Design method of pre-time controller for second order system with disturbance based on power integral

Abstract

The application discloses a pre-time controller design method for a second-order system with disturbance based on power integral, which comprises the following steps: collecting operation parameters of a second-order system and constructing a system dynamics equation; constructing a time-varying segmented function based on effective control gains of the system in different periods; constructing a target function of the system based on a power integral algorithm, and constructing an error function of the system based on the target function; and designing a pre-time controller for the second-order system with disturbance in combination with the time-varying segmented function and the power integral error function, so as to realize effective control of the system in different periods. The pre-time controller for the second-order system is designed, so that the system can reach stability within any given time, the system stable time does not need to depend on the initial state of the system, the rapidity, stability and accuracy of the system can be ensured and the jumping problem can not occur, the disturbance existing in the second-order system can be effectively handled, and the pre-time controller has good robustness.

Description

Technical Field

[0001] This invention relates to the technical field of second-order system control, and in particular to a design method for a pre-time controller of a second-order system with disturbance based on power integral. Background Technology

[0002] Second-order systems are the most common system models in real life. Drones, intelligent cars, robots, and other intelligent machines are all second-order system models. Therefore, controller design for second-order systems has always been a very important research direction in the field of control. Ensuring the control effect of the system and enabling it to achieve stability quickly and accurately requires a focused exploration of the system's convergence rate. Researchers in the field of control have achieved significant results in this area.

[0003] For a long time, system stability was often an asymptotic outcome, meaning the system could achieve stability in an infinite amount of time. However, for situations with strict convergence time requirements, asymptotic time control is insufficient. Therefore, some researchers proposed finite-time control strategies, which guarantee system stability within a finite time. However, their convergence time depends on the initial state, leading to very long convergence times when the initial state is far from the system's stable point. In many practical scenarios, this control effect is meaningless. Fixed-time control strategies were proposed to improve upon finite-time control strategies, as their convergence time can be explicitly calculated and is independent of the system's initial state.

[0004] However, in many practical control scenarios, it is necessary to ensure that the system can reach stability within any given time. Preset-time control strategies were proposed to address such applications, enabling the system to reach stability within a preset time. However, in practical application, they also reveal certain drawbacks. When applied to complex systems, the system is prone to abrupt changes at the piecewise function switching points, resulting in sudden increases in control input. In practical applications, saturation limiting methods are often used to control system input, making excessively large control inputs impractical. Therefore, this abrupt change problem inherent in preset-time control strategies spurred the development of pre-time control strategies. Summary of the Invention

[0005] The purpose of this section is to outline some aspects of embodiments of the present invention and to briefly describe some preferred embodiments. Simplifications or omissions may be made in this section, as well as in the abstract and title of this application, to avoid obscuring the purpose of these documents; however, such simplifications or omissions should not be construed as limiting the scope of the invention.

[0006] In view of the aforementioned existing problems, the present invention is proposed.

[0007] Therefore, the technical problem solved by the present invention is that the existing technology has the problem that the system stabilization time depends on the initial state of the system and has poor stability.

[0008] To solve the above-mentioned technical problems, the present invention provides the following technical solution: a design method for a pre-time controller of a second-order system with disturbance based on power integral, comprising:

[0009] Collect the operating parameters of the second-order system and construct the system dynamic equations;

[0010] Based on the effective control gain of the system at different time periods, a time-varying piecewise function is constructed.

[0011] Based on the power-law integral algorithm, the objective function of the system is constructed, and the error function of the system is constructed based on the objective function;

[0012] A pre-time controller for a second-order system with disturbances is designed by combining a time-varying piecewise function and a power integral error function to achieve effective control of the system at different time periods.

[0013] As a preferred embodiment of the design method for a pre-time controller of a second-order perturbation system based on power integral as described in this invention, the dynamic equation is mathematically modeled based on the dynamic characteristics of the second-order perturbation system.

[0014] As a preferred embodiment of the design method for a pre-time controller of a second-order perturbation system based on power-law integrals as described in this invention, the dynamic equation is expressed as:

[0015]

[0016] Where x1 represents the position state, x2 represents the velocity state, u represents the control input of the system, and d represents the disturbance experienced by the system.

[0017] As a preferred embodiment of the design method for a pre-time controller of a second-order system with disturbance based on power integral as described in this invention, the time-varying piecewise function provides corresponding gain values ​​according to different time points of the system.

[0018] As a preferred embodiment of the design method for a pre-time controller of a second-order perturbation system based on power integral as described in this invention, the function value continuously increases before the system reaches a preset time to enable the system to quickly stabilize; after the system reaches a preset time, the function value remains unchanged to ensure that the system can maintain stability.

[0019] As a preferred embodiment of the design method for a pre-time controller of a second-order perturbation system based on power-law integrals as described in this invention, the time-varying piecewise function is defined as follows:

[0020] The preset time for the system to reach stability is defined as T. p The control parameters are a and b, respectively, and the time-varying piecewise function... Represented as:

[0021]

[0022] Among them, T p ≥0, a>0, b>0, t is the system running time.

[0023] As a preferred embodiment of the design method for a second-order perturbation system lead time controller based on power integral of the present invention, wherein the system objective function is the objective condition for the system to achieve lead time stability.

[0024] As a preferred embodiment of the design method for a pre-time controller of a second-order perturbation system based on power-law integration described in this invention, the objective function of the system is defined as:

[0025] When the system's velocity state x2 equals the system's target state x2 * At that time, the system can achieve lead time stability, and the target state of the system is x2. * Represented as:

[0026]

[0027] Where m>0 represents the power of the time-varying piecewise function, α is the fractional order of the controller, α is the fraction of odd to odd, and 0 <a<1。

[0028] As a preferred embodiment of the design method for a pre-time controller of a second-order perturbation system based on power-law integration described in this invention, the error function of the system is expressed as:

[0029]

[0030] When the system achieves lead time stability, the power-integral error function ξ is 0, i.e., x² = x². * The system can achieve the target state.

[0031] As a preferred embodiment of the design method for a pre-time controller of a second-order perturbation system based on power-law integration described in this invention, the controller is represented as:

[0032]

[0033] Where l is the upper bound of the system disturbance, which satisfies d≤l, and sign is the sign function, expressed as:

[0034]

[0035] The beneficial effects of the present invention are as follows: The present invention provides a design method for a pre-time controller of a second-order system with disturbance based on power integral. By designing a pre-time controller for a second-order system, the system can reach stability within any given time, and the system stability time does not depend on the initial state of the system. At the same time, it ensures the speed, stability and accuracy of the system and does not have abrupt change problems. It can effectively handle the disturbances present in the second-order system and has good robustness. Attached Figure Description

[0036] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the following description of the embodiments will be briefly introduced. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. Wherein:

[0037] Figure 1 This is a flowchart illustrating the design method of a pre-time controller for a second-order system with perturbation based on power integral, according to an embodiment of the present invention.

[0038] Figure 2 The diagram shows the state effect of the system under the action of the power-integral-based pre-time controller, which is an example of the design method of the pre-time controller of a second-order system with disturbance described in an embodiment of the present invention.

[0039] Figure 3 The diagram shows the state effect of the system under the action of the power-integral-based pre-time controller, which is a case study of the design method of the pre-time controller for a second-order system with disturbance described in an embodiment of the present invention.

[0040] Figure 4 The diagram shows the state effect of the system under the action of the power integral-based pre-time controller, which is an example of the design method of the pre-time controller for a second-order perturbation system based on power integral as described in one embodiment of the present invention. Detailed Implementation

[0041] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the protection scope of the present invention.

[0042] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and those skilled in the art can make similar extensions without departing from the spirit of the invention. Therefore, the invention is not limited to the specific embodiments disclosed below.

[0043] 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 phrase "in one embodiment" appearing in different places in this specification does not necessarily refer to the same embodiment, nor is it a single or selective embodiment that is mutually exclusive with other embodiments.

[0044] This invention is described in detail with reference to the schematic diagrams. When detailing the embodiments of this invention, for ease of explanation, the cross-sectional views illustrating the device structure may be partially enlarged, not adhering to the usual scale. Furthermore, the schematic diagrams are merely examples and should not be construed as limiting the scope of protection of this invention. In actual fabrication, the three-dimensional spatial dimensions of length, width, and depth should be included.

[0045] Furthermore, in the description of this invention, it should be noted that the terms "upper," "lower," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. These terms are used solely for the convenience of describing the invention and for simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. In addition, the terms "first," "second," or "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0046] Unless otherwise explicitly specified and limited, the terms "installation," "connection," and "joining" in this invention should be interpreted broadly. For example, they can refer to fixed connections, detachable connections, or integral connections; similarly, they can refer to mechanical connections, electrical connections, or direct connections, or indirect connections through an intermediate medium, or internal connections between two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0047] Example 1

[0048] Reference Figure 1This is the first embodiment of the present invention, which provides a design method for a pre-time controller of a second-order system with perturbation based on power integral, characterized in that it includes:

[0049] S1: Collect the operating parameters of the second-order system and construct the system dynamic equations;

[0050] Furthermore, the dynamic equations are mathematically modeled based on the dynamic characteristics of a second-order perturbed system. The dynamic equations are expressed as:

[0051]

[0052] Where x1 represents the position state, x2 represents the velocity state, u represents the control input of the system, and d represents the disturbance experienced by the system.

[0053] S2: Construct a time-varying piecewise function based on the effective control gain of the system at different time periods;

[0054] Furthermore, the time-varying piecewise function provides corresponding gain values ​​based on different points in time the system is at. Before the preset time, the function value continuously increases to help the system quickly reach stability; after the preset time, the function value remains unchanged to ensure the system can maintain stability.

[0055] Furthermore, the time-varying piecewise function is defined as follows:

[0056] The preset time for the system to reach stability is defined as T. p The control parameters are a and b, respectively, and the time-varying piecewise function... Represented as:

[0057]

[0058] Among them, T p ≥0, a>0, b>0, t is the system running time.

[0059] S3: Based on the power-law integral algorithm, construct the objective function of the system, and based on the objective function, construct the error function of the system;

[0060] Furthermore, the system objective function is the objective condition for the system to achieve pre-time stability. The system objective function is defined as follows:

[0061] When the system's velocity state x2 equals the system's target state x2 * At that time, the system can achieve lead time stability, and the target state of the system is x2. * Represented as:

[0062]

[0063] Where m>0 represents the power of the time-varying piecewise function, α is the fractional order of the controller, α is the fraction of odd to odd, and 0 <a<1。

[0064] Furthermore, the error function of the system is expressed as:

[0065]

[0066] When the system achieves lead time stability, the power-integral error function ξ is 0, i.e., x² = x². * The system can achieve the target state.

[0067] S4: Design a pre-time controller for a second-order system with disturbance by combining a time-varying piecewise function and a power integral error function to achieve effective control of the system at different time periods.

[0068] Furthermore, the controller is represented as:

[0069]

[0070] Where l is the upper bound of the system disturbance, which satisfies d≤l, and sign is the sign function, expressed as:

[0071]

[0072] It should be noted that the proof process for the pre-time controller of a second-order perturbation system based on power integral designed in this invention to enable the system to converge within the pre-time is as follows:

[0073] First, the following lemmas need to be given:

[0074] Lemma 1. For any real-valued function x, y, there exists a real number p satisfying 0 < p ≤ 1 and p = p1 / p2, where p1 and p2 are both positive odd numbers. Then |x p -y p |≤2 1-p |xy| p Established;

[0075] Lemma 2. For any real-valued functions x and y and r(x, y) > 0, there exist c > 0 and d > 0 such that x and y satisfy the following inequalities:

[0076]

[0077] Lemma 3. Suppose a and b are non-negative real numbers, and there exist p and q satisfying p > 1 and 1 / p + 1 / q = 1, then

[0078]

[0079] And if and only if a p=b q When the equality holds, the equation is true.

[0080] Lemma 4. For the system If there exists a positive, continuous differential function V(t) that satisfies the following relationship:

[0081]

[0082] in, Parameters a>0, b>0, α∈(0,1), T p Let μ be a preset time that can be arbitrarily given, and μ satisfy the following expression:

[0083]

[0084] The system is then globally time-stable.

[0085] Choose Lyapunov functions as

[0086]

[0087] Taking the derivative with respect to V0, we can obtain

[0088]

[0089] According to Lemma 1 and Lemma 2, we can obtain...

[0090]

[0091] Among them, let but

[0092]

[0093] Taking the derivative with respect to V1, we can obtain

[0094]

[0095] Then, by rearranging the three equations, we can obtain...

[0096]

[0097]

[0098] x2 as defined in step (1.3) * The expression can be obtained

[0099]

[0100] Substituting it in, we can get

[0101]

[0102] Among them, let Then, by arranging, we can obtain

[0103]

[0104] From the expression of the time-varying piecewise function designed in step (1.2), we can obtain After substitution, we can obtain

[0105]

[0106] From Lemma 3 we can obtain

[0107]

[0108] Substituting the above equation, we get...

[0109]

[0110] Will By rearranging the above formula, we can obtain

[0111]

[0112] Substituting the pre-time controller into the above equation yields...

[0113]

[0114] According to the definition of V1, we can obtain...

[0115]

[0116] The following definitions are given.

[0117]

[0118] It can be obtained

[0119] V≤k(x1 α+1 +ξ α+1 V β ≤k β (x1 α+1 +ξ α+1 ) β ≤k β (x1 2α +ξ 2α (28)

[0120] V and Combining

[0121]

[0122] Right now

[0123]

[0124] According to Lemma 4, the designed controller can achieve lead time stability of the system.

[0125] Example 2

[0126] Reference Figure 2 —4 is an embodiment of the present invention, which provides a design method for a pre-time controller of a second-order system with disturbance based on power integral. In order to verify the beneficial effects of the present invention, scientific demonstration is carried out through comparative experiments and implementation effects.

[0127] Case 1: The parameters of the pre-time controller are set as a = 1, b = 2, α = 5 / 7, m = 0.8, the relevant initial states of the system are set as x1(0) = 2, x2(0) = 4, and the preset time of the system is set to T. p =1, the perturbation is given as d = 2tanh(t), and its upper limit can be given as l = 2. The simulation results are as follows. Figure 2 As shown.

[0128] Case 2: The parameters of the pre-time controller are set as a = 1, b = 2, α = 5 / 7, m = 1, the relevant initial states of the system are set as x1(0) = 30, x2(0) = 10, and the preset time of the system is set to T. p =3, the perturbation is given as d = 2tanh(t), and its upper limit can be given as l = 2. The simulation results are as follows. Figure 3 As shown.

[0129] Case 3: The parameters of the pre-time controller are set as a = 1, b = 2, α = 5 / 7, m = 1, the relevant initial states of the system are set as x1(0) = -10, x2(0) = 5, and the preset time of the system is set to T. p =1.5, the perturbation is given as d = 2tanh(t) + 0.5sin(t), and its upper limit can be given as l = 2.5. The simulation results are as follows. Figure 4 As shown.

[0130] As can be seen from the above three sets of cases, the controller designed by the power integral-based second-order perturbation system pre-time controller design method provided by the present invention can enable the system to reach stability within any given time and has good robustness.

[0131] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for designing a pre-time controller for a second-order system with perturbation based on power-law integration, characterized in that: include, Collect the operating parameters of the second-order system and construct the system dynamic equations; Based on the effective control gain of the system at different time periods, a time-varying piecewise function is constructed. Based on the power-law integral algorithm, the objective function of the system is constructed, and the error function of the system is constructed based on the objective function; A pre-time controller for a second-order system with disturbance is designed by combining a time-varying piecewise function and a power-integral error function to achieve effective control of the system at different time periods. The time-varying piecewise function is defined as follows: The preset time for the system to reach stability is defined as follows: The control parameters are respectively and The time-varying piecewise function Represented as: in, , t is the system runtime; The system objective function is the objective condition for the system to achieve pre-time stability. The objective function of the system is defined as follows: When the system speed state Equal to the system's target state At that time, the system can achieve lead time stability, and the target state of the system... Represented as: in, For the time-varying piecewise function, For the fractional order of the controller, For the fraction of an odd number to an odd number, and ; The error function of the system is expressed as: When the system achieves lead time stability, the power-integral error function =0 The system can achieve the target state.

2. The design method for a pre-time controller of a second-order system with perturbation based on power integration as described in claim 1, characterized in that: The dynamic equations are mathematically modeled based on the dynamic characteristics of a second-order system with perturbations.

3. The design method for a pre-time controller of a second-order system with perturbation based on power integral as described in claim 2, characterized in that: The dynamic equation is expressed as: in, Indicates the position status. Indicates speed status. Indicates the system's control input, This indicates the disturbance experienced by the system.

4. The design method for a pre-time controller of a second-order system with perturbation based on power integral as described in claim 3, characterized in that: The time-varying piecewise function provides corresponding gain values ​​according to different time points of the system.

5. The design method for a pre-time controller of a second-order system with perturbation based on power integral as described in claim 4, characterized in that: Also includes: Before the preset time, the function value will continuously increase to enable the system to quickly reach stability; after the preset time, the function value will remain unchanged to ensure that the system can maintain stability.

6. The design method for a pre-time controller of a second-order system with perturbation based on power integral as described in claim 1, characterized in that: The controller is represented as: in, This is the upper bound of the system disturbance, which satisfies , For symbolic functions, it is represented as: 。

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