Command filtering backstepping control method for flexible joint manipulator based on fuzzy observer

Abstract

The present invention belongs to the technical field of position tracking control of flexible joint manipulators, and specifically discloses a command filtering backstepping control method for flexible joint manipulators based on a fuzzy observer. The method proposes a command filtering fuzzy control method for a flexible joint manipulator system with unknown model parameters by using command filtering and fuzzy observer technology, and realizes tracking control of the desired trajectory of the system. A fuzzy observer is designed to estimate the connecting rod angular velocity and motor angular velocity of the flexible manipulator; fuzzy adaptive technology is used to solve the uncertainty of model parameters to varying degrees; at the same time, the command filtering technology is applied to solve the computational complexity problem in the controller design process; and the convergence of all variables in the flexible joint manipulator control system is proved by the Lyapunov control principle. The observer designed by the present invention has a good observation effect, and solves the computational complexity problem in the controller design process, with a smaller trajectory tracking error and a stronger system robustness.

Description

Technical Field

[0001] The invention belongs to the technical field of position tracking control of flexible joint mechanical arms, and in particular relates to a flexible joint mechanical arm instruction filtering backstepping control method based on a fuzzy observer. Background Art

[0002] In recent years, due to the rapid development of robotics technology, flexible joint manipulators have been widely used in aerospace, workpiece assembly, national defense and military industry, and education industries. Combined with the rapid development of machine vision technology, they have high research value in manufacturing and medical industries. Traditional rigid manipulators are difficult to solve a large number of complex high-precision control problems. A large number of studies and practices have shown that if the joint flexibility of the manipulator is ignored during the design process, the manipulator will be greatly restricted when performing high-precision operations. Compared with traditional rigid manipulators, flexible manipulators have obvious continuity and multiple degrees of freedom on the basis of harmonic reducers. When facing complex and multi-obstacle environments and refined operation requirements, they are flexible and accurate in action, and have a wide range of joint torsion. However, the flexible manipulator system is a strongly coupled high-order nonlinear system, and these characteristics limit the control performance of the manipulator. Therefore, the precise control problem of flexible manipulators has become a research hotspot in the field of robotics today. Control methods include singular perturbation method, intelligent control, variable structure control, feedback linearization method, backstepping control, etc.

[0003] Among them, backstepping control is a scheme based on recursive Lyapunov function, which is an effective nonlinear system control framework. Based on this method, it has unique advantages in dealing with nonlinear control problems. Backstepping control is widely used in the control field of flexible manipulators. However, when the virtual control function is continuously derived, the traditional backstepping method used in the controller design process will have the problem of "computational explosion", which will increase the amount of calculation, resulting in great limitations in the application of the backstepping control strategy of flexible manipulators. Later, the dynamic surface control method was proposed to solve the "computational explosion", but the filtering error of the dynamic surface control will affect the control accuracy of the system. Further, the command filtering technology was proposed, and the interference of the filtering error on the control accuracy was reduced by combining the error compensation technology. In addition, in some control methods, in order to achieve more accurate control of the running flexible manipulator system, it is necessary to accurately understand the various state variables in the running system, but some state variables are unmeasurable. When you want to observe the velocity term of the flexible manipulator, you need to use a sensor, but the sensor is easily disturbed during measurement.

[0004] In summary, there are unknown items in the current flexible joint robot arm system. The connecting rod speed and motor angular velocity state are unpredictable, the joint position tracking is complex to derive in the backstepping design, and the large error causes safety accidents and other technical problems. Summary of the invention

[0005] In view of the above technical problems existing in the prior art, the present invention proposes a flexible joint robot arm command filtering backstepping control method based on fuzzy observer, so as to perform precise position tracking control on the flexible joint robot arm system.

[0006] In order to achieve the above object, the present invention adopts the following technical scheme:

[0007] Step 1. Establish a mathematical model of the flexible joint manipulator dynamics system considering uncertainty;

[0008] Step 2. Design a fuzzy observer to estimate the unmeasurable link velocity and motor angular velocity in the flexible joint robot model, and use a fuzzy logic system to handle different degrees of uncertainty in the system;

[0009] Step 3. Design a command filter backstepping controller, construct a second-order filter and error compensation signal to solve the computational complexity problem of the virtual control law in the backstepping design;

[0010] Step 4. Select the Lyapunov function for derivation, and then prove that the control system of the flexible joint robot arm command filtering control method based on fuzzy observer designed in step 3 is Lyapunov stable.

[0011] The present invention has the following advantages:

[0012] (1) The method of the present invention designs an observer to estimate the connecting rod velocity and motor angular velocity of the flexible manipulator, which solves the problem that the manipulator velocity term is unmeasurable and improves the reliability of the system.

[0013] (2) The method of the present invention combines command filtering technology and error compensation mechanism, solves the computational complexity problem in the controller design process, eliminates the influence of filtering errors, and no longer needs to consider the requirement of high-order differentiability of the desired signal, thereby achieving better tracking effect and improving the control accuracy of the system.

[0014] (3) The method of the present invention takes into account the uncertain factors in the system and uses a fuzzy logic system to deal with the uncertainty of the flexible robotic arm system, thereby improving the robustness of the system.

[0015] (4) The observer designed by the method of the present invention avoids the use and installation of sensors in actual systems and reduces maintenance requirements, thereby solving the problem that speed sensors are susceptible to interference during measurement, and is more conducive to practical applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 It is an overall schematic diagram of the system overall controller and the system composite controlled object in an embodiment of the present invention.

[0017] Figure 2It is a tracking response curve diagram of the joint angle position and the expected angle position after adopting the method of the present invention.

[0018] Figure 3 It is a response curve diagram of the tracking error of the joint angle position and the expected angle position after adopting the method of the present invention.

[0019] Figure 4 This is a tracking curve diagram of the connecting rod velocity term estimated by the fuzzy observer after adopting the method of the present invention.

[0020] Figure 5 The following is a tracking curve diagram of the motor angular velocity term estimated by the fuzzy observer after adopting the method of the present invention.

[0021] Figure 6 The figure is a control input response curve diagram of the system after adopting the method of the present invention.

[0022] Figure 7 This is a curve diagram of the tracking error response of the position when dealing with different degrees of uncertainty B after adopting the method of the present invention.

[0023] Figure 8 To deal with different degrees of uncertainty after using the method of the present invention Tracking error response curve of position at time . DETAILED DESCRIPTION

[0024] The embodiment of the present invention describes a command filtering backstepping control method for a flexible joint robot arm based on a fuzzy observer. The method is aimed at a flexible joint robot arm system with unknown model parameters, utilizes command filtering and fuzzy observer technology, proposes a command filtering fuzzy control strategy, and realizes tracking control of the desired trajectory of the system. The method includes the following steps: first, a fuzzy observer is designed to estimate the connecting rod angular velocity and motor angular velocity of the flexible robot arm; second, fuzzy adaptive technology is used to solve the uncertainty of model parameters; at the same time, the command filtering technology is applied to solve the computational complexity problem in the controller design process; and the convergence of all variables in the flexible joint robot arm control system is proved by the Lyapunov control principle.

[0025] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments:

[0026] Figure 1 It is the overall schematic diagram of the system overall controller and the composite controlled object of the flexible joint robot system, where: Figure 1 It involves the command filter backstepping controller, the fuzzy observer part and the filter compensation mechanism.

[0027] The control process of the method of the present invention is as follows: for the four output quantities of the joint position, speed, and angular position and speed of the motor side of the flexible robotic arm system, the control signal is input into the controller; since the speed term is unmeasurable, it is input into the fuzzy observer to generate an estimate of the speed term, and then the estimate of the speed term is sent to the controller; the difference generated after comparing with the given target position is sent to the controller, and the estimated error of the joint angle position and the motor angular position is sent to the controller; the control signal of the command filter backstepping controller is input into the flexible robotic arm system to achieve control of the system.

[0028] The flexible joint robot arm command filtering backstepping control method based on fuzzy observer specifically includes the following steps:

[0029] Step 1. Establish a mathematical model of the flexible joint robot arm dynamics system considering uncertainties.

[0030] First, the dynamic equation of the flexible joint manipulator is given, as shown in formula (1):

[0031]

[0032] Among them, q, denote the position, velocity and acceleration of the connecting rod, respectively, and q m , They represent the angular position, speed and acceleration of the motor respectively, M represents the mass of the connecting rod, l represents the length of the connecting rod, g is the acceleration due to gravity, represents the friction term, i.e. the unknown interference term, K represents the joint stiffness coefficient, J and B represent the inertia and natural damping term of the actuator respectively, u represents the motor control input, and y represents the output of the system; set and B contain uncertainties.

[0033] Let the state variable x 1 =q, x 3 =q m , The dynamic equation, formula (1), is rewritten as:

[0034]

[0035] To simplify the dynamic equation, formula (1), the following variables are defined, as shown in formula (3):

[0036]

[0037] Based on formula (3), formula (2) is further rewritten as:

[0038]

[0039] Assume h2 and h 4 is a known positive constant, f 2 and f 4 is a continuous function with unknown terms and has a continuous and bounded first-order derivative. Step 2. Design a fuzzy observer to estimate the link velocity and motor angular velocity of the flexible joint robot.

[0040] Let formula (4) Further formula (5) is obtained:

[0041]

[0042] Among them, Z 1 =[x 1 ,x 2 ,x 3 ,x 4 ] T .

[0043]

[0044] From the definition of fuzzy logic system, we know: Among them, θ i is the optimal parameter vector, Represents θ i The estimate, For x i The estimated value of the state.

[0045] express With θ i As function variables, express The estimate, Indicates that the variable is Function block, express estimates;

[0046] Represents state variables With θ i The function that is simplified together is, express Estimates.

[0047] For an arbitrarily given positive number ε, the fuzzy adaptive approximation theorem gives:

[0048]

[0049] in, Indicates that the variable in the function is Z 1 And the coefficient is θ iT Function block, δ i (Z 1 )≤|ε|.

[0050] Based on formula (5) and combining the above formula The observer of formula (4) is designed as:

[0051] In the formula, express The error of the observer is defined as e = [e 1 ,e 2 ,e 3 ,e 4 ] T , where e j Represents the estimation error, j = 1, 2, 3, 4; estimation error

[0052] D 1 , D 2 , D 3 , D 4 is a positive number to be designed;

[0053]

[0054] in, express The estimate, express Estimates.

[0055] Assume a function f i (·) satisfies the following inequality: is a normal number, i=2,4; we get in, Represents Z 1 Estimates.

[0056] make in, is an indeterminate normal number; express Estimates.

[0057] but get ε represents an arbitrarily small positive number.

[0058] Suppose there is a symmetric matrix Q = Q T > 0, then there will be a symmetric matrix P = P T >0 satisfies A TP+PA=-Q.

[0059] Step 3. According to the command filtering and backstepping method, a command filtering backstepping control method for the flexible joint manipulator based on the fuzzy observer is designed. The specific process is as follows:

[0060] The formula for defining the command filter is as follows:

[0061]

[0062] Among them, z o,1 represents the first-order variable of the command filter, z o,2 Represents the second-order variable of the command filter.

[0063] x o,c =z o,1 and are the two outputs of the command filter; α o is the input of the command filter, α o represents the virtual control function; the initial state of the instruction filter α o (0) = z o,1 (0), z o,2 (0)=0.

[0064] When the system of formula (4) starts to run, if the input signal satisfies: Then for any μ>0, there exist ζ∈(0,1] and ω n ≥0, such that |x o,c -α o |≤μ,o=1,2,3, All are bounded.

[0065] ρ 1 represents the upper bound of the first-order derivative of the virtual control function, ρ 2 Represents the upper bound of the second-order derivative of the virtual control function.

[0066] The error variable is defined by the principle of backstepping:

[0067]

[0068] Among them, z j represents the error variable, j = 1, 2, 3, 4; x d For a given expected signal, x o,c is the output signal of the command filter, is the estimated value of the velocity term.

[0069] The error compensation signal is defined as:

[0070]

[0071] Among them, ξj represents the error compensation signal, v j Represents the compensated error tracking signal.

[0072] The specific forms of the virtual control law and the filtered error compensation signal will be given in the following design process.

[0073] Step 3.1. Select the Lyapunov function as V 0 =e T Pe, the derivative is:

[0074]

[0075] From Young's inequality we get:

[0076] 2e T PF≤||e|| 2 +(m 2 2 +m 4 2 )||P 2 ||e|| 2 (11)

[0077]

[0078] Among them, m 2 、m 4 represents an arbitrarily small positive number; Substituting formula (11) and formula (12) into formula (10), we obtain:

[0079]

[0080] Among them, λ min (Q) is the smallest eigenvalue of Q.

[0081] Step 3.2. Select the Lyapunov function of the first subsystem The derivative is:

[0082]

[0083] Due to e 2 Unknown, obtained from Young's inequality:

[0084]

[0085] Design virtual control law α 1 and compensation signal for:

[0086]

[0087] Among them, k 1is a positive constant. Substituting formula (15) and formula (16) into formula (14), we get:

[0088]

[0089] Step 3.3. Select the Lyapunov function The derivative is:

[0090]

[0091] in, is θ 2 The estimate, represents the adaptive law, λ 2 is a positive constant, h 2 is a positive constant. From Young's inequality we get:

[0092]

[0093] Substituting formula (19) into formula (18), we get:

[0094]

[0095] Design virtual control law α 2 , compensation signal and adaptive law for:

[0096]

[0097] Among them, k 2 is a positive constant, n 2 is a positive constant, substituting formula (21) into formula (20) to obtain:

[0098]

[0099] Step 3.4. Select the Lyapunov function The derivative is:

[0100]

[0101] Due to e 4 Unknown, obtained from Young's inequality:

[0102]

[0103] Design virtual control law α 3 and compensation signal for:

[0104]

[0105] Among them, k3 is a positive constant. Substituting formula (24) and formula (25) into formula (23), we get:

[0106]

[0107] Step 3.5. Select the Lyapunov function The derivative is:

[0108]

[0109] Among them, λ 4 is a positive constant; is θ 4 The estimate, represents the adaptive law. From Young’s inequality we get:

[0110]

[0111] Substituting formula (28) into formula (27), we get:

[0112]

[0113] Design the real control law u and compensation signal and adaptive law for:

[0114]

[0115] Among them, k 4 、n 4 are all positive constants. Substituting formula (30) into formula (29), we get:

[0116]

[0117] Step 4. Perform stability analysis on the command filtering backstepping control method of the flexible joint manipulator based on fuzzy observer. Select V = V 4 , from formula (31):

[0118]

[0119] According to Young's inequality:

[0120]

[0121]

[0122] Substituting formula (13), formula (33) and formula (34) into (32), we obtain:

[0123]

[0124] Among them, λ min (Q)-3-(m 2 2 +m 4 2 )||P|| 2 >0,

[0125]

[0126]

[0127] Among them, λ max (P) represents the maximum eigenvalue of vector P; further obtained from formula (35):

[0128]

[0129] Formula (36) shows that v j and All of them belong to compact sets:

[0130] Therefore, all signals of the robot system are bounded, and the error compensation signal ξ of the filter j satisfy in Because z 1 =v 1 +ξ 1 And 1 Bounded, tracking error z 1 is bounded, and we get:

[0131]

[0132] Among them, min j (k j ) represents k j The minimum value in .

[0133] Through the above analysis, we know that v j , j 、z j are all bounded, so for a given parameter k j 、m i 、n i After that, keep a constant and choose a large enough i and a sufficiently small ε to ensure that the tracking error z 1 Small enough.

[0134] In order to verify the effectiveness of the proposed control method, the flexible robotic arm control system selects the following parameters for simulation:

[0135] M=0.25kg, L=0.45m, g=9.8m / s 2 , K = 5N·m / rad, J = 5×10 -4 m / s 2 ,B=0.01,B 1 =0.01.

[0136] Select the friction term as

[0137] Select reference signal x d =sin(t).

[0138] Select the controller parameters as: k 1 =k 2 =k 3 =k 4 =10,b 2 =b 4 =0.5,n 2 =n 4 =3.

[0139] Select the observer parameters as: D 1 =D 3 =100,D 2 =D 4 =10000, select filter parameter ω n =1100,ζ=0.8.

[0140] At the same time, in order to prove the effectiveness of the proposed fuzzy logic system in dealing with uncertain terms in the system, the friction terms with uncertainty are changed respectively under the premise that the other parameters remain unchanged. Compare with B for simulation.

[0141] When B = 0.01, take when Take B = 0.01; 0.05; 0.1 respectively. The simulation results are as follows Figures 2 to 8 As shown:

[0142] Figure 2 Reflects the lower angle position x of the present invention 1 For the expected trajectory x d The tracking effect, Figure 3 is the position tracking error x 1 -x d ,Depend on Figure 2 and Figure 3 It can be seen that the joint position can track the given target trajectory well and has high control accuracy.

[0143] Figure 4 and Figure 5It reflects the estimation effect of the connecting rod angular velocity state and the motor angular velocity state under the method of the present invention, which is Figure 4 and Figure 5 It can be seen that the observer designed by the method of the present invention can effectively estimate the actual state and the observation effect is good.

[0144] Figure 6 The control input response curve of the system under the method of the present invention is reflected. Figure 6 It can be seen that the control law curve responds smoothly, without large fluctuations, and the control effect is stable.

[0145] Figure 7 and Figure 8 It reflects the system's ability to cope with different degrees of uncertainty B and The response curve of the tracking error of the position when Figure 7 and Figure 8 It can be seen that the method of the present invention can handle different degrees of uncertainty in the system model without almost changing the tracking effect of the system.

[0146] For a flexible joint robot system with uncertainties, the method of the present invention can well realize tracking control of the target trajectory with a small tracking error. The designed observer has a good observation effect. The control method used can well handle different degrees of uncertainty in the system model, and the system has strong robustness.

[0147] In summary, no matter from the theoretical stability proof or from the simulation results, the method of the present invention has achieved the expected goals of improving the control effect, reducing the computational complexity and meeting the effective estimation of the speed term under the uncertainty of the model parameters.

[0148] Of course, the above description is only a preferred embodiment of the present invention, and the present invention is not limited to the above embodiments. It should be noted that all equivalent substitutions and obvious deformation forms made by any technician familiar with the field under the guidance of this specification fall within the essential scope of this specification and should be protected by the present invention.

Claims

1. Command filtering backstepping control method for flexible joint manipulator based on fuzzy observer, It is characterized in that The steps include: Step 1. Establish a mathematical model of the flexible joint manipulator dynamics system considering uncertainties; First, the dynamic equation of the flexible joint manipulator is given, as shown in formula (1): Among them, q, denote the position, velocity and acceleration of the connecting rod, respectively, and q m , They represent the angular position, speed and acceleration of the motor respectively, M represents the mass of the connecting rod, l represents the length of the connecting rod, g is the acceleration due to gravity, represents the friction term, i.e. the unknown interference term, K represents the joint stiffness coefficient, J and B represent the inertia and natural damping term of the actuator respectively, u represents the motor control input, and y represents the output of the system; set and B contain uncertainty; Let the state variable x 1 =q, x 3 =q m , The dynamic equation, formula (1), is rewritten as: To simplify the dynamic equation, formula (1), the following variables are defined, as shown in formula (3): Based on formula (3), formula (2) is further rewritten as: Assume h 2 and h 4 is a known positive constant, f 2 and f 4 is a continuous function with unknown terms and a continuous and bounded first-order derivative; Step 2. Design a fuzzy observer to estimate the link velocity and motor angular velocity of the flexible joint robot arm; Let formula (4) Further formula (5) is obtained: Among them, Z 1 =[x 1 ,x 2 ,x 3 ,x 4 ] T ; As defined by the fuzzy logic system: Among them, θ i is the optimal parameter vector, Represents θ i The estimate, For x i The estimated value of the state; express With θ i As function variables, express The estimate, Indicates that the variable is Function block, express estimates; Represents state variables With θ i The function that is simplified together is, express estimates; For an arbitrarily given positive number ε, the fuzzy adaptive approximation theorem gives: in, Indicates that the variable in the function is Z 1 And the coefficient is θ i T Function block, δ i (Z 1 )≤|ε|; Based on formula (5) and combining the above formula The observer of formula (4) is designed as: In the formula, express The error of the observer is defined as e = [e 1 ,e 2 ,e 3 ,e 4 ] T , where e j Represents the estimation error, j = 1, 2, 3, 4; estimation error D 1 、D 2 、D 3 、D 4 is a positive number to be designed; obtain in, express The estimate, express estimates; Assume a function f i (·) satisfies the following inequality: is a normal number, i=2,4; we get in, Represents Z 1 estimates; make in, is an indeterminate normal number; express estimates; but get ε represents an arbitrarily small positive number; Suppose there is a symmetric matrix Q = Q T > 0, then there will be a symmetric matrix P = P T >0 satisfies A T P+PA=-Q; Step 3. According to the command filtering and backstepping method, a command filtering backstepping control method for the flexible joint manipulator based on the fuzzy observer is designed. The specific process is as follows: The formula for defining the command filter is as follows: Among them, z o,1 represents the first-order variable of the command filter, z o,2 represents the second-order variable of the command filter; x o,c =z o,1 and are the two outputs of the command filter; α o is the input of the command filter, α o represents the virtual control function; the initial state of the instruction filter α o (0) = z o,1 (0), z o,2 (0) = 0; When the system of formula (4) starts to run, if the input signal satisfies: Then for any μ>0, there exist ζ∈(0,1] and ω n ≥0, such that |x o,c -α o |≤μ,o=1,2,3, All are bounded; ρ 1 represents the upper bound of the first-order derivative of the virtual control function, ρ 2 represents the upper bound of the second-order derivative of the virtual control function; The error variable is defined by the principle of backstepping: Among them, z j represents the error variable, j = 1, 2, 3, 4; x d For a given expected signal, x o,c is the output signal of the command filter, is the estimated value of the speed term; The error compensation signal is defined as: Among them, ξ j represents the error compensation signal, v j represents the compensation error tracking signal; Step 4. Perform stability analysis on the command filtering backstepping control method of the flexible joint robot arm based on fuzzy observer.

2. The method for backstepping control of flexible joint manipulator command filtering based on fuzzy observer according to claim 1, It is characterized in that In step 3, the specific forms of the virtual control law and the filter error compensation signal are given in the following design process; Step 3.

1. Select the Lyapunov function as V 0 =e T Pe, the derivative is: From Young's inequality we get: 2e T PF≤||e|| 2 +(m 2 2 +m 4 2 )||P|| 2 ||e|| 2 (11) Among them, m 2 、m 4 represents an arbitrarily small positive number; Substituting formula (11) and formula (12) into formula (10), we obtain: Among them, λ min (Q) is the smallest eigenvalue of Q; Step 3.

2. Select the Lyapunov function of the first subsystem The derivative is: Due to e 2 Unknown, obtained from Young's inequality: Design virtual control law α 1 and compensation signal for: Among them, k 1 is a positive constant. Substituting formula (15) and formula (16) into formula (14), we get: Step 3.

3. Select the Lyapunov function The derivative is: in, is θ 2 The estimate, represents the adaptive law, λ 2 is a positive constant, h 2 is a positive constant; from Young's inequality we get: Substituting formula (19) into formula (18), we get: Design virtual control law α 2 , compensation signal and adaptive law for: Among them, k 2 is a positive constant, n 2 is a positive constant, substituting formula (21) into formula (20) to obtain: Step 3.

4. Select the Lyapunov function The derivative is: Due to e 4 Unknown, obtained from Young's inequality: Design virtual control law α 3 and compensation signal for: Among them, k 3 is a positive constant. Substituting formula (24) and formula (25) into formula (23), we get: Step 3.

5. Select the Lyapunov function The derivative is: Among them, λ 4 is a positive constant; is θ 4 The estimate, represents the adaptive law; from Young’s inequality we get: Substituting formula (28) into formula (27), we get: Design the real control law u and compensation signal and adaptive law for: Among them, k 4 、n 4 are all positive constants. Substituting formula (30) into formula (29), we get:

3. The method for backstepping control of flexible joint manipulator command filtering based on fuzzy observer according to claim 2, It is characterized in that In step 4, the stability analysis process is as follows: Select V = V 4 , from formula (31): According to Young's inequality: Substituting formula (13), formula (33) and formula (34) into (32), we obtain: Among them, l min (Q)-3-(m 2 2 +m 4 2 )||P|| 2 >0, Among them, λ max (P) represents the maximum eigenvalue of vector P; further obtained from formula (35): Formula (36) shows that v j and All of them belong to compact sets: Therefore, all signals of the robotic arm system are bounded, and the error compensation signal ξ of the filter j satisfies where since z 1 = v 1 + ξ 1 and ξ 1 is bounded, the tracking error z 1 is bounded, and thus it can be obtained that: Among them, min j (k j ) represents k j The minimum value in ; Through the above analysis, we know that v j , j 、z j are all bounded, so for a given parameter k j 、m i 、n i After that, keep a constant and choose a large enough i and a sufficiently small ε to ensure that the tracking error z 1 Small enough.

Images