A logarithmic terminal sliding mode control method for nonlinear systems

Abstract

The present invention proposes a logarithmic terminal sliding mode control method for nonlinear systems, which belongs to the technical field of sliding mode control. Based on the designed logarithmic terminal sliding mode surface, the present invention obtains a first-order logarithmic terminal sliding mode control strategy and a super-torsion logarithmic terminal sliding mode control strategy, and obtains a logarithmic terminal sliding mode collaborative control architecture for nonlinear systems, which realizes multi-objective optimization of jitter suppression, global fast convergence and explicit sliding time, and can improve the global convergence speed of the system state on the sliding mode surface while weakening the sliding mode control jitter, and explicitly gives the sliding time of the system state on the sliding mode surface. Through the explicit analytical expression of the sliding time parameter, the strict mathematical correspondence between the controller gain coefficient and the system dynamic performance is established, which provides a quantitative theoretical basis for parameter setting in industrial sites and significantly enhances the system robustness under complex working conditions.

Description

Technical Field

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

[0002] As a nonlinear robust control method, sliding mode control technology is widely used in spacecraft tether deployment control due to its strong robustness to system parameter perturbations, external disturbances and model uncertainties.

[0003] Sliding mode control technology faces two core challenges: convergence rate optimization and jitter suppression. Terminal sliding mode control technology achieves system state convergence within a finite time by introducing a fractional-order state feedback mechanism, effectively solving the asymptotic convergence problem of traditional sliding mode control. High-order sliding mode control methods (typically represented by the supertorsion algorithm) significantly weaken the jitter phenomenon while retaining the robustness of the sliding mode by constructing the differential characteristics of the continuous control quantity. However, in engineering practice, there is still a theoretical bottleneck in achieving both fast convergence and low jitter control at the same time.

[0004] Research shows that the non-singular terminal sliding mode controller has a convergence rate decay phenomenon in the far region of the equilibrium point. The root cause is the insufficient nonlinear gain caused by the fractional-order sliding surface design. More importantly, the second-order derivative of this type of control law presents a non-integrable nonlinear phase component, which makes the super-torsion algorithm parameter tuning based on Lyapunov stability analysis invalid and it is difficult to establish a globally stable composite control architecture. It is worth noting that when a linear sliding surface is combined with a super-torsion algorithm, although the effective suppression of jitter can be achieved, the convergence characteristics of the system state on the sliding surface degenerate into an exponential form, resulting in a significant extension of the transition process time.

[0005] This contradiction essentially stems from the inherent conflict between the sliding mode dynamics and the convergence mechanism: the finite-time convergence characteristics of the terminal sliding mode depend on the introduction of strong nonlinear terms, while the chattering suppression of the high-order sliding mode requires the continuous differentiability of the control law.

[0006] In response to the above technical bottlenecks, the applicants have proposed logarithmic sliding surfaces (Dong et al., IEEE / ASMETMech 2022)

[0007] (1.1)

[0008] and practical terminal sliding surface (Dong et al., IEEE TIE 2023)

[0009] (1.2)

[0010] By constructing a nonlinear switching function with adaptive gain characteristics, Pareto optimization of convergence speed and jitter intensity is achieved in the neighborhood of the equilibrium point. The core mechanism is that the asymmetric gain characteristics of the logarithmic term can dynamically adjust the amplitude of the equivalent control quantity, while the finite time convergence mechanism of the practical terminal structure realizes phase trajectory contraction through state-related exponential parameters.

[0011] However, the existing schemes still have essential limitations in terms of global performance optimization: both the logarithmic sliding manifold and the practical terminal sliding manifold can establish high gain near the equilibrium point, but when the system state is far away from the equilibrium point, the nonlinear gain of both types of sliding manifolds shows a gradient attenuation characteristic, resulting in a convergence rate significantly lower than the linear sliding benchmark under large deviation conditions. Although the global dynamic characteristics can be enhanced by introducing linear compensation terms, the resulting hybrid sliding surface will destroy the integrability conditions of the differential equation, making the derivation of the analytical solution of the sliding mode time parameter fall into the dilemma of no explicit solution of the Lyapunov equation. In particular, the multi-order nonlinear coupling terms in the hybrid sliding manifold will induce the non-smooth characteristics of the hyperplane geometric structure, resulting in the failure of the time estimation method based on homogeneity theory.

[0012] In addition, the inventor proposed a terminal sliding surface construction method based on log-hyperbolic tangent function in Chinese invention patents CN202010692622.0 and CN202010693051.2. This technology constructs a composite sliding manifold with continuous second-order derivatives by integrating the gain adaptive characteristics of the logarithmic function and the asymmetric saturation characteristics of the hyperbolic tangent function, and theoretically achieves the coordinated optimization of finite-time convergence and chattering suppression. However, in-depth analysis shows that such schemes still have theoretical defects: although the Lyapunov stability can be used to prove that the system state enters the neighborhood of the equilibrium point within a finite time, the derivation of the explicit analytical expression of the convergence time encounters essential obstacles. This is mainly because the nonlinear coupling of the hyperbolic tangent function and the logarithmic term causes the sliding mode differential equation to exhibit strong non-autonomous characteristics, and the existence of its analytical solution cannot meet the prerequisite of the Picard-Lindelöf theorem. More importantly, the time-varying curvature characteristics of the sliding phase trajectory in the approaching stage destroy the applicable basis of the homogeneity theory, making the scaling law relied on in the traditional finite-time estimation method invalid. The unanalyzability of convergence time leads to a lack of quantitative guidance for controller parameter tuning in engineering practice. In actual applications, trial and error adjustments can only be made through numerical simulation. Moreover, in the presence of unmodeled dynamic conditions, the actual convergence time of the system may deviate by an order of magnitude from the theoretical prediction, which restricts the industrial reliability of this method.

[0013] In summary, there is no effective solution to the problem of how to improve the global convergence speed of the system state on the sliding surface while weakening the chattering of the sliding mode control and explicitly giving the sliding time of the system state on the sliding surface. Summary of the invention

[0014] In order to improve the global convergence speed of the system state on the sliding mode surface while weakening the chattering of the sliding mode control, and explicitly give the sliding time of the system state on the sliding mode surface, the present invention proposes a logarithmic terminal sliding mode control method for nonlinear systems.

[0015] The technical solution of the present invention is:

[0016] The logarithmic terminal sliding mode control method for a nonlinear system comprises the following steps:

[0017] Step 1: Set up a nonlinear second-order system with uncertainty:

[0018]

[0019] in are the state variables of the nonlinear second-order system, yes The first derivative of yes The first-order derivative of and and their respective first-order and second-order derivatives can be measured; and is a known nonlinear function, is an unknown nonlinear perturbation, is the control signal of the nonlinear second-order system, is the output of the nonlinear second-order system;

[0020] Step 2: Design the logarithmic terminal sliding surface based on the nonlinear second-order system established in step 1:

[0021]

[0022] in is the designed logarithmic terminal sliding surface; ln [ • ] is the natural logarithm function; is a constant parameter set according to performance requirements, and ; and are all positive odd numbers, and ;

[0023] Step 3: Design the first-order logarithmic terminal sliding mode control strategy and the super-torsion logarithmic terminal sliding mode control strategy based on the logarithmic terminal sliding mode surface designed in step 2:

[0024] The first-order logarithmic terminal sliding mode control strategy designed according to the logarithmic terminal sliding mode surface is:

[0025]

[0026] in is the first-order logarithmic terminal sliding mode control output; is a known nonlinear function The reciprocal of is a symbolic function; is the upper bound of external disturbance; is an intermediate variable;

[0027] The super-torsion logarithmic terminal sliding mode control strategy designed according to the logarithmic terminal sliding mode surface is:

[0028]

[0029] in It is the output of super-torsion logarithmic terminal sliding mode control; is the intermediate variable, for The derivative of for The absolute value of and is the set control parameter;

[0030] Step 4: When the sliding surface When the state variable of the nonlinear second-order system is established, In limited time Converges to a small neighborhood near the equilibrium point, where the finite time The display expression is:

[0031]

[0032] in is the logarithmic integral function, Sliding surface The state parameters of the nonlinear second-order system when is a tiny neighborhood around the equilibrium point.

[0033] Furthermore, the intermediate variable for:

[0034] .

[0035] Furthermore, the nonlinear perturbation Satisfy constraints , is the upper bound of external disturbances, yes The absolute value of .

[0036] Furthermore, the nonlinear second-order system is a spacecraft attitude control system in a spacecraft tether deployment control application.

[0037] Furthermore, the uncertainty includes environmental uncertainty and dynamic modeling uncertainty.

[0038] In addition, the present invention also provides an electronic device and a readable storage medium:

[0039] An electronic device comprises a processor and a memory, wherein the memory is used to store one or more programs;

[0040] When the one or more programs are executed by the processor, the above method is implemented.

[0041] A readable storage medium stores a computer program, and when the computer program is executed by a processor, the above method is implemented.

[0042] Beneficial effects:

[0043] The present invention proposes a logarithmic terminal sliding mode control method for nonlinear systems. Based on the designed logarithmic terminal sliding mode surface, a first-order logarithmic terminal sliding mode control strategy and a super-torsion logarithmic terminal sliding mode control strategy are obtained, and a logarithmic terminal sliding mode collaborative control architecture for nonlinear systems is obtained. Multi-objective optimization of chattering suppression, global fast convergence and explicit sliding time is achieved. The global convergence speed of the system state on the sliding mode surface can be improved while weakening the sliding mode control chattering, and the sliding time of the system state on the sliding mode surface is explicitly given. Through the explicit analytical expression of the sliding time parameter, a strict mathematical correspondence between the controller gain coefficient and the system dynamic performance is established, which provides a quantitative theoretical basis for parameter setting in industrial sites and significantly enhances the system robustness under complex working conditions.

[0044] Additional aspects and advantages of the present invention will be given in part in the following description and in part will be obvious from the following description, or will be learned through practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] The above and / or additional aspects and advantages of the present invention will become apparent and easily understood from the description of the embodiments in conjunction with the following drawings, in which:

[0046] Figure 1 : Phase plane diagram of the terminal sliding surface, the logarithmic sliding surface, the practical terminal sliding surface and the logarithmic terminal sliding surface proposed by the present invention; wherein is the terminal sliding surface, is the logarithmic sliding surface, is the practical terminal sliding surface, is the logarithmic terminal sliding surface;

[0047] Figure 2 : The process of the system state parameters of Example 1 reaching the sliding surface under the first-order logarithmic terminal sliding mode control strategy; wherein the abscissa is time and the ordinate is the logarithmic terminal sliding surface Corresponding numerical value;

[0048] Figure 3 : The process in which the system state of Example 1 slides along the sliding surface to a small neighborhood near the equilibrium point under the first-order logarithmic terminal sliding mode control strategy; wherein the horizontal axis is time and the vertical axis is the system state parameter Corresponding numerical value;

[0049] Figure 4 : Output curve of Example 1 under the first-order logarithmic terminal sliding mode control strategy; wherein the horizontal axis is time and the vertical axis is the output of the first-order logarithmic terminal sliding mode control strategy Corresponding numerical value;

[0050] Figure 5 : The process of the system state parameters of Example 1 reaching the sliding surface under the action of the hypertorsion logarithmic terminal sliding mode control strategy; wherein the abscissa is time and the ordinate is the logarithmic terminal sliding surface Corresponding numerical value;

[0051] Figure 6 : Under the action of the hypertorsion logarithmic terminal sliding mode control strategy, the system state of Example 1 slides along the sliding mode surface to a small neighborhood near the equilibrium point; wherein the horizontal axis is time and the vertical axis is the system state parameter Corresponding numerical value;

[0052] Figure 7 : Output curve of Example 1 under the action of the super-torque logarithmic terminal sliding mode control strategy; wherein the horizontal axis is time and the vertical axis is the output of the super-torque logarithmic terminal sliding mode control Corresponding numerical value;

[0053] Figure 8 : The process of the system state parameters of Example 2 reaching the sliding surface under the action of the hypertorsion logarithmic terminal sliding mode control strategy; wherein the abscissa is time and the ordinate is the corresponding value of the logarithmic terminal sliding surface;

[0054] Fig. 9 : The process in which the system state of Example 2 slides along the sliding surface to a small neighborhood near the equilibrium point under the action of the hypertorsion logarithmic terminal sliding mode control strategy; wherein the abscissa is time and the ordinate is the value corresponding to the attitude quaternion state parameter;

[0055] Fig.10: The output curve of Example 2 under the action of the super-torsion logarithmic terminal sliding mode control strategy; wherein the horizontal axis is time and the vertical axis is the value of the three-dimensional control quantity. DETAILED DESCRIPTION

[0056] Embodiments of the present invention are described in detail below. The embodiments are exemplary and intended to be used to explain the present invention, but should not be construed as limiting the present invention.

[0057] Embodiment 1:

[0058] This embodiment proposes a logarithmic terminal sliding mode control method for a nonlinear system, which can improve the global convergence speed of the system state on the sliding mode surface while weakening the sliding mode control chattering, and explicitly gives the sliding time of the system state on the sliding mode surface. Specifically, the method includes the following steps:

[0059] Step 1: Establish a nonlinear second-order system with uncertainty. For the application of spacecraft tether deployment control, the uncertainty here includes environmental uncertainty, such as space environment disturbances, electromagnetic effects, space debris, etc., and also includes dynamic modeling uncertainty, such as complex multi-body coupling and parameter time-varying.

[0060] The nonlinear second-order system with uncertainty is expressed as:

[0061] (1.3)

[0062] in are the state variables of the nonlinear second-order system, yes The first derivative of yes The first-order derivative of and and their respective first-order and second-order derivatives can be measured; for a second-order system, Usually refers to information such as position and angle. Usually refers to information such as speed and angular velocity. In this embodiment, the initial value of the state variable is: , . and is a known nonlinear function. In this embodiment, , . is an unknown nonlinear perturbation, but Satisfy constraints , is the upper bound of the external disturbance. In this embodiment, . is the control signal of the nonlinear second-order system, is the output of the nonlinear second-order system.

[0063] Step 2: Design the logarithmic terminal sliding surface based on the nonlinear second-order system established in step 1:

[0064] (1.4)

[0065] in is the designed logarithmic terminal sliding surface; ln [ • ] is the natural logarithm function; is a constant parameter set according to performance requirements, and In this embodiment, the value is ; and are all positive odd numbers, and In this embodiment, the value is , .

[0066] Step 3: Design a first-order logarithmic terminal sliding mode control strategy and a super-torsion logarithmic terminal sliding mode control strategy based on the logarithmic terminal sliding mode surface designed in step 2.

[0067] First-order logarithmic terminal sliding mode control strategy:

[0068] The first-order logarithmic terminal sliding mode control strategy designed according to the logarithmic terminal sliding mode surface is:

[0069] (1.5)

[0070] in is the first-order logarithmic terminal sliding mode control output; is a known nonlinear function The reciprocal of is a symbolic function; is the upper bound of the external disturbance. ; For the intermediate variable:

[0071]

[0072] Under the action of the first-order logarithmic terminal sliding mode control strategy, the state variables of the nonlinear second-order system The process of reaching the sliding surface is as follows Figure 2 As shown, the state variables of the nonlinear second-order system are The process of converging to a small neighborhood of the equilibrium point is as follows Figure 3 The output curve of the first-order logarithmic terminal sliding mode control strategy is shown in Figure 4 The results show that the first-order logarithmic terminal sliding mode control strategy can make the state variables of the nonlinear second-order system It converges to a small neighborhood near the equilibrium point within a finite time; and the output curve of the first-order logarithmic terminal sliding mode control strategy is smooth, with no tendency to suddenly change to infinity (i.e., no singular terms).

[0073] Super-torsion logarithmic terminal sliding mode control strategy:

[0074] The super-torsion logarithmic terminal sliding mode control strategy designed according to the logarithmic terminal sliding mode surface is:

[0075] (1.6)

[0076] in It is the output of super-torsion logarithmic terminal sliding mode control; is the intermediate variable, for The derivative of for The absolute value of and is the set control parameter, which is taken as , .

[0077] Under the action of the hypertorsion logarithmic terminal sliding mode control strategy, the state variables of the nonlinear second-order system The process of reaching the sliding surface is as follows Figure 5 As shown, the state variables of the nonlinear second-order system are The process of converging to a small neighborhood of the equilibrium point is as follows Figure 6 The output curve of the super-torque logarithmic terminal sliding mode control strategy is shown in Figure 7 The results show that the hypertorsion logarithmic terminal sliding mode control strategy can make the state variables of the nonlinear second-order system It converges to a tiny neighborhood near the equilibrium point within a finite time; moreover, the output curve of the hyper-torsion logarithmic terminal sliding mode control strategy is smooth, with no tendency to suddenly change to infinity (i.e., no singular terms), and the hyper-torsion logarithmic terminal sliding mode control strategy has smaller jitter than the first-order logarithmic terminal sliding mode control strategy, thereby weakening the jitter effect of the sliding mode control.

[0078] Step 4: Both the first-order logarithmic terminal sliding mode control strategy and the super-torsion logarithmic terminal sliding mode control strategy can make the sliding surface Reachable within a limited time; once The state variables of the nonlinear second-order system are In limited time

[0079] (1.7)

[0080] Converges to a small neighborhood near the equilibrium point In which is the logarithmic integral function, Sliding surface The state parameters of the nonlinear second-order system when It can be seen that the first-order logarithmic terminal sliding mode control strategy and the super-torsion logarithmic terminal sliding mode control strategy proposed in this embodiment can explicitly give the sliding time of the system state on the sliding surface, which is conducive to the controller parameter setting in a reasonable manner in engineering practice and improves the system reliability.

[0081] For the above finite time, here is the proof:

[0082] when hour,

[0083] So the state variable of the nonlinear second-order system The time to reach the equilibrium point can be expressed as:

[0084]

[0085] Finally, we get .

[0086] The following is an explanation using the traditional second-order linear supertorsion sliding mode control strategy as a comparison:

[0087] The traditional second-order linear supertorsion sliding mode control strategy is:

[0088] (1.8)

[0089] in

[0090] (1.9)

[0091] pass Figure 1 It can be seen that the first-order logarithmic terminal sliding mode control strategy and the super-torsion logarithmic terminal sliding mode control strategy proposed in the present invention can make the state parameters of the nonlinear second-order system It converges to a tiny neighborhood near the equilibrium point within a finite time (i.e., reaches a certain accuracy within a finite time, not an infinite time); while the traditional second-order linear hypertorsion sliding mode control strategy can only ensure that the system state converges to the equilibrium point when the time is infinite.

[0092] Embodiment 2:

[0093] This embodiment aims at the problem of spacecraft attitude control during the tether deployment process of a spacecraft, and uses the logarithmic terminal sliding mode control method proposed in the present invention. The specific process is as follows:

[0094] Consider a spacecraft attitude control model described by attitude quaternions:

[0095] (1.10)

[0096] in, , is the scalar part of the spacecraft attitude quaternion, q = [ q 1 q 2 q 3 ] T is the vector part of the spacecraft attitude quaternion, and are its first-order derivative and second-order derivative respectively, q × = [ 0 − q 3 q 2 q 3 0 − q 1 − q 2 q 1 0 ] ; is the angular velocity of the spacecraft, is its first-order derivative, The definition of same; is the external bounded disturbance to the spacecraft; It is the control signal acting on the spacecraft; is the moment of inertia of the spacecraft. Obviously, the attitude control model (1.10) can be transformed into the following nonlinear system:

[0097] (1.11)

[0098] in: , , , , .

[0099] Then, based on the nonlinear system (1.11), the hypertorsion logarithmic terminal sliding mode control strategy is designed using the logarithmic terminal sliding mode surface to achieve the final effect. Figure 8-10 As shown, it shows that the components of the spacecraft attitude can quickly reach the sliding surface, and then quickly slide along the sliding surface to a small neighborhood near the equilibrium point within a finite time, and the control signal overshoot is small.

[0100] Although the embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention without departing from the principles and intent of the present invention.

Claims

1. A logarithmic terminal sliding mode control method for a nonlinear system, characterized in that: The following steps are involved: Step 1: Set up a nonlinear second-order system with uncertainty: in are the state variables of the nonlinear second-order system, yes The first derivative of yes The first-order derivative of and and their respective first-order and second-order derivatives can be measured; and is a known nonlinear function, is an unknown nonlinear perturbation, is the control signal of the nonlinear second-order system, is the output of a nonlinear second-order system; the nonlinear second-order system is a spacecraft attitude control system in a spacecraft tether deployment control application; Step 2: Design the logarithmic terminal sliding surface based on the nonlinear second-order system established in step 1: in is the designed logarithmic terminal sliding surface; is the natural logarithm function; is a constant parameter set according to performance requirements, and ; and are all positive odd numbers, and ; Step 3: Design the first-order logarithmic terminal sliding mode control strategy and the super-torsion logarithmic terminal sliding mode control strategy based on the logarithmic terminal sliding mode surface designed in step 2: The first-order logarithmic terminal sliding mode control strategy designed according to the logarithmic terminal sliding mode surface is: in is the first-order logarithmic terminal sliding mode control output; is a known nonlinear function The reciprocal of is a symbolic function; is the upper bound of external disturbance; is an intermediate variable; The super-torsion logarithmic terminal sliding mode control strategy designed according to the logarithmic terminal sliding mode surface is: in It is the output of super-torsion logarithmic terminal sliding mode control; is the intermediate variable, for The derivative of for The absolute value of and is the set control parameter; Step 4: When the sliding surface When the state variable of the nonlinear second-order system is established, In limited time It converges to a small neighborhood near the equilibrium point, where the explicit expression for the finite time is: in is the logarithmic integral function, Sliding surface The state parameters of the nonlinear second-order system when is a tiny neighborhood around the equilibrium point.

2. The logarithmic terminal sliding mode control method for a nonlinear system according to claim 1, characterized in that: Intermediate variables for: 。 3. A logarithmic terminal sliding mode control method for a nonlinear system according to claim 1 or 2, characterized in that: The nonlinear perturbation Satisfy constraints , is the upper bound of external disturbances, yes The absolute value of .

4. The logarithmic terminal sliding mode control method for a nonlinear system according to claim 1, characterized in that: The uncertainties include environmental uncertainties and dynamic modeling uncertainties.

5. An electronic device, comprising a processor and a memory, wherein the memory is used to store one or more programs; characterized in that: When the one or more programs are executed by the processor, the method described in any one of claims 1 to 4 is implemented.

6. A readable storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, the method described in any one of claims 1 to 4 is implemented.

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