CN104142627A - Networked brushless direct current motor time-delay compensation and control method using active-disturbance-rejection control technology - Google Patents

Abstract

The invention discloses a networked brushless direct current motor time-delay compensation and control method using the active-disturbance-rejection control technology. The method comprises the steps of (1) establishing a brushless direct current motor control system model containing time-varying network-induced delay, wherein the brushless direct current motor control system is described as a discrete time linear time-varying system containing single-step input time lag, and then the uncertain dynamic state, caused by time-varying delay, of the system is partially described as the additive noise of the system; (2) designing an extended state observer which is used for estimating the uncertain dynamic state caused by time-varying delay as a part of total disturbance; (3) conducting compensation on the time-varying delay item in the brushless direct current motor control system. The method can be used for effectively conducting real-time estimation and compensation on the uncertain dynamic state caused by time-varying network-induced delay by means of the extended state observer, and can well restrain the uncertainty caused by time delay, internal and external disturbance of the system and model uncertainty.

Description

Networked brushless direct current motor time delay compensation and control method adopting active disturbance rejection control technology

Technical Field

The invention is applied to the field of networked motion control, relates to the control problem of a brushless direct current motor based on an industrial network, and particularly relates to an effective real-time control method for eliminating the influence of network induced time delay on the performance of a brushless direct current motor control system.

Background

With the development of power electronic technology and microelectronic technology, a new type of adjustable speed motor, i.e. a brushless dc motor, has been developed. Due to the advantages of low noise, high efficiency, simple structure, long service life, quick response, larger starting torque and the like, the magnetic-field-type magnetic-. With the continuous enlargement of industrial production scale and the continuous improvement of the safety requirement of the production process, the traditional motor control system can not meet the actual requirement more and more. The development of modern network control technology makes it possible to replace traditional control mode with networked control.

The networked control system is a closed-loop control system formed by connecting a sensor, a controller and an actuator by a communication network instead of a traditional point-to-point connection mode. Compared with the traditional control system, the remote control system has various advantages, such as remote control, reduction of system wiring, convenience in installation and maintenance, system information integration and sharing and the like. However, introducing a communication network into the control loop also brings new problems, and since the information transmission method using time division multiplexing is limited to the carrying capacity and limited bandwidth of the network, the situations such as collision and retransmission of information inevitably occur, so that a time delay is generated in the transmission process of information in the control system, and the time delay varies with the variation of the network load, and is time-varying and uncertain.

Network-induced delay is generally divided into long delay (greater than one system sampling period) and short delay (less than one system sampling period), wherein the frequency of the long delay is not high in an actual system, the long delay has a great influence on a motion control system, and the delay is generally reduced as much as possible in engineering by improving a network protocol and a structure, but the short delay is often unavoidable.

The invention mainly considers how to reduce or even eliminate the influence of network induced time delay on the performance of a brushless direct current motor control system, and at present, common processing methods comprise a robust control method, a Smith predictor compensation method, a time-drive-based artificial time delay prolonging method and the like. The robust control method does not need to accurately know the size of network delay, and the robust controller has good anti-interference capacity but high conservation. The Smith predictor method compensates for the time delay by using a prediction model, but the Smith predictor method has high requirement on the accuracy of a motor model, which is difficult to realize in practice. The time-driven actuator-based artificial delay method can convert time-varying delay into constant delay, facilitates the design of a controller, but can cause the control input of a system not to be updated in time, and reduces the control performance of the system.

Disclosure of Invention

In order to overcome the defects that the existing control method cannot accurately estimate the uncertain dynamics caused by the time-varying delay and has weak anti-interference capability, the invention designs the delay compensation and control strategy of the networked brushless direct current motor control system by adopting the active disturbance rejection technology, can effectively estimate and compensate the uncertain dynamics caused by the time-varying network induced delay in real time by using the extended state observer, and has strong inhibition capability on the uncertainty caused by the delay, the internal and external disturbance of the system and the model uncertainty.

The technical scheme adopted by the invention for solving the technical problems is as follows:

a networked brushless DC motor time delay compensation and control method adopting an active disturbance rejection control technology comprises the following steps:

step 1) establishing a brushless direct current motor control system model containing time-varying network induced time delay.

Considering the condition that the network induced delay is less than a sampling period, a networked brushless direct current motor control system is described as a discrete time linear time varying system with one-step input time lag, and further, an uncertain dynamic part of the system caused by the time varying delay is described as additive noise of the system, and the specific process comprises the following steps:

1.1) establishing a linear transfer function model of a brushless direct current motor control system

The brushless direct current motor control system is composed of a current loop and a rotating speed loop, the current loop and the rotating speed loop can be described by a first-order linear model, a second-order system model of the brushless direct current motor control system is obtained through the series connection of the current loop and the rotating speed loop, and the transfer function of the second-order system model is as follows:

$\frac{N\left(s\right)}{U\left(s\right)}=\frac{b}{s^2+\mathrm{as}+c}---\left(1\right)$

wherein, <math><mrow> <mi>a</mi> <mo>=</mo> <mfrac> <mi>r</mi> <mrow> <mi>L</mi> <mo>-</mo> <mi>M</mi> </mrow> </mfrac> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mfrac> <mrow> <mi>&pi;</mi> <msub> <mi>K</mi> <mi>e</mi> </msub> </mrow> <mrow> <mn>30</mn> <mi>J</mi> <mrow> <mo>(</mo> <mi>L</mi> <mo>-</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <mi>c</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msubsup> <mi>K</mi> <mi>e</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mi>J</mi> <mrow> <mo>(</mo> <mi>L</mi> <mo>-</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow></math> r is the resistance of the stator phase winding, L is the self inductance of the winding, M is the mutual inductance between the two phase windings, K_eIs the electromotive force coefficient, J is the rotational inertia of the motor, n(s) is the laplace transform of the motor speed, and u(s) is the laplace transform of the branch voltage that conducts the two phases. For the convenience of controller design, the transfer function model shown in equation (1) is converted into a state space model as follows:

wherein x is₁Is the rotational speed, x, of the DC motor₂The acceleration of the dc motor is u, which is a control quantity, i.e. the branch voltage for conducting two phases.

1.2) obtaining a motor control system model under the influence of time-varying network induced time delay

When data packets are transmitted in the network, there is a time delay from the sensor to the controller and from the controller to the actuatorAndrespectively representing the time delay experienced by the transmission of the measurement signal from the sensor to the controller and the time delay experienced by the transmission of the control quantity from the controller to the actuator, the total network-induced time delay of the control loop is thenBecause the time delay is less than a sampling period, the control input voltage u (t) of the direct current motor in one period is composed of two parts, one part is the control input voltage u (k-1) obtained by calculation in the previous period, and the other part is the control input voltage u (k-1) obtained by calculation in the current periodAn input voltage u (k) and having the form:

<math><mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>&le;</mo> <mi>t</mi> <mo><</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>&le;</mo> <mi>t</mi> <mo><</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <mi>T</mi> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow></math>

where T is the sampling period, T_kRepresenting the k-th sampling instant. Thus, discretized according to equations (2) and (3)The model of the brushless direct current motor control system with time-varying network induced time delay is as follows:

<math><mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>c</mi> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <mi>c</mi> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow></math>

e is to be^-aTAfter approximation by 1-aT, equation (4) can be converted to:

<math><mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <mo>(</mo> <mo>-</mo> <mrow> <mi>a</mi> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>c</mi> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <mi>c</mi> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow></math>

delay tau by time variation in equation (5)_kThe resulting time-varying dynamics use a new state variable x₃(k) Is shown, i.e. <math><mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>ax</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mi>b</mi> <mi>aT</mi> </mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mrow> <mi>T</mi> <mo>-</mo> <mi>&tau;</mi> </mrow> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mrow></math> And orderThe networked brushless dc motor control system model represented by equation (5) can thus be expanded to a third order system model as follows:

$\left\{\begin{array}{c}{x}_1(k+1)=x_1\left(k\right)+{\mathrm{Tx}}_2\left(k\right)\\ x_2(k+1)=x_2\left(k\right)+T(x_3\left(k\right)+\frac{b}{\mathrm{aT}}u\left(k\right))\\ x_3(k+1)=x_3\left(k\right)+\mathrm{Tw}\left(k\right)\end{array}\right.---\left(6\right)$

wherein x is₁(k+1)、x₂(k+1)、x₃(k +1) is the motor rotation speed x₁(k) Motor acceleration x₂(k) New expansion state quantity x₃(k) The value of the next sampling instant of (a);

step 2) designing an extended state observer for estimating uncertainty caused by network-induced time delay in a brushless direct current motor control system;

and 3) designing a networked brushless direct current motor active disturbance rejection controller with an extended state observer to realize real-time compensation of time-varying network induced time delay and real-time control of motor rotating speed.

Further, in the step 2), the extended state observer designing process includes:

2.1) design extended State observer

The extended state observer for estimating three state variables in a system (6) has the form:

<math><mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>fe</mi> <mo>=</mo> <mi>fal</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mi>&delta;</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>fe</mi> <mn>1</mn> <mo>=</mo> <mi>fal</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mn>0.25</mn> <mo>,</mo> <mi>&delta;</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&beta;</mi> <mn>01</mn> </msub> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&beta;</mi> <mn>02</mn> </msub> <mi>fe</mi> <mo>+</mo> <mfrac> <mi>b</mi> <mi>aT</mi> </mfrac> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mrow> <mo>-</mo> <mi>&beta;</mi> </mrow> <mn>03</mn> </msub> <mi>fe</mi> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow></math>

wherein, e (k)) Is the difference between the reference value of the motor speed and the estimated value of the actual speed, i.e. the error quantity of the motor speed, z₁(k) Is to the motor speed x₁(k) Estimate of (b), z₂(k) Is to the motor acceleration x₂(k) Estimate of (b), z₃(k) Is to the new expansion state quantity x₃(k) H is the integration step. fal (e (k),0.25, δ) is a nonlinear function, which is shown in formula (8). Delta, beta₀₁、β₀₂、β₀₃Is a set of parameters to be set, and beta is a set of parameters to be set according to the design principle of a high-gain state observer to ensure certain estimation accuracy₀₁、β₀₂、β₀₃A larger, generally larger, upper bound on noise or disturbances can be achieved.

<math><mrow> <mi>fal</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>&delta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mfrac> <mrow> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> <msup> <mi>&delta;</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>a</mi> </mrow> </msup> </mfrac> </mtd> <mtd> <mo>|</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>&le;</mo> <mi>&delta;</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>|</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mi>a</mi> </msup> <mi>sign</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>|</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>></mo> <mi>&delta;</mi> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow></math>

Wherein a is a power exponent, δ is the interval length of the linear segment, sign () is a sign function, and a specific expression is shown in formula (9).

$\mathrm{sign}\left(x\right)=\left\{\begin{array}{cc}1& x>0\\ 0& x=0\\ -1& x<0\end{array}\right.---\left(9\right)$

2.2) estimation of uncertainty caused by time-varying network-induced delay

The designed extended state observer can estimate the rotating speed of the brushless direct current motor, the acceleration of the motor and the newly extended uncertain quantity caused by network-induced time delay, and as can be seen from the formula (7), the designed extended state observer can estimate the uncertain dynamics containing time-varying time delay in the system as sum disturbance. The key and difficulty for compensating the network induced delay lies in accurately estimating the uncertain dynamics caused by the time-varying delay, and no effective result aiming at the problem exists at present.

Still further, in the step 3), a compensation process for a time-varying delay term in the networked brushless dc motor control system:

3.1) arranging the transition process. The process is that the motor is referenced to the rotating speed v, and a differential signal v of the approximate rotating speed of the rotating speed v is obtained through a tracking differentiator₂Simultaneously, a transition signal v of the rotating speed v can be obtained₁Smoothing the jump speed signal to prevent overshoot, and the following differentiator is given by the formula (10).

$\left\{\begin{array}{c}e\left(k\right)=v_1\left(k\right)-v\left(k\right)\\ \mathrm{fh}=\mathrm{fhan}(e\left(k\right),v_2\left(k\right),r,h_0)\\ v_1\left(k\right)=v_1\left(k\right)+{\mathrm{hv}}_2\left(k\right)\\ v_2\left(k\right)=v_2\left(k\right)+\mathrm{hfh}\end{array}\right.---\left(10\right)$

Wherein r is a fast tracking factor, h is an integration step length, h₀As filter factor, fhan (e (k), v₂(k),r,h₀) For the steepest control synthesis function, fhan (x)₁,x₂R, h) the specific expression is as follows:

<math><mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>d</mi> <mo>=</mo> <mi>rh</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>hd</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>hx</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>=</mo> <msqrt> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>8</mn> <mi>r</mi> <mo>|</mo> <mi>y</mi> <mo>|</mo> </msqrt> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>=</mo> <mi></mi> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <mi></mi> <mfrac> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>d</mi> <mo>)</mo> </mrow> <mn>2</mn> </mfrac> <mi>sign</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi></mi> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>y</mi> <mo>|</mo> <mo>></mo> <msub> <mi>d</mi> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>y</mi> <mi>h</mi> </mfrac> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>y</mi> <mo>|</mo> <mo>&le;</mo> <msub> <mi>d</mi> <mn>0</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mi>fhan</mi> <mo>=</mo> <mi></mi> <mo>-</mo> <mfenced open='{' close='' separators=' '> <mtable> <mtr> <mtd> <mi>rsign</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>a</mi> <mo>|</mo> <mo>></mo> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mfrac> <mi>a</mi> <mi>d</mi> </mfrac> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>a</mi> <mo>|</mo> <mo>&le;</mo> <mi>d</mi> </mtd> </mtr> </mtable> <mi></mi> </mfenced> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow></math>

3.2) estimating the uncertainty caused by the network-induced delay by using an extended state observer. Obtaining a new extended state quantity x by an extended state observer₃(k) Is estimated z₃(k) Amount of new dilation x₃(k) The dynamic state estimation method not only contains uncertain dynamics caused by time-varying time delay, but also contains internal and external disturbances, and the extended state observer estimates the dynamic state as total disturbance.

3.3) control System uncertainty Compensation and control Law setup due to time delayAnd (6) counting. This process yields two error quantities, i.e. e₁(k)＝v₁(k)-z₁(k) And e₂(k)＝v₂(k)-z₂(k) In that respect The control quantity u can be calculated through the nonlinear combination module₀(k) The calculation process is shown in formula (12).

$\left\{\begin{array}{c}{e}_1\left(k\right)=v_1\left(k\right)-z_1\left(k\right),e_2\left(k\right)=v_2\left(k\right)-z_2\left(k\right)\\ u_0\left(k\right)=-\mathrm{fhan}(e_1\left(k\right),e_2\left(k\right),r,h)\\ u\left(k\right)=\mathrm{aT}(u_0\left(k\right)-z_3\left(k\right))/b\end{array}\right.---\left(12\right)$

In order to compensate for uncertainties and internal and external disturbances in the system caused by time-varying delays, the control quantity u is obtained₀(k) Minus z₃(k) Obtaining new control quantities, i.e.The compensation process can offset all total disturbance terms containing time-varying time delay in the system, so that the system is converted into a pure integral linear system, and the influence of the time-varying time delay on the system performance is eliminated.

Compared with the prior art, the invention has the advantages that:

1. can accurately estimate the system uncertain dynamics caused by network induced time delay

The extended state observer designed by the invention can estimate the state quantity of the brushless direct current motor control system in real time, namely z in the system₁(k) For motor speed x₁(k) Estimate of (b), z₂(k) To motor acceleration x₂(k) Estimate of (b), z₃(k) Uncertainty-containing state quantities x for new expansions₃(k) Is estimated. Therefore, the designed extended state observer can estimate the uncertain dynamics caused by time-varying network induced time delay in the system as the sum disturbance, and the problem of accurately estimating the uncertain dynamics caused by time-varying time delay is effectively solved.

2. Compensation for uncertainty dynamics caused by time-varying network induced delay

Disturbance sum estimate z obtained using extended state observer₃(k) The control law shown in the formula (12) can be reconstructed, and the control law can be used for offsetting the total disturbance of all time-varying network induced delay dynamics in the system, so that the system is converted into a pure integral linear system, and the influence of the time-varying network induced delay on the system performance is eliminated.

3. Has strong anti-interference capability

z₃(k) The newly expanded system state quantity contains uncertain dynamics caused by time-varying delay and internal and external disturbances, and the expanded state observer estimates the dynamic state quantity as the total disturbance so as to compensateIn the compensation process, internal and external disturbances are compensated while the uncertain dynamics caused by time-varying network induced time delay are compensated, and the influence of the internal and external disturbances on the system performance is eliminated.

Drawings

Fig. 1 is a block diagram of a networked dc motor control system with time delay.

Fig. 2 is a signal timing diagram of a networked control system with time-varying short delays.

Fig. 3 is a block diagram of the structure of the active disturbance rejection controller.

Fig. 4 is a graph of motor speed tracking with time delay compensation.

FIG. 5 is a graph of motor speed tracking with the addition of external disturbances.

Detailed Description

In order to make the technical scheme and the design idea of the present invention clearer, the following detailed description is made with reference to the accompanying drawings.

Referring to fig. 1 to 5, a networked brushless dc motor delay compensation and control method using an active disturbance rejection control technology includes the following steps:

step 1) establishing a brushless direct current motor control system model containing time-varying network induced time delay.

Considering the condition that the network induced delay is less than a sampling period, a networked brushless direct current motor control system is described as a discrete time linear time varying system with one-step input time lag, and further, an uncertain dynamic part of the system caused by the time varying delay is described as additive noise of the system, and the specific process comprises the following steps:

1.1) establishing a linear transfer function model of a brushless direct current motor control system

The brushless direct current motor control system is composed of a current loop and a rotating speed loop, the current loop and the rotating speed loop can be described by a first-order linear model, a second-order system model of the brushless direct current motor control system is obtained through the series connection of the current loop and the rotating speed loop, and the transfer function of the second-order system model is as follows:

$\frac{N\left(s\right)}{U\left(s\right)}=\frac{b}{s^2+\mathrm{as}+c}---\left(13\right)$

wherein, <math><mrow> <mi>a</mi> <mo>=</mo> <mfrac> <mi>r</mi> <mrow> <mi>L</mi> <mo>-</mo> <mi>M</mi> </mrow> </mfrac> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mfrac> <mrow> <mi>&pi;</mi> <msub> <mi>K</mi> <mi>e</mi> </msub> </mrow> <mrow> <mn>30</mn> <mi>J</mi> <mrow> <mo>(</mo> <mi>L</mi> <mo>-</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <mi>c</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msubsup> <mi>K</mi> <mi>e</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mi>J</mi> <mrow> <mo>(</mo> <mi>L</mi> <mo>-</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow></math> r is the resistance of the stator phase winding, L is the self inductance of the winding, M is the mutual inductance between the two phase windings, K_eIs the electromotive force coefficient, J is the rotational inertia of the motor, n(s) is the laplace transform of the motor speed, and u(s) is the laplace transform of the branch voltage that conducts the two phases. For the convenience of controller design, the transfer function model shown in equation (1) is converted into a state space model as follows:

wherein x is₁Is the rotational speed, x, of the DC motor₂The acceleration of the dc motor is u, which is a control quantity, i.e. the branch voltage for conducting two phases.

1.2) obtaining a motor control system model under the influence of time-varying network induced time delay

When data packets are transmitted in the network, there is a time delay from the sensor to the controller and from the controller to the actuatorAndrespectively representing the time delay experienced by the transmission of the measurement signal from the sensor to the controller and the time delay experienced by the transmission of the control quantity from the controller to the actuator, the total network-induced time delay of the control loop is thenBecause the time delay is less than a sampling period, the control input voltage u (t) of the direct current motor in one period is composed of two parts, one part is the control input voltage u (k-1) calculated in the previous period, and the other part is the current periodA calculated control input voltage u (k) and having the form:

<math><mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>&le;</mo> <mi>t</mi> <mo><</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>&le;</mo> <mi>t</mi> <mo><</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <mi>T</mi> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow></math>

where T is the sampling period, T_kRepresenting the k-th sampling instant. Thus, according to the formulae (2) and(3) the discretized model of the brushless direct current motor control system with the time-varying network induced time delay is as follows:

<math><mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow></math>

e is to be^-aTAfter approximation by 1-aT, equation (4) can be converted to:

<math><mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>ax</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow></math>

delay tau by time variation in equation (5)_kThe resulting time-varying dynamics use a new state variable x₃(k) Is shown, i.e. <math><mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>ax</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mi>b</mi> <mi>aT</mi> </mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mrow></math> And orderThe networked brushless dc motor control system model represented by equation (5) can thus be expanded to a third order system model as follows:

$\left\{\begin{array}{c}{x}_1(k+1)=x_1\left(k\right)+T{x}_2\left(k\right)\\ x_2(k+1)=x_2\left(k\right)+T(x_3\left(k\right)+\frac{b}{\mathrm{aT}}u\left(k\right))\\ x_3(k+1)=x_3\left(k\right)+\mathrm{Tw}\left(k\right)\end{array}\right.---\left(18\right)$

wherein x is₁(k+1)、x₂(k+1)、x₃(k +1) is the motor rotation speed x₁(k) Motor acceleration x₂(k) New expansion state quantity x₃(k) The value of the next sampling instant of (a);

step 2) designing an extended state observer for estimating uncertainty caused by network-induced time delay in a brushless direct current motor control system;

and 3) designing a networked brushless direct current motor active disturbance rejection controller with an extended state observer to realize real-time compensation of time-varying network induced time delay and real-time control of motor rotating speed.

Further, in the step 2), the extended state observer designing process includes:

2.1) design extended State observer

The extended state observer for estimating three state variables in a system (6) has the form:

<math><mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>fe</mi> <mo>=</mo> <mi>fal</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mi>&delta;</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>fe</mi> <mn>1</mn> <mo>=</mo> <mi>fal</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mn>0.25</mn> <mo>,</mo> <mi>&delta;</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&beta;</mi> <mn>01</mn> </msub> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&beta;</mi> <mn>02</mn> </msub> <mi>fe</mi> <mo>+</mo> <mfrac> <mi>b</mi> <mi>aT</mi> </mfrac> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>&beta;</mi> <mn>03</mn> </msub> <mi>fe</mi> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow></math>

wherein e (k) is the difference between the reference value of the motor rotation speed and the estimated value of the actual rotation speed, i.e. the error amount of the motor rotation speed, z₁(k) Is to the motor speed x₁(k) Estimate of (b), z₂(k) Is to the motor acceleration x₂(k) Estimate of (b), z₃(k) Is to the new expansion state quantity x₃(k) H is the integration step. fal (e (k),0.25, delta) isThe non-linear function is specifically shown in formula (8). Delta, beta₀₁、β₀₂、β₀₃Is a set of parameters to be set, and beta is a set of parameters to be set according to the design principle of a high-gain state observer to ensure certain estimation accuracy₀₁、β₀₂、β₀₃A larger, generally larger, upper bound on noise or disturbances can be achieved.

<math><mrow> <mi>fal</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>&delta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mfrac> <mrow> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> <msup> <mi>&delta;</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>a</mi> </mrow> </msup> </mfrac> </mtd> <mtd> <mo>|</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>&le;</mo> <mi>&delta;</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>|</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mi>a</mi> </msup> <mi>sign</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>|</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>></mo> <mi>&delta;</mi> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow></math>

Wherein a is a power exponent, δ is the interval length of the linear segment, sign () is a sign function, and a specific expression is shown in formula (9).

$\mathrm{sign}\left(x\right)=\left\{\begin{array}{cc}1& x>0\\ 0& x=0\\ -1& x<0\end{array}\right.---\left(21\right)$

2.2) estimation of uncertainty caused by time-varying network-induced delay

The designed extended state observer can estimate the rotating speed of the brushless direct current motor, the acceleration of the motor and the newly extended uncertain quantity caused by network-induced time delay, and as can be seen from the formula (7), the designed extended state observer can estimate the uncertain dynamics containing time-varying time delay in the system as sum disturbance. The key and difficulty for compensating the network induced delay lies in accurately estimating the uncertain dynamics caused by the time-varying delay, and no effective result aiming at the problem exists at present.

Still further, in the step 3), a compensation process for a time-varying delay term in the networked brushless dc motor control system:

3.1) arranging the transition process. The process is that the motor is referenced to the rotating speed v, and a differential signal v of the approximate rotating speed of the rotating speed v is obtained through a tracking differentiator₂Simultaneously, a transition signal v of the rotating speed v can be obtained₁Smoothing the jump speed signal to prevent overshoot, and the following differentiator is given by the formula (10).

$\left\{\begin{array}{c}e\left(k\right)=v_1\left(k\right)-v\left(k\right)\\ \mathrm{fh}=\mathrm{fhan}(e\left(k\right),v_2\left(k\right),r,h_0)\\ v_1(k+1)=v_1\left(k\right)+h{v}_2\left(k\right)\\ v_2(k+1)=v_2\left(k\right)+\mathrm{hfh}\end{array}\right.---\left(22\right)$

Wherein r is a fast tracking factor, h is an integration step length, h₀As filter factor, fhan (e (k), v₂(k),r,h₀) For the steepest control synthesis function, fhan (x)₁,x₂R, h) the specific expression is as follows:

<math><mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>d</mi> <mo>=</mo> <mi>rh</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>hd</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>+</mo> <mi>h</mi> <msub> <mi>x</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>=</mo> <msqrt> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>8</mn> <mi>r</mi> <mo>|</mo> <mi>y</mi> <mo>|</mo> </msqrt> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi></mi> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>d</mi> <mo>)</mo> </mrow> <mn>2</mn> </mfrac> <mi>sign</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>y</mi> <mo>|</mo> <mo>></mo> <msub> <mi>d</mi> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>y</mi> <mi>h</mi> </mfrac> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>y</mi> <mo>|</mo> <mo>&le;</mo> <msub> <mi>d</mi> <mn>0</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mi>fhan</mi> <mo>=</mo> <mo>-</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>rsign</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>a</mi> <mo>|</mo> <mo>></mo> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mfrac> <mi>a</mi> <mi>d</mi> </mfrac> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>a</mi> <mo>|</mo> <mo>&le;</mo> <mi>d</mi> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow></math>

3.2) estimating the uncertainty caused by the network-induced delay by using an extended state observer. Obtaining a new extended state quantity x by an extended state observer₃(k) Is estimated z₃(k) Amount of new dilation x₃(k) The dynamic state estimation method not only contains uncertain dynamics caused by time-varying time delay, but also contains internal and external disturbances, and the extended state observer estimates the dynamic state as total disturbance.

3.3) control system uncertainty compensation caused by time delay and control law design. This process yields two error quantities, i.e. e₁(k)＝v₁(k)-z₁(k) And e₂(k)＝v₂(k)-z₂(k) In that respect The control quantity u can be calculated through the nonlinear combination module₀(k) The calculation process is shown in formula (12).

$\left\{\begin{array}{c}{e}_1\left(k\right)=v_1\left(k\right)-z_1\left(k\right),e_2\left(k\right)=v_2\left(k\right)-z_2\left(k\right)\\ u_0\left(k\right)=-\mathrm{fhan}(e_1\left(k\right),e_2\left(k\right),r,h)\\ u\left(k\right)=\mathrm{aT}(u_0\left(k\right)-z_3\left(k\right))/b\end{array}\right.---\left(24\right)$

In order to compensate for uncertainties and internal and external disturbances in the system caused by time-varying delays, the control quantity u is obtained₀(k) Minus z₃(k) Obtaining new control quantities, i.e.The compensation process can offset all total disturbance terms containing time-varying delay in the system, so that the system is converted into a pure integral linear system, and the time-varying delay is eliminated from influencing the system performanceThe influence of (c).

As shown in figure 1, due to the introduction of the network, the data has time delay in the transmission process, mainly including the time delay from the sensor to the controllerAnd time delay between controller to actuatorControlling the total delay of the loopThe sensor node is driven by time, and the output rotating speed of the direct current motor is sampled at a fixed sampling period T. The controller node and the actuator (brushless DC motor) node are event driven, and when data reaches the controller, the control quantity is calculated and transmitted to the brushless DC motor system.

As shown in fig. 2, when data is transmitted through a wireless network, there is a time delay from the sensor to the controller and from the controller to the actuator, and when the brushless dc motor outputs data, since the sensor is time-driven, the rotation speed of the brushless dc motor is sampled by a period T through the sensor, so as to obtain a discrete rotation speed value sequence. When the sampled rotating speed data reach the controller, a certain time delay exists between the rotating speed data and the controller, and the time delay is time-varying and uncertain, but the size of the time delay is limited within a sampling period; when the controller calculates the control quantity and transmits the control quantity data to the brushless direct current motor, time delay still exists between the control quantity data and the brushless direct current motor, and the time delay is time-varying and uncertain; when the data reaches the brushless DC motor, the time delay of the data is the sum of the two time delays, and the sum of the time delays is limited within a sampling period, so that the control quantity applied to the object is actually formed by two parts, wherein one part is the control quantity calculated in the previous period, and the other part is the control quantity calculated in the current period.

As shown in fig. 3, studied in this patentThe object is a brushless DC motor, when testing the effectiveness of the active disturbance rejection method, a reference value of the motor speed can be given, and two quantities are obtained after a scheduled transition process, wherein one quantity is a transition signal v of the reference value of the motor speed₁(k) The other is a differential signal v of a reference value of the motor speed₂(k) In that respect The rotating speed value and the control quantity output by the motor are used as the input of the extended state observer, so that the extended state observer can respectively measure the rotating speed x of the motor₁(k) Motor acceleration x₂(k) And the sum of disturbances x₃(k) Three state quantities are estimated, and the corresponding three estimated values are z₁(k)、z₂(k)、z₃(k) In that respect From this the error of the motor speed and the differential of the error, i.e. e, can be calculated₁(k)＝v₁(k)-z₁(k) And e₂(k)＝v₂(k)-z₂(k) In that respect The control quantity u can be calculated by nonlinear combination of the two obtained error signals as shown in formula (12)₀(k) In order to eliminate the dynamics of uncertainty caused by time-varying delay in the system, the control quantity u is obtained₀(k) Minus z₃(k) Obtain new control value voltageWhen the control quantity is applied to the object, all uncertain dynamics caused by time-varying time delay in the system can be counteracted, so that the system is converted into a pure integral linear system.

As shown in fig. 4, it can be seen from the figure that, by applying the active disturbance rejection control method, even if there is a time-varying network-induced delay, the output rotation speed of the brushless dc motor control system still has a fast response time, and the tracking curve is stable without overshoot and steady-state error, which indicates that the uncertain dynamics generated by the network-induced delay are effectively compensated, and the motor control system is basically not affected by the time-varying delay on its performance.

As shown in fig. 5, it can be seen from the graph that even in the presence of external disturbance, the motor rotation speed can still quickly track the given reference value, no obvious jitter occurs after the reference value is reached, and uncertain dynamics caused by time-varying network induced delay can still be effectively compensated in the presence of external disturbance.

Claims (3)

1. A networked brushless direct current motor time delay compensation and control method adopting an active disturbance rejection control technology is characterized in that: the method comprises the following steps:

step 1) establishing a brushless direct current motor control system model containing time-varying network induced time delay, describing a networked brushless direct current motor control system as a discrete time linear time varying system with one-step input time delay, and further describing a system uncertain dynamic part caused by the time-varying time delay as additive noise of the system, wherein the method comprises the following processes:

1.1) establishing a linear transfer function model of a brushless direct current motor control system

The transfer function of the second-order system model of the brushless direct current motor control system is as follows:

$\frac{N\left(s\right)}{U\left(s\right)}=\frac{b}{s^2+\mathrm{as}+c}---\left(1\right)$

wherein, <math> <mrow> <mi>a</mi> <mo>=</mo> <mfrac> <mi>r</mi> <mrow> <mi>L</mi> <mo>-</mo> <mi>M</mi> </mrow> </mfrac> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mfrac> <mrow> <mi>&pi;</mi> <msub> <mi>K</mi> <mi>e</mi> </msub> </mrow> <mrow> <mn>30</mn> <mi>J</mi> <mrow> <mo>(</mo> <mi>L</mi> <mo>-</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <mi>c</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msubsup> <mi>K</mi> <mi>e</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mi>J</mi> <mrow> <mo>(</mo> <mi>L</mi> <mo>-</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </math> r is the resistance of the stator phase winding, L is the self inductance of the winding, M is the mutual inductance between the two phase windings, K_eIs an electromotive force coefficient, J is the rotational inertia of the motor, N(s) is the Laplace transform of the motor rotation speed, and U(s) is the Laplace transform of the branch voltage for conducting two phases; converting the transfer function model shown in the formula (1) into a state space model as follows:

wherein x is₁Is the rotational speed, x, of the DC motor₂The acceleration of the direct current motor is used, u is a control quantity, namely, the branch voltage of two phases is conducted;

1.2) obtaining a motor control system model under the influence of time-varying network induced time delay

When data packets are transmitted in the network, there is a time delay from the sensor to the controller and from the controller to the actuatorAndrespectively representing the time delay experienced by the transmission of the measurement signal from the sensor to the controller and the time delay experienced by the transmission of the control quantity from the controller to the actuator, the total network-induced time delay of the control loop is thenBecause the time delay is less than one sampling period, the control input voltage u (t) of the direct current motor in one period is composed of two parts, one part is the control input voltage u (k-1) calculated in the previous period, the other part is the control input voltage u (k) calculated in the current period, and the control input voltage u (k) has the following form:

<math> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>&le;</mo> <mn>1</mn> <mo>&lt;</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>&le;</mo> <mi>t</mi> <mo>&lt;</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <mi>T</mi> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </math>

where T is the sampling period, T_kRepresents the kth sampling instant; therefore, according to the equations (2) and (3), the discretized model of the brushless direct current motor control system containing the time-varying network induced time delay is as follows:

<math> <mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> </math>

e is to be^-aTAfter approximation by 1-aT, equation (4) is converted to:

<math> <mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>ax</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </math>

delay tau by time variation in equation (5)_kThe resulting time-varying dynamics use a new state variable x₃(k) Is shown, i.e. <math> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>ax</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mi>b</mi> <mi>aT</mi> </mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mi>aT</mi> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>aT</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>bu</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>cx</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </math> And make an orderThe networked brushless dc motor control system model represented by equation (5) is thus expanded to a third order system model as follows:

$\left\{\begin{array}{c}{x}_1(k+1)=x_1\left(k\right)+{\mathrm{Tx}}_2\left(k\right)\\ x_2(k+1)=x_2\left(k\right)+T(x_3\left(k\right)+\frac{b}{\mathrm{aT}}u\left(k\right))\\ x_3(k+1)=x_3\left(k\right)+\mathrm{Tw}\left(k\right)\end{array}\right.---\left(6\right)$

wherein x is₁(k+1)、x₂(k+1)、x₃(k +1) is the motor rotation speed x₁(k) Motor acceleration x₂(k) New expansion state quantity x₃(k) The value of the next sampling instant of (a);

step 2) designing an extended state observer for estimating uncertainty caused by network-induced time delay in a brushless direct current motor control system;

and 3) designing a networked brushless direct current motor active disturbance rejection controller with an extended state observer to realize real-time compensation of time-varying network induced time delay and real-time control of motor rotating speed.

2. The networked brushless direct current motor delay compensation and control method adopting the active disturbance rejection control technology according to claim 1, wherein in the step 2), the extended state observer designing process includes:

2.1) design extended State observer

The extended state observer for estimating three state variables in a system (6) has the form:

<math> <mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>fe</mi> <mo>=</mo> <mi>fal</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mi>&delta;</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>fe</mi> <mn>1</mn> <mo>=</mo> <mi>fal</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mn>0.25</mn> <mo>,</mo> <mi>&delta;</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&beta;</mi> <mn>01</mn> </msub> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&beta;</mi> <mn>02</mn> </msub> <mi>fe</mi> <mo>+</mo> <mfrac> <mi>b</mi> <mi>aT</mi> </mfrac> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>h</mi> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>&beta;</mi> <mn>03</mn> </msub> <mi>fe</mi> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein e (k) is the difference between the reference value of the motor rotation speed and the estimated value of the actual rotation speed, i.e. the error of the motor rotation speed, z₁(k) Is to the motor speed x₁(k) Estimate of (b), z₂(k) Is to the motor acceleration x₂(k) Estimate of (b), z₃(k) Is to the new expansion state quantity x₃(k) H is the integration step; fal (e (k),0.25, delta) is a nonlinear function, specifically as shown in formula (8), delta, beta₀₁、β₀₂、β₀₃A group of parameters to be set;

<math> <mrow> <mi>fal</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>&delta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mfrac> <mrow> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> <msup> <mi>&delta;</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>a</mi> </mrow> </msup> </mfrac> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>&le;</mo> <mi>&delta;</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>|</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mi>a</mi> </msup> <mi>sign</mi> <mrow> <mo>(</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>e</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>></mo> <mi>&delta;</mi> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein a is a power exponent, δ is an interval length of a linear segment, sign () is a sign function, and a specific expression is shown as formula (9):

$\mathrm{sign}\left(x\right)=\left\{\begin{array}{cc}1& x>0\\ 0& x=0\\ -1& x<0\end{array}\right.---\left(9\right)$

2.2) estimation of uncertainty caused by time-varying network-induced delay

The designed extended state observer can estimate the rotating speed and the acceleration of the brushless direct current motor and the newly expanded uncertain quantity caused by network-induced time delay, and as can be seen from the formula (7), the designed extended state observer can estimate the uncertain dynamics containing time-varying time delay in the system as sum disturbance; the key and difficulty in compensating the network induced delay lies in accurately estimating the uncertain dynamics caused by the time-varying delay, and the uncertain dynamics caused by the time-varying delay can be estimated as a part of the total disturbance by adopting the extended state observer.

3. The networked brushless direct current motor delay compensation and control method adopting the active disturbance rejection control technology according to claims 1 and 2, wherein in the step 3), the compensation process for the time-varying delay term in the networked brushless direct current motor control system is as follows:

3.1) arranging a transition process: the process is that the motor is referenced to the rotating speed v, and a differential signal v of the approximate rotating speed of the rotating speed v is obtained through a tracking differentiator₂While obtaining a transition signal v of the rotating speed v₁Smoothing the jump speed signal to prevent overshoot, wherein the following differentiator is given by the formula (10):

$\left\{\begin{array}{c}e\left(k\right)=v_1\left(k\right)-v\left(k\right)\\ \mathrm{fh}=\mathrm{fhan}(e\left(k\right),v_2\left(k\right),r,h_0)\\ v_1(k+1)=v_1\left(k\right)+{\mathrm{hv}}_2\left(k\right)\\ v_2(k+1)=v_2\left(k\right)+\mathrm{hfh}\end{array}\right.---\left(10\right)$

wherein r is a fast tracking factor, h is an integration step length, h₀As filter factor, fhan (e (k), v₂(k),r,h₀) For the steepest control synthesis function, fhan (x)₁,x₂R, h) the specific expression is as follows:

<math> <mrow> <mfenced open='{' close='-'> <mtable> <mtr> <mtd> <mi>d</mi> <mo>=</mo> <mi>rh</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>hd</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>+</mo> <mi>h</mi> <msub> <mi>x</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>=</mo> <msqrt> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>8</mn> <mi>r</mi> <mo>|</mo> <mi>y</mi> <mo>|</mo> </msqrt> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>d</mi> <mo>)</mo> </mrow> <mn>2</mn> </mfrac> <mi>sign</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>y</mi> <mo>|</mo> <mo>></mo> <msub> <mi>d</mi> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>y</mi> <mi>h</mi> </mfrac> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>y</mi> <mo>|</mo> <mi></mi> <mo>&le;</mo> <msub> <mi>d</mi> <mn>0</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mi>fhan</mi> <mo>=</mo> <mo>-</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mi>rsign</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>a</mi> <mo>|</mo> <mo>></mo> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mfrac> <mi>a</mi> <mi>d</mi> </mfrac> <mo>,</mo> </mtd> <mtd> <mo>|</mo> <mi>a</mi> <mo>|</mo> <mo>&le;</mo> <mi>d</mi> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow> </math>

3.2) estimating uncertainty caused by network induced delay by using an extended state observer; obtaining a new extended state quantity x by an extended state observer₃(k) Is estimated z₃(k)，x₃(k) The system comprises uncertain dynamics caused by time-varying time delay and internal and external disturbances, and an extended state observer estimates the dynamic disturbance as a sum disturbance;

3.3) control system uncertainty compensation and control law design caused by time delay, and two error quantities are obtained in the process, namely e₁(k)＝v₁(k)-z₁(k) And e₂(k)＝v₂(k)-z₂(k) The control quantity u can be calculated through nonlinear combination₀(k) The calculation process is shown as formula (12):

$\left\{\begin{array}{c}{e}_1\left(k\right)=v_1\left(k\right)-z_1\left(k\right){,e}_2\left(k\right)=v_2\left(k\right)-z_2\left(k\right)\\ u_0\left(k\right)=-\mathrm{fhan}(e_1\left(k\right),e_2\left(k\right),r,h)\\ u\left(k\right)=\mathrm{aT}(u_0\left(k\right)-z_3\left(k\right))/b\end{array}\right.---\left(12\right)$

at the obtained control quantity u₀(k) Minus z₃(k) Obtaining new control quantities, i.e.The compensation process can offset all the total disturbance terms containing time-varying time delay in the system, so that the system is converted into a pure integral linear system.