Steering-by-wire system steering angle tracking control method based on internal model
By adopting an internal model-based angle tracking control method for steer-by-wire systems, the problems of tracking accuracy and robustness of steer-by-wire systems are solved, achieving high-precision angle tracking and interference suppression, thereby improving vehicle handling stability and driving safety.
Patent Information
- Application Number
- CN202511390385.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-09-26
- Publication Date
- 2025-12-12
AI Technical Summary
The tracking accuracy of steer-by-wire systems is easily affected by factors such as temperature and wear, and complex external disturbances can cause deviations between the reference steering angle and the actual steering angle, affecting vehicle handling stability and driving safety. Existing PID control has insufficient robustness and low control accuracy.
The method for steering steer-by-wire system angle tracking control based on internal model is proposed. By establishing a mathematical model, it is described as a global robust output regulation problem. The internal model and state feedback controller are designed, and the final controller is constructed to achieve angle tracking control.
It achieves high-precision angle tracking and interference suppression under conditions of parameter uncertainty and complex interference, thereby improving the robustness and control accuracy of the steer-by-wire system.
Smart Images
Figure CN121106482A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of steering angle tracking control technology for steer-by-wire systems, specifically to a steering angle tracking control method for steer-by-wire systems based on an internal model. Background Technology
[0002] As the automotive industry accelerates its move towards intelligentization, steer-by-wire systems have become the next-generation steering technology that automotive R&D personnel worldwide are focusing on. Compared to traditional steering systems that rely on mechanical connections to transmit steering commands, steer-by-wire systems achieve a revolutionary breakthrough through electrical signal transmission and precise control: an angle sensor installed on the steering column collects the driver's steering intentions in real time and converts them into electrical signals, which are then transmitted to the central control center; after receiving the reference steering angle from the control center, the steer-by-wire system drives the actuator to complete the steering action, without the need for mechanical intervention throughout the process. This not only significantly optimizes the vehicle's spatial layout but also provides a flexible and adaptable technological foundation for steering control in intelligent driving scenarios.
[0003] However, in practical applications, the tracking accuracy of steer-by-wire systems faces two major challenges: firstly, the system parameters are easily affected by factors such as temperature and wear; secondly, complex external disturbances cause deviations between the reference steering angle and the actual steering angle, directly affecting vehicle handling stability and driving safety. While existing technologies such as traditional PID control are simple in structure and easy to implement, they are unable to address these two challenges, exhibiting insufficient robustness and low control accuracy. Summary of the Invention
[0004] The purpose of this invention is to provide a steering angle tracking control method for a steer-by-wire system based on an internal model, so as to solve the above-mentioned defects.
[0005] To achieve the above objectives, the present invention provides the following technical solution:
[0006] This invention proposes a steering angle tracking control method for a steer-by-wire system based on an internal model, comprising the following steps:
[0007] S1. Establish a mathematical model of a steer-by-wire system driven by a DC motor;
[0008] S2. The steering angle tracking control problem of the steer-by-wire system is described as a global robust output adjustment problem;
[0009] S3. Design an internal model to transform the global robust output regulation problem of the system into a global robust stabilization problem of an augmented system composed of the system and the internal model.
[0010] S4. Design a state feedback controller to solve the global robust stabilization problem of the augmented system; construct the final controller to realize the steering angle tracking control of the steer-by-wire system.
[0011] Preferably, in step S1, the system model establishment process of the DC motor-driven steer-by-wire system is as follows:
[0012] S11. The mathematical model of the DC motor is as follows:
[0013]
[0014] In equation (1), u is the duty cycle of the PWM, and V bat It is the battery voltage, L a For armature inductance, i a R is the armature current. a K is the armature resistance. e K is the back electromotive force coefficient. T It is the torque coefficient, I m and c m For the inertia and damping of the motor rotor, θ m T is the angular displacement of the motor. mf For internal friction of the motor, T m This is the output torque of the motor;
[0015] S12. The mathematical model of the transmission mechanism is as follows:
[0016]
[0017] In equation (2), I T Let c be the moment of inertia of the transmission mechanism. T For the damping of the transmission mechanism, i T η is the transmission ratio of the transmission mechanism. T For the efficiency of the transmission mechanism, T Tf For internal friction of the transmission mechanism; T out and θ out These are steering torque and output angle, respectively. The derivative representing the angular displacement of the motor. The second derivative represents the angular displacement of the motor;
[0018] S13. Combining equations (1) and (2), the dynamic equations of the steer-by-wire system are obtained as follows:
[0019]
[0020] In equation (3), I equ =I m +I T c is the system equivalent inertia of the motor. equ =c m +c T The system equivalent damping of the transmission mechanism. Let i be the load torque of the system. aIt represents the derivative of electric current.
[0021] Preferably, step S2 includes the following steps:
[0022] S21. Assuming the reference signal θ d and load torque T L It can be generated by the following external systems:
[0023]
[0024] In equation (4), τ represents the state variable of the external system. Let A1 denote the derivative of τ, C2 denote the system matrix of the external system, C3 denote the constant matrix of the reference angle, and C4 denote the constant matrix of the load torque.
[0025] S22, Definition Given the system angular velocity, the state variable in the steer-by-wire system is x1 = θ. m x2=ω m x3=i a Disturbance make and Formula (3) can be changed to the following form:
[0026]
[0027] In equation (5), Denotes the derivative of x1. Denotes the derivative of x². Let x3 be the derivative;
[0028] S23. Establish the angle tracking error equation for the steer-by-wire system:
[0029] e = x1 - C2τ (6),
[0030] In equation (6), e represents the angle tracking error of the steer-by-wire system;
[0031] S24. Considering the system parameter perturbations caused by uncertainties, define the uncertain parameters. in The nominal value of the system, ∈R 7 ;
[0032] Combining equations (4), (5), and (6), we obtain the following compact form:
[0033]
[0034] In equation (7),
[0035] S25. At this point, the angle tracking control problem of system (3) has been described as the global robust output regulation problem of system (7), and its control objective is to ensure that the tracking error asymptotically approaches 0 while ensuring that the closed-loop system trajectory starting from any initial value is bounded.
[0036] Preferably, step S3 is as follows:
[0037] S31. Solve the following regulator equation:
[0038]
[0039] In equation (8), x(τ,∈) and u(τ,∈) are the steady-state state and steady-state input, respectively. The obtained steady-state solution is as follows:
[0040] x1(τ,∈)=C2τ
[0041] x2(τ,∈)=C2A1τ
[0042]
[0043] S32. Let g(x,u) = col(x2,x3,u), and use g i (x,u) represents the i-th element in g(x,u), where i = 1, 2, 3. Construct the following steady-state generator to produce a steady-state solution:
[0044]
[0045] In equation (9), (Φ i ,Ψ i ) represents a pair of observable matrices;
[0046] S33. Select a pair of controllable matrices (M) i N i ), where M i Let T be a Hurwitz matrix, i = 1, 2, 3, such that the nonsingular matrix T i It satisfies the following Sylvester equation:
[0047] T i Φ i -M i T i =N i Ψ i (10);
[0048] S34. The inner mold is designed in the following form:
[0049]
[0050] In equation (11), η1 represents the first state variable of the internal model, η2 represents the second state variable of the internal model, and η3 represents the third state variable of the internal model. The derivative of η1 is represented. The derivative of η² is given by... The derivative of η³ is given by [the expression].
[0051] S35. Perform the following coordinate transformation and input transformation:
[0052]
[0053] In the formula, and z i These are the transformed state variables, i = 1, 2, 3. It is the transformed control input;
[0054] The following error equation is obtained:
[0055]
[0056] in, This represents the first derivative of z1 with respect to time. express The first derivative with respect to time, This represents the first derivative of z2 with respect to time. express The first derivative with respect to time, This represents the first derivative of z3 with respect to time. express The first derivative with respect to time; and we have:
[0057]
[0058]
[0059] d9=Ψ1T1 -1 (b1N1-M1N1-N1Ψ1T1 -1 N1)
[0060]
[0061] d 16 =b2Ψ1T1 -1
[0062]
[0063] d 19 =b2Ψ1T1 -1 N1
[0064]
[0065] S36, The global robust output regulation problem of system formula (7) has been transformed into the global robust stabilization problem of system (13).
[0066] Preferably, step S4 is as follows:
[0067] S41. Construct a state feedback controller:
[0068]
[0069] In equation (14), k1 > 0, k2 > 0, k3 > 0. This represents the first state variable of the state feedback controller. This represents the second state variable of the state feedback controller. This represents the third state variable of the state feedback controller. This represents the control input after coordinate transformation;
[0070] S42, Order For the X1 subsystem, make Where P1 is a positive definite symmetric matrix satisfying I is the identity matrix, and m1 is a specific positive number; there exists a sufficiently large gain k1 that satisfies the following inequality:
[0071]
[0072] S43. For the X2 subsystem, let Where P2 is a positive definite symmetric matrix satisfying m2 and m3 are specific positive numbers; there exists a sufficiently large gain k2 that satisfies the following inequality:
[0073]
[0074] In equation (16), l is a specific positive number;
[0075] S44, Final Order Where P3 is a positive definite symmetric matrix satisfying m4 and m5 are specific positive numbers; there exists a sufficiently large gain k3 that satisfies the following inequality:
[0076]
[0077] S45. From (11), (12) and (14), the final controller can be obtained in the following form:
[0078]
[0079] The beneficial effects of this invention are as follows:
[0080] This invention discloses an angle tracking control method for a steer-by-wire system based on an internal model. Addressing the challenges of parameter uncertainty and complex interference during vehicle steering, a state feedback controller based on an internal model is designed to achieve angle tracking control of the steer-by-wire system. This method offers high-precision angle tracking and interference suppression performance, and allows for the assumption that all system parameters are unknown. Attached Figure Description
[0081] Figure 1 This is a schematic diagram illustrating the principle framework of the steering angle tracking control of the steer-by-wire system based on the internal mold of the present invention.
[0082] Figure 2 This is a diagram showing the steering angle tracking curve of the steer-by-wire system of the present invention.
[0083] Figure 3 This is a diagram showing the steering angle error of the steer-by-wire system of the present invention.
[0084] Figure 4 This is a graph showing the duty cycle of the PWM control input for the steer-by-wire system of this invention. Detailed Implementation
[0085] The present invention will be further described below with reference to the embodiments. It should be noted that these are merely examples and descriptions of the inventive concept. Those skilled in the art can make various modifications or additions to the specific embodiments described or use similar methods to replace them, as long as they do not deviate from the inventive concept or exceed the scope defined in the claims, they should all be considered to fall within the protection scope of the present invention.
[0086] Example 1:
[0087] Figure 1 This is a schematic diagram illustrating the principle framework of the steering angle tracking control of the steer-by-wire system based on the internal model of the present invention. Figure 1 As shown, the present invention proposes a steering angle tracking control method for a steer-by-wire system based on an internal model, comprising the following steps:
[0088] S1. Establish a mathematical model of a steer-by-wire system driven by a DC motor.
[0089] The system model establishment process for the DC motor-driven steer-by-wire system is as follows:
[0090] S11. The mathematical model of the DC motor is as follows:
[0091]
[0092] In equation (1), u is the duty cycle of the PWM, and V bat It is the battery voltage, L a For armature inductance, i aR is the armature current. a K is the armature resistance. e K is the back electromotive force coefficient. T It is the torque coefficient, I m and c m For the inertia and damping of the motor rotor, θ m T is the angular displacement of the motor. mf For internal friction of the motor, T m This refers to the output torque of the motor.
[0093] S12. The mathematical model of the transmission mechanism is as follows:
[0094]
[0095] In equation (2), I T Let c be the moment of inertia of the transmission mechanism. T For the damping of the transmission mechanism, i T η is the transmission ratio of the transmission mechanism. T For the efficiency of the transmission mechanism, T Tf For internal friction of the transmission mechanism; T out and θ out These are steering torque and output angle, respectively. The derivative representing the angular displacement of the motor. The second derivative represents the angular displacement of the motor;
[0096] S13. Combining equations (1) and (2), the dynamic equations of the steer-by-wire system are obtained as follows:
[0097]
[0098] In equation (3), I equ =I m +I T c is the system equivalent inertia of the motor. equ =c m +c T The system equivalent damping of the transmission mechanism. The load torque of the system, It represents the derivative of electric current.
[0099] S2. The steering angle tracking control problem of the steer-by-wire system is described as a global robust output adjustment problem. The specific steps are as follows:
[0100] S21. Assuming the reference signal θ d and load torque T L It can be generated by the following external systems:
[0101]
[0102] In equation (4), τ represents the state variable of the external system. Let A1 denote the derivative of τ, C2 denote the system matrix of the external system, C3 denote the constant matrix of the reference angle, and C4 denote the constant matrix of the load torque.
[0103] S22, Definition Given the system angular velocity, the state variable in the steer-by-wire system is x1 = θ. m x2=ω m x3=i a Disturbance make and Formula (3) can be changed to the following form:
[0104]
[0105] In equation (5), Denotes the derivative of x1. Denotes the derivative of x². Let x3 be the derivative;
[0106] S23. Establish the angle tracking error equation for the steer-by-wire system:
[0107] e = x1 - C2τ (6),
[0108] In equation (6), e represents the angle tracking error of the steer-by-wire system.
[0109] S24. Considering the system parameter perturbations caused by uncertainties, define the uncertain parameters. in The nominal value of the system, ∈R 7 ;
[0110] Combining equations (4), (5), and (6), we obtain the following compact form:
[0111]
[0112] In equation (7),
[0113] S25. At this point, the angle tracking control problem of system (3) has been described as the global robust output regulation problem of system (7), and its control objective is to ensure that the tracking error asymptotically approaches 0 while ensuring that the closed-loop system trajectory starting from any initial value is bounded.
[0114] S3. Design the internal model to transform the global robust output regulation problem of the system into a global robust stabilization problem of an augmented system composed of the system and the internal model. Specifically:
[0115] S31. Solve the following regulator equation:
[0116]
[0117] In equation (8), x(τ,∈) and u(τ,∈) are the steady-state state and steady-state input, respectively. The obtained steady-state solution is as follows:
[0118] x1(τ,∈)=C2τ
[0119] x2(τ,∈)=C2A1τ
[0120]
[0121] S32. Let g(x,u) = col(x2,x3,u), and use g i (x,u) represents the i-th element in g(x,u), where i = 1, 2, 3. Construct the following steady-state generator to produce a steady-state solution:
[0122]
[0123] In equation (9), (Φ i ,Ψ i ) is a pair of observable matrices.
[0124] S33. Select a pair of controllable matrices (M) i N i ), where M i Let T be a Hurwitz matrix, i = 1, 2, 3, such that the nonsingular matrix T i It satisfies the following Sylvester equation:
[0125] T i Φ i -M i T i =N i Ψ i (10).
[0126] S34. The inner mold is designed in the following form:
[0127]
[0128] In equation (11), η1 represents the first state variable of the internal model, η2 represents the second state variable of the internal model, and η3 represents the third state variable of the internal model. The derivative of η1 is represented. The derivative of η² is given by... The derivative of η³ is given by [the expression].
[0129] S35. Perform the following coordinate transformation and input transformation:
[0130]
[0131] In the formula, and z i These are the transformed state variables, i = 1, 2, 3. It is the transformed control input.
[0132] The following error equation is obtained:
[0133]
[0134] in, Let z1 represent the first derivative of z1 with respect to time (i.e., the rate of change of z1). express The first derivative with respect to time (i.e.) (rate of change) This represents the first derivative of z² with respect to time (i.e., the rate of change of z²). express The first derivative with respect to time (i.e.) (rate of change) Let z3 represent the first derivative of z3 with respect to time (i.e., the rate of change of z3). express The first derivative with respect to time (i.e.) (rate of change); and we have:
[0135]
[0136]
[0137] d9=Ψ1T1 -1 (b1N1-M1N1-N1Ψ1T1 -1 N1)
[0138]
[0139] d 16 =b2Ψ1T1 -1
[0140]
[0141] d 19 =b2Ψ1T1 -1 N1
[0142]
[0143] S36, The global robust output regulation problem of system formula (7) has been transformed into the global robust stabilization problem of system (13).
[0144] S4. Design a state feedback controller to solve the global robust stabilization problem of the augmented system; construct the final controller to realize the steering angle tracking control of the steer-by-wire system. Details are as follows:
[0145] S41. Construct a state feedback controller:
[0146]
[0147] In equation (14), k1 > 0, k2 > 0, k3 > 0. This represents the first state variable of the state feedback controller. This represents the second state variable of the state feedback controller. This represents the third state variable of the state feedback controller. This represents the control input after coordinate transformation;
[0148] S42, Order For the X1 subsystem, make Where P1 is a positive definite symmetric matrix satisfying I is the identity matrix, and m1 is a specific positive number; there exists a sufficiently large gain k1 that satisfies the following inequality:
[0149]
[0150] S43. For the X2 subsystem, let Where P2 is a positive definite symmetric matrix satisfying m2 and m3 are specific positive numbers; there exists a sufficiently large gain k2 that satisfies the following inequality:
[0151]
[0152] In equation (16), l is a specific positive number.
[0153] S44, Final Order Where P3 is a positive definite symmetric matrix satisfying m4 and m5 are specific positive numbers; there exists a sufficiently large gain k3 that satisfies the following inequality:
[0154]
[0155] S45. From (11), (12) and (14), the final controller can be obtained in the following form:
[0156]
[0157] Figure 1This is a schematic diagram illustrating the principle framework of the steering angle tracking control method for a steer-by-wire system based on an internal model, as described in this invention. To verify the effectiveness of the proposed method, the control effect of the state feedback controller is simulated and verified.
[0158] The standard parameters of the selected steer-by-wire system are shown in Table 1:
[0159] Table 1. Standard parameters of steer-by-wire system
[0160]
[0161]
[0162] The system's uncertain parameters are selected as follows:
[0163]
[0164] The controller parameters are designed as follows:
[0165]
[0166] Reference signal Disturbance The external system parameters are as follows:
[0167]
[0168] Set the initial system state value to θ. m =0 rad, ω m =0 rad / s,i a =0A.
[0169] Using the above parameters, the steer-by-wire system is controlled using the internal model-based steering angle tracking control method of this invention, resulting in the following: Figure 2-4 The simulation results are shown. Figure 2 , Figure 3 These are, respectively, the steering angle tracking curve and the steering angle error curve of the steer-by-wire system of the present invention; as follows: Figure 2 , Figure 3 As shown, under the influence of complex disturbances, the method of the present invention can achieve good corner tracking performance and suppress this type of load torque disturbance. Figure 4 This is the control input PWM duty cycle curve of the steer-by-wire system of the present invention; from Figure 4 It can be seen that the duty cycle range of the PWM control input of the steer-by-wire system is [-20%, 20%], which verifies the practical feasibility of the present invention.
[0170] This invention discloses an angle tracking control method for a steer-by-wire system based on an internal model. Addressing the challenges of parameter uncertainty and complex interference during vehicle steering, a state feedback controller based on an internal model is designed to achieve angle tracking control of the steer-by-wire system. This method offers high-precision angle tracking and interference suppression performance, and allows for the assumption that all system parameters are unknown.
[0171] The above is an exemplary description of the invention. Obviously, the specific implementation of the invention is not limited to the above-described manner. Any non-substantial improvement made using the inventive concept and technical solution of the invention, or the direct application of the inventive concept and technical solution to other situations without modification, is within the protection scope of the invention.
Claims
1. A steering angle tracking control method for a steer-by-wire system based on an internal model, characterized in that, Includes the following steps: S1. Establish a mathematical model of a steer-by-wire system driven by a DC motor; S2. The steering angle tracking control problem of the steer-by-wire system is described as a global robust output adjustment problem; S3. Design an internal model to transform the global robust output regulation problem of the system into a global robust stabilization problem of an augmented system composed of the system and the internal model. S4. Design a state feedback controller to solve the global robust stabilization problem of augmented systems; The final controller is obtained, enabling the steering angle tracking control of the steer-by-wire system.
2. The steering angle tracking control method for a steer-by-wire system based on an internal model according to claim 1, characterized in that, In step S1, the system model establishment process of the DC motor-driven steer-by-wire system is as follows: S11. The mathematical model of the DC motor is as follows: In equation (1), u is the duty cycle of the PWM, and V bat L is the battery voltage. a For armature inductance, i a R is the armature current. a K is the armature resistance. e K is the back electromotive force coefficient. T I is the torque coefficient. m For the inertia of the motor rotor, c m θ is the damping of the motor rotor. m T is the angular displacement of the motor. mf For internal friction of the motor, T m This is the output torque of the motor; S12. The mathematical model of the transmission mechanism is as follows: In equation (2), I T Let c be the moment of inertia of the transmission mechanism. T For the damping of the transmission mechanism, i T η is the transmission ratio of the transmission mechanism. T For the efficiency of the transmission mechanism, T Tf For internal friction of the transmission mechanism; T out and θ out These are steering torque and output angle, respectively. The derivative representing the angular displacement of the motor. The second derivative represents the angular displacement of the motor; S13. Combining equations (1) and (2), the dynamic equations of the steer-by-wire system are obtained as follows: In equation (3), I equ =I m +I T c is the system equivalent inertia of the motor. equ =c m +c T The system equivalent damping of the transmission mechanism. The load torque of the system, It represents the derivative of electric current.
3. The steering angle tracking control method for a steer-by-wire system based on an internal model according to claim 2, characterized in that, The specific steps of step S2 are as follows: S21. Assuming the reference signal θ d and load torque T L It can be generated by the following external systems: In equation (4), τ represents the state variable of the external system. Let A1 denote the derivative of τ, C2 denote the system matrix of the external system, C3 denote the constant matrix of the reference angle, and C4 denote the constant matrix of the load torque. S22, Definition Given the system angular velocity, the state variable in the steer-by-wire system is x1 = θ. m x2=ω m x3=i a Disturbance make and Formula (3) can be changed to the following form: In equation (5), Denotes the derivative of x1. Denotes the derivative of x². Let x3 be the derivative; S23. Establish the angle tracking error equation for the steer-by-wire system: e = x1 - C2τ (6), In equation (6), e represents the angle tracking error of the steer-by-wire system; S24. Considering the system parameter perturbations caused by uncertainties, define the uncertain parameters. in The nominal value of the system, ∈R 7 ; Combining equations (4), (5), and (6), we obtain the following compact form: In equation (7), S25. At this point, the angle tracking control problem of system (3) has been described as the global robust output regulation problem of system (7), and its control objective is to ensure that the tracking error asymptotically approaches 0 while ensuring that the closed-loop system trajectory starting from any initial value is bounded.
4. The steering angle tracking control method for a steer-by-wire system based on an internal model according to claim 3, characterized in that, The specific steps in S3 are as follows: S31. Solve the following regulator equation: In equation (8), x(τ,∈) and u(τ,∈) are the steady-state state and steady-state input, respectively. The obtained steady-state solution is as follows: x1(τ,∈)=C2τ x2(τ,∈)=C2A1τ S32. Let g(x,u) = col(x2,x3,u), and use g i (x,u) represents the i-th element in g(x,u), where i = 1, 2, 3. Construct the following steady-state generator to produce a steady-state solution: In equation (9), (Φ i ,Ψ i ) represents a pair of observable matrices; S33. Select a pair of controllable matrices (M) i N i ), where M i Let T be a Hurwitz matrix, i = 1, 2, 3, such that the nonsingular matrix T i It satisfies the following Sylvester equation: T i F i -M i T i =N i P i (10); S34. The inner mold is designed in the following form: In equation (11), η1 represents the first state variable of the internal model, η2 represents the second state variable of the internal model, and η3 represents the third state variable of the internal model. The derivative of η1 is represented. The derivative of η² is given by... The derivative of η³ is given by [the expression]. S35. Perform the following coordinate transformation and input transformation: In the formula, and z i These are the transformed state variables, i = 1, 2, 3. It is the transformed control input; The following error equation is obtained: in, This represents the first derivative of z1 with respect to time. express The first derivative with respect to time, This represents the first derivative of z2 with respect to time. express The first derivative with respect to time, This represents the first derivative of z3 with respect to time. express The first derivative with respect to time; and we have: S36, The global robust output regulation problem of system formula (7) has been transformed into the global robust stabilization problem of system (13).
5. The steering angle tracking control method for a steer-by-wire system based on an internal model according to claim 4, characterized in that, The S4 step is as follows: S41. Construct a state feedback controller: In equation (14), k1 > 0, k2 > 0, k3 > 0. This represents the first state variable of the state feedback controller. This represents the second state variable of the state feedback controller. This represents the third state variable of the state feedback controller. This represents the control input after coordinate transformation; S42, Order For the X1 subsystem, make Where P1 is a positive definite symmetric matrix satisfying I is the identity matrix, and m1 is a specific positive number; there exists a sufficiently large gain k1 that satisfies the following inequality: S43. For the X2 subsystem, let Where P2 is a positive definite symmetric matrix satisfying m2 and m3 are specific positive numbers; there exists a sufficiently large gain k2 that satisfies the following inequality: In equation (16), l is a specific positive number; S44, Final Order Where P3 is a positive definite symmetric matrix satisfying m4 and m5 are specific positive numbers; there exists a sufficiently large gain k3 that satisfies the following inequality: S45. From (11), (12) and (14), the final controller can be obtained in the following form: