2.1

Dynamics model of quadrotor UAV

The dynamics model of quadrotor UAV¹² is established in the following equation:

In the last equation, x, y, z represent the spatial coordinates of the three directions of the quadrotor; φ, θ, ψ represent pitch, roll, yaw three attitude angles; m is the total mass of the quadrotor body, I_x, I_y, I_z are moment of inertia of the body rotating about the x, y, z axis; d is the distance between the quadrotor body’s center of mass and its rotor axis; g is the acceleration of gravity at the earth’s surface; U₁, U₂, U₃, U₄are virtual control quantity in the control process.

2.2

Fractional order PID controller

The definition of fractional derivative mainly has two following formations.

The fractional derivative of Caputo¹³ is defined as:

where α is the fractional order, n – 1 ≤ α < n, n 𝜖 N. Г(n – α)is Gamma function.

The fractional integral of Riemann-Liouville¹³ is defined as:

where α is the fractional order, 0 ≤ α < 1.

The calculation of fractional calculus has the following properties:

In 1999, Podlubny proposed a fractional order PID controller (FOPID) in the form of ΡΙ^λD^μ. The transfer function has the following form:

where K_p is proportional gain, K_i is the integral gain, K_d is differential gain, λ is the order of integration, μ is the differential order, and 0 ≤ λ, μ ≤ 2. When λ, μ are both 1, equation is the traditional integer order PID controller. The structure of FOPID controller and integer order PID controller is roughly the same, but the introduction of λ, μ adds two adjustable parameters, which makes the controller design more flexible and has better system performance.

Based on the advantages of fractional order PID with multiple degrees of freedom and the nonlinear and strong coupling characteristics of quadrotor UAV system, the fractional order PID control strategy above is adopted to achieve more accurate and more stable control.

3.1

Position controller design

From equation (1) we can observe that the position subsystem is an underactuation system with output of x, y, z, a virtual control quantity U₁is used to control the position quantity in three directions, in order to control the underactuation system at this position, control quantities u_x, u_y, u_z, in three directions of the position are defined, as shown in equation (7):

The expected values given in three directions are x_d, y_d, z_d respectively, the position tracking error in the three directions is set as:

Therefore, the design of location subsystem FOPID is as follows:

where, K_p, K_i, K_d are proportional coefficient, integral coefficient and differential coefficient of the FOPID controller of the position subsystem, λ is the order of integration, μ is the differential order.

3.2

Attitude controller design

It can be easily concluded from equation (2) that the expected input of the attitude controller of the inner ring is determined by the output of the position controller, the output of the position controller is the position control quantity u_x, u_y, u_zin the three directions set in the previous section. Where the expected value of yaw angle 𝜓_d is given directly from the outside, therefore, the expected values of roll angle and pitch angle can be calculated from position control quantities u_x, u_y, u_z,which is given by equation (1):

The tracking errors of the three attitude angles are as follows:

Therefore, the attitude subsystem FOPID controller is designed as:

where K_p, K_i, K_d are proportional coefficient, integration coefficient, differential coefficient of attitude subsystem FOPID controller, λ is the order of integration, μ is the order of differential.

3.3

The tuning of FOPID parameters

Due to huge amount of FOPID controller parameters, the selection of controller parameters directly affects the control effect. If the trial-and-error method is used to select FOPID controller parameters, it has great limitations and precision control is difficult. Therefore, it is particularly important to select an appropriate intelligent optimization algorithm for parameter tuning. Particle swarm optimization (PSO) algorithm is a commonly used optimization algorithm. It simulates the biological mechanism of nature and uses each individual in the population to cooperate with each other to search for the best solution. Compared with genetic algorithm (GA), PSO algorithm has no complex operations such as crossover and variation, and is easier to express and implement.

For the quadrotor UAV control system in this paper, the specific implementation steps of the algorithm are as follows:

Based on the above ideas, the simulated annealing strategy and compression factor are added on the basis of the standard PSO algorithm. The main steps are as follows:

4.1

Standard PSO algorithm

The initial parameters of particle swarm optimization are as follows: Number of populations N is 100, the spatial dimension d is 5, maximum iteration M is 100, self-learning factor c₁ is 1.5, group learning factor c₂is 1.5, for a, b in the fitness function, considering that quadrotor UAV should not only ensure small tracking error, but also ensure small control energy and fast response speed in the realistic flight process, the weight of tracking error a and rise time b are respectively 0.6 and 0.4, and c is 0.5. The reference values of both horizontal and vertical directions are 5. Running the algorithm, the obtained IOPID parameters and FOPID parameters are shown in Table 1, and simulation results are shown in Figures 2-5.

Figure 2.

Convergence process.

Figure 3.

Horizontal response curve.

Figure 4.

Vertical response curve.

Figure 5.

Loop tracking.

Table 1.

IOPID and FOPID parameters based on PSO tuning.

Note: “*” means no value.

Figure 2 is the PSO optimization convergence process curve, and from the figure it can be seen that the algorithm completes convergence during the 46th iteration. Figure 3 is the horizontal response process of the quadrotor UAV, the dotted line is the traditional integer order PID control, and the solid line is the fractional order PID control. It can be seen from the response curve that the quadrotor UAV controlled by the integer order PID has a large overshoot obviously in the horizontal control. However, the quadrotor UAV system controlled by fractional order PID has an obvious improvement in overshoot. In the vertical direction, FOPID is superior to IOPID in response speed, as shown in Figure 4. Figure 5 shows threedimensional annular trajectory tracking, indicating that the quadrotor UAV system controlled by FOPID can complete the trajectory tracking test stably and has better control effect than the traditional integer order PID.

4.2

SA-PSO algorithm

According to the algorithm combined with the strategy in Section 3, running the SA-PSO algorithm, the new FOPID parameters obtained are shown in Table 2. The system is simulated again, and the convergence process and simulation results are shown in Figures 6-9 by compared with FOPID control optimized by standard PSO in Section 4.1.

Figure 6.

Convergence process.

Figure 7.

Horizontal response curve.

Figure 8.

Vertical response curve.

Figure 9.

Loop tracking.

Table 2.

FOPID parameters based on SA-PSO tuning.

The simulation image show that the PSO based on simulated annealing algorithm greatly reduces the convergence time, and the accuracy is better, specifically embodies in quadrotor UAV system improved the response speed further under the condition of less overshoot, and improved the dynamic performance of the system again, so the simulated annealing PSO in the field of quadrotor UAV system control is worthy of application.