sliding surface s. d : s d =z-x3 Formula 25; Step S3.2, design the auxiliary variables of the variable speed approaching Terminal sliding mode disturbance observer as follows: In the formula, k3, k4, and k5 are design parameters of the disturbance observer; p and q are finite-time convergence factors, both of which are positive odd numbers, and p <q; ε2 is the power-order convergence coefficient of the variable-speed approach-Terminal sliding mode disturbance observer, 0 < ε2 < 1; sgn(s d ) is a sign function with respect to the sliding surface of the observer; sig(s d ) is the activation function that adjusts the rate of change of the variable speed sliding mode disturbance observer approaching the Terminal. It can be quickly adjusted when the system is subjected to rapidly changing disturbances, so as to achieve accurate observation of the lumped disturbances of the system; τ2 is the contraction coefficient of the activation function; The dynamic representation of the sliding surface of the variable-speed approach-Terminal sliding mode disturbance observer is as follows: Step S3.3, design the variable speed approaching Terminal sliding mode interference observer as In the formula, For variable speed approaching Terminal sliding mode interference observer.

4. The method for speed change approach and disturbance compensation control of a pump-controlled electro-hydraulic steering system according to claim 3, characterized in that: The aforementioned variable-speed approach terminal sliding mode disturbance observer can accurately observe the lumped disturbance of a pump-controlled electro-hydraulic steering system that changes rapidly and over a wide range, and can converge to the actual lumped disturbance of the system within a finite time. The specific proof method is as follows: Define the observer Lyapunov function V2: Differentiating with respect to V2, we get: By choosing a sufficiently large k4, the following inequality always holds. The expression can then be rewritten as: Therefore, the sliding surface of the variable-speed approach terminal sliding mode disturbance observer can converge to the origin in a finite time, and the convergence time is: In the formula, t0 is the initial time, and T is the final approach time; Combining the above formula, we get As can be seen from the equation, if the auxiliary variable s can converge to the equilibrium point within a certain time, then the approach error of the observer can also converge within the effective time.

5. The method for speed change approach and disturbance compensation control of a pump-controlled electro-hydraulic steering system according to claim 3, characterized in that: The design of the incomplete disturbance compensation variable speed approach sliding mode controller in step S4 is as follows: The incomplete disturbance compensation variable speed approach sliding mode controller consists of an equivalent control law, a variable speed sliding mode approach law, and an incomplete disturbance compensator, expressed as: u = u eq + u sw + u d Formula 36; In the formula, u is the incomplete disturbance compensation variable speed approach sliding mode controller; u eq This is the equivalent control law; u sw For variable speed sliding mode reaching law; u d For incomplete disturbance compensators; their expressions are as follows: In the formula, κ is the disturbance compensation coefficient, 0≤κ≤1. To fully utilize the robustness of disturbance compensation control, an activation function based on steering error is designed to adaptively adjust the disturbance compensation coefficient according to changes in steering angle error. Its specific expression is as follows: In the formula, κ0 is the initial disturbance compensation coefficient; τ3 is the contraction factor of the disturbance compensation coefficient; The compensation coefficient κ of the incomplete disturbance compensation variable speed approach sliding mode controller gradually increases with the increase of steering error, thereby improving the robustness of the pump-controlled electro-hydraulic steering system; the larger the value of τ3, the faster the adjustment rate of the compensation coefficient κ, enabling the controller to respond to changes in disturbance in a timely manner.

6. The method for speed change approach and disturbance compensation control of a pump-controlled electro-hydraulic steering system according to claim 5, characterized in that: The incomplete disturbance compensation variable speed approach sliding mode controller can control the system sliding surface and steering error within a smaller attraction domain. The proof method is as follows: Substituting the expression into the equation, we get... The formula can be further rewritten as: The variable-speed sliding mode approach law will asymptotically converge to the origin, therefore 1-ε1sig(1-|s|)=1-ε1′, since Therefore, the following inequality holds. |s|≥(1-κ)D / k1 Formula 44; at this time If true, V1 will asymptotically converge to V1≤0.5[(1-κ)D / k1] 2 , making |s|≤(1-κ)D / k1 Formula 45; Using the same method, we obtain the following inequality. In the formula, ε′1 is the coefficient of the power term at the final moment of the variable speed sliding mode approach law, and ε′1=ε1|sig(s)|; As can be seen from the equation, the rotation angle error e1 is obtained by connecting two low-pass filters in series with s: In the formula, p is the Laplace operator, p = d / dt; λ is the control bandwidth of the sliding surface; The convergence region of the sliding surface and angular error of the pump-controlled electro-hydraulic steering system is obtained by combining the above formula.

7. The method for speed change approach and disturbance compensation control of a pump-controlled electro-hydraulic steering system according to claim 5, characterized in that: The method to prove the stability of the entire closed-loop system is as follows: Define the Lyapunov function for closed-loop control of the pump-controlled electro-hydraulic steering system: Differentiating the expression, we get: In the formula, The formula can be further rewritten as: In the formula, Solving the differential equation yields In the formula, V(0) is the initial value of the Lyapunov function V.

8. The method for speed change approach and disturbance compensation control of a pump-controlled electro-hydraulic steering system according to claim 7, characterized in that: The variable speed approach and disturbance compensation control method can improve the control accuracy of the system and ensure system stability without requiring an accurate model and disturbance upper limit of the pump-controlled electro-hydraulic steering system. The proof that the angular error of the pump-controlled electro-hydraulic steering system can converge within a finite time is as follows: As can be seen from the equation, by using an incomplete disturbance compensation variable speed approach sliding mode controller, the pump control steering error will be expressed as a function of speed. The exponential curve converges to the region of convergence in time T. r .

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