Motor control technology based on fractional active disturbance rejection

For non-linear, strongly coupled, multivariable and naturally unstable motor control systems to realize the rapid return of the motor to the original equilibrium point without overshoot, the traditional PID controller algorithm is prone to defects such as large overshoot and long response time. The study proposes fractional order active disturbance rejection control for the motor self-balancing control method. A novel controller composed of a fractional order controller and an active disturbance rejection controller is designed and its control characteristics are analyzed and simulated. The experimental results show that compared with the traditional PID control, the controller has better dynamic performance, steady-state performance and anti-interference ability so that the motor can reach the equilibrium position quickly, stably and accurately. The illustrations of the experimental verification have been modified and explained.


Introduction
For nonlinear, strongly coupled, multivariable and naturally unstable motor control systems, an inverted pendulum motion model has also become an ideal platform for verifying various control algorithms, which has important theoretical significance.
At present, commonly used self-balancing algorithms include traditional control algorithms and intelligent control algorithms.Traditional control algorithms mainly use the PID [1] algorithm, state feedback and pole allocation method, etc., and mainly deal with known linear systems; The linearization of the pendulum model will lead to inaccurate models.The control effect cannot be optimal.Moreover, the dynamic response of the traditional control algorithm is not ideal and the antiinterference ability is poor.Intelligent control mainly adopts fuzzy control [2] and adaptive control [3], which can deal with nonlinear unknown systems.Also, it does not require the establishment of system mathematical models and improves dynamic responsiveness and anti-interference ability.
This article applies modern control theory active disturbance rejection control [4] technology and fractional order [5] to the inverted pendulum control system.By retaining the core concept of classic PID control, three processes are improved.Using fractional calculus, the orders of integer order calculus are replaced by fractions to improve the flexibility and robustness of the entire system, thereby eliminating the excessive dependence of integer order [6] PD controllers on the observation accuracy of extended state observers [7].At the same time, the improved control algorithm is simple, the software running cycle is short and the code portability is strong.It has the characteristics of fast To obtain the model of the inverted pendulum [8], firstly, the labels used in the force analysis and establishment of the equation are uniformly described, as shown in Table 1.Dynamic analysis based on the Newton-Euler method [9]: Force analysis on car body wheels: Force analysis of the car body: sin cos Combining the above equations, the nonlinear mathematical model of the inverted pendulum can be obtained: ( ) To simplify the model and locally linearize the equilibrium point, since the inverted pendulum mainly considers the balance performance, the state variable is taken as the only quantity and the parameters are brought into it.We can get: 26.4141 0.4468 The inverted pendulum car body uses a brushless DC motor [10] to provide electromagnetic torque.The equation of the electromagnetic torque is: Flux linkage equation: Voltage equation: In the formula,  are the stator voltage, current and flux linkage components transformed to the synchronous rotating dq coordinate system respectively.s R is the stator winding resistance; q L and d L are the stator inductance of the dq axis respectively. is the electrical angular velocity of the rotor.r  is the flux linkage produced by the permanent magnet.n p is the number of pole pairs of the motor rotor.p d dt = is the differential operator.For the surface-mounted rotor structure, dq L L L == .By using the vector control technology of 0 d i = , the torque of the motor can be obtained: The parameters of the motor are selected as rated speed 3500 min nr = ; moment of inertia   17) is put into Equation (10), considering the input disturbance The fractional order PD  controller is a generalized form of the traditional integer order PD.It contains differential order  , where  is a parameter that can vary continuously from 0 to 1.By continuously adjusting the changes in the differential order, the controller can achieve ideal control performance.During this process, the proportional feedback coefficient remains fixed.The control block diagram is shown in Figure 2.
( ) The transfer function is:  is the fractional order factor of the differential term, 01   .

ADRC design
Active disturbance rejection controller is mainly to establish linear extended states observers (LESO).The extended state of LESO is used to estimate the sum of the internal and external disturbances of the system in real-time, compensating in the control network, changing the loop into an integrator series structure and performing fractional order control.According to Equation 18, it can be seen that the objects controlled by the inverted pendulum system are the angle of the car body and its angular velocity.Controlling these two variables within an effective range is the result we want.The angle and angular velocity have a first-order derivative relationship.Therefore, combined with the motor equations, this system is a second-order system.The common second-order system state equation is constructed, as shown in Equation ( 21): Expanding the total disturbance f received by the system into a new state 3 x , the corresponding expanded state observer design is shown in Equation ( 22):  and 3  are adjusted so that states 1 z and 2 z of the extended state observer can track 1 x and 2 x respectively.3 z is the estimator of 3 x , which is the total disturbance.The parameter range satisfies the following relationship. is 3 to 5 times the bandwidth of the expected closed-loop system.Then the control law ( ) is generated which can simplify the original system to an integral series type.It is 0 yu  .0 u is the control quantity obtained by fractional order PD  .

Control model
The control system is the combination of PD  controller and active disturbance rejection as a feasible strategy.The essence of linear active disturbance rejection control technology is to design a linear state observer to observe the disturbance of the compensation system and simplify the control object into a series system to facilitate the feedback controller control.However, the linear active disturbance rejection control technology uses the proportional derivative controller as the feedback.The differential link of the integer order can improve the stability of the system, but it will amplify highfrequency noise and affect the control performance.Compared with the effect of integer order fractional order differential can weigh the effect of differential by adjusting fractional order  to vary from 0 to 1.The fractional differential has good attenuation for high-frequency noise signals, so the differential signal can be directly extracted from the feedback signal.When the error signal change rate changes, the fractional differential response curve does not change suddenly.This shows that the controller can respond slowly to sudden changes in system differential parameters, which is more conducive to controller control.Therefore, the system can achieve a strong robust control effect.The block diagram of the control system is shown in Figure 3.The input is a given position signal.After making a difference with the state variable obtained by the extended state observer, the control quantity 0 u is obtained after a PD  link.After subtracting the disturbance estimated by the state observer, the obtained control quantity u is used as q.The shaft voltage is input into the brushless DC motor.The brushless DC motor provides torque to keep the car in balance.
Fractional order active disturbance rejection control system model.
Adjusting the fractional differential order can make the system obtain better dynamics and stability.After observation, the fractional order is selected as 0.7 order, then the fractional differential tracker can be designed as: 1.7 10 10 2.4 Linearly extended state observer design: Equation ( 18) is changed into Equation ( 25 f represents system disturbance including model internal disturbance and external disturbance. Letting ( ) f be expanded into a state, the expanded state observer is established as: The nonlinear feedback rate is: In the equation, ( ) ( ) ( ) vt is the reference angle after the reference angle passes through the fractional order differential tracker.
Then the control law is: ( ) ( )

Experimental results
In the Matlab environment, an inverted pendulum control system is built according to the control model shown in Figure 3.It mainly includes the motor model, extended state observer model and fractional differential model.The control system is verified through modeling and simulation.Taking

 =
, the control effect is shown in Figure 4.Among them, the system balance target position is 0-degree position.The initial angle deviation starts from 2 degrees.Comparing the PD  and PID control effects, the PD  can return to the equilibrium position more quickly without overshooting.The simulation results prove that in response to the step signal, the existence of fractional differential order allows the system to pass the overshoot period more smoothly.The response speed is improved compared to traditional PD control.Therefore, this control system can better make the inverted pendulum reach the equilibrium point.By applying a load disturbance at the equilibrium position of the system, it can be measured that the PD  system can achieve position tracking more quickly without oscillation.The results are shown in Figure 5.It can be seen from the simulation results that when the vehicle body encounters unpredictable large load disturbances during operation, the control system has good automatic disturbance rejection performance.Through the estimation and integration of the disturbance error, the control amount of the motor is quickly compensated so that the motor can quickly return to the equilibrium position.At the same time, the effect of fractional differential prevents the motor from frequent overshoot oscillations.The results reflect that the system is an excellent active disturbance rejection control system.

Conclusions
This paper mainly aims at nonlinear, strongly coupled, multivariable and naturally unstable motor control systems.It establishes a fractional-order active disturbance rejection controller for this system.All motor disturbances are estimated through the expanded state observer.The fractional order is used as the feedback controller to eliminate the dependence of the traditional linear active disturbance rejection controller on the observation accuracy of the expanded state observer.This system is also an integral series system.After continuous accumulation of interference errors, the motor can respond quickly to eliminate error interference.The motor can run more smoothly.The experimental results prove that the control system has strong robustness and anti-interference performance.
In the future, this system will be beneficial to be applied to robot transportation, automated office spaces and small scooters used by people to travel in complex and changeable road conditions to help save energy and reduce emissions.

Figure 1 .
Figure 1.Schematic diagram of inverted pendulum and wheel force analysis diagram.
)etis the error value between the controller given and the actual value, which can be used as the input of the fractional order controller; ( ) Cs is the fractional order of the conduction function of the controller shown in the dashed box; ( ) Gs is a transfer function for the controlled object; P K , d K are proportional and differential gain;

Figure 4 .
Figure 4. Comparison chart of step response of fractional order and PID control system.

Table 1 .
Definition and value of parameters.