Statistical complexity measurement-based fractional order system stochastic resonance analysis method

Abstract

The invention belongs to the field of structural dynamics analysis, and particularly relates to a statistical complexity measurement-based fractional order system stochastic resonance analysis method, which comprises the following steps of: 1, solving a time sequence of system response; 2, counting a time sequence of continuous residence time intervals of the system; 3, calculating a probability distribution function of the residence time interval; and 4, calculating the statistical complexity of the system.

Description

Technical Field

[0001] The present application belongs to the field of structural dynamics analysis, and specifically relates to a random resonance analysis method for fractional-order systems based on statistical complexity measurement. Background Art

[0002] Stochastic resonance is essentially the behavior of noise enhancing system response. It mainly describes the synergistic effect of weak periodic signals input into nonlinear systems and random forces in the system output. That is, at a certain optimal noise intensity value, the signal-to-noise ratio of the system output reaches the maximum value, and the noise plays a positive role in the system.

[0003] In recent years, fractional derivatives have received extensive attention in the study of some "complex" engineering and physical phenomena, such as stress relaxation of viscoelastic materials, anomalous diffusion of particles, turbulence and fractal problems, etc. However, the research on fractional-order random dynamic systems has just started, and the research on fractional-order stochastic resonance is relatively scarce. How to measure the fractional-order stochastic resonance phenomenon is currently a difficult and key issue.

[0004] Due to the power-law characteristics of fractional-order systems, which exhibit memory properties of past states in time (non-Markov properties), existing methods for measuring stochastic resonance of fractional-order systems cannot accurately measure the internal dynamic behavior of the system. In view of this, this application is proposed. Summary of the invention

[0005] The purpose of this application is to provide a random resonance analysis method for fractional-order systems based on statistical complexity measurement, so as to more accurately measure the internal dynamic behavior of fractional-order systems and provide a reference for regulating the random resonance effect of fractional-order systems.

[0006] The technical solution of this application is:

[0007] A method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement, comprising:

[0008] Step 1: Obtain the time series of system response;

[0009] Step 2: Count the time series of the continuous residence time intervals of the system;

[0010] Step 3, calculating the probability distribution function of the dwell time interval;

[0011] Step 4: Calculate the statistical complexity of the system.

[0012] According to at least one embodiment of the present application, in the above-mentioned method for analyzing stochastic resonance of a fractional-order system based on statistical complexity measurement, in step 1, a fractional-order numerical solution is used to obtain a time series of system response.

[0013] According to at least one embodiment of the present application, in the above-mentioned method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement, step 1 is specifically as follows:

[0014] Starting from the original Grünwald-Letnikov definition of fractional derivatives, we can write the discrete equivalent of fractional derivatives:

[0015]

[0016] in,

[0017] h is the step length;

[0018] p is the fractional derivative, usually 0 <p<1;

[0019] x(t l-j ) is t l-j Time series of time points;

[0020] D p [x(t l )] is t l The fractional derivative at a time point, where t l = lh is the time point, l is the number;

[0021] is the j-th fractional binomial coefficient, specifically:

[0022]

[0023]

[0024] The fractional-order system is iteratively solved to obtain the time series of the system response.

[0025] According to at least one embodiment of the present application, in the above-mentioned method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement, step 2 is specifically as follows:

[0026] According to the potential function diagram of the system, the stable and unstable points of the potential function are obtained, so as to calculate the residence time interval of the particle transition between potential wells, and obtain the time series of the continuous residence time intervals of the system {x t :t=1,…M}.

[0027] According to at least one embodiment of the present application, in the above-mentioned method for analyzing stochastic resonance of a fractional-order system based on statistical complexity measurement, in step three, the probability distribution function of the dwell time interval is calculated using the Bandt-Pompe algorithm.

[0028] According to at least one embodiment of the present application, in the above-mentioned method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement, step three is specifically as follows:

[0029] P = {p(π)};

[0030]

[0031] in,

[0032] P is the probability distribution function of the dwell time interval;

[0033] p(π) is the probability distribution function of the dwell time interval reconstructed into elements;

[0034] D is the embedding dimension;

[0035] M is the length of the dwell time interval sequence;

[0036] # indicates the corresponding quantity.

[0037] According to at least one embodiment of the present application, in the above-mentioned method for analyzing stochastic resonance of a fractional-order system based on statistical complexity measurement, in step three, 3≤D≤7.

[0038] According to at least one embodiment of the present application, in the above-mentioned method for analyzing stochastic resonance of a fractional-order system based on statistical complexity measurement, in step 3, M>>D! .

[0039] According to at least one embodiment of the present application, in the above-mentioned random resonance analysis method of fractional-order systems based on statistical complexity measurement, in step four, the Shannon entropy value of the system is calculated according to the definition formula of information theory measurement, and then the statistical complexity of the system is calculated.

[0040] According to at least one embodiment of the present application, in the above-mentioned method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement, step 4 is specifically as follows:

[0041] C[P]=H[P]·Q[P];

[0042] in,

[0043] C[P] is the statistical complexity of the system;

[0044] H[P] is the relative standard Shannon entropy;

[0045] Q[P] is the degree of imbalance.

[0046] According to at least one embodiment of the present application, in the above-mentioned method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement, in step 4, H[P]=S[P] / S max , 0≤H[P]≤1;

[0047]

[0048] S max =S[Pe ];

[0049] P e ={1 / N,…1 / N};

[0050] in,

[0051] S[P] is the standard Shannon entropy;

[0052] p i is the i-th element in the probability distribution function P;

[0053] N is the number of elements in the probability distribution function P;

[0054] P e is the uniform probability distribution function;

[0055] S max is the maximum Shannon entropy.

[0056] According to at least one embodiment of the present application, in the above-mentioned method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement, in step 4, Q[P]=Q0·D s [P,P e ];

[0057]

[0058] D s [P] = {S[(P + P e ) / 2]-S[P] / 2-S[P e ] / 2};

[0059] in,

[0060] Q[P] is the degree of imbalance, which measures the distance of the system from equilibrium;

[0061] Q0 is a normalization constant, and 0≤Q[P]≤1;

[0062] D s [P,P e ] is the distance between the probability distribution of the system’s reachable states and the uniform distribution.

[0063] This application has at least the following beneficial technical effects:

[0064] A random resonance analysis method for fractional-order systems based on statistical complexity measurement is provided, which can accurately measure the internal dynamic behavior of the system with statistical complexity, improve the efficiency of measuring the random resonance of fractional-order systems, and provide a feasible means for future research on the random resonance phenomenon of fractional-order bistable systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1It is a schematic diagram of a random resonance analysis method of a fractional-order system based on statistical complexity measurement provided in an embodiment of the present application;

[0066] Figure 2 is a system potential function diagram provided in an embodiment of the present application;

[0067] Figure 3 It is a schematic diagram of the stochastic resonance of the system provided in the embodiment of the present application.

[0068] In order to better illustrate the present embodiment, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of the actual product. In addition, the drawings are only used for illustrative purposes and should not be construed as limitations on the present application. DETAILED DESCRIPTION

[0069] In order to make the technical solution and advantages of the present application clearer, the technical solution of the present application will be described in further detail in detail and in detail with reference to the accompanying drawings. It can be understood that the specific embodiments described here are only partial embodiments of the present application, which are only used to explain the present application, not to limit the present application. It should be noted that, for the convenience of description, only the parts related to the present application are shown in the accompanying drawings, and other related parts can refer to the general design.

[0070] In addition, unless otherwise defined, the technical terms or scientific terms used in the description of this application shall have the usual meanings understood by those of ordinary skill in the art to which this application belongs. The words used in the description of this application to indicate orientation are only used to indicate relative directions or positional relationships. When the absolute position of the described object changes, its relative positional relationship may also change accordingly. The word "include" used in the description of this application indicates that the elements or objects appearing before the word include the elements or objects listed after the word and their equivalents, but does not exclude other elements or objects.

[0071] In addition, it should be noted that, unless otherwise clearly specified and limited, the words "installation", "connection" and similar terms used in the description of this application should be understood in a broad sense. For example, the connection can be a fixed connection or a detachable connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium. Technical personnel in the field can understand its specific meaning in this application according to the specific circumstances.

[0072] A method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement, such as Figure 1 shown.

[0073] Step 1: Obtain the time series of system response.

[0074] The fractional order numerical solution is used to obtain the time series of system response.

[0075] Starting from the original Grünwald-Letnikov definition of fractional derivatives, the discrete equivalent form of fractional derivatives is as follows:

[0076]

[0077] in,

[0078] h is the step length;

[0079] p is the fractional derivative, usually 0 <p<1;

[0080] x(t l-j ) is t l-j Time series of time points;

[0081] D p [x(t l )] is t l The fractional derivative at a time point, where t l = lh is the time point, l is the number;

[0082] is the j-th fractional binomial coefficient, as follows:

[0083]

[0084]

[0085] Therefore, using the above formula, the fractional-order system can be iteratively solved to obtain the time series of the system response.

[0086] Step 2: Count the time series of the continuous residence time intervals of the system.

[0087] According to the potential function diagram of the system, the stable and unstable points of the potential function are obtained, so as to calculate the residence time interval of the particle transition between potential wells, and obtain the time series of the continuous residence time intervals of the system {x t :t=1,…M}.

[0088] Step 3: Calculate the probability distribution function of the dwell time interval.

[0089] The Bandt-Pompe algorithm is used to calculate the probability distribution function of the residence time interval. This method mainly considers the temporal causal relationship of the system time series, and its main idea is based on the process of attractor reconstruction.

[0090] For the dwell time interval time series {x t :t=1,…M} and embedding dimension D>1, for any time t, select corresponding continuous D data at time {t, t-1,…, t-(D-1)}: Then rearrange the D data to obtain an "ordered pattern" about the ordinal number D:

[0091]

[0092] Let π=(r0,r1,…,r (D-1) ) represents the rearrangement of (0,1,…D-1). In order to ensure the uniqueness of the arrangement in the above formula, when Season i <r i-1 Established, that is, each corresponds to a unique permutation π.

[0093] Therefore, for all D! possible permutations π of the ordinal number D, the time series {x t :t=1,…M}, we can get the corresponding probability distribution function of “ordered pattern” P={p(π)}:

[0094]

[0095] in,

[0096] # indicates the corresponding quantity, 3≤D≤7, and the length of the time series of the dwell time interval must meet the condition M>>D! .

[0097] Step 4: Calculate the statistical complexity of the system.

[0098] According to the definition of information theory measurement, the Shannon entropy value of the system is calculated, and then the statistical complexity of the system is calculated.

[0099] The standard Shannon entropy is defined as:

[0100]

[0101] in,

[0102] S[P] is the standard Shannon entropy;

[0103] p i is the i-th element in the probability distribution function P;

[0104] N is the number of elements in the probability distribution function P.

[0105] When S[P]=0, Shannon entropy is the smallest, indicating that the system is in an ordered state, that is, the amount of information obtained by the system is the largest. When S[P]=lnN, Shannon entropy is the largest, indicating that the system is in a disordered state and the amount of information obtained by the system is the smallest.

[0106] Furthermore, the relative standard Shannon entropy can be defined as

[0107] H[P]=S[P] / S max

[0108] S max =S[P e ]=lnN

[0109] P e ={1 / N,…1 / N};

[0110] in:

[0111] H[P] is the relative standard Shannon entropy, 0≤H[P]≤1;

[0112] P e is the uniform probability distribution function;

[0113] S max is the maximum Shannon entropy.

[0114] The imbalance degree is used to measure the distance between the probability distribution of the reachable state of the system and the uniform distribution, as follows:

[0115] Q[P]=Q0·D s [P,P e ]

[0116]

[0117] in,

[0118] Q[P] is the degree of imbalance, which measures the distance of the system from equilibrium;

[0119] Q0 is a normalization constant, and 0≤Q[P]≤1;

[0120] D s [P,P e ] is the distance between the probability distribution of the reachable state of the system and the uniform distribution, which can generally be defined by Jensen divergence, as follows:

[0121] D s [P,P e ]={S[(P+P e ) / 2]-S[P] / 2-S[P e ] / 2};

[0122] Then the statistical complexity of the system is:

[0123] C[P]=H[P]·Q[P];

[0124] in,

[0125] C[P] is the statistical complexity of the system.

[0126] The statistical complexity C[P] can systematically measure the internal dynamic behavior of the fractional-order system and provide a reference for regulating the stochastic resonance effect of the fractional-order system.

[0127] After obtaining the statistical complexity C[P] of the system, the accuracy of the statistical complexity in measuring stochastic resonance can be verified by further numerically calculating the simulated signal-to-noise ratio. Specifically, the accuracy of the statistical complexity in measuring stochastic resonance can be verified by calculating the signal-to-noise ratio of the system. The signal-to-noise ratio of the system is calculated as follows:

[0128]

[0129] in,

[0130] SNR is the signal-to-noise ratio of the system;

[0131] S(w) is the power spectrum density of the input signal;

[0132] N(w) is the power spectrum of the noise at the signal frequency w.

[0133] In a specific example, the stochastic resonance analysis method for fractional-order systems based on statistical complexity measurement disclosed in the above embodiment is implemented as follows.

[0134] Consider a fractional-order bistable system driven by a periodic signal under Gaussian white noise excitation. The system equation is as follows:

[0135] D q x(t)=-U'0(x)+Asin(ωt)+ξ(t),0 <q<1;

[0136] Among them, r, s represents the damping coefficient, the fractional derivative D q x(t) uses the Caputo definition and satisfies the following:

[0137]

[0138] Wherein, ξ(t) is Gaussian white noise, satisfying <ξ(t)>=0, <ξ(t)ξ(0)>=2Dδ(t); F(t) is a periodic signal, F(t)=Acos(ωt), A is the signal amplitude, and ω is the signal frequency.

[0139] The potential function of the fractional-order system is V(x) = -ax 2 / 2+bx 4 / 4, the selected parameters are a=1, b=1. Figure 1 As shown, the potential function has two stable points and an unstable point x m =0, the potential barrier energy height of the system is △V=1 / 4.

[0140] Firstly, the solution of the system equation is obtained by the fractional-order Euler algorithm. Then, the residence time series of particles transitioning between potential wells is calculated. Finally, the Bandt–Pompe algorithm is used to extract the probability distribution of the residence time series to finally obtain the transformation curve of statistical complexity and standard Shannon entropy. In the process, the embedding dimension D=6 and the residence time series interval is M=50000.

[0141] The stochastic resonance case when the fractional derivative α = 0.93 is considered, such as Figure 2 As shown in the figure, when the signal amplitude A = 0.2 and the frequency ω = 0.03 are taken, the statistical complexity, standard Shannon entropy and signal-to-noise ratio all show non-monotonic curves. As the noise intensity increases, the statistical complexity first increases to the peak and then decreases, while the standard Shannon entropy shows the opposite trend, first decreasing to the trough and then increasing. This non-monotonic behavior indicates that the system state first goes from disorder to order, and then from order to disorder. The complexity of the fractional-order system presents a maximum value at a certain noise intensity, and the output signal level reaches the maximum at the same time, which indicates that random resonance occurs at a certain noise intensity. The accuracy of the statistical complexity method is further verified by numerically simulating the signal-to-noise ratio. The conclusion shows that random resonance can occur in the fractional-order overdamped bistable system at an appropriate noise intensity.

[0142] The random resonance analysis method of a fractional-order system based on statistical complexity measurement disclosed in the above embodiment can accurately measure the internal dynamic behavior of the system with statistical complexity, improve the efficiency of measuring the random resonance of the fractional-order system, and provide a feasible means for future research on the random resonance phenomenon of the fractional-order bistable system.

[0143] So far, the technical solution of the present application has been described in conjunction with the preferred embodiments shown in the accompanying drawings. Those skilled in the art should understand that the scope of protection of the present application is obviously not limited to these specific embodiments. Without departing from the principles of the present application, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions after these changes or substitutions will fall within the scope of protection of the present application.

Claims

1. A method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement, characterized in that: include: Step 1: Obtain the time series of system response; Step 2: Count the time series of the continuous residence time intervals of the system; Step 3, calculating the probability distribution function of the dwell time interval; Step 4: Calculate the statistical complexity of the system.

2. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 1 is characterized in that: In step 1, a fractional order numerical solution is used to obtain the time series of the system response.

3. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 2 is characterized in that: Step 1 is as follows: Starting from the original Grünwald-Letnikov definition of fractional derivatives, we can write the discrete equivalent of fractional derivatives: in, h is the step length; p is the fractional derivative, usually 0 <p<1; x(t l-j ) is t l-j Time series of time points; D p [x(t l )] is t l The fractional derivative at a time point, where t l = lh is the time point, l is the number; is the j-th fractional binomial coefficient, specifically: The fractional-order system is iteratively solved to obtain the time series of the system response.

4. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 3 is characterized in that: Step 2 is as follows: According to the potential function diagram of the system, the stable and unstable points of the potential function are obtained, so as to calculate the residence time interval of the particle transition between potential wells, and obtain the time series of the continuous residence time intervals of the system {x t :t=1,…M}.

5. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 4 is characterized in that: In step 3, the probability distribution function of the dwell time interval is calculated using the Bandt-Pompe algorithm.

6. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 5, characterized in that: Step three is as follows: P = {p(π)}; in, P is the probability distribution function of the dwell time interval; p(π) is the probability distribution function of the dwell time interval reconstructed into elements; D is the embedding dimension; M is the length of the dwell time interval sequence; # indicates the corresponding quantity.

7. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 6 is characterized in that: In step 3, 3≤D≤7.

8. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 7 is characterized in that: In step three, M>>D! .

9. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 8, characterized in that: In step 4, according to the definition of information theory measurement, the Shannon entropy value of the system is calculated, and then the statistical complexity of the system is calculated.

10. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 9, characterized in that: Step 4 is as follows: C[P]=H[P]·Q[P]; in, C[P] is the statistical complexity of the system; H[P] is the relative standard Shannon entropy; Q[P] is the degree of imbalance.

11. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 10, characterized in that: In step 4, H[P]=S[P] / S max , 0≤H[P]≤1; S max =S[P e ]; P e ={1 / N,…1 / N}; in, S[P] is the standard Shannon entropy; p i is the i-th element in the probability distribution function P; N is the number of elements in the probability distribution function P; P e is the uniform probability distribution function; S max is the maximum Shannon entropy.

12. The method for analyzing stochastic resonance of fractional-order systems based on statistical complexity measurement according to claim 11, characterized in that: In step 4, Q[P] = Q0·D s [P,P e ]; D s [P]={S[(P+P e ) / 2]-S[P] / 2-S[P e ] / 2}; in, Q[P] is the degree of imbalance, which measures the distance of the system from equilibrium; Q0 is a normalization constant, and 0≤Q[P]≤1; D s [P,P e ] is the distance between the probability distribution of the system’s reachable states and the uniform distribution.

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