Linear-nonlinear tracking differentiator backstepping control method

Abstract

The invention discloses a backstepping control method for a linear-nonlinear tracking differentiator, and belongs to the field of signal tracking and processing, and the method comprises the steps: constructing an excitation function R2SS based on Relu-2Softsign-Sigmoid; designing a linear-nonlinear tracking differentiator based on the constructed excitation function; a differentiator based on linear-nonlinear tracking is used for replacing a differential process occurring during backstepping control design. The differentiator is simple in structure and convenient in parameter selection, flutter can be effectively eliminated, rapidity and stability of signal convergence can be considered at the same time, 'complex explosion 'is eliminated, trajectory tracking precision is improved, convergence speed is increased, and loss in the control process is reduced.

Description

Technical Field

[0001] The invention belongs to the field of signal tracking and processing, and in particular relates to a linear-nonlinear tracking differentiator backstepping control method. Background Art

[0002] Backstepping control is a classic method for dealing with high-order nonlinear system control problems. It makes the design process of control scalar functions and controllers systematic and structured through reverse design, and there is no chattering, so it has been widely used. In the backstepping control design process, the quality of position feedback signals, speed feedback signals, etc. is the key to improving control accuracy, which is related to signal filtering and differentiation. In most cases, the detection of system position signals, speed signals, etc. is completed by various position sensors and speed sensors, but sensors not only increase hardware costs and structural complexity, but also have noise interference problems.

[0003] A series of differential signals such as velocity and acceleration can be calculated based on the position signal using numerical differentiation. Traditionally, differential signals are approximated using the difference method, but this method requires multiple continuous derivatives when used in backstepping control, which can easily lead to the "complexity explosion" problem. When affected by signal noise, it will reduce the estimation accuracy of the signal differential, thereby affecting the control effect of the backstepping controller.

[0004] In order to suppress signal noise and avoid "complexity explosion", past studies usually use filters and various signal processing methods to pre-process the signal, such as dynamic surface control (DSC), command filter backstepping control and tracking differentiator (TD). The main feature of DSC is the introduction of a first-order filter in the backstepping design process to replace the differential operator in each virtual controller design step. However, the DSC scheme does not consider the impact of the filter error on the control system; the command filter backstepping control method introduces a second-order filter of the virtual input at each step, but the transient performance of the filtered signal is not very good, and when the input signal frequency becomes larger, the performance of its differential process will decrease; the tracking differentiator can enoughImprove the transient performance of the filtered signal and the differential performance at higher frequencies. It can complete the process of tracking and differentiating the real-time signal without relying on the controlled object model. It uses the integration process to replace the differentiation process in the traditional numerical differentiation method, avoiding the "complexity explosion" while achieving the signal filtering effect. With the in-depth research on the tracking differentiator technology in recent years, various improved tracking differentiators such as linear tracking-differentiator (LTD) and high-speed tracking-differentiator (HSTD) have been gradually developed to improve the dynamic performance of the differentiator. However, the tracking functions of most current tracking differentiators cannot take into account the rapidity and stability of signal convergence at the same time. When adjusting the convergence trend near the equilibrium point, it may cause problems with the rapidity of signal convergence or its stability. Summary of the invention

[0005] In order to solve the above technical problems, the present invention proposes a linear-nonlinear tracking differentiator backstepping control (Backstepping Control Of Linear-Nonlinear Tracking Differentiator Based On Relu-2Softsign-Sigmoid Function, SL-NTDBSC) method, which is based on the Relu-2Softsign-Sigmoid function and replaces the traditional differentiation process with a linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid excitation function, which has the advantages of simple structure, convenient parameter selection, effective elimination of chatter, and both rapidity and stability of signal convergence. The method is combined with backstepping to avoid the problem of "complexity explosion" and alleviate the vanishing of its gradient, which will further improve the effect of extracting differentials by the tracking differentiator, thereby ultimately improving the trajectory tracking of the backstepping control.

[0006] To achieve the above purpose, the technical solution adopted by the present invention is as follows:

[0007] A linear-nonlinear tracking differentiator backstepping control method comprises the following steps:

[0008] Step 1: Construct the activation function R2SS based on Relu-2Softsign-Sigmoid:

[0009] ,

[0010] In the formula, is the independent variable of the activation function R2SS. The activation function based on Relu-2Softsign-Sigmoid acts as a linear unit on the positive semi-axis, and its derivative is always 1; it acts as a nonlinear function on the negative semi-axis, and its derivative does not have a mutation at the zero point and gradually tends to 0. e represents an exponential function.

[0011] Step 2: Design a linear-nonlinear tracking differentiator based on the constructed excitation function;

[0012] Step 3: Use a linear-nonlinear tracking differentiator to replace the differentiation process that occurs in the backstepping control design.

[0013] The beneficial effects of the present invention are:

[0014] The present invention uses a linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid excitation function to replace the traditional differentiation process, and can improve the transient performance of the filtered signal and the differentiation performance at higher frequencies; reduce the number of parameters, alleviate the burden of parameter adjustment workload, and at the same time take into account the rapidity and stability of signal convergence, so that the gradient vanishing is alleviated, the differential extraction effect is better, and the trajectory tracking of the backstepping control is more accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 It is the control structure diagram of the linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid function;

[0016] Figure 2 A structural diagram of a backstepping control of a linear-nonlinear tracking differentiator of the present invention;

[0017] Figure 3 The figure is a comparison diagram of the trajectory tracking results of the present invention and the backstepping control based on command filtering, wherein (a) is the trajectory tracking curve, and (b) is the trajectory tracking error curve;

[0018] Figure 4 The figure is a comparison diagram of the tracking results of the differential signal between the present invention and the prior art, wherein (a) is the trajectory tracking curve, and (b) is the trajectory tracking error curve. DETAILED DESCRIPTION

[0019] The specific implementation modes of the present invention are described in detail below with reference to the accompanying drawings.

[0020] The present invention proposes a linear-nonlinear tracking differentiator backstepping control method. By using a linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid excitation function to replace the traditional differentiation process, the emergence of "complexity explosion" is avoided, and the advantages of the linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid excitation function, such as simple structure, fewer parameters, effective elimination of jitter, good dynamic response, and taking into account the rapidity and stability of signal convergence, the accuracy of trajectory tracking of backstepping control is further improved.

[0021] First, the specific form of the improved activation function is proposed as follows:

[0022] ,

[0023] In the formula, It is the independent variable of the improved excitation function; the improved Relu-2Softsign-Sigmoid excitation function acts as a linear unit on the positive semi-axis, and its derivative is always 1; the improved excitation function acts as a nonlinear function on the negative semi-axis, and its derivative does not have a mutation at the zero point and gradually tends to 0.

[0024] Secondly, a linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid function is designed using the improved excitation function. The specific form is:

[0025] ,

[0026] in, and Respectively and The first time derivative of represents the output signal after filtering, represents the extracted differential signal, R represents the time constant, a and b represent the first and second intermediate parameters of the amplitude, represents the input signal of the linear-nonlinear tracking differentiator based on the improved excitation function, i.e., the tracked signal, Represents the improved Relu-2Softsign-Sigmoid activation function; Figure 1 Shown is the control structure diagram of the linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid activation function.

[0027] The tracking differentiator is divided into two action stages under the influence of the improved activation function:

[0028] When improving the independent variable of the activation function When , the tracking differentiator system is far away from the equilibrium point, and the tracking differentiator takes a linear form:

[0029] ,

[0030] When improving the independent variable of the activation function When , the tracking differentiator system is close to the equilibrium point, and the tracking differentiator takes a nonlinear form:

[0031] ,

[0032] Therefore, for any tracked signal and any given finite time ,satisfy:

[0033]

[0034] Right now It will fully approach the tracked signal within any given finite time T ,based on ,but It will also converge to the tracked signal within any given finite time T The generalized differential of the tracked signal is completed and the differential of the tracked signal Trajectory tracking.

[0035] Next, we will conduct a specific design of the backstepping control of the linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid excitation function, such as Figure 2 As shown in the figure, each differentiation process is implemented using a linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid excitation function, and the stability backstepping control design is performed on each subsystem in turn based on the Lyapunov stability method.

[0036] The backstepping control design process is divided into n subsystems, and the n subsystems correspond one-to-one to the n-order controlled objects:

[0037] ,

[0038] In the formula, represents the state quantity of the first, second, ..., nth subsystem, represents the control law, where It also represents the output of the system, represents a nonlinear function;

[0039] remember , is the system virtual input quantity, that is, the state quantity of the first, second, ..., nth subsystem The expected value of; design Lyqpunov functions for n subsystems respectively , the backstepping control law is obtained as:

[0040] ,

[0041] in, is the differential signal obtained using a linear-nonlinear tracking differentiator The approximate differential of represents the adjustable parameters of the nth subsystem, Represents the tracking error of the nth subsystem.

[0042] Among them, for a continuous n-order system, if it is possible to construct A scalar function with continuous first-order partial derivatives , , and an attraction region around the origin of the state space , so that all non-zero states The following conditions must be met:

[0043] ,

[0044] ,

[0045] For any non-zero ; For any non-zero The original system equilibrium state exist The domain is gradually stable.

[0046] The above method is explained below using a third-order system as an example.

[0047] A specific control object is introduced for detailed description. The state space equation of the control object is:

[0048] ,

[0049] in, , , is the state quantity, is a constant, is the control law, and the input is , the output is .

[0050] Next, we design the backstepping control, replacing each differential process involved in the backstepping with the differential signal extracted by the tracking differentiator, and it becomes: form.

[0051] First, clarify the goal: Infinitely close to ;

[0052] Assume tracking error :

[0053] ,

[0054] Define the Lyapunov function :

[0055] ,

[0056] ,

[0057] In order for the system to be stable, , that is, negative definite;

[0058] Observe the above formula, as long as Approaching The above requirements can be met, among which represents an adjustable parameter greater than zero;

[0059] ,

[0060] At this time set for tracking object, and because the system itself ,Right now:

[0061] ,

[0062] Tracking Error for:

[0063] ,

[0064] Redefine the Lyapunov function :

[0065] ,

[0066] right Perform a derivative operation:

[0067] ,

[0068] In order for the system to be stable, , that is, negative definite;

[0069] Observe the above formula, as long as Approaching The above requirements can be met, among which represents an adjustable parameter greater than zero;

[0070] ,

[0071] set up for tracking object, and because the system itself ,Right now:

[0072] ,

[0073] Tracking Error for:

[0074] ,

[0075] Redefine the Lyapunov function :

[0076] ,

[0077] right Perform a derivative operation:

[0078] ,

[0079] If you want to ensure system stability, you need , that is, negative definite;

[0080] even though:

[0081] ,

[0082] in, represents an adjustable parameter greater than zero, we have:

[0083] ,

[0084] Calculating the above formula, we can get the control law:

[0085] .

[0086] Because the error is exponentially asymptotically stable, the control law designed above can ensure that the original nonlinear system is exponentially asymptotically stable.

[0087] In order to clearly illustrate the advantages of the backstepping control of the tracking differentiator based on the Relu-2Softsign-Sigmoid excitation function in the present invention, the following comparative description is made:

[0088] The trajectory tracking of the backstepping control based on command filtering and the backstepping control based on the tracking differentiator of the Relu-2Softsign-Sigmoid excitation function in the present invention are compared, as shown in FIG. Figure 3 As shown in the figure (a), it can be seen that both can achieve trajectory tracking, but the backstepping control based on command filtering cannot keep up in the initial stage. From the trajectory tracking errors of the two in figure (b), it can be more obvious that the backstepping control based on command filtering has a larger error, with a large overshoot at the beginning, and there is still periodic error jitter after stabilization. This shows that the trajectory tracking effect of the backstepping control based on the tracking differentiator of the Relu-2Softsign-Sigmoid excitation function of the present invention is significantly better than that of the backstepping control based on command filtering.

[0089] In order to more effectively illustrate the excellent trajectory tracking performance of the backstepping control of the linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid excitation function in the present invention, it is compared with the backstepping control combined with other tracking differentiators. The formula of the tracking differentiator used for comparison is as follows:

[0090] Contrast Tracking Differentiator (NTD):

[0091] ,

[0092] in, is the hyperbolic tangent function, All are constant parameters.

[0093] Contrast Tracking Differentiator (LTD):

[0094] ,

[0095] in, is a constant parameter.

[0096] Contrast Tracking Differentiator (HSTD):

[0097] ,

[0098] in, is the symbolic function, All are constant parameters.

[0099] The relevant parameter values ​​of the linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid excitation function introduced in the present invention and other comparative tracking differentiators are shown in Table 1.

[0100] Table 1

[0101]

[0102] From the information in Table 1, we can see that the number of parameters of the linear-nonlinear tracking differentiator based on the Relu-2Softsign-Sigmoid activation function is smaller than that of other contrast tracking differentiators, which makes the overall parameter adjustment of backstepping control easier.

[0103] Figure 4 The tracking results of the differential signal of the backstepping control based on the linear-nonlinear tracking differentiator (R2SSL-NTDBSC) of the present invention and the backstepping control based on the linear tracking differentiator (LTDBSC), the backstepping control based on the full-range fast tracking differentiator (HSTDBSC), and the backstepping control based on the new nonlinear-linear tracking differentiator (NTDBSC) are compared. Figure 4 It can be clearly seen from Figures (a) and (b) that the trajectory tracking error of R2SSL-NTDBSC is the smallest, and there is no excessive jitter and periodic error, achieving error minimization within this range.

[0104] In order to quantitatively see the improvement effect, three indicators are selected to prove it, as shown in Table 2 below:

[0105] Table 2

[0106]

[0107] ITAE in Table 2 is the integral of the absolute value of the error multiplied by the time term over time, which can reflect both the size of the error (control accuracy) and the speed of error convergence, taking into account both control accuracy and convergence speed. ISE is the integrated square error, which indicates the suppression of large errors in the transition process. ISU is the integral of the absolute value of time and control law. From the perspective of the ITAE improvement effect of R2SSL-NTDBSC relative to other backstepping, R2SSL-NTDBSC has the best control accuracy and the fastest convergence speed; from the perspective of the ISE improvement effect of R2SSL-NTDBSC relative to other backstepping, R2SSL-NTDBSC has almost no large errors in the transition process; from the perspective of the ISU improvement effect of R2SSL-NTDBSC relative to other backstepping, R2SSL-NTDBSC has reduced energy loss.

[0108] The specific embodiments described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A linear-nonlinear tracking differentiator backstepping control method, characterized in that: The steps include: Step 1: Construct the activation function R2SS based on Relu-2Softsign-Sigmoid: ， In the formula, is the independent variable of the activation function R2SS. The activation function based on Relu-2Softsign-Sigmoid acts as a linear unit on the positive semi-axis, and its derivative is always 1; it acts as a nonlinear function on the negative semi-axis, and its derivative does not have a mutation at the zero point and gradually tends to 0. e represents an exponential function. Step 2: Design a linear-nonlinear tracking differentiator based on the constructed excitation function; Step 3: Use a linear-nonlinear tracking differentiator to replace the differentiation process that occurs in the backstepping control design.

2. A linear-nonlinear tracking differentiator backstepping control method according to claim 1, characterized in that: The specific form of the linear-nonlinear tracking differentiator in step 2 is: ， in, and Respectively and The first time derivative of represents the output signal after filtering, represents the extracted differential signal, R represents the time constant, a and b represent the first and second intermediate parameters of the amplitude, represents the input signal of the linear-nonlinear tracking differentiator, i.e., the tracked signal; When improving the independent variable of the activation function When , the tracking differentiator system is far away from the equilibrium point, and the tracking differentiator takes a linear form: ， When improving the independent variable of the activation function When , the tracking differentiator system is close to the equilibrium point, and the tracking differentiator takes a nonlinear form: ， For any tracked signal and any given finite time ,satisfy: ， Right now It will fully approach the tracked signal within any given finite time T ,based on ,but It will also converge to the tracked signal within any given finite time T The generalized differential of the tracked signal is completed and the differential of the tracked signal Trajectory tracking.

3. A linear-nonlinear tracking differentiator backstepping control method according to claim 1, characterized in that: The step 3 uses a linear-nonlinear tracking differentiator to replace the differentiation process that occurs during the backstepping control design, including: The backstepping control design process is divided into subsystems, Subsystems and The controlled objects correspond one to one: ， In the formula, represents the state quantity of the first, second, ..., nth subsystem, represents the control law, where It also represents the output of the system, represents a nonlinear function; remember , is the system virtual input quantity, that is, the state quantity of the first, second, ..., nth subsystem expected value; Design Lyqpunov functions for each subsystem , the backstepping control law is obtained as: ， in, is the differential signal obtained using a linear-nonlinear tracking differentiator The approximate differential of represents the adjustable parameters of the nth subsystem, Represents the tracking error of the nth subsystem.

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