THE 3-DOF helicopter system is a benchmark laboratory

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1 Vol:8, No:8, 14 LQR Based PID Controller Design for 3-DOF Helicoter System Santosh Kr. Choudhary International Science Index, Electrical and Information Engineering Vol:8, No:8, 14 waset.org/publication/ Abstract In this article, LQR based PID controller design for 3DOF helicoter system is investigated. The 3-DOF helicoter system is a benchmark laboratory model having strongly nonlinear characteristics and unstable dynamics which make the control of such system a challenging task. This article first resents the mathematical model of the 3DOF helicoter system and then illustrates the basic idea and technical formulation for controller design. The aer exlains the simle aroach for the aroximation of PID design arameters from the LQR controller gain matrix. The simulation results show that the investigated controller has both static and dynamic erformance, therefore the stability and the quick control effect can be obtained simultaneously for the 3DOF helicoter system. Keywords 3DOF helicoter system, PID controller, LQR controller, modeling, simulation. I. INTRODUCTION THE 3-DOF helicoter system is a benchmark laboratory model for theoretical study on helicoter controls and verification of the control algorithm. The helicoter is a comlex mechanical system with strongly nonlinear characteristics and has oen-loo unstable dynamics which make the control of such system a challenging task. The urose of the control is to regulate desired itch and roll ositions as well as angular travel seed of the rototye 3DOF helicoter model. The 3-DOF helicoter system (see Fig DOF Helicoter System [1 Fig. 1) consists of base uon which an arm is mounted. The arm carries the helicoter body on one end and a balance block on the other end. The arm can roll about an itch axis as well as swivel about a travel axis. Sensors that are mounted on these axes allow measuring the roll and travel of the arm. The helicoter body is mounted at the end of the arm and is free to Santosh Kr. Choudhary is with the Deartment of Instrumentation and Control Engineering, Manial Institute of Technology, Manial, Karnataka, India, ( santosh.kumar@manial.edu). swivel about a itch axis. The itch angle is measured via third sensor. Two motors with roellers mounted on the helicoter body can generate a force roortional to the voltage alied to the motors.the force generated by the roellers can cause the helicoter body to lift off the ground. The urose of the balance block (counterweight) is to reduce the ower requirements on the motors. The 3DOF helicoter system control involves linearization of the nonlinear dynamics about a set of re-selected equilibrium conditions within the flight enveloe. Based on the obtained linear models, classical single inut single-outut (SISO) techniques with a PID controller are widely used [ [5, etc. In another control aroach, otimal tracking strategy using Linear quadratic regulator (LQR) control for helicoter model was roosed in [6. Fuzzy and PID combined control used for an unmanned helicoter was discussed in [7. Conventional PID controller s tuning methods seem inadequate for achieving better control erformance because of bad tuning of arameters therefore these situations give strong motivation to LQR based PID control strategy for helicoter model. In this article, LQR based PID controller design for 3DOF helicoter system is investigated. The aer begins with mathematical modeling of the 3DOF helicoter system and then controller design methodology is briefly discussed to deal with both erformance and stability for the system. The design roblem is then dealt with finding a LQR controller gain matrix, which gives a control solution. Finally an aroximation method is suggested for finding the design arameters for PID controller from the obtained LQR controller gain matrix. The whole rocedure involves selecting several arameters and the comutation is simle, so it serves as a PID tuning method for 3DOF helicoter system. The simulation results show that the method is easy to use and the resulting PID controller has good time-domain erformance and robustness of the system. The rest of the article is organized as follows: In Section II, we resent a mathematical modelling of helicoter system. Section III illustrates basic ideas and technical formulations for controller design. Section IV discuss about the results and simulation analysis and Section V concludes the aer with some remarks and conclusion. II. MATHEMATICAL MODELLING OF HELICOPTER SYSTEM DYNAMICS Understanding the flight behavior has become essential to ensure control. Helicoter models are now well established [8, International Scholarly and Scientific Research & Innovation 8(8) scholar.waset.org/ /

2 Vol:8, No:8, 14 [9 within the reach of many fields (academic and commercial uroses). Indeed, the ability to describe and exlain the various henomena involved and interacting in the helicoter dynamics has a large imact in ractice. The aim of modeling is then to evaluate and control as soon as ossible. C. Travel Axis Model A. Pitch Axis Model International Science Index, Electrical and Information Engineering Vol:8, No:8, 14 waset.org/publication/ Fig.. Schematic Diagram for Pitch Axis Model Consider the diagram in Fig.. Assuming the roll is zero, then the itch axis dynamics of 3-DOF helicoter system can be modelled as: J e = l 1 F h l 1 G = l 1 (F 1 + F ) l 1 G = K c l 1 (V 1 + V ) T g J e = K c l 1 V s T g (1) where is the itch angle, J e = m h l 1 + m b l is the moment of inertia of the system about itch axis, m b is the mass of balance blocks, m h is the total mass of two roeller motor, V 1 and V are the voltages alied to the front and back motors resulting in force F 1 and F, K c is the force constant of the motor-roeller combination, l 1 is the distance from the ivot oint to the roeller motor, l is the distance from the ivot oint to the balance blocks, T g is the effective gravitational torque due to the mass differential G about the itch axis. Fig. 3. B. Roll Axis Model Schematic Diagram for Roll Axis Model Consider the diagram in Fig. 3. If the force generated by the left motor is higher than the force generated by the right motor, the helicoter body will be overturned clockwise. The roll axis dynamics of 3-DOF helicoter system can be modelled as: J = F 1 l F l = K c l (V 1 V ) J = K c l V d () where is the roll angle, J is the moment of inertia of the system about the roll axis, l is the distance from the roll axis to either motor. Fig. 4. Schematic Diagram for Travel Axis Model When the roll axis is tilted and overturned, the horizontal comonent of G will cause a torque about that the travel axis which results in an acceleration about the travel axis. Assume the body has roll u by an angle as shown in Fig. 4. The travel axis dynamics of 3-DOF helicoter system can be modelled as: J t ṙ = G sin()l 1 J t ṙ = K sin()l 1 (3) where J t is the moment of inertia of the system about the travel axis, r is the travel rate in rad/sec, K is the force required to maintain the helicoter in flight and is aroximately G, sin() is the trigonometric sin of the roll angle. Remark 1: If the roll angle is zero, no force is transmitted along the travel axis. A ositive roll causes a negative acceleration in the travel direction. Remark : To design a seed controller for the travel axis, we consider the model (3) but for a osition controller, we need to consider the travel differential equation as:j t r = K sin()l 1. D. State Sace Model of 3 DOF Helicoter system The simlified state sace dynamic model of 3 DOF helicoter system relies on the following assumtions: All angles are sufficiently small so that the linear aroximation is valid. The couling dynamics is neglected. The gravitational torque T g is neglected. The frictions forces are neglected. Under the above assumtions, (1), () and (3) for the overall motion of the helicoter can be effectively reduced to these following equations: = Kcl1 J e U 1 = Kcl J U (4) ṙ = Gl1 J t where control inuts U 1 and U are given by { U1 = Vs+V d U = Vs V d (5) International Scholarly and Scientific Research & Innovation 8(8) scholar.waset.org/ /

3 Vol:8, No:8, 14 International Science Index, Electrical and Information Engineering Vol:8, No:8, 14 waset.org/publication/ Remark 3: We calculate the inut voltage U 1 and U for each motor. Finally, using (4), the sate sace model of the helicoter is formulated as below: 1 1 K cl 1 K cl 1 ṗ = J e J e [ ṗ + K cl r Gl 1 Kcl U1 J J U J ξ t r 1 ξ γ 1 γ (6) 1 [ U1 = 1 ṗ + (7) U r 1 r ξ γ d dt where, and r are the itch angle, roll angle and travel rate of the 3-DOF helicoter system resectively and ξ = and γ = r. TABLE I PHYSICAL PARAMETERS OF 3-DOF HELICOPTER SYSTEM Symbol Physical unit Numerical values J e Kg.m J t Kg.m J Kg.m.319 G N l 1 m.88 l m.35 l m.17 K c N/V 1 In order to obtain the linear 3-DOF state sace model, we consider the hysical arameter s value from Table I. III. CONTROLLER DESIGN This section is dedicated to design three PID controllers to allow us to command a desired itch osition, roll angle and travel rate of 3-DOF helicoter system. In this work, PID controller s design arameters are aroximated from the LQR controller gain matrix. A. PID Equation If c, c and r c are the desired itch angle, desired roll angle and desired travel rate of helicoter system, we can exress the PID controllers in the following form to meet closed loo exectations: Pitch Control Equation: V s = K ( c )+K d + K i ( c ) dt (8) Roll Control Equation: V d = K ( c )+K d ṗ (9) Travel Control Equation: c = K r (r r c )+K ri (r r c ) dt (1) Remark 4: In order to achieve a desired travel rate r c,we design a controller to command a desired roll angle c. B. LQR Controller Linear-quadratic-regulator (LQR) is a art of the otimal control strategy [1 which has been widely develoed and used in various alications. LQR design is based on the selection of feedback gains K such that the cost function or erformance index J is minimized. This ensures that the gain selection is otimal for the cost function secified. An advantage of using the LQR otimal control scheme is that system designed will be stable and robust, excet in the case where the system is not controllable. In this design method, the system must be described by state sace model: {ẋ = Ax + Bu (11) y = Cx + Du The erformance index J is defined as: J = (x T Qx + u T Ru)dt (1) Where Q is ositive definite (or semi-ositive definite) Hermitian or a real symmetric matrix. and R is ositive definite Hermitian or real symmetric matrix. These two matrices Q and R are often known as weighting matrices. The LQR otimal control rincile [1 for the system defined in (11) is to determine matrix K that gives the otimal control vector u(t) = Kx(t) (13) such that erformance index J is minimized. The control gain matrix K is given by K = R 1 B T P (14) and P can be found by solving the continuous time algebraic Riccati equation: A T P + PA PBR 1 B T P + Q = (15) The block diagram showing the LQR control system configuration is shown in Fig. 5. Fig. 5. LQR Control Systems Configuration The crucial and difficult task in the LQR controller design International Scholarly and Scientific Research & Innovation 8(8) scholar.waset.org/ /

4 Vol:8, No:8, 14 International Science Index, Electrical and Information Engineering Vol:8, No:8, 14 waset.org/publication/ is a choice of the weighting matrices. We generally select weighting matrices Q and R to satisfy exected erformance criterion. The different Q and R values give a different system resonse. The system will be more robust to disturbance and the settling time will be shorter if Q is larger (in a certain range). But there is no straightforward way to select these weighting matrices and it is usually done through an iterative simulation rocess. In this article, we aly the Bryson s rule for weighting matrix selection and select the matrices Q and R for the 3-helicoter system in the following manner: Q = ρ α 1 (max) α (max) α 3 ( max) α 4 (ṗmax) α 5 (rmax).. =...1 R = ρ [ 1 = 1 where: β 1 (U 1max ) β (U max ) α 6 (ξmax) α 7 (γmax) (16) (17) max, max, max, ṗ max,r max,ξ max and γ max reresent the resective largest desired resonse for that comonent of the states. U 1max and U max are maximum accetable control voltage inuts for actuator signal. 7 αi =1and =1are used to add an additional i=1 βi i=1 relative weighting on the various comonents of the state and control resectively. ρ is used as the last relative weighting between the control and state enalties which gives a relatively concrete way to discuss the relative size of Q and R and their ratio Q/R. The synthesis of a state feedback controller K is obtained according to the LQR control system configuration shown in the Fig. 5. Using MATLAB code K=lqr(A,B,Q,R), we obtain controller gain as follows: K = [ C. PID Aroximation (18) In this subsection PID controller s design arameter K,K i and K d are aroximated from the LQR controller gain matrix (18). First, we write the LQR otimal control law using (13) and (18) for the 3-DOF helicoter system as : [ U1 U = [ ṗ r ξ γ [ k11 k 1 k 13 k 14 k 15 k 16 k 17 = x(t) (19) k 1 k k 3 k 4 k 5 k 6 k 7 Analyzing carefully (19), we can obtain: [ U1 U = [ k 11 k 1 k 13 k 14 k 15 k 16 k 17 k 11 k 1 k 13 k 14 k 15 k 16 k 17 Now, the sum of the rows of () results in: U 1 + U = (k 11 +k 13 +k 16 ξ) = (k 11 +k 13 +k 16 ṗ r ξ γ ) dt () (1) The full state feedback results in a controller that feedback two voltages, therefore (1) can be written as: V s = k 11 ( c ) k 13 k 16 ( c ) dt () Equations (8) and () have the same structure, this means that the gains we obtain from LQR design can still be used for the itch PID controller. Thus, comaring () with (8), we can get the following itch PID controller design arameters: K = k 11 K d = k 13 (3) K i = k 16 Similarly, the difference of the rows of () results in: U 1 U = k 1 k 14 ṗ k 15 (r r c) k 17 γ V d = k 1 k 14 ṗ k 15 (r r c) k 17 Now, using PID equation (1) in (9), we can get, V d = K + K d ṗ K K r (r r c) K K ri (r r c) dt (4) (r r c) dt (5) Equations (4) and (5) have exactly the same structure. The TABLE II PID DESIGN GAINS VALUE PID arameters Relationshi Absolute Value K k K d k K i k 16. K k K d k K r k 15 k K ri k 17 k means that we can design an LQR controller and still maintain the same structure we used reviously. International Scholarly and Scientific Research & Innovation 8(8) scholar.waset.org/ /

5 Vol:8, No:8, 14 International Science Index, Electrical and Information Engineering Vol:8, No:8, 14 waset.org/publication/ On comaring (4) and (5), the feedback gains for the controller are obtained from the LQR gains as: K = k 1 K d = k 14 K r = k15 k 1 (6) K ri = k17 k 1 The design we have erformed, resulting in the controller gain values and it is shown in the Table II and got PID controller, PD controller and PI controller for itch, roll and travel axis model resectively of 3DOF helicoter system. Remark 5: Due to the minimum-hase requirement of a PID controller, the signs of the roortional gain, the integral gain and the differential gain must be the same. Therefore, absolute gain values are considered to avoid the roblem of different signs. IV. SIMULATION AND RESULTS ANALYSIS We now resent the erformance analysis for 3DOF helicoter feedback system through MATLAB simulation. Simulations are carried out for different axis: itch, roll and travel axis. The control objective is to get good erformances in dynamics as well as in statics. In the simulation, the reference signal for the itch, travel and roll angles are changed between to to simulate the demands given by the ilot. Also samling time and simulation time used are.1 and 6 seconds resectively. Pitch Angle (degree) Fig Linear Simulation Results Time (seconds) PID control simulation of the itch axis model B. Roll Axis Model Simulation The aim of this art is to design a controller that will allow us to command the roll movement. The tracking curve of the reference inut signal to PD control of the roll axis model is shown in the Fig. 7. From the figure it can be seen that steady state have been comletely obtained. The system has small overshoot which is settled within a fraction of a second. Roll Angle (degree) Fig. 7. Travel Angle (degree) Time (seconds) Fig PD control simulation of the roll axis model Time (seconds) PI control simulation of the travel axis model A. Pitch Axis Model Simulation The control objective is to minimize the error between the desired itch ositions with the helicoter itch axis osition. The Fig. 6 illustrates the tracking curve of reference inut with PID controller for itch axis model of the 3DOF helicoter system. Simulation result shows that PID control for itch axis has negligible overshoot and shorter settling time and is able to track the desired resonse, which is considerable system erformance. C. Travel Axis Model Simulation This section is dedicated to control of the travel rate of the helicoter by means of PI controller. The Fig. 8 shows the control curve of helicoter travel rate. It is cleared from the simulation result that PI control of the travel axis model is able to track desired resonse and has no overshoot. The simulation also shows that there is negligible steady state tracking error but overall tracking erformance looks accetable. International Scholarly and Scientific Research & Innovation 8(8) scholar.waset.org/ /

6 Vol:8, No:8, 14 V. CONCLUSION In this work, LQR based PID controller design for 3DOF helicoter system is investigated. The 3DOF helicoter dynamical equations along with state sace model are first resented in the aer and then the basic ideas and technical formulations for designing a PID controller are briefly illustrated. This aer resents a simle aroach for designing a PID controller based on the aroximation of design arameters from the LQR controller gain matrix. The investigated controller has both static and dynamic erformance, therefore the stability and the quick control effect can be obtained simultaneously. The robustness of the designed controller is suerior in terms of less overshoot, short settling time and stable tracking of reference inuts. Santosh Kr. Choudhary received the M.Tech in Astronomy & Sace Engineering in 9 from Manial University, Manial, Karnataka and B.Sc(Hons) and M.Sc in Mathematics from Magadh University, Bodh-Gaya, Bihar, India in 1 and 3 resectively. He is currently working as an assistant rofessor in MIT, Manial University, Manial, India. He is also the life member of Indian Society of Technical Education (ISTE), Indian Society of Systems for Science and Engineering (ISSE), Asian Control Association (ACA), Society for Industrial and Alied Mathematics (SIAM) and International Association of Engineers (IAENG). His area of research interests includes control theory, scientific comuting, robust & otimal control, modeling & simulation, sacecraft dynamics & control, industrial & alied mathematics and orbital mechanics. International Science Index, Electrical and Information Engineering Vol:8, No:8, 14 waset.org/publication/ ACKNOWLEDGEMENTS The author would like to acknowledge the suort of Manial Institute of Technology, Manial University, Manial, India. REFERENCES [1 3DOF Helicoter Exeriment manual, Googol Technology Ltd., Kowloon, Hon Kong, 9. [ H. Rios, A. Rosales, A. Ferreira, and A. Davilay, Robust regulation for a 3-dof helicoter via sliding-modes control and observation techniques, in American Control Conference (ACC), 1, June 1, [3 S. Lee, C. Ha, and B. Kim, Adative nonlinear control system design for helicoter robust command augmentation, Aerosace Science and Technology, vol. 9, no. 3, , 5. [4 J. Gao, X. Xu, and C. He, A study on the control methods based on 3-dof helicoter model. journal of Comuters, vol. 7, no. 1, , 1. [5 A. Boubakir, S. Labiod, F. Boudjema, and F. Plestan, Design and exerimentation of a self-tuning id control alied to the 3dof helicoter, Archives of Control Sciences, vol. 3, no. 3, , 13. [6 Z. Liu, Z. Choukri El Haj, and H. Shi, Control strategy design based on fuzzy logic and lqr for 3-dof helicoter model, in Intelligent Control and Information Processing (ICICIP), 1 International Conference on, Aug 1, [7 E. N. Sanchez, H. Becerra, and C. Velez, Combining fuzzy and id control for an unmanned helicoter, in Fuzzy Information Processing Society, 5. NAFIPS 5. Annual Meeting of the North American, June 5, [8 M. Boukhnifer, A. Chaibet, and C. Larouci, H-infinity robust control of 3-dof helicoter, in Systems, Signals and Devices (SSD), 1 9th International Multi-Conference on, March 1, [9 Z. Yujia, N. Mohamed, N. Hazem, and A. Yasse, Model redictive control of a 3-dof helicoter system using successive linearization, International Journal of Engineering, Science and Technology, vol., no. 31,. 9 19, 1. [1 D. S. Naidu, Otimal Control Systems. CRC Press,. International Scholarly and Scientific Research & Innovation 8(8) scholar.waset.org/ /