Finite-time tracking control method for fractional-order multi-Euler-Lagrange systems

Abstract

The present invention discloses a finite-time tracking control method for a fractional-order multi-Euler-Lagrange system, and designs a suitable non-singular terminal sliding mode control NTSM so that the state of the multi-Euler-Lagrange system converges within a finite time, including: establishing a multi-Euler-Lagrange dynamic equation with external disturbance based on a multi-double-link mechanical arm model, defining joint position error and joint velocity error, introducing the concept of Caputo fractional-order calculus to obtain the fractional-order local form of the two, and simplifying the two to a global form with the help of Kronecker product; designing a new finite-time controller under NTSM control, citing the Lyapunov stability theory of the fractional-order multi-Euler-Lagrange system; designing a class of NTSM functions, and constructing a suitable Lyapunov function, proving from a theoretical perspective that the state of the fractional-order system converges within a finite time, and finally realizing the finite-time tracking control of the multi-Euler-Lagrange system. The present invention designs a new finite-time controller based on the NTSM function, which can avoid the occurrence of singularity and chattering in the multi-double-link mechanical arm system.

Description

Technical Field

[0001] The present invention relates to the field of finite-time tracking control of fractional-order multi-agent systems, and in particular to a finite-time tracking control method for fractional-order multi-Euler-Lagrange systems. Background Art

[0002] In recent years, with the continuous combination of artificial intelligence technology and distributed control methods, robotic arms have developed rapidly with their unique advantages and have been widely used in different fields for production operations. In actual production, robotic arm systems are often affected by various uncertain nonlinear factors, which puts higher and higher requirements on the control of robotic arms. According to different production operation requirements, the appearance of the robotic arm and the predetermined motion trajectory that meet the actual situation can be designed, so that it can not only operate in toxic, harmful or dangerous environments, thereby protecting the life, health and safety of workers, but also replace people to perform repetitive and single tasks, realize automatic operation, reduce unnecessary labor, and greatly improve production efficiency.

[0003] However, the tracking control problem of multiple Euler-Lagrange systems is an important part of multi-agent tracking control, and therefore has received extensive attention and research from scholars at home and abroad. At present, the main research methods include tracking control methods based on SM, PID, adaptive, Backstepping, and robustness. Because SM variable structure has the advantages of fast response speed and strong robustness, SM control is often used to solve the tracking control problem of multiple Euler-Lagrange systems. In the existing literature, the tracking control problem of multiple Euler-Lagrange systems with model uncertainty and external interference, considering external interference and internal uncertainty of the system, and the system containing obstacles has been studied.

[0004] When selecting SM control, it is necessary to consider how to avoid singularity and chattering. When designing an SM controller, the SM controller often tends to infinity due to unreasonable selection of SM function, resulting in singularity. For this, it is necessary to select an NTSM function to avoid singularity; and the finite time controller designed based on NTSMC can avoid chattering.

[0005] Therefore, the present invention will establish multiple Euler-Lagrange dynamic equations (with external disturbance w) based on the multi-double-link mechanical arm model i (t)), defines the joint position error Joint velocity error The concept of Caputo fractional calculus is introduced to obtain the fractional local form of the two, and then the Kronecker product is used to simplify the two into a global form; a new finite-time controller τ is designed under NTSM control. i (t), citing the Lyapunov stability theory of fractional-order multi-Euler-Lagrange systems; designing a class of NTSM functions and constructing a suitable Lyapunov function, and proving from a theoretical point of view that the states of fractional-order multi-Euler-Lagrange systems converge in a finite time. Summary of the invention

[0006] Purpose of the invention: In view of the problems that need to be considered when selecting SM control pointed out in the background technology, the present invention discloses a finite-time tracking control method for a fractional-order multi-Euler-Lagrange system, designs a new finite-time controller under NTSM control, and quotes the Lyapunov stability theory of the fractional-order multi-Euler-Lagrange system; designs a class of NTSM functions, and constructs a suitable Lyapunov function, and proves from a theoretical perspective that the state of the fractional-order multi-Euler-Lagrange system converges within a finite time.

[0007] Technical solution: The present invention provides a finite time tracking control method for a fractional-order multi-Euler-Lagrange system, and designs a suitable non-singular terminal sliding mode NTSM so that the state of the multi-Euler-Lagrange system converges within a finite time, including the following steps:

[0008] Step 1: Establish multiple Euler-Lagrange dynamic equations based on the multi-double-link robotic arm system model. The dynamic equations include external disturbances w i (t) and give the properties and assumptions satisfied by the multi-link manipulator system;

[0009] Step 2: Define joint position errors Joint velocity error The introduction of Caputo fractional calculus will Rewritten in fractional local form By using the Kronecker product, Rewritten in fractional global form

[0010] Step 3: Based on the established multi-Euler-Lagrange dynamic equations, design an effective finite-time controller τ under non-singular terminal sliding mode control NTSMC i (t);

[0011] Step 4: Introduce the Lyapunov stability theory suitable for fractional-order multi-Euler-Lagrange systems and define the conditions for achieving the distributed tracking control objective in finite time;

[0012] Step 5: Select a class of non-singular terminal sliding mode NTSM functions and construct a suitable Lyapunov function to prove from a theoretical perspective that the state of the fractional-order multi-Euler-Lagrange system converges in a finite time.

[0013] Furthermore, the multi-Euler-Lagrange system in step 1 is composed of n double-link robotic arms, and the dynamic equation of the i-th double-link follower robotic arm is shown in formula (1):

[0014]

[0015] in, are the joint angular velocity, joint angular velocity, and joint angular acceleration of the i-th double-link follower robot, are the symmetric positive definite inertia matrix, centripetal Coriolis matrix, and gravity vector of the i-th follower manipulator, are the generalized control torque and external disturbance of the ith follower manipulator, respectively;

[0016] The multi-Euler-Lagrange system satisfies the following properties and assumptions:

[0017] Property 1: Boundedness, there exist positive constants k1, k2, k3 such that

[0018]

[0019] Among them, I n is the n-order unit matrix, i=1,2,…,n;

[0020] Property 2: Antisymmetry, is an antisymmetric matrix. For any given vector have

[0021]

[0022] Where i = 1, 2, ..., n;

[0023] Assumption 1: Model uncertainty and external disturbance w in a multi-Euler-Lagrange system i (t) is bounded, that is,

[0024] ‖M -1 (q)w i (t)‖≤w sup (4)

[0025] Among them, w sup is the known minimum supremum, and w sup >0.

[0026] Furthermore, the multi-Euler-Lagrange system in step 2 defines the joint position error Joint velocity error As shown in formula (5) and (6) respectively:

[0027]

[0028]

[0029] in, are the leader’s joint angle and joint angular velocity, respectively;

[0030] With the help of Caputo's fractional calculus concept and combined with equation (1), equations (5) and (6) can be rewritten as:

[0031]

[0032]

[0033] With the help of Kronecker product, the fractional-order local forms of equations (7) and (8) are rewritten into fractional-order global forms, as shown in equations (9) and (10), respectively:

[0034]

[0035]

[0036] Among them, the specific definitions of each global variable are as follows:

[0037]

[0038]

[0039] M(q)=diag{M1(q1),M2(q2),…,M n (q n )},

[0040]

[0041]

[0042] Furthermore, in step 3, based on the established multi-Euler-Lagrange dynamic equations, an effective finite time controller τ is designed under NTSMC. i (t) is shown in formula (11):

[0043]

[0044] Where τ(t) is τ i (t), sign(s(t)) is the sign function of s(t), s(t) is the NTSM function, β>0, η is a non-negative number, p1 and p2 are both odd numbers, and p2<p1<2p2 is satisfied. The specific definitions are as follows:

[0045]

[0046]

[0047] in, are the leader’s joint angular velocity and joint angular acceleration respectively;

[0048] In order to simplify the proof process, (11) is further rewritten as:

[0049]

[0050] Where τ=τ(t), s=s(t).

[0051] Furthermore, the Lyapunov stability theory suitable for fractional-order multi-Euler-Lagrange systems cited in step 4 is shown in equations (13)-(14):

[0052] Let x(t) = 0 be a non-autonomous fractional-order system

[0053] The equilibrium point, if there exists a Lyapunov function V(t) and a non-negative number η, satisfies the following formula:

[0054]

[0055] Then (13) is said to be asymptotically stable;

[0056] The conditions for achieving the distributed tracking control target in a finite time are defined as shown in equations (15)-(16):

[0057]

[0058]

[0059] Among them, q i (t) is the joint angular velocity of the i-th dual-link follower robot, is the joint angular velocity of the i-th two-link follower robot, q0(t) is the joint angle of the leader, is the joint angular velocity of the leader. To simplify the proof process, q i (t) q0(t), Simplified to q i , q0,

[0060] Furthermore, the NTSM function selected in step 5 is shown in formula (17):

[0061]

[0062] For multiple Euler-Lagrange systems, the NTSM function (17) satisfies:

[0063] s(t)=0 (18)

[0064] Then the state q of the multi-Euler-Lagrange system is Converges in finite time;

[0065] The constructed Lyapunov function is shown in formula (19):

[0066]

[0067] Beneficial effects:

[0068] The present invention selects an NTSM function and designs a new finite time controller based on the NTSM function, which can avoid the occurrence of singularity and chattering in the multi-double-link robot system. First, based on the multi-double-link robot model, a multi-Euler-Lagrange dynamic equation (with external disturbance w i (t)); then define the joint position error Joint velocity error The concept of Caputo fractional calculus is introduced to obtain the fractional local form of the two, and then the Kronecker product is used to simplify the two into fractional global form. Then a new finite-time controller τ is designed under NTSM control. i (t), citing the Lyapunov stability theory of fractional-order multi-Euler-Lagrange systems; finally, a class of NTSM functions is designed, and a suitable Lyapunov function is constructed to prove that the states of fractional-order multi-Euler-Lagrange systems converge in finite time, that is, the fractional-order multi-Euler-Lagrange systems with external disturbances under the original NTSMC can achieve finite-time tracking control. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 The present invention provides a flowchart of a finite-time tracking control method for fractional-order multi-Euler-Lagrange systems with external disturbances under NTSMC. DETAILED DESCRIPTION

[0070] The present invention will be further explained below in conjunction with the accompanying drawings.

[0071] The present invention discloses a finite time tracking control method for a fractional order multi-Euler-Lagrange system with external disturbance under NTSMC, which is used to realize tracking control of the state of the multi-Euler-Lagrange system with external disturbance in a finite time. Figure 1 , including the following steps:

[0072] Step 1: Establish multiple Euler-Lagrange dynamic equations based on the multi-double-link robotic arm system model. The dynamic equations include external disturbances w i (t), and give the properties and assumptions satisfied by the multi-double-link manipulator system.

[0073] The multi-Euler-Lagrange system consists of n two-link manipulators, and the dynamic equation of the i-th two-link follower manipulator is shown in formula (1):

[0074]

[0075] in, are the joint angular velocity, joint angular velocity, and joint angular acceleration of the i-th double-link follower robot, are the symmetric positive definite inertia matrix, centripetal Coriolis matrix, and gravity vector of the i-th follower manipulator, are the generalized control torque and external disturbance of the i-th follower robot, respectively.

[0076] In the present invention, the multi-Euler-Lagrange system satisfies the following properties and assumptions:

[0077] Property 1: (Boundedness) There exist positive constants k1, k2, k3 such that

[0078]

[0079] Among them, I n is the n-order unit matrix, i=1,2,…,n.

[0080] Property 2: (Antisymmetry) is an antisymmetric matrix. For any given vector have

[0081]

[0082] Among them, i=1,2,…,n.

[0083] Assumption 1: Model uncertainty and external disturbance w in a multi-Euler-Lagrange system i (t) is bounded, that is,

[0084] ||M -1 (q)w i (t)||≤w sup

[0085] Among them, w sup is the known minimum supremum, and w sup >0.

[0086] Step 2: Define joint position errors Joint velocity error Introducing the concept and proof method of Caputo fractional calculus, Rewritten in fractional local form By using the Kronecker product, Rewritten in fractional global form

[0087] Multiple Euler-Lagrange systems define joint position errors Joint velocity error As shown in formula (2) and (3) respectively:

[0088]

[0089]

[0090] in, are the leader’s joint angles and joint angular velocities, respectively.

[0091] The concepts of Caputo fractional calculus introduced are as follows:

[0092] (1) (Caputo integral) The α (α>0)-order Caputo fractional integral of a continuously differentiable function f(t) is defined as shown in formula (4):

[0093]

[0094] Where n=[α]+1.

[0095] (2) (Caputo differential) The α (α>0)-order Caputo fractional differential of a continuously differentiable function f(t) is defined as shown in formula (5):

[0096]

[0097] Where n = [α] + 1, f (n) (·) is the nth-order derivative of f(·), Γ(·) is the Gamma function, and Satisfies the formula: Γ(p+1)=pΓ(p), p∈R.

[0098] Caputo fractional differential has the following properties:

[0099] (1) Linear operation properties:

[0100]

[0101] (2) Superposition:

[0102]

[0103] Among them, a and b are arbitrary constants, g(t) and h(t) are continuous functions, n-1<p<n, p∈R, 0<q<1.

[0104] With the help of Caputo's fractional calculus concept and combined with equation (1), equations (2) and (3) can be rewritten as:

[0105]

[0106]

[0107] With the help of Kronecker product, the fractional-order local forms of equations (6) and (7) are rewritten into fractional-order global forms, as shown in equations (8) and (9), respectively:

[0108]

[0109]

[0110] Among them, the specific definitions of each global variable are as follows:

[0111]

[0112]

[0113] M(q)=diag{M1(q1),M2(q2),…,M n (q n )},

[0114]

[0115]

[0116] Step 3: Based on the established multi-Euler-Lagrange dynamic equations, design an effective finite-time controller τ under NTSMC i (t).

[0117] Based on the established multi-Euler-Lagrange dynamic equations, an effective finite-time controller τ is designed under NTSMC. i (t) is shown in formula (10):

[0118]

[0119] Where τ(t) is τ i The global form of (t), sign(s(t)) is the sign function of s(t), s(t) is the NTSM function, β>0, η is a non-negative number, w sup Same as assumption 1, p1 and p2 are both odd numbers, and satisfy p2<p1<2p2, The specific definitions are as follows:

[0120]

[0121]

[0122] in, are the leader’s joint angular velocity and joint angular acceleration respectively.

[0123] To simplify the proof process, (10) is further rewritten as:

[0124]

[0125] Where τ=τ(t), s=s(t).

[0126] Step 4: Introduce the Lyapunov stability theory for fractional-order multi-Euler-Lagrange systems and define the conditions for achieving the distributed tracking control objective in finite time.

[0127] The Lyapunov stability theory suitable for fractional-order multi-Euler-Lagrange systems is shown in equations (12)-(13):

[0128] Lemma 1: Let x(t) = 0 be a non-autonomous fractional-order system

[0129]

[0130] If there exists a Lyapunov function V(t) and a non-negative number η, the following equation is satisfied:

[0131]

[0132] Then (12) is said to be asymptotically stable.

[0133] The conditions for achieving the distributed tracking control goal in a finite time are defined as shown in equations (14)-(15):

[0134]

[0135]

[0136] Among them, q i (t) is the joint angular velocity of the i-th dual-link follower robot, is the joint angular velocity of the i-th two-link follower robot, q0(t) is the joint angle of the leader, is the joint angular velocity of the leader. To simplify the proof process, q i (t) q0(t), Simplified to q i , q0,

[0137] Step 5: Select a type of NTSM surface and construct a suitable Lyapunov function to prove theoretically that the state of the fractional-order multi-Euler-Lagrange system converges in a finite time.

[0138] A type of NTSM function is selected as shown in formula (16):

[0139]

[0140] Lemma 2: For the multi-Euler-Lagrange system in (1), the NTSM function (16) satisfies

[0141] s(t)=0(17)

[0142] Then the state of the multi-Euler-Lagrange system is Converges in finite time

[0143] The constructed Lyapunov function is shown in formula (18):

[0144]

[0145] In order to facilitate the proof of the theoretical process, the following adjustments are made: V(t) and s(t) are replaced by V and s respectively, and the Caputo fractional differential symbol is Abbreviated as

[0146] The fractional differential result of formula (18) is shown in formula (19):

[0147]

[0148] The fractional differentiation of equation (16) is shown in equation (20):

[0149]

[0150] Substituting formula (20) into formula (19) yields:

[0151]

[0152] Substituting formula (9) into formula (21), we can obtain:

[0153]

[0154] Using equations (2) and (8), equation (22) can be rewritten as:

[0155]

[0156] in,

[0157] Substituting the controller (11) into equation (23), we can obtain:

[0158]

[0159] According to Assumption 1, equation (24) can be rewritten as:

[0160]

[0161] The p1 and p2 designed in the present invention are both odd numbers and satisfy p2<p1<2p2, and then we can get so β and η are both greater than 0, and |s| is a non-negative number.

[0162] So we can get:

[0163]

[0164] Therefore, according to Lemma 1 and Lemma 2, it can be obtained that the error dynamic system of the fractional-order multi-Euler-Lagrange system (1) with external disturbance studied in the present invention is stable under the action of the finite-time controller (11) designed based on NTSMC, and can achieve tracking control objectives (14) and (15), so that the original fractional-order multi-Euler-Lagrange system is asymptotically stable, and the system state can converge within a finite time.

[0165] The above embodiments are only for illustrating the technical concept and features of the present invention, and their purpose is to enable people familiar with the technology to understand the content of the present invention and implement it accordingly, and they cannot be used to limit the protection scope of the present invention. Any equivalent transformation or modification made according to the spirit of the present invention should be included in the protection scope of the present invention.

Claims

1. A finite-time tracking control method for fractional-order multi-Euler-Lagrange systems, characterized in that: Designing a suitable non-singular terminal sliding mode NTSM makes the state of the multi-Euler-Lagrange system converge within a finite time, including the following steps: Step 1: Establish multiple Euler-Lagrange dynamic equations based on the multi-double-link robotic arm system model. The dynamic equations include external disturbances w i (t) and give the properties and assumptions satisfied by the multi-link manipulator system; The multi-Euler-Lagrange system consists of n two-link manipulators, and the dynamic equation of the i-th two-link follower manipulator is shown in formula (1): Among them, q i , are the joint angular velocity, joint angular velocity, and joint angular acceleration of the i-th double-link follower robot, are the symmetric positive definite inertia matrix, centripetal Coriolis matrix, and gravity vector of the i-th follower manipulator, are the generalized control torque and external disturbance of the ith follower manipulator, respectively; The multi-Euler-Lagrange system satisfies the following properties and assumptions: Property 1: Boundedness, there exist positive constants k1, k2, k3 such that Among them, I n is the n-order unit matrix, i=1,2,···,n; Property 2: Antisymmetry, is an antisymmetric matrix. For any given vector have Where, i=1,2,···,n; Assumption 1: Model uncertainty and external disturbance w in a multi-Euler-Lagrange system i (t) is bounded, that is, ||M -1 (q)w i (t)||≤w sup (4) Among them, w sup is the known minimum supremum, and w sup >0; Step 2: Define joint position errors Joint velocity error The introduction of Caputo fractional calculus will Rewritten in fractional local form By using the Kronecker product, Rewritten in fractional global form Multiple Euler-Lagrange systems define joint position errors Joint velocity error As shown in formula (5) and (6) respectively: Among them, q0, are the leader’s joint angle and joint angular velocity, respectively; With the help of Caputo's fractional calculus concept and combined with equation (1), equations (5) and (6) can be rewritten as: With the help of Kronecker product, the fractional-order local forms of equations (7) and (8) are rewritten into fractional-order global forms, as shown in equations (9) and (10), respectively: Among them, the specific definitions of each global variable are as follows: M(q)=diag{M1(q1),M2(q2),···,M n (q n )}, Step 3: Based on the established multi-Euler-Lagrange dynamic equations, design an effective finite-time controller τ under non-singular terminal sliding mode control NTSMC i (t); Design an efficient finite-time controller τ under NTSMC i (t) is shown in formula (11): Where τ(t) is τ i (t), sign(s(t)) is the sign function of s(t), s(t) is the NTSM function, β>0, η is a non-negative number, p1 and p2 are both odd numbers, and p2<p1<2p2 is satisfied. The specific definitions are as follows: in, are the leader’s joint angular velocity and joint angular acceleration respectively; To simplify the proof process, (11) is further rewritten as: Where, τ=τ(t), s=s(t); Step 4: Introduce the Lyapunov stability theory suitable for fractional-order multi-Euler-Lagrange systems and define the conditions for achieving the distributed tracking control objective in finite time; Step 5: Select a class of non-singular terminal sliding mode NTSM functions and construct a suitable Lyapunov function to prove theoretically that the state of the fractional-order multi-Euler-Lagrange system converges in a finite time; A type of NTSM function is selected as shown in formula (17): For multiple Euler-Lagrange systems, the NTSM function (17) satisfies: s(t)=0 (18) Then the state q of the multi-Euler-Lagrange system is Converges in finite time; The constructed Lyapunov function is shown in formula (19):

2. The finite time tracking control method for fractional order multi-Euler-Lagrange system according to claim 1, characterized in that: The Lyapunov stability theory suitable for fractional-order multi-Euler-Lagrange systems cited in step 4 is shown in equations (13)-(14): Let x(t) = 0 be a non-autonomous fractional-order system The equilibrium point, if there exists a Lyapunov function V(t) and a non-negative number η, satisfies the following formula: Then (13) is said to be asymptotically stable; The conditions for achieving the distributed tracking control target in a finite time are defined as shown in equations (15)-(16): Among them, q i (t) is the joint angular velocity of the i-th dual-link follower robot, is the joint angular velocity of the i-th two-link follower robot, q0(t) is the joint angle of the leader, is the joint angular velocity of the leader. To simplify the proof process, q i (t) q0(t), Simplified to q i , q0,

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