阅读说明：本技术 一种基于二阶超螺旋滑模的多通道直流电机系统控制方法 (Multichannel direct current motor system control method based on second-order supercoiled sliding mode ) 是由 李猛 徐龙宇 陈勇 陈章勇 于 2019-09-20 设计创作，主要内容包括：本发明公开了一种基于二阶超螺旋滑模的直流电机系统控制方法,含有多通道与外界扰动的直流电机系统模型建立以及二阶超螺旋滑模控制算法的设计。包括考虑外界扰动的多通道直流电机系统建模、外界扰动估计、二阶超螺旋滑模滑模观测器设计、二阶超螺旋滑模控制器设计以及系统稳定性证明。本发明针对具有外界干扰的多通道直流电机系统,建立了服从马尔可夫跳变的切换系统模型,并对扰动进行估计。针对其系统特性,设计并求解了一种基于二阶超螺旋滑模的控制器,并对其控制的稳定性加以分析和证明。本发明能够有效解决系统模型在服从马尔可夫跳变下,多通道直流电机系统的稳定控制。(The invention discloses a direct current motor system control method based on a second-order supercoiled sliding mode, which comprises establishment of a direct current motor system model with multiple channels and external disturbance and design of a second-order supercoiled sliding mode control algorithm. The method comprises the steps of multi-channel direct current motor system modeling considering external disturbance, external disturbance estimation, design of a second-order supercoiled sliding mode observer, design of a second-order supercoiled sliding mode controller and system stability verification. The invention aims at a multi-channel direct current motor system with external interference, establishes a switching system model complying with Markov jump, and estimates disturbance. Aiming at the system characteristics, a controller based on a second-order supercoiled sliding mode is designed and solved, and the control stability is analyzed and proved. The method can effectively solve the problem of stable control of a multi-channel direct current motor system under the condition that a system model obeys Markov jump.)

1. A direct current motor system control method based on a second-order super-spiral sliding mode is characterized by comprising the following steps:

(1) establishing a Markov-compliant channel switching model aiming at the multichannel problem of the direct current motor system;

(2) estimating disturbance aiming at the problem of external interference, and designing and solving a second-order supercoiled sliding-mode observer;

(3) and aiming at the system characteristics, a sliding mode controller based on second-order super-spiral is designed and solved, and the stability of the sliding mode controller is analyzed.

2. The direct current motor system control method based on the second-order supercoiled sliding mode according to claim 1, characterized in that the establishment of the multichannel direct current motor system model considering the external disturbance is as follows:

multiple transmission channels y for a DC motor system₁(k),y₂(k),...,y_n(k) Is in accordance with a random transmission protocol xi (k) epsilon { y₁(k),y₂(k),...,y_n(k) And selecting only one channel for data transmission each time.

3. The method of claim 2, wherein only one channel is selected at a time for data transmission, and wherein: the selection of the channel and the switching of the channel accord with Markov jump rules and satisfy the condition probability distribution: prob { ξ (k +1) ═ j | ξ (k) ═ i } - } pi_ij(k) Wherein, in the step (A),

4. the direct current motor system control method based on the second-order supercoiled sliding mode according to the claim 2, characterized in that, each state variable of the system considers the injected external random disturbance w (k), and the disturbance is bounded.

5. The direct current motor system control method based on the second-order supercoiled sliding mode according to claim 1, characterized in that the external disturbance is estimated as: and estimating the external disturbance according to a one-step time delay method, wherein the injected external random disturbance is bounded, and the obtained disturbance estimation value is also bounded.

6. The control method of the direct current motor system based on the second-order supercoiled sliding mode according to claim 1, characterized in that the design of the second-order supercoiled sliding mode observer is as follows: for the state variable x₁(k +1) and x₂(k +1) by two non-linear terms z₁，z₂Correcting to obtain observed valueAnd

7. the method for controlling a direct current motor system based on a second-order supercoiled sliding mode according to claim 1, characterized in that the pair of two nonlinear terms z₁And z₂And (5) correcting: non-linear term z₁And z₂Is a key item of the supercoiling algorithm:and z₂＝k₃sign(e₁(k))+k₄e₁(k) In that respect Wherein k is₁，k₂，k₃And k₄To the parameters to be adjusted, e₁(k) And e₂(k) The errors of the two state variables and the observed value are respectively.

8. The control method of the direct current motor system based on the second-order supercoiled sliding mode according to claim 1, characterized in that the second-order supercoiled sliding mode controller is designed as follows: designing a sliding mode surface according to the error epsilon (k) of the actual speed and the reference speed of the motor system, wherein the sliding mode surface is designed as follows:and other forms having proportional, integral (sum), differential (difference). Wherein the content of the first and second substances,λ₁＞0，λ₂＞0，λ₃＞0，and

9. the method according to claim 1, wherein the second-order supercoiled sliding mode-based dc motor system is designed according to a control law of u (k) to u_eq(k)+u_s(k) In that respect Wherein u is_eq(k) To an equivalent control law, u_s(k) Is a supercoiled control term, which is in the general form of a second order supercoiled.

10. The method for controlling the system of the direct current motor based on the second-order supercoiled sliding mode according to claim 1, characterized by further comprising the following steps of analyzing and proving the control stability: from the accessibility of the sliding-mode surface s (k) and the limited time convergence. Sliding mode surface accessibility is expressed as: and | s (k +1) | is less than or equal to | s (k) |, the motion track is attenuated, and the motion track gradually converges to the sliding mode surface. The finite time convergence is: and after a finite time T, the motion trail converges to the quasi-sliding mode bandwidth.

Technical Field

The invention belongs to the technical field of multi-channel direct current motor system control, and particularly relates to a direct current motor system control method based on a second-order supercoiled sliding mode, which is established by a direct current motor system model containing multiple channels and external disturbance.

Background

The direct current motor has the advantages of high reliability, simplicity, easy use, high speed tracking performance, good dynamic characteristics and the like, so that the direct current motor becomes a research hotspot of scholars, and is widely applied to various electric power equipment, including satellites, aerospace, robot control and the like. With the development of industrial information modernization, remote control and networked control of multiple direct current motor systems and multiple transmission channels gradually become emerging hot spots. Therefore, it is important to establish a channel switching model and to quickly and effectively control the motor system. For the multi-channel problem, the document "Robust control of network systems with variable communication capabilities and application to a semi-active delivery system" (xunyan Yin, Lixian Zhang, Yanzheng Zhu, Changhong Wang, zhajian Li, IEEE/ASME Transactions on mechanics, 2016,21 (2094): 2097-. However, the control algorithm is not optimized. The literature [ "Design of an integral sub-sampled mode controller for the robust motion control of robust controllers" (Antonella Ferrara, Gian Paolo Incremena, IEEE Transactions on control Systems Technology,2015,23(6):2316 and 2325) ] researches an integral sub-optimal second order sliding mode control algorithm, which is a continuous controller different from the traditional sliding mode control algorithm, and can greatly reduce the jitter problem caused by the discontinuity of control signals, thereby being applied to industrial control. However, the control of continuity generates a large amount of useless data, increases transmission pressure, and is not suitable in the context of networked control.

Disclosure of Invention

The invention aims to overcome the defects of the prior art and provides a direct current motor system control method based on a second-order supercoiled sliding mode, so as to reduce transmission pressure and be suitable for the background of networked control.

In order to achieve the above object, the present invention provides a dc motor system control method based on a second-order supercoiled sliding mode, which is characterized by comprising the following steps:

(1) establishing a Markov-compliant channel switching model aiming at the multichannel problem of the direct current motor system;

(2) estimating disturbance aiming at the problem of external interference, and designing and solving a second-order supercoiled sliding-mode observer;

(3) and aiming at the system characteristics, a sliding mode controller based on second-order super-spiral is designed and solved, and the stability of the sliding mode controller is analyzed.

The object of the invention is thus achieved.

The invention relates to a direct current motor system control method based on a second-order supercoiled sliding mode, which considers the modeling of a multichannel direct current motor system with external disturbance and a control algorithm based on the second-order supercoiled sliding mode, and comprises the establishment of a multichannel direct current motor system model with external disturbance, the estimation of the external disturbance, the design of a second-order supercoiled sliding mode observer, the design of a second-order supercoiled sliding mode controller and the proof of control stability.

Drawings

Fig. 1 is a schematic diagram of a specific embodiment of a dc motor system control method based on a second-order supercoiled sliding mode according to the present invention.

Detailed Description

The following description of the embodiments of the present invention is provided in order to better understand the present invention for those skilled in the art with reference to the accompanying drawings. It is to be expressly noted that in the following description, a detailed description of known functions and designs will be omitted when it may obscure the subject matter of the present invention.

Considering the multi-channel direct current motor system model of external disturbance, for a plurality of transmission channels { y ] of the direct current motor system₁(k),y₂(k),...,y_n(k) Is in accordance with a random transmission protocol xi (k) epsilon { y₁(k),y₂(k),...,y_n(k) And selecting only one channel for data transmission each time. The selection of the channel and the switching of the channel accord with Markov jump rules and satisfy the condition probability distribution: prob { ξ (k +1) ═ j | ξ (k) ═ i } ═ b ═ j ═ ξ (k) ═ j ═π_ij(k) Wherein, in the step (A),

the method comprises the steps of considering an external disturbance of a multi-channel direct current motor system model, considering injected external random bounded disturbance w (k) for each state variable of the system, estimating the external disturbance according to a one-step time delay method, and obtaining a disturbance estimation value which is bounded because the injected external random disturbance is bounded.

Design of second-order supercoiled sliding-mode observer for state variable x₁(k +1) and x₂(k +1) by two non-linear terms z₁And z₂Correcting to obtain observed valueAndnon-linear term z₁And z₂Is a key term of the supercoiling algorithm, z₁＝k₁|e₁(k)|^1/2 sign(e₁(k))+k₂e₁(k) And z₂＝k₃sign(e₁(k))+k₄e₁(k) And the like. Wherein k is₁，k₂，k₃And k₄To the parameters to be adjusted, e₁(k) And e₂(k) The errors of the two state variables and the observed value are respectively.

Designing a second-order super-spiral sliding mode controller, designing a sliding mode surface according to an error epsilon (k) of an actual speed and a reference speed of a motor system, wherein the sliding mode surface is as follows:and other forms having proportional, integral (sum), differential (difference). Wherein the content of the first and second substances,λ₁＞0，λ2₂＞0，λ₃＞0，the control law is u (k) u_eq(k)+u_s(k) In that respect Wherein u is_eq(k) To an equivalent control law, u_s(k) Is a supercoiled control term, which is in the general form of a second order supercoiled.

The analysis on the control stability proves that the method is characterized in that: from the accessibility of the sliding-mode surface s (k) and the limited time convergence. And the accessibility of the sliding mode surface is expressed as | s (k +1) | less than or equal to | s (k) |, the motion track is attenuated, and the motion track gradually converges to the sliding mode surface. The finite time convergence is that after the finite time T, the motion track converges to the quasi-sliding mode bandwidth.

The following describes the technical solution of the present invention in detail by taking a NetCon network control brushless dc motor platform as an example and combining with the accompanying drawings.

As shown in FIG. 1, the invention relates to the establishment of a direct current motor system model containing multiple channels and external disturbance and the design of a second-order super-spiral sliding mode control algorithm. The method comprises the steps of multi-channel direct current motor system modeling considering external disturbance, external disturbance estimation, design of a second-order supercoiled sliding mode observer, design of a second-order supercoiled sliding mode controller and system stability verification.

Model building

The working state space equation of the direct current motor system can be expressed as follows:

wherein x (k) ═ x₁(k) x₂(k)]^T∈RⁿFor the state variables of the system, u (k) e R^mFor control input, y_i(k)∈R^qDenotes the output of the ith channel, w (k) ═ w₁(k) w₂(k)]^T∈R^wIs an external perturbation of the implant.

According to the random communication protocol, only one channel xi (k) e { y } is in each data transmission process₁(k),y₂(k),...,y_n(k) The data of the channel is selected to transmit the data, and the channel selection process is obeyedA markov jump.

Interference estimation

The external disturbance is estimated by a one-step time delay method,can be obtained by combining the formula (1)

Wherein the content of the first and second substances,representing the disturbance estimation error. Therefore, the temperature of the molten metal is controlled,

definitions Δ x (k) ═ x (k) — x (k-1) and Δ u (k) ═ u (k) — u (k) (k-1), then

Since x (k) and u (k) are bounded, soIs also bounded.

Observer design and certification

In order to restrain external disturbance and observe state variables, a sliding mode observer based on a second-order supercoiling algorithm is provided. In the state space equation, the error of the system state variable from the observed quantity is defined as follows:

wherein the content of the first and second substances,for observed state variables, assumptionsThen (1) can be written as:

then, the second-order supercoiled sliding-mode observer can be designed as

WhereinAnd z₂＝k₃sign(e₁(k))+k₄e₁(k) Two correction terms. Further, in conjunction with (5), system state errors may be written as

Where w (k) is bounded, k₁，k₂，k₃And k₄Is the gain that needs to be adjusted. Error of observation e₁(k +1) and e₂(k +1) eventually converges to 0, by which the estimated state variable can be considered equal to the actual state variable of the system, i.e. it is assumed thatThe stability of the observer is demonstrated below.

Given a suitable gain parameter k₁，k₂，k₃And k₄If a positive definite matrix X is present, X^T＞0，Q＝Q^T> 0 andso that the following inequality holds

The observer can be considered to be stable and the trajectory of the observation error will converge to an intra-sphere Y centered at the origin_r＝e:||e||²＜R_d，Indicating the radius of the sphere. Wherein the content of the first and second substances,

and (3) proving that: the following Lyapunov equation was introduced

V(k)＝e^T(k)Xe(k) (10)

Thus, the

ΔV(k+1)＝V(k+1)-V(k)＝e^T(k+1)Xe(k+1)-e^T(k)Xe(k) (11)

Introduce inequalityThen there is

Thus, (12) can be restated as:

thus, the system can be expressed as

Wherein the content of the first and second substances,m₂＝k₁k₃f₁₂，

according to the inequality introduced above, Q ═ Q, if any^T> 0 satisfyThe term in equation (15) can be further expressed as

Wherein

(16) The formula can be further deduced as

Namely, it isThus, it is possible to prevent the occurrence of,by recursion, can derive

The above equation shows that the observation error is convergent, passing through the finite step k,n is a finite number. And because ofThe trajectory indicating the observation error converges into the sphere.

Controller design and certification

The output error is defined as ∈ (k) ═ ζ (k) -y_r(k) (19)

Where ζ (k) represents the actual output speed of the DC motor system, y_r(k) Is the reference velocity.

The discrete sliding-mode function is defined as:

whereinAnd λ₁＞0，λ₂＞0，λ₃> 0 andis a gain parameter; τ is an accumulated variable, andthis parameter ensures that only the most recent ten data will be accumulated (if any), effectively reducing system instability due to residual errors.

According to the formula (20), s (k +1) can be derived as

When the sliding mode enters the equilibrium state s (k +1) ═ 0, thenCombining equations (19) and (20), the equivalent control input can be calculated as

The design control law is u (k) u_eq(k)+u_s(k) (23)

Wherein u is_s(k) That is, a nonlinear control input in the form of a supercoiled algorithm in which the specific form is

Wherein gamma is₁> 0 and gamma₂> 0 is the gain constant that needs to be designed.

Control stability demonstration

If the sliding mode motion satisfies the inequality | s (k +1) | is less than or equal to | s (k) |, then for the discrete system (1) and the equivalent control law (23), the motion track of the system can be converged into the quasi-sliding mode bandwidth after a finite step length.

And (3) proving that:

by substituting the expressions (22) and (23) into the expression (25), the compound

Further on

When s (k) is not less than 0,

substitution (27) to obtain

s(k+1)-s(k)＝(λ₁+λ₃)CBΔu_s(k) (29)

Thus, the

Thus, it is possible to obtain

Wherein λ is₁,λ₃,Γ₁≧ 0, so if there is s (k) -s (k-1) ≦ 0, then there is s (k +1) -s (k) ≦ 0.

And because of

Since e (0) ═ e (-1) ═ 0, so

Whereinλ₁,λ₃,y_r(1) Is more than or equal to 0, so that s (1) -s (0) is less than or equal to 0. Recursion can be carried out to obtain s (2) -s (1) is less than or equal to 0, s (3) -s (2) is less than or equal to 0, …, s (k +1) -s (k) is less than or equal to 0. When s (k) is less than or equal to 0, s (k +1) -s (k) is more than or equal to 0 in the same way. By combining the two conditions, | s (k +1) | is less than or equal to | s (k) |, and the accessibility verification of the sliding mode is finished.

When s (k) is not less than 0, s (1) -s (0) is not more than 0 as verified above. Thus s (1) -s (0) ≦ s (1)^1/2-s(0)^1/2Less than or equal to 0. Definition of(λ₁+λ₃)C_iBΓ₁＝δ，Is bounded, then

Can be obtained after pushing

And s (k +1) -s (k) is less than or equal to 0

Supposing that the step length is tau > 0, through a finite step k, the sliding mode motion enters a balanced state, and the convergence time isWhen s (k) is less than or equal to 0, the convergence time can be derived in the same wayThus, the authentication is complete.

Although illustrative embodiments of the present invention have been described above to facilitate the understanding of the present invention by those skilled in the art, it should be understood that the present invention is not limited to the scope of the embodiments, and various changes may be made apparent to those skilled in the art as long as they are within the spirit and scope of the present invention as defined and defined by the appended claims, and all matters of the invention which utilize the inventive concepts are protected.