Asynchronous quantized control method for a single-link robotic arm

Abstract

The present invention discloses an asynchronous quantized control method for a single-link robotic arm, belonging to the field of single-link robotic arms and networked control technologies. The method comprises: establishing a networked Markov jump system based on the motion equation of the single-link robotic arm; establishing an asynchronous state feedback controller; establishing an asynchronous quantizer; establishing a closed-loop system based on the networked Markov jump system, the asynchronous state feedback controller, and the asynchronous quantizer; generating a criterion for the closed-loop system based on the Lyapunov function method; and optimizing and outputting the feedback gain of the asynchronous state feedback controller based on the criterion for the closed-loop system. The present invention can not only be applied to single-link robotic arms to ensure the random stability and passivity of the single-link robotic arm, but also helps reduce the transmission burden of communication channels.

Description

Technical Field

[0001] The present invention relates to the technical field of single-link robotic arms and networked control, and in particular to an asynchronous quantitative control method for a single-link robotic arm. Background Art

[0002] Since the beginning of the 21st century, with the advancement of network technology and computer control, the requirements for control accuracy and performance have also been increasing; however, an important challenge brought about by this progress is the higher requirements for the accuracy of system modeling; in practical fields, control systems are often disturbed by internal factors and external environment, which may lead to system structure changes, parameter mutations and other phenomena; for these complex practical systems, traditional single differential equations or difference equations are often difficult to accurately describe their dynamic characteristics, so hybrid systems came into being; the networked Markov jump model, as a special form of hybrid system, consists of multiple subsystems, and the operating order of each subsystem is determined by the Markov chain; the single-link robotic arm is a common component that is easily affected by external environment such as temperature and humidity, so with the help of Markov jump theory, the single-link robotic arm is modeled as a more accurate networked Markov jump model.

[0003] In traditional feedback control architectures, the controlled object and controller are typically placed in the same factory building or within a small area and connected via point-to-point dedicated lines. However, with the expansion of industrial scale and advancements in space technology, traditional wiring design methods face challenges such as high cost, difficult maintenance, poor scalability, and low reliability. Furthermore, with the increasing demand for remote, remote, and spatial control, the geographical limitations of traditional control architectures have become increasingly prominent. Consequently, traditional point-to-point connections are no longer sufficient to meet the rapidly evolving needs of today's society. The rapid advancement of computer and communication technologies has led to the integration of networking, communication, and control, resulting in a new architecture: the networked control architecture. In this type of control architecture, system components, including sensors, actuators, and controllers, are connected via a communication network to enable real-time data exchange and control signal transmission. Networked control architectures offer many advantages in practical applications, such as ease of installation and maintenance, low system construction costs, and flexible and scalable applications. Consequently, researchers have conducted extensive research on networked Markov jump systems, achieving a series of successful results.

[0004] Efficient information exchange between components in a networked Markov jump model is key to ensuring the normal operation of the system. Ideally, continuous information exchange between nodes through a communication network can maximize system stability. However, in practical engineering, the bandwidth and throughput of communication networks are often limited. This means that when transmitting massive data packets, transmission delays, packet loss, or confusion are likely to occur, further affecting system performance. To address these problems, an effective method is to quantize the transmitted signal, converting the continuous signal into multiple discrete values, thereby reducing the size of the data packet in the communication channel, reducing the transmission burden, and conserving network communication resources. However, after introducing a quantizer into the networked Markov jump model, the quantized data will inevitably deviate from the original data to a certain extent, resulting in quantization error, which may lead to information loss and reduced signal transmission accuracy. In some cases, even a small quantization error can have a negative impact on system performance and even cause system instability. Therefore, it is particularly important to design a quantitative control strategy without affecting system stability and performance.

[0005] Currently, there are patents for networked Markov jump model quantization control methods. For example, the Chinese invention patent with an application publication date of September 29, 2023 and application publication number CN116810830A discloses a dynamic event triggered control method for a single-link robotic arm system based on a dynamic uniform quantizer; the method includes establishing a Markov model of the single-link robotic arm system based on Markov jump system theory; designing a dynamic uniform quantizer to improve control accuracy by quantizing input to reduce communication bandwidth; and designing a state feedback controller to control the bounded stability of the single-link robotic arm system.

[0006] For example, a Chinese invention patent with an authorization publication date of December 19, 2023 and an authorization publication number of CN116160455B discloses a dynamic event triggering and quantitative control method for a single-arm manipulator under multi-channel attacks; the method includes establishing a state space equation that obeys Markov jump transition based on the mechanical model of the single-arm manipulator; designing an asynchronous output feedback controller based on a dynamic event triggering mechanism and a quantization strategy for multi-channel random deception attacks; based on the Lyapunov theory, establishing sufficient conditions to ensure the random stability and strict dissipative performance of the closed-loop control system of the single-arm manipulator, thereby solving the matrix of the feedback gain of the asynchronous state feedback controller.

[0007] The quantizers used in the above patents are all mode-independent. This type of quantizer has a simple and easy structure, but it completely ignores the system modal information and is highly conservative. In networked control scenarios, due to network-induced phenomena such as packet loss and communication transmission delay, it is difficult for the quantizer to obtain accurate information about the system modality in real time, and modal mismatch may occur. Therefore, how to design asynchronous quantizers and asynchronous state feedback controllers is an issue worthy of attention. Summary of the Invention

[0008] The object of the present invention is to provide an asynchronous quantitative control method for a single-link robotic arm to solve the problems raised in the above background technology.

[0009] In order to solve the above technical problems, the present invention provides the following technical solutions:

[0010] An asynchronous quantitative control method for a single-link robotic arm, the method comprising:

[0011] S1: Based on the motion equation of a single-link robotic arm, a networked Markov jump system is established;

[0012] S2: Establish an asynchronous state feedback controller;

[0013] S3: Establish asynchronous quantizer;

[0014] S4: Establish a closed-loop system based on a networked Markov jump system, an asynchronous state feedback controller, and an asynchronous quantizer;

[0015] S5: Generate the criterion of closed-loop system based on Lyapunov function method;

[0016] S6: Based on the criteria of the closed-loop system, optimize and output the feedback gain of the asynchronous state feedback controller.

[0017] The specific process of establishing a networked Markov jump system is as follows:

[0018] Establish the equation of motion for a single-link robotic arm:

[0019] in, represents the angular acceleration of the robotic arm, represents the angular velocity of the robot arm, v(t) represents the angle of the robot arm, represents the second component of the control input u(t), w(t) represents the external disturbance parameter, G represents the acceleration of gravity, L represents the length of the manipulator, M r(t) Indicates the mass of the load, J r(t)represents the moment of inertia, r(t) represents the Markov chain parameter, D represents the friction coefficient of the robot arm rotation, and t represents the independent variable of the time node;

[0020] Based on Markov jump, the controlled output equation of the single-link robotic arm is established:

[0021]

[0022] Where z(t) represents the controlled output, B2[r(t)] represents the second state parameter matrix of the control input, and D2[r(t)] represents the second state parameter matrix of the external disturbance. represents the first component of the control input u(t);

[0023] Select x(t)=v(t), and Then the state equation of the controlled output of the single-link robotic arm is:

[0024]

[0025] Based on the first-order Euler approximation formula And the step size H = 0.1, and h∈{1, 2, ..., n u}, n u is a positive integer, discretize the motion equation of the single-link manipulator, and obtain the discretized networked Markov jump system:

[0026]

[0027] in, And x(k) represents the angle of the robot arm v(t) = the angle of the robot arm after x(t) is discretized, Indicates the angular velocity of the robotic arm The angular velocity of the discretized robotic arm, represents the component of the control input u(k) after the control input u(t) is discretized, z(k) represents the controlled output after the controlled output z(t) is discretized, w(k) represents the external disturbance parameter after the external disturbance parameter w(t) is discretized, and r(k) represents the Markov chain parameter after the Markov chain parameter r(t) is discretized.

[0028] The specific process of establishing an asynchronous state feedback controller is as follows:

[0029] The control input u(t) is discretized into the control input u(k) = K[σ(k)]x(k); (2)

[0030] Where K[σ(k)] is the feedback gain of the asynchronous state feedback controller to be determined, σ(k)∈{r(k)|k∈[1,Q]}, and Q represents the total number of discretized time nodes.

[0031] The specific process of establishing an asynchronous quantizer is as follows:

[0032] The control input u(t) is discretized into the following:

[0033]

[0034] Among them, f h [σ(k),u h (k)] represents the logarithmic quantizer, u h (k) represents the hth component of the discretized control input u(k);

[0035] And the logarithmic quantizer f h [σ(k),u h (k)]:

[0036]

[0037] Among them, r lh [σ(k)] represents the output of the quantizer, and l represents the cutoff value of the quantizer, and l is a non-negative integer, ρ hσ(k) represents the quantization density and ρ hσ(k) ∈(0, 1), hσ(k) represents the mode of the quantizer, Represents the quantization density when the demarcation point value is l, r0 represents the initial quantization parameter of the quantizer and r0>0,

[0038] Based on the logarithmic quantizer and the sector boundary method, where the upper boundary of the sector is 1+δ hσ(k) , the lower boundary of the sector is 1-δ hσ(k) , then the quantized output expression of the logarithmic quantizer is:

[0039] f h [σ(k),u h (k)]=Δ hσ(k) u h (k)+u h (k); (3)

[0040] Among them, Δ hσ(k) ∈[-δ hσ(k) , δ hσ(k )],0<δ hσ(k) <1, from (2) and (3) we can get the asynchronous quantizer as:

[0041]

[0042] Where I represents the identity matrix, H σ(k) =[Δ hσ(k) |h∈{1,2,...,n u}] and |Δ hσ(k) |≤δ hσ(k) .

[0043] The specific process of establishing a closed-loop system is as follows:

[0044] Substituting the asynchronous quantizer (4) obtained in S3 into the networked Markov jump model (1) obtained in S1, we get the closed-loop system:

[0045]

[0046] in, A i =A[r(k)], i=r(k), H q =H σ(k) , K φ =K[σ(k)], q=φ=σ(k), B 1i , B 2i , D 1i , D 2i Both represent the preset system parameter matrix.

[0047] The specific process of generating the criterion of the closed-loop system is as follows:

[0048] When w(k)=0, if:

[0049]

[0050] If it holds, the closed-loop system is said to be stochastically stable;

[0051] Under zero initial conditions, if there exists a scalar γ>0 such that:

[0052]

[0053] If it holds, the closed-loop system is said to be passive, where E{·} represents the expected value;

[0054] Based on the Lyapunov function: V[k, x(k), r(k)] = x T (k)P[r(k)]x(k), we get:

[0055] E{ΔV(k)}=E{x T (k+1)P[r(k+1)]x(k+1)}-E{x T(k)P[r(k)]x(k)}=E{x T (k+1)P j x(k+1)}-E{x T (k)P i x(k)};(7)

[0056] in, ΔV(k) represents the unit increment of the robot arm’s angle, π ij Denotes the transition probability, and substituting the closed-loop system (5) into (7), we obtain:

[0057]

[0058] Among them, u iφ represents the conditional probability of the controller, and X i All represent system parameters, m=σ(k),

[0059] When w(k)=0, equation (8) is written as:

[0060]

[0061] When the positive definite matrix R iφq So that:

[0062]

[0063] If it holds, then according to (9) and (10), we can get:

[0064]

[0065] in, λ max >0, P>0, N∈[1,r(k)];

[0066] When ρ≥0, (11) can be obtained:

[0067]

[0068] like:

[0069]

[0070] Established, let ρ tend to ∞, we get: Then the closed-loop system (5) is stochastically stable;

[0071] Under zero initial conditions, according to (5) and (8), we get:

[0072]

[0073] If there exists a matrix F iφq So that:

[0074]

[0075] If established, we get:

[0076]

[0077] like:

[0078]

[0079]

[0080] If it holds, then according to (6), the closed-loop system (5) is passive;

[0081] Given a scalar γ>0, if there exists a matrix R iφq 、F iφq 、P i >0 so that inequalities (12), (13) and (14) hold, then the closed-loop system is said to be stochastically stable and passive.

[0082] The specific process of optimizing and outputting the feedback gain of the asynchronous state feedback controller is as follows:

[0083] Select R iφ is a positive definite matrix;

[0084] Applying Schur complement to inequalities (12) and (13) respectively, we obtain:

[0085]

[0086]

[0087] in,

[0088]

[0089]

[0090]

[0091]

[0092] definition

[0093] Inequality (15) is multiplied on the left by and right multiplication get:

[0094]

[0095] in:

[0096] G φ Expressed as an additional matrix, C i is represented as a system parameter and is represented by roll out

[0097] when:

[0098]

[0099] When it is established, then (17) is established:

[0100] Based on the logarithmic quantizer definition:

[0101]

[0102] in:

[0103]

[0104] Applying Schuldt's complement to (18) and combining it with (19) yields:

[0105]

[0106] in,

[0107] Then, given the quantization parameter δ hq ∈(0,1),γ>0, if there exists a scalar ε>0 and a matrix F iφq , G φ If (14), (16) and (20) hold, the feedback gain of the asynchronous state feedback controller is:

[0108] Compared with the prior art, the beneficial effects achieved by the present invention are: based on the motion equation of a single-link robotic arm, a networked Markov jump system is established; an asynchronous state feedback controller is established; an asynchronous quantizer is established; based on the networked Markov jump system, the asynchronous state feedback controller and the asynchronous quantizer, a closed-loop system is established; based on the Lyapunov function method, a criterion for the closed-loop system is generated; based on the criterion of the closed-loop system, the feedback gain of the asynchronous state feedback controller is optimized and output; the present invention can not only be applied to a single-link robotic arm to ensure the random stability and passivity of the single-link robotic arm, but also helps to reduce the transmission burden of the communication channel; considering network-induced phenomena such as data packet loss and communication transmission delay, it is difficult for the controller and quantizer to obtain accurate information on the system mode in real time, and there is a mode mismatch. In view of this, the present invention proposes an asynchronous quantizer and feedback gain to ensure that the closed-loop system has random stability and passivity. BRIEF DESCRIPTION OF THE DRAWINGS

[0109] The accompanying drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation of the present invention. In the accompanying drawings:

[0110] Figure 1 This is a flow chart of an asynchronous quantitative control method for a single-link robotic arm of the present invention;

[0111] Figure 2 This is a framework diagram of a networked Markov jump system for an asynchronous quantized control method of a single-link manipulator according to the present invention;

[0112] Figure 3 This is a modal jump diagram of a single-link arm system in an embodiment of an asynchronous quantized control method for a single-link manipulator arm of the present invention;

[0113] Figure 4 1 is a controller mode jump diagram in an embodiment of an asynchronous quantized control method for a single-link manipulator according to the present invention;

[0114] Figure 5 1 is a modal jump diagram of a quantizer in an embodiment of an asynchronous quantization control method for a single-link manipulator of the present invention;

[0115] Figure 6 is a state trajectory diagram of a single-link arm system in an embodiment of an asynchronous quantized control method of a single-link manipulator arm of the present invention;

[0116] Figure 7 This is a control input trajectory diagram in an embodiment of an asynchronous quantized control method for a single-link robotic arm of the present invention. DETAILED DESCRIPTION

[0117] 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.

[0118] See also Figure 1-7 , the present invention provides a technical solution:

[0119] An asynchronous quantitative control method for a single-link robotic arm, the method comprising:

[0120] S1: Based on the motion equation of a single-link robotic arm, a networked Markov jump system is established;

[0121] S2: Establish an asynchronous state feedback controller;

[0122] S3: Establish asynchronous quantizer;

[0123] S4: Establish a closed-loop system based on a networked Markov jump system, an asynchronous state feedback controller, and an asynchronous quantizer;

[0124] S5: Generate the criterion of closed-loop system based on Lyapunov function method;

[0125] S6: Based on the criteria of the closed-loop system, optimize and output the feedback gain of the asynchronous state feedback controller.

[0126] The specific process of establishing a networked Markov jump system is as follows:

[0127] Establish the equation of motion for a single-link robotic arm:

[0128] in, represents the angular acceleration of the robotic arm, represents the angular velocity of the robot arm, v(t) represents the angle of the robot arm, represents the second component of the control input u(t), w(t) represents the external disturbance parameter, G represents the acceleration of gravity, L represents the length of the manipulator, M r(t) Indicates the mass of the load, J r(t) represents the moment of inertia, r(t) represents the Markov chain parameter, D represents the friction coefficient of the robot arm rotation, and t represents the independent variable of the time node;

[0129] Based on Markov jump, the controlled output equation of the single-link robotic arm is established:

[0130]

[0131] Where z(t) represents the controlled output, B2[r(t)] represents the second state parameter matrix of the control input, and D2[r(t)] represents the second state parameter matrix of the external disturbance. represents the first component of the control input u(t);

[0132] Select x(t)=v(t), and Then the state equation of the controlled output of the single-link robotic arm is:

[0133]

[0134] Based on the first-order Euler approximation formula And the step size H = 0.1, and h∈{1, 2, ..., n u}, n u is a positive integer, discretize the motion equation of the single-link manipulator, and obtain the discretized networked Markov jump system:

[0135]

[0136] in, C[r(k)]=[1 0];x(k) represents the angle of the robot arm v(t)=x(t) is the angle of the robot arm after being discretized, Indicates the angular velocity of the robotic arm The angular velocity of the discretized robotic arm, represents the component of the control input u(k) after the control input u(t) is discretized, z(k) represents the controlled output after the controlled output z(t) is discretized, w(k) represents the external disturbance parameter after the external disturbance parameter w(t) is discretized, and r(k) represents the Markov chain parameter after the Markov chain parameter r(t) is discretized.

[0137] The specific process of establishing an asynchronous state feedback controller is as follows:

[0138] The control input u(t) is discretized into the control input u(k) = K[σ(k)]x(k); (2)

[0139] Where K[σ(k)] is the feedback gain of the asynchronous state feedback controller to be determined, σ(k)∈{r(k)|k∈[1,Q]}, and Q represents the total number of discretized time nodes.

[0140] The specific process of establishing an asynchronous quantizer is as follows:

[0141] The control input u(t) is discretized into the following:

[0142]

[0143] Among them, f h [σ(k),u h (k)] represents the logarithmic quantizer, u h (k) represents the hth component of the discretized control input u(k);

[0144] And the logarithmic quantizer f h [σ(k),u h (k)]:

[0145]

[0146] Among them, r lh [σ(k)] represents the output of the quantizer, and l represents the cutoff value of the quantizer, and l is a non-negative integer, ρ hσ(k) represents the quantization density and ρ hσ(k) ∈(0, 1), hσ(k) represents the mode of the quantizer, Represents the quantization density when the demarcation point value is l, r0 represents the initial quantization parameter of the quantizer and r0>0,

[0147] Based on the logarithmic quantizer and the sector boundary method, where the upper boundary of the sector is 1+δ hσ(k) , the lower boundary of the sector is 1-δ hσ(k) , then the quantized output expression of the logarithmic quantizer is:

[0148] f h [σ(k),u h (k)]=Δ hσ(k) u h (k)+u h (k); (3)

[0149] Among them, Δ hσ(k) ∈[-δ hσ(k) , δ hσ(k) ],0<δ hσ(k) <1, from (2) and (3) we can get the asynchronous quantizer as:

[0150]

[0151] Where I represents the identity matrix, H σ(k) =[Δ hσ(k) |h∈{1,2,...,n u}] and |Δ hσ(k) |≤δ hσ(k) .

[0152] The specific process of establishing a closed-loop system is as follows:

[0153] Substituting the asynchronous quantizer (4) obtained in S3 into the networked Markov jump model (1) obtained in S1, we get the closed-loop system:

[0154]

[0155] in, A i =A[r(k)], i=r(k), H q =H σ(k) , K φ =K[σ(k)], q=φ=σ(k), B 1i , B 2i , D 1i , D 2i Both represent the preset system parameter matrix.

[0156] The specific process of generating the criterion of the closed-loop system is as follows:

[0157] When w(k)=0, if:

[0158]

[0159] If it holds, the closed-loop system is said to be stochastically stable;

[0160] Under zero initial conditions, if there exists a scalar γ>0 such that:

[0161]

[0162] If it holds, the closed-loop system is said to be passive, where E{·} represents the expected value;

[0163] Based on the Lyapunov function: V[k, x(k), r(k)] = x T (k)P[r(k)]x(k), we get:

[0164] E{ΔV(k)}=E{x T (k+1)P[r(k+1)]x(k+1)}-E{x T (k)P[r(k)]x(k)}=E{x T (k+1)P j x(k+1)}-E{x T (k)P i x(k)};(7)

[0165] in, ΔV(k) represents the unit increment of the robot arm’s angle, π ijDenotes the transition probability, and substituting the closed-loop system (5) into (7), we obtain:

[0166]

[0167] Among them, u iφ represents the conditional probability of the controller, and X i All represent system parameters, m=σ(k),

[0168] When w(k)=0, equation (8) is written as:

[0169]

[0170] When the positive definite matrix R iφq So that:

[0171]

[0172] If it holds, then according to (9) and (10), we can get:

[0173]

[0174] in, λ max >0, P>0, N∈[1,r(k)];

[0175] When ρ≥0, (11) can be obtained:

[0176]

[0177] like:

[0178]

[0179] Established, let ρ tend to ∞, we get: Then the closed-loop system (5) is stochastically stable;

[0180] Under zero initial conditions, according to (5) and (8), we get:

[0181]

[0182] If there exists a matrix F iφq So that:

[0183]

[0184] If established, we get:

[0185]

[0186] like:

[0187]

[0188]

[0189] If it holds, then according to (6), the closed-loop system (5) is passive;

[0190] Given a scalar γ>0, if there exists a matrix R iφq 、F iφq 、P i >0 so that inequalities (12), (13) and (14) hold, then the closed-loop system is said to be stochastically stable and passive.

[0191] The specific process of optimizing and outputting the feedback gain of the asynchronous state feedback controller is as follows:

[0192] Select R iφ is a positive definite matrix;

[0193] Applying Schur complement to inequalities (12) and (13) respectively, we obtain:

[0194]

[0195]

[0196] in,

[0197]

[0198]

[0199]

[0200]

[0201] definition

[0202] Inequality (15) is multiplied on the left by and right multiplication get:

[0203]

[0204] in:

[0205] G φ Expressed as an additional matrix, C i is represented as a system parameter and is represented by roll out

[0206] when:

[0207]

[0208] When it is established, then (17) is established:

[0209] Based on the logarithmic quantizer definition:

[0210]

[0211] in:

[0212]

[0213] Applying Schuldt's complement to (18) and combining it with (19) yields:

[0214]

[0215] in,

[0216] Then, given the quantization parameter δ hq ∈(0,1),γ>0, if there exists a scalar ε>0 and a matrix F iφq , G φ If (14), (16) and (20) hold, the feedback gain of the asynchronous state feedback controller is:

[0217] First, consider the equation of motion for a single-link robotic arm: Among them, M i is the mass of the load, J i is the moment of inertia, g = 0.81 is the acceleration due to gravity, L = 0.5 is the length of the manipulator, D = 2 represents the friction coefficient of the manipulator rotation, and i = {1, 2} is the Markov chain; the controlled output equation of the single-link manipulator is expressed as:

[0218]

[0219] Secondly, through discretization, the following discrete-time networked Markov jump system model is obtained:

[0220]

[0221] The specific system parameters are:

[0222] h=0.1,C1=C2=[1 0],B 21 =[0.1 1], B 22 =[0.9 0.5], D21 =0.01, D 22 =0.02, M1=J1=1, M2=J2=2;

[0223] In addition, the transition probability matrix π and the conditional probability matrices U, Σ are set as:

[0224]

[0225] and the error bound of the logarithmic quantizer is fixed as:

[0226]

[0227] Set γ0 = 1 and According to the performance optimization algorithm proposed in step S7, the optimal passive performance index γ can be obtained by solving the linear matrix inequality using MATLAB and YALMIP and MOSEK toolboxes. min =0.0065 and the corresponding feedback gain of the asynchronous state feedback controller is:

[0228]

[0229] In the simulation, the initial state value is set to x(0) = [0.28 0.17] T And the external disturbance is fixed to w(k) = 0.9 k sin(k).

[0230] Figure 2 The framework of networked Markov jump systems is presented. Figure 3-5 The modal switching of the controlled system, controller, and quantizer is depicted respectively. From the figure, it can be seen that there is a modal mismatch between the controlled system and the controller, and between the controlled system and the quantizer. Figure 6-7 The state and control input trajectories of the quantized closed-loop system are shown respectively. Obviously, as time k→∞, their trajectories tend to zero, which means that the designed asynchronous quantized controller can ensure the stability of the system.

[0231] The working principle of this invention is based on the equations of motion of a single-link robotic arm system and the establishment of a networked Markov jump model. In networked control scenarios, due to network-induced phenomena such as packet loss and communication transmission delay, the controller and quantizer have difficulty obtaining accurate information about the system's modes in real time, resulting in modal mismatch. Given this, this method aims to design an asynchronous controller and quantizer to ensure that the closed-loop system has stochastic stability and passivity.

[0232] It should be noted that, in this document, relational terms such as first and second, etc., are used only to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "comprises," "comprising," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that includes a list of elements includes not only those elements but also other elements not explicitly listed, or elements inherent to such process, method, article, or apparatus.

[0233] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art will be able to modify the technical solutions described in the aforementioned embodiments or substitute equivalents for some of the technical features. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included within the scope of protection of the present invention.

Claims

1. An asynchronous quantitative control method for a single-link robotic arm, characterized in that: The method includes: S1: Based on the motion equation of a single-link robotic arm, a networked Markov jump system is established; S2: Establish an asynchronous state feedback controller; S3: Establish asynchronous quantizer; S4: Establish a closed-loop system based on a networked Markov jump system, an asynchronous state feedback controller, and an asynchronous quantizer; S5: Generate the criterion of closed-loop system based on Lyapunov function method; S6: Based on the criterion of the closed-loop system, optimize and output the feedback gain of the asynchronous state feedback controller; The specific process of establishing the networked Markov jump system is as follows: Establish the equation of motion for a single-link robotic arm: in, represents the angular acceleration of the robotic arm, represents the angular velocity of the robot arm, v(t) represents the angle of the robot arm, represents the second component of the control input u(t), w(t) represents the external disturbance parameter, G represents the acceleration of gravity, L represents the length of the manipulator, M r(t) Indicates the mass of the load, J r(t) represents the moment of inertia, r(t) represents the Markov chain parameter, D represents the friction coefficient of the robot arm rotation, and t represents the independent variable of the time node; Based on Markov jump, the controlled output equation of the single-link robotic arm is established: Where z(t) represents the controlled output, B2[r(t)] represents the second state parameter matrix of the control input, and D2[r(t)] represents the second state parameter matrix of the external disturbance. represents the first component of the control input u(t); Select x(t)=v(t), and Then the state equation of the controlled output of the single-link robotic arm is: Based on the first-order Euler approximation formula And the step size H = 0.1, and h∈{1, 2, ..., n u }, n u is a positive integer, discretize the motion equation of the single-link manipulator, and obtain the discretized networked Markov jump system: in, C[r(k)]=[1 0];x(k) represents the angle of the robot arm v(t)=x(t) is the angle of the robot arm after being discretized, Indicates the angular velocity of the robotic arm The angular velocity of the discretized robotic arm, represents the component of the control input u(k) after the control input u(t) is discretized, z(k) represents the controlled output after the controlled output z(t) is discretized, w(k) represents the external disturbance parameter after the external disturbance parameter w(t) is discretized, and r(k) represents the Markov chain parameter after the Markov chain parameter r(t) is discretized; The specific process of establishing the asynchronous state feedback controller is as follows: The control input u(t) is discretized to obtain the control input u(k)=K[σ(k)]x(k); (2) Where K[σ(k)] is the feedback gain of the asynchronous state feedback controller to be determined, σ(k)∈{r(k)|k∈[1,Q]}, and Q represents the total number of discretized time nodes; The specific process of establishing the asynchronous quantizer is as follows: The control input u(t) is discretized into the control input: Among them, f h [σ(k),u h (k)] represents the logarithmic quantizer, u h (k) represents the hth component of the discretized control input u(k); And the logarithmic quantizer f h [σ(k),u h (k)]: Among them, r lh [σ(k)] represents the output of the quantizer, and l represents the cutoff value of the quantizer, and l is a non-negative integer, ρ hσ(k) represents the quantization density and ρ hσ(k) ∈(0, 1), hσ(k) represents the mode of the quantizer, Represents the quantization density when the demarcation point value is l, r0 represents the initial quantization parameter of the quantizer and r0>0, Based on the logarithmic quantizer and the sector boundary method, where the upper boundary of the sector is 1+δ hσ(k) , the lower boundary of the sector is 1-δ hσ(k) , then the quantized output expression of the logarithmic quantizer is: f h [σ(k),u h (k)]=Δ hσ(k) in h (k)+u h (k); (3) Among them, Δ hσ(k) ∈[-δ hσ(k) , δ hσ(k )],0<δ hσ(k) <1, the asynchronous quantizer can be obtained from equations (2) and (3): Where I represents the identity matrix, H σ(k) =[Δ hσ(k) |h∈{1,2,...,n u }] and |Δ hσ(k )|≤δ hσ(k) .

2. The asynchronous quantitative control method of a single-link robotic arm according to claim 1, characterized in that: The specific process of establishing the closed-loop system is as follows: Substituting the asynchronous quantizer (4) obtained in S3 into the networked Markov jump model (1) obtained in S1, we get the closed-loop system: in, A i =A[r(k)], i=r(k), H q =H σ(k) , K φ =K[σ(k)], q=φ=σ(k), B 1i , B 2i , D 1i , D 2i Both represent the preset system parameter matrix.

3. The asynchronous quantitative control method of a single-link robotic arm according to claim 2, characterized in that: The specific process of generating the criterion of the closed-loop system is as follows: When w(k)=0, if: If it holds, the closed-loop system is said to be stochastically stable; Under zero initial conditions, if there exists a scalar γ>0 such that: If it holds, the closed-loop system is said to be passive, where E{·} represents the expected value; Based on the Lyapunov function: V[k, x(k), r(k)] = x T (k)P[r(k)]x(k), we get: in, ΔV(k) represents the unit increment of the robot arm’s angle, π ij Denotes the transition probability, and substituting the closed-loop system (5) into formula (7) yields: Among them, u iφ represents the conditional probability of the controller, and X i All represent system parameters, m=σ(k), When w(k)=0, equation (8) is written as: When the positive definite matrix R iφq So that: If it holds, then according to formula (9) and inequality (10), we can get: in, λ max >0, P>0, N∈[1,r(k)]; When ρ≥0, inequality (11) yields: like: Established, let ρ tend to ∞, we get: Then the closed-loop system (5) is stochastically stable; Under zero initial conditions, according to the closed-loop system (5) and equation (8), we get: If there exists a matrix F iφq So that: If established, we get: like: If it holds, then according to inequality (6), the closed-loop system (5) is passive; Given a scalar γ>0, if there exists a matrix R iφq 、F iφq 、P i >0 so that inequalities (12), (13) and (14) hold, then the closed-loop system is said to be stochastically stable and passive.

4. The asynchronous quantitative control method of a single-link robotic arm according to claim 3, characterized in that: The specific process of optimizing and outputting the feedback gain of the asynchronous state feedback controller is as follows: Select R iφ is a positive definite matrix; Applying Schur complement to inequalities (12) and (13) respectively, we obtain: in, definition Inequality (15) is multiplied on the left by and right multiplication get: in: G φ Expressed as an additional matrix, C i is represented as a system parameter and is represented by roll out when: When , inequality (17) holds: Based on the logarithmic quantizer definition: in: Applying Schur complement to inequality (18) and combining it with inequality (19) yields: in, Then, given the quantization parameter δ hq ∈(0,1),γ>0, if there exists a scalar ε>0 and a matrix F iφq , G φ If inequalities (14), (16) and (20) hold, the feedback gain of the asynchronous state feedback controller is:

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