A new criterion for Chua's circuit system based on fault-tolerant synchronous control

Abstract

This paper discloses a new criterion for Chua's circuit systems based on fault-tolerant synchronous control. This method considers a Chua's circuit system model containing unknown system states and fault signals under the interference of external noise. Furthermore, a projection network is designed to reduce the dimensionality of the system state space through clustering and aggregation, and all nodes are divided into measurable and unmeasurable clusters according to whether they are measurable or unmeasurable. An average intermediate observer is used to estimate unknown components, and a corresponding fault-tolerant controller is constructed to compensate for unknown actuator failures, ensuring that the estimated error system is consistently and ultimately bounded, thereby achieving synchronization of the error system.

Description

Technical Field

[0001] The present invention relates to a fault-tolerant synchronous control method, and in particular to a new criterion of Chua's circuit system based on fault-tolerant synchronous control. Background Art

[0002] Many dynamic models for Chua's circuit systems are constructed only under the assumption that the system state and fault signals are known. However, in real applications, the actual system state and the actual fault signals are unknown. Furthermore, conventional observers are prohibitively expensive for Chua's circuit systems with a large number of nodes. Therefore, addressing the fault tolerance issues of Chua's circuit systems with a large number of nodes is crucial for both ensuring system stability and reducing estimation costs. Based on this understanding, researchers have conducted extensive research on Chua's circuit systems and achieved a series of successful results. Summary of the Invention

[0003] The purpose of the present invention is to propose a new criterion for Chua's circuit system based on fault-tolerant synchronous control, which can effectively reduce the estimated cost of Chua's circuit system.

[0004] The specific technical solution of the present invention is as follows: A new criterion for Chua's circuit system based on fault-tolerant synchronous control includes the following steps:

[0005] Under the interference of external noise, a Cai circuit system model containing unknown system state and fault signal is established, and its system output function is given

[0006] Design a projection network to reduce the dimension of the system state space through clustering and aggregation, and divide all nodes into measurable clusters and unmeasurable clusters according to whether they are measurable or unmeasurable.

[0007] Establish an average intermediate observer to estimate the unknown components and construct the corresponding fault-tolerant controller to compensate for unknown actuator failures

[0008] Using Lyapunov stability theory, a sufficient criterion is established to ensure that H ∞ Ultimately bounded in terms of performance

[0009] For the Cai circuit system in reference [1], the following dynamic model is established:

[0010]

[0011] Where x(t) represents the unknown system state, A represents the coupling matrix, B, C, D, and F represent the known parameter matrices, u(t) represents the control input, I(t) represents the unknown system fault, J(t) represents the external input, and y(t) represents the system output.

[0012] Assume that the system state is divided into where x1(t)=[x1(t), x2(t), ..., x r (t)] T is a cluster of measurable nodes, x2(t)=[x r+1 (t), x r+2 (t),...,x n (t)] T A cluster of unmeasurable nodes

[0013] Since x1(t) is known, we only need to estimate the average state of the unmeasurable nodes.

[0014] By aggregating unmeasurable states, a projection network can be established

[0015]

[0016] in is the projection state, P is the projection matrix satisfying

[0017]

[0018] To handle unknown states and actuator failures, an intermediate variable is introduced

[0019] ψ(t)=I(t)-L f x(t)

[0020] Among them L f is the estimated gain

[0021] Then the proportional-integral intermediate observer is designed as

[0022]

[0023] make Then the estimated error system can be obtained

[0024]

[0025] Therefore, the control input u(t) is constructed as

[0026]

[0027] This control scheme can ensure that the system is uniformly bounded and stable. The proof process is as follows:

[0028] C001: Select the following Lyapunov function:

[0029]

[0030] C002: where Q1, Q2, Q3 are any positive definite matrices;

[0031] C003: Calculate the first derivative of V(t), where:

[0032]

[0033] C004: Further, we can get:

[0034]

[0035] C005: Therefore

[0036]

[0037] C006: Among them

[0038]

[0039] C007: It should be noted that the gains Q1, Q3, L f is unknown, then some coupling terms cannot be solved directly by the YALMIP toolbox;

[0040] C008: To solve this problem, congruence transformation is introduced, that is, there is a suitable matrix W i (i=1, 2, 3, 4, 5) such that Q1B=BW1, W2=W1K, Q1D=DW3, W4=W3L f , W5D=DL f ;

[0041] C009: Then, C006 can be transformed into

[0042]

[0043] C010: where ε1, ε2, and ε3 are positive constants

[0044] C011: Based on Lyapunov stability theory, when have if Then the system is uniformly eventually bounded. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1-4 Dynamic trajectory diagram of unknown state and tracking state of unmeasurable nodes;

[0046] Figure 5-8 Dynamic trajectory diagram for unknown faults and tracking faults of unmeasurable nodes; DETAILED DESCRIPTION

[0047] The present invention is further illustrated below with reference to specific examples. It should be understood that these examples are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, modifications of various equivalent forms of the present invention made by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0048] A new criterion for Chua's circuit system based on fault-tolerant synchronous control includes the following steps:

[0049] Step 1: Set various system parameters;

[0050] Step 2: Construct a projection network and intermediate variables based on the unknown states and unknown fault signals of the unmeasurable nodes;

[0051] Step 3: Set up the average intermediate observer;

[0052] Step 4: Set up the estimation error system;

[0053] Step 5: Verify whether the estimated error system is consistently ultimately bounded.

[0054] An embodiment of the present invention is described below:

[0055] Consider a class of fault-tolerant synchronous control Chua's circuit system, whose corresponding dynamic models are:

[0056]

[0057] The dynamic trajectory of the unknown state and tracking state of the unmeasurable node under the average intermediate observer is as follows: Figure 1-4 As shown, the dynamic trajectory of unknown faults and tracking faults of unmeasurable nodes under the average intermediate observer is as follows: Figure 5-8 shown

[0058] References

[0059] [1] J. Zhu, G. Yang. Robust distributed fault estimation for a network of dynamical systems. IEEE Transactions on Control of Network Systems 2016; 5: 14-22.

Claims

1. A new criterion for Chua's circuit system based on fault-tolerant synchronous control, characterized in that: The following steps are involved: Under the interference of external noise, a Chua circuit system model containing unknown system states and fault signals is established, and its system output function is given. Design a projection network to reduce the dimensionality of the system state space through clustering and aggregation, and divide all nodes into measurable clusters and unmeasurable clusters according to whether they are measurable or unmeasurable. An average intermediate observer is established to estimate the unknown components, and a corresponding fault-tolerant controller is constructed to compensate for unknown actuator failures. Under the interference of external noise, the dynamic model of the Chua circuit system containing unknown system state and fault signal is established as follows: y(t)=Cx(t), Where x(t) represents the unknown system state, A represents the coupling matrix, B, C, D, and F represent the known parameter matrices, u(t) represents the control input, I(t) represents the unknown system fault, J(t) represents the external input, and y(t) represents the system output. Assume that the system state is divided into where x1(t)=[x1(t),x2(t),...,x r (t)] T is a cluster of measurable nodes, x2(t)=[x r+1 (t),x r+2 (t),...,x n (t)] T It is a cluster of unmeasurable nodes; Since x1(t) is known, we only need to estimate the average state of the unmeasurable nodes. By aggregating unmeasurable states, a projection network can be established in is the projection state, P is the projection matrix satisfying To handle unknown states and actuator failures, an intermediate variable is introduced; ψ(t)=I(t)-L f x(t) Among them L f is the estimated gain; Then the proportional-integral intermediate observer is designed as make Then the estimated error system can be obtained Therefore, the control input u(t) is constructed as Using Lyapunov stability theory, a sufficient criterion is established to ensure that H ∞ The proof of the uniformly bounded performance is as follows: B001: Select the following form of Lyapunov function: Where Q1, Q2, Q3 are any positive definite matrices; B002: Calculate the first derivative of V(t): B003: On the other hand B004: To ensure H ∞ The performance is uniformly bounded, and the following inequality is established in, Oh 1,1 =2Q1A+Q1BKP+C T C,Ω 1,2 =-Q1BKP+Q1DL f , Oh 1,3 =Q1D,Ω 1,4 =Q1F,Ω 2,2 =2Q2(A+L f )+2Q2LC,Ω 2,3 =Q2D-Q3(L f A+L f DL f ), Oh 2,4 =Q2F,Ω 3,3 =Q3R -1 Q3-2Q3L f D,Oh 3,4 =-Q3L f F,Ω 4,4 =-ρ 2 E B005: It should be noted that the gains Q1, Q3, L f is unknown, then some coupling terms cannot be solved directly by the YALMIP toolbox; B006: To solve this problem, congruence transformation is introduced, that is, there is a suitable matrix W i , i=1,2,3,4,5, so that Q1B=BW1,W2=W1K,Q1D=DW3,W4=W3L f ,W5D=DL f ; B007: Then, B004 can be transformed into [(Q1B-BW1) T (Q1B-BW1)]<∈1E, [(Q1D-DW3) T (Q1D-DW3)]<∈2E, [(W5D-DL f ) T (W5D-DL f )]<∈3E Among them, ε1, ε2, ε3 are positive constants; B008: Based on Lyapunov stability theory, when have if Then the system is uniformly eventually bounded.

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