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Research Article
 

LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor



S. Faiz Ahmed, Kushsairy Kadir and M. Kamran Joyo
 
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ABSTRACT

Quad-rotor unmanned aerial vehicles have become one of the prominent type of rotorcraft which has been researched massively during the last decade. This study addresses the issue regarding control of quad-rotor under noisy IMU and GPS measurements. Several solutions are provided to resolve the issue of proper controlling of position and altitude of quad-rotor under uncertainties such as noisy measurements but still they are not fully succeeded. This study presents LQR technique for the longitudinal motion control of quad-rotor under noisy sensor measurements. The proposed control technique is simulated on MATLAB. The results of the study show that the applied control technique is effective for altitude and position control of quad-rotor specifically under noisy conditions.

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  How to cite this article:

S. Faiz Ahmed, Kushsairy Kadir and M. Kamran Joyo, 2016. LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor. Journal of Applied Sciences, 16: 588-593.

DOI: 10.3923/jas.2016.588.593

URL: https://scialert.net/abstract/?doi=jas.2016.588.593
 
Received: June 03, 2016; Accepted: August 22, 2016; Published: November 15, 2016



INTRODUCTION

The UAVs have become an eminent subject in the last past years. Rise in their demand is due to their capability of carrying payloads with no life risk. Their ability of spying makes them ideal for military applications. Forces deploy them predominantly for special purpose applications. Their abilities are also utilized in civil applications such as police task force, firefighting, surveillance of pipelines, search and rescue operations, nuclear plant inspections and geo weather surveillance.

Their exist variety of UAVs with different capabilities. Quad-rotor is one of the prominent amongst the rotorcraft type UAVs. It has a simple and unique structural arrangement with distinct functionality as compared to helicopter UAVs. Like helicopters, quad-rotor can hover, Vertically Take Off and Land (VTOL) and can manoeuvre autonomously. Rather than its simple design this UAV has severe issues regarding its dynamic controls due to its nonlinearity. The altitude and position controlling has been an issue over the years. This article introduces a control technique for altitude and position control of a quad-rotor. The proposed control design also minimizes the effect of sensor noises.

In the recent years, altitude and position controlling of quad-rotor has remained a problem due to the constraints of unstable kinematics and dynamics. However, some of the techniques of control were developed in this field. Dynamic Surface Control (DSC) method was used for altitude control and position control1. The PID based position control was applied for position control system of quad-rotor2. Sliding mode and PID control techniques were also implemented onthequad-rotor system3. Adaptive neural controller was used for handling attitude control of quad-rotor4. Stabilizing control laws were presented by sliding mode and backstepping approach5. Fuzzy logic control was implemented on position, altitude and attitude control of quad-rotor6. Backstepping method was used for altitude, attitude and position control of a quad-rotor, the results of this technique showed a flexible control structure. Further quad-rotor was able to perform autonomous hovering with altitude control and autonomous take off and landing7. This study presents a control technique based on LQR controller for altitude and position stabilization. The proposed algorithm is simulated on (MATLAB).

MATERIALS AND METHODS

Quad-rotor modelling: The quad-rotor UAV has four fixed pitch rotors mounted on the ends of a simple cross frame structure as shown in Fig. 1.

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
Fig. 1:Quad-rotor body

In this figure a pair of motors (Q1 and Q3) rotates clockwise while other pair of motors (Q2 and Q4) rotates counter-clockwise direction to counters the effect of opposite torques produced by motors. By changing the combination and rotor speeds of these motors quad-rotor can perform moves in pitch, roll, yaw, hover, take off and landing positions. Pitch and roll movements can be achieved by changing the speed of any one pair of motors while other motor pair speeds remain constant. Yaw movement is done by altering the speed of both motors pairs in quad-rotor and to perform vertical take off and landing tasks, the rotation speed of all four rotors gradually increased and decreased, respectively.

The motors which rotate quad-rotors propellers are brushless DC motors and its dynamics cab be define by its aerodynamics and geometric structure described in8:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(1)

where, KM and τ are motor constant and motor time constant, respectively.

To understand quad-rotor dynamic model, two frames must be defined as a reference which are earth inertial frame (E frame) and quad-rotor fixed-body frame (F frame) as shown in Fig. 2. The quad-rotor orientation and translation vectors are define by three Euler angles ΩT = (φ, θ, ψ) and vector qT = (x, y, z) in the inertial frame. The transformation of vectors of the fixed frame to the inertial frame is given by the resultant transformation matrix of z, y and x axis.

The movements of quad-rotor are achieved by changing the combination and varying the speeds of the motors described as:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
Fig. 2:Quad-rotor body with earth and body frame

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(2)

where, u1, u2, u3 and u4 are the thrust force, roll torque, pitch torque and yaw torque, respectively.

Finally the mathematical modeling of quad-rotor is extracted from above mentioned frames defined in the study9,10:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(3)

Proposed controller design: In this study optimed gain LQR controller is presented for better stabalization and improving the flight quality specially for position and altitude controlling of quad-rotor under uncertanity conditions such sensor and system noises which makes systm unstable11,12.

Figure 3 shows the overall block diagram of proposed control algorithm for stabilizing quad-rotor position/altitude control. The control law for LQR is derived and shown in Fig. 3. The system defined as:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
Fig. 3:LQR based controller

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(4)

The state-feedback controller can be defined as:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(5)

Further LQG state-estimator is constructed:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(6)

The output feedback controller can be obtained by using estimated state rather than true state.

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(7)

Hence, the control law can be defined as:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(8)

The output equation will become:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(9)

Altitude stabilization controller: The altitude controller ensures that the quad-rotorfly at desired distance from the ground. While evaluating altitude, only z-axis equation from Eq. 3 is taken into account with an addition of noise effect on it while other axes are considered to be constant.

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(10)

For hovering condition, φ and θ are kept equivalent to 0 while g and Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor are constant values at each instant, ‘ζ’ is the noise factor which is generated by the sensors. Taking constants as Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotorand taking laplace transformation the system will become:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(11)

By adding rotor dynamics, the system becomes:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(12)

The close loop transfer function can thus be defined as:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(13)

Position controller: Position controller, controls quad-rotor maneuvering in x and y direction as per desired action, these movements are obtain by rolling and pitching the quad-rotor, respectively. For position controlling only X and Y equations will be taken in account from Eq. 3 with considering the system and sensor noise effect on it:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(14)

Table 1:Control and system parameters
Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor

For position controlling we take Ψ= 0 and by considering rotor dynamics and propeller rotation action on translational motion equations and also using small angle approximation and then write it in laplase domain, we can get:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(15)

The close loop transfer function can thus be defined as:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(16)

Similar approach for y-axis may be followed in order to develop the system transfer function:

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
(17)

RESULTS AND DISCUSSION

In the simulations a complete flight path followed by quad-rotor is demonstrated. Noise is introduced in the system which is due to the GPS and SONAR sensors which are measuring position and altitude, respectively. Under noisy conditions quad-rotor response becomes uncertain and challenging for controller to understand the abrupt response. Figure 4 shows the response of a quad-rotor equipped with LQR controller under noisy conditions. The noise added in the system is also shown in Fig. 4. By applying the proposed LQR controller the response is improved and noise is eliminated to high extent.

For simulation purposes the system and control parameters are chosen as in Table 1.

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor

Image for - LQR Based Controller Design for Altitude and Longitudinal Movement of Quad-rotor
Fig. 4(a-c): LQR controller response in x, y and z direction under noisy data measurements

CONCLUSION

In this study the issue regarding quad-rotor position and altitude controlling is elaborated. Due to the effect of system and sensor noises the performance of quad-rotor is degraded. The LQR control technique is proposed inorder to suppress the noises and follow the trajectory smoothly. It was observed that under noisy conditions quad-rotor was able to maneuver smoothly and noises were suppressed by controller itself.

REFERENCES
1:  Lee, K.U., Y.H. Yun, W. Chang, J.B. Park and Y.H. Choi, 2011. Modeling and altitude control of quad-rotor UAV. Proceedings of the 11th International Conference on Control, Automation and Systems, October 26-29, 2011, Gyeonggi-do, South Korea, pp: 1897-1902.

2:  Joyo, M.K., S.F. Ahmed, D. Hazry, M.H. Tanveer and F.A. Warsi, 2013. Position controller design for quad-rotor under perturbed condition. WULFENIA J., 20: 178-189.
Direct Link  |  

3:  Xu, R. and U. Ozguner, 2006. Sliding mode control of a quadrotor helicopter. Proceedings of the 45th IEEE Conference on Decision and Control, December 13-15, 2006, San Diego, CA., pp: 4957-4962.

4:  Shaiful, Z and D. Hazry, 2011. Adaptive neural controller implementation in autonomous mini aircraft quadrotor (AMAC-Q) for attitude control stabilization. Proceedings of the IEEE 7th International Colloquium on Signal Processing and its Applications, March 4-6, 2011, Penang, Malaysia -.

5:  Bouadi, H., M. Bouchoucha and M. Tadjine, 2008. Sliding mode control based on backstepping approach for an UAV type-quadrotor. Int. J. Applied Math. Comput. Sci., 4: 12-17.
Direct Link  |  

6:  Raza, S.A. and W. Gueaieb, 2010. Intelligent Flight Control of an Autonomous Quadrotor. In: Motion Control, Casolo, F. (Ed.). InTech, Croatia, ISBN-13: 9789537619558.

7:  Bouabdallah, S. and R. Siegwart, 2007. Full control of a quad-rotor. Proceedings of the IEEE International Conference on Intelligent Robots and Systems, October 29-November 2, 2007, San Diego, CA., USA., pp: 153-158.

8:  Guenard, N., T. Hamel and L. Eck, 2006. Control laws for the tele operation of an unmanned aerial vehicle known as an x4-flyer. Proceedings of the International Conference on Intelligent Robots and Systems, October 9-15, 2006, Beijing, China, pp: 3249-3254.

9:  Tanveer, M.H., S.F. Ahmed, D. Hazry, F.A. Warsi and M.K. Joyo, 2013. Stabilized controller design for attitude and altitude controlling of quad-rotor under disturbance and noisy conditions. Am. J. Applied Sci., 10: 819-831.
CrossRef  |  Direct Link  |  

10:  Joyo, M.K. D. Hazry, S.F. Ahmed, M.H. Tanveer, F.A. Warsi and A.T. Hussain, 2013. Altitude and horizontal motion control of quadrotor UAV in the presence of air turbulence. Proceedings of the IEEE Conference on Systems, Process and Control, December 13-15, 2013, Kuala Lumpur, pp: 16-20.

11:  Tanveer, M.H., D. Hazry, S.F. Ahmed, M.K. Joyo and F.A. Warsi et al., 2014. NMPC-PID based control structure design for avoiding uncertainties in attitude and altitude tracking control of quad-rotor (UAV). Proceedings of the IEEE 10th International Colloquium on Signal Processing and its Applications, March 7-9, 2014, Kuala Lumpur, pp: 117-122.

12:  Warsi, F.A., D. Hazry, S.F. Ahmed, M.K. Joyo, M.H. Tanveer, H. Kamarudin and Z.M. Razlan, 2014. Yaw, pitch and roll controller design for fixed-wing UAV under uncertainty and perturbed condition. Proceedings of the IEEE 10th International Colloquium on Signal Processing and its Applications, March 7-9, 2014, Kuala Lumpur, pp: 151-156.

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