Logarithmic terminal sliding mode control method, device and equipment for a nonlinear system

Abstract

The present invention discloses a logarithmic terminal sliding mode control method, device and equipment for a non-linear system, which relates to the technical field of sliding mode control. Aiming at the control problem of a non-linear second-order system, the present invention combines a terminal sliding mode surface and a natural logarithmic function to propose a logarithmic terminal sliding mode surface. Based on the terminal sliding mode surface, the proposed logarithmic terminal sliding mode surface accelerates the convergence rate of the state of the non-linear second-order system when it is far from the equilibrium point. At the same time, considering that the non-linear second-order system often contains unknown matching non-linear disturbances, which will affect the stability and control accuracy of the system, a sliding mode disturbance observer is designed based on the proposed logarithmic terminal sliding mode surface, realizing the fast convergence of the disturbance estimation error, which is convenient for the subsequent design of the controller. Finally, a stable first-order logarithmic terminal sliding mode controller is designed based on the Lyapunov stability theory, realizing the global fast convergence of the state of the non-linear second-order system to the equilibrium point, and improving the convergence rate and control accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of sliding mode control, and particularly relates to a logarithmic terminal sliding mode control method, device, and equipment for a nonlinear system. Background Art

[0002] Currently, as an advanced control method with strong robustness and simple structure, sliding mode control has been widely applied in fields such as nonlinear drive systems and robots.

[0003] The classic linear sliding mode surface can only asymptotically converge to the equilibrium point at an exponential rate, having a satisfactory convergence rate when far from the equilibrium point, but a very slow convergence rate near the equilibrium point, and the system state cannot converge within a finite time. The terminal sliding mode surface and the non-singular terminal sliding mode surface introduce a nonlinear terminal attractor to form an infinite gain near the equilibrium point, achieving finite-time convergence, but the convergence rate is still slow when the system state is far from the equilibrium point; although the fast non-singular terminal sliding mode surface accelerates the convergence rate by introducing an additional nonlinear term, it still cannot achieve global fast convergence. At the same time, when using a sliding mode observer to estimate the lumped uncertainty, it is also expected that the estimation error of the observer can converge quickly along the sliding mode surface. Both of them require the sliding mode surface to achieve global fast convergence. Summary of the Invention

[0004] Based on this, it is necessary to provide a logarithmic terminal sliding mode control method, device, and equipment for a nonlinear system in view of the above technical problems.

[0005] The present invention adopts the following technical solutions:

[0006] The present invention provides a logarithmic terminal sliding mode control method for a nonlinear system. First, a nonlinear second-order system to be controlled is modeled. Then, based on the terminal sliding mode surface of the state of the nonlinear second-order system, a logarithmic terminal sliding mode surface is constructed by embedding the terminal attractor into the natural logarithmic function. Thus, a disturbance observer that enables the observation error to satisfy the fast convergence condition is constructed based on the logarithmic terminal sliding mode surface to observe the lumped uncertainty in the nonlinear second-order system. Finally, based on the logarithmic terminal sliding mode surface and the disturbance observer, a first-order logarithmic terminal sliding mode controller is constructed based on the Lyapunov stability theory, and the nonlinear second-order system is controlled by the first-order logarithmic terminal sliding mode controller.

[0007] The present invention provides a logarithmic terminal sliding mode control device for a nonlinear system, including:

[0008] A modeling module, configured to model a nonlinear second-order system to be controlled;

[0009] A sliding mode surface construction module, which is used to embed a terminal attractor into a natural logarithm function according to a terminal sliding mode surface based on the state of a nonlinear second-order system to construct a logarithmic terminal sliding mode surface;

[0010] An observer construction module, which is used to construct a disturbance observer that enables the observation error to satisfy the fast convergence condition based on the logarithmic terminal sliding mode surface, and observe the lumped uncertainty in the nonlinear second-order system;

[0011] A control module, which is used to construct a first-order logarithmic terminal sliding mode controller based on the logarithmic terminal sliding mode surface and the disturbance observer according to the Lyapunov stability theory, and control the nonlinear second-order system through the first-order logarithmic terminal sliding mode controller.

[0012] The present invention provides a computer-readable storage medium, and the storage medium stores a computer program, and when the computer program is executed by a processor, the logarithmic terminal sliding mode control method of the above-mentioned nonlinear system is realized.

[0013] The present invention provides a computer device, which includes a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the program, the logarithmic terminal sliding mode control method of the above-mentioned nonlinear system is realized.

[0014] The above at least one technical solution adopted by the present invention can achieve the following beneficial effects:

[0015] Aiming at the control problem of a nonlinear second-order system, the present invention combines a terminal sliding mode surface and a natural logarithm function to propose a logarithmic terminal sliding mode surface. Based on the terminal sliding mode surface, the convergence rate of the state of the nonlinear second-order system when it is far from the equilibrium point is accelerated. At the same time, considering that the nonlinear second-order system often contains unknown matching nonlinear disturbances, which will affect the stability and control accuracy of the system. Therefore, a sliding mode disturbance observer is designed based on the proposed logarithmic terminal sliding mode surface to achieve the fast convergence of the disturbance estimation error, which is convenient for the subsequent design of the controller. Finally, a stable first-order logarithmic terminal sliding mode controller is designed based on the Lyapunov stability theory to achieve the global fast convergence of the state of the nonlinear second-order system to the equilibrium point, improving the convergence rate and control accuracy. Description of the Drawings

[0016] The drawings described herein are used to provide a further understanding of the present invention, and constitute a part of the present invention. The schematic embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0017] Figure 1 It is a schematic flow chart of a logarithmic terminal sliding mode control method for a nonlinear system provided by the present invention;

[0018] Figure 2 Schematic diagram of phase plane with different sliding surfaces provided by the present invention;

[0019] Figure 3 Schematic diagram of motor speed curve under different control strategies provided by the present invention;

[0020] Figure 4 Schematic diagram of controller output voltage curve under different control strategies provided by the present invention;

[0021] Figure 5 Schematic diagram of current response curve under different control strategies provided by the present invention;

[0022] Figure 6 Schematic diagram of logarithmic terminal sliding mode control device for a nonlinear system provided by the present invention. Detailed implementation manners

[0023] To make the objectives, technical solutions and advantages of the present invention clearer, the technical solutions of the present invention will be clearly and completely described below in conjunction with specific embodiments of the present invention and the corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without making creative efforts shall fall within the protection scope of the present invention.

[0024] The current sliding surface cannot achieve the global fast convergence of sliding mode control. At the same time, when using a sliding mode observer to estimate the lumped uncertainty, similar to sliding mode control, it is also expected that the estimation error of the uncertainty can quickly and smoothly converge to the equilibrium point.

[0025] The present invention nests the terminal attractor into the natural logarithmic function, proposes a logarithmic terminal sliding surface with global fast finite-time convergence, and analytically solves the time required for the system state to slide to zero. Based on the improved logarithmic terminal sliding surface, a corresponding disturbance observer and controller are designed. This control method can make the system state converge to a small neighborhood near the equilibrium point within a finite time.

[0026] The following will, in conjunction with the drawings, detail the technical solutions provided by each embodiment of the present invention.

[0027] Figure 1 Schematic diagram of the flow of a logarithmic terminal sliding mode control method for a nonlinear system in the present invention, which specifically includes the following steps:

[0028] S101: Model the nonlinear second-order system to be controlled.

[0029] S102: Based on the terminal sliding mode surface of the state of the nonlinear second-order system, embed the terminal attractor into the natural logarithm function to construct a logarithmic terminal sliding mode surface.

[0030] S103: Based on the logarithmic terminal sliding mode surface, construct a disturbance observer that enables the observation error to meet the fast convergence condition, and observe the lumped uncertainty in the nonlinear second-order system.

[0031] S104: According to the logarithmic terminal sliding mode surface and the disturbance observer, construct a first-order logarithmic terminal sliding mode controller based on the Lyapunov stability theory, and control the nonlinear second-order system through the first-order logarithmic terminal sliding mode controller.

[0032] For the convenience of description, only the server is used as the execution entity for description below. The server mentioned in the present invention may be a server set up on the service platform, or a device such as a desktop computer or a notebook computer that can execute the solution of the present invention.

[0033] Generally, when the server controls the nonlinear second-order system, it can first establish a nonlinear second-order system model including the matching disturbance: .

[0034] Among them, and are different state variables of the nonlinear second-order system, and their respective first and second derivatives are measurable. and are the derivative functions of the state variables and respectively. and are both known nonlinear functions. is the unknown matching disturbance, which is smooth, continuous and slowly varying, and its derivative with respect to time can be approximately ignored. is the control signal of the nonlinear second-order system.

[0035] Based on the established nonlinear second-order system model above, a global fast logarithmic terminal sliding mode surface with an analytical convergence time is designed. Specifically, the logarithmic terminal sliding mode surface can be constructed based on the terminal sliding mode surface and the natural logarithm through the following formula: , .

[0036] Among them, is the logarithmic terminal sliding mode surface. , , and are constant parameters set according to the control performance requirements, satisfying , , , and . and are both odd constants.

[0037] To accurately compensate for the matching disturbances in the nonlinear second-order system , a disturbance observer can be constructed by the following formula to observe the unknown matching disturbances in the nonlinear second-order system: , .

[0038] where is the disturbance observer, is the adaptive function updated according to the adaptive law , , , and are constant parameters set according to the control performance requirements, satisfying , , and , and are both odd constants. Define the observation error as , and we can get , where .

[0039] Since and are both greater than 0, thus is positive definite, and the observation error asymptotically converges. We can use to estimate .

[0040] Finally, based on the designed logarithmic terminal sliding mode surface and disturbance observer above, a first-order logarithmic terminal sliding mode controller can be constructed by the following formula based on the Lyapunov stability theory:

[0041] ;

[0042] .

[0043] where is the output of the first-order logarithmic terminal sliding mode control, is the reciprocal of the known nonlinear function , represents the sign function, is the upper bound of the observation error of the disturbance observer.

[0044] Consider the Lyapunov functions and , and their derivatives can be expressed as and respectively.

[0045] When When , the system state can reach the sliding mode surface within a finite time.

[0046] When , after the disturbance observer converges to , , the system state asymptotically converges to the narrow boundary layer near the logarithmic terminal sliding mode surface , and then it can be obtained that , when the state of the nonlinear second-order system satisfies the condition , the state of the nonlinear second-order system can converge to the logarithmic terminal sliding mode surface within a finite time.

[0047] When the logarithmic terminal sliding mode surface is established, the state variables of the nonlinear second-order system converge to the equilibrium point within a finite time , and the analytical expression of the finite convergence time is: .

[0048] Among them, is the state of the nonlinear second-order system when the logarithmic terminal sliding mode surface is established.

[0049] Based on the logarithmic terminal sliding mode control method of the nonlinear system shown in Figure 1 , the present invention aims at the control problem of the nonlinear second-order system, and combines the terminal sliding mode surface and the natural logarithmic function to propose a logarithmic terminal sliding mode surface, which, based on the terminal sliding mode surface, accelerates the convergence rate of the state of the nonlinear second-order system when it is far from the equilibrium point. At the same time, considering that the nonlinear second-order system often contains unknown matching nonlinear disturbances, which will affect the stability and control accuracy of the system, a sliding mode disturbance observer is designed based on the proposed logarithmic terminal sliding mode surface to achieve the rapid convergence of the disturbance estimation error, facilitating the subsequent design of the controller. Finally, a stable first-order logarithmic terminal sliding mode controller is designed based on the Lyapunov stability theory to achieve the global rapid convergence of the state of the nonlinear second-order system to the equilibrium point, improving the convergence rate and control accuracy.

[0050] As a comparison, the expression form of the traditional terminal sliding mode surface is: .

[0051] Among them is defined in the same way as in the logarithmic terminal sliding mode surface. The controller designed based on the traditional terminal sliding mode surface has the same form as the controller designed based on the logarithmic terminal sliding mode surface: .

[0052] Among them, .

[0053] When the terminal attractor index remains consistent, the logarithmic terminal sliding surface proposed by the present invention has better dynamic performance than the traditional terminal sliding surface globally. And when the parameter is satisfied, the logarithmic terminal sliding surface degenerates into a fast terminal sliding surface. However, compared with the fast terminal sliding surface, it does not increase the slope near the equilibrium point when improving the convergence rate far from the equilibrium point, avoiding the noise included in Figure 2 from being amplified. As shown in

[0054] , it is a schematic diagram of the phase plane of a different sliding surface in the present invention.

[0055] The control strategy based on the logarithmic terminal sliding surface of the present invention produces a more satisfactory control effect than the control strategy based on the traditional terminal sliding surface, which is reflected in aspects such as smaller overshoot, faster convergence speed, and higher control accuracy. The control strategy proposed by the present invention is singular when the system state approaches 0. To solve this problem, it can be solved in various ways. For example, when , is replaced with . This does not affect the effect of the present invention, and the system state will still quickly enter the narrow boundary layer of the sliding surface and converge to a small neighborhood near the equilibrium point with a high gain .

[0056] When applying the logarithmic terminal sliding mode control method for the nonlinear system provided by the present invention, it is not necessary to execute according to the order of the steps shown in Figure 1 . The specific execution order of each step can be determined as needed, and the present invention does not limit this.

[0057] In addition, the present invention also provides an embodiment of applying the logarithmic terminal sliding mode control method for the nonlinear system of the present invention. Taking a permanent magnet synchronous motor system driven by a field-oriented control method as an example, its mathematical model is expressed as: .

[0058] Among them, is the angular velocity of the motor, , , , , and respectively represent the direct-axis current, quadrature-axis current, direct-axis voltage, quadrature-axis voltage, direct-axis inductance, and quadrature-axis inductance. is the stator winding, , is the number of pole pairs, is the rotor flux, is the viscous friction coefficient, The disturbances caused by temperature rise and magnetic flux leakage.

[0059] Considering the speed tracking problem of a permanent magnet synchronous motor, define as the desired speed, and the speed tracking error can be expressed as and . Then the above mathematical model can be rewritten as: .

[0060] Among them, , obviously, it has the same form as the nonlinear second-order system proposed in the present invention, and the above control strategy can be used for the speed tracking control of the permanent magnet synchronous motor.

[0061] For the speed tracking control of an actual 4-pole 24V permanent magnet synchronous motor, the actual relevant physical parameters are as follows: , , , , , , .

[0062] When applying the method of the present invention, the parameters in the logarithmic terminal sliding mode surface are taken as: , , ; the parameters in the disturbance observer are taken as: , ; the parameters in the first-order logarithmic terminal sliding mode controller are taken as: . In practical applications, in order to make the designed controller output a smooth voltage signal, can be used to replace the sign function . To ensure the fairness of comparison, the parameters in the traditional terminal sliding mode surface and controller are kept consistent with the above parameters.

[0063] Applying the proposed logarithmic terminal sliding mode control strategy and the traditional terminal sliding mode control strategy to the speed tracking problem of the permanent magnet synchronous motor, the speed tracking curves are as Figure 3 shown, Figure 3 is a schematic diagram of the motor speed curve under a different control strategy in the present invention. When the parameters in the two controllers are the same, it can be found that the first-order logarithmic terminal sliding mode controller proposed in the present invention will produce a more satisfactory overshoot, drive the motor rotor to reach near the desired speed faster, and have higher steady-state accuracy. When the parameters of the traditional terminal sliding mode controller are changed to , although the large switching term will reduce the overshoot and speed tracking error, the control quality is still not as good as that of the present invention. As Figure 4 and Figure 5 shown, they are the output voltages of these two controllers and the resulting current responses, where,Figure 4 Schematic diagram of the output voltage curve of the controller under different control strategies in the present invention Figure 4 The left side corresponds to the classical terminal sliding mode controller Figure 4 The right side corresponds to the first-order logarithmic terminal sliding mode controller of the present invention Figure 5 Schematic diagram of the current response curve under different control strategies in the present invention Figure 5 The left side corresponds to the classical terminal sliding mode controller Figure 5 The right side corresponds to the first-order logarithmic terminal sliding mode controller of the present invention. Obviously, the voltage signal output by the controller based on the logarithmic terminal sliding mode has a more sensitive response to the lumped disturbance

[0064] The above is the logarithmic terminal sliding mode control method for nonlinear systems provided by one or more embodiments of the present invention. Based on the same idea, the present invention also provides a corresponding logarithmic terminal sliding mode control device for nonlinear systems, as Figure 6 shown

[0065] Figure 6 Schematic diagram of a logarithmic terminal sliding mode control device for nonlinear systems provided by the present invention, including:

[0066] A modeling module 201 for modeling the nonlinear second-order system to be controlled

[0067] A sliding mode surface construction module 202 for constructing a logarithmic terminal sliding mode surface by embedding a terminal attractor into the natural logarithmic function according to the terminal sliding mode surface based on the state of the nonlinear second-order system

[0068] An observer construction module 203 for constructing a disturbance observer that makes the observation error satisfy the fast convergence condition based on the logarithmic terminal sliding mode surface to observe the lumped uncertainty in the nonlinear second-order system

[0069] A control module 204 for constructing a first-order logarithmic terminal sliding mode controller based on the logarithmic terminal sliding mode surface and the disturbance observer according to the Lyapunov stability theory, and controlling the nonlinear second-order system through the first-order logarithmic terminal sliding mode controller

[0070] For the specific limitations of the logarithmic terminal sliding mode control device for nonlinear systems, reference can be made to the limitations of the logarithmic terminal sliding mode control method for nonlinear systems in the above text, which will not be elaborated here. Each module in the above logarithmic terminal sliding mode control device for nonlinear systems can be implemented in whole or in part by software, hardware and their combination. The above modules can be embedded in the processor of the computer device in hardware form or independent of it, or stored in the memory of the computer device in software form, so that the processor can call and execute the operations corresponding to the above modules

[0071] The present invention also provides a computer-readable storage medium storing a computer program, which can be used to execute the above-mentioned Figure 1 logarithmic terminal sliding mode control method for the nonlinear system provided.

[0072] The present invention also provides a computer device. At the hardware level, the computer device includes a processor, an internal bus, a network interface, a memory, and a non-volatile memory. Of course, it may also include other hardware required for other services. The processor reads the corresponding computer program from the non-volatile memory into the memory and then runs it to implement the above-mentioned Figure 1 logarithmic terminal sliding mode control method for the nonlinear system provided.

[0073] Those of ordinary skill in the art can understand that all or part of the processes in the methods of the above embodiments can be completed by instructing relevant hardware through a computer program. The computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it may include the processes of the embodiments of the above methods. Among them, any reference to a memory, storage, database, or other medium used in the embodiments provided by the present invention may include at least one of non-volatile and volatile memories. The non-volatile memory may include a read-only memory (ROM), magnetic tape, floppy disk, flash memory, or optical memory, etc. The volatile memory may include a random access memory (RAM) or an external cache memory. By way of illustration and not limitation, RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM), etc.

[0074] The technical features of the above embodiments can be combined arbitrarily. For the sake of concise description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as the scope recorded in the present invention.

Claims

1. A logarithmic terminal sliding mode control method for a nonlinear system, characterized in that: include: Modeling of the nonlinear second-order system to be controlled; According to the terminal sliding surface based on the state of the nonlinear second-order system, the terminal attractor is embedded into the natural logarithmic function, and the logarithmic terminal sliding surface is constructed by the following formula: ， ， ； Based on the logarithmic terminal sliding surface, a disturbance observer is constructed to make the observation error meet the fast convergence condition, and the lumped uncertainty in the nonlinear second-order system is observed. According to the logarithmic terminal sliding mode surface and the disturbance observer, a first-order logarithmic terminal sliding mode controller is constructed based on the Lyapunov stability theory by the following formula: ， ， ， ， And the nonlinear second-order system is controlled by a first-order logarithmic terminal sliding mode controller; in, is the logarithmic terminal sliding surface, , , and is a constant parameter set according to the control performance requirements, and and is an odd constant, is a state variable The derivative of is a first-order logarithmic terminal sliding mode controller, is the disturbance observer, is the upper bound of the observation error of the disturbance observer, and are different state variables of the nonlinear second-order system, and is a known nonlinear function, is a symbolic function.

2. The logarithmic terminal sliding mode control method for a nonlinear system according to claim 1, characterized in that: The modeling of the nonlinear second-order system to be controlled specifically includes: The nonlinear second-order system to be controlled is modeled as follows: ； in, and are different state variables of the nonlinear second-order system, and The state variables and The derivative function of and is a known nonlinear function, is the unknown aggregate uncertainty, is the control signal of the nonlinear second-order system.

3. The logarithmic terminal sliding mode control method for a nonlinear system according to claim 1, characterized in that: The method of constructing a disturbance observer based on the logarithmic terminal sliding surface so that the observation error satisfies the fast convergence condition and observing the lumped uncertainty in the nonlinear second-order system specifically includes: Based on the logarithmic terminal sliding surface, the disturbance observer is constructed by the following formula to observe the lumped uncertainty in the nonlinear second-order system: ， ， ， ； in, is the disturbance observer, According to the adaptive law Updated adaptive function, , , and is a constant parameter set according to the control performance requirements, and and is an odd constant, and are different state variables of the nonlinear second-order system, and is a known nonlinear function, is the control signal of the nonlinear second-order system.

4. A logarithmic terminal sliding mode control device for a nonlinear system based on the logarithmic terminal sliding mode control method for a nonlinear system according to any one of claims 1 to 3, characterized in that: include: A modeling module for modeling the nonlinear second-order system to be controlled; A sliding surface construction module, used for embedding a terminal attractor into a natural logarithmic function to construct a logarithmic terminal sliding surface according to a terminal sliding surface based on a state of a nonlinear second-order system; The observer construction module is used to construct a disturbance observer based on the logarithmic terminal sliding surface so that the observation error meets the fast convergence condition and observe the lumped uncertainty in the nonlinear second-order system; The control module is used to construct a first-order logarithmic terminal sliding mode controller based on the logarithmic terminal sliding mode surface and the disturbance observer based on Lyapunov stability theory, and control the nonlinear second-order system through the first-order logarithmic terminal sliding mode controller.

5. A computer-readable storage medium, characterized in that: The storage medium stores a computer program, and when the computer program is executed by a processor, the method according to any one of claims 1 to 3 is implemented.

6. A computer device, characterized in that: The method comprises a memory, a processor and a computer program stored in the memory and executable on the processor, wherein when the processor executes the program, the method according to any one of claims 1 to 3 is implemented.

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