Sliding mode control method and device for networked semi-Markov jump system

Abstract

The invention discloses a sliding mode control method and device for a networked semi-Markov jump system, and belongs to the technical field of sliding mode control, and the method specifically comprises the steps: building a state space model of the networked semi-Markov jump system, and building a transfer rate matrix under the condition that the element part of the transfer rate matrix is unknown; when the sliding mode gain matrix in the semi-Markov jump system meets a positive definite condition, generating a sliding mode control law; on the basis of the sliding mode control law, obtaining a sufficient condition of reachability of a sliding mode surface and a stability condition of the semi-Markov jump system; and solving the stability condition to realize sliding mode control of the networked semi-Markov jump system. According to the method, the adaptability, the reliability and the anti-interference capability of the control system are enhanced by establishing the control system model which is more practical and considering the condition that the element part of the transfer rate matrix is unknown.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of motor control, in particular to a sliding mode control method and device for networked semi-Markov jump system. BACKGROUND

[0002] Multi-modal, dwell time non-exponential distribution and resource-limited networked communication are three significant features of modern complex engineering systems. Therefore, research on networked semi-Markov jump system is of great value to establish a more realistic control system model and achieve higher control requirements.

[0003] Networked semi-Markov jump system has been widely used in flexible link robot, unmanned vertical take-off and landing aircraft and satellite radar due to its unique advantages in describing complex systems with multi-modal and network communication constraints. Since the transition rate matrix of the system is difficult to obtain in most cases in practical systems, the networked semi-Markov jump system with partially unknown elements in the transition rate matrix is more realistic, and it is of great significance and application value to research and design the control framework of the system. SUMMARY

[0004] The present application proposes a sliding mode control method and device for networked semi-Markov jump system with partially unknown elements in the transition rate matrix. The semi-Markov jump system considered in the present application generalizes the dwell time from the exponential distribution of Markov system to arbitrary distribution, establishes a more realistic control system model, and considers the case of partially unknown elements in the transition rate matrix, thereby enhancing the adaptability, reliability and anti-interference ability of the control system.

[0005] To achieve the above purpose, the present application provides the following solutions: In a first aspect, the present application proposes a sliding mode control method for networked semi-Markov jump system, comprising the following steps: establishing a state space model of the networked semi-Markov jump system and a transition rate matrix under the condition of partially unknown elements in the transition rate matrix; generating a sliding mode control law when the sliding mode gain matrix in the semi-Markov jump system satisfies the positive definite condition; obtaining a sufficient condition for the reachability of the sliding mode surface and a stability condition of the semi-Markov jump system based on the sliding mode control law; solving the stability condition to achieve sliding mode control of the networked semi-Markov jump system.

[0006] Further, the process of establishing a state space model of the networked semi-Markov jump system specifically comprises: ; in, x ( t ) is the state of a semi-Markov jump system; y ( t () is the system's measurement output; w ( t ) is an external disturbance to the system; u ( t () is the system's control input; It is a fixed coefficient matrix; The modes of the semi-Markov jumping system are represented and generated by the following generator: ; Indicates the length of stay Let represent the transition rate of the system from mode p at time t to mode q at time t+l, and we have: s represents the value of the system modes, the set of mode values. , It represents an infinitesimal quantity.

[0007] Furthermore, the steps for establishing the transfer rate matrix when some elements are unknown specifically include: Set the transfer rate matrix as follows: ; in, Indicates that the transfer rate is unknown, for , It is A much smaller number, representing the deviation of elements in the transfer rate matrix, given a small positive number. And satisfy ; Based on the transfer rate matrix, define a set. and set ,gather ,gather ; for ,show It is unknown. ,in, express The lower bound, ,express The upper boundary.

[0008] Furthermore, in a semi-Markov jump system, when the sliding mode gain matrix satisfies the positive definite condition, the process of generating the sliding mode control law specifically includes: The sliding surface equation of a semi-Markov jumping system is expressed as: ; Among them, the sliding mode gain matrix G satisfy ; The sliding mode control law is as follows: In the formula, Describes the 2-norm of a matrix. Represents a symbolic function. Represents a given positive constant.

[0009] Furthermore, the process of obtaining sufficient conditions for the reachability of the sliding surface based on the aforementioned sliding mode control law specifically includes: Give the Lyapunov functional ; ; according to Conclusion: ; Get When hour, That is, when When the sliding surface accessibility is satisfied, the formula is as follows: This represents the largest eigenvalue of the matrix.

[0010] Furthermore, the process of obtaining the stability conditions of the semi-Markov jump system based on the sliding mode control law specifically includes: Based on the sliding mode control law, for a given positive parameter , ,matrix G When there exists a positive definite matrix Set the condition matrix A semi-Markov transition system satisfies the following mode, which gives it the following properties: Performance indicators Stochastic stability: Pattern (1): When hour: ; in ; Pattern (2): When hour, ; in ; Pattern (3): When At that time, there exist positive numbers. a satisfy : ; in ; In the above formula, the constant matrix I represents the identity matrix. These represent the deviation elements in the transfer rate matrix. The upper realm, Used to estimate the unknown elements in the system transfer rate matrix under mode (3) .

[0011] Furthermore, solving the stability conditions to achieve sliding mode control of the networked semi-Markov jump system includes solving for the given parameters. Substitute the system matrix into the condition matrix The weight matrix is ​​obtained by solving, and sliding mode control of the networked semi-Markov jump system is realized.

[0012] In a second aspect, the present invention also proposes a sliding mode control device for a networked semi-Markov transition system to perform the sliding mode control method as described in the first aspect, comprising: The model building module is used to build the state space model of the networked semi-Markov transition system and to build the transition rate matrix when some elements of the transition rate matrix are unknown. The parameter generation module is used to generate a sliding mode control law when the sliding mode gain matrix of a semi-Markov jump system satisfies the positive definite condition. The condition determination module is used to obtain the sufficient conditions for the reachability of the sliding surface and the stability conditions of the semi-Markov jump system based on the sliding mode control law. The control output module is used to solve the stability conditions and realize sliding mode control of the networked semi-Markov jump system.

[0013] Thirdly, the present invention also proposes a readable storage medium storing a computer program, which, when executed by a processor, implements the steps of the method as described in the first aspect.

[0014] Fourthly, the present invention also proposes a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the method as described in the first aspect.

[0015] The beneficial effects of this invention are as follows: The semi-Markov jump system of this invention extends the residence time from the exponential distribution of Markov systems to any distribution, enabling the establishment of control system models that are more realistic.

[0016] This invention considers the case where some elements in the system's mode transition rate matrix are unknown. By designing a sliding surface and a sliding mode control law, and using Lyapunov stability theory, it derives that the system possesses... The stochastic stability of performance and sufficient conditions for the reachability of the sliding surface.

[0017] By combining sliding mode control with a networked semi-Markov transition system with partially unknown transfer rates, the required control performance can be achieved. This approach is suitable for various types of control systems and application scenarios, such as chemical control, robotic arm control, or drone systems.

[0018] The algorithm in this invention is simple to operate and implement, and the system has strong adaptability, anti-interference and reliability. Attached Figure Description

[0019] Figure 1 This is a flowchart of the sliding mode control method for the networked semi-Markov jump system of the present invention; Figure 2 This is a structural block diagram of the tunnel diode circuit according to an embodiment of the present invention; Figure 3 This is a simulation diagram of the state response of the open-loop system of the tunnel diode circuit according to an embodiment of the present invention; Figure 4 This is a schematic diagram of the state response simulation of the tunnel diode circuit closed-loop system according to an embodiment of the present invention; Figure 5 This is a simulation diagram of the sliding surface dynamic response of the tunnel diode circuit according to an embodiment of the present invention; Figure 6 This is a simulation diagram of the system mode switching of the tunnel diode circuit according to an embodiment of the present invention.

[0020] Figure 7 This is a schematic diagram of the sliding mode control device for the networked semi-Markov transition system of the present invention. Detailed Implementation

[0021] The present application will now be described in further detail with reference to the accompanying drawings. It should be noted that the following specific embodiments are only used to further illustrate the present application and should not be construed as limiting the scope of protection of the present application. Those skilled in the art can make some non-essential improvements and adjustments to the present application based on the above application content.

[0022] like Figure 1 As shown, this embodiment provides a sliding mode control method for a networked semi-Markov jump system, including the following steps: S100. Establish the state-space model of the networked semi-Markov transition system, and establish the transition rate matrix when some elements of the transition rate matrix are unknown.

[0023] In this embodiment, the process of establishing the state-space model of the networked semi-Markov transition system specifically includes: ; in, x ( t ) is the state of a semi-Markov jump system; y ( t () is the system's measurement output; w ( t ) is an external disturbance to the system; u ( t () is the system's control input; It is a fixed coefficient matrix; The modes of the semi-Markov jumping system are represented and generated by the following generator: ; Indicates the length of stay Let represent the transition rate of the system from mode p at time t to mode q at time t+l, and we have: s represents the value of the system modes, the set of mode values. .

[0024] Considering that some elements of the system's transfer rate matrix are unknown, the transfer rate matrix as follows: ; In the formula, This represents an unknown transfer rate, and for , It is A much smaller number, representing the deviation of elements in the transfer rate matrix, given a small positive number. And satisfy For the purposes of subsequent content, a set is defined. and set ,gather , In particular, for ,show It is unknown, therefore a lower bound is set. The upper bound is represented as: , from Take the value from the middle.

[0025] S200. When the sliding mode gain matrix in a semi-Markov jump system satisfies the positive definite condition, a sliding mode control law is generated.

[0026] In this embodiment, the sliding surface equation of the semi-Markov jump system is expressed as: ; Among them, the sliding mode gain matrix G satisfy ,Right now Zhengding; The sliding mode control law is as follows: In the formula, Describes the 2-norm of a matrix. Represents a symbolic function. Represents a given positive constant.

[0027] S300. Based on the sliding mode control law, obtain the sufficient conditions for the reachability of the sliding surface and the stability conditions of the semi-Markov jump system.

[0028] In this embodiment, a Lyapunov functional is given. ; ; according to We can conclude that: ; It can be concluded that when hour, ;when When the sliding surface accessibility is satisfied, the formula is as follows: This represents the largest eigenvalue of the matrix.

[0029] In this embodiment, the definition is as follows: , , T Represents the transpose of a matrix. I The symbol represents the identity matrix, and the asterisk (*) represents a symmetric term in the matrix. Y -1 "Represents a matrix Y The reverse, Represents the empty set. Represents mathematical expectation, Representative matrix Y The minimum and maximum eigenvalues.

[0030] Based on the sliding mode control law designed by S200, for a given positive parameter , ,matrix G When there exists a positive definite matrix Condition matrices can be designed. The system has the following pattern: Performance indicators Stochastic stability: Pattern (1): When Time, i.e., the current mode of the system. p In the transfer rate matrix The p Partial transfer rate of row elements It is known (including diagonal elements) (is known), the p The transfer rate of the remaining part of the row element It is unknown: ; in ; Pattern (2): When Time, i.e., the current mode of the system. p In the transfer rate matrix The p Partial transfer rate of row elements It is unknown (including diagonal elements). (It is unknown), the first p The transfer rate of the remaining part of the row element It is known that: ; in ; Pattern (3): When At that time, there exist positive numbers. a satisfy That is, the current mode of the system. p In the transfer rate matrix The p All transfer rates in row elements It is unknown: ; in ; These represent the deviation elements in the transfer rate matrix. The upper realm, Used to estimate the unknown elements in the system transfer rate matrix under mode (3) .

[0031] In this embodiment, a Lyapunov function is given. The expression for the weak infinity operator is as follows: ; Represents the symbol for the weak infinity operator. Represents a positive mathematical variable. Expressing expectations, Let Lyapunov function be represented.

[0032] Using the aforementioned weak infinitesimal operator, we have: ; For pattern (1): when Time, i.e., the current mode of the system. p In the transfer rate matrix The p Partial transfer rate of row elements It is known (including diagonal elements) (is known), the p The transfer rate of the remaining part of the row element It is unknown: definition ,when hour, We can obtain the following formula: ; ; For pattern (2): when Time, i.e., the current mode of the system. p In the transfer rate matrix The p Partial transfer rate of row elements It is unknown (including diagonal elements). (It is unknown), the first p The transfer rate of the remaining part of the row element It is known that: definition ,when hour, We can obtain the following formula: ; ; For pattern (3): when At that time, there exist positive numbers. a satisfy That is, the current mode of the system. p In the transfer rate matrix The p All transfer rates in row elements It is unknown: definition Given a constant Through formula To estimate the unknown transfer rate This yields the following formula: ; ; Therefore, we get: ; ; Where vector superscript T To represent the transpose of a matrix, These represent the largest and smallest eigenvalues ​​of the matrix, respectively.

[0033] The system achieves the desired state based on the above equation. Performance indicators It is stochastically stable.

[0034] S400. Solve the stability conditions to achieve sliding mode control of the networked semi-Markov jump system.

[0035] In this embodiment, solving the stability condition to achieve sliding mode control of the networked semi-Markov jump system includes solving for the given parameters. Substitute the system matrix into the condition matrix The weight matrix is ​​obtained by solving, and sliding mode control of the networked semi-Markov jump system is realized.

[0036] This invention can be applied to many related fields and is suitable for various types of control systems and application scenarios, such as chemical control, robotic arm control, or drone systems, and can improve the system's adaptability, anti-interference and reliability.

[0037] In this embodiment, experimental examples are used to verify the proposed design method. Since tunnel diode circuits are widely used in military applications (radar), satellites, and other scenarios, this embodiment considers... Figure 2 A typical tunnel diode circuit system is shown in the figure, with C, L, R 1. R 2 represents capacitor, inductor, resistor 1, and resistor 2, respectively. These represent capacitor current, inductor current, capacitor voltage, diode voltage, circuit disturbance, and ideal voltage source, respectively. Considering the disturbance, the physical model is as follows: ; In terms of practical physical meaning, among which Control input u ( t From an ideal voltage source U ( t To represent external disturbances The system model parameters are as follows: ; ; Given system parameters , , , The following matrix is ​​obtained by solving: ; As can be seen from the simulation data, Figure 3 Without control input, the system state is divergent, but with sliding mode control, as shown below... Figure 4 The system state is convergent. The state response curve of the capacitor voltage. The state response curve representing the inductor current, such as Figure 5 As shown, the sliding surface is reachable within a finite time. Figure 6 What is shown is the modal transition diagram of the system.

[0038] In one embodiment, this embodiment provides a sliding mode control device 70 for a networked semi-Markov transition system to perform the sliding mode control method as described above, including: Model building module 701 is used to build the state space model of the networked semi-Markov transition system and to build the transition rate matrix when some elements of the transition rate matrix are unknown. The parameter generation module 702 is used to generate a sliding mode control law when the sliding mode gain matrix of the semi-Markov jump system satisfies the positive definite condition. Condition determination module 703 is used to obtain the sufficient conditions for the reachability of the sliding surface and the stability conditions of the semi-Markov jump system based on the sliding mode control law. The control output module 704 is used to solve the stability conditions and realize sliding mode control of the networked semi-Markov jump system.

[0039] In one embodiment, a readable storage medium stores a computer program that, when executed by a processor, implements the steps of the above-described method.

[0040] In one embodiment, a computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the method described above.

[0041] Using the aforementioned readable storage medium and computer equipment, a state-space model of a networked semi-Markov transition system is established, and a transition rate matrix is ​​established when some elements of the transition rate matrix are unknown. In a semi-Markov transition system, a sliding mode control law is generated when the sliding mode gain matrix satisfies the positive definite condition. Based on the sliding mode control law, sufficient conditions for the reachability of the sliding surface and stability conditions of the semi-Markov transition system are obtained. The stability conditions are solved to achieve sliding mode control of the networked semi-Markov transition system. The algorithm in this invention is simple to operate and implement, and the system exhibits strong adaptability, anti-interference capabilities, and reliability.

[0042] In this invention, unless otherwise explicitly specified and limited, the terms "installation," "connection," "linking," and "fixing," etc., should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral part; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication of two components or the interaction between two components. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0043] In this invention, unless otherwise explicitly specified and limited, "above" or "below" the second feature can mean that the first feature is in direct contact with the second feature, or that the first feature is in indirect contact with the second feature through an intermediate medium. Furthermore, "above," "over," and "on top" of the second feature can mean that the first feature is directly above or diagonally above the second feature, or simply that the first feature is at a higher horizontal level than the second feature. "Below," "below," and "under" the second feature can mean that the first feature is directly below or diagonally below the second feature, or simply that the first feature is at a lower horizontal level than the second feature.

[0044] The embodiments described above are merely examples of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. A sliding mode control method for a networked semi-Markov jump system, characterized in that, Includes the following steps: Establish a state-space model of a networked semi-Markov transition system, and establish the transition rate matrix when some elements of the transition rate matrix are unknown; In a semi-Markov jump system, when the sliding mode gain matrix satisfies the positive definite condition, a sliding mode control law is generated. Based on the sliding mode control law, the sufficient conditions for the reachability of the sliding surface and the stability conditions of the semi-Markov jump system are obtained respectively. Solving the stability conditions enables sliding mode control of the networked semi-Markov jump system.

2. The sliding mode control method for a networked semi-Markov jump system according to claim 1, characterized in that, The process of establishing a state-space model for a networked semi-Markov transition system specifically includes: ； in, It is the state of a semi-Markov transition system; It is the system's measurement output; It is an external disturbance to the system; It is the system's control input; It is a fixed coefficient matrix; The modes of the semi-Markov jumping system are represented and generated by the following generator: ； Indicates the duration of stay. Let represent the transition rate of the system from mode p at time t to mode q at time t+l, and we have: s represents the value of the system modes, the set of mode values. , It represents an infinitesimal quantity.

3. The sliding mode control method for a networked semi-Markov jump system according to claim 2, characterized in that, The specific steps for constructing the transfer rate matrix when some elements are unknown include: Set the transfer rate matrix as follows: ； in, Indicates that the transfer rate is unknown, for , It is A much smaller number, representing the deviation of elements in the transfer rate matrix, given a small positive number. ,satisfy ; Based on the transfer rate matrix, define a set. and set ,gather ,gather ; for ,show It is unknown. ,in, express The lower bound, ,express The upper boundary.

4. The sliding mode control method for a networked semi-Markov jump system according to claim 3, characterized in that, In a semi-Markov jump system, when the sliding mode gain matrix satisfies the positive definite condition, the process of generating the sliding mode control law specifically includes: The sliding surface equation of a semi-Markov jumping system is expressed as: ； Among them, the sliding mode gain matrix G satisfy ,Right now Zhengding; The sliding mode control law is as follows: In the formula, Describes the 2-norm of a matrix. Represents a symbolic function. Represents a given positive constant.

5. The sliding mode control method for a networked semi-Markov jump system according to claim 4, characterized in that, The process of obtaining sufficient conditions for the reachability of the sliding surface based on the aforementioned sliding mode control law specifically includes: Give the Lyapunov functional ; ； according to Conclusion: ； When hour, That is, when When the sliding surface accessibility is satisfied, the formula is as follows: This represents the largest eigenvalue of the matrix.

6. The sliding mode control method for a networked semi-Markov jump system according to claim 5, characterized in that, The process of obtaining the stability conditions of a semi-Markov jump system based on the sliding mode control law specifically includes: Based on the sliding mode control law, for a given positive parameter , ,matrix G When there exists a positive definite matrix Set the condition matrix A semi-Markov transition system satisfies the following mode, which gives it the following properties: Performance indicators Stochastic stability: Pattern (1): When hour: ； in: ； Pattern (2): When hour, ； in: ； Pattern (3): When At that time, there exist positive numbers. a satisfy : ； in: ； In the above formula, the constant matrix I represents the identity matrix. These represent the deviation elements in the transfer rate matrix. The upper realm, Used to estimate the unknown elements in the system transfer rate matrix under mode (3) .

7. The sliding mode control method for a networked semi-Markov jump system according to claim 5, characterized in that, Solving the stability conditions to achieve sliding mode control of the networked semi-Markov jump system includes solving for the given parameters. Substitute the system matrix into the condition matrix The weight matrix is ​​obtained by solving, and sliding mode control of the networked semi-Markov jump system is realized.

8. A sliding mode control device for a networked semi-Markov transition system, for performing the sliding mode control method as described in any one of claims 1-7, characterized in that, include: The model building module is used to build the state space model of the networked semi-Markov transition system and to build the transition rate matrix when some elements of the transition rate matrix are unknown. The parameter generation module is used to generate a sliding mode control law when the sliding mode gain matrix of a semi-Markov jump system satisfies the positive definite condition. The condition determination module is used to obtain the sufficient conditions for the reachability of the sliding surface and the stability conditions of the semi-Markov jump system based on the sliding mode control law. The control output module is used to solve the stability conditions and realize sliding mode control of the networked semi-Markov jump system.

9. A readable storage medium storing a computer program, characterized in that, When the stored computer program is executed by a processor, it implements the steps of the method as described in any one of claims 1-7.

10. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the method as described in any one of claims 1-7.

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