Finite time control method of multi-planar motor, medium and equipment

Abstract

In order to solve the problem of cooperative control of planar motors in given finite time, the invention provides a finite time control method of a multi-planar motor, a medium and equipment, and relates to the technical field of motor control, the method comprises the following steps: establishing a heterogeneous multi-agent system controlled by the multi-planar motor, including n followers and one leader; constructing all-drive system mathematical models of the leader and the follower, and designing a consensus control protocol based on the all-drive system mathematical models of the leader and the follower; according to the full-drive system mathematical models of the leader and the follower, collecting state information of the leader, and designing a distributed observer for the follower to estimate the state information of the leader in a given finite time; and detecting an estimation error of the distributed observer, and designing a distributed controller to realize multi-planar motor control. The control target can be achieved within the preset time by adopting finite time control.

Description

Technical Field

[0001] The present invention relates to the technical field of motor control, and particularly relates to a finite-time control method, medium, and device for a multi-plane motor. Background Art

[0002] A planar motor is a precision displacement and flexible transmission system different from traditional motors or traditional mechanical structures. It has the ability of omnidirectional movement in six degrees of freedom, with a suspension height of about 10 mm and a position accuracy that can reach 0.005 mm. It adopts its innovative magnetic levitation drive technology, which can realize efficient and modular handling of materials, greatly reducing maintenance costs and energy consumption. In the current industrial automation field, mature complex production lines bring a relatively large burden to the equipment investment of enterprises, and it is very difficult to improve the process and production line. The magnetic levitation planar motor on which this patent is based can, to a certain extent, achieve flexible production and promote the advancement of industrial automation towards intelligent manufacturing.

[0003] The tracking control of a multi-plane motor is a type of multi-agent control scheme. The main research area in this field is a leader-follower consensus control. First, it is necessary to achieve the observation of the motion dynamics of the leader, that is, a distributed observer. Currently, the technical solutions of distributed observers are mainly divided into two types: centralized observers and distributed observers. The centralized observer scheme relies on a central processing unit and can theoretically achieve high precision. However, it needs to transmit a large amount of data, resulting in an extremely large communication load and easily causing bandwidth bottleneck problems. The distributed observer scheme, on the other hand, adopts a completely different strategy. In this scheme, each follower has the ability to independently process part of the information, and they only need to interact with adjacent nodes, thus effectively reducing the communication load. It is more adaptable and flexible in practical applications.

[0004] Currently, most of the existing distributed observer-related results assume that the leader input is bounded and available to all followers, which is too idealistic. Some existing observer designs include sign functions, which will cause chattering phenomena. The current scheme only observes the four-dimensional state of the leader, and its access to the six degrees of freedom is not comprehensive enough, lacking the observation of the R(Y) and R(Z) states. Although it can meet the requirements of production line transportation tasks, its portability is not enough. Summary of the Invention

[0005] The purpose of the present invention is to propose a finite-time control method for a multi-plane motor to solve the cooperative control problem of a planar motor within a given finite time, including the following steps:

[0006] S1. Establish a heterogeneous multi-agent system for multi-plane motor control, including n followers and 1 leader;

[0007] S2. Construct the mathematical model of the fully actuated system for the leader and the follower, and design a consensus control protocol based on the mathematical model of the fully actuated system for the leader and the follower;

[0008] S3. According to the mathematical model of the fully actuated system for the leader and the follower, collect the state information of the leader, and design a distributed observer for the follower to estimate the state information of the leader within a given finite time;

[0009] S4. Detect the estimation error of the distributed observer and design a distributed controller to achieve multi-plane motor control.

[0010] Furthermore, the mathematical model of the dynamic fully actuated system of the leader is:

[0011]

[0012] where represents the mathematical model of the m0-order dynamic fully actuated system of the leader, t represents time, and u0(t) represents the external input of the desired trajectory of the leader;

[0013] The mathematical model of the dynamic fully actuated system of the follower is:

[0014]

[0015] where represents the mathematical model of the mi-order dynamic fully actuated system of the i-th follower, k = 1, 2, u k i(t) represents the controller input of the i-th follower, and d i (t) represents the external disturbance of the i-th follower. i (t) represents the external disturbance of the i-th follower.

[0016] Furthermore, the communication of the heterogeneous multi-agent system is represented by a directed graph. The node set V = {1, 2,..., n} represents n followers, and the edge set represents the communication between agent pairs. The relationship matrix A = [a ij ij], if the i-th agent can receive the data of the j-th agent, a ij ij = 1, otherwise, a ij ij = 0.

[0017] Furthermore, the consensus control protocol is expressed as:

[0018] For the system there exists a positive continuously differentiable function and class function such that:

[0019] α1(|x|) ≤ V(x(t)) ≤ α2(|x|),

[0020]

[0021] The stability of the system is defined as global stability within a finite time.

[0022] Among them, α i represents the bound of the Lyapunov function. represents a family of functions, α1() represents the lower bound of the Lyapunov function, α2() represents the upper bound of the Lyapunov function, x(t) represents the state of the system at time t, V() represents the Lyapunov function of the system, x represents the state of the system, g(t) represents a time-varying control gain function that satisfies the same characteristics as v(t), v(t) represents the time-varying control gain function, γ and represent adjustable system control parameters.

[0023] v(t) satisfies the following conditions:

[0024] v(t) is a non-increasing positive function, with an initial value of v(0) and a maximum value of v(t p ), v)t p ) = ξ, and when t ≥ t p then v(t) = ξ, where t p is a constant.

[0025] Furthermore, the state information of the leader includes: position, velocity, and acceleration.

[0026] Furthermore, the distributed observer is expressed as:

[0027]

[0028] Among them, represents the estimated value of x(k) by the i-th follower. represents the estimated value of x(k) by the j-th follower. represents the estimated value of x(m0 - 1) by the i-th follower, m0 represents the order of the full-driven system mathematical model of the leader, x(k) represents the state of the k-th order of the system leader, m represents the total order of the leader, ∈0 represents an adjustable system control parameter, w i,k represents the error of the i-th agent at the k-th order, a ij represents the correlation relationship between the i-th agent and the j-th agent, b i represents a positive constant, x 0,k represents the system state of the k-th order of the leader.

[0029] Furthermore, the estimation error of the distributed observer converges to a residual set, and the residual set is:

[0030]

[0031] where, Ω0 represents the remaining set, and e i,k represents the estimation error of the i-th follower with respect to the leader, and λ min (H) represents the minimum eigenvalue of the H matrix, and H represents the information exchange matrix of the heterogeneous multi-agent system, represents a given bounded positive constant of the system. The present invention also provides a computer-readable storage medium storing a computer program, and when the computer program is executed by a processor, the finite-time control method of the multi-plane motor described above is implemented.

[0032] The present invention also provides an electronic device, including a processor and a memory, the processor is interconnected with the memory, wherein the memory is used to store a computer program, the computer program includes computer-readable instructions, and the processor is configured to call the computer-readable instructions to execute the finite-time control method of the multi-plane motor described above.

[0033] The beneficial effects brought by the technical solution provided by the present invention are as follows:

[0034] The present invention regards the multi-plane motor control as a multi-agent system, constructs a fully actuated system of leaders and followers, simplifies the design of the controller; designs a distributed observer for the followers to estimate the state information of the leader within a given finite time; the estimation error of the distributed observer is bounded, and a distributed controller is designed to achieve the multi-plane motor control. The present invention adopts finite-time control and can reach the control target within a preset time. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 is a flowchart of the finite-time control method of the multi-plane motor according to an embodiment of the present invention;

[0036] Figure 2 is a block diagram of an electronic device in an exemplary embodiment of an embodiment of the present invention;

[0037] Figure 3 is a communication topology diagram of agents in an example of an embodiment of the present invention;

[0038] Figure 4 is the trajectory tracking estimation error of the followers at a specified time according to an embodiment of the present invention;

[0039] Figure 5 is the angle tracking estimation error of the followers at a specified time according to an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0040] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments of the present invention will be further described below in conjunction with the accompanying drawings.

[0041] The present invention studies a cooperative control scheme based on planar motors, including the tracking of multiple planar motors to a single planar motor. In industrial automation production lines or logistics management, the present invention can be used to track the positions of goods or processed workpieces. For example, in an automobile manufacturing workshop, a robotic arm needs to cooperatively track the positions of automobile parts to complete tasks such as welding and assembly. By accurately tracking the positions of the parts, the robot can improve processing accuracy and production efficiency, and ensure the consistency of product quality.

[0042] The flowchart of the finite-time control method for multi-planar motors according to an embodiment of the present invention is as Figure 1 , and specifically includes the following steps:

[0043] S1. Establish a heterogeneous multi-agent system for multi-planar motor control, including n followers and 1 leader.

[0044] The tracking of multiple planar motors to a single planar motor is a type of multi-agent control scheme. Among them, the leader consists of m0 orders, and the followers consist of m1 and m2 orders.

[0045] S2. Construct the mathematical models of the fully actuated systems of the leader and the followers, and design a consensus control protocol based on the mathematical models of the fully actuated systems of the leader and the followers.

[0046] The mathematical model of the dynamic fully actuated system of the leader is:

[0047]

[0048] Among them, represents the mathematical model of the dynamic fully actuated system of the m0-order leader, t represents time, and u0(t) represents the external input of the leader's required trajectory.

[0049] The mathematical model of the dynamic fully actuated system of the follower is:

[0050]

[0051] Among them, represents the mathematical model of the dynamic fully actuated system of the ith m k -order follower, k = 1, 2, u i (t) represents the controller input of the ith follower, and d i (t) represents the external disturbance of the ith follower.

[0052] The communication of the heterogeneous multi-agent system is represented by a directed graph. The node V = {1, 2,..., n} represents n followers, the edge set ε represents the communication between agent pairs, and the relationship matrix A = [a ij ∈ R n×n , if the ith agent and the jth agent communicate, aij = 1, otherwise, a ij = 0. Define the Laplacian matrix L = D - A, where D = diag(v1, v2,..., v n ), N i represents the neighbors of the i-th agent. Define a directed graph which consists of the directed graph G, node 0, and the directed edges from the leader 0 to G, assuming is included in a spanning tree with a leader.

[0053] Consensus control problem:

[0054] Definition 1: For the system (1) If there exists a controller such that for any |x(0)| ≤ Δ0, where Δ0 is a positive scalar;

[0055] where t p is an arbitrarily pre-set time, is a constant, Δ is a positive scalar, x(0) represents the state of the system at time 0, and x represents the system state. If the above conditions are satisfied, the system is called locally practically PT stable, and if x(0) ∈ R, it is called globally practically PT (finite-time) stable.

[0056] Lemma 2: For the system there exists a positive continuously differentiable function and class function i can take the value 1 or 2, such that:

[0057] α1(|x|) ≤ V(x(t)) ≤ α2(|x|),

[0058]

[0059] where α i represents the bounds of the Lyapunov function, represents a family of functions, α1() represents the lower bound of the Lyapunov function, α2() represents the upper bound of the Lyapunov function, x(t) represents the state of the system at time t, V() represents the Lyapunov function of the system, which is a positive continuously differentiable function, x represents the state of the system, g() represents a time-varying control gain function with the same characteristics as v(), and is replaced by a positive constant in the present invention, γ and represent adjustable system control parameters, v(t) represents the time-varying control gain function, and v(t) satisfies the following conditions:

[0060] v(t) is a non-increasing positive function with an initial value of v(0) and a maximum value of v(t p ), v(t p) = ξ, and when t ≥ t p then v(t) = ξ, where t p is a constant representing a t p stable moment.

[0061] The stability of the system is called global use stability within a finite time.

[0062] Proof:

[0063] Transform the formula of Lemma 2 above to get:

[0064]

[0065] is an intermediate variable,

[0066] Integrating both sides simultaneously gives:

[0067]

[0068] From Lemma 2, the inequality can be rewritten as:

[0069]

[0070] So there is the following inequality:

[0071]

[0072] So there is:

[0073]

[0074] The consensus control problem is thus proven.

[0075] S3. According to the mathematical model of the fully actuated systems of the leader and followers, collect the state information of the leader and design a distributed observer for the followers to estimate the state information of the leader within a given finite time.

[0076] The state information of the leader includes: position, velocity, acceleration. The distributed observer is expressed as:

[0077]

[0078] Among them, represents the estimated value of x(k) by the i-th follower, represents the estimated value of x(k) by the j-th follower, denotes the estimated value of the \(i\)-th follower for \(x(m_0 - 1)\), where \(m_0\) represents the order of the full - drive system mathematical model of the leader, \(x(k)\) represents the state of the \(k\)-th order of the system leader, \(m\) represents the total order of the leader, \(\epsilon_0\) represents the adjustable system control parameter, \(w\) i,k denotes the error of the \(i\)-th agent at the \(k\)-th order, \(a\) ij denotes the association relationship between the \(i\)-th agent and the \(j\)-th agent, \(b\) i denotes a positive constant, \(x\) 0,k represents the system state of the \(k\)-th order of the leader.

[0079] The estimation error of the distributed observer converges to a residual set, and the residual set is:[[]]

[0080]

[0081] where \(\Omega_0\) represents the residual set, \(e\) i,k denotes the estimation error of the \(i\)-th follower for the leader, \(\lambda\) min \((H)\) represents the minimum eigenvalue of the \(H\) matrix, and \(H\) represents the information - exchange matrix of the heterogeneous multi - agent system. denotes the upper bound of the Lyapunov function. denotes the given bounded positive constant of the system.

[0082] Proof: Design the Lyapunov function

[0083] Then the derivative of the Lyapunov function can be obtained

[0084]

[0085] Then by the following lemma: For any two functions \(\mu_1\), \(\mu_2\), there exists a positive constant \(\alpha\) satisfying

[0086] Finally, we get:[[]]

[0087]

[0088] Also by Lemma 2, we can get So finally the error converges to a residual set.

[0089] S4. Detect the estimation error of the distributed observer and design a distributed controller to achieve multi - plane motor control.

[0090] In an exemplary embodiment, it includes a computer - readable storage medium. The computer - readable storage medium stores a computer program, and when the computer program is executed by a processor, it implements the above - mentioned finite - time control method for the multi - plane motor.

[0091] Please refer toFigure 2 , in an exemplary embodiment, it further includes an electronic device, which includes at least one processor, at least one memory, and at least one communication bus.

[0092] Wherein, a computer program is stored on the memory, the computer program includes computer-readable instructions, and the processor calls the computer-readable instructions stored in the memory through the communication bus to execute the above-mentioned finite-time control method of the multi-plane motor.

[0093] Taking four-agent planar motors as an example for simulation, where there is one leader and three followers, and the communication topology is as Figure 3 shown. Assume that the leader motor has a fourth-order dynamics, and its position, velocity, and acceleration are observed and tracked. Define the reference trajectory as follows. First, define the initial state and input of the leader motor:

[0094]

[0095] u0 = 0.0625sin(0.5*t),

[0096] Then define the estimated reference table 1 of each follower for the initial state of the leader.

[0097] Table 1

[0098]

[0099] Design δ0 = 0.001, ∈0 = 50, and design T = 0.5s. The reference of the trajectory tracking estimation error of the follower at the specified time can be obtained Figure 4 . It can be seen that all the estimated errors converge within the closed set [0, 0.02] within 0.5s, and the result proves that the distributed observer has good observation performance. The reference of the angle tracking estimation error of the follower at the specified time Figure 5 . It can be found that all four agents track to the target trajectory within the specified time, and the tracking error converges within the closed set [-0.05, 0.05] and remains unchanged within 2s. The simulation example verifies that the distributed controller of this design scheme can control the error.

[0100] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present invention. Various modifications to these embodiments will be obvious to those skilled in the art, and the general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to these embodiments shown herein, but rather will be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A finite-time control method for a multi-planar motor, characterized in that, Including the following steps: S1. Establish a heterogeneous multi-agent system for multi-plane motor control, including n followers and 1 leader; S2. Construct the mathematical model of the fully actuated systems of the leader and followers, and design a consensus control protocol based on the mathematical models of the fully actuated systems of the leader and followers; S3. According to the mathematical models of the fully actuated systems of the leader and followers, collect the state information of the leader, and design a distributed observer for the followers to estimate the state information of the leader within a given finite time; S4. Detect the estimation error of the distributed observer and design a distributed controller to achieve multi-plane motor control.

2. The finite-time control method of a multi-plane motor according to claim 1, characterized in that The mathematical model of the dynamic fully actuated system of the leader is: Among them, represents the mathematical model of the fully actuated system of the leader dynamics of order m0, t represents time, and u0(t) represents the external input of the leader's desired trajectory; The mathematical model of the dynamic fully actuated system of the followers is: Among them, represents the mathematical model of the fully actuated system of the i-th m k -th order follower dynamics, k = 1, 2, u i (t) represents the controller input of the i-th follower, d i (t) represents the external disturbance of the i-th follower.

3. A finite-time control method for a multi-planar motor according to claim 1, characterized in that, The communication of a heterogeneous multi-agent system is represented by a directed graph, where the nodes \(V = \{1, 2, \ldots, n\}\) represent \(n\) followers, the edge set represents the communication between agent pairs, and the relationship matrix \(A=[a ij \). If the \(i\)-th agent can receive data from the \(j\)-th agent, \(a ij = 1\); otherwise, \(a ij = 0\).

4. A finite-time control method for a multi-planar motor according to claim 2, characterized in that, The consensus control protocol is expressed as: For the system there exists a positive continuously differentiable function and class-$\mathcal{K}$ function $\alpha$ i $\in\mathcal{K}$ ∝ , for $i\in[1,2]$, such that: α1(|x|)≤V(x(t))≤α2(|x|), It is said that the stability of the system is globally stable within a finite time; where, α i represents the bound of the Lyapunov function, K ∝ represents a class of function families, α1() represents the lower bound of the Lyapunov function, α2() represents the upper bound of the Lyapunov function, x(t) represents the state of the system at time t, V() represents the Lyapunov function of the system, x represents the state of the system, g(t) represents a time-varying control gain function that satisfies the same characteristics as v(t), v(t) represents the time-varying control gain function, γ and represent adjustable system control parameters; v(t) satisfies the following conditions: v(t) is a non-increasing positive function with an initial value of v(0) and a maximum value of v(t p ), v(t p ) = ξ, and v(t) = ξ when t ≥ t p , where t p is a constant.

5. A finite-time control method for a multi-planar motor according to claim 1, characterized in that, The state information of the leader includes: position, velocity, acceleration.

6. The finite-time control method of a multi-planar motor according to claim 1, characterized in that, The distributed observer is expressed as: Among them, represents the estimated value of \(x(k)\) by the \(i\)-th follower, represents the estimated value of \(x(k)\) by the \(j\)-th follower, represents the estimated value of \(x(m_0 - 1)\) by the \(i\)-th follower, where \(m_0\) represents the order of the full - drive system mathematical model of the leader, \(x(k)\) represents the state of the \(k\)-th order of the system leader, \(m\) represents the total order of the leader, \(\epsilon_0\) represents the adjustable system control parameter, \(w\) i,k represents the error of the \(i\)-th agent at the \(k\)-th order, \(a\) ij represents the correlation relationship between the \(i\)-th agent and the \(j\)-th agent, \(b\) i represents a positive constant, \(x\) 0,k represents the system state of the \(k\)-th order of the leader.

7. A finite-time control method for a multi-planar motor according to claim 6, characterized in that The estimation error of the distributed observer converges to a residual set, and the residual set is: where, $\Omega_0$ represents the remaining set, and $e$ i,k represents the estimation error of the $i$-th follower with respect to the leader, and $\lambda$ min (H) represents the minimum eigenvalue of the matrix $H$, and $H$ represents the information exchange matrix of the heterogeneous multi-agent system, represents a bounded positive constant given by the system.

8. A computer-readable storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, it implements the method according to any one of claims 1-7.

9. An electronic device, characterized in that, Including a processor and a memory, the processor is interconnected with the memory, wherein the memory is used to store a computer program, the computer program includes computer-readable instructions, and the processor is configured to call the computer-readable instructions to execute the method according to any one of claims 1-7.

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