SELF-TUNING FUZZY-PID CONTROL FOR THE ATTITUDE STABILIZATION OF A QUAD-ROTOR AIRCRAFT
Nguyen Xuan Vinh, Woon Kyung Baek, and Inpil Kang.
CAMEL Lab, School of Mechanical Engineering, Pukyong National University, Korea.
ABSTRACT
This paper describes a self-tuning fuzzy-PID attitude control algorithm for a new application for a quad-rotor aircraft (QRA). Firstly, a classical PD control algorithm was applied as the simple simulation approach for the attitude control. Secondly, a direct fuzzy controller is applied for the attitude control. Then the simulation results of the direct fuzzy controller were observed for the structure, attribute and performance. Finally, a self-tuning fuzzy-gain of PID controller was proposed as a new algorithm for attitude control of the QRA. For the implementation of these approaches, some attitude control architectures were simulated in Matlab. The proposed control algorithm was validated for the QRA attitude stabilization using Matlab.
1. 0B0BINTRODUCTION
Quad-rotor aircraft (QRA) is known as the advantage helicopter with the ability of VTOL (vertical take-off and landing), highly maneuverable and extremely stable. The attitude control of quad-rotors has been the main concern topic of many researches over the last years. Many papers related to the attitude control of quad-rotor have been published with different algorithms. Some quad-rotor aircrafts and VTOL testing systems are designed including the fully dynamic modelling [1-4]. For applying classical approaches such as PID, LQ and a direct adaptive control, various simulations and several tests were proposed in [5]. Dynamic feedback controllers of Euler angles and wind parameters estimation were shown with simulation results [6-7]. Recently, modern controllers were promptly applied for attitude control of the QRA. Back-stepping and sliding-mode techniques were utilized with average experimental results [8-9]. Recent papers are interested in real-time attitude stabilization by apply Lyapunov functions [10-12]. To improve the abilities of the QRA, visual feed systems including on-board cameras were used as a primary sensor for trajectory tracking controls [13].
This paper presents a new development of attitude control for a QRA by using a self-tuning fuzzy-PID controller. The PID controller is known as the most popular controller in industrial processes with simple structure and robust performance in the wide range of operating conditions. On the other hand, fuzzy control provides a convenient method for construction nonlinear controllers via the use of heuristic information. Self-tuning fuzzy-PID controller is a combination of wisely used PID control and fuzzy controller known as suitable method for many applications [14-16]. This approach tunes the parameters of PID control by implementation of fuzzy control rules. The fuzzy-PID hybrid controller shows better control simulation performance in transient state with faster settling time than PD and fuzzy controllers. Therefore, it is suitable with the attitude control of a quad-rotor aircraft.
In this paper, some real-time attitude control architectures are developed in simulations. A classical control algorithm is presented as the simple application in Section 3. For improvement the control performances, a basic fuzzy and a self-tuning fuzzy-PID controller are proposed to control the attitude of the QRA in Section 4. Simulation results are showed in Section 5 and conclusion remarks are described in Section 6.
2. 1B1BQUAD-ROTOR AIRCRAFT CONFIGURATION
A QRA is described as a kind of helicopter which has four propellers attached in a rigid cross frame as show in Fig. 1. Two pairs of propellers turn in opposite directions. The pair propellers (front and rear) are running in the same direction (counter clockwise) when two other propellers (left and right) have the clockwise direction.
Fig. 1: Quad-rotor configuration.
This configuration permits the simple controller design and reduces the gyroscopic effects. By changing lift forces, the moving of the aircraft is based on the velocities of each propeller. Up (down) motion is controlled by increase (decrease) same quantity of four propellers’ speeds. Changing left and right propeller’s speeds conversely produce the roll rotation. Similar, the pitch rotation is obtained by changing front and rear propellers velocities. Thus, the forward (backward) motion of the aircraft is generated by increase (decrease) the rear rotor’s speed and decrease (increase) the front rotor’s with maintenance of the total thrust. By the same principle, the left (right) motion is given by roll motion when the total thrust is kept. Finally, the yaw rotation is generated by control reactive moments created from propellers.
Let denotes an inertial frame and denotes a body fixed frame of the aircraft as show as in Fig. 1. The dynamical model described in [5], slightly modified in [12], is given as follow:
(1)
(2)
(3)
(4)
, (5)
where vectordenotes the position of the center of mass of the body-fixed frameand vector denotes the linear velocity of the center of mass of the aircraft and denotes the angular velocity of the quad-rotor expressed in frame. The orientation of the rigid body is given by a rotation matrix. Vector denotes the symmetric positive definite constant inertial matrix around the center of mass. The torque, moment of inertial and the velocity of motorare defined as,and, respectively, whendenotes the gyroscopic effect due to rigid body rotation.
The total thrust applied to the aircraft is
(6)
where is the lift-force generated by the rotor in free air,andare two parameters depending on air’s density, the shape, radius, pitch angle of the rotor blades (see [17] for more details).
The gyroscopic torque of the aircraft due to propellers change in orientation is obtained by
(7)
The airframe torques generated by rotors are following:
(8)
where is the distant between the center of mass of the rigid body to the center of rotors.
In order to make the multiple controllers, the gyroscopic effects from equation (4) can be neglected. Finally, the control inputs of the system are series ofwhich describe the torque produced in equation (8).
To obtain the simulation results described in section 5, a DraganFly Quad-rotor was examined to get required parameters display in Table I.
Table I. Quad-rotor parameters
Symbol / Description / Value / Unitsg
m
d
Ir
Ifx
Ify
Ifz
b
k / Gravity
Mass
Distance
Rotor Inertial
Roll Inertial
Pitch Inertial
Yaw Inertial
Proportionality Constant
Proportionality Constant / 9.81
0.45
0.225
11 x 10-3
4.86 x 10-3
4.86 x 10-3
8.8x 10-3
4.95 x 10-5
1.87 x 10-6 / m/s2
kg
m
kg.m2
kg.m2
kg.m2
kg.m2
3. 2B2BPID CONTROLLER
The PID controller is the best-known in industry as simple control structure, robust performance and easy to design in a wide range of operating conditions.
Fig. 2: PID controller structure.
PID controller has the formal transfer function following form:
(9)
where,andare proportional, integral, derivative gains respectively.
A PD controller was applied as the first step to test a control for the attitude of the aircraft at hover situation.
The PD controller for each orientation angle is introduced from equation (8):
(10)
where whenare the errors of angles for roll, pitch and yaw rotation, respectively.
The PD control parameters were tuned after several simulations on Simulink, Matlab. For these simulations, the dynamic modelling in Equation (4) was used and the results showed in section 5.
4. 3B3BFUZZY CONTROL
This section describes the development of a control strategy for stabilizing the QRA by using fuzzy controllers. As the especially application of knowledge-based system control algorithm, fuzzy control provides a convenient method for constructing nonlinear controllers by using heuristic information. In fuzzy control, linguistic description of human knowledge in controlling and attribute of plant are represented as fuzzy rules of relations.
4.1 6B6BA direct PD-fuzzy controller
For the first approach, direct PD-fuzzy controllers were applied for each orientation angles and its structure is described in Fig. 3. The fuzzy logic controller has two inputs and the first one is the angle error e(t) and another one is the derivation of error de(t). The outputs of fuzzy controller are rotor torques described in Equation (8) for each controller.
Fig. 3: Direct Fuzzy control structure.
The triangular membership functions of these fuzzy set are showed in Fig. 4-7 with linguistic levels assigned as follow: NB: negative big; NSB: negative small big; NM: negative medium; NS: negative small; ZE: zero; PS: positive small; PM: positive medium; PSB: positive small big; PB: positive big and μ is the grade of the membership function.
Fig. 4: Membership functions of e(t).
Fig. 5: Membership functions of de(t).
Fig. 6: Membership functions of Torque (Roll/Pitch).
Fig. 7: Membership functions of Torque (Yaw).
Fig. 8: Fuzzy control rules of Torques.
The maximum error, derivation of error and torques were chosen from the specification of the QRA’s structure and parameters. Relationships between inputs and output are established in 25 rules showed as surface-plotting in Fig. 8. Following these specific membership functions, the Max-Min fuzzy reasoning method were set to determine the output concomitant with applying the centroid method for defuzzification.
As the simulation results presented in Fig. 17 and 18, the fuzzy control provides impressible response for stabilization of the QRA. The fuzzy controller shows smother responses than PD controller but its settling time is larger than PD controller. More discussions are represented in Section 6.
4.2 7B7BA self-tuning fuzzy-PID controller
Having the good results from direct fuzzy controller, a new strategy for the attitude controller is proposed with a self-tuning fuzzy-PID control algorithm. This approach is the combination of a conventional fuzzy and PID controller to combine the advantages of two control responses. The structure of self-tuning fuzzy-PID control is described in Fig. 9. In this approach, the PID parameters are automatically tuned by fuzzy inference [14-16] based on the errorand the derivation of error. The fuzzy controller determines the PID andwithin the inertial parameter boundaries.
Fig. 9: Self-tuning fuzzy-PID control structure.
Assume thatare ranged in respectively. Following the parameters of the QRA in Table I, the PID parameters are determined as: for roll, pitch and yaw rotation.
Linear transformation for and (see [14] for more details) are given by:
(11)
where the parameters determined from fuzzy control and must be standardize in the interval [0,1]. The parameter is examined base on the attribution of the tuningandvalues followingand. The memberships functions ofandare plotted in Fig. 10-12 with triangular linguistic variables. The linguistic levels assigned forare defined as follow: S: Small; L: Low; M: Medium; H: High; B: Big. Fuzzy tuning rules for are presented on Table II, III, and IV with the membership functions ofandthose are similar with Fig. 4 and 5, respectively.
Fig. 10: Membership functions for KP, KD
Fig. 11: Membership functions for β.
Table II. Fuzzy tuning rules of.
de(t)NB / NM / NS / ZE / PS / PM / PB
e(t) / NB / B / B / B / B / B / B / B
NM / MB / MB / B / B / B / MB / MB
NS / M / MB / MB / B / MB / MB / M
ZE / S / MS / MS / M / MS / MS / S
PS / M / MB / MB / B / MB / MB / M
PM / MB / MB / B / B / B / MB / MB
PB / B / B / B / B / B / B / B
Table III. Fuzzy tuning rules of.
de(t)NB / NM / NS / ZE / PS / PM / PB
e(t) / NB / S / S / S / S / S / S / S
NM / M / MS / S / S / S / MS / M
NS / MB / M / MS / MS / MS / M / MB
ZE / B / MB / M / M / M / MB / B
PS / MB / M / MS / MS / MS / M / MB
PM / M / MS / S / S / S / MS / M
PB / S / S / S / S / S / S / S
Table IV. Fuzzy tuning rules of.
de(t)NB / NM / NS / ZE / PS / PM / PB
e(t) / NB / B / B / B / B / B / B / B
NM / M / H / B / B / B / H / M
NS / L / M / H / B / H / M / L
ZE / S / L / M / H / M / L / S
PS / L / M / H / B / H / M / L
PM / M / H / B / B / B / H / M
PB / B / B / B / B / B / B / B
Fig. 12: Fuzzy control rules of KP.
Fig. 13: Fuzzy control rules of KD.
Fig. 14: Fuzzy control rules of β.
5. 4B4BSIMULATION RESULTS
To verify the properties of the above controllers in the attitude control for QRA, the dynamic modelling described in Equation (4) was used with the parameters of Table I. To implement these approaches, some attitude control architectures were designed with Simulink. These controllers were tested in numerous simulations. The goal of the controllers is maintaining the QRA within the initial angle and to return back to the hovering position. Attitude control simulations were also simulated with the bound desired angles. Nevertheless, the results have the same specific attributes. Consequently, the results with the desired angle are not shown in this paper.