A variable gain fractional-order fast terminal sliding mode control method and system

Abstract

The present invention discloses a variable gain fractional order fast terminal sliding mode control method and system, which relates to the field of sliding mode control technology, including combining fractional order calculus theory and nonlinear function to construct a new variable gain fractional order fast terminal sliding mode surface, avoiding singularity problems, and further improving the convergence speed of the system in the sliding stage; using inverse hyperbolic sine function to design a new power reaching law, so that the system has a faster reaching speed in the approaching stage; designing a fast terminal sliding mode disturbance observer to quickly and accurately observe the total disturbance in the nonlinear system, and compensate the observed disturbance value to the controller in real time; combining the sliding surface, reaching law and disturbance observer to obtain a new variable gain fractional order fast terminal sliding mode controller. The variable gain fractional order fast terminal sliding mode control method proposed by the present invention has good dynamic response characteristics, faster convergence speed and higher control accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of sliding mode control, and in particular to a variable gain fractional order fast terminal sliding mode control method and system. Background Art

[0002] Sliding mode control is highly robust to system uncertainty disturbances, and has the advantages of simple structure and easy implementation. It is widely used in the control of nonlinear systems. Since the terminal sliding mode control introduces nonlinear functions in the sliding surface, the system state can converge to 0 in a finite time, but the convergence speed is slow when the system state is far from the equilibrium state. In order to improve the convergence speed of the system, some scholars have proposed fast terminal sliding mode control. However, negative exponential terms will appear in the process of derivation of the terminal sliding surface, which may cause the control amount to be infinite, which is the so-called singularity problem.

[0003] Fractional-order terminal sliding mode control combines the advantages of fractional-order calculus theory and terminal sliding mode control. Compared with general terminal sliding mode control, it can improve the convergence speed and control accuracy of the system state by utilizing the hereditary characteristics and memory of fractional-order calculus operators. However, general fractional-order terminal sliding mode control has singularity and chattering problems. In addition, the convergence speed and control accuracy still need to be further improved. Summary of the invention

[0004] In view of the problems existing in the existing fast terminal sliding mode control, the present invention is proposed.

[0005] Therefore, the problem to be solved by the present invention is how to provide a variable gain fractional-order fast terminal sliding mode control method to solve the problems of low control accuracy and long convergence time when a class of second-order nonlinear systems has uncertain disturbances.

[0006] In order to solve the above technical problems, the present invention provides the following technical solutions:

[0007] In the first aspect, an embodiment of the present invention provides a variable gain fractional-order fast terminal sliding mode control method, which includes constructing a variable gain fractional-order fast terminal sliding mode surface for a second-order nonlinear system containing uncertain disturbances by combining fractional-order calculus theory and nonlinear functions; designing a power reaching law using an inverse hyperbolic sine function; building a fast terminal sliding mode disturbance observer to observe the total disturbance in the second-order nonlinear system, and compensating the total disturbance in real time to a variable gain fractional-order fast terminal sliding mode controller; combining the variable gain fractional-order fast terminal sliding mode surface, the power reaching law and the fast terminal sliding mode disturbance observer to obtain a variable gain fractional-order fast terminal sliding mode controller; and completing the anti-disturbance control of a class of second-order nonlinear systems through the variable gain fractional-order fast terminal sliding mode controller.

[0008] As a preferred solution of the variable gain fractional-order fast terminal sliding mode control method of the present invention, the equation of the second-order nonlinear system containing uncertain disturbance is as follows:

[0009]

[0010] Among them, x1 and x2 represent the state variables of the system, x=[x1,x2] T , f(x) and g(x)≠0 are continuous functions, u is the control input, and d is the total disturbance;

[0011] The total disturbance d and its first-order differential Both are bounded and satisfy

[0012]

[0013] Among them, L1 and L2 are positive constants.

[0014] As a preferred solution of the variable gain fractional-order fast terminal sliding mode control method of the present invention, the specific formula of the nonlinear function is as follows:

[0015]

[0016] Where x is the state variable of the system, x=[x1,x2] T , c1, c2, η are constants;

[0017] Taking the first-order differential of the nonlinear function H(x), we get:

[0018]

[0019] in, is the first-order differential of the nonlinear function H(x), c1, c2, η are constants, x is the state variable of the system, x=[x1,x2] T .

[0020] As a preferred solution of the variable gain fractional-order fast terminal sliding mode control method of the present invention, the specific formula of the variable gain fractional-order fast terminal sliding mode surface is as follows:

[0021]

[0022] Among them, x1 is the state variable of the system, D is the fractional calculus operator, μ, λ, r1, r2, β are constants, and sig(·) is a custom function, that is, sig(·) * =|· * sign(·), sign(·) is the sign function, and H(x1) is a nonlinear function.

[0023] As a preferred solution of the variable gain fractional-order fast terminal sliding mode control method of the present invention, the power reaching law is designed by using an inverse hyperbolic sine function, and the specific formula is as follows:

[0024]

[0025] Where s is the sliding mode function, is the first-order differential of the sliding mode function, asinh(·) is the inverse hyperbolic sine function, sign(·) is the sign function, and k1, k2, k3, l, σ1, σ2, and θ are constants.

[0026] As a preferred solution of the variable gain fractional-order fast terminal sliding mode control method of the present invention, the specific formula of the fast terminal sliding mode disturbance observer is as follows:

[0027]

[0028] Among them, x2 is the state variable of the system, d is the total disturbance, and are the estimated values ​​of x2 and d, respectively. Ω and I are auxiliary variables. sig(·) is a custom function, i.e., sig(·) * =|· * sign(·), sign(·) is the sign function, κ1, κ2, α1, α2, ξ1, ξ2, ξ3, β1, β2 are constants.

[0029] As a preferred solution of the variable gain fractional-order fast terminal sliding mode control method of the present invention, the specific formula of the variable gain fractional-order fast terminal sliding mode controller is as follows:

[0030]

[0031] in, is the estimated value of the total disturbance d, μ, λ, β, r1, r2, k1, k2, k3, l, σ1, σ2 are constants, asinh(·) is the inverse hyperbolic sine function, sig(·) is a custom function, and sig(·) is a constant. * =|· * sign(·), sign(·) is the sign function, is the first-order differential of the nonlinear function H(x1), H(x1) is a nonlinear function, s is the sliding mode function, and x1 is the state variable of the system.

[0032] In the second aspect, an embodiment of the present invention provides a variable gain fractional-order fast terminal sliding mode control system, which includes a mode surface construction module, which is used to construct a variable gain fractional-order fast terminal sliding mode surface for a second-order nonlinear system containing uncertain disturbances, combining fractional-order calculus theory and nonlinear functions; a power reaching law design module, which is used to design a power reaching law using an inverse hyperbolic sine function; a disturbance compensation module, which is used to build a fast terminal sliding mode disturbance observer, observe the total disturbance in the second-order nonlinear system, and compensate the total disturbance in real time to a variable gain fractional-order fast terminal sliding mode controller; a sliding mode controller module, which is used to combine the variable gain fractional-order fast terminal sliding mode surface, the power reaching law and the fast terminal sliding mode disturbance observer to obtain a variable gain fractional-order fast terminal sliding mode controller; and a control module, which is used to complete the anti-disturbance control of a class of second-order nonlinear systems through a variable gain fractional-order fast terminal sliding mode controller.

[0033] In a third aspect, an embodiment of the present invention provides a computer device, comprising a memory and a processor, wherein the memory stores a computer program, wherein: when the computer program instructions are executed by the processor, the steps of the variable gain fractional-order fast terminal sliding mode control method as described in the first aspect of the present invention are implemented.

[0034] In a fourth aspect, an embodiment of the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein: when the computer program instructions are executed by a processor, the steps of the variable gain fractional-order fast terminal sliding mode control method as described in the first aspect of the present invention are implemented.

[0035] The beneficial effects of the present invention are as follows: based on the existing fast terminal sliding surface, the present invention combines fractional-order theory and nonlinear functions to construct a novel variable gain fractional-order fast terminal sliding surface to solve the singularity problem and improve the convergence speed of the system in the sliding stage; based on the inverse hyperbolic sine function, a novel power reaching law is designed to effectively improve the convergence speed of the system in the approaching stage; a fast terminal sliding mode disturbance observer is designed to quickly and accurately estimate the total disturbance of the system and compensate it in the controller; a variable gain fractional-order fast terminal sliding mode controller is designed by combining the novel variable gain fractional-order fast terminal sliding surface, the novel power reaching law and the fast terminal sliding mode disturbance observer, so that the system has faster convergence characteristics, effectively avoids singularity problems, and has good robustness to uncertain disturbances; compared with the prior art, the variable gain fractional-order fast terminal sliding mode control algorithm proposed by the present invention has good dynamic response characteristics and higher control accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0037] Figure 1 A schematic flow chart of a variable gain fractional-order fast terminal sliding mode control method according to an embodiment of the present invention;

[0038] Figure 2 Under the variable gain fractional-order fast terminal sliding mode control method described in an embodiment of the present invention, the system tracks the desired trajectory x d = 2sin(0.5πt) when the disturbance and its estimated value response curve;

[0039] Figure 3 The variable gain fractional-order fast terminal sliding mode control method according to an embodiment of the present invention is a system tracking the desired trajectory x d =2sin(0.5πt) response curve diagram of the system state variable x1 under the two control methods;

[0040] Figure 4 The variable gain fractional-order fast terminal sliding mode control method according to an embodiment of the present invention is a system tracking the desired trajectory x d =2sin(0.5πt) response curve diagram of error variable e1 under two control methods;

[0041] Figure 5 Under the variable gain fractional-order fast terminal sliding mode control method described in an embodiment of the present invention, the system tracks the desired trajectory x d =2 when the disturbance and its estimated value response curve;

[0042] Figure 6 The variable gain fractional-order fast terminal sliding mode control method according to an embodiment of the present invention is a system tracking the desired trajectory x d =2, schematic diagram of the response curve of the system state variable x1 under the two control methods;

[0043] Figure 7 The variable gain fractional-order fast terminal sliding mode control method according to an embodiment of the present invention is a system tracking the desired trajectory x d Schematic diagram of the response curve of the error variable e1 under the two control methods when =2. DETAILED DESCRIPTION

[0044] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the specific implementation methods of the present invention are described in detail below in conjunction with the accompanying drawings.

[0045] In the following description, many specific details are set forth to facilitate a full understanding of the present invention, but the present invention may also be implemented in other ways different from those described herein, and those skilled in the art may make similar generalizations without violating the connotation of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed below.

[0046] Secondly, the term "one embodiment" or "embodiment" as used herein refers to a specific feature, structure, or characteristic that may be included in at least one implementation of the present invention. The term "in one embodiment" that appears in different places in this specification does not necessarily refer to the same embodiment, nor does it refer to a separate or selective embodiment that is mutually exclusive with other embodiments.

[0047] Example 1

[0048] Reference Figure 1 , which is the first embodiment of the present invention, and provides a variable gain fractional-order fast terminal sliding mode control method, comprising:

[0049] The method of the present invention is applied to the control problem of an inverted pendulum. By controlling this type of inverted pendulum, it is used to verify that the control method proposed by the present invention has superior control performance. The control problem of the inverted pendulum is to make the pendulum reach an equilibrium position as quickly as possible without excessive oscillation, angle and speed. When the pendulum reaches the desired position, the system can overcome uncertainty disturbances and always remain in a stable position. The mathematical model of an inverted pendulum system can be:

[0050]

[0051] in,

[0052]

[0053]

[0054] Among them, x1 is the actual position of the pendulum, x2 is the pendulum speed (i.e. the angular velocity of the pendulum), d is the total disturbance, g is the acceleration of gravity, m is the mass of the ball, and m c is the mass of the pendulum, L is the length of the pendulum, and u is the control input.

[0055] Preferably, g = 9.8 m / s 2 ,m=0.1kg,m c =1kg,L=0.5, assuming the total disturbance d=sin(10x1)+cos(x2), the initial state of the system is [x1,x2]=[1,0.5].

[0056] According to the mathematical model of the inverted pendulum system, its error state equation can be:

[0057]

[0058] Among them, e1 is the error between the actual pendulum position and the expected pendulum position, e2 is the first-order differential of e1, and d is the total disturbance, which includes modeling error and external interference signal.

[0059] It should be noted that the total disturbance d and its first-order differential are bounded and satisfy:

[0060]

[0061] Among them, D1 and D2 are positive constants.

[0062] It should be noted that the control objective is to design the controller so that the error e1 between the actual swing arm position and the desired swing arm position and the first-order differential e2 of the error e1 approach zero as quickly as possible, that is, e1→0, e2→0.

[0063] S1: For the second-order nonlinear system with uncertain disturbances, a variable gain fractional-order fast terminal sliding surface is constructed by combining fractional-order calculus theory and nonlinear functions.

[0064] It should be noted that the definition formula of the nonlinear function H(e1) is as follows:

[0065]

[0066] Among them, e1 is the error between the actual pendulum position and the expected pendulum position, c1, c2, η are constants, and to ensure the stability of the system, c1, c2, η meet the following selection rules:

[0067] 0<c1<1, c2>0, c1+c2≥1, η>0

[0068] Furthermore, taking the first-order differential of the nonlinear function H(e1), we obtain:

[0069]

[0070] in, is the first-order differential of the nonlinear function H(e1), c1, c2, η are constants, e1 is the error between the actual pendulum position and the expected pendulum position, and e2 is the first-order differential of e1.

[0071] Preferably, combining nonlinear functions and fractional-order calculus theory, the expression of variable-gain fractional-order fast terminal sliding mode surface s is:

[0072]

[0073]

[0074] μ>0,λ>0,r1>1,0<r2<1,0<β<1

[0075] Where e1 is the error between the actual pendulum position and the expected pendulum position, that is, e1 = x1-x d , D β-1 is a fractional calculus operator, μ, λ, r1, r2, β are constants, and sig(·) is a custom function, that is, sig(·) * =|· * sign(·), sign(·) is the sign function.

[0076] Preferably, the traditional fast terminal sliding mode control has a singularity problem, and the singularity problem is avoided by introducing the fractional order calculus theory into the design of the terminal sliding surface. At the same time, by introducing the nonlinear function into the design of the sliding surface, the system state can be guaranteed to have a faster convergence characteristic regardless of whether it is in the large error stage or the small error stage, and the parameter fragility problem can be avoided, thereby reducing the system jitter.

[0077] S2: Design a power reaching law using the inverse hyperbolic sine function.

[0078] Preferably, the power approaching law is designed using an inverse hyperbolic sine function, and the expression is:

[0079]

[0080] k1>0,k2>0,k3>0,l>0,θ>0,σ1>0,0<σ2<1

[0081] Where s is the sliding mode function, is the first-order differential of the sliding mode function, asinh(·) is the inverse hyperbolic sine function, sign(·) is the sign function, and k1, k2, k3, l, σ1, σ2, and θ are constants.

[0082] Preferably, the novel power approaching law function proposed in the present invention is composed of three terms, wherein the first term plays a leading role when the sliding mode function s approaches the sliding mode surface, and the second term and the third term play a leading role when the sliding mode function s is far away from the sliding mode surface. By reasonably selecting parameters, the system state can have a faster approaching speed whether it is far away from the sliding mode surface or close to the sliding mode surface, and can effectively suppress the sliding mode chattering.

[0083] S3: Build a fast terminal sliding mode disturbance observer to observe the total disturbance in the second-order nonlinear system, and compensate the total disturbance in real time into a variable gain fractional-order fast terminal sliding mode controller.

[0084] Preferably, for the inverted pendulum system, in order to observe the total disturbance in the nonlinear system, the construction formula of the fast terminal sliding mode disturbance observer is as follows:

[0085]

[0086] Among them, x2 is the swing speed, d is the total disturbance, and are the estimated values ​​of x2 and d, respectively. I is an auxiliary variable, and sig(·) is a custom function, i.e., sig(·) * =|·| * sign(·), sign(·) is the sign function, κ1, κ2, α1, α2, ξ1, ξ2, β1, β2 are constants.

[0087] It should be noted that κ1>0, κ2>0, α1>1, 0<α2<1, ξ1>0, ξ2>0, ξ3>0, β1>1, 0<β2<1.

[0088] In addition, there are often uncertain disturbances in second-order nonlinear systems. By introducing a fast terminal sliding mode disturbance observer, the total disturbance in the system can be observed quickly and accurately, and the observed value can be compensated to the controller in real time to further suppress the disturbance and enhance the robustness of the system.

[0089] S4: Combining the variable gain fractional-order fast terminal sliding mode surface, the power reaching law and the fast terminal sliding mode disturbance observer, a variable gain fractional-order fast terminal sliding mode controller is obtained.

[0090] Preferably, the variable gain fractional-order fast terminal sliding mode surface is differentiated to obtain:

[0091]

[0092] Where e1 is the error between the actual pendulum position and the expected pendulum position, e2 is the first-order differential of e1, is the first-order differential of e2, sig(·) is a custom function, that is, sig(·) * =|·| * sign(·), sign(·) is the sign function, is the first-order differential of the nonlinear function H(e1), H(e1) is a nonlinear function, D is the calculus operator, μ, λ, β, r1, r2 are constants.

[0093] Furthermore, in order to ensure that the system can converge quickly and reduce chattering, the power reaching law is used to obtain:

[0094]

[0095] Where, e1 is the error between the actual pendulum position and the expected pendulum position, e2 is the first-order differential of e1, is the first-order differential of e2, sig(·) is a custom function, that is, sig(·) * =|·| * sign(·), is the first-order differential of the nonlinear function H(e1), H(e1) is a nonlinear function, D is, μ, λ, β, r1, r2 are constants, s is the sliding mode function, is the first-order differential of the sliding mode function, asinh(·) is the inverse hyperbolic sine function, sign(·) is the sign function, and k1, k2, k3, l, σ1, σ2, and θ are constants.

[0096] Furthermore, substituting the error state equation of the inverted pendulum system into the power reaching law, we obtain:

[0097]

[0098] Preferably, the specific formula of the variable gain fractional-order fast terminal sliding mode controller is as follows:

[0099]

[0100] in, is the estimated value of the disturbance value d, μ, λ, β, r1, r2 are constants, k1, k2, k3, l, σ1, σ2 are constants, asinh(·) is the inverse hyperbolic sine function, sig(·) is a custom function, that is, sig(·) * =|· * sign(·), sign(·) is the sign function, is the first-order differential of the nonlinear function H(e1), where H(e1) is a nonlinear function.

[0101] S5: The anti-disturbance control of a second-order nonlinear system is completed through a variable gain fractional-order fast terminal sliding mode controller.

[0102] It should be noted that the inverted pendulum system has the characteristics of nonlinearity, instability, multivariable and strong coupling. As the controlled object of the control system, many abstract control concepts such as system stability, system convergence speed and system anti-interference ability can be intuitively expressed through the inverted pendulum as an example. In order to facilitate the understanding of the inverted pendulum control problem by non-technical personnel, it is also necessary to explain in this embodiment that the control problem of the inverted pendulum is essentially that the inverted pendulum system automatically transfers from a stable equilibrium state to another equilibrium state under the action of an external force. In this process, it is required to start the swing quickly, but overshoot is not desired. Since the relationship between input and output is nonlinear, many commonly used linear control methods are not applicable; therefore, the present invention introduces fractional calculus theory and nonlinear functions to design a new variable gain fractional order fast terminal sliding surface; uses inverse hyperbolic sine function to design a new power reaching law; builds a fast terminal sliding interference observer; and constructs a variable gain fractional order fast terminal sliding controller by sliding surface, reaching law and observer.

[0103] Preferably, the method of the present invention is aimed at a second-order nonlinear system containing modeling errors and external disturbances, and constructs a new variable gain fractional-order fast terminal sliding mode controller. The controller designs a new variable gain fractional-order fast terminal sliding mode surface, which can further improve the convergence speed of the system state while solving the singularity problem of the terminal sliding mode surface. The convergence speed is faster than that of the ordinary non-singular fast terminal sliding mode surface; the controller also designs a new power reaching law, which has a faster reaching speed compared with the fast power reaching law and the double power reaching law; at the same time, the controller introduces a fast terminal sliding mode disturbance observer to estimate the total disturbance, and compensates the observed disturbance value in the controller, thereby enhancing the robustness of the system.

[0104] Furthermore, the present embodiment also provides a variable gain fractional-order fast terminal sliding mode control system, including a mode surface construction module, which is used to construct a variable gain fractional-order fast terminal sliding mode surface for a second-order nonlinear system containing uncertain disturbances, combining fractional-order calculus theory and nonlinear functions; a power reaching law design module, which is used to design a power reaching law using an inverse hyperbolic sine function; a disturbance compensation module, which is used to build a fast terminal sliding mode disturbance observer, observe the total disturbance in the second-order nonlinear system, and compensate the total disturbance in real time to a variable gain fractional-order fast terminal sliding mode controller; a sliding mode controller module, which is used to combine the variable gain fractional-order fast terminal sliding mode surface, the power reaching law and the fast terminal sliding mode disturbance observer to obtain a variable gain fractional-order fast terminal sliding mode controller; and a control module, which is used to complete the anti-disturbance control of a class of second-order nonlinear systems through a variable gain fractional-order fast terminal sliding mode controller.

[0105] This embodiment also provides a computer device, which is applicable to the case of a variable gain fractional-order fast terminal sliding mode control method, and includes a memory and a processor; the memory is used to store computer executable instructions, and the processor is used to execute the computer executable instructions to implement the variable gain fractional-order fast terminal sliding mode control method proposed in the above embodiment.

[0106] The computer device may be a terminal, and the computer device includes a processor, a memory, a communication interface, a display screen and an input device connected via a system bus. The processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The communication interface of the computer device is used to communicate with an external terminal in a wired or wireless manner, and the wireless manner can be achieved through WIFI, an operator network, NFC (near field communication) or other technologies. The display screen of the computer device may be a liquid crystal display screen or an electronic ink display screen, and the input device of the computer device may be a touch layer covering the display screen, or a key, trackball or touchpad provided on the housing of the computer device, or an external keyboard, touchpad or mouse, etc.

[0107] This embodiment further provides a storage medium on which a computer program is stored. When the program is executed by a processor, the variable gain fractional-order fast terminal sliding mode control method proposed in the above embodiment is implemented.

[0108] In summary, based on the existing fast terminal sliding surface, the present invention combines fractional-order theory and nonlinear functions to construct a new variable gain fractional-order fast terminal sliding surface to solve the singularity problem and improve the convergence speed of the system in the sliding stage; a new power reaching law is designed based on the inverse hyperbolic sine function, which can effectively improve the convergence speed of the system in the approaching stage; a fast terminal sliding mode disturbance observer is designed to quickly and accurately estimate the total disturbance of the system and compensate it to the controller; a variable gain fractional-order fast terminal sliding mode controller is designed by combining the new variable gain fractional-order fast terminal sliding surface, the new power reaching law and the fast terminal sliding mode disturbance observer, so that the system has faster convergence characteristics, effectively avoids singularity problems, and has good robustness to uncertain disturbances; compared with the prior art, the variable gain fractional-order fast terminal sliding mode control algorithm proposed in the present invention has good dynamic response characteristics and higher control accuracy.

[0109] Example 2

[0110] Reference Figure 2 to Figure 7This is the second embodiment of the present invention. In order to better verify and illustrate the technical effects adopted in the method of the present invention, this embodiment selects a common non-singular fast terminal sliding mode control method and the method of the present invention for comparative testing, and compares the test results by means of scientific demonstration to verify the real effect of the method of the present invention.

[0111] Specifically, the non-singular fast terminal sliding surface is designed as follows:

[0112] s=e2+μsig(e1) r1 +Z(e1)

[0113]

[0114] Furthermore, the non-singular fast terminal sliding mode control adopts the double power reaching law as follows:

[0115]

[0116] Furthermore, the non-singular fast terminal sliding mode controller is designed as follows:

[0117]

[0118]

[0119] It should be noted that the parameters of the non-singular fast terminal sliding mode controller are set as follows:

[0120] μ=λ=2, r1=1.5, r1=0.4, k1=k2=10, σ1=1.5, σ2=0.6, δ=0.01.

[0121] The fast terminal sliding mode disturbance observer provided by the present invention is set with the following parameters:

[0122] κ1=κ2=5, α1=β1=1.5, α2=β2=0.5,

[0123] The parameters of the variable gain fractional-order fast terminal sliding mode controller provided by the present invention are set as follows:

[0124] μ=λ=θ=2, r1=1.5, r2=0.4, β=0.9, c1=c2=0.5, eta=k1=k2=10,

[0125] σ1=1.5, σ2=0.6, ι=5.

[0126] The sliding surface, reaching law, disturbance observer and controller expression formula of the variable gain fractional-order fast terminal sliding mode control method provided by the present invention are shown in Example 1. The sliding surface, reaching law and controller expression formula of the non-singular fast terminal sliding mode control method are shown in Example 2. The control parameters of the non-singular fast terminal sliding mode control method and the method of the present invention are respectively input into the Matlab / Simulink environment for simulation operation.

[0127] Reference Figure 2 , is to track the expected trajectory x d =2sin(0.5πt), a fast terminal sliding mode observer is used to observe the disturbance, and the response curve of the disturbance and its observed value is calculated according to Figure 2 From the illustration, it can be seen that the fast terminal sliding mode observer can estimate the disturbance quickly and accurately.

[0128] Reference Figure 3 , is to track the expected trajectory x d =2sin(0.5πt), the response curves of the system state variable x1 under the two control methods. Figure 4 , is to track the expected trajectory x d =2sin(0.5πt), the response curves of the system error variable e1 under the two control methods. Figure 3 and Figure 4 It can be seen that the variable gain fractional-order fast terminal sliding mode control method proposed in the present invention has a faster convergence rate and a smaller steady-state error compared with the non-singular fast terminal sliding mode control method.

[0129] Reference Figure 5 , is the tracking expected position x d = 2, the fast terminal sliding mode observer is used to observe the disturbance, and the response curve of the disturbance and its observed value is calculated according to Figure 5 From the illustration, it can be seen that the fast terminal sliding mode observer can estimate the disturbance quickly and accurately.

[0130] Reference Figure 6 , is the tracking expected position x d =2, the response curve of the system state variable x1 under the two control methods. Figure 7 , is the tracking expected position x d =2, the response curves of the system error variable e1 under the two control methods. Figure 6 and Figure 7 It can be seen that the variable gain fractional-order fast terminal sliding mode control method proposed in the present invention has a faster convergence rate and a smaller steady-state error compared with the non-singular fast terminal sliding mode control method.

[0131] Preferably, the method of the present invention can make the system state variables converge quickly to the expected values, effectively overcome the influence of modeling errors and external interference signals, and has good robustness and stability. Compared with the existing non-singular fast terminal sliding mode control method, the control method designed by the present invention has a faster convergence speed and higher control accuracy.

[0132] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the spirit and scope of the technical solutions of the present invention, which should all be included in the scope of the claims of the present invention.

Claims

1. A variable gain fractional-order fast terminal sliding mode control method, characterized in that: include, For the second-order nonlinear system with uncertain disturbances, a variable gain fractional-order fast terminal sliding surface is constructed by combining fractional-order calculus theory and nonlinear functions. Design power reaching law using inverse hyperbolic sine function; A fast terminal sliding mode disturbance observer is constructed to observe the total disturbance in the second-order nonlinear system, and the total disturbance is compensated in real time into a variable gain fractional-order fast terminal sliding mode controller; Combining the variable gain fractional order fast terminal sliding mode surface, the power reaching law and the fast terminal sliding mode disturbance observer, a variable gain fractional order fast terminal sliding mode controller is obtained; The anti-disturbance control of a second-order nonlinear system is completed by the variable gain fractional-order fast terminal sliding mode controller; The equation of the second-order nonlinear system with uncertain disturbance is as follows: Among them, x1 and x2 represent the state variables of the system, x=[x1,x2] T , f(x) and g(x)≠0 are continuous functions, u is the control input, and d is the total disturbance; The total disturbance d and its first-order differential Both are bounded and satisfy |d|≤L1, Among them, L1 and L2 are positive constants; The specific formula of the nonlinear function is as follows: Where x is the state variable of the system, x=[x1,x2] T , c1, c2, η are constants; Taking the first-order differential of the nonlinear function H(x), we get: in, is the first-order differential of the nonlinear function H(x), c1, c2, η are constants, x is the state variable of the system, x=[x1,x2] T ; The specific formula of the variable gain fractional-order fast terminal sliding mode surface is as follows: Among them, x1 is the state variable of the system, D is the fractional calculus operator, μ, λ, r1, r2, β are constants, sig(·) is a custom function, and sig(·) * =|·| * sign(·), sign(·) is the sign function, H(x1) is the nonlinear function; The power reaching law is designed by using the inverse hyperbolic sine function, and the specific formula is as follows: Where s is the sliding mode function, is the first-order differential of the sliding mode function, asinh(·) is the inverse hyperbolic sine function, sign(·) is the sign function, k1, k2, k3, l, σ1, σ2, θ are constants; The specific formula of the fast terminal sliding mode disturbance observer is as follows: Among them, x2 is the state variable of the system, d is the total disturbance, and are the estimated values ​​of x2 and d, respectively. Ω and I are auxiliary variables. sig(·) is a custom function. * =|·| * sign(·), sign(·) is the sign function, κ1, κ2, α1, α2, ξ1, ξ2, ξ3, β1, β2 are constants; The specific formula of the variable gain fractional-order fast terminal sliding mode controller is as follows: in, is the estimated value of the total disturbance d, μ, λ, β, r1, r2, k1, k2, k3, l, σ1, σ2 are constants, asinh(·) is the inverse hyperbolic sine function, sig(·) is a custom function, and sig(·) is a constant. * =|·| * sign(·), sign(·) is the sign function, is the first-order differential of the nonlinear function H(x1), H(x1) is a nonlinear function, s is the sliding mode function, and x1 is the state variable of the system.

2. A variable gain fractional order fast terminal sliding mode control system, based on the variable gain fractional order fast terminal sliding mode control method according to claim 1, characterized in that: Also includes, The sliding surface construction module is used to construct variable gain fractional-order fast terminal sliding surface for second-order nonlinear systems with uncertain disturbances by combining fractional-order calculus theory and nonlinear functions; Power reaching law design module, used to design power reaching law using inverse hyperbolic sine function; A disturbance compensation module is used to build a fast terminal sliding mode disturbance observer, observe the total disturbance in the second-order nonlinear system, and compensate the total disturbance in real time into a variable gain fractional-order fast terminal sliding mode controller; A sliding mode controller module, used for combining the variable gain fractional order fast terminal sliding mode surface, the power reaching law and the fast terminal sliding mode disturbance observer to obtain a variable gain fractional order fast terminal sliding mode controller; The control module is used to complete the anti-disturbance control of a second-order nonlinear system through the variable gain fractional-order fast terminal sliding mode controller.

3. A computer device comprising a memory and a processor, wherein the memory stores a computer program, wherein: When the processor executes the computer program, the steps of the variable gain fractional-order fast terminal sliding mode control method according to claim 1 are implemented.

4. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the variable gain fractional-order fast terminal sliding mode control method according to claim 1 are implemented.

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