Double Closed Loop Coordinated Control of Quadrotor Based on PID / LADRC

Double Closed Loop Coordinated Control of Quadrotor Based on PID / LADRC

Qingsong Jiao

Tianjin University of Technology and Education Tianjin 300222, PR China

Abstract- For control system of a quadrotor aircraft with under-driven, strong coupling and nonlinear characteristics. This paper presents a double closed loop collaborative control method of PID / LADRC to control the position and attitude of the control system. Among them, the attitude loop is the inner loop of the control system, and the linear automatic disturbance rejection control method is adopted. The position loop is the outer loop of the control system, and the PID control method is adopted. Due to the under-driven nature of the control system of quadrotors aircraft itself, the expected values of pitch and roll angle are given by the outer loop controller. Finally, through the MATLAB simulation of the control system, and then to verify the effectiveness of the control system of quadrotor using the PID / LADRC double closed-loop coordinated control method.

Keywords: Quadrotors; Position Control; Attitude Control; Automatic Disturbance Rejection Control; PID Control

1. INTRODUCTION

   In recent years, the continuous development and advancement in the field of control have promoted the development of the quadrotors aircraft. In terms with the traditional unmanned aerial vehicle, the quadrotors aircraft have the pros of small size, light weight and flexible maneuver performance. Therefore, a considerable number of domestic and foreign universities and researcher's attention. As the quadrotors aircraft can achieve translation and rotation during the movement, and then can do pitch, roll, yaw, hover and vertical movement, before and after the movement and lateral movement of the six basic movement states. Through the different combinations of the six basic movement states, it can make any trajectories and motions in any three-dimensional space and is widely used in civil and military areas. For example, on the military side, the quadrotors aircraft can perform control tasks such as detecting battlefields, targeting and tracking targets, and delivering weapons as carriers; On the civilian side, the quadrotors aircraft can be used in aerial photography, environmental surveys and search and rescue after a major disaster [1], and can carry on scientific experiments by carrying a variety of devices.

   Due to the typical features of under-driven, strong coupling and non-linearity, the quadrotors aircraft bring tremendous difficulties to the control of quadrotor aircraft [2]. At the same time, the mass of quadrotors aircraft is lighter, so during the movement in the extremely vulnerable to external environment and other factors. Thus, we need to be strong anti-interference performance in the design of the controller.

   The continuous development and advancement of control theory has given rise to an increasing number of control methods being applied to the analysis and design of the quadrotors control system. In the meantime, the control system can be controlled more optimally by using the above control methods. For example, PID control [3-4]adaptive control

   [5], fuzzy control [6], sliding mode control [7-8]and ADRC [9]. Simple sliding mode control has the significant pros of strong robustness. When the control system is disturbed by the outside world, the steady state error will be generated, which will result in lower control accuracy. The adaptive control is generally used in conjunction with other methods, and through the integrated control technology to achieve the stability of its control system. In the case of model uncertainty and external disturbance, adaptive fuzzy control based on fuzzy CMAC is applied to the control system. At the same time, adaptive parameters are used to make the control system have good steady state performance.

   In this paper, based on a dual closed-loop control method combining PID and LADRC to control the position and attitude of the quadrotors, the inner loop adopts the method of LADRC to control the attitude of the aircraft, while the outer loop adopts the method of PID control to control the position of the aircraft, the controller has strong anti-jamming performance and strong robustness.

2. THE QUADROTOR AIRCRAFT DYNAMICS MODEL AND ANALYSIS

   As the quadrotor aircraft is a nonlinear, multivariable and strongly coupled under-driven system, in order to further control such under-actuated control systems, such as the quadcopter, firstly,the dynamics model of the control system must be established. In order to establish the dynamics model of quad-rotor control system, first and foremost,we must select the appropriate coordinate system, which is the ground coordinate system and the airframe coordinate system. When the quadrotor aircraft is flying, the constant change of the three Euler angles will directly result in the change of the flight state of the quadrotor aircraft in the airframe coordinate system, but the ground coordinate system will remain unchanged all the time. Four-rotor aircraft flying attitude in the air is usually reflected by three Euler angles. which is rolling angle , pitch angle and yaw angle .Euler angle is used to accurately determine the role of rigid body at a fixed point where the specific location. The selected fixed point is the origin of the airframe coordinate system, and the three attitude angles – roll

   angle , pitch angle and yaw angle are used to determine

   the coordinate of the axis of motion in the spatial direction. It can also indicate the rigid body around its corresponding rotation angle of the shaft. In order to describe the attitude of quadrotors aircraft more clearly, three Euler angles can be used to express the concrete conversion relationship between attitude matrix and Euler rotation, and then the concrete conversion relation between ground coordinate system and body coordinate system can be derived. Fig. 1 shows the relationship between the ground coordinate system and the airframe coordinate system.

   xbThe horizontal plane 4

   U1 Fi m

   i 1

   x' y

   U 2 F1 F2 F3 F4 I1

   (6)

   g U F F F F I

   3 1 2 3 4 2

   x

   g U 4 C F1 F2 F3 F4 I3

   y'

   O

   yb

   The vertical plane

   zb

   z'

   Where U1 is the control input acting on the quadrotors in the z-axis direction; U 2 and U 3 denote the control inputs of the pitch angle and roll angle respectively; U 4 denotes

   the yaw moment; C denotes the force-to-torque conversion factor .

   Therefore, we can draw:

   x U1 cos sin cos +sin sin K1 x m

   z y U1 cos sin sin sin cos K2 y m

   (7)

   g

   Figure 1 the relationship between the ground coordinate system and the

   z U

   1 cos cos g K3 z m

   airframe coordinate system

   The ground coordinate system to the airframe coordinate system conversion matrix as (1) shown, that is

   The quadrotors body in rotation, the moment balance equation can be drawn:

   I1 l F1 F2 F3 F4 K4

   cos cos

   R sin cos

   cos sin sin sin sin

   sin sin sin cos cos

   cos sin cos sin sin

   sin sin cos sin cos

   (1)

   I2 F1 F2 +F3 F4 K5

   (8)

   I3 C F1 F2 F3 F4 K6

   sin

   cos sin

   cos cos

   In orderto further simplify the dynamic model of the qu adrotors control system, according to Newton's second law, th e dynamic equations of the quadrotors can be expressed as:

   Combined with equation(6) can be drawn:

   =U2 lK5 I2

   U3 lK4 I1

   (9)

   F m dv

   (2)

   U K I

   dt

   4 6 3

   M dH

   (3)

   In conclusion, the nonlinear dynamics model of the

   dt quadrotor obtained is:

   In equation (2) and (3), F is the resultant force of

   quadrotors aircraft flying in space , m is the mass of the quad-

   x U1 (cos sin cos sin sin ) K1 x / m

   y U (sin sin cos sin cos ) K y / m

   copter itself, v is the speed of the quadrotors in space motion,

   1 2

   z U1 (cos cos ) g K3 z / m

   M is the external moment of rotation of the quadrotors, and H is the absolute moment of momentum of the quadrotors with respect to the ground coordinate system.

   The four-rotor aircraft in flight during the specific force

   U lK / I

   2 5 2

   U lK / I

   3 4 1

   (10)

   situation is as follows: that the four rotor generated by the lift Fi i 1, 2,3, 4 , The body's own gravity mg and air resistance, quadrotors aircraft lift in the body coordinate

   4

   U4 K6 / I3

   In the dynamics model of quadrotor aircraft,

   x, y, z represents the position of the quadrotor aircraft;

   , , represents the attitude of the aircraft, that is, roll

   system as follows:

   F Fi , The force components along

   i1

   angle, pitch angle and yaw angle; Ui i 1, 2,3, 4 indicates

   the x-axis, y-axis and z-axis in three directions are:

   the amount of control; Ki is the drag coefficient ; Ii represents

   4 the moment of inertia of each axis; m is the mass of the

   Fx cos sin cos + sin sin Fi

   aircraft body itself; l is the distance between the center of

   F cos sin sin sin cos

   i1

   4

   F

   (4)

   mass of the aircraft body and the axis of rotation of the

   y i

   4

   i1

   Fz cos cos Fi

   rotor; g

   is the gravitational acceleration of the earth's surface.

3. THE DESIGN OF CONTROL SYSTEM

   i1

   And because the x-axis, y-axis, z-axis acceleration are:

   1. The Structure of Control System

      x Fx K1 x m

      y 2

      y F K y m

      z F mg K z

      m

      (5)

      Because the four-rotor aircraft dynamics model has four control inputs and six control outputs, the control system has a strong coupling, highly nonlinear.The rotation speed of the four rotors plays a decisive role in the three position

      z 3

      In reference [11], the control inputs of the quadrotors are defined as follows:

      coordinates of the four-rotor and the three position coordinates of the centroid in the inertial coordinate system. The change of the three position coordinates of the quadruped's center of mass in the inertial coordinate system will give rise to the change of the three attitude angles, but the change of the three attitude angles will not cause the changes of the coordinates of the three positions. According to the strong coupling between

      the three position coordinates of the quadrocopter's center of mass in the inertial coordinate system and the three attitude angles of the quadrotors, most of the control methods adopted by the quad-rotors are the double closed-loop coordinated control method. Double closed-loop control system is

      observation of the total disturbance, w is the disturbance and

      u is the amount of control acting on the object.

      According to the mathematical model of the control system of quadrotors, a LADRC is designed. Because its attitude angle satisfies the following relation:

      composed of inner and outer loop, the outer ring is a position

      control subsystem, the use of PID control method, and the

      =U

      2 lK5 I1

      inner loop is the attitude control subsystem, the use of linear

      U3 lK4 I2

      (17)

      automatic disturbance rejection control. Fig. 2 shows the block diagram of a double closed-loop control system composed of

      U

      4 K6 I3

      PID control and linear automatic disturbance rejection control.

      Therefore, you can order

      The output of (x, y, z)

      =U3 lK4

      I1 1

      The command signal generator

      (xd , yd , zd )

      The PID U1

      controller

      The position control subsystem

      position

      U2 lK5

      U K

      I

      2

      2

      I +

      (18)

      4 6 3 3

      Linear automatic disturbance rejection controller

      d U2 ,U3 ,U4

      The output of

      The attitude control subsystem

      attitude (,,)

      Among them, i i 1, 2,3

      is the uncertainty of the

      system internal disturbance, the actual flight process also need

      Figure 2 The double closed loop control system structure

      to consider the external disturbance.

      di (i 1, 2, 3)

      ,let

      fi i di ,

      fi be the total perturbation of the control system,

   2. The Design of Outer Loop Pid Controller

      The position control loop of the quadrotors can be divided into two independent parts: height control and horizontal position control. Height control and horizontal position control

      are shown in Equations. (11), (12) and (13), respectively

      and

      fi can lead, then

      U3 lK4

      U2 lK5

      I1 f1 I2 f2

      (19)

      U K I f

      z U1 cos cos g K3 z m

      (11)

      4 6 3 3

      1 2

      x U1 cos sin cos sin sin K1x m y U sin sin cos cos sin K y m

      (12)

      (13)

      Take pitch channel as an example, design a LADRC,

      Set x1 , x2 , x3 f1 . x3 is the expansion state

      variable of pitch angle , so =U

      • lK

      I f

      can be

      Supposing that the expected value of the position

      coordinate x, y, z is xd , yd , zd , according to PID control algorithm can be obtained:

      rewritten as

      x1 x2

      3 4 1 1

      z K

      z z K

      z z dt K z z

      (14)

      pz d ix d dx d

      x2 x3 lK4

      I1 x2 U3

      (20)

      x K px xd x Kix xd xdt Kdx xd x

      (15)

      x f

      3 1

      y K py yd y Kiy yd y dt Kdy yd y (16)

   3. The Design of Inner Loop Linear Automatic Disturbance Rejection Controller

      Linear expansion of the pitch channel state observer design:

      Let z1 x1 e , for its pitch channel linear expansion

      Linear automatic disturbance controller LADRC is

      state observer design, the process is as follows:

      mainly composed of linear PD controller, linear expansion

      z1 z2 1 e

      2

      2

      3

      state observer LESO and error compensation control

      z z e a z

      • b U

      (21)

      law LNESEF . Second-order linear automatic disturbance

      z e

      3

      2

      3 3

      rejection controller schematic diagram shown in Fig. 3.

      Where

      zi i 1, 2,3

      is the observed value of

      PD

      W

      V0

      The disturbance

      xi

      i 1, 2,3, i

      i 1, 2,3 is the observer gain, and the

      The disturbance

      u0 u ypoles of the extended state observer are all configured to

      w , where a is the observer bandwidth, and 3w

      0 1 0

      1/b0

      3w2

      w3

      a lK I

      0.00192 ,

      2 0

      3 0

      4 1

      LESO

      z

      z3 b b0 1 .

      1

      2 The design of linear state error feedback controller for

      Figure 3 The schematic of second-order linear automatic disturbance rejection

      pitch channel NLSEF

      controller

      Based on the feedback error signal, the design of a

      U3

      y z3 b

      (22)

      LADRC is independent of the exact math model of the control

      system. The input signal is V0 , which is given as a state

      y KP (d z1 ) Kd z2

      reference.

      z1 and

      z2 are the observations of the linear

      dilatometer, z3 is the dilatational state variable, that is, the

      Where d

      represents the expected value of pitch

      angle, K P and Kd both represent the gain of the pitch channel

      controller, and let

      K p

      2

      w

      c

      , Kd

      2wc ,

      where

      wc represents the bandwidth of the controller

      and y represents the amount of feedback control.

      By the same token, the design of the LADRC for the roll path and the yaw path can be performed.

4. SIMULATION RESULTS

   The body parameters of a quadrotor aircraft are given by reference [10], and the body parameters of a quadrotor aircraft

   Figure 5 The response curve of position coordinates

   are set as:

   K1 K2 K3 0.010

   K4 K5 K6 0.012

   I1 I2 1.25 I3 2.5 m 2kg l 0.2m g 9.8m / s2

   Proportional coefficient K p , integral coefficient Ki

   and

   differential coefficient Kd values of PID controller can be

   obtained by trial and error method, while the design of the main parameters of the LADRC, so that the LADRC bandwidth: w0 28 ,the pitch, yaw and roll the observer gain of

   these three channel

   1 1 1 3w0 84

   1 , 2

   , 3

   respectively,the value is

   Figure 6 The response curve of attitude angle

   Through the simulation of quadruped rotor dual-loop

   3w2 2352 ,

   w3 21952 . Let

   2 2 2 0

   3 3 3 0

   coordinated control system by MATLAB/ simulink.As you

   controller bandwidth

   wc 2.8 , the parameters of PD

   can see, the response curve of the position coordinate x, y, z

   controller are configured as kd kd kd 2wc 5.6

   k k k w2 7.84 b b b b 1 .

   can reach the expected value in about 8 seconds and remains unchanged at all times, at the same time, the response curve of

   p p p c

   0 0 0 0

   its yaw angle can reach a constant value in about 3 seconds

   According to the above derivation process, a block diagram of the PID / LADRC double closed-loop coordinated control system of the quad-rotor aircraft control system can be set up, as shown in Fig. 4.

   Figure 4 PID / LADRC double closed-loop coordinated control simulation module diagram

   By running the control system simulation block diagram in Fig. 4, we can get the curve of the position coordinates and attitude angle of the quad-rotor, as shown in Fig.5 and Fig.6.

   and remains unchanged, The response curve is almost no overshoot, while the pitch angle and roll angle can be restored to the desired value of 0 degrees after a period of time, which can make the quadrotors aircraft operating conditions remain stable, It is demonstrated that the coordinated control of quadrotors with PID and linear automatic disturbance rejection control can make the position response and attitude response have good dynamic performance with fast performance and stable performance. The experimental results show the effectiveness of the dual closed-loop control which combines the PID control of quad-rotor with the linear automatic disturbance rejection control.

5. CONCLUSION

   In this paper, a quad-rotor aircraft of four-input and six- output is proposed, which is based on PID and LADRC. The inner loop is designed with a linear automatic disturbance rejection controller control the attitude of the quadrotors aircraft, and the outer loop is designed PID controller to control the position of the quadrotors aircraft. For the linear automatic disturbance rejection controller, the linear state of

   expansion observer is a core part of the system, which can estimate and compensate the total disturbance generated by the quadcopter in real time, according to the simulation results in this paper, we can know that the designed PID / LADRC double closed-loop cooperative controller can make sure the control system of quadrotors aircraft can track and control the position and attitude better, and the control system has strong robustness.

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