A direct current motor driving single-link mechanical arm system control method based on dynamic event triggering

Abstract

This invention discloses a control method for a singular Markov system of a DC motor-driven single-link robotic arm based on dynamic event triggering. Addressing the complex control challenges posed by the singular characteristics of the system caused by the extremely small armature inductance, random fault jumps in motor parameters, and the incomplete knowledge of the system's mode transition rates (general transition rates) during actual robotic arm operation, this invention proposes a fault-tolerant control strategy. First, a singular Markov jump dynamic model of the robotic arm with parameter jumps is established. Second, an internal dynamic variable is introduced to design a dynamic event-triggered communication mechanism, dynamically adjusting the data transmission frequency according to real-time fluctuations in the system state, effectively saving network communication bandwidth. Subsequently, considering the general transition rate matrix, a mode-dependent closed-loop controller is constructed, and sufficient conditions to ensure asymptotic stability of the system are given. Furthermore, this invention rigorously proves that the dynamic triggering mechanism has a strictly positive minimum trigger interval, completely eliminating the Zeno phenomenon. This invention not only effectively tolerates sudden hardware failures of the DC motor but also significantly reduces communication energy consumption while ensuring high-precision, high-reliability, and stable control of the robotic arm system, possessing strong engineering and physical feasibility.

Description

Technical Field

[0001] This invention relates to a method for achieving stable control of a DC motor-driven single-link robotic arm, specifically to a general transfer rate singular Markov system control method based on dynamic event triggering. Background Technology

[0002] DC motor-driven single-link robotic arms, as the most basic motion unit of industrial robots, are widely used in precision manufacturing, automated assembly, and other fields. This robotic arm is essentially a complex electromechanical coupling system. In practical modeling, since the armature inductance of the motor is usually extremely small, neglecting it will result in a "singular" characteristic where differential and algebraic equations coexist in the system's dynamics model. Furthermore, under long-term complex operating conditions, robotic arms often encounter hardware failures such as sudden resistance changes due to motor coil overheating or sudden bearing wear. These random abrupt changes in parameters are typically precisely described mathematically by Markov jump systems. However, most existing research on singular Markov robotic arm systems assumes the transfer rate matrix is ​​completely known. In real engineering environments, limited by measurement technology, sensor costs, or severe interference, the system often contains some unknown or difficult-to-measure transfer rates. Ignoring this reality and directly applying existing methods will lead to overly conservative controller designs, or even complete failure in actual physical systems.

[0003] On the other hand, modern robotic arm systems typically operate within communication networks with limited bandwidth. Traditional periodic sampling control generates a large amount of redundant signals, easily leading to network congestion and energy waste. Although existing event-triggered mechanisms can save communication resources to some extent, they mostly employ static triggering conditions with fixed parameters, lacking the flexibility to dynamically adapt to the system in real time, resulting in suboptimal utilization of network resources. More critically, in complex systems combining singular characteristics, parameter jumps, and incompletely known transfer rates, existing triggering strategies often struggle to rigorously eliminate the "Zeno phenomenon" in mathematical theory, which can cause controller hardware jamming or damage in practical engineering. Therefore, designing a control method that can dynamically save communication bandwidth and completely avoid the Zeno phenomenon is particularly crucial and important for robotic arm singular Markov systems with general transfer rates. Summary of the Invention

[0004] The purpose of this invention is to propose a singular Markov system control method for a DC motor-driven single-link robotic arm based on dynamic event triggering. This method effectively solves the fault-tolerant control problem of the system when the transfer rate is not fully known. While significantly reducing the network communication load, it strictly ensures the asymptotic stability of the electromechanical system and completely avoids the Zeno phenomenon.

[0005] The specific technical solution of the present invention is as follows: A method for achieving stable control of a DC motor-driven single-link robotic arm, comprising the following steps:

[0006] Based on the theory of singular Markov jump systems, the following dynamic model is established for the DC motor driven single-link robotic arm system in reference [1]:

[0007]

[0008] In the formula, x1(t), x2(t), and x3(t) represent the angle, angular velocity, and armature current, respectively; g represents the acceleration due to gravity; l represents the pendulum length; J represents the moment of inertia; and B... f K represents the viscous friction damping coefficient. t K represents the motor torque constant. e R(η) represents the back electromotive force constant. t ) represents the armature resistance, and u(t) represents the control input;

[0009] The DC motor-driven single-link robotic arm system is modeled as a singular Markov jump system with a general transfer rate, as follows:

[0010]

[0011] In the formula, A δ(t) B 6(t) C δ(t) G δ(t) H δ(t) Let w(t) be a known system matrix of appropriate dimension, w(t) be a known nonlinear perturbation, and δ(t) represent a Markov random process whose values ​​take the form of a finite set S = {1, 2, ..., N}. For simplification, let δ(t) = i. δ(t) has the following properties:

[0012]

[0013] In the formula, Δ>0, π ij It is the transition rate from mode i to mode j, and satisfies:

[0014]

[0015] Furthermore, when π ij When there is general uncertainty, the transfer rate matrix can be expressed as

[0016]

[0017] In the formula, and Δ ij It is known that Δij ∈[-α ij , α ij , α ij is known. Assume is a singular matrix with rank rank(E) = r < n. Define the full column rank matrix E L , E R ∈R n×(n-r) , satisfying EE L E R T = E, and define the matrix satisfying RE = 0, ES = 0 for later use;

[0018] To effectively save the network communication bandwidth resources of the DC motor-driven single-link manipulator system and strictly avoid the occurrence of Zeno phenomenon, a signal transmission strategy based on a dynamic event-triggering mechanism is designed by introducing internal dynamic variables:

[0019] t k+1 = inf{t > t k |η(t) + θ(σx T (t)Ω δ(t) x(t) - e T (t)Ω δ(t) e(t)) ≤ 0}

[0020] where the triggering error e(t) = x(t k ) - x(t), t ∈ [t k , t k+1 ), θ > 0, 0 < σ < 1, t k represents the triggering moment, t k+1 represents the next triggering moment, Ω i represents the weight matrix, η(t) represents the internal dynamic variable, which is simplified to η later and assumed to satisfy:

[0021]

[0022] where the initial condition η (0) = η0 ≥ 0 and λ > 0;

[0023] To control the DC motor-driven single-link manipulator system, an event-triggered mode-dependent state feedback controller is constructed: u(t) = K δ(t) x(t k ), t ∈ [t k , t k+1 ), where K δ(t) is the mode-dependent gain matrix to be determined for the state feedback controller. According to the triggering error e(t) = x(t k)-x(t), thus obtaining the closed-loop control system:

[0024] Next, we prove the absence of the Zeno phenomenon: based on e(t) = x(t) k )-x(t), obviously we can obtain

[0025] because Then there exist real numbers ∈ = ∈(t) > 0 such that So,

[0026]

[0027] in

[0028] Considering the conditions for triggering dynamic events, we have:

[0029]

[0030] in Because (|||x(t) k )||-||e(t)||)2≤||x(t k )-e(t)|| 2 The following inequalities are necessary conditions for the event to be triggered:

[0031] So,

[0032]

[0033] Based on the above formula, we can obtain

[0034]

[0035] Since 0 < σ < 1, it can be guaranteed that τ > 0, thus proving that there is no Zeno phenomenon.

[0036] The control method for a singular Markov system of a DC motor-driven single-link robotic arm based on dynamic event triggering as described in claim 1 is characterized by introducing vertex separation technology to handle the uncertainty problem of the transfer rate of the singular Markov jump system, and controlling a single-link robotic arm system to be bounded and stable by designing a state feedback controller. The specific steps are as follows:

[0037] Choose the following form of Lyapunov function

[0038] V(t) = x T (t)E T P i Ex(t)+η(t)

[0039] In the formula, i represents the mode, and P i It is a Lyapunov variable matrix.

[0040] Order Λ i =P i E+R T Φ i S T A ci =A i +B i K i Calculate the derivative of V(t), where:

[0041]

[0042]

[0043] Therefore, we have:

[0044]

[0045]

[0046] choose

[0047]

[0048] right Shure supplementation

[0049]

[0050] Furthermore, a fixed-point separation technique is introduced to address the uncertainty in singular Markov transition rates, as shown in the following lemma:

[0051]

[0052] Slack variables are introduced;

[0053] Furthermore, construct the matrix.

[0054]

[0055]

[0056] In the formula,

[0057]

[0058] definition right Contract modification get:

[0059]

[0060] In the formula,

[0061] It is easy to obtain from the constructed matrix inequality This proves to be true.

[0062] Combining vertex separation techniques, we can finally obtain the following formula:

[0063]

[0064]

[0065] Attached Figure Description

[0066] Figure 1 This is a flowchart of a method according to an embodiment of the present invention;

[0067] Figure 2 This is a state diagram of a DC motor-driven single-link robotic arm system using the method proposed in this invention under state feedback control, as shown in the embodiment.

[0068] Figure 3 The example uses a singular Markov mode transition diagram based on the method proposed in this invention;

[0069] Figure 4 This is an example of an event triggering diagram using the method proposed in this invention; Detailed Implementation

[0070] 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.

[0071] A control method for a singular Markov system of a DC motor-driven single-link robotic arm based on dynamic event triggering includes the following steps;

[0072] Step 1: Set the initial values ​​of various parameters, including the initial state of the robotic arm system x(0), the initial value of the Markov mode, and the initial value of the internal dynamic variables η(0);

[0073] Step 2: Based on the current state of the system, update the internal dynamic variable η(t) in real time.

[0074] Step 3: Verify the dynamic event triggering condition using the internal dynamic variable η(t) and the latest sampled state of the system. If the triggering condition is met, update the triggering time t. kand triggering state x(t) k );

[0075] Step 4: Utilize the latest trigger to output x(t) k Given the current Markov jump mode, calculate and update the mode-dependent controller input u(t). k );

[0076] Step 5: Input the updated controller into u(t) k The signal is transmitted to the zero-order hold (ZOH) and held until the next trigger time.

[0077] Step 6: Apply the held control signal to the controlled object of the DC motor-driven single-link robotic arm. Considering the general transfer rate, solve the singular Markov electromechanical coupling dynamics equation and update the actual continuous state x(t) of the system:

[0078] Step 7: Repeat steps 2, 3, 4, 5, and 6, continuously monitoring the triggering conditions until the system runtime ends. During this process, strictly avoid the Zeno phenomenon, and finally achieve asymptotic stability of the robotic arm system.

[0079] An embodiment of the present invention is described below:

[0080] Consider a single-link robotic arm system driven by a DC motor. Its corresponding dynamic model is as follows:

[0081]

[0082] System parameters are set as follows:

[0083] C1 = [10.51], H1 = 0.1

[0084] C2 = [1 - 0.1 - 0.1], H2 = 0.1

[0085] The simulation results demonstrate that the general transfer rate singular Markov system control method based on dynamic event triggering has good hardware fault tolerance control and network bandwidth saving effect for the DC motor driven single-link robotic arm system. It also strictly avoids the Zeno phenomenon and improves the overall stability, security and network resource utilization of the system.

[0086] Based on the above Figure 2 It can be seen that the joint angle, angular velocity, and armature current of the DC motor-driven single-link robotic arm system are all in an asymptotically stable state under the control of this method. Figure 3 , 4These represent the Markov mode transitions and dynamic event triggering time intervals that occur during system operation, respectively. It can be observed in real time that the system state can still converge smoothly when the robotic arm encounters a sudden motor parameter transition fault, and the triggering time interval is strictly greater than zero. This can effectively save network bandwidth and accurately verify that no Zeno phenomenon occurs.

[0087] References

[0088] [1]Wang G, Bo H.Stabilization of singular Markovian jump systems bygenerally observer-based controllers[J]. Asian Journal of Control, 2016, 18(1): 328-339.

[0089] [2]Guan C, Fei Z, Feng Z, et al.Stability and stabilization of singularMarkovian jump systems by dynamic event-triggered control strategy[J].Nonlinear Analysis: Hybrid Systems, 2020, 38: 100943.

[0090] [3]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 method for achieving stable control of a DC motor-driven single-link robotic arm, characterized in that, Includes the following steps: Based on the theory of singular systems and Markov jumps, a singular Markov electromechanical coupling dynamic model of a DC motor-driven single-link robotic arm is established, and a system model with a general transfer rate is also considered. By introducing internal dynamic variables, a threshold-adaptive and adjustable dynamic event-triggered communication mechanism is designed. The network communication transmission frequency is reduced by dynamically filtering the sampled signal, and it is rigorously proven that there is no Zeno phenomenon in the system. To address the uncertainty caused by unknown elements in a general transfer rate matrix, a modally dependent state feedback controller is designed to control a singular Markov system of a DC motor-driven single-link robotic arm to achieve asymptotic stability. Based on the singular Markov jump system theory, the following dynamic model is established for the DC motor driven single-link robotic arm system in reference [1]: In the formula, x1(t), x2(t), and x3(t) represent the angle, angular velocity, and armature current, respectively; g represents the acceleration due to gravity; l represents the pendulum length; J represents the moment of inertia; and B... f K represents the viscous friction damping coefficient. t K represents the motor torque constant. e R(η) represents the back electromotive force constant. t ) represents the armature resistance, and u(t) represents the control input; Furthermore, the DC motor-driven single-link robotic arm system is modeled as a singular Markov jump system model with a general transfer rate, as follows: In the formula, A δ(t) B δ(t) C δ(t) G δ(t) H δ(t) Let w(t) be a known system matrix of appropriate dimension, w(t) be a known nonlinear perturbation, and δ(t) represent a Markov random process whose values ​​take the form of a finite set S = {1, 2, ..., N}. For simplification, let δ(t) = i. δ(t) has the following properties: In the formula, Δ>0, π ij It is the transition rate from mode i to mode j, and satisfies: Where, when π ij When there is general uncertainty, the transfer rate matrix can be expressed as: In the formula, and Δ ij It is known that Δ ij ∈[-α ij α ij ], α ij It is known. Assume... It is a singular matrix whose rank rank(E) = r < n. Define a column full-rank matrix E. L E R ∈R n×(n-r) Satisfying E L E R T =E, and define the matrix. Ensure RE = 0 and ES = 0 for future use; To effectively conserve network communication bandwidth resources in the DC motor-driven single-link robotic arm system and strictly avoid the Zeno phenomenon, a signal transmission strategy based on a dynamic event triggering mechanism is designed by introducing internal dynamic variables. t k+1 =inf{t>t k |η(t)+θ(σx T (t)Ω δ(t) x(t)-e T (t)Ω δ(t) e(t))≤0} Where the triggering error e(t) = x(t) k )-x(t), t∈[t k , t k+1 ), θ>0, 0<σ<1, t k Indicates the trigger time, t k+1 Indicates the next trigger time, Ω i Let denot be the weight matrix, and η(t) represent the internal dynamic variables, which will be simplified to η and assumed to satisfy: The initial conditions are η(0) = η0 ≥ 0 and λ > 0; To control the DC motor-driven single-link robotic arm system, an event-triggered modal-dependent state feedback controller is constructed: u(t) = K δ(t) x(t k ), t∈[t k , t k+1 ), where K δ(t) The mode-dependent gain matrix to be determined for the state feedback controller is based on the trigger error e(t) = x(t). k )-x(t), thus obtaining the closed-loop control system: Next, we prove the absence of the Zeno phenomenon: based on e(t) = x(t) k )-x(t), obviously we can obtain t∈[t k , t k+1 ); because Then there exist real numbers ∈ = ∈(t) > 0 such that So, in Considering the conditions for triggering dynamic events, we have: in Because (||x(t) k )||-||e(t)||) 2 ≤||x(t k )-e(t)|| 2 The following inequalities are necessary conditions for the event to be triggered: So, Based on the above formula, we can obtain Since 0 < σ < 1, it can be guaranteed that τ > 0, thus proving that there is no Zeno phenomenon.

2. The control method for a singular Markov system of a DC motor-driven single-link robotic arm based on dynamic event triggering as described in claim 1, characterized in that, Vertex separation technology is introduced to address the uncertainty in the transition rate of singular Markov transition systems. A state feedback controller is designed to control 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. V(t)=x T (t)E T P i Ex(t)+η(t) B002: In the formula, i represents the mode, P i It is a Lyapunov variable matrix. B003; Let Λ i =P i E+R T Φ i S T A ci =A i +B i K i Calculate the derivative of V(t), where: Therefore, we have: B004: Select B005: Yes Shure supplementation B006: Further utilize vertex separation techniques to address the uncertainty in the transition rates of singular Markov jump systems, as shown in the following lemma: B007: Slack variables are introduced; B008: Further, construct the matrix. In the formula, B009: Definition right Contract modification get: In the formula, B0010: It is easy to obtain from the constructed matrix inequality This proves to be true; B0011: Combining vertex separation technology, the following formula can be obtained:

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