A new criterion for passivity and dissipativity of noise reduction system based on fractional order theory

Abstract

This invention discloses a novel criterion for the passivity and dissipativeness of noise reduction systems based on fractional-order theory. The method first establishes a discrete-time fractional-order generalized Cohen-Grossberg neural network model of the noise reduction system based on fractional-order difference equation theory. Then, it gives a definition of discrete-time fractional-order passivity based on Lyapunov functionals and designs a control input consisting of state variables with and without time delays, enabling the noise reduction system to reach a passive state, thereby reducing noise. Finally, it utilizes the dissipativeness problem to control the network system to achieve closed-loop stability under different initial conditions. When applied to noise reduction systems, this method reduces system noise and improves the operating environment.

Description

Technical Field

[0001] This invention relates to a passive and dissipative control method, specifically to a new criterion for the passive and dissipative properties of a noise reduction system based on fractional-order theory. Background Technology

[0002] Many dynamic models used in noise reduction systems are based on continuous-time operations. However, in practical applications, noise reduction systems are not always operational. Using difference equations, more accurate system models of discrete-time noise reduction systems can be established. Furthermore, noise reduction systems may experience performance degradation due to time delays and other factors. Therefore, discussing the passivity and dissipation properties of noise reduction systems is crucial for ensuring system stability. Based on this idea, researchers have conducted extensive research on the passivity and dissipation properties of discrete-time noise reduction systems, achieving a series of results.

[0003] Furthermore, current denoising systems are typically characterized using integer-order differential equations. However, integer-order calculus is insufficient for describing the memory and genetic properties of network systems. Therefore, introducing fractional-order difference equation theory into discrete-time denoising systems is particularly important. Summary of the Invention

[0004] The purpose of this invention is to propose a new criterion for the passivity and dissipation of noise reduction systems based on fractional-order theory, which can effectively improve the performance of noise reduction.

[0005] The specific technical solution of this invention is as follows: A new criterion for the passivity and dissipation of a noise reduction system based on fractional-order theory, comprising the following steps:

[0006] Based on the theory of discrete-time fractional difference equations, a discrete-time fractional generalized Cohen-Grossberg neural network model for the noise reduction system is established.

[0007] Based on the discrete-time fractional difference equation theory, the following dynamic model is established for the noise reduction system in reference [1]:

[0008]

[0009] In the formula, β represents the fractional-order system order, z(t) represents the system state variable, p(z(t)) represents the amplification function, h(z(t)) represents the behavior function, I(t) represents the external input, U(t) represents the control input, f(·) represents the activation function, α1 and α2 represent the synaptic connections of the network, η represents the discrete time delay, and y(t) represents the system output. A, D, D1, and D2 are known system matrices of appropriate dimensions. p(z(t)), h(z(t)), and f(t) have the following properties:

[0010]

[0011] Furthermore, using the definition of passivity for integer-order systems in discrete time, we give a definition of discrete-time fractional-order passivity based on Lyapunov functionals. The specific steps are as follows:

[0012] First, regarding the inequalities Do both sides get

[0013]

[0014] Next, for any time t≥0, we have and therefore

[0015] Right now

[0016] Finally, according to the definition of passivity, for the energy function V(t), if the inequality is satisfied... Therefore, a discrete-time fractional-order system is passive.

[0017] Furthermore, a control input consisting of state variables with and without time delay is designed to enable the discrete-time network system to reach a passive state, i.e., the network system achieves internal stability at the energy level, thereby reducing noise. The specific steps are as follows:

[0018] Construct the following control input:

[0019] U(t)-K1z(t)-K2z(t-η)

[0020] In the formula, K1 and K2 are both constant matrices. This control scheme can guarantee the uniformly bounded stability of the system, and the proof is as follows:

[0021] C001: Select the energy function in the following form:

[0022] V(t) = z T (t)Qz(t),

[0023] C002: In the formula, z(t) represents the state variables of the system, Q represents the positive definite matrix, and T represents the transpose;

[0024] C003: Calculate the Nabla fractional sum of V(t), where:

[0025]

[0026] C004: Further, we can obtain:

[0027]

[0028] C005: Further, we can obtain:

[0029]

[0030]

[0031]

[0032] C006: Combining C004-C005, we can obtain:

[0033] C007: In the formula

[0034] C008: Further, when Sometimes,

[0035] C009: According to C008, the noise reduction system is passive under the definition of discrete-time fractional-order passivity based on the Lyapunov functional used in this invention.

[0036] Furthermore, the dissipation problem is utilized to control the network system to achieve stability under closed-loop conditions under different initial conditions. The specific steps are as follows:

[0037] C010: Select the energy function in the following form:

[0038] V(t) = z T (t)Qz(t),

[0039] C011: In the formula, z(t) represents the state variables of the system, Q represents the positive definite matrix, and T represents the transpose;

[0040] C012: Calculate the Nabla fractional sum of V(t), where:

[0041]

[0042] C013: Further, we can obtain:

[0043]

[0044] C014: Combining C012 and C013, we get:

[0045] in

[0046] C015: Further, when Ω < 0, we can obtain:

[0047]

[0048] C016: Consider the system Define x * (t)=x(t)-b * So there are Where b * =b / a

[0049] C017: Applying a discrete Laplace transform to both sides of the above equation, we obtain: Right now

[0050] Where Δ(s) = s β +a, d(s)=s β-1 x * (0)

[0051] C018: Next, we will prove that the characteristic equation det(Δ(s))=0 has no purely imaginary roots.

[0052] C019: First, assume that det(Δ(s)) = 0 has purely imaginary roots, that is, For any If ξ > 0, then Or ζ < 0, then

[0053] C020: will Substituting into the characteristic equation det(Δ(s))=0, we can obtain sβ +a=0

[0054] C021: Considering the real and imaginary parts separately, we can obtain

[0055]

[0056] C022: Therefore,

[0057] C023: Based on C022, we can obtain That is, the characteristic equation has no purely real solutions. Therefore, the assumption in C019 is invalid.

[0058] C024: Matrix M = -a has negative eigenvalues, that is...

[0059] C025: Furthermore, in B016, we consider that the zero solution of the system is Lyapunov globally asymptotically stable. Therefore, x(t)-b * →0 for t→∞. In other words, for any ε>0, there exists T>0 such that x(t)<b * +ε, where x(t)>0.

[0060] C026: Based on the comparison principle of discrete-time fractional-order systems, we can obtain 0≤V(t)≤x(t).

[0061] C027: Furthermore, for t > T, V(t) < b * +ε, that is, for t>T, V(t)≤b * = b / a. That is, for t > T, ||z(t)||1 ≤ b * .

[0062] C028: Therefore, there exists T > 0 such that for That is, the noise reduction system in this invention is dissipative. Attached Figure Description

[0063] Figure 1 The above is an energy relationship diagram of a noise reduction system using the method proposed in this invention under controlled input conditions, as shown in the example.

[0064] Figure 2 The following is a dynamic response diagram of a noise reduction system using the method proposed in this invention under conditions of no external input, as shown in the example.

[0065] Figure 3 The following is a dynamic response diagram of a noise reduction system using the method proposed in this invention under no control input conditions, as shown in the example.

[0066] Figure 4The following is a dynamic response diagram of a noise reduction system using the method proposed in this invention under conditions of external input and control input;

[0067] Figure 5-8 The following are dynamic response diagrams of the noise reduction system using the method proposed in this invention under different initial states in the embodiments. Detailed Implementation

[0068] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading the present invention, any modifications of the present invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0069] A new criterion for the passivity and dissipation of a noise reduction system based on fractional-order theory includes the following steps:

[0070] Step 1: Set the various system parameters;

[0071] Step 2: Solve for the positive definite matrix Q that satisfies the passivity condition;

[0072] Step 3: Set the initial state of the system;

[0073] Step 4: Based on the relationship between the input and output systems in the definition of passivity, determine 2I. T (t)y(t)+λI T Is I(t) positive?

[0074] Step 5: Remove external inputs to the system and verify whether the system is stable under control input only;

[0075] Step 6: Remove the system's control inputs and verify whether the system is stable under external input only;

[0076] Step 7: Verify whether the system achieves stability under the combined effect of external input and control input;

[0077] Step 8: Reset all system parameters;

[0078] Step 9: Solve for the positive definite matrix Q and positive definite diagonal matrices R1 and R2 that satisfy the dissipative property conditions;

[0079] Step 10: Set several different initial system states;

[0080] Step 11: Verify whether the system is stable within a certain range under different initial states.

[0081] An embodiment of the present invention is described below:

[0082] Considering the noise reduction system, its corresponding dynamic model is:

[0083]

[0084]

[0085] The energy relationship diagram of the noise reduction system under controlled input conditions is as follows: Figure 1 As shown, the dynamic response diagram of the noise reduction system under no external input condition is as follows: Figure 2 As shown, the dynamic response diagram of the noise reduction system under no control input condition is as follows: Figure 3 As shown, the dynamic response diagram of the noise reduction system under conditions of external input and control input is as follows: Figure 4 As shown in the figure, the proposed method based on fractional-order theory achieves a passive state when applied to the noise reduction system. The dynamic response diagrams of the noise reduction system under different initial states are shown in the figure. Figure 5-8 As shown, the proposed method based on fractional-order theory achieves a dissipative state when applied to the noise reduction system, eventually stabilizing within a fixed range.

[0086] References [1]Zheng P, Zhang J, Tang W.Color image associative memory on a class of Cohen-Grossberg networks.Pattern Recogn 2010;43(10):3255-3260.

Claims

1. A new criterion for passivity and dissipativity of noise reduction systems based on fractional order theory, characterized in that, Includes the following steps: Based on the theory of discrete-time fractional difference equations, the specific steps for establishing a discrete-time fractional generalized Cohen-Grossberg neural network model for the noise reduction system are as follows: Based on the discrete-time fractional difference equation theory, the following dynamic model is established for the noise reduction system in reference [1]: In the formula, β represents the order of the fractional-order system, z(t) represents the system state variable, p(z(t)) represents the amplification function with z(t) as the independent variable, h(z(t)) represents the behavior function with z(t) as the independent variable, I(t) represents the external input, U(t) represents the control input, f(·) represents the activation function, α1 and α2 represent the synaptic connections of the network, η represents the discrete time delay, y(t) represents the system output, and A, D, D1, and D2 are known system matrices of appropriate dimensions; the amplification function p(z(t)), the behavior function h(z(t)), and the activation function f(t) have the following properties: |f(z2)-f(z1)|≤δ|z2-z1 |, z1≠z2 Using the definition of passivity for integer-order systems in discrete time, we give a definition of passivity for discrete-time fractional-order systems based on Lyapunov functionals. Design a control input consisting of state variables with and without time delay, and use the proposed discrete-time fractional-order passivity definition to control the network system to achieve internal stability at the energy level, thereby reducing noise; The dissipation problem is used to control the stability of the network system under different initial conditions in a closed-loop system.

2. The new criteria of passivity and dissipativity for a noise reduction system based on fractional order theory according to claim 1, characterized in that, Using the definition of passivity for integer-order systems in discrete time, we present a definition of discrete-time fractional-order passivity based on Lyapunov functionals. The specific steps are as follows: First, take the inequality and divide both sides by to get Where V(t) represents the energy function; Next, for any time t≥0, we have and therefore Right now Finally, according to the definition of passivity, for the energy function V(t), if it satisfies the inequality then the discrete-time fractional-order system is passive.

3. The new criteria of passivity and dissipativity for a fractional order theory based noise reduction system according to claim 2, characterized in that, Design a control input consisting of state variables with and without time delays to enable the discrete-time network system to reach a passive state, i.e., the network system achieves internal stability at the energy level, thereby reducing noise. The specific steps are as follows: Construct the control input U(t) as follows. U(t) = K1z(t) - K2z(t-η) In the formula, K1 and K2 are both constant matrices; the control input U(t) guarantees the uniform bounded stability of the system, and the proof is as follows: B001: Select the energy function in the following form: V(t) = z T (t) Qz(t) B002: In the formula, z(t) represents the state variables of the system, Q represents the positive definite matrix, and T represents the transpose; B003: Calculate the Nabla fractional sum of V(t). B004: Further considering the input and output, we can obtain: B005: By scaling, we can obtain: B006: Combining B003 and B004, we can obtain: B007: In the formula B008: When time, there B009: According to B008, the noise reduction system is passive under the discrete-time fractional-order passivity definition based on the Lyapunov functional used in this invention.

4. The new criteria of passivity and dissipativity for a noise reduction system based on fractional theory according to claim 3, characterized in that, The specific steps for using dissipation problems to control the stability of a network system under different initial conditions in a closed-loop system are as follows: B010: Select the energy function in the following form: V(t) = z T (t) Qz(t) B011: In the formula, z(t) represents the state variables of the system, Q represents the positive definite matrix, and T represents the transpose; B012: Calculate the Nabla fractional sum of V(t). B013: In the formula, z(t) represents the system's state variables, Q represents the positive definite matrix, and T represents the transpose; further, we can obtain: B014: Combining B012 and B013, we can obtain: B015: When Available: B016: For the system If x exists * (t)=x(t)-b * So there are Where b * =b / a; B017: Taking the discrete Laplace transform on both sides of the above equation, we have i.e. S β x * (s)-s β-1 x * (0) = -ax * (s) Δ(s)x * (s) = d(s) where Δ(s) = s β + a, d(s) = s β-1 x * (0) B018: Next, we will prove that the characteristic equation det(Δ(s))=0 has no purely imaginary roots; B019: Assume that the characteristic equation det(Δ(s)) = 0 has a pure imaginary root, i.e. If ξ > 0, then If, on the contrary, ξ < 0, then B020: will be Substituting the characteristic equation det(Δ(s)) = 0, we get s β +a=0 B021: Considering the real and imaginary parts separately, we can obtain B022: Thus, available B023: Based on B022, we have i.e. the characteristic equation has no real solution; then, the assumption in B019 is not true; B024: Assume that the matrix M = -a has negative eigenvalues, i.e. B025: Furthermore, the zero solution of the system in B010 is Lyapunov globally asymptotically stable; therefore, for t→∞, we have x(t)-b * →0, meaning that for any ε>0, there exists T>0 such that x(t)<b * +ε; B026: Based on the discrete-time fractional order comparison principle, we can obtain 0≤V(t)≤x(t); B027: Furthermore, for t > T, V(t) < b * +ε; Similarly, we can obtain V(t)≤b if t>T. * =b / a, then for t>T we have ||z(t)||1≤b * ; B028: Therefore, the noise reduction system is dissipative.

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