A dynamic event-triggered control method for single-link manipulator system based on dynamic uniform quantizer

Abstract

This invention discloses a dynamic event-triggered control method for a single-link robotic arm system based on a dynamic uniform quantizer. The method first establishes a Markov model of the single-link robotic arm system based on Markov jump system theory, considering the system model with a general transition rate. Next, a dynamic uniform quantizer is designed to improve control accuracy and reduce bandwidth usage by quantizing the input. A threshold-adjustable and modally dependent dynamic event-triggered mechanism is designed to reduce communication transmission frequency, and the absence of the Zeno phenomenon is proven. Finally, vertex separator processing is introduced to address the uncertainty in the Markov transition rate. A state feedback controller is designed to control the bounded stability of the single-link robotic arm system. When applied to a single-link robotic arm system, this method ensures the normal operation of the system under disturbances.

Description

Technical Field

[0001] This invention relates to a dynamic event-triggered control method, specifically a dynamic event-triggered control method for a single-link robotic arm system based on a dynamic uniform quantizer. Background Technology

[0002] Many real-world systems exhibit random fluctuations in their structure and parameters. These random fluctuations often arise from random failures and repairs of system components, changes in internal interconnections, sudden environmental changes, and variations in the operating point range of nonlinear systems after linearization. Single-link robotic arms are highly susceptible to factors such as temperature, humidity, and electromagnetic interference. Using Markov-based theory of jumping systems, a more accurate system model of the single-link robotic arm can be established. Furthermore, single-link robotic arms often struggle to detect measurement faults masked by interference during operation, which can lead to performance degradation. State feedback control, as a crucial component of robust control, is essential for improving system safety and reliability. State feedback involves feeding back any state of the system to the input in a specific proportion, combining it with the system's reference input to form a control law, which serves as the control input for the controlled system. Based on this concept, researchers have conducted extensive research on the design of state feedback controllers, achieving a series of results.

[0003] Furthermore, newly built intelligent control systems typically employ networked control methods that are easy to install and highly scalable. However, due to limited network bandwidth, data packets inevitably experience delays, packet loss, and timing discrepancies during network transmission. Event-triggered control becomes particularly important in reducing communication transmission frequency without sacrificing ideal stability and performance. In an event-triggered control environment, communication transmission only occurs when preset conditions are violated. Secondly, quantitative control becomes crucial in reducing bandwidth consumption without sacrificing ideal stability and performance.

[0004] However, the aforementioned research mainly focuses on static quantization control and static event-triggered control, offering limited reduction in network load. Therefore, it is necessary to employ dynamic uniform quantizers to save bandwidth usage and dynamic event-triggered strategies to conserve communication resources. Summary of the Invention

[0005] The purpose of this invention is to propose a dynamic event triggering control method for a single-link robotic arm system based on a dynamic uniform quantizer, which can effectively reduce communication transmission frequency, improve control accuracy, and reduce bandwidth usage.

[0006] The specific technical solution of the present invention is as follows: A dynamic event triggering control method for a single-link robotic arm system based on a dynamic uniform quantizer, comprising the following steps:

[0007] Based on the Markov jump system theory, the following dynamic model is established for the single-link robotic arm system in reference [1]:

[0008]

[0009] The above dynamic model can be modeled as the following state-space expression:

[0010]

[0011] In the formula, θ(t) represents the angle of the robotic arm, u(t) represents the control input, w(t) represents the external disturbance, M represents the mass of the load, J represents the moment of inertia, g represents the gravitational acceleration, L represents the length of the robotic arm, and D(t) represents the uncertainty coefficient of viscous friction.

[0012] The single-link robotic arm system is modeled as a Markov transition system with a general transfer rate, as follows:

[0013]

[0014] In the formula, A(r(t)), ΔA(r(t)), B(r(t)), C(r(t)), and D(r(t)) are known system matrices of appropriate dimensions, and g(t, r(t), x(t)) is a known nonlinear perturbation. For simplification, let A... i = A(r(t)), and other matrices are similarly abbreviated. r(t) is a right-continuous Markov process that takes values ​​from a finite set cS = {1, 2, ..., N}. r(t) has the following properties:

[0015]

[0016] In the formula, π ij It is the transition rate from mode i to mode j, and satisfies:

[0017]

[0018] Furthermore, when π in Λ ij When there is general uncertainty, the transfer rate matrix Λ can be expressed as:

[0019]

[0020] In the formula, and Δ ij It is known that Δ ij ∈[-δ ij δ ij ], δ ij "?" represents a known element, while "?" represents an unknown element.

[0021] Furthermore, a dynamic uniform quantizer is designed to improve control accuracy and reduce bandwidth usage by quantizing the input. The specific steps are as follows:

[0022] First, the dynamic uniform quantizer is designed as follows:

[0023]

[0024] In the formula, the quantization sensitivity parameter τ(t) satisfies:

[0025]

[0026] In the formula, T * ,k,v and It is a scalar, q(·) represents the floor function, and t0 represents the initial time.

[0027] Furthermore, a dynamic event triggering mechanism with adjustable threshold and modality dependence is designed to reduce the communication transmission frequency. The specific steps are as follows:

[0028] Establish the following dynamic event-triggered sampling mechanism:

[0029] t k+1 =inf{t|t>t k ,[x(t) k )-x(t)] T Ψ i [x(t k )-x(t)]≥σ i (txt k )] T Ψ i [x(t k )]}

[0030] In the formula, x(t) k )-x(t) represents the input trigger error, t k Indicates the last trigger time, t k+1 Indicates the next triggering time, Ψ i Let σ represent the weighted matrix. i (t) is a dynamically triggered parameter and is updated by the following formula:

[0031] σ i (t)=σ im -σ im tanh(f(t))

[0032]

[0033] In the formula, σ im These represent the upper bounds of the trigger threshold, and ρ(t) is a scalar.

[0034] Next, we will prove that there is no Zeno phenomenon:

[0035]

[0036] a=(||A i ||+β max ||M i ||||N i ||+||L i ||)

[0037] c = (||A i ||+β max ||M i ||||N i ||+α max ||B i K i ||+||L i ||)||x(t k )||+α max ||B i K i ||τ cm +||D i ||w m

[0038] In the formula, F i (t) represents the time-varying function, e(t) represents the triggering error, and α max ,β max ,κ,||L i || is a scalar, M i N i Given a matrix, K i Indicates the controller gain;

[0039]

[0040]

[0041]

[0042]

[0043] This proves that there is no Zeno phenomenon. Wherein, λ max (Ψi), λ min (Ψ i ) represent the maximum and minimum values ​​of the eigenvalues ​​of the weighted matrix, respectively.

[0044] Furthermore, a vertex separator is introduced to address the uncertainty in the Markov transition rate. A state feedback controller is designed to ensure the bounded stability of a single-link robotic arm system. The specific steps are as follows:

[0045] C001: In the formula, the Lyapunov function of the following form is selected:

[0046] V(x(t),i)=V1(x(t),i)+V2(x(t),i)

[0047] V1(x(t), i) = x T (t)P i x(t)

[0048]

[0049] C002: In the formula, i represents the mode, η is a scalar, P is the Lyapunov variable matrix, and ||U i x(τ)|| denotes the weighted matrix, ||g i (x(τ)), where τ|| represents the nonlinear perturbation;

[0050] C003: Calculate the derivative of V(t) and consider event triggering, where:

[0051]

[0052]

[0053] C004: In the formula, e τc (t) represents the quantization error. It is a scalar, and ||||2 represents the 2-norm;

[0054] C005: Based on the above transformation, consider H ∞ The performance can be obtained by the following formula:

[0055]

[0056]

[0057] C006: Select ξ T (t)=[x T (t)g T (x(t))e T (t)w T (t)];

[0058] C007: In the formula, z(t) represents the control strategy, and γ represents H. ∞ Performance level indicators;

[0059] C008: Furthermore, a vertex separator is introduced to address the uncertainty in Markov transition rates, as shown in the lemma below:

[0060]

[0061]

[0062] C009: Further, Represents the vertex separator. Represents a set whose transition rate is unknown:

[0063] C010: Furthermore, by substituting the vertex separation method into the stability analysis, we can finally obtain the following equation:

[0064]

[0065]

[0066]

[0067]

[0068]

[0069]

[0070] C011: In the formula, It is a 6x6 matrix, Δ ij ∈[-δ ij δ ij ], This represents a set whose transition rate is unknown. The single-link robotic arm system is bounded and stable under the state feedback controller designed in this invention. Attached Figure Description

[0071] Figure 1 This is a flowchart of a method according to an embodiment of the present invention;

[0072] Figure 2 This is a state diagram of a single-link robotic arm system using the method proposed in this invention under state feedback control, as shown in the embodiment.

[0073] Figure 3 The Markov mode transition diagram used in this embodiment employs the method proposed in this invention;

[0074] Figure 4 This is an example of an event triggering diagram using the method proposed in this invention;

[0075] Figure 5 This is a graph showing the change in modality-dependent event triggering thresholds using the method proposed in this invention in an embodiment. Detailed Implementation

[0076] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading the present invention, any modifications of the present invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0077] like Figure 1 As shown, a dynamic event-triggered control method for a single-link robotic arm system based on a dynamic uniform quantizer includes the following steps:

[0078] Step 1: Set initial values ​​for each parameter;

[0079] Step 2: Update the threshold parameter σ i (t);

[0080] Step 3: Use the threshold parameter σ i Verify the event triggering condition with system state x(t), and update the triggering state x(t). k );

[0081] Step 4: Use trigger to output x(t) k ), update controller input u(t) k );

[0082] Step 5: Quantize the input u(t) using a quantizer. k ), to obtain f(u(t) k ));

[0083] Step 6: Add f(u(t) k The data is transmitted to the ZOH and then to the controlled object.

[0084] Step 7: Repeat steps 3, 4, 5, and 6 until the runtime ends.

[0085] An embodiment of the present invention is described below:

[0086] Consider a single-link robotic arm system from reference [1], whose corresponding dynamic model is:

[0087]

[0088]

[0089] Figure 1 This is a flowchart of a method according to an embodiment of the present invention; applying the proposed method, the state of the single-link robotic arm system under state feedback control is as follows: Figure 2 As shown, the Markov mode transition is as follows Figure 3 As shown, the event triggering graph is as follows: Figure 4 As shown, the event triggering threshold changes under modality dependency as follows: Figure 5 As shown, the proposed dynamic uniform quantizer effectively reduces bandwidth usage, and the proposed event-triggered mechanism effectively reduces communication transmission.

[0090] References

[0091] [1]Cao Z, Niu Y, Song J. Finite-time sliding-mode control of Markovianjump cyber-physical systems against randomly occurring injection attacks[J]. IEEE Transactions on Automatic Control, 2019, 65(3): 1264-1271.

[0092] [2]Li LW, Yang G H.Stabilisation of Markov jump systems with input quantisation and general uncertain transition rates[J]. IET Control Theory&Applications, 2017, 11(4): 516-523.

Claims

1. A dynamic event-triggered control method for a single-link manipulator system based on a dynamic uniform quantizer, characterized in that, Includes the following steps: Based on the Markov jump system theory, a Markov model of a single-link robotic arm system is established, and a system model with a general transfer rate is also considered. Design a dynamic uniform quantizer to improve control accuracy and reduce bandwidth usage by quantizing the input; A threshold-adjustable and modality-dependent dynamic event triggering mechanism is designed to reduce the communication transmission frequency, and it is proven that there is no Zeno phenomenon. To address the uncertainty in Markov transition rates, a vertex separator is introduced. A state feedback controller is designed to ensure the bounded stability of a single-link robotic arm system. The specific steps are as follows: B001: Choose a Lyapunov function of the following form: B002: In the formula, Represents mode, It is a scalar. It is a Lyapunov variable matrix. Represents a weighted matrix. Indicates nonlinear perturbation; B003: Calculation The derivative of the event and considering event triggering, where: B004: In the formula, Indicates quantization error. , It is a scalar. Represents the 2-norm; B005: Based on the above transformation, consider The performance can be obtained by the following formula: B006: Select B007: In the formula, Indicates the control strategy. express Performance level indicators; B008: Furthermore, a vertex separator is introduced to address the uncertainty in Markov transition rates, as shown in the following lemma: B009: Further, , , Represents the vertex separator. Represents a set whose transition rate is unknown: B010: Furthermore, substituting the vertex separation method into the stability analysis, we can finally obtain the following equation: B011: In the formula, It is a 6x6 matrix. , Represents a set whose transition rate is unknown; The single-link robotic arm system is bounded and stable under the designed state feedback controller.

2. The dynamic event triggering control method for a single-link robotic arm system based on a dynamic uniform quantizer according to claim 1, characterized in that, Based on the Markov transition system theory, a Markov transition model of a single-link robotic arm with a general transfer rate is established: The dynamic equations of a single-link robotic arm are as follows: The above dynamic equations can be modeled as the following state-space expression: In the formula, Indicates the angle of the robotic arm. Indicates control input, Indicates external disturbance. The mass of the load, Indicates the moment of inertia. Represents gravitational acceleration. Indicates the length of the robotic arm. The uncertainty factor representing viscous friction; The single-link robotic arm system is modeled as a Markov transition system with a general transfer rate, as follows: In the formula, , , , , It is a known system matrix of appropriate dimensions. It is a known nonlinear perturbation; for simplicity, let Other matrices are similarly abbreviated; It is a right-continuous Markov process and originates from a finite set. Take the value from; It has the following properties: In the formula, , , From modality To mode The transfer rate, and satisfies: Furthermore, when In When there is general uncertainty, the transfer rate matrix It can be represented as In the formula, and It is known. , "?" represents a known element, while "?" represents an unknown element.

3. The dynamic event triggering control method for a single-link robotic arm system based on a dynamic uniform quantizer according to claim 2, characterized in that, A dynamic uniform quantizer was designed to improve control accuracy and reduce bandwidth usage by quantizing the input. The specific steps are as follows: First, the dynamic uniform quantizer is designed as follows: Quantization sensitivity parameters satisfy: In the formula, , , and It is a scalar. This represents the floor function. Indicates the initial time.

4. The dynamic event triggering control method for a single-link robotic arm system based on a dynamic uniform quantizer according to claim 3, characterized in that, Design a dynamic event triggering mechanism with adjustable threshold and modality dependence to reduce communication transmission frequency. The specific steps are as follows: Establish the following dynamic event-triggered sampling mechanism: In the formula, Indicates input trigger error. Indicates the last time it was triggered. Indicates the next trigger time. Represents a weighted matrix. It is a dynamically triggered parameter and is updated by the following formula: In the formula, These represent the upper bounds of the trigger threshold, It is a scalar; Next, we will prove that there is no Zeno phenomenon: In the formula, Represents a time-varying function. Indicates triggering error. , , , It is a scalar. , It is a known matrix. Indicates the controller gain; The above equation proves that there is no Zeno phenomenon, where, , These represent the maximum and minimum values ​​of the eigenvalues ​​of the weighted matrix, respectively.

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