Fixed-Time Algebraic Parameter Estimation Algorithm for Chua's Circuit System

Abstract

The present invention discloses a fixed-time algebraic parameter estimation algorithm for a Chua circuit system. Based on the constructed fixed-time algebraic parameter estimation algorithm, the fixed-time algebraic parameters of the predicted output are estimated. The fixed-time algebraic parameter estimation algorithm is constructed as follows: according to the Chua circuit system, a nonlinear mathematical model is established; the Volterra integral operator is used to determine the kernel function; the fixed-time stability is determined; and based on the kernel function and the nonlinear mathematical model, the fixed-time algebraic parameter estimation algorithm is constructed. The present invention can achieve accurate estimation of parameters within a fixed time without depending on the initial value of the system when only the output signal is known; by cleverly selecting the kernel function in the Volterra integral operator, the influence of the system initial value can be effectively eliminated, and at the same time, the calculation of the derivative of the output of the Chua circuit system is avoided; different from the traditional adaptive parameter estimation algorithm, the algebraic parameter estimation algorithm proposed by the present invention can achieve fast and accurate estimation of parameters.

Description

Technical Field

[0001] The invention relates to a fixed-time algebraic parameter estimation algorithm for a Chua circuit system, and belongs to the technical field of parameter estimation for a Chua circuit system. Background Art

[0002] As a very important nonlinear dynamic phenomenon in differential dynamics, chaos has attracted extensive attention from scholars due to its applications in secure communications, laser systems, electronic chemistry, and neurophysiology. Since the development of chaos theory has gradually entered the right track, people have been trying to observe chaos, an elusive dynamic phenomenon, with visible methods. As a result, the study of nonlinear circuits in electricity has gradually come into people's attention and has become an important research topic in the process of uncovering the mechanism of branches and chaos. The study of chaos in nonlinear circuits has a history of more than 30 years. With the unremitting efforts of researchers, many circuits for the purpose of studying chaos mechanisms have been constructed. At the same time, researchers have also conducted close research on the chaos phenomena generated in some practical circuits.

[0003] In 1983, Professor Leon Chua of the University of California designed the first Chua circuit that can simulate chaotic phenomena through experiments. This is a landmark design that has opened up a new path for the application of chaotic behavior in the real world. The Chua circuit is a third-order autonomous circuit that can produce double-scroll chaos. By giving a piecewise linear negative resistance a combination of different forms of parameters, it can produce particularly rich branching and chaotic behavior. The Chua circuit is the simplest autonomous circuit that can cause chaotic phenomena. Any chaotic phenomenon generated in a third-order autonomous system can be simulated by the Chua circuit system. The Shilnikov theorem makes a rigorous proof of the chaotic form generated by the Chua system. Based on this, people can observe and study various mechanisms of branching and chaos through the form of circuits. Therefore, the study of the Chua circuit has aroused the interest of many researchers, and the Chua system has gradually become a standard model in the study of chaotic phenomena.

[0004] The initial research on the Chua circuit system needed to assume that the parameters of the system were precisely known. Considering that in actual circuit design, due to various internal and external factors such as component aging, uncertain interference, and measurement costs, it is often difficult to accurately or directly obtain some parameters in the circuit. Therefore, how to solve the parameter estimation problem of the Chua circuit system with unknown parameters becomes particularly important, and the problem will become more difficult when only partial state variables of the system are available. To solve this problem, scholars have proposed various different methods, such as methods based on delay embedding and methods based on control theory. Among them, the methods based on control theory can estimate unknown parameters from hidden variables through methods such as observer design and system identification, and thus have been more and more widely used.

[0005] The convergence speed of the estimated parameters is a very crucial performance index in the process of parameter estimation. Most of the literatures involved in the previous text can only achieve the asymptotic estimation of parameters. Asymptotic estimation can only ensure that the estimation error of the parameters tends to zero when time approaches infinity, so a faster estimation speed cannot be obtained. From the perspective of time optimization, the estimation method that makes the estimation error converge in finite time is the time-optimal estimation algorithm. In addition, the finite-time estimation algorithm usually has a fractional power term, making the finite-time parameter estimation often have better robustness and anti-disturbance ability compared with the traditional asymptotic estimation. It is precisely because of the many advantages of finite-time parameter estimation that the research on finite-time parameter estimation of chaotic systems has received more and more attention from researchers and many positive results have been achieved.

[0006] Whether it is the traditional asymptotic parameter estimation algorithm or the finite-time estimation algorithm that has been widely studied in recent years, its convergence time depends on the initial estimation error of the parameters and increases with the increase of the initial estimation error. To overcome this defect, the concept of fixed-time stability has been proposed in the relevant literature. Fixed-time stability retains many advantages such as fast convergence speed and high convergence accuracy of finite-time stability, and also has the excellent characteristic that the convergence time does not depend on the initial value, so it has become a research hotspot at home and abroad. In recent years, problems such as fixed-time synchronization and fixed-time control of chaotic systems have been extensively studied, but there are few reports on the research of fixed-time parameter estimation problems of chaotic systems. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to overcome the defects of the prior art and provide a fixed-time algebraic parameter estimation algorithm for the Chua circuit system based on the Volterra integral operator. This algorithm can achieve accurate parameter estimation within a fixed time without relying on the system initial value when only the output signal is known. By cleverly selecting the kernel function in the Volterra integral operator, the present invention can effectively eliminate the influence of the system initial value and avoid calculating the derivative of the output of the Chua circuit system. In addition, different from the traditional adaptive parameter estimation algorithm, the algebraic parameter estimation algorithm proposed by the present invention can achieve fast and accurate parameter estimation.

[0008] To achieve the above object, the present invention provides a fixed-time algebraic parameter estimation algorithm for the Chua circuit system, including:

[0009] Predict the output fixed-time algebraic parameters based on the constructed fixed-time algebraic parameter estimation algorithm;

[0010] Construct the fixed-time algebraic parameter estimation algorithm through the following steps:

[0011] Establish a non-linear mathematical model according to the Chua circuit system;

[0012] Use the Volterra integral operator to determine the kernel function K(t,τ);

[0013] Determine the fixed-time stability;

[0014] Construct and obtain the fixed-time algebraic parameter estimation algorithm according to the kernel function K(t,τ) and the non-linear mathematical model.

[0015] Preferably, according to the Chua circuit system, construct a simplified non-linear model and establish a dimensionless non-linear mathematical model, including: The expression of the simplified non-linear model of the Chua circuit system is:

[0016]

[0017] In the formula, R is the linear resistance, is the voltage of the capacitor C1, is the voltage of the capacitor C2, i l is the current passing through the inductor L; is a function of the capacitor voltage, The expression is:

[0018]

[0019] In formula (2), m0, m1 and B p are three fixed constants of the diode;

[0020] Construct a non - linear mathematical model using the dimensionless form;

[0021] The expression of the non - linear mathematical model is:

[0022]

[0023] In the formula, \(f(x1)=ax1 + b(|x1 + 1|-|x1 - 1|)\) (4),

[0024]

[0025] In the formula, \(y1\) and \(y2\) represent the outputs of formula (3); \(m0\), \(m1\) and \(B\) p are three fixed constants of the Chua diode, is the voltage of capacitor \(C1\), is the voltage of capacitor \(C2\), \(i\) l is the current passing through inductor \(L\).

[0026] Preferably, use the Volterra integral operator to determine the kernel function \(K(t,\tau)\), including:

[0027] Define as a locally square - integrable function in the Hilbert space from \(R\) ≥0 to \(R\), \(R\) represents the real numbers, \(R\) ≥0 represents non - negative real numbers, and the locally square - integrable function \(f(t)\) is mapped and defined through the Volterra integral operator:

[0028]

[0029] In the formula, \(K(·,·):R×R→R\) is the Hilbert - Schmidt kernel function;

[0030] Based on the Leibniz differential rule, \([V\) K \(f](t)\) is generated by the following differential equation:

[0031]

[0032] In the formula, \(K(t,t)\) is the kernel function, and \(\xi0\) is a preset initial value constant;

[0033] Define \(f\) (i) (t), \(i\in Z\) + as the \(i\) - th derivative of \(f(t)\), \(Z\) + is a positive integer, then \([V\) K \(f\) (i) (t) expands to:

[0034]

[0035] In the formula, \(K\) (i)(·,·) represents the i-th derivative of K(·,·) with respect to the second variable;

[0036] The selection of the kernel function K(t,τ) determines the mapping of the Volterra integral operator [V K f](t) effect, and the selected kernel function is:

[0037] In formula (9), ω h , ω>0 are all preset parameters;

[0038] Expanding formula (9) gives:

[0039]

[0040] where f p (τ), p = 0, 1, 2 satisfies:

[0041]

[0042] In the formula, q = {0, 1, 2};

[0043] Based on formula (11), calculate and obtain

[0044]

[0045] Based on formula (10) and formula (12), obtain the derivative of the kernel function with respect to:

[0046]

[0047] Obtain for There is:

[0048] K (i) (t,t) = 0, K (i) (t,0) = 0 (14).

[0049] Preferably, determine the fixed-time stability, including:

[0050] The Chua circuit system is:

[0051]

[0052] In the formula, f(0) = 0, x(t) ∈ R n , f(x(t)): R n →R n is a continuous function from the domain n-dimensional space R n to the range n-dimensional space R n in, x0 = x(t0) represents the initial value of the Chua circuit system, and t0 represents the initial time;

[0053] Denote the solution of formula (15) as x(t, x0);

[0054] If there exists a fixed time T independent of the initial value x0 of the Chua's circuit system such that for any initial state x0, holds, then it is determined that the equilibrium point x = 0 of the Chua's circuit system is globally fixed-time stable.

[0055] Preferably, before constructing the fixed-time algebraic parameter estimation algorithm, the following processing is performed on the Chua's circuit system:

[0056] Transform formula (3) into the form of the output of the Chua's circuit system and its derivative:

[0057]

[0058] where y3 = |y1 + 1| - |y1 - 1|,

[0059] The parameters θ1, θ2, and θ3 satisfy:

[0060] θ1 = -β(1 + a), θ2 = β, θ3 = -bβ (17);

[0061] Transform the estimation of β, a, b, γ into the estimation of θ1, θ2, θ3, γ;

[0062] Perform the integral operator V K operation on both sides of formula (16):

[0063]

[0064] where y1(t), y2(t) are the outputs of formula (3);

[0065] Substitute formula (14) into formula (18) to obtain:

[0066]

[0067] Substitute formula (19) into formula (18) to obtain:

[0068]

[0069] Design a parameter estimator based on formula (20);

[0070] For Construct a differential equation:

[0071]

[0072] where, Obtained from formula (13);

[0073] If the kernel function K(t, τ) satisfies formula (9), then For Holds;

[0074] Obtained from the differential equation Therefore, formula (20) is rewritten as:

[0075]

[0076] Where

[0077]

[0078] Where, v1(t) and v2(t) satisfy the persistent excitation assumption.

[0079] Preferably, before constructing the fixed-time algebraic parameter estimation algorithm, the following persistent excitation assumption is set:

[0080] Assumption 1: The functions v1(t) and v2(t) satisfy the persistent excitation condition, and there exist constants r > 0, T0 > 0 such that the following inequality holds for Holds:

[0081]

[0082] In the formula, I 3×3 Is a 3×3 identity matrix;

[0083] Multiply formula (23) on the left by v1 T (t) and v2(t) respectively, and we get:

[0084]

[0085] Where

[0086]

[0087] For both sides of formula (26), perform the integral operator Operation, where K g = e -g(t-τ) , g > 0, and we get:

[0088]

[0089] Where

[0090] The Chua circuit system generates variables s 1,f (t), s 2,f (t), v 1,f (t) and v 2,f(t):

[0091]

[0092] Preferably, according to the kernel function K(t, τ) and the non - linear mathematical model, a fixed - time algebraic parameter estimation algorithm is constructed, including: The expression of the fixed - time algebraic parameter estimation algorithm is:

[0093]

[0094] The parameters ε, g satisfy:

[0095] Wherein, represents the previous value of the parameter , represents the previous value of the parameter , min{eig(v 1,f (t))} and min{eig(v 2,f (t))} respectively represent the minimum eigenvalue of v 1,f (t) and v 2,f (t).

[0096] An electronic device, including a memory, a processor, and a computer program stored on the memory and executable on the processor, characterized in that when the processor executes the program, the steps of the method described in any one of the above are implemented.

[0097] A computer - readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the steps of the method described in any one of the above are implemented.

[0098] The beneficial effects achieved by the present invention:

[0099] 1. First, compared with the existing parameter estimation strategies based on asymptotic convergence and finite - time convergence, the fixed - time parameter estimation algorithm proposed by the present invention not only ensures a fast convergence speed and high convergence accuracy, but also has the characteristic that the convergence time does not depend on the initial estimation error;

[0100] 2. In addition, the Volterra integral operator used in the present invention can effectively eliminate the influence of the system initial value and avoid calculating the derivative of the system output, so that the proposed algorithm only needs to use the output information of the system, thus effectively reducing the use of sensors and effectively reducing the implementation cost of the algorithm;

[0101] 3. Finally, in the existing literature, in order to obtain enough equations to eliminate the influence of unknown variables, several different kernel functions need to be selected, while the algorithm proposed by the present invention only needs to select one kernel function in addition to being able to achieve fixed - time parameter estimation, thus greatly reducing the algorithm complexity. Brief Description of the Drawings

[0102] Figure 1 is the schematic diagram of the Chua circuit system of the present invention;

[0103] Figure 2 is the three-dimensional diagram of the chaotic attractor of the Chua circuit system of the present invention;

[0104] Figure 3 is the chaotic attractor diagram projected on the x1Ox2 plane of the Chua circuit system of the present invention;

[0105] Figure 4 is the chaotic attractor diagram projected on the x2Ox3 plane of the Chua circuit system of the present invention;

[0106] Figure 5 is the chaotic attractor diagram projected on the x1Ox3 plane of the Chua circuit system of the present invention;

[0107] Figure 6 is for the present invention with adaptive initial values and the curve diagram of the true value and estimated value of β;

[0108] Figure 7 is for the present invention with adaptive initial values and the curve diagram of the true value and estimated value of γ;

[0109] Figure 8 is for the present invention with adaptive initial values and the curve diagram of the true value and estimated value of a;

[0110] Figure 9 is for the present invention with adaptive initial values and the curve diagram of the true value and estimated value of b;

[0111] Figure 10 is for the present invention with adaptive initial values and the curve diagram of the true value and estimated value of β;

[0112] Figure 11 is for the present invention with adaptive initial values and the curve diagram of the true value and estimated value of γ;

[0113] Figure 12 is for the present invention with adaptive initial values and the curve diagram of the true value and estimated value of a;

[0114] Figure 13 is for the present invention with adaptive initial values and the curve diagram of the true value and estimated value of n;

[0115] Figure 14This is the flowchart of the present invention. Detailed implementation manners

[0116] The following embodiments are only used to illustrate the technical solutions of the present invention more clearly, and cannot be used to limit the protection scope of the present invention.

[0117] Embodiment 1

[0118] The fixed-time algebraic parameter estimation algorithm for the Chua circuit system includes the following steps:

[0119] Step 1: Establish a nonlinear mathematical model according to the Chua circuit system;

[0120] Step 2: Use the Volterra integral operator to determine the kernel function K(t,τ);

[0121] Step 3: Determine the fixed-time stability according to the kernel function K(t,τ) and the nonlinear mathematical model, and construct a fixed-time algebraic parameter estimation algorithm;

[0122] Based on the constructed fixed-time algebraic parameter estimation algorithm, predict and output the fixed-time algebraic parameters.

[0123] Step 1 includes:

[0124] The Chua circuit system includes three energy storage elements (one inductor L and two capacitors C1, C2), a linear resistor R, and a nonlinear resistor called the Chua diode. The simplified nonlinear mathematical model of the Chua circuit system is as follows:

[0125]

[0126] In formula (1), R is the linear resistor, is the voltage of C1, is the voltage of C2, and i l is the current passing through the inductor L; is the function of the voltage to represent the current passing through the nonlinear resistor. This nonlinear function is described by the following odd-symmetric piecewise linear function:

[0127]

[0128] In formula (2), m0, m1, and B p are three fixed constants of the Chua diode;

[0129] Hypothesis 1: During the experiment, in the case of many variables or an inability to fully reproduce the model, through the method of dimensionlessization, by substituting with a suitable variable, some or all of the units of an equation involving physical quantities can be removed to simplify the experiment or calculation purposes.

[0130] Equation (1) is transformed into a dimensionless form:

[0131]

[0132] In Equation (3), y1 and y2 represent the outputs of the Chua circuit system; f(x1) is specifically expressed as Equation (4); β, γ, a, b, x1, x2, x3 are specifically expressed as Equation (5);

[0133] f(x1) = ax1 + b(|x1 + 1| - |x1 - 1|) (4)

[0134]

[0135] Step 2 is specifically as follows:

[0136] Define as R ≥0 to a locally square-integrable function in the Hilbert space from R to R, where R represents the real numbers, and R ≥0 represents non-negative real numbers. The locally square-integrable function f(t) is mapped and defined through the Volterra integral operator:

[0137]

[0138] In Equation (6), K(·,·): R×R→R is a Hilbert-Schmidt kernel function;

[0139] For Equation (6), it can be better calculated in the form of a differential equation. Based on the Leibniz differential rule, [V K f](t) is generated by the following differential equation:

[0140]

[0141] where K(t,t) is the kernel function and ξ0 is a preset initial value constant;

[0142] Define f (i) (t), i∈Z + as the i-th derivative of f(t), and Z + is a positive integer. Then [V K f (i) (t) expands to:

[0143]

[0144] where \(K\) (i) (·,·) represents the \(i\)-th derivative of \(K(·,·)\) with respect to the second variable;

[0145] The selection of the kernel function \(K(t,τ)\) determines the mapping effect of the Volterra integral operator \([V K f](t)\), and the kernel function selected in the present invention is:

[0146]

[0147] In formula (9), \(\omega\) h , \(\omega>0\) are all preset parameters;

[0148] Expanding formula (9) gives:

[0149]

[0150] where \(f p (τ), p = 0, 1, 2 satisfy:

[0151]

[0152] In the formula, \(q=\{0, 1, 2\}\);

[0153] Based on formula (11), calculate and obtain

[0154]

[0155] Based on formula (10) and formula (12), obtain the derivative of the kernel function with respect to

[0156]

[0157] Based on formula (12) and formula (13), obtain for there is:

[0158] K (i) (t, t)=0, \(K (i) (t, 0)=0\ (14).

[0159] In step 3, determining the fixed-time stability specifically includes the following steps:

[0160] Property 1: Consider the following Chua circuit system:

[0161]

[0162] where \(f(0)=0\), \(x(t)\in R n , \(f(x(t)):R n \to R n is the domain \(n\)-dimensional space \(Rn to the range of an n-dimensional space R n a continuous function in, where x0 = x(t0) represents the initial value of the Chua circuit system, and t0 represents the initial moment, which is a fixed value;

[0163] Property 2: The solution of formula (15) is denoted as x(t, x0): If there exists a fixed time T that does not depend on the initial value x0 of the Chua circuit system, such that for any initial state x0, there is holds, then the equilibrium point x = 0 of formula (15) is globally fixed-time stable.

[0164] Before constructing the fixed-time algebraic parameter estimation algorithm, the following processing needs to be done on the Chua circuit system:

[0165] For the convenience of designing the parameter estimator, rewrite formula (3) in the form of the output of the Chua circuit system and its derivative:

[0166]

[0167] where y3 = |y1 + 1| - |y1 - 1|, and the parameters θ1, θ2, and θ3 satisfy:

[0168] θ1 = -β(1 + a), θ2 = β, θ3 = -bβ (17)

[0169] The estimation of β, a, b, γ is transformed into the estimation of θ1, θ2, θ3, γ;

[0170] Perform the integral operator V K operation on both sides of formula (16):

[0171]

[0172] In formula (18), y1(t), y2(t) are known, but their derivatives are unknown, so is unknown; Expand formula (18) and substitute formula (14) into formula (18) to get:

[0173]

[0174] Substitute formula (19) into formula (18) to get:

[0175]

[0176] In Equation (20), except for the parameters θ1, θ2, θ3, and γ, all other variables are known, so they can be used to design a parameter estimator. However, directly calculating the Volterra integral operator mapping in Equation (20) according to Equation (6) is very cumbersome. Calculating the required variables in Equation (20) using differential equations is more convenient for practical applications.

[0177] Property 1: For Construct a differential equation:

[0178]

[0179] where is obtained from Equation (13); if the kernel function K(t, τ) satisfies Equation (9), then For holds;

[0180] Obtained from Property 1 Therefore, Equation (20) is rewritten as:

[0181]

[0182] For the convenience of parameter estimator design, Equation (22) is rewritten as:

[0183]

[0184] where

[0185]

[0186] where v1(t) and v2(t) satisfy the persistent excitation assumption.

[0187] Before constructing a fixed-time algebraic parameter estimation algorithm, the following persistent excitation assumption is set:

[0188] Assumption 1: The functions v1(t) and v2(t) satisfy the persistent excitation condition, that is, there exist constants r > 0 and T0 > 0 such that the following inequality holds for holds:

[0189]

[0190] where I 3×3 is a 3×3 identity matrix;

[0191] According to Assumption 1, one of the two equalities in Equation (23) is in vector form and the other is in scalar form. Therefore, multiply both equations on the left by v1 T (t) and v2(t) to obtain:

[0192]

[0193] Among them

[0194]

[0195] Integrate both sides of formula (26) with the integral operator operation, where K g = e -g(t-τ) , g > 0, to obtain:

[0196]

[0197] Among them The variable s 1,f (t), s 2,f (t), v 1,f (t) and v 2,f (t) are generated by the following Chua circuit system:

[0198]

[0199] The fixed-time algebraic parameter estimation algorithm in step 3 is specifically as follows:

[0200] Theorem 1: The estimated values of the parameters θ, γ Satisfy:

[0201]

[0202] Among them Represents the previous value of the parameter , Represents the previous value of the parameter , where each variable is given by formulas (21), (24), (27), and (29), min{eig(v 1,f (t))} and min{eig(v 2,f (t))} respectively represent the minimum eigenvalue of v 1,f (t) and v 2,f (t).

[0203] If assumption 1 is satisfied, and the parameters ε, g satisfy

[0204]

[0205] Then for any t ≥ T0, it satisfies

[0206]

[0207] That is The estimation of the parameters θ, γ can be achieved at a fixed time T0.

[0208] The proof of Theorem 1 is as follows:

[0209] From the definition of v 1,f (t) and Hypothesis 1, it can be obtained that when t ≥ T0:

[0210]

[0211] Similarly, it can be obtained that:

[0212]

[0213] From (33) and (34), it can be seen that when t ≥ T0, v 1,f (t) and v 2,f (t) are invertible and satisfy

[0214]

[0215] Therefore, from formula (28), it can be known that when t ≥ T0, holds, and thus Theorem 1 is proved.

[0216] Example 2

[0217] The Chua system circuit is modeled according to the attached Figure 1 of the specification, and the parameter selection is as follows:

[0218] x1(0) = -0.9, x2(0) = -0.15, x3(0) = 1.47

[0219] γ = 27, β = 15.6, a = -5 / 7, b = -3 / 14

[0220] The chaotic attractor of the system and its projections on each coordinate plane are as shown in the attached Figure 2 of the specification. The fixed-time algebraic parameter estimation algorithm is designed according to Theorem 1, and the parameter selection is as shown in Table 1 of the specification.

[0221] Table 1

[0222]

[0223] To verify the fixed-time convergence of the algorithm, two different adaptive initial values are selected respectively:

[0224] (C.1):

[0225] (C.2):

[0226] to conduct the simulation verification of the algorithm.

[0227] When k = 10 -12 , T0 = 0.6s, the convergence time T of the parameter estimator can be calculated max≈1.4 s, select the adaptive initial value The simulation results are shown in the appendix Figures 6 - 9 as follows. From Figures 6 - 9 it can be seen that the algorithm proposed by the present invention can accurately estimate the system parameters within 1.4 seconds. When a large adaptive initial value is selected The simulation results are shown in the appendix Figures 10 - 13 as follows. From the appendix Figures 10 - 13 it can be seen that even under a large initial state, the algorithm of the present invention can still accurately estimate the system within 1.4 seconds, thus verifying the fixed-time convergence property of the algorithm.

[0228] This application is described with reference to the flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or block in the flowchart and / or block diagram can be implemented by computer program instructions, and the combination of the processes and / or blocks in the flowchart and / or block diagram can also be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing devices generate a device for implementing the functions specified in one process Figure 1 or multiple processes and / or blocks Figure 1 or multiple blocks.

[0229] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer-readable memory generate a manufactured article including an instruction device, and the instruction device implements the functions specified in one process Figure 1 or multiple processes and / or blocks Figure 1 or multiple blocks.

[0230] These computer program instructions can also be loaded onto a computer or other programmable data processing device, so that a series of operation steps are executed on the computer or other programmable device to generate a computer-implemented process, and thus the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in one process Figure 1 or multiple processes and / or blocks Figure 1 or multiple blocks.

[0231] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the technical principle of the present invention, several improvements and modifications can be made, and these improvements and modifications should also be regarded as the protection scope of the present invention.

Claims

1. Fixed-time algebraic parameter estimation algorithm for Chua's circuit system, characterized in that, Including: Predict the output fixed-time algebraic parameters based on the constructed fixed-time algebraic parameter estimation algorithm; Construct the fixed-time algebraic parameter estimation algorithm, which is realized through the following steps: Establish a non-linear mathematical model according to the Chua circuit system; Determine the kernel function using the Volterra integral operator ; Determine the fixed-time stability; According to the kernel function and the non-linear mathematical model, a fixed-time algebraic parameter estimation algorithm is constructed and obtained; According to the Chua circuit system, construct a simplified non-linear model and establish a dimensionless non-linear mathematical model, including: The expression of the simplified non-linear model of the Chua circuit system is: （1） wherein, R is a linear resistor, is the capacitor voltage, is the capacitor voltage, is the current passing through the inductor L; is a function of the capacitor voltage, The expression is: （2） In formula (2) , and are three fixed constants of the diode; Use the dimensionless form to construct a non-linear mathematical model; The expression of the non-linear mathematical model is: （3）， In the formula, (4), （5）， In the formula, and represent the output of formula (3); , and are three fixed constants of the Chua diode, is the voltage of capacitor , is the voltage of capacitor , is the current passing through inductor L; Determine the kernel function using the Volterra integral operator , including: Definition is to a locally square-integrable function in the Hilbert space, denotes a real number, denotes a non-negative real number, the locally square-integrable function is defined by mapping through the Volterra integral operator: （6）， In the formula, : Hilbert - Schmidt kernel function; Based on Leibniz's differential rule, which is generated by the following differential equation: （7）， In the formula, is the kernel function, is the preset initial value constant; Definition is of order derivative, is a positive integer, then expands to: （8）， In the formula, represents the n-th derivative with respect to the second variable; Kernel function The selection of which determines the mapping of the Volterra integral operator effect, and the selected kernel function is as follows: （9）， In Formula (9), are all preset parameters; Expanding formula (9) gives: （10）， Among them Satisfy: （11） ， In the formula, q = {0, 1, 2}; Based on Equation (11), it is calculated that :[[]]END]] （12）， Based on formula (10) and formula (12), obtain the derivative of the kernel function with respect to: （13）， Obtain for , there is: （14）； Determine the fixed-time stability, including: The Chua circuit system is: （15）， In the formula, is a continuous function from the domain n-dimensional space to the range n-dimensional space , and represents the initial value of the Chua circuit system, represents the initial time. Denote the solution of formula (15) as ; If there exists a fixed time that does not depend on the initial value of the Chua circuit system , such that for any initial state , there is holding, then it is determined that the equilibrium point of the Chua circuit system is globally fixed-time stable; Before constructing the fixed-time algebraic parameter estimation algorithm, perform the following processing on the Chua circuit system: Transform formula (3) into the form of the output of the Chua circuit system and its derivative: （16）， Among them, , Parameter , and satisfy: （17）； Convert the estimate of to an estimate of ; Integrate both sides of formula (16) with the integral operator operation: （18）， In the formula, is the output of formula (3); Substitute formula (14) into formula (18) to obtain: （19） Substitute formula (19) into formula (18) to obtain: （20）， Design a parameter estimator based on formula (20); For , construct a differential equation: （21） wherein, obtained by formula (13); If the kernel function satisfies formula (9), then for it holds; Obtained from the differential equation , so formula (20) is rewritten as: （23）， Where （24）， Among them, and satisfy the continuous incentive hypothesis; Before constructing the fixed-time algebraic parameter estimation algorithm, set the following persistent excitation assumption: Hypothesis 1: Function and satisfy the continuous incentive condition. There exists a constant such that the following inequality holds for : （25）， where I 3×3 is the identity matrix; Left-multiply both sides of Equation (23) by and respectively, and we get: （26）， Where （27）， Integrate both sides of formula (26) with the integral operator and perform the operation, where , to obtain: （28）， Among them, , , , ; The Chua's circuit system generates variables , , and : （29）； According to the kernel function and the non-linear mathematical model, a fixed-time algebraic parameter estimation algorithm is constructed, including: The expression of the fixed-time algebraic parameter estimation algorithm is: （30）， Parameter , satisfies: (31), Among them, represents the previous value of parameter , represents the previous value of parameter , and respectively represent and the minimum eigenvalue.

2. An electronic device, comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, characterized in that, When the processor executes the program, it realizes the steps of the fixed-time algebraic parameter estimation algorithm of the Chua circuit system described in any one of claims 1.

3. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it realizes the steps of the fixed-time algebraic parameter estimation algorithm of the Chua circuit system described in any one of claims 1.

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