A single-link robotic arm system modeling and dual-asynchronous event-triggered control design method

Abstract

The present invention discloses a single-link robotic arm system modeling and dual-asynchronous event-triggered control design method; the present invention considers the random fluctuations of the payload mass and moment of inertia of the single-link robotic arm system, system interference, and bandwidth limitation and time lag in the signal transmission process, and uses Semi-Markov theory to establish a single-link robotic arm model to solve the dual-asynchronous event-triggered control problem of the single-link robotic arm system; the specific steps are as follows: first, considering the random changes in payload mass and moment of inertia, a model reflecting the state changes of the single-link robotic arm system is established based on Semi-Markov theory. Then, an event generator and controller that are asynchronous with the system modal information are designed. Furthermore, by constructing a Lyapunov function that depends on the system mode and the residence time within the mode, the closed-loop single-link robotic arm system is analyzed for stability with the help of a probability density function, and finally the obtained system random stability conditions and controller existence conditions are expressed by a set of linear matrix inequalities LMI.

Description

Technical Field

[0001] The present invention relates to a single-link robotic arm system technology, and in particular to a single-link robotic arm system modeling and dual-asynchronous event triggering control design method, and more specifically to a single-link robotic arm system modeling and dual-asynchronous event triggering control design method based on Semi-Markov theory. Background Art

[0002] A manipulator, a component of a robotic system, is a computer-controlled mechanical arm. It is widely used in industries such as transportation, aerospace, and industrial manufacturing. Manipulators can perform repetitive or dangerous tasks, reducing labor costs, improving production efficiency, and ensuring process safety. However, the variations in payload mass and moment of inertia in single-link manipulator systems are often random and multimodal, making system modeling and state control difficult. Currently, most research on the stability analysis of single-link manipulator systems is based on Markov theory. The primary tool for stochastic stability analysis of single-link manipulator systems is Lyapunov stability theory. By constructing Lyapunov functions, stability conditions and controller existence conditions for single-link manipulators are derived based on linear matrix inequalities (LMIs). However, in Markov theory, the dwell time of modes has a memoryless exponential distribution, and the transition rate is independent of time. This results in a certain degree of conservatism in the system stability conditions and controller design schemes derived from Markov theory. At the same time, the various components of the robotic arm or different robotic arms usually need to realize data transmission and remote control through the network, and are accompanied by external disturbances, time delays, data packet loss and other phenomena. This puts higher requirements on the mastery of modal information between the system, event generator and controller, the utilization of limited network resources and the realization of system stability control. Therefore, it is urgent to propose a more accurate modeling method and a less conservative stability criterion to ensure the effective control of the operating state of the single-link robotic arm system. Summary of the Invention

[0003] The purpose of the present invention is to provide a single-link robotic arm system modeling and dual asynchronous event triggering control design method to solve the problems existing in the above-mentioned prior art.

[0004] The present invention provides a single-link robotic arm system modeling and dual-asynchronous event-triggered control design method based on Semi-Markov theory, comprising the following steps:

[0005] (1) Considering the random changes of payload mass and moment of inertia during the modeling process of the single-link manipulator system, a single-link manipulator system model is established;

[0006]

[0007] Where, is the system state variable; Control output for the system; It is the control input signal of the system; is the disturbance of the system; η(t) is the Semi-Markov jump process that describes the random changes of the payload mass and moment of inertia in the system; let η(t) = m, m = 1, 2, A m , B m , C m and D m is a system parameter matrix with appropriate dimensions.

[0008] (2) Design asynchronous event triggering conditions and controllers;

[0009] (2.1) Based on the Semi-Markov random process, a single-link robotic arm system is constructed, and the asynchronous event triggering conditions are designed as follows:

[0010] e T (t)Ω α(t) e(t)≤δ α(t) x T (t k d)Ω α(t) x(t k d)

[0011] Where, e(t)=x(t k d+cd)-x(t k d), x(t k d+cd) represents the current sampling data, x(t k d) represents the latest trigger data, δ α(t) ∈[0,1) is a parameter related to the system mode, Ω α(t) >0 indicates the event trigger weight matrix that needs to be determined, and α(t) indicates the modal information actually received by the event trigger condition. Let α(t) = p, p = 1, 2, η(t) and α(t) be asynchronous and meet the conditional probability P r {α(t)=p|η(t)=m}=ψ mp ,0≤ψ mp ≤1,

[0012] The holding interval of the zero-order holder is where t k+1 d=t k d+cd, Consider data delay in network transmission where 0≤τ(t)≤τ M, the event triggering condition is transformed into the following form:

[0013] e T (t)Ω p e(t)≤δ p [x(t-τ(t))-e(t)] T Ω p [x(t-τ(t))-e(t)]

[0014] (2.2) Further, the asynchronous event trigger controller is designed as follows:

[0015] u(t)=K β(t) x(t k d)

[0016] Where K β(t) is the controller gain to be determined, and β(t) represents the modal information actually received by the controller. Let β(t) = q, q = 1, 2, η(t) and β(t) be asynchronous and satisfy the conditional probability P r {β(t)=q|η(t)=m}=μ mq , 0≤μ mq ≤1,

[0017] (2.3) Combining the single-link robotic arm system model with the asynchronous event triggering conditions and controller, a closed-loop single-link robotic arm system model is obtained:

[0018]

[0019] (3) Construct a Lyapunov function that depends on the system modal changes and the dwell time within the mode; provide a stability criterion for the single-link manipulator system, and then implement the stability analysis of the single-link manipulator system;

[0020] Consider a single-link robotic arm system model. All conversion rates depend on the dwell time h. When the system mode changes, h is updated to 0. For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a series of positive definite matrices P with appropriate dimensions m ,Q,R,Ω p , a positive constant δ p , τ M ,γ, matrix K q This makes the following matrix inequality hold:

[0021]

[0022] in,

[0023]

[0024] I represents the identity matrix, then the single-link robotic arm system is said to be stochastically stable and satisfies H ∞ performance.

[0025] (4) Based on the derived stability criterion of the single-link manipulator system and related lemmas, the existence conditions of the asynchronous controller are derived, and the controller gain that ensures the stability of the single-link manipulator system is obtained, and then the asynchronous controller is designed;

[0026] For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a set of positive definite matrices P with appropriate dimensions m , Z, Q, R, Ω p , matrix Y q , and the positive constant δ p , τ M , σ, ε1, ε2, such that the following inequality holds:

[0027]

[0028] in,

[0029]

[0030] The single-link robotic arm system is said to be stochastically stable and satisfies H ∞ performance, and determine the controller gain as: K q =OS -1 M -1 SO T Y q .

[0031] (5) Controller performance test;

[0032] The linear matrix inequality toolbox in Matlab is used to determine whether the system parameter matrix under all given modes satisfies the controller design conditions given in step (4). If so, it is determined that the single-link robotic arm system is stochastically stable based on the designed asynchronous controller.

[0033] Beneficial effects: Compared with the existing technology, the present invention has the following advantages: 1. By introducing the Semi-Markov process, the random changes of the payload mass and moment of inertia of the single-link robotic arm system can be described more accurately and reasonably; 2. By performing stability analysis and controller design on the established single-link robotic arm system model based on the Semi-Markov theory, the dual-asynchronous event triggering control problem of the single-link robotic arm system can be solved. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0035] Figure 1 is a flow chart of the steps of the present invention;

[0036] Figure 2 Schematic diagram of a single-link robotic arm system;

[0037] Figure 3 This is a modal change diagram of a single-link robotic arm system, event triggering conditions, and controller based on Semi-Markov theory;

[0038] Figure 4 It is the release time and release interval diagram of the event triggering condition;

[0039] Figure 5 This is the system state response diagram of the single-link robotic arm closed-loop system based on Semi-Markov theory;

[0040] Figure 6 This is the system state response diagram of the single-link robotic arm open-loop system based on Semi-Markov theory. DETAILED DESCRIPTION

[0041] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0042] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0043] like Figure 1 As shown, a single-link robotic arm system modeling and dual asynchronous event-triggered control design method based on Semi-Markov theory includes the following steps:

[0044] (1) Modeling of a single-link robotic arm system based on Semi-Markov theory

[0045] Combine Figure 2 The single-link robotic arm system shown in the figure can be described in the following form:

[0046]

[0047] Where θ(t) represents the angular position of the manipulator, L represents the length of the manipulator, M represents the mass of the payload, g represents the acceleration due to gravity, J represents the moment of inertia, ν(t) represents the viscous friction coefficient, and u(t) represents the control input.

[0048] Considering the random variation of payload mass and moment of inertia, the single-link manipulator model is established through a Semi-Markov process with two modes. Based on the Semi-Markov theory, considering the existence of external disturbance ω(t), the state space model of the single-link manipulator system is established as:

[0049]

[0050] Where m (η(t) = m) represents the Semi-Markov process and takes values ​​of 1 and 2, L takes a value of 0.5 m, M1 takes a value of 0.12 kg, M2 takes a value of 0.39 kg, and g takes a value of 9.81 m / s 2 ν(t)=ν(0) takes the value of 2kg·m 2 / s, and the value of J1 is 6.67 kg·m 2 , J2 is 10kg·m 2 The variation of this Semi-Markov process is based on the following transition probabilities:

[0051]

[0052] Where, Furthermore, the system parameters can be expressed as:

[0053]

[0054] (2) Asynchronous event triggering conditions and controller design

[0055] For the single-link robotic arm system, the asynchronous event triggering conditions are designed as follows:

[0056] e T (t)Ω α(t) e(t)≤δ α(t) x T (t k d)Ω α(t) x(t k d) (3)

[0057] Where, e(t)=x(t k d+cd)-x(t k d), x(tk d+cd) represents the current sampling data, x(t k d) represents the latest trigger data, δ α(t) ∈[0,1) is a parameter related to the system mode, Ω α(t) >0 indicates the event trigger weight matrix that needs to be determined, and α(t) indicates the modal information actually received by the event trigger condition. Let α(t) = p, p = 1, 2, η(t) and α(t) be asynchronous and meet the conditional probability P r {α(t)=p|η(t)=m}=ψ mp ,0≤ψ mp ≤1,

[0058] The holding interval of the zero-order holder is where t k+1 d=t k d+cd, Consider data delay in network transmission where 0≤τ(t)≤τ M , the event triggering condition is transformed into the following form:

[0059] e T (t)Ω p e(t)≤δ p [x(t-τ(t))-e(t)] T Ω p [x(t-τ(t))-e(t)] (4)

[0060] Furthermore, the asynchronous event triggering controller is designed as follows:

[0061] u(t)=K β(t) x(t k d) (5)

[0062] Where K β(t) is the controller gain to be determined, and β(t) represents the modal information actually received by the controller. Let β(t) = q, q = 1, 2, η(t) and β(t) be asynchronous and satisfy the conditional probability P r {β(t)=q|η(t)=m}=μ mq , 0≤μ mq ≤1,

[0063] Combining the single-link robotic arm system model with the asynchronous event triggering conditions and controller, a closed-loop single-link robotic arm system model is obtained:

[0064]

[0065] (3) Stability criterion method for single-link robotic arm system

[0066] Criteria: Consider a single-link robotic arm system model. All conversion rates depend on the dwell time h. When the system mode changes, h is updated to 0. For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a series of positive definite matrices P with appropriate dimensions m ,Q,R,Ω p , a positive constant δ p , τ M ,γ, matrix K q This makes the following matrix inequality hold:

[0067]

[0068] in,

[0069]

[0070] I represents the identity matrix, * represents the symmetric terms in the matrix, then the single-link manipulator system is said to be stochastically stable and satisfies H ∞ performance.

[0071] Proof: For the single-link robotic arm system model, choose a Lyapunov function that depends on the system modal changes and the dwell time within the mode:

[0072]

[0073] Where, P m is a positive definite matrix to be determined.

[0074] Then, we can find the infinitesimal operator for equation (8) and use the cumulative distribution function and probability density function to get:

[0075]

[0076] By using Jensen's inequality, we can get:

[0077]

[0078] Considering the event triggering condition (4), we can get:

[0079]

[0080] Where, Based on formula (7), the system satisfies H ∞ Performance indicators.

[0081] When ω(t) = 0, based on formula (7), we can get:

[0082]

[0083] Where, matrix It can be deduced from formula (7). Then, it can be deduced:

[0084]

[0085] According to the Dynkin formula, we can get:

[0086]

[0087] Finally, when T approaches infinity, we can get:

[0088]

[0089] Get the certificate.

[0090] (4) Controller gain solution

[0091] First, the lemma used to solve the event triggering related parameters and controller gains is given.

[0092] Lemma 1: Given real matrices A, B, C, X, W1, W2, W3 of appropriate dimensions, there exists a matrix P = P T >0 makes:

[0093]

[0094] If and only if there exists a scalar σ>0, the matrix P=P T > 0, the matrix Z is such that:

[0095]

[0096] Lemma 2: For a full-rank matrix B, rank(B) = n u , The singular value decomposition of B can be described as Among them O·O T =I,V·V T =I. Matrix Z T >0, Existence Matrix Make If and only if the following conditions hold:

[0097]

[0098] Criteria: For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a series of positive definite matrices P with appropriate dimensions m , Z, Q, R, Ω p , matrix Y q , and the positive constant δp , τ M , σ, ε1, ε2, such that the following inequality holds:

[0099]

[0100] in,

[0101]

[0102] The single-link robotic arm system is said to be stochastically stable and satisfies H ∞ performance, and the controller gain can be determined as: K q =OS -1 M -1 SO T Y q .

[0103] Proof: By Lemma 1, inequality (7) is equivalent to:

[0104]

[0105] in,

[0106]

[0107] Multiply both sides of inequality (13) by the matrix diag{I,I,I,I,I,I,Z,I}, considering the inequality Go to process item-ZR -1 Z, definition According to Lemma 2, for exist Make make We can get: Then we can get (11) and (12).

[0108] The embodiments of the present invention are described below:

[0109] Single-link robotic arm system Figure 2 As shown, the modeled related system matrix is ​​given as follows:

[0110] D1=D2=[0 1],τ M =0.01, δ1=0.1, δ2=0.2ε1=ε2=0.5, φ 12 = φ 21 =0.5, σ=0.01

[0111] Set the perturbation to:

[0112]

[0113] Given the conditional probability matrix Γ=[ψ mp ] and Υ=[μ mq ]as follows:

[0114]

[0115] Based on the above parameters, the method of the present invention is used to simulate and test the single-link robotic arm system. Figure 3 Describes the single-link robotic arm system based on Semi-Markov theory, event triggering conditions, and modal changes of the controller. Figure 4 Describes the release time and release interval of the event trigger scheme, Figure 5 Depicts the system state response of a single-link robotic arm closed-loop system based on Semi-Markov theory. Figure 6 The system state response of a single-link manipulator open-loop system based on Semi-Markov theory is depicted.

[0116] The embodiments described above are merely descriptions of preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by persons skilled in the art should fall within the scope of protection defined by the claims of the present invention.

Claims

1. A single-link robotic arm system modeling and dual-asynchronous event-triggered control design method, characterized in that: The following steps are involved: Step S1: Establish a single-link robotic arm system model; Step S2: Designing asynchronous event triggering conditions and controllers, and combining them with the single-link robotic arm system model to obtain a closed-loop single-link robotic arm system model; Step S3: Given a random stability condition for the single-link robotic arm system model; Step S4: solving the asynchronous controller gain and designing the asynchronous controller; Step S5: controller performance test; In step S2, the asynchronous event triggering conditions are designed as follows: e T (t)Ω α(t) e(t)≤δ α(t) x T (t k d)Ω α(t) x(t k d) Where, e(t)=x(t k d+cd)-x(t k d), x(t k d+cd) represents the current sampling data, x(t k d) represents the latest trigger data, δ α(t) ∈[0,1) is a parameter related to the system mode, Ω α(t) >0 indicates the event trigger weight matrix that needs to be determined, α(t) indicates the modal information actually received by the event trigger condition; let α(t) = p, p = 1, 2, η(t) and α(t) be asynchronous and satisfy the conditional probability P r {α(t)=p|η(t)=m}=ψ mp ,0≤ψ mp ≤1, The holding interval of the zero-order holder is where t k+1 d=t k d+cd, Consider data delay in network transmission where 0≤τ(t)≤τ M , the event triggering condition is transformed into the following form: e T (t)Ω p e(t)≤δ p [x(t-τ(t))-e(t)] T Ω p [x(t-τ(t))-e(t)]。 2. A single-link robotic arm system modeling and dual asynchronous event-triggered control design method according to claim 1, characterized in that: The single-link robotic arm system model is established in step S1 as follows: Where, is the system state variable; Control output for the system; It is the control input signal of the system; is the disturbance of the system; η(t) is the Semi-Markov jump process that describes the random changes of the payload mass and moment of inertia in the system; let η(t) = m, m = 1, 2, A m , B m , C m and D m is a system parameter matrix with appropriate dimensions.

3. The single-link robotic arm system modeling and dual-asynchronous event-triggered control design method according to claim 1 is characterized in that: In step S2, the asynchronous event triggering controller is designed as follows: u(t)=K β(t) x(t k d) Where K β(t) is the controller gain to be determined, β(t) represents the modal information actually received by the controller; let β(t) = q, q = 1, 2, η(t) and β(t) be asynchronous and satisfy the conditional probability P r {β(t)=q|η(t)=m}=μ mq , 0≤μ mq ≤1, 4. A single-link robotic arm system modeling and dual-asynchronous event-triggered control design method according to claim 3, characterized in that: In step S2, the single-link robotic arm system model is combined with the asynchronous event triggering condition and the controller to obtain a closed-loop single-link robotic arm system model:

5. The single-link robotic arm system modeling and dual-asynchronous event-triggered control design method according to claim 4 is characterized in that: The step S3 is specifically as follows: Consider a single-link robotic arm system model. All conversion rates depend on the dwell time h. When the system mode changes, h is updated to 0. For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a series of positive definite matrices P with appropriate dimensions m ,Q,R,Ω p , a positive constant δ p , τ M ,γ, matrix K q This makes the following matrix inequality hold: in, φ mn (h) represents the transfer rate, I represents the unit matrix, and * represents the symmetric term in the matrix. The single-link manipulator system is said to be stochastically stable and satisfies H ∞ performance.

6. The single-link robotic arm system modeling and dual-asynchronous event-triggered control design method according to claim 5 is characterized in that: The step S4 is specifically as follows: For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a set of positive definite matrices P with appropriate dimensions m , Z, Q, R, Ω p , matrix Y q , and the positive constant δ p , τ M ,`,σ,ε1,ε2, so that the following inequality holds: in, φ mn represents the lower limit of the transfer rate, represents the upper limit of the transfer rate, then the single-link robotic arm system is said to be stochastically stable and satisfies H ∞ performance, and determine the controller gain as: K q =OS -1 M -1 SO T Y q , O, S, M are matrices and satisfy the following for matrix B m =B, the singular value decomposition is matrix 7. The single-link robotic arm system modeling and dual-asynchronous event-triggered control design method according to claim 6 is characterized in that: The step S5 is specifically as follows: The linear matrix inequality toolbox in Matlab is used to determine whether the system parameter matrix under all given modes satisfies the controller design conditions given in step S4. If so, it can be determined that the single-link robotic arm system is stochastically stable based on the designed asynchronous event triggering conditions and controller.

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