Sliding mode control method based on fast damping approaching law

Abstract

The application belongs to the automatic control technical field and is based on a sliding mode control method of a fast damping approaching law; the sliding mode control method comprises the following steps: establishing a state space equation of a second-order nonlinear system; designing a sliding mode variable matrix; designing a fast damping approaching law; analyzing the fast damping approaching law characteristics; and designing a sliding mode controller based on the fast damping approaching law; the application has the beneficial effects that: through the analysis of a traditional exponential approaching law and a power approaching law, a fast damping approaching law is designed, compared with the existing approaching law, the form is simple, the number of parameters to be adjusted is small, the time for the system state variable to reach the sliding surface is further shortened, the chattering amplitude of the control signal output by the controller is further weakened, and based on the designed fast damping approaching law, a sliding mode control method is proposed, the influence of the initial position of the system on the controller is reduced, and the saturation of the controller is effectively prevented.

Description

Technical Field

[0001] This invention relates to a sliding mode control method based on a fast vibration reduction approach law, which belongs to the field of automatic control technology. Background Technology

[0002] The article "Power system instability and chaos" points out that voltage collapse in power systems is closely related to chaotic oscillations. In many cases, eliminating chaos can help avoid operational failures and improve system performance.

[0003] In recent years, extensive research has been conducted on the control of chaotic systems, resulting in the proposal of numerous controller design methods, such as adaptive control, sliding mode control, feedback linearization, and inversion methods. Among these, sliding mode control is widely popular due to its fast dynamic response and robustness to system disturbances.

[0004] Sliding mode control is a special type of variable structure control that allows the system's state variables to move along a predetermined "sliding mode" trajectory. Because the sliding mode can be designed and is independent of object parameters and disturbances, sliding mode control offers advantages such as fast response, insensitivity to parameter changes and disturbances, no need for online system identification, and simple physical implementation. However, a drawback of this method is that once the state trajectory reaches the sliding surface, it is difficult to strictly slide along the sliding surface to the equilibrium point; instead, it traverses back and forth on both sides of the sliding surface. This characteristic is called discontinuous switching, which causes system chattering. Since eliminating chattering also eliminates its disturbance rejection capability, chattering is inevitable and cannot be eliminated; it can only be mitigated to a certain extent. Chattering has become a prominent obstacle to the practical application of sliding mode control in systems.

[0005] Numerous studies, both domestically and internationally, have addressed the problem of chattering, with many scholars proposing solutions from various perspectives. These include quasi-sliding mode methods, reaching law methods, and dynamic sliding mode methods. Among these, Professor Gao Weibing's reaching law method can both ensure the motion quality during the sliding mode arrival process and reduce high-frequency jitter in the control signal.

[0006] Currently, the approach law method can be divided into two aspects: traditional approach law and new approach law.

[0007] I. Traditional Approach Law

[0008] Traditional laws of convergence include the constant-rate convergence law, the exponential convergence law, and the power-law convergence law. The exponential convergence law is a prime example. For example, Professor Gao Weibing effectively reduced chattering by adjusting the parameters k and ε of the approach law, but a large ε would cause chattering; Lin improved the traditional vibration reduction approach law by analyzing the arrival time and chattering degree of the traditional approach law, shortening the arrival time and chattering amplitude, and improving the stability of the system.

[0009] II. New Approach Law

[0010] To address the shortcomings of traditional convergence laws and their various applications, scholars have designed new convergence law methods.

[0011] Mehran combined fixed-time theory with the reaching law, making the arrival time independent of the initial position, but the reaching law is complex and the parameters are difficult to tune; Parijat combined the minimum difference equation with the reaching law, achieving the main purpose of reducing chattering, but did not optimize the arrival time; Deepika replaced the switching function with a continuous hyperbolic function, which reduced the chattering caused by the discontinuity of the switching function, but when the system state variable reached the sliding surface, the presence of the hyperbolic sine function enhanced the chattering of the system.

[0012] It is evident that the future research direction of the approach law method will be characterized by its simplicity, small number of parameter tunings, short time for system state variables to reach the sliding surface, and small chattering amplitude of the controller output signal. Summary of the Invention

[0013] To address the shortcomings of existing reaching law methods and to target future research directions in reaching law methods, this invention proposes a sliding mode control method based on a fast vibration reduction reaching law, aiming to achieve the following technical objectives: the reaching law is simple in form, the number of parameters to be tuned is relatively small, the time for the system state variables to reach the sliding surface is further shortened, and the chattering amplitude of the control signal output by the controller is further reduced.

[0014] The objective of this invention is achieved as follows:

[0015] The sliding mode control method based on the fast vibration reduction reaching law is characterized by the following steps:

[0016] Step a: Establish the state-space equations of the second-order nonlinear system.

[0017]

[0018] in, This is the system state variable matrix; H(X) = [0, H2(X)] T The zero matrix in H(X) The nonlinear part of the system A and B are both coefficient matrices of the system, and: in, B = [0, B2] T The zero matrix in B Full rank matrix Controller output

[0019] Step b: Design the sliding mode variable matrix S

[0020] S=K1X1+K2X2+Γσ(X1)

[0021] Among them, the sliding mode variable matrix X1 and X2 are two submatrices of the system state variable matrix X, respectively. K1, K2, and Γ are all sliding mode variable parameter matrices. Full rank matrix σ(X1)=[|x1| τ sign(x1),...,|x n-m | τ sign(x n-m )] T , 0 < τ < 1, x1...x n-m For system state variables, sign() is the sign function;

[0022] Step c: Design a fast vibration reduction approach law

[0023]

[0024]

[0025] The approach law parameters are: 0 < η < 1; γ < 0; |γ| < η; μ = α|s0|; k1 > 0, s0 is the value of the sliding mode variable s at t = 0;

[0026] Step d: Analyze the characteristics of the rapid vibration reduction reaching law.

[0027] The time it takes for the system state variables to reach the sliding surface is:

[0028]

[0029] The jitter amplitude output by the controller is:

[0030]

[0031] Where T is the system sampling period;

[0032] Step e: Design a sliding mode controller based on a fast vibration reduction reaching law.

[0033] By making the expression for the derivative of the sliding mode variable with respect to time equal to the expression for the fast damping reaching law, we obtain the expression for the sliding mode controller as follows:

[0034]

[0035] Where K = [K1, K2], Us =[u s1 ,...,u sm ] T , It is a constant;

[0036]

[0037] Where i = 1, ..., m.

[0038] The beneficial effects of this invention are as follows:

[0039] First, by analyzing the traditional exponential and power-law approach laws, a fast vibration reduction approach law was designed. Compared with the existing approach laws, it is simpler in form and requires fewer parameters to be tuned; the time for the system state variables to reach the sliding surface is further shortened; and the chattering amplitude of the control signal output by the controller is further reduced.

[0040] Secondly, based on the fast vibration reduction approach law designed in this invention, a sliding mode control method is proposed, which reduces the influence of the initial position of the system on the controller and effectively prevents the controller from saturating. Attached Figure Description

[0041] Figure 1 This is a flowchart of the sliding mode control method based on the fast vibration reduction reaching law of the present invention;

[0042] Figure 2 This is a simulation block diagram of the method of the present invention applied to the Lü's chaotic system;

[0043] Figure 3 The figure shows the simulation results of the system state variable trajectory in the Lü's chaotic system using the sliding mode control method based on the vibration reduction reaching law.

[0044] Figure 4 The simulation results of the system state variable trajectory in the Lü's chaotic system using the method of the present invention are shown in the figure.

[0045] Figure 5 The figure shows the simulation results of the sliding mode variable trajectory in the Lü's chaotic system using the sliding mode control method based on the vibration reduction reaching law.

[0046] Figure 6 The simulation results of the sliding mode variable trajectory in the Lü's chaotic system using the method of the present invention are shown in the figure.

[0047] Figure 7 The figure shows the simulation results of the controller output in the Lü's chaotic system using the sliding mode control method based on the vibration reduction reaching law.

[0048] Figure 8 The figure shows the simulation results of the controller output in the Lü's chaotic system using the method of the present invention. Detailed Implementation

[0049] The specific embodiments of the present invention will now be described in further detail with reference to the accompanying drawings.

[0050] The flowchart of the sliding mode control method based on the fast vibration reduction reaching law in this specific implementation is as follows: Figure 1 As shown, it includes the following steps:

[0051] Step a: Establish the state-space equations of the second-order nonlinear system.

[0052]

[0053] in, This is the system state variable matrix; H(X) = [0, H2(X)] T The zero matrix in H(X) The nonlinear part of the system A and B are both coefficient matrices of the system, and: in, B = [0, B2] T The zero matrix in B Full rank matrix Controller output

[0054] Step b: Design the sliding mode variable matrix S

[0055] S=K1X1+K2X2+Γσ(X1)

[0056] Among them, the sliding mode variable matrix X1 and X2 are two submatrices of the system state variable matrix X, respectively. K1, K2, and Γ are all sliding mode variable parameter matrices. Full rank matrix σ(X1)=[|x1| τ sign(x1),...,|x n-m | τ sign(x n-m )] T , 0 < τ < 1, x1...x n-m For system state variables, sign() is the sign function;

[0057] Step c: Design a fast vibration reduction approach law

[0058]

[0059]

[0060] The approach law parameters are: 0 < η < 1; γ < 0; |γ| < η; μ = α|s0|; k1 > 0, s0 is the value of the sliding mode variable s at t = 0;

[0061] Step d: Analyze the characteristics of the rapid vibration reduction reaching law.

[0062] The time it takes for the system state variables to reach the sliding surface is:

[0063]

[0064] The jitter amplitude output by the controller is:

[0065]

[0066] Where T is the system sampling period;

[0067] Step e: Design a sliding mode controller based on a fast vibration reduction reaching law.

[0068] By making the expression for the derivative of the sliding mode variable with respect to time equal to the expression for the fast damping reaching law, we obtain the expression for the sliding mode controller as follows:

[0069]

[0070] Where K = [K1, K2], U s =[u s1 ,...,u sm ] T , It is a constant;

[0071]

[0072] Where i = 1, ..., m.

[0073] To more effectively demonstrate the superiority of the sliding mode control method based on the fast vibration reduction reaching law proposed in this invention, the designed controller was applied to a Lübe chaotic system, and MATLAB / Simulink simulation experiments were conducted, such as... Figure 2 As shown. This experiment uses the ode45 solver in Simulink, with a variable step size, a maximum step size of 0.0001s, and a simulation time of 2s. The expression for the Lü's chaotic system is:

[0074]

[0075] The initial state of the system is [1.5, 0.8, -3], and the system parameters are [a1, a2, a3]. T =[36,20,3] T .

[0076] The sliding surface is defined by S = K1X1 + K2X2 + Γσ(X1), with relevant parameters K1 = [0.1, 0.1]. T The identity matrix K2 = R 2 ×2 Γ = [0.1, 0.1] T , τ=0.5.

[0077] The controller adopts:

[0078]

[0079] The relevant parameters are: identity matrix B2 = R 2×2 , k1=4, eta=0.2, α=0.01, γ=-0.06.

[0080] Figure 3 The figure shows the simulation results of the system state variable trajectory in the Lü's chaotic system using the sliding mode control method based on the vibration reduction reaching law. Figure 4 The figure shows the simulation results of the system state variable trajectory in a Lübe chaotic system using a sliding mode control method based on a fast vibration reduction reaching law. The vibration reduction reaching law is...

[0081]

[0082]

[0083] The relevant parameters are: k1 = 4, η = 0.2, γ = 0.05, μ = 3.3.

[0084] Depend on Figure 3 and Figure 4 It can be seen that the convergence time of the system state variables is significantly shortened.

[0085] Figure 5 The figure shows the simulation results of the sliding mode variable trajectory in the Lü's chaotic system using the sliding mode control method based on the vibration reduction reaching law. Figure 6 The figure shows the simulation results of the sliding mode variable trajectory in the Lü's chaotic system using the sliding mode control method based on the fast vibration reduction reaching law.

[0086] Depend on Figure 5 and Figure 6 It can be seen that the rapid vibration reduction approach law has a significant advantage in terms of arrival time.

[0087] Figure 7 The figure shows the simulation results of the controller output in the Lü's chaotic system using the sliding mode control method based on the vibration reduction reaching law. Figure 8 The figure shows the simulation results of the controller output in the Lü's chaotic system using the sliding mode control method based on the fast vibration reduction reaching law.

[0088] Depend on Figure 7and Figure 8 It can be seen that the chattering amplitude output by the sliding mode controller is significantly reduced under the action of the rapid damping approach law.

[0089] The sliding mode control method based on the fast vibration reduction reaching law of this invention has better robustness than the traditional reaching law sliding mode control method. It can effectively reduce the chattering of the system and accelerate the convergence speed of the system.

Claims

1. A sliding mode control method based on a fast vibration reduction reaching law, characterized in that, Includes the following steps: Step a: Establish the state-space equations of the second-order nonlinear system. in, This is the system state variable matrix; H(X) = [0, H2(X)] T The zero matrix in H(X) The nonlinear part of the system A and B are both coefficient matrices of the system, and: in, B = [0, B2] T The zero matrix in B Full rank matrix Controller output Step b: Design the sliding mode variable matrix S S=K1X1+K2X2+Γσ(X1) Wherein, the sliding mode variable matrix S=[s1,...,s m ] T , X1 and X2 are two submatrices of the system state variable matrix X, i.e., X = [X1, X2] T , K1, K2, and Γ are all sliding mode variable parameter matrices. Full rank matrix σ(X1)=[|x1| τ sign(x1),...,|x n-m | τ sign(x n-m )] T , 0 < τ < 1, x1...x n-m For system state variables, sign() is the sign function; Step c: Design a fast vibration reduction approach law The approach law parameters are: 0 < η < 1; γ < 0; |γ| < η; μ = α|s0|; k1 > 0, s0 is the value of the sliding mode variable s at t = 0; Step d: Analyze the characteristics of the rapid vibration reduction reaching law. The time it takes for the system state variables to reach the sliding surface is: The jitter amplitude output by the controller is: Where T is the system sampling period; Step e: Design a sliding mode controller based on a fast vibration reduction reaching law. By making the expression for the derivative of the sliding mode variable with respect to time equal to the expression for the fast damping reaching law, we obtain the expression for the sliding mode controller as follows: Where K = [K1, K2], U s =[u s1 ,...,u sm ] T , It is a constant; Where i = 1, ..., m.

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