A simple intermittent quantization control method for Chua's circuit system

Abstract

This invention discloses a simple intermittent quantization control method for Chua's circuit systems. This method first designs an indirect signal-based estimation algorithm to estimate unknown states and disturbances in the network system. Then, two threshold functions are designed to construct an aperiodic intermittent control strategy. A uniform quantization controller quantizes the system's control inputs to reduce the transmission of redundant control signals. When applied to Chua's circuit systems, this method achieves excellent unknown state and disturbance estimation results and effectively conserves system control resources.

Description

Technical Field

[0001] The present invention relates to an intermittent quantization control method, and in particular to a simple intermittent quantization control method of a Chua circuit system. Background Art

[0002] With the advancement of globalization, various network systems, such as the Internet, power grids, transportation networks, ecological networks, and social networks, are closely intertwined with our daily lives. The effective operation of these network systems is crucial for maintaining global economic stability, safeguarding national security, and promoting scientific and technological progress. Synchronization is ubiquitous in these networks, with problems such as the spread of computer viruses and the stability of power systems being directly related to network synchronization. As a typical chaotic network, the control and synchronization of Chua's circuit networks has been a hot topic in nonlinear dynamics and chaos theory. By controlling the chaotic behavior of Chua's circuits, it is possible to achieve a transition from a chaotic state to a periodic or steady orbit, which is of great significance for understanding and utilizing chaotic phenomena. Furthermore, in practical applications, systems are inevitably affected by external interference, environmental noise, component failures, and other factors. These factors can lead to abnormal operation or even loss of control. Therefore, studying the synchronization of complex dynamic networks subject to disturbances has important theoretical significance and application prospects.

[0003] Furthermore, with the widespread use of digital computers, the study of signal quantization in control systems has garnered considerable attention. Quantizing signals before transmission can effectively reduce the transmission of redundant signals, thereby conserving system resources. Intermittent control offers advantages in reducing control costs and channel congestion. It operates only within the controlled interval, rather than the entire time interval, making it easy to implement in engineering practice. Furthermore, intermittent control can be periodic or aperiodic, with the latter having a non-fixed control phase, providing greater flexibility in control strategies. Therefore, studying intermittent quantized control for complex network systems plays an important role in reducing control costs and improving system performance.

[0004] The Chua circuit model has been widely studied as a prototype electronic system. Its typical circuit structure has made it a ubiquitous example of a real-world chaotic system. By applying the laws of electromagnetism, the Chua circuit can be accurately modeled, providing a practical system model for studying intermittent quantized synchronization. Summary of the Invention

[0005] The purpose of the present invention is to propose a simple intermittent quantization control method for Chua's circuit system, which can effectively save control costs and improve system performance.

[0006] The specific technical solution of the present invention is as follows: A simple intermittent quantization control method of the Chua circuit system is characterized by comprising the following steps:

[0007] Design an indirect signal-based estimation algorithm to estimate the unknown states and disturbances of Chua's circuit system;

[0008] Based on the estimated state and disturbance, a quantized input distributed intermittent controller is designed, which includes estimated state feedback and disturbance compensation.

[0009] Based on Lyapunov stability theorem, Young's inequality and mathematical induction, the uniformly bounded stability conditions of the estimation error system and the synchronization error system are obtained, and the synchronization of Chua's circuit system is realized.

[0010] The Chua circuit system is described as follows:

[0011]

[0012] Where C1 and C2 are capacitors, L1 is an inductor, R is a resistor, R0 is a series resistor on the power branch, v1(t) and v2(t) are the voltages across C1 and C2, i3(t) is the current flowing through L1, and f(v1(t)) is the nonlinear function of the Chua diode, which is expressed as f(v1(t))=G b1 v1(t)+0.5(G a1 -G b1 )(|v1(t)+1|-|v1(t)-1|), G a1 and G b1 are the parameters of the Chua diode;

[0013] Let x i1 (t) = v i1 (t), x i2 (t) = v i2 (t), x i3 (t) = i i3 (t), i = 1,…,N, the Chua circuit system can be written as follows:

[0014]

[0015] Its synchronization target system is defined as:

[0016]

[0017] in,

[0018]

[0019]

[0020] N=5 is the number of nodes, x i (t) represents the state of the i-th node and is unknown, s(t) is the leader state, f(x i (t)) and f(s(t)) represent the nonlinear function of the i-th node and the nonlinear function of the synchronization target, respectively, and assuming that f(·) satisfies the Lipschitz condition, u i (t) represents the control input of the i-th node, Q(u i (t)) represents the quantized u i (t), d i (t) represents a type of differentiable perturbation of the i-th node, and its upper bound is The upper bound of the derivative is δ, y i (t) and y s (t) represents the output of the i-th node and the output of the synchronization target, A, B, C are the known parameter matrices of the system, Γ and L represent the internal coupling matrix and external coupling matrix of the system, respectively;

[0021] Define the indirect signal η of the i-th node i (t) = d i (t)-L b x i (t), the estimation algorithm for the unknown state and disturbance of the system is developed as follows:

[0022]

[0023] in, is x i The estimated value of (t), d i The estimated value of (t), is η i The estimated value of (t), L a , L b and L c is the gain matrix of the estimation algorithm, D is the gain matrix, is the integral term of the estimation error;

[0024] make Then the estimated error system can be obtained as follows:

[0025]

[0026] in,

[0027] Using Kronecker product operator The augmented form of the closed-loop estimation error system can be obtained:

[0028] Among them, IN is the N-dimensional identity matrix; E x (t) is e xi (t) in matrix form; F(E x (t)) is f(e xi (t)) in matrix form; E η (t) is e ηi (t) is the matrix form of D(t); D(t) is the matrix form of d(t); yes The matrix form of

[0029] Next, design a multi-channel uniform quantizer as follows:

[0030]

[0031] Among them, α 1i (t) and α 2i (t) represent the encoder and decoder parameters of the i-th channel, respectively, satisfying and α 1i and α 2i Represents α 1i (t) and α 2i The lower bound of (t), and Represents α 1i (t) and α 2i The upper bound of (t), round(·) is the rounding function;

[0032] Define the ratio of the encoder and decoder parameters of the i-th channel as β i (t), which can be decomposed into known and unknown parts We can get: Q(u i (t))=β i (t)[u i (t)+e qi (t)]

[0033] in, e q =q(u i (t))-u i (t) represents the quantization error.

[0034] According to the definition of the rounding function, e qi The infinite norm of (t) |e qi (t)| ∞ Satisfy the following properties:

[0035]

[0036] Let e i (t) = x i (t)-s(t) and The closed-loop error system is as follows:

[0037]

[0038] in,

[0039] Assumptions According to the above quantization scheme and the estimation algorithm proposed in claim 1, the intermittent control strategy is designed as follows:

[0040]

[0041] in, for |α 2i (t)| ∞ An upper bound of , sign(·) is the sign function, represents the pseudo-inverse of the parameter matrix C, k i is the feedback gain of the ith node, t∈[t k ,s k ] and t∈[s k ,t k+1 ] represent the kth working cycle and rest cycle of the controller, t k is the starting time of the kth working cycle, s k is the starting time of the kth rest period.

[0042] t k and s k Determined by the following conditions:

[0043]

[0044] Where inf{·} represents the infimum of the selected condition, exp(·) is the exponential function, α1, α2, β1, β2 are threshold parameters, and α1>α2>0, β2>β1>0;

[0045] Therefore, the closed-loop error system can be rewritten as:

[0046]

[0047] Using Kronecker product operator The augmented form of the closed-loop synchronization error system can be obtained:

[0048]

[0049] Among them, I Nis the N-dimensional identity matrix, E(t) is e i (t), F(E(t)) is In matrix form, U(t) is u i (t) in matrix form, E q (t) is e qi (t) in matrix form.

[0050] C001: Rewrite the estimated error system as follows:

[0051]

[0052] in,

[0053]

[0054] C002: Select a suitable Lyapunov function;

[0055]

[0056] Among them, the positive definite matrix P1, P2, P3 are positive definite matrices, and Q1 is a given constant matrix;

[0057] C003: Taking the derivative of C002, we get:

[0058]

[0059] C004: According to the Lipschitz condition, we can get:

[0060]

[0061] In the formula, according to Young's inequality, we can get:

[0062]

[0063] Where ∈ is an arbitrary positive constant.

[0064] C005: Based on C004, we can further sort out the following:

[0065]

[0066] Where,

[0067] in,

[0068]

[0069] C006: Ensuring the negative definiteness of Θ means that the estimated error system is eventually uniformly bounded and stable and eventually converges to

[0070] C007: Now consider the stability of the synchronization error system and select a suitable Lyapunov function;

[0071]

[0072] Where P4 is a positive definite matrix;

[0073] C008: First, analyze the stability of the system when the controller is working, that is, t∈[t k ,s k ];

[0074] The derivative of C007 under the working state of the controller can be obtained:

[0075]

[0076] C008: Order According to the Lipschitz condition, C007 is sorted as follows:

[0077]

[0078] Where:

[0079]

[0080] C009: Command According to C008, we can get:

[0081]

[0082] In the formula, according to Young's inequality and ||ξ(t)||<B1, we can get:

[0083]

[0084] Where a and ∈1 are any positive constants,

[0085] C010: If but

[0086] C011: The following is an analysis of the stability when the controller is not working; taking the derivative of C007 when the controller is not working, we can get:

[0087]

[0088] C012: According to the Lipschitz condition, we can get:

[0089]

[0090] in,

[0091] C013: Based on the results in C010 and C012, we can obtain:

[0092]

[0093] C014: According to mathematical induction, we can conclude that:

[0094]

[0095] Among them, l∈(0,1) represents the average control rate, N0>0 is an elastic coefficient;

[0096] C015: Based on the above conclusions, the synchronization error system can converge if the following conditions are met:

[0097]

[0098] The Cai circuit system achieves synchronization under the proposed intermittent strategy. BRIEF DESCRIPTION OF THE DRAWINGS

[0099] Figure 1 This is the structure diagram of Cai's circuit;

[0100] Figure 2 This is a diagram showing the chaotic effect of Cai's circuit network;

[0101] Figure 3 is the state estimation error graph of the network system;

[0102] Figure 4 and Figure 5 is the disturbance state and disturbance state estimation diagram;

[0103] Figure 6 is the disturbance estimation error map;

[0104] Figure 7 and Figure 8 It is the input diagram of intermittent quantization control; DETAILED DESCRIPTION

[0105] 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.

[0106] like Figure 1As shown in FIG, a simple intermittent quantization control method of the Chua circuit network includes the following steps:

[0107] Step 1: Set the initial values ​​of various parameters;

[0108] Step 2: Sampling sensor output y i (t);

[0109] Step 3: Construct an estimation algorithm to estimate the unknown state and disturbance of the system;

[0110] Step 4: Design a controller using the estimated state and disturbance of the system;

[0111] Step 5: quantify the set control input;

[0112] Step 6: Set up intermittent control strategy;

[0113] Step 7: Verify whether the estimation error of the proposed estimation algorithm converges to zero;

[0114] Step 8: Verify whether the network system is synchronized;

[0115] An embodiment of the present invention is described below:

[0116] Consider the system parameters shown in the following table:

[0117] symbol value symbol value symbol value <![CDATA[C1]]> 1F <![CDATA[R0]]> 0.008Ω <![CDATA[G a1 ]]> -1.39386 <![CDATA[C2]]> 0.11F <![CDATA[L1]]> 0.06H R 1Ω <![CDATA[G b1 ]]> -0.7559

[0118] L c =3.5890;

[0119] Figure 1 This is the structure diagram of Cai's circuit; Figure 2 This is a diagram showing the chaotic effect of Cai's circuit network; Figure 3 This is the state estimation error diagram of the network system. It can be seen that the proposed estimation algorithm has a good effect on the estimation of the system state; Figure 4 and Figure 5 is the disturbance state and disturbance state estimation diagram; Figure 6 This is the disturbance estimation error diagram. It can be seen that the proposed estimation algorithm has a good effect on the estimation of system disturbances. Figure 7 and Figure 8 This is the intermittent quantized control input diagram. The quantization effect is obvious, and it can be seen that the control input acts intermittently.

Claims

1. A simple intermittent quantization control method for Chua's circuit system, characterized in that: The following steps are involved: Design an indirect signal-based estimation algorithm to estimate the unknown states and disturbances of the Chua circuit system. The specific steps are as follows: The Chua circuit system is described as follows: Where C1 and C2 are capacitors, L1 is an inductor, R is a resistor, R0 is a series resistor in the power branch, v1(t) and v2(t) are the voltages across C1 and C2, i3(t) is the current flowing through L1, and f(v1(t)) is the nonlinear function of the Chua diode, which is expressed as f(v1(t))=G b1 v1(t)+0.5(G a1 -G b1 )(|v1(t)+1|-|v1(t)-1|), G a1 and G b1 are the parameters of the Chua diode; Let x i1 (t) = v i1 (t), x i2 (t) = v i2 (t), x i3 (t) = i i3 (t), i = 1,…,N, the Chua circuit system can be written as follows: Its synchronization target system is defined as: in, N=5 is the number of nodes, x i (t) represents the state of the i-th node and is unknown, s(t) is the synchronization target state, f(x i (t)) and f(s(t)) represent the nonlinear function of the i-th node and the nonlinear function of the synchronization target, respectively, and assuming that f(·) satisfies the Lipschitz condition, u i (t) represents the control input of the i-th node, Q(u i (t)) represents the quantized u i (t), d i (t) represents a type of differentiable perturbation of the i-th node, and its upper bound is The upper bound of the derivative is δ, y i (t) and y s (t) represents the output of the i-th node and the output of the synchronization target, A, B, C are the known parameter matrices of the system, Γ and L represent the internal coupling matrix and external coupling matrix of the system, respectively; Define the indirect signal η of the i-th node i (t) = d i (t)-L b x i (t), the estimation algorithm for the unknown state and disturbance of the system is developed as follows: in, is x i The estimated value of (t), d i The estimated value of (t), is η i The estimated value of (t), L a , L b and L c is the gain matrix of the estimation algorithm, D is the gain matrix, is the integral term of the estimation error; make Then the estimated error system can be obtained as follows: in, Using Kronecker product operator The augmented form of the closed-loop estimation error system can be obtained: Among them, I N is the N-dimensional identity matrix, E x (t) is e xi (t) in matrix form, F(E x (t))Yes The matrix form of E η (t) is e ηi (t), D(t) is the matrix form of d(t), yes The matrix form of Based on the estimated state and disturbance, a quantized input distributed intermittent controller is designed, which includes estimated state feedback and disturbance compensation. The consistent ultimate bounded stability conditions of the estimation error system and the synchronization error system are obtained based on Lyapunov stability theorem, Young's inequality and mathematical induction, and the synchronization of Chua's circuit system is realized.

2. The simple intermittent quantization control method of the Chua circuit system according to claim 1, characterized in that: Based on the estimated state and disturbance, a quantized input distributed intermittent controller is designed, which includes estimated state feedback and disturbance compensation. The specific steps are as follows: Design a multi-channel uniform quantizer as follows: Among them, α 1i (t) and α 2i (t) represent the encoder and decoder parameters of the i-th channel, respectively, satisfying and α 1i and α 2i Represents α 1i (t) and α 2i The lower bound of (t), and Represents α 1i (t) and α 2i The upper bound of (t), round(·) is the rounding function; Define the ratio of the encoder and decoder parameters of the i-th channel as β i (t), which can be decomposed into known and unknown parts We can get: Q(u i (t))=β i (t)[u i (t)+e qi (t)] in, e q =q(u i (t))-u i (t) represents the quantization error. According to the definition of the rounding function, e qi The infinite norm of (t) |e qi (t)| ∞ Satisfy the following properties: Let e i (t) = x i (t)-s(t) and The closed-loop error system is as follows: in, Assumptions According to the above quantization scheme and the estimation algorithm, the intermittent control strategy is designed as follows: in, t∈[t k ,s k ] and t∈[s k ,t k+1 ] represent the kth working cycle and rest cycle of the controller, k i is the feedback gain of the i-th node, t k is the starting time of the kth working cycle, s k is the starting time of the kth rest period, for |α 2i (t)| ∞ A known upper bound of , sign(·) is the sign function, Represents the pseudo-inverse of the parameter matrix C, P4 is a positive definite matrix determined by the following conditions: Where inf{·} represents the infimum of the selected condition, exp(·) is the exponential function, α1, α2, β1, β2 are threshold parameters, and α1>α2>0, β2>β1>0; Its closed-loop error system can be rewritten as: Using Kronecker product operator The augmented form of the closed-loop synchronization error system can be obtained: Among them, I N is the N-dimensional identity matrix, E(t) is e i (t), F(E(t)) is In matrix form, U(t) is u i (t) in matrix form, E q (t) is e qi (t) in matrix form.

3. The simple intermittent quantization control method of the Chua circuit system according to claim 1, characterized in that: Based on the Lyapunov stability theorem, Young's inequality, and mathematical induction, we obtain the uniformly bounded stability conditions for the estimation error system and the synchronization error system, and achieve the synchronization of the Chua circuit system. The proof process is as follows: B001: Rewrite the estimated error system into the following form: in, B002: Select a suitable Lyapunov function: Among them, the positive definite matrix P1, P2, P3 are positive definite matrices, and Q1 is a given constant matrix; B003: Taking the derivative of B002, we get: In the formula, according to Young's inequality, we can get: Where,∈ is an arbitrary positive constant; B004: Based on B003, we can further sort out the following: Where, in, B005: Ensure the negative definiteness of Θ, and the estimated error system is uniformly bounded and stable and eventually converges to B006: Now consider the stability of the synchronization error system and select a suitable Lyapunov function: in, P4 is a positive definite matrix; B007: First, analyze the stability of the system when the controller is working, that is, t∈[t k ,s k ], the derivative of B006 under the working state of the controller can be obtained: B008: Order According to the Lipschitz condition, B007 is sorted as follows: Where: B009: Command According to B008, we can get: In the formula, according to Young's inequality and ||ξ(t)||<B1, we can get: Where a and ∈1 are any positive constants, B010: If but B011: The following is an analysis of the stability when the controller is not working; taking the derivative of Equation B006 when the controller is not working, we can obtain: B012: According to the Lipschitz condition, we can get: in, B013: Based on the results in B010 and B012, we can get: B014: According to mathematical induction, we can conclude that: in, l∈(0,1) represents the average control rate, N0>0 is an elastic coefficient; B015: Combining B005 and B014, the synchronization error system can converge if the following conditions are met: The Cai circuit system achieves synchronization under the proposed intermittent strategy.

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