Terminal sliding mode control method based on arc tangent function and application thereof

Abstract

The invention discloses a terminal sliding mode control method based on an arc tangent function and application of the terminal sliding mode control method. The traditional terminal sliding mode control is fast in convergence, but has a singular problem. The traditional nonsingular terminal sliding mode control solves the singular problem at the cost of sacrificing the convergence time. Therefore, the arc tangent function and the power function are introduced on the basis of the terminal sliding mode, the novel non-singular fast terminal sliding mode control method is constructed, and compared with a traditional terminal sliding mode control method, the method is continuous at a balance point, the singular problem is solved, and the method has the advantages that the method is simple and convenient to operate. Compared with the traditional linear sliding mode control, integral sliding mode control and nonsingular terminal sliding mode control, the method ensures that the speed regulation system has higher convergence speed, higher tracking precision and stronger anti-interference performance.

Description

Technical Field

[0001] The present invention relates to the technical field of motor control, and more specifically, to a terminal sliding mode control method based on an inverse tangent function and an application thereof. Background Art

[0002] Permanent Magnet Synchronous Motor (PMSM) is widely used in industrial fields such as robots, electric vehicles, and wind power generation due to its good torque performance, wide speed range, and fast dynamic response. At present, field-oriented control (FOC) technology is widely used in permanent magnet synchronous motor speed regulation systems. This technology generally uses proportional integral (PI) control. Because the PI control method has a simple structure and is easy to implement, it has become the mainstream method for motor control. However, in actual PMSM applications, since uncertain interferences such as parameter changes and external interferences will affect the control performance of the system, it is difficult to meet the application of PMSM in high-performance occasions by using only linear control methods such as PI control.

[0003] In order to meet the high-performance application of PMSM, domestic and foreign scholars have proposed many nonlinear control methods, such as adaptive control, fuzzy control, predictive control, sliding mode control (SMC), etc. Due to the large-scale application of digital signal processors, SMC with discontinuous high-frequency switching terms stands out among these nonlinear control methods due to its strong anti-interference ability and insensitivity to system uncertainty.

[0004] However, sliding mode control is not perfect. The traditional sliding mode control method uses a linear sliding surface, which has few parameters and is easy to implement, but has a large overshoot without constraints and can only guarantee the asymptotic convergence of the controlled output; although the traditional terminal sliding mode control method has a faster response speed, its singularity problem will reduce the tracking accuracy, while the non-singular terminal sliding mode control method has a slower response speed. Therefore, in order to improve the tracking accuracy, response speed, and anti-interference performance of the permanent magnet synchronous motor speed control system, it is necessary to seek an advanced control method to achieve fast and high-precision response of the permanent magnet synchronous motor under interference conditions. Summary of the invention

[0005] In order to overcome the above defects in the prior art, the present invention provides a terminal sliding mode control method based on an inverse tangent function and its application.

[0006] The above technical objectives of the present invention are achieved through the following technical solutions: a terminal sliding mode control method based on an inverse tangent function and its application, comprising the following steps:

[0007] On the basis of the hyperbolic tangent terminal sliding surface, a new non-singular fast terminal sliding surface based on the inverse hyperbolic tangent function is constructed, and its expression is as follows:

[0008] s AT =x2+λ|x1| 1-ε arctan(h|x1| ε )sgn(x1) (1)

[0009] Where ε, λ and h are real numbers greater than zero, and 0<ε<1;

[0010] When AT = 0, the state error function of formula (1) quickly reaches the neighborhood near the zero point within a certain period of time; the convergence time t1 of the proposed sliding surface can be obtained as follows:

[0011]

[0012] Furthermore, the expression of the hyperbolic tangent terminal sliding surface is as follows:

[0013] s TH =x2+λx1 1-q / p tanh(hx1 q / p ) (3)

[0014] Where q, p, λ and h are real numbers greater than zero, p and q are both odd numbers, and p>q, 0<ε<1.

[0015] When the system variable converges from x0 to a value x(t1) near zero, and x(t1)≠0, integrating both sides of equation (3) yields

[0016]

[0017] Application of the terminal sliding mode control method described in claim 1 in a permanent magnet synchronous motor speed control system.

[0018] The application of the terminal sliding mode control method described in claim 1 in the design of a speed controller for a permanent magnet synchronous motor speed control system.

[0019] 3. Further, the specific method of speed controller design is as follows:

[0020] S1. Assuming that the permanent magnet synchronous motor model is in an ideal state, the stator voltage equation can be obtained as follows:

[0021]

[0022] The stator flux equation is:

[0023]

[0024] Combining equation (5) and equation (6), we can get:

[0025]

[0026] The mechanical motion equation of the interior permanent magnet synchronous motor is:

[0027]

[0028] Its electromagnetic torque equation is:

[0029]

[0030] where u q and u d Represent the current of q-axis and d-axis respectively; i q and i d Represent the current of q-axis and d-axis respectively; R is the stator resistance of the motor, L d , L q are the d-axis and q-axis inductances respectively; ψ d and ψ q is the stator flux of d-axis and q-axis; ψ f is the permanent magnet flux of the motor; ω e is the electrical angular velocity, ω e is the mechanical angular velocity, and ω e =n p +ω m ;n p is the number of motor pole pairs; J is the moment of inertia; B is the damping coefficient; T L is the load torque;

[0031] S2. Considering the parameter changes, the surface mounted permanent magnet synchronous motor (L d =L q ) can be expressed as:

[0032]

[0033] where ΔB, ΔJ and Δψ are the system parameters B, J and ψ f The amount of uncertainty caused;

[0034] Simplifying formula (10) we can get:

[0035]

[0036] where F is the total disturbance, and

[0037] The state function of the permanent magnet synchronous motor speed error is defined as:

[0038]

[0039] By taking the derivative of equation (12), the motion equation of the permanent magnet synchronous motor with disturbance can be obtained:

[0040]

[0041] S3, using the inverse tangent terminal sliding mode to derive it, and combining the exponential reaching law and the motor motion equation, the control law of the speed controller can be obtained as follows:

[0042]

[0043] S4. In formula (14), when x1 gradually approaches 0, arctan(h|x1| ε ) / |x1| ε The limit when it approaches 0 is h, thus solving the singularity problem of the traditional terminal sliding surface.

[0044] In summary, the present invention has the following beneficial effects: the sliding mode control method based on the inverse tangent terminal function proposed in the present invention can improve the control performance of the system while avoiding singularity and has higher tracking accuracy.

[0045] The invention proposes a sliding mode control method based on the inverse tangent terminal function, which can improve the response speed, tracking accuracy and anti-interference ability of the system while avoiding singularity, and reduce the overshoot when tracking the step speed. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 is a control block diagram of an inverse tangent terminal sliding mode speed controller in Embodiment 2 of the present invention;

[0047] Figure 2 and Figure 3 is a step speed response diagram of six control methods in Example 3 of the present invention;

[0048] Figure 4 and Figure 5 It is the sinusoidal speed response diagram of six control methods in Example 3 of the present invention. DETAILED DESCRIPTION

[0049] The following is combined with Figure 1-5 The present invention is described in further detail.

[0050] Example 1: Analysis of linear sliding surface:

[0051] The traditional linear sliding surface is shown below:

[0052] s L =x2+cx1 (15)

[0053] Where c is a constant greater than zero. x1 is a state variable, and

[0054] When the system state variables converge to the sliding surface, s L =0, formula (16) can be rewritten as

[0055]

[0056] Integrate both sides of equation (16). When the system state variable converges from x0 to a value x(t1) near zero, and x(t1)≠0, we can get

[0057]

[0058] By analyzing equation (17), we can see that if x(t1) is infinitely close to zero, the convergence time t1 will also be infinitely extended. Therefore, the traditional linear sliding surface will make the system state variables converge asymptotically to zero in infinite time.

[0059] (2) The traditional terminal sliding mode control method adopts the following sliding surface:

[0060] s T =x2+α|x1| r sgn(x1) (18)

[0061] Where r and α are any positive real numbers, and 0 <r<1。

[0062] When T = 0, integrating both sides of equation (18) yields the convergence time

[0063]

[0064] From the analysis of formula (19), we can see that increasing α or decreasing r can reduce the convergence time.

[0065] (3) Based on equation (17), a non-singular terminal sliding mode surface is obtained:

[0066] s N =x1+β|x2| γ sgn(x2) (20)where γ and β are any positive real numbers, and 0<β<1, 1<γ<2.

[0067] When N = 0, integrating both sides of equation (20) yields the convergence time:

[0068]

[0069] By analyzing formula (21), it can be seen that increasing α can reduce the convergence time. Compared with formula (17), both formula (19) and formula (21) can converge within a finite time. This shows that the terminal sliding control method can make the system state variables converge within a finite time. By observing formulas (19) and (21), it can be seen that when the initial state x0 is large, the system converges slowly.

[0070] Consider a second-order nonlinear system:

[0071]

[0072] where x=[x1,x2] T , is the system state variable; u is the system output signal, f(x) and g(x) are continuous functions of the system, d(x, t) is uncertainty and external disturbance, and d(x, t) ≤ M. The main goal of this paper is to design an appropriate control output u so that the system output state variable can track the reference instruction as quickly as possible with high accuracy.

[0073] Combining formulas (15) and (22), the control law based on the linear sliding surface can be obtained as follows:

[0074]

[0075] According to the Lyapunov function V = 0.1s 2 , after taking the derivative, we can get

[0076]

[0077]

[0078] In formula (24), if k1 ≥ D, the designed control law can converge within a finite time.

[0079] Combining formulas (18) and (24), the control law based on terminal sliding mode control can be obtained as follows:

[0080]

[0081] Where k1 is the coefficient of the constant velocity reaching law.

[0082] According to the Lyapunov function V = 0.1s 2 , and combined with formula (25), we can get

[0083]

[0084] Equation (26) shows that the system state will converge to zero in a finite time. However, there is a singular term x1 in equation (25) r-1x2, when the state variable tends to 0, this item will tend to infinity, which will increase the error of the system state variable and limit the development of terminal sliding mode control in the field of high-precision servo to a certain extent.

[0085] Combining formulas (20) and (22), the control law based on non-singular terminal sliding mode control can be obtained as follows:

[0086]

[0087] According to the Lyapunov function V = 0.1s 2 , and combined with formula (27), we can get

[0088]

[0089] From equation (28), we can see that the singular terms have been eliminated. However, as the s function gradually approaches the equilibrium point, x2 will also gradually decrease. When βγ|x2| γ-1 When <1, compared with equation (26), it can be seen that the convergence speed of non-singular terminal sliding mode control is slower than that of terminal sliding mode control when approaching the equilibrium point. This shows that the elimination of singularity by non-singular terminal sliding mode control comes at the expense of convergence rate.

[0090] In order to solve the above problems of sliding surface, a terminal sliding surface based on hyperbolic tangent function is proposed in the prior art, and its expression is as follows:

[0091] s TH =x2+λx1 1-q / p tanh(hx1 q / p ) (29)

[0092] Where q, p, λ and h are real numbers greater than zero, p and q are both odd numbers, and p>q, 0<ε<1.

[0093] When the system state variable converges from x0 to a value x(t1) near zero, and x(t1)≠0, integrating both sides of equation (29) yields

[0094]

[0095] The above analysis shows that the sliding surface has the following shortcomings: (1) From a structural point of view, the traditional sliding surface has only two parameters, while the sliding surface has four parameters to be designed; (2) The sliding surface will only converge to the neighborhood near the equilibrium point, and the convergence time will not converge to the equilibrium point, which prolongs the convergence time to a certain extent; (3) The tracking accuracy of the speed controller designed based on the sliding surface is low.

[0096] Therefore, the present application proposes a novel non-singular fast terminal sliding surface based on the inverse hyperbolic tangent function on the basis of the hyperbolic tangent terminal sliding surface. This sliding surface can avoid singularity and improve the control performance of the system, and specifically includes the following steps:

[0097] On the basis of the hyperbolic tangent terminal sliding surface, a new non-singular fast terminal sliding surface based on the inverse hyperbolic tangent function is constructed, and its expression is as follows:

[0098] s AT =x2+λ|x1| 1-ε arctan(h|x1| ε )sgn(x1) (1)

[0099] Where ε, λ and h are real numbers greater than zero, and 0<ε<1;

[0100] When AT = 0, the state error function of formula (1) quickly reaches the neighborhood near the zero point within a certain period of time; the convergence time t1 of the proposed sliding surface can be obtained as follows:

[0101]

[0102] Proof: When s AT =0, formula (1) can be rewritten as

[0103] x2+λ|x1| 1-ε arctan(h|x1| ε sgn(x1)=0 (31)

[0104] Select the Lyapunov function Taking the derivative of V and substituting equation (31) into it, we can obtain

[0105]

[0106] Since inequality (32) holds, it means that the controlled system can converge to the equilibrium point in a finite time. The convergence time is calculated below. When the system state variable converges from x0 to a value x(t1) near zero, and x(t1)≠0, integrate both sides of equation (31) when x1>0, and we get

[0107]

[0108] Since equation (31) is an odd function, the absolute value of the state variable should be added in equation (33) to prove the convergence time of the device. From the analysis of equation (33), it can be seen that the proposed new sliding surface can converge to the neighborhood of the equilibrium point in a finite time, and its convergence time is shorter than that of the hyperbolic tangent terminal sliding surface.

[0109] Example 2: Speed ​​Controller Design:

[0110] S1. Assuming that the permanent magnet synchronous motor model is in an ideal state, the stator voltage equation can be obtained as follows:

[0111]

[0112] The stator flux equation is:

[0113]

[0114] Combining equation (5) and equation (6), we can get:

[0115]

[0116] The mechanical motion equation of the interior permanent magnet synchronous motor is:

[0117]

[0118] Its electromagnetic torque equation is:

[0119]

[0120] where u q and u d Represent the current of q-axis and d-axis respectively; i q and i d Represent the current of q-axis and d-axis respectively; R is the stator resistance of the motor, L d , L q are the d-axis and q-axis inductances respectively; ψ d and ψ q is the stator flux of d-axis and q-axis; ψ f is the permanent magnet flux of the motor; ω e is the electrical angular velocity, ω e is the mechanical angular velocity, and ω e =n p ω m ;n p is the number of motor pole pairs; J is the moment of inertia; B is the damping coefficient; T L is the load torque;

[0121] S2. Considering the parameter changes, the surface mounted permanent magnet synchronous motor (L d =L q ) can be expressed as:

[0122]

[0123] where ΔB, ΔJ and Δψ are the system parameters B, J and ψ f The amount of uncertainty caused;

[0124] Simplifying formula (10) we can get:

[0125]

[0126] where F is the total disturbance, and

[0127] The state function of the permanent magnet synchronous motor speed error is defined as:

[0128]

[0129] By taking the derivative of equation (12), the motion equation of the permanent magnet synchronous motor with disturbance can be obtained:

[0130]

[0131] S3, using the inverse tangent terminal sliding mode surface to derive it, and combining the exponential reaching law and the motor motion equation, the control law of the speed controller can be obtained as follows:

[0132]

[0133] S4. In formula (14), when x1 gradually approaches 0, arctan(h|x1| ε ) / |x1| ε The limit when it approaches 0 is h, thus solving the singularity problem of the traditional terminal sliding surface. Figure 1 Shown is the control block diagram of the inverse tangent terminal sliding mode speed controller.

[0134] Example 3: Simulation analysis:

[0135] Build a permanent magnet synchronous motor MATLAB simulation and compare the speed tracking performance of linear sliding mode control, integral sliding mode control, terminal sliding mode control, non-singular terminal sliding mode control, hyperbolic tangent sliding mode control, and the proposed inverse tangent sliding mode control. The parameters of the permanent magnet synchronous motor are shown in Table 1 below.

[0136] Table 1 Permanent magnet synchronous motor parameters

[0137]

[0138] Figure 2 and Figure 3 The control performance diagrams of the six control methods tracking the step speed are shown below. The reference speed is 2000rpm and the step rated load is applied at t=1s. Figure 2It can be seen that the linear sliding mode control and the integral sliding mode control will produce a large overshoot at startup. Although the overshoot of the non-singular terminal sliding mode control is smaller than that of the above two control methods, its convergence time is slower than them. The overshoot of the hyperbolic tangent terminal sliding mode surface is the smallest, followed by the inverse tangent terminal sliding mode control. Although the inverse tangent terminal sliding mode control method has a large overshoot, its convergence speed is faster than other methods. According to the local enlarged figure, when the external load enters, the integral sliding mode surface has a slightly stronger ability to resist load disturbances than the proposed inverse tangent terminal sliding mode control, but the recovery time is 1.43 times that of the inverse tangent terminal sliding mode control, and the steady-state accuracy after the disturbance is applied is also weaker than it, which shows that the proposed inverse tangent terminal sliding mode control has better dynamic and steady-state performance. In addition, the enlarged figure shows that the dynamic performance and load disturbance resistance of the non-singular terminal sliding mode control are the worst. The terminal sliding mode control has the largest steady-state error due to its singularity. The convergence speed, steady-state accuracy and anti-interference ability of the hyperbolic tangent terminal sliding mode control are all at a medium level. This shows that the inverse tangent terminal sliding mode control can effectively improve the convergence speed, anti-interference ability and speed tracking accuracy. Figure 3 It can be seen that due to the large overshoot, the electromagnetic torque curves of integral sliding mode control and linear sliding mode control will also have large fluctuations. Table 2 below shows the performance parameters of the six control methods. The speed RMSE calculation range is 1.25s to 1.5s.

[0139] Table 2 Step speed response performance parameters of six control methods

[0140]

[0141]

[0142] In order to further prove that the inverse tangent terminal sliding mode control proposed in this application has high tracking accuracy, Figure 4 and Figure 5 Six control methods are given to track the sinusoidal speed and the simulation control performance diagram under full load. At this time, the reference speed is 2000+100*sin(8πt). Figure 4 It can be seen that the non-singular terminal sliding mode control method has a certain phase delay. Although the linear sliding mode control, integral sliding mode control, terminal sliding mode control, and hyperbolic tangent terminal sliding mode control methods have no phase delay, their maximum and minimum values ​​exceed the reference speed. Figure 5 is the speed tracking error of the six control methods. The proposed inverse tangent terminal sliding mode control has the best tracking effect, and the fluctuation of tracking accuracy is within the range of [-9,6].

[0143] This specific embodiment is merely an explanation of the present invention and is not a limitation of the present invention. After reading this specification, those skilled in the art may make non-creative modifications to the present embodiment as needed. However, as long as they are within the scope of the claims of the present invention, they are protected by the patent law.

Claims

1. A terminal sliding mode control method based on inverse tangent function, characterized in that: The following steps are involved: On the basis of the hyperbolic tangent terminal sliding surface, a new non-singular fast terminal sliding surface based on the inverse hyperbolic tangent function is constructed, and its expression is as follows: s AT =x2+λ|x1| 1-ε arctan(h|x1| ε )sgn(x1) (1) Where ε, λ and h are real numbers greater than zero, and 0<ε<1; When AT = 0, the state error function of formula (1) quickly reaches the neighborhood near the zero point within a certain period of time; the convergence time t1 of the proposed sliding surface can be obtained as follows:

2. The terminal sliding mode control method based on the inverse tangent function according to claim 1 is characterized in that: The expression of the hyperbolic tangent terminal sliding surface is as follows: s TH =x2+λx1 1-q / p tanh(hx1 q / p ) (3) Where q, p, λ and h are real numbers greater than zero, p and q are both odd numbers, and p>q, 0<ε<1. When the system variable converges from x0 to a value x(t1) near zero, and x(t1)≠0, integrating both sides of equation (3) yields 3. Application of the terminal sliding mode control method described in claim 1 in a permanent magnet synchronous motor speed control system.

4. Application of the terminal sliding mode control method according to claim 1 in the design of a speed controller for a permanent magnet synchronous motor speed control system.

5. The application of the terminal sliding mode control method according to claim 4 in the design of a speed controller for a permanent magnet synchronous motor speed control system is characterized in that: The specific method of speed controller design is as follows: S1. Assuming that the permanent magnet synchronous motor model is in an ideal state, the stator voltage equation can be obtained as follows: The stator flux equation is: Combining equation (5) and equation (6), we can get: The mechanical motion equation of the interior permanent magnet synchronous motor is: Its electromagnetic torque equation is: where u q and u d Represent the current of q-axis and d-axis respectively; i q and i d Represent the current of q-axis and d-axis respectively; R is the stator resistance of the motor, L d , L q are the d-axis and q-axis inductances respectively; ψ d and ψ q is the stator flux of d-axis and q-axis; ψ f is the permanent magnet flux of the motor; ω e is the electrical angular velocity, ω e is the mechanical angular velocity, and ω e =n p ω m ;n p is the number of motor pole pairs; J is the moment of inertia; B is the damping coefficient; T L is the load torque; S2. Considering the parameter changes, the surface mounted permanent magnet synchronous motor (L d =L q ) can be expressed as: where ΔB, ΔJ and Δψ are the system parameters B, J and ψ f The amount of uncertainty caused; Simplifying formula (10) we can get: where F is the total disturbance, and The state function of the permanent magnet synchronous motor speed error is defined as: By taking the derivative of equation (12), the motion equation of the permanent magnet synchronous motor with disturbance can be obtained: S3, using the inverse tangent terminal sliding mode to derive it, and combining the exponential reaching law and the motor motion equation, the control law of the speed controller can be obtained as follows: S4. In formula (14), when x1 gradually approaches 0, arctan(h|x1| ε ) / |x1| ε The limit when it approaches 0 is h, thus solving the singularity problem of the traditional terminal sliding surface.

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