Self-adaptive robust control method for energy-saving forklift pump control steering system

Abstract

The invention provides a self-adaptive robust control method of an energy-saving forklift pump control steering system, and belongs to the technical field of electro-hydraulic servo control. The self-adaptive robust control method comprises the steps that a mathematical model of the energy-saving forklift pump control steering system is established; based on the mathematical model of the pump control steering system, designing a nonlinear disturbance observer, estimating disturbance of the system and compensating the disturbance; and designing an adaptive robust controller based on the mathematical model of the pump control steering system and the nonlinear observer. Precise observation of steering resistance is realized through a nonlinear disturbance observer, uncertain parameters are dynamically tracked in combination with a parameter projection adaptive law, a composite control strategy of parameter adaptive and robust items is adopted, parameter changes are actively adapted, dependence on strong gain adjustment is not needed, system buffeting is reduced, and control flexibility and practicability are improved.

Description

Technical Field

[0001] This invention relates to the field of electro-hydraulic servo control technology, and more specifically to an adaptive robust control method for an energy-saving forklift pump-controlled steering system. Background Technology

[0002] Distributed pump control systems are gradually replacing energy-intensive centralized valve-controlled steering systems because they can eliminate energy losses. However, pump control systems have problems such as strong hydraulic nonlinearity, drastic load changes, and leakage, which lead to uncertainties and uncertain nonlinearities in parameters, making it difficult to guarantee position servo accuracy.

[0003] To address the above issues, Chinese invention patent CN 118545646A discloses a backstepping sliding mode control method for a heavy-duty forklift pump-controlled lifting system based on a nonlinear observer. Based on the backstepping method, it designs a nonlinear robust sliding mode position servo controller with disturbance compensation, using sliding mode terms to handle nonlinear uncertainties, thus solving the problem of difficult position servo precision control in heavy-duty forklifts. However, this control method, with its backstepping sliding mode controller, requires adjusting the sliding mode gain to balance disturbance suppression and system chattering, resulting in insufficient control flexibility. Furthermore, it does not centrally define uncertain parameters, only handling parameter uncertainties in a general way through sliding mode terms, failing to achieve quantitative estimation and dynamic tracking of parameters, and thus cannot meet the high-precision steering angle control requirements of energy-saving forklifts.

[0004] In addition, the mathematical models in the above patents do not take into account the leakage coefficient of hydraulic pumps, and their fit with the leakage characteristics of actual steering systems is insufficient, resulting in limited control accuracy. Summary of the Invention

[0005] The purpose of this invention is to provide an adaptive robust control method for an energy-saving forklift pump-controlled steering system, enabling the forklift pump-controlled steering system to achieve asymptotic tracking steady-state performance and realize efficient and high-precision steering control of the energy-saving forklift.

[0006] The present invention adopts the following technical solution: An adaptive robust control method for an energy-saving forklift pump-controlled steering system includes the following steps: Step 1: Establish a mathematical model of the energy-saving forklift pump-controlled steering system; Step 2: Based on the mathematical model of the pump-controlled steering system, design a nonlinear disturbance observer to estimate the system disturbance and compensate for it. Step 3: Based on the mathematical model and nonlinear observer of the pump-controlled steering system, design an adaptive robust controller, specifically including:

[0007] 3.1 First Design Error ; 3.2 The second design error , ; 3.3 The third design error ;in: , and For the defined state variables, For target turning signal, This is the gain coefficient. For virtual control laws; 3.4 Adaptive Law for Design Parameters: definition express The estimated value, This represents the estimation error. and Representing the maximum and minimum values ​​of an uncertain parameter, the projection of a discontinuous parameter is as follows: ; The adaptive law of parameters is expressed as: ; in, It is a diagonal matrix representing the adaptive gain; Represents an adaptive function; This represents parametric projection; by using an adaptive law that satisfies the projection rules, we will have: (1); where, for θ The matrix transpose of the estimation error.

[0008] Furthermore, step 1 above is detailed as follows: The mathematical model of the system is established based on the following assumptions: the hydraulic cylinder has only internal leakage and no external leakage; the load on the hydraulic cylinder is an inertial load and no elastic load; and the flow equation of the bidirectional fixed displacement pump is as follows. ; In the formula: This refers to the displacement of a bidirectional fixed displacement pump. The rotational angular velocity of the bidirectional fixed displacement pump, The leakage coefficient of the hydraulic pump; The flow continuity equation for a hydraulic cylinder is: ; In the formula: Q 1 and Q 2 represents the flow rate of the two chambers of the hydraulic cylinder, respectively; A 1 and A 2 represents the effective area of ​​the two chambers of the hydraulic cylinder piston, which, assuming they are the same, is set as 2. A p ; x L This represents piston displacement;C i The internal leakage coefficient of the hydraulic cylinder; P 1 and P 2 represents the pressure in the two chambers of the hydraulic cylinder; V 1 and V 2 represents the volume of the two chambers of the hydraulic cylinder; β e It is the bulk modulus of elasticity; Define load pressure as P L = P 1- P 2, and the load flow is defined as Q L = ( Q 1+ Q 2) / 2, then by combining the flow continuity equations of the hydraulic cylinders, we get: ; In the formula: V t This refers to the total volume of the two chambers of the hydraulic cylinder; According to Newton's second law, the equation for the balance between the output force and the load of a hydraulic cylinder is: ; In the formula: m This refers to the total mass of the piston and its load, which is then factored onto the piston. B The viscous damping coefficient of the piston and load; f d For items that are difficult to model accurately, including external disturbance forces and unmodeled frictional forces; Define state variables x0 = [x1, x2, x3] T =[x L , x(•) L , P L ] T From the above formulas, the state-space equations of the pump-controlled steering system can be obtained as follows: ; consider m , B , , C i , It is a parameter that causes uncertainty, defined The total leakage coefficient is defined as follows: the set of uncertain parameters is defined as follows: , , , , ; The state-space equations of the above pump-controlled steering system can be simplified to the following expression: .

[0009] Furthermore, step 2 above is detailed as follows: The formula for the linear disturbance observer is: ;

[0010] In the formula: For the nonlinear function to be designed, For the gain of the nonlinear observer, it should satisfy... ; The observation error of the nonlinear disturbance observer is defined as: ; Assuming disturbance The dynamic characteristics of the observer change slowly, that is: The dynamic equation of the observer error system is: ; choose , As a constant, design .

[0011] Furthermore, step 3 above specifically includes:

[0012] Furthermore, the output of the nonlinear interference observer is sent to the gain adjustment module to convert the observed interference into control of the corresponding input channel, specifically: Gain is ,have to , This is a compensation item.

[0013] Furthermore, in steps 3.1 and 3.2, when When it approaches 0, It will also inevitably tend towards 0, making The tendency toward 0 is equivalent to making Approaching 0, the design goal becomes designing a control law to make tending towards 0, The dynamic equation can be expressed as: In the formula The derivative of the virtual control input;

[0014] for Design virtual control law : In the formula , These are the nonlinear gain 1 of the stable feedback term and the nonlinear gain 1 of the robust term, respectively;

[0015] Selecting Lyapunov functions: Taking its derivative, we get: (2);

[0016] in For nonlinear error, in the formula: ; design The following conditions must be met for calming: (3); As a robust controller to manage various uncertainties in the system model, it is derived from equations (2) and (3): ; Let be an arbitrarily small positive number; if Precise tracking ,Right now System tracking error It will eventually move into a domain that can be manually defined over time.

[0017] Furthermore, in step 3.3, The dynamic equation can be expressed as: ; Design control law : ; for The differentiable part can be represented as: ; , These are the nonlinear gain 2 of the stable feedback term and the nonlinear gain 2 of the robust term, respectively;

[0018] in: ; ; ; In the formula, .

[0019] Furthermore, it also includes step 4, using Lyapunov stability theory to prove the stability of the adaptive robust controller, and obtaining the result that the system tracking error is asymptotically stable.

[0020] Furthermore, the specific proof process for step 4 above is as follows: Selecting Lyapunov functions: ;

[0021] right Differentiating, we get: ;

[0022] In the formula: ; design The following conditions must be met for calming: (5);

[0023] For any arbitrarily small positive number, we can derive from equations (4) and (5): (6); definition: ;

[0024] As long as the design , Greater than 0, satisfying If positive definite, then: ; From the above formula, we can see that Globally bounded, that is Since the signals in the closed-loop system are consistent and eventually bounded, the closed-loop system is stable.

[0025] Furthermore, if the system only has parameter uncertainties, the Lyapunov function is defined as follows: In the formula , is a diagonal matrix representing the adaptive gain of the parameters;

[0026] Differentiating it, we get: ;

[0027] From equations (1) and (6), we get:

[0028] .

[0029] It can be concluded that when the system only has parameter uncertainty, the controller can still achieve asymptotic tracking performance.

[0030] As can be seen from the above description of the present invention, compared with the prior art, the present invention has the following advantages: 1. This invention achieves accurate observation of steering resistance through a nonlinear disturbance observer, combines parameter projection adaptive law to dynamically track uncertain parameters, and adopts a composite control strategy of parameter adaptation and robust terms to actively adapt to parameter changes without relying on strong gain adjustment, thereby reducing system chattering and improving control flexibility and practicality.

[0031] 2. The parameter adaptive law of the present invention ensures that the estimated value is bounded through the projection rule, avoiding the contradiction of gain adjustment in sliding mode control. All signals of the closed-loop system are consistent and eventually bounded. Moreover, it can achieve asymptotic tracking when only parameter uncertainty exists, effectively reducing the impact of drastic changes in steering resistance, external interference and unmodeled dynamics.

[0032] 3. This invention addresses the steering resistance load characteristics of pump-controlled steering systems in energy-saving forklifts by establishing a mathematical model that includes the hydraulic pump leakage coefficient, defining the total leakage coefficient and the vector of lumped uncertain parameters, which better reflects the actual working characteristics of pump-controlled steering systems and improves the control accuracy of pump-controlled steering systems. Attached Figure Description

[0033] Figure 1 This is a schematic diagram of the adaptive robust control principle of the present invention.

[0034] Figure 2 This is a simplified schematic diagram of the energy-saving forklift pump-controlled steering system of the present invention.

[0035] Figure 3 This is a control input curve diagram of the system under the adaptive robust control action of the present invention.

[0036] Figure 4 This is a comparison curve of the tracking error of the system under the action of adaptive robust control based on nonlinear observer, adaptive robust control without observer, and PID control according to the present invention.

[0037] Figure 5 This is a graph showing the observation curves of the nonlinear observer of this invention.

[0038] Figure 6 This is a tracking error curve diagram of the present invention. Detailed Implementation

[0039] Specific embodiments of the present invention will now be described with reference to the accompanying drawings. Many details are described below to provide a comprehensive understanding of the invention; however, those skilled in the art will not need these details to implement the invention. Well-known components, methods, and processes will not be described in detail below.

[0040] Combination Figure 1 and Figure 2 The present invention discloses an adaptive robust control method for an energy-saving forklift pump-controlled steering system, comprising the following steps: Step 1: Establish a mathematical model for the energy-saving forklift pump-controlled steering system, as follows: The mathematical model of the system is established based on the following assumptions: the hydraulic cylinder has only internal leakage and no external leakage; the load on the hydraulic cylinder is an inertial load and has no elastic load.

[0041] The flow equation for a bidirectional fixed displacement pump is: (1)

[0042] In the formula: The displacement of the bidirectional fixed displacement pump is m. 3 / rad; The rotational angular velocity of the bidirectional fixed displacement pump is rad / s. This represents the leakage coefficient of the hydraulic pump.

[0043] The flow continuity equation for a hydraulic cylinder is: (2)

[0044] In the formula: Q 1 and Q 2 represents the flow rate of the two chambers of the hydraulic cylinder, in m³. 3 / s; A 1 and A 2 represents the effective area of ​​the two chambers of the hydraulic cylinder piston, which, assuming they are the same, is set as 2. A p m 2 ; x L Let the piston displacement be m; C i The internal leakage coefficient of the hydraulic cylinder is (m 3 / s) / Pa; P 1 and P 2 represents the pressure in the two chambers of the hydraulic cylinder, in Pa; V 1 and V 2 represents the volume of the two chambers of the hydraulic cylinder, in meters. 3 ; β e Let be the bulk modulus, Pa.

[0045] Define load pressure as P L = P 1- P 2, and the load flow is defined as Q L = ( Q 1+ Q 2) / 2, then combining equation (1) and equation (2) yields: (3)

[0046] In the formula:V t The total volume of the two chambers of the hydraulic cylinder is in meters. 3 .

[0047] According to Newton's second law, the equation for the balance between the output force and the load of a hydraulic cylinder is: (4)

[0048] In the formula: m The total mass of the piston and load, expressed in kg, is the total mass of the piston and load applied to the piston. B is the viscous damping coefficient of the piston and load, N / (m / s); f d For items that are difficult to model accurately, including external disturbance forces and unmodeled frictional forces.

[0049] Define state variables x0 = [x1, x2, x3] T =[x L , x(•) L , P L ] T Then, from equations (1) to (4), the state-space equations of the pump-controlled steering system can be obtained as follows: (5)

[0050] consider It is a parameter that causes uncertainty, defined The total leakage coefficient is defined as follows: the set of uncertain parameters is defined as follows: ,in: , , , .

[0051] The state space of the above pump-controlled steering system can be simplified to the following expression: (6)

[0052] Step 2: Based on the mathematical model of the pump-controlled steering system above, design a nonlinear disturbance observer, as follows: A nonlinear disturbance observer is used to estimate and compensate for disturbances in the pump-controlled steering system, which has the following form: (7)

[0053] In the formula: For the nonlinear function to be designed, For the gain of the nonlinear observer, it should satisfy... .

[0054] The observation error of the nonlinear disturbance observer is defined as: (8)

[0055] Due to disturbance Since there is no prior data, it is assumed that its dynamic characteristics change slowly relative to the observer, that is: (9)

[0056] Considering equation (8), the dynamic equation of the observer error system is: (10)

[0057] As can be seen from equation (10), by making appropriate selections This can cause the observation error of the observer to converge exponentially.

[0058] choose , As a constant, design (11)

[0059] The output of the nonlinear interference observer is sent to the gain adjustment module, which converts the observed interference into control of the corresponding input channel. From the system, we know that: (12)

[0060] Therefore, the gain is ,have to , This is a compensation item.

[0061] Step 3: Based on the mathematical model of the pump-controlled steering system and the nonlinear disturbance observer, design an adaptive robust controller, as follows: 3.1 First Design Error .

[0062] 3.2 The second design error , .

[0063] 3.3 The third design error .

[0064] in: For target turning signal, This is the gain coefficient. This is a virtual control law.

[0065] 3.4 Definition express The estimated value, This represents the estimation error. and Representing the maximum and minimum values ​​of an uncertain parameter, a projection of a discontinuous parameter can be given as follows: (13)

[0066] The adaptive law of parameters can be expressed as: (14)

[0067] in: It is a diagonal matrix representing the adaptive gain; Let represent an adaptive function. For any adaptive function... By satisfying the adaptive law of the projection rule, all of them will have the following properties: (15); among which, for θ The matrix transpose of the estimation error.

[0069] In steps 3.1 and 3.2, the first error is designed. The second error From the system's characteristics, we know that the first equation has no uncertainty, and the first and second equations can be combined in the design. According to linear system theory, when... When it approaches 0, It will also inevitably tend towards 0, therefore making The tendency toward 0 is equivalent to making Approaching 0, the design goal becomes designing a control law to make It tends towards 0.

[0070] The dynamic equation can be expressed as: (16); where The derivative of the virtual control input;

[0071] for Design virtual control law : (17);

[0072] In the formula: For model compensation terms, For linearly stable feedback terms, For robust control terms; , These are the nonlinear gain 1 of the stable feedback term and the nonlinear gain 1 of the robust term, respectively.

[0073] Selecting Lyapunov functions: (18)

[0074] Differentiating it, we get: (19)

[0075] in For nonlinear error, in the formula: (20)

[0076] design The following conditions must be met for calming: (twenty one)

[0077] From equation (21), we can see that This serves as a robust controller to manage various uncertainties in the system model. Therefore, from equation (19) and the stabilization condition (21), we can derive: (twenty two)

[0078] For any small positive number. Precise tracking ,Right now System tracking error , It will eventually move into a domain that can be manually defined over time.

[0079] In step 3.3, the third error is designed. In order to make tending towards 0, The dynamic equation can be expressed as: (twenty three)

[0080] Design control law : (twenty four)

[0081] for The differentiable part can be represented as: (25); , These are the nonlinear gain 2 of the stable feedback term and the nonlinear gain 2 of the robust term, respectively;

[0082] in: (26) (27) (28)

[0083] In the formula: .

[0084] Step 4: Apply Lyapunov stability theory to prove the stability of the adaptive robust controller based on nonlinear observations, obtaining the result that the system tracking error is asymptotically stable. Details are as follows: For equation (6) of the above system, under the assumption that the conditions are met, using equation (7) of the nonlinear disturbance observer and equation (24) of the control law, all signals in the closed-loop system are bounded, therefore the system is stable. The proof is as follows: Selecting Lyapunov functions: (29)

[0085] right Differentiating, we get:

[0086] (30)

[0087] In the formula:

[0088] (31)

[0089] design The following conditions must be met for calming: (32)

[0090] It is a positive number that can be arbitrarily small. From equation (32), we know... To provide a robust controller to govern the various uncertainties of the system model, we can derive the following from equation (30) and the stabilization condition (32): (33)

[0091] definition: (34)

[0092] As long as the design , Greater than 0, satisfying If positive definite, then: (35)

[0093] From equation (35), we can see that Globally bounded, that is Since the signals in a closed-loop system are consistent and eventually bounded, the closed-loop system is stable.

[0094] If, after a certain moment, the system only has parameter uncertainty, then the Lyapunov function is defined as follows: (36); In the formula , is a diagonal matrix representing the adaptive gain of the parameters;

[0095] Differentiating it, we get: (37)

[0096] From condition (15) and equation (33), we get:

[0097] (38)

[0098] Therefore, it can be concluded that when the system only has parameter uncertainty, the controller can still achieve asymptotic tracking performance.

[0099] The following are specific implementation examples of the present invention: To evaluate the performance of the designed controller, the physical parameters of the energy-saving forklift pump-controlled steering servo system in the simulation are shown in Table 1: Table 1 System Physical Parameters

[0100] Given the system's desired instruction: Reference Figure 3 and Figure 4 The following controllers are compared in the simulation: Nonlinear disturbance observer adaptive robust controller: , , ,V=diag([5e -19 ,1e -12 ,1e 16 ,1e -12 ]); Observer .

[0101] Adaptive Robust Controller: , , ,V=diag([5e -19 ,1e -12 ,1e 16 ,1e -12 ]).

[0102] Set up a PID controller with P=0.1 and I=1.5e. 5 , D=0.

[0103] Tracking error, nonlinear observers such as Figure 5 and Figure 6 .from Figure 4 It can be seen that the maximum error of the adaptive robust control position servo control of the energy-saving forklift based on the nonlinear observer is 1.217e. -4m It has higher accuracy and superior tracking performance than PID and observerless adaptive robust control.

[0104] The above are merely specific embodiments of the present invention, but the design concept of the present invention is not limited thereto. Any non-substantial modifications made to the present invention using this concept shall be considered as infringing upon the protection scope of the present invention.

Claims

1. An adaptive robust control method for an energy-saving forklift pump-controlled steering system, characterized in that, Includes the following steps: Step 1: Establish a mathematical model of the energy-saving forklift pump-controlled steering system; Step 2: Based on the mathematical model of the pump-controlled steering system, design a nonlinear disturbance observer to estimate the system disturbance and compensate for it. Step 3: Based on the mathematical model and nonlinear observer of the pump-controlled steering system, design an adaptive robust controller, specifically including: 3.1 First Design Error , For target turning signal; 3.2 The second design error , ; 3.3 The third design error ;in: , and For the defined state variables, This is the gain coefficient. For virtual control laws; 3.4 Adaptive Law for Design Parameters: definition express The estimated value, This represents the estimation error. and Representing the maximum and minimum values ​​of an uncertain parameter, the projection of a discontinuous parameter is as follows: The adaptive law for parameters is expressed as: ; in, It is a diagonal matrix representing the adaptive gain; Represents an adaptive function; This represents parametric projection; by using an adaptive law that satisfies the projection rules, we will have: ； in, for θ The matrix transpose of the estimation error.

2. The adaptive robust control method for an energy-saving forklift pump-controlled steering system as described in claim 1, characterized in that, Step 1 is described in detail as follows: The mathematical model of the system is established based on the following assumptions: the hydraulic cylinder has only internal leakage and no external leakage; the load on the hydraulic cylinder is an inertial load and no elastic load; and the flow equation of the bidirectional fixed displacement pump is: ； In the formula: This refers to the displacement of a bidirectional fixed displacement pump. The rotational angular velocity of the bidirectional fixed displacement pump, The leakage coefficient of the hydraulic pump; The flow continuity equation for a hydraulic cylinder is: ； In the formula: Q 1 and Q 2 represents the flow rate of the two chambers of the hydraulic cylinder, respectively; A 1 and A 2 represents the effective area of ​​the two chambers of the hydraulic cylinder piston, which, assuming they are the same, is set as 2. A p ; x L This represents piston displacement; C i The internal leakage coefficient of the hydraulic cylinder; P 1 and P 2 represents the pressure in the two chambers of the hydraulic cylinder; V 1 and V 2 represents the volume of the two chambers of the hydraulic cylinder; β e It is the bulk modulus of elasticity; Define load pressure as P L = P 1- P 2, and the load flow is defined as Q L = ( Q 1+ Q 2) / 2, then by combining the flow continuity equations of the hydraulic cylinders, we get: ； In the formula: V t This refers to the total volume of the two chambers of the hydraulic cylinder; According to Newton's second law, the equation for the balance between the output force and the load of a hydraulic cylinder is: ； In the formula: m This refers to the total mass of the piston and its load, which is then factored onto the piston. B The viscous damping coefficient of the piston and load; f d For items that are difficult to model accurately, including external disturbance forces and unmodeled frictional forces; Define state variables x0 = [x1, x2, x3] T =[x L , x(•) L , P L ] T From the above formulas, the state-space equations of the pump-controlled steering system can be obtained as follows: ； consider m , B , , C i , It is a parameter that causes uncertainty, defined The total leakage coefficient is defined as follows: the set of uncertain parameters is defined as follows: , , , , ; The state-space equations of the above pump-controlled steering system can be simplified to the following expression: 。 3. The adaptive robust control method for an energy-saving forklift pump-controlled steering system as described in claim 2, characterized in that, Step 2 is described in detail below: The formula for the linear disturbance observer is: ； In the formula: For the nonlinear function to be designed, For the gain of the nonlinear observer, it should satisfy... ; The observation error of the nonlinear disturbance observer is defined as: ; Assuming disturbance The dynamic characteristics of the observer change slowly, that is: The dynamic equation of the observer error system is: ; choose As a constant, design .

4. The adaptive robust control method for an energy-saving forklift pump-controlled steering system as described in claim 3, characterized in that, The output of the nonlinear disturbance observer is sent to the gain adjustment module to convert the observed disturbance into control of the corresponding input channel, specifically: Gain is ,have to , This is a compensation item.

5. The adaptive robust control method for an energy-saving forklift pump-controlled steering system as described in claim 4, characterized in that: In steps 3.1 and 3.2, when When it approaches 0, It will also inevitably tend towards 0, making The tendency toward 0 is equivalent to making Approaching 0, the design goal becomes designing a control law to make tending towards 0, The dynamic equation can be expressed as: In the formula The derivative of the virtual control input; for Design virtual control law : In the formula , These are the nonlinear gain 1 of the stable feedback term and the nonlinear gain 1 of the robust term, respectively; Selecting Lyapunov functions: Taking its derivative, we get: ； in For nonlinear error, in the formula: ； design The following conditions must be met for calming: ； As a robust controller to manage various uncertainties in the system model, it is derived from equations (2) and (3): ； Let be an arbitrarily small positive number; if Precise tracking ,Right now System tracking error , It will eventually move into a domain that can be manually defined over time.

6. The adaptive robust control method for an energy-saving forklift pump-controlled steering system as described in claim 1, characterized in that: In step 3.3, The dynamic equation can be expressed as: ； Design control law : ； for The differentiable part can be represented as: ; , These are the nonlinear gain 2 of the stable feedback term and the nonlinear gain 2 of the robust term, respectively; in: ； ； ； In the formula, .

7. The adaptive robust control method for an energy-saving forklift pump-controlled steering system as described in claim 6, characterized in that: It also includes step 4, using Lyapunov stability theory to prove the stability of the adaptive robust controller, and obtaining the result that the system tracking error is asymptotically stable.

8. The adaptive robust control method for an energy-saving forklift pump-controlled steering system as described in claim 7, characterized in that, The specific proof process for step 4 is as follows: Selecting Lyapunov functions: ; right Differentiating, we get: ； In the formula: ; design The following conditions must be met for calming: ; For any arbitrarily small positive number, we can derive from equations (4) and (5): （6）； definition: ; As long as the design , Greater than 0, satisfying If positive definite, then: ； From the above formula, we can see that Globally bounded, that is Since the signals in the closed-loop system are consistent and eventually bounded, the closed-loop system is stable.

9. The adaptive robust control method for an energy-saving forklift pump-controlled steering system as described in claim 8, characterized in that, If the system only has parameter uncertainties, the Lyapunov function is defined as follows: ; In the formula , is a diagonal matrix representing the adaptive gain of the parameters; Differentiating it, we get: ； From equations (1) and (6), we get: ； The above demonstrates that the system possesses progressive tracking performance.

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