Application research on improved genetic algorithm and active disturbance rejection control on quadcopters

Abstract

For nonlinear, strongly coupled, underdriven quadcopters in the context of modeling complexity and demanding performance requirements for the controller, this paper proposes a strategy based on an improved genetic algorithm to optimize the active disturbance rejection control (ADRC) controller. To make the quadcopter continue to fly stably in a complex environment, the dynamics model of the quadcopter was firstly established, the mathematical model was simplified according to the real world, and the ADRC controller of the quadcopter was designed. Given a large number of ADRC controller parameters, the difficulty of manual tuning and obtaining the optimal control effect, and the shortcomings of the genetic algorithm in solving the problem of local optimal and precocious convergence, a control strategy based on improved genetic algorithm to optimize ADRC’s parameters is proposed to improve the genetic diversity in the population and enhance the adaptability of individuals to the environment, ITAE (Integral-of-Time-multiple Absolute Error) evaluation index is selected as the fitness value. Finally, the model of the control system is built according to the real aircraft. The application results prove that the altitude, attitude of the quadcopter are controlled stably, and it is verified that the control strategy based on the improved genetic algorithm optimizing ADRC has faster rapidity, stronger tracking performance, and robustness in altitude, attitude control of the quadcopter, which has greater practical application value.

Introduction

Due to their low cost and high mobility, quadcopters have been widely used in many applications in recent years, such as radar localization, forest detection, geological exploration, search and rescue, aerial photography, and ground object tracking.^1,2 The quadcopter flight control system acts as the vehicle’s control hub, because the airframe is nonlinear, strongly coupled, and under-driven, it is challenging to accurately mathematically model it and is susceptible to outside interference. ³ As a result, how to efficiently control and optimize the quadcopter’s design is a hot topic and challenging area of research in the current global multi-rotor flight control field. ⁴

At this stage, there are certain research results in the altitude, attitude control of quadcopters. For these circumstances, a cascaded PID controller is proposed to solve the given trajectory tracking task of the quadcopters. ⁵ Alkamachi and Erçelebi ⁶ considered both the fast response and stability of the system in the objective function, and used the genetic algorithm to optimize the parameters of the proposed cascaded PID controller, which proves the effectiveness of the proposed strategy in achieving the best performance in tracking error, settling time, and peak time. Sarhan and Qin ⁷ proposed a robust adaptive PID controller combining fuzzy method and neural network to improve the processing ability of external wind disturbances and quadcopters parametric uncertainty. The PID control strategy is to design the corresponding PID control strategy separately after independent multiple channels of the control system, which has the advantages of simple principle, mature control technology and easy to realize, but it has the shortcomings such as poor portability and weak stability, which leads to the difficulty of obtaining the ideal control effect of the quadrotor. The quadcopter controller based on Backstepping decomposes the system into subsystems that do not exceed the order of the system, and then designs the Lyapunov function for each subsystem to obtain the control rate, which has the advantages of low overshoot and fast adjustment. Liu et al. ⁸ proposed a robust cascaded trajectory tracking controller based on the backstepping method and robust compensation theory, which uses adaptive sliding mode control to deal with the time delay. The proposed control approach effectively reduces the influence of uncertainty disturbance and time delay. Vahdanipour and Khodabandeh ⁹ proposed a fractional sliding mode controller based on the backstepping method to attenuate the effects of wind disturbances and changes in load and inertial momentum. Zhang et al. ¹⁰ proposed a predictive sliding mode cascade control method to solve the modeling uncertainty and external disturbance problem of cross-domain motion of coaxial hybrid aerial underwater vehicle (HAUV). In addition, some scholars have focused on the fault-tolerant formation control of a group of quadrotors.^11,12 A variety of control strategies for the flight motion system of quadrotors. Although the linear control strategies such as PID and LQR are simple, it is difficult to meet the requirements of interference sensitivity and non-equilibrium point stability, and the control effect is not good as expected. Dynamic inverse control and sliding mode control are nonlinear control strategies. Among them, the effect of dynamic inverse control is highly dependent on the mathematical model of the controlled object, and the adaptability to the uncertainty of the model parameters is poor. Although sliding mode control does not rely on accurate mathematical models, it can cause problems with the jitter of the system. ¹³ Therefore, for the flight motion control of quadrotors, it is necessary to adopt a control method that can effectively suppress uncertainty and external interference. ADRC controller is a robust control method for solving uncertainty, which estimates and compensates for external disturbance, is independent of the model of the controlled object, and is not sensitive to changes in system parameters. As one of the robust control methods used to address uncertainty, the ADRC was first proposed in 1998.^14,15 Based on the disturbance observer, a boundary anti-disturbance control strategy was developed,^16–18 which effectively suppressed the vibration and improved the accuracy of the manipulator system. To improve the immunity of the vehicle, scholars have started to use ADRC technique to design rotor controllers. Yang et al. ¹⁹ proposed a set of attitude solution algorithm based on ADRC control, which solves the strong coupling of the vehicle and the error problem generated by modeling, and has strong immunity and high control efficiency under the premise of fast response and no overshoot. Ahmed et al. ²⁰ developed a quadcopter tracking control scheme combining standard sliding mode control and disturbance observer, which introduced a matrix related to disturbance in the system model, considered matching and mismatched disturbance, and effectively improved the anti-disturbance performance of the system. Wu et al. ²¹ proposed an ADRC control strategy based on PSO algorithm to solve the problem of difficulty in setting ADRC parameters. This strategy improves the disturbance rejection ability and the control accuracy of aircraft in dynamic environment. An ADRC parameter tuning method based on genetic algorithm is proposed^22,23 and experimental results show that the strategy effectively improves the stability accuracy and robustness of the system in multi-source disturbance environment. In addition, Abadi et al. ²⁴ designed a feedback controller in the ADRC controller to eliminate unknown external disturbances and ensure accurate trajectory tracking in the quadcopter system. In order to obtain the best control effect and solve the problem of difficult tuning, this paper proposes a strategy to improve the genetic algorithm to optimize the parameters of the ADRC controller, which not only improves the altitude, attitude control accuracy of the quadcopter, but also solves the problem that the traditional genetic algorithm is easy to fall into the local optimization.

Compared with the existing research results, the main contributions of this paper are threefold: (1) when the quadcopter encounters unknown disturbances during flight, the trajectory tracking performance is inaccurate. Therefore, we designed a second-order nonlinear ADRC controller. (2) In view of the difficulty of parameter tuning of the ADRC controller, we propose an improved genetic algorithm, which solves the problem that the traditional algorithm is easy to fall into local optimization and premature convergence to a certain extent, and can improve the control accuracy while improving the system immunity performance. (3) The ADRC strategy based on the improved genetic algorithm has been applied and verified in the tracking and disturbance rejection experiments of quadcopters, showing excellent tracking performance and disturbance rejection performance, which proves the effectiveness of the proposed strategy.

This article is organized as follows. In section 2, a dynamic model of the quadcopter is established. In section 3, a second-order nonlinear ADRC controller for altitude, attitude control of quadcopters is developed. In section 4, an improved genetic algorithm is proposed and applied to the parameter tuning of the ADRC controller. The application validation and discussion of ADRC control strategy based on improved genetic algorithm for quadcopters is carried out in section 5 followed by a conclusion in section 6.

Dynamic modeling

To realize the position attitude control of the quadcopter, it is necessary to establish the mathematical model for each flight state first. The quadcopter is a device with four independently controlled motors driving corresponding propellers to provide all the power of the vehicle, four rigid cantilevers and four propellers in a “Cruciform” or “X” shape. ⁵ In this paper, we choose the “Cruciform” structure to build the vehicle dynamics model. To facilitate the analysis of the attitude, position, velocity and acceleration of the quadrotor in space, the inertial OXYZ and airframe coordinate systems O_BX_BY_BZ_B are established as shown in Figure 1.

Figure 1.

Inertial coordinate system and airframe coordinate system: (a) inertial coordinate system and (b) airframe coordinate system.

The starting position of the quadcopter is the coordinate origin O of the inertial coordinate system, the pointing of the nose is the positive direction of the X-axis, the direction of the horizontal movement of the vehicle to the left is the positive direction of the Y-axis, and the vertical movement of the vehicle is the positive direction of the Z-axis; the body coordinate system O_BX_BY_BZ_B is a coordinate system fixed on the body of the vehicle that remains unchanged: the direction from the center point to rotor 1 is the positive direction of the X_B-axis, the direction from the center point to rotor 2 is the positive direction of the Y_B-axis, the positive direction of the Z_B-axis is perpendicular to the O_BX_BY_B plane and is determined by the right-hand rule. The spatial description of the three attitude angles determined in the airframe coordinate system is shown in Figure 2.

Figure 2.

Schematic diagram of attitude angle and attitude motion: (a) rolling motion, (b) pitching motion, and (c) yawing motion.

The quadcopter is a 6-degree of freedom system with 4 inputs and 6 outputs. The inputs for a quadrotor are four rotor speeds, and the outputs are 3 attitude angles and 3 positions in 3D coordinate system. The motion of quadcopter includes: vertical flight, pitch, roll and yaw motion, according to the division of its motion, the corresponding four basic control methods are: vertical flight control, pitch control, roll control and yaw control.

Establishing a reasonably accurate dynamics model of the quadcopter is the primary prerequisite for the attitude control of the flight position. Therefore, it is usually necessary to make some reasonable assumptions about the vehicle: (1) the gravitational acceleration is constant during the flight and the four motors of the vehicle are identical; (2) the mass of the quadcopter remains constant during the flight; (3) the quadcopter is considered as a rigid body with constant size and shape during the motion and force; 4)the quadcopter is “dec” structure is absolutely symmetrical, the mass distribution is uniform, the center of gravity of the body and the far point of the body coordinate system coincide, and the center of mass is located in the center of the vehicle. Based on the above assumptions, the nonlinear dynamic model of the quadcopter is^4,6:

$$\{\begin{array}{c}\stackrel{··}{x}=\frac{(\sin \psi \sin \varphi +\cos \psi \sin \theta \cos \varphi )u_1}{m}\hfill \\ \stackrel{··}{y}=\frac{(\sin \psi \sin \theta \cos \varphi -\cos \psi \sin \varphi )u_1}{m}\hfill \\ \stackrel{··}{z}=\frac{\left(\cos \varphi \cos \theta \right)u_1}{m}-g\hfill \\ \stackrel{··}{\varphi }=\frac{u_2-\stackrel{·}{\theta }\stackrel{·}{\psi }(I_{\mathrm{zz}}-I_{\mathrm{yy}})}{I_{\mathrm{xx}}}\hfill \\ \stackrel{··}{\theta }=\frac{u_3-\stackrel{·}{\varphi }\stackrel{·}{\psi }(I_{\mathrm{xx}}-I_{\mathrm{zz}})}{I_{\mathrm{yy}}}\hfill \\ \stackrel{··}{\psi }=\frac{u_4-\stackrel{·}{\varphi }\stackrel{·}{\theta }(I_{\mathrm{yy}}-I_{\mathrm{xx}})}{I_{\mathrm{zz}}}\hfill \end{array}$$

(1)

Where: $\varphi$ , $\theta$ , $\psi$ are the angle of traverse, pitch and yaw respectively; $I_{\mathrm{xx}}$ , $I_{\mathrm{yy}}$ , $I_{\mathrm{zz}}$ are the rotational inertia around X, Y and Z axes respectively; m is the mass of the vehicle; g is the gravitational acceleration of the location; $u_1~u_4$ is the control virtual volume of the altitude, traverse, pitch and yaw channels respectively, whose expressions are:

$$\{\begin{array}{c}{u}_1=F_1+F_2+F_3+F_4=c_t{\mathrm{\Omega }}_1^2+c_t{\mathrm{\Omega }}_2^2+c_t{\mathrm{\Omega }}_3^2+c_t{\mathrm{\Omega }}_4^2\hfill \\ u_2=M_h=l(F_4-F_2)=l(c_t{\mathrm{\Omega }}_4^2-c_t{\mathrm{\Omega }}_2^2)\hfill \\ u_3=M_f=l(F_3-F_1)=l(c_t{\mathrm{\Omega }}_3^2-c_t{\mathrm{\Omega }}_1^2)\hfill \\ u_4=M_p=c_{x}l(F_2+F_4-F_1-F_3)\hfill \\ =c_{x}l(c_t{\mathrm{\Omega }}_2^2+c_t{\mathrm{\Omega }}_4^2-c_t{\mathrm{\Omega }}_1^2-c_t{\mathrm{\Omega }}_3^2)\hfill \end{array}$$

(2)

Where: $M_h$ , $M_f$ , $M_p$ are the moments of roll, pitch and yaw motion respectively; ${\mathrm{\Omega }}_1~{\mathrm{\Omega }}_4$ , $F_1~F_4$ correspond to the rotational speed and the thrust or lift generated by the four propellers of the vehicle; l is the distance from the center axis of the vehicle to the center of the propellers; $c_t$ , $c_x$ are the lift and drag coefficients of the propellers respectively.

ADRC controller

Controller structure

The control quantity of the second-order ADRC controller takes the form of u = (u₀−z₃)/b₀. The controller input is v, the output is u, and the feedback is the controlled object output y. The tracking differentiator TD, the expansive state observer ESO, and the nonlinear state error feedback NLSEF are described in (3), (4), and (5). ²²

$$\{\begin{array}{c}\begin{array}{c}\mathrm{fh}=\mathrm{fhan}(v_1-v,v_2,r_0,h)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{v}}_1=v_2}\end{array}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{v}}_2=\mathrm{fh}}\end{array}$$

(3)

$$\{\begin{array}{c}e=z_1-y\hfill \\ {\stackrel{·}{z}}_1=z_2-{\beta }_{01}e\hfill \\ {\stackrel{·}{z}}_2=z_3-{\beta }_{02}\mathrm{fal}(e,{\alpha }_{01},\delta )+b_{0}u\hfill \\ {\stackrel{·}{z}}_3=-{\beta }_{03}\mathrm{fal}(e,{\alpha }_{02},\delta )\hfill \end{array}$$

(4)

$$u_0={\beta }_1\mathrm{fal}(e_1,{\alpha }_1,\delta )+{\beta }_2\mathrm{fal}(e_2,{\alpha }_2,\delta )$$

(5)

In (3), the $\mathrm{fhan}(·)$ function is the integrated function of the fastest control of the discrete system, and its specific algorithm is shown in (6); in (4) and (5) $\mathrm{fal}(·)$ is a non-linear function, ²³ and its specific description is shown in (7), and $0<{\alpha }_1<1<{\alpha }_2$ .

$$\mathrm{fhan}(x_1,x_2,r,h)=\{\begin{array}{c}\begin{array}{c}d=r{h}^2\hfill \\ {\alpha }_0=h{x}_2\hfill \\ p=x_1+{\alpha }_0\hfill \\ {\alpha }_1=\sqrt{d(d+8\left|p\right|)}\hfill \\ {\alpha }_2={\alpha }_0+\frac{{\alpha }_1-d}{2}\mathrm{sign}\left(p\right)\hfill \\ S_y=\frac{\mathrm{sign}(p+d)-\mathrm{sign}(p-d)}{2}\hfill \\ \alpha =({\alpha }_0+p-{\alpha }_2)S_y+{\alpha }_2\hfill \end{array}\hfill \\ \multicolumn{1}{c}{S_{\alpha }=\frac{\mathrm{sign}(\alpha +d)-\mathrm{sign}(\alpha -d)}{2}}\\ \multicolumn{1}{c}{\mathrm{fhan}(x_1,x_2,r,h)=-r[\frac{\alpha }{d}-\mathrm{sign}\left(\alpha \right)]}\\ \multicolumn{1}{c}{S_{\alpha }-r\mathrm{sign}\left(\alpha \right)}\end{array}$$

(6)

$$\mathrm{fal}(e,\alpha ,\delta )=\{\begin{array}{c}{\left|e\right|}^{\alpha }\mathrm{sign}\left(e\right),\left|e\right|>\delta \hfill \\ \frac{e}{{\delta }^{(1-\alpha )}},\left|e\right|\le \delta \hfill \end{array}$$

(7)

Altitude – Attitude control

z, $\varphi$ , $\theta$ , and $\psi$ can be obtained by (1) for all the virtual quantities $u_1~u_4$ , while x, y position coordinates can be expressed by $u_1$ and attitude angle. Therefore, z, $\varphi$ , $\theta$ , and $\psi$ are selected as the controlled variables. According to the definition of the virtual quantity $u_1~u_4$ , it can be determined that: the controller can be regarded as a combination of four independent channels of the ADRC controller.

Controller parameter setting

The adjustable parameters of the second-order tracking differentiator are the speed factor $r_0$ and the integration step $h$ : the larger the parameter $r_0$ , the better rapidity, but too large will make the tracking differentiator signal increase, making $v_2$ show a very large amplitude spikes, which will amplify the noise of the input signal, resulting in overshoot or even oscillation; the larger the parameter $h$ , the smaller the static error, the smaller the overshoot will be, but too small will lead to a slow rise, weakening the rapidity of $r_0$ . The adjustable parameters of the third-order dilated state observer are the observer gain coefficient ${\beta }_{01}~{\beta }_{03}$ , the internal parameters ${\delta }_0$ and ${\alpha }_{01}~{\alpha }_{02}$ of the ${\alpha }_{01}$ function and the compensation factor $b_0$ . It is pointed out that ${\alpha }_{01}$ and ${\alpha }_{02}$ take fixed values ²⁵ : ${\alpha }_{01}=0.5$ and ${\alpha }_{02}=0.25$ . Considering the actual model of the quadcopter, this paper takes ${\delta }_0=0.01$ ; ${\beta }_{01}$ and $1/h$ are of the same order of magnitude, too small is not good for rapidity, too large will bring oscillation or even make the system diverge; ${\beta }_{02}$ too small will diverge, too large will have high-frequency noise; ${\beta }_{03}$ too small is not conducive to system tracking, reducing the tracking speed, too large will make the system oscillate; compensation factor $b_0$ in this design to take $b_0=1$ . Second-order nonlinear state error feedback control law adjustable parameters are gain coefficient ${\beta }_1~{\beta }_2$ and $\mathrm{fal}(·)$ function internal parameters $\delta$ and ${\alpha }_1~{\alpha }_2$ , ${\alpha }_1$ and ${\alpha }_2$ generally take ${\alpha }_1=0.75$ , ${\alpha }_2=1.25$ ; $\delta$ generally choose between $0.01~0.1$ , too large will produce oscillation, and smaller values basically do not affect the output, this paper takes $\delta =0.01$ ; The larger the ${\beta }_1$ , the smaller the error, but it will weaken the rapidity of the system; the larger the ${\beta }_2$ , it will enhance the rapidity of the system, but too large will make the system oscillate. The second-order ADRC controller has a total of 14 adjustable parameters, which is much more cumbersome compared to the three adjustable parameters of the conventional PID. The parameters of the second order ADRC controller are as follows, for TD module, $h$ is 0.001, $r_0$ is 800. For NESO module, ${\alpha }_{01}$ is 0.5, ${\alpha }_{02}$ is 0.25, ${\beta }_{01}$ , ${\beta }_{02}$ , and ${\beta }_{03}$ are greater than 0, $b_0$ is 1, ${\delta }_0$ is 0.01. For NLSEF module, ${\alpha }_1$ is 0.75, ${\alpha }_2$ is 1.25, both ${\beta }_1$ and ${\beta }_2$ are greater than 0, and $\delta$ is 0.01.

ADRC controller based on improved genetic algorithm

ADRC controller based on genetic algorithm

Genetic algorithm (GA) originated from computer simulation studies on biological systems and is a stochastic global search and optimization method developed to mimic the evolutionary mechanisms of living organisms in nature. In this paper, a genetic algorithm is used to rectify ${\beta }_1$ , ${\beta }_2$ , ${\beta }_{01}$ , ${\beta }_{02}$ , and ${\beta }_{03}$ of the second-order nonlinear ADRC controller for altitude, traverse roll angle, pitch angle, and yaw angle.

First of all, we have to determine the upper and lower limits of the parameters to be rectified, by the observer gain ${\beta }_{01}$ and $1/h$ for the same order of magnitude, so the lower limit of the generated random numbers are 100 and 1000 respectively; for the observer gain ${\beta }_{02}$ , ${\beta }_{03}$ is not specified, the upper limit is taken as 10,000 and the lower limit is 0 in this study, which can cover the existing parameters of the ADRC controller with high probability; Similarly, the gain coefficients ${\beta }_1$ and ${\beta }_2$ for the nonlinear state error feedback are taken as $0~1000$ and $0~100$ , respectively.

Determining the fitness function, the comprehensive evaluation index of the controller’s rectification performance effect uses the time times absolute error integral ITAE index, and the form of the fitness function is:

$$\mathrm{fitness}={\int }_0^{\infty }t\left|e\left(t\right)\right|\mathrm{dt}$$

(8)

The symbol $J$ is used to denote the fitness function $\mathrm{fitness}$ , and $1/J$ is used to denote the fitness of an individual in the environment. The process of adaptive controller parameter tuning of genetic algorithm is as follows: (1) Calculate the fitness value of each individual in this generation, and encode the 5 chromosomes of each individual to obtain a 5-dimensional binary number. (2) Arrange all individuals in this generation according to the fitness value from lowest to highest, and compare the fitness values of two individuals in turn by competitive method, and the higher fitness value can copy itself into the next generation. (3) Single-point crossover of each chromosome in the above arrangement, and if the crossover probability is greater than the set value, the genes with different expressions will be reversed. If the crossover probability is greater than the set value, the genes with different expressions will be reversed and the offspring with the duplicate itself will be the next generation of the population. (4) Perform a single point gene mutation for each chromosome in the individual, and when the mutation probability is greater than the set value, invert the gene for that locus and change its expression type. (5) Enter the fitness value calculation for the next generation, and repeat the athletic replication, crossover, and mutation operations to obtain all individuals in the last generation population. (6) Sort the fitness of all individuals in the last generation, and the one with the greatest value is the one with the greatest fitness, and the five chromosome decoding to obtain the optimal values of the 5 integer parameters. The block diagram of the GA optimized ADRC based on the above analysis is shown in Figure 3.

Figure 3.

GA optimized ADRC block diagram.

The height, roll angle, pitch angle and yaw angle controllers are rectified respectively, all need to rectify 5 parameters, so the initialized population individuals are 5-dimensional, and the genetic algorithm model parameters are given below: the population size is 30, which is recorded as $S$ ; The probability of crossover variation is 0.9, which is denoted as $P_c$ ; the probability of genetic variation is 0.1, which is denoted as $P_m$ ; and the number of evolutionary iterations is 100, which is denoted as $G$ .

Improved genetic algorithm

In the basic genetic algorithm, the population size and the number of evolutionary iterations set values to facilitate the calculation and obtain high-quality results in a short period of time, and if the population size and the number of iterations are dynamically changing, it may lead to a continuous expansion or contraction of the population, resulting in early convergence or falling into a local optimum.

Early convergence is the emergence of super individuals in the population early in the genetic algorithm, which refer to as “super babies” in this study. The fitness value of the super babies far exceeds the fitness value of other existing individuals in the population, which makes the superbaby find an absolute proportion in the population and always survive and even reproduce super or sub-super offspring after many iterations. Caught in the local optimum that is, if the population size is small, although a larger fitness individual is finally found, it is not the global optimal solution, which also makes the model not have the feature of searching for the global optimum.

The choice of crossover probability and genetic variation probability is crucial to the behavior and performance of the algorithm, which directly affects the convergence of the algorithm. If the crossover probability is too large, individuals with high fitness values will be easily destroyed, but too small will lead to slow population renewal and stagnation. The value of variation probability is also very delicate, taking too large a value, the genetic algorithm is not essentially different from the random search algorithm; on the contrary, taking too small a value, it is not easy to generate new individual institutions. Combined with the above mentioned analysis, this paper proposes a new parameter setting method that sets the genetic variation probability as a dynamic parameter, the higher the fitness of the individual, the higher the probability of genetic variation, and it is different from the genetic algorithm proposed ²² and the design scheme of the swarm optimization algorithms proposed.^26,27

In theory, the higher the rate of variation, the higher the degree of adaptation, and the greater the coefficient of variation of an organism, the more genetic diversity it has and the more adaptable it is to its environment. The probability of genetic variation is $P_m$ with a fixed value of 0.1, but through the above analysis, the probability of genetic variation is linked to the fitness of its individuals, and is specified as follows: the inverse of the fitness value of individuals in the population is arranged in descending order, and the greater the inverse of the fitness value, the lower the fitness value, and taking into account that the difference in the probability of genetic variation between individuals is not conducive to the diversity of the population and its adaptability to the environment if the difference is too great. The higher the fitness value, the lower the fitness value. According to the principle that the higher the fitness, the lower the probability of genetic variation, the highest probability of genetic variation $P_{m|max}=0.09966$ , the lowest probability of genetic variation $P_{m|min}=0.0900$ , and the probabilities of other genetic variation within this interval are evenly distributed.

Application validation

The model parameters of the quadcopter in this study are shown in Table 1, and the structure of the control system model is shown in Figure 4.

Table 1.

Quadcopter parameters.

Parameters	Description	Value	Unit
$m$	Quadrotor mass	$0.25$	$\mathrm{kg}$
$g$	Gravity acceleration	$9.8$	$\mathrm{m}/{\mathrm{s}}^2$
$l$	Distance from the center of the quadrotor to the center of the propeller	$0.198$	$\mathrm{m}$
$c_t$	Lift factor	$3.13\times {10}^{-5}$	$\mathrm{N}/{\mathrm{s}}^2$
$c_x$	Resistance factor	$7.5\times {10}^{-7}$	$\mathrm{N}/{\mathrm{s}}^2$
$I_{\mathrm{xx}}$	$X$ -axis rotational inertia	$2.353\times {10}^{-3}$	$\mathrm{kg}·{\mathrm{m}}^2$
$I_{\mathrm{yy}}$	$Y$ -axis rotational inertia	$2.353\times {10}^{-3}$	$\mathrm{kg}·{\mathrm{m}}^2$
$I_{\mathrm{zz}}$	$Z$ -axis rotational inertia	$5.262\times {10}^{-2}$	$\mathrm{kg}·{\mathrm{m}}^2$

Figure 4.

The structure diagram of the control system model.

In Figure 4, v-input is the set value, that is, the system input; u-output is the output of the controller; and y-fbk is the feedback of the system, that is, the output of the controlled object. The traditional genetic algorithm, the genetic algorithms proposed in different literatures, and the genetic algorithm proposed in this article are used to tunning the ADRC altitude, attitude controllers, respectively. The controller parameters for each channel under the four ADRC controller tunning strategies are shown in Tables 2 and 3.

Table 2.

Table of ADRC controller parameters for altitude and roll channels based on different genetic algorithms.

Module parameters	Altitude channel				Roll channel
	GA-ADRC	GA-ADRC-22	GA-ADRC-26	IGA-ADRC	GA-ADRC	GA-ADRC-22	GA-ADRC-26	IGA-ADRC
${\beta }_{01}$	743.67	439.75	307.50	489.38	875.12	723.88	572.98	731.64
${\beta }_{02}$	2549.52	9407.30	5905.36	9156.41	9786.48	7635.38	8527.80	8898.04
${\beta }_{03}$	66.32	38.05	2014.90	6.04	960.56	877.12	1350.96	1993.84
${\beta }_1$	74.83	861.14	991.84	935.59	84.63	38.57	67.95	44.15
${\beta }_2$	9.57	97.81	89.52	89.29	3.53	5.49	2.12	4.13

Table 3.

Table of ADRC controller parameters for PITCH and YAW channels based on different genetic algorithms.

Module parameters	Pitch channel				Yaw channel
	GA-ADRC	GA-ADRC-22	GA-ADRC-26	IGA-ADRC	GA-ADRC	GA-ADRC-22	GA-ADRC-26	IGA-ADRC
${\beta }_{01}$	264.72	802.16	639.19	429.62	986.88	944.40	995.76	774.46
${\beta }_{02}$	5428.19	9518.92	8642.46	8435.49	2955.53	5552.04	9525.57	9249.50
${\beta }_{03}$	948.37	2481.65	2448.50	2011.92	523.35	0.65	40.96	2.68
${\beta }_1$	21.22	55.22	30.20	66.88	442.41	374.05	371.73	358.59
${\beta }_2$	0.56	3.39	1.24	2.07	3.17	0.02	0.60	0.84

In Table 5, GA-ADRC is the ADRC controller based on genetic algorithm. GA-ADRC-22²² and GA-ADRC-26²⁶ are genetic algorithms proposed in literatures. IGA-ADRC is the ADRC controller based on improved genetic algorithm, and “I” is Improved.

In this work, the PID controller is used for comparison reference, and the time domain is:

$$u\left(t\right)=K_{p}e\left(t\right)+K_i{\int }_0^{t}e\left(\tau \right)d\tau +K_d\frac{\mathrm{de}\left(t\right)}{\mathrm{dt}}$$

(9)

Where $e\left(t\right)$ is the error function, that is, the input of PID, $u\left(t\right)$ is the output of PID, and $K_p$ , $K_i$ , $K_d$ are the proportional, integral, and differential coefficients, respectively. The critical proportionality method is used to adjust the PID parameters, and the PID controller parameters of each channel are obtained as shown in Table 4.

Table 4.

parameters of each channel PID controller.

Channel name	$K_p$	$K_i$	$K_d$
Altitude channel	18	0.1	0.1
Rolling channel	9	12	0.5
Pitching channel	9	12	0.5
Yawning channel	100	0.1	0.1

Response experiment

The initial and expected values of attitude angle and altitude are 0 and 1, respectively. Figure 5 shows the step response curves under the PID control strategy and the four ADRC control strategies. The system response performance indicators are shown in Table 5. There are mainly overshoot ${\sigma }_p$ , transition process time $t_s$ , rise time $t_r$ and peak time $t_p$ , and optimal individual fitness values $J$ .

Figure 5.

Response experiment curves.

Table 5.

Step response performance index of each channel under three control strategies.

Controller	Channel	${\sigma }_p$	$t_s$	$t_r$	$t_p$	$J$
PID	Altitude	13.4801%	0.4549	0.4887	0.7186
	Roll	4.9102%	0.1537	0.1752	0.2482
	Pitch	2.1082%	0.1567	0.1855	0.2346
	Yaw	0.0040%	0.1012	0.2571	0.7536
GA-ADRC	Altitude	0.4317%	0.1232	0.2555	0.3396	142.0455
	Roll	0%	0.0708			0.0129
	Pitch	0%	0.1082			0.0067
	Yaw	0.1895%	0.0838	0.1777	0.3306	0.0084
GA-ADRC-22	Altitude	0.0386%	0.0615	0.0821	0.1144	0.0039
	Roll	0%	0.1128			0.0028
	Pitch	0%	0.1009			0.0035
	Yaw	0.001%	0.0637	0.1232	0.4455	0.0024
GA-ADRC-26	Altitude	0.2395%	0.0640	0.0767	0.1105	0.0027
	Roll	0%	0.0825			0.0030
	Pitch	0%	0.1387			0.0031
	Yaw	0%	0.0684			0.0034
IGA-ADRC	Altitude	0.1568%	0.0640	0.0750	0.0811	0.0014
	Roll	0%	0.1339			0.0024
	Pitch	0%	0.0882			0.0015
	Yaw	0%	0.0699			0.0015

The results are analyzed in conjunction with Figure 5 and Table 5. For altitude channel control of quadrotor, the overshoot under PID control is very large, and the long transition process time and peak time make the system response slow and unsatisfactory control. The fitness value performs poorly under GA-ADRC, and the analysis of the cause of this may be caused by falling into the local optimum. The system overshoots under the IGA-ADRC control strategy proposed in this article are both better than GA-ADRC and GA-ADRC-26, while slightly higher than GA-ADRC-22. However, the transition process time of the proposed strategy in this article is better than that of GA-ADRC-22. For roll channel control, the strategy proposed by GA-ADRC-22 exhibits superior system response speed, but the transition process time is longer. The strategy proposed in this paper also shows some fast system response capability.GA-ADRC-26 is slightly better than GA-ADRC. For pitch and yaw channel control, IGA-ADRC shows excellent response capability with zero overshoot and has shorter transition process time and peak time. Both GA-ADRC and GA-ADRC-22 have overshoots in yaw channel control, and the overshoot is smaller under GA-ADRC-22. Although GA-ADRC-22 and GA-ADRC-26 may perform better in one performance metric in the control of a particular channel, the IGA-ADRC strategy, which calculates the ITAE metrics for individual fitness, is considered globally and has a smaller $J$ value.

Altitude attitude tracking experiment

The initial values of altitude, traverse angle, pitch angle and yaw angle are set as $z_0=0\mathrm{m}$ , ${\varphi }_0=0°$ , ${\theta }_0=0°$ and ${\psi }_0=0°$ respectively, and the tracked attitude and altitude are square waves with amplitude of 2 and period of 2 s. The total time is 5 s. The system output curve results are shown in Figure 6.

Figure 6.

Tracking experiment curves.

From the application results, the tracking curve overshoot for attitude, altitude control of the quadcopter using ADRC controller is very small or basically no overshoot. In terms of altitude control, the transition time is greatly reduced by GA-ADRC compared with PID control, and the control effect is slightly improved by IGA-ADRC compared with GA-ADRC-22 and GA-ADRC-26. In terms of attitude control, the attitude tracking signal under PID control also has nearly zero overshoot, but the advantages of GA-ADRC-22 and GA-ADRC-26 in tracking rapidity are more obvious and IGA-ADRC is slightly improved. According to Figure 6, the roll and yaw channels need to controlled better controller parameters with the help of improved genetic algorithm to obtain better controller parameters. The improved genetic algorithm has more significant performance improvement compared to GA-ADRC, and IGA-ADRC has a slight improvement in rapidity compared to GA-ADRC-22 and GA-ADRC-26.

Disturbance rejection experiment

The quadcopter works in a complex environment where external disturbances are always present, including air resistance, free wind, etc. To simulate the actual operation of the quadrotor and to consider the sudden changes in external disturbances, the disturbance rejection experiments are conducted.

A square wave signal with amplitude 2, period 1 s, and duty cycle 50% is added before each channel, and the action starts at 1 s with an action time of 4 s. The attitude angle, initial value of altitude, and target expectation values of the quadrotor are as described in the response experiment. The system output curve results are as Figure 7.

Figure 7.

Disturbance rejection experiment curves.

The disturbance rejection experiments show that the ADRC controller can reduce the effect of perturbations on the system to a very low range compared to the PID controller. In the altitude and pitch channels, IGA-ADRC and GA-ADRC-22 have very significant improvement in disturbance rejection performance compared to GA-ADRC and GA-ADRC-26, while IGA-ADRC has a more significant advantage. In addition, in the traverse and yaw channels, the ADRC control strategies of the two improved genetic algorithms are similar to the strategy proposed in this paper, without too significant improvement, instead, GA-ADRC performs less effectively in the yaw channel. However, if the external influence is large, it can be inferred that the ADRC controller with better parameters must have a greater advantage in disturbance rejection. The experimental results show that the IGA-ADRC has more superior disturbance rejection performance against external uncertainty perturbations. What makes ADRC more advantageous in the convenience of disturbance rejection performance is that the extended state observer of this controller has the ability of disturbance observation and compensation.

Conclusion

In this article, an altitude, attitude ADRC control strategy for a quadcopter is proposed. The ADRC controller is able to realize more accurate tracking. In addition, to address the problem that the ADRC controller has many parameters, which are difficult to be adjusted manually and difficult to get the optimal control effect, as well as the shortcomings of the traditional genetic algorithm in early convergence and falling into the problem of local optimization, an improved genetic algorithm-based method for adjusting the parameters of ADRC controller is proposed in order to improve the genetic diversity in the population and the adaptability of the individuals to the environment.

The experimental results show that the ADRC control strategy is able to realize the accurate control of the quadcopter, and although the ADRC controller has cumbersome parameter rectification, its ability to compensate for the perturbation is very outstanding, and it has superior robustness and control response compared with other control strategies or schemes. The improved genetic algorithm proposed in the article can effectively solve the problems of premature convergence and falling into local optimum. In addition, the control strategy is also verified in tracking and perturbation suppression experiments to have faster rapidity, stronger tracking performance and robustness in altitude, attitude control of quadrotor, validating the effectiveness of the proposed strategy. Our future work includes the development of machine learning or reinforcement learning schemes for the ADRC controller to maximize the adaptability of the system in complex environments, as well as the improvement of the internal modular structure of the second-order nonlinear ADRC controller to make it more robust and fast-response.