October  2017, 13(4): 2067-2091. doi: 10.3934/jimo.2017032

Comparison of GA and PSO approaches for the direct and LQR tuning of a multirotor PD controller

Faculty of Engineering and Architecture, Cittadella Universitaria -Enna, Italy

* Corresponding author

Received  September 2015 Revised  October 2016 Published  April 2017

In the present paper different approaches for flight control stabilization tuning of a prototype of Unmanned Aerial Vehicle (UAV) are investigated. The mathematical model of a multirotor airframe is presented and its dynamical system is described by means of Newton-Euler equations for a rigid body. The proposed controller systems for the stabilization of the altitude and the attitude of the multirotor around its hovering configuration are based on Proportional Derivative (PD) regulator tuned by optimization techniques such as Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) or, for the linearized system, by means of Linear Quadratic Regulator (LQR). In this case, PSO and GA are applied to set entries of LQR scheme. Control flight simulation results have been compared in terms of minimization of work carried out by the control actions. Performed simulations pointed out that all approaches lead to zero the error of the position along gravity acceleration direction, stop the rotation of UAV around body axes and stabilize the multirotor. The direct PSO tuning of the PD parameters (PD-PSO) shows better results in terms of errors, performance and speed of convergence.

Citation: Valeria Artale, Cristina L. R. Milazzo, Calogero Orlando, Angela Ricciardello. Comparison of GA and PSO approaches for the direct and LQR tuning of a multirotor PD controller. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2067-2091. doi: 10.3934/jimo.2017032
References:
[1]

Y. M. Al-YounesM. A. Al-Jarrah and A. A. Jhemi, Linear vs. nonlinear control techniques for a quadrotor vehicle, Mechatronics and its Applications (ISMA), 2010 7th International Symposium on. IEEE, (2010), 1-10.   Google Scholar

[2]

K. H. AngG. Chong and Y. Li, PID control system analysis, design, and technology, IEEE Transactions on Control Systems Technology, 13 (2005), 559-576.   Google Scholar

[3]

V. Artale, C. L. Milazzo, C. Orlando and A. Ricciardello, Genetic algorithm applied to the stabilization control of a hexarotor, AIP Conference Proceedings, 1648 (2015), 780003. Google Scholar

[4]

R. AruneshwaranS. SureshJ. Wang and T. K. Venugopalan, Neural adaptive fight controller for ducted-fan UAV performing nonlinear maneuver, IEEE Symposium on Computational Intelligence for Security and Defense Applications (CISDA), (2013), 51-56.   Google Scholar

[5]

S. BouabdallahP. Murrieri and R. Siegwart, Design and control of an indoor micro quadrotor, Proceedings of IEEE International Conference on Robotics and Automation, 5 (2004), 4939-4398.   Google Scholar

[6]

S. BouabdallahA. Noth and R. Siegwart, PID vs LQ control techniques applied to an indoor micro quadrotor, proceedings of IEEE International Conference on Intelligent Robots and Systems IROS, (2004), 2451-2456.   Google Scholar

[7]

H. BoubertakhS. Bencharef and S. Labiod, PSO-based PID control design for the stabilization of a quadrotor, Systems and Control (ICSC), 2013 3rd International Conference on, (2013), 514-517.   Google Scholar

[8]

P. CastilloR. Lozano and A. Dzul, Stabilization of a mini rotorcraft with four rotors, Control Systems, IEEE, 25 (2005), 45-55.  doi: 10.1109/MCS.2005.1550152.  Google Scholar

[9]

F. G. CastellanosN. MarchandS. Lesecq and J. Delamare, Bounded attitude stabilization: Real-time application on four-rotor mini-helicopter, Proceedings of the 17th World Congress The International Federation of Automatic Control, (2008), 3167-3173.   Google Scholar

[10]

H. ChenK. Chang and C. S. Agate, UAV path planning with tangent-plus-lyapunov vector field guidance and obstacle avoidance, IEEE Transactions on Aerospace and Electronic Systems, 49 (2013), 840-856.   Google Scholar

[11]

M. CutlerN. K. UreB. Michini and J. P. How, Comparison of fixed and variable pitch actuators for agile quadrotors, AIAA Conf. on Guidance, Navigation and Control, Portland, OR, (2011), 2011-6406.   Google Scholar

[12]

M. Cutler and J. P. How, Actuator constrained trajectory generation and control for variable-pitch quadrotors, AIAA Guidance, Navigation, and Control Conference (GNC), (2012), 2012-4777.   Google Scholar

[13]

M. A. Duarte-Mermoud and R. A. Prieto, Performance index for quality response of dynamical systems, ISA Transactions, 43 (2004), 133-151.   Google Scholar

[14]

G. A. Garcia and S. Keshmiri, Nonlinear Model Predictive Controller for Navigation, Guidance and Control of a Fixed-Wing UAV, AIAA Guidance, Navigation, and Control Conference, 2011. Google Scholar

[15]

D. GreerP. McKerrow and J. Abrantes, Robots in urban search and rescue operation, Proceedings of Australasian Conference on Robotics and Automation ACRA, (2002), 25-30.   Google Scholar

[16]

J. M. Herrero, X. Blasco, M. Martínez and J. V. Salcedo, Optimal PID Tuning With Genetic Algorithms for Non Linear Process Models, IFAC 15th Triennial World Congress, 2002. Google Scholar

[17]

H. A. IsmailM. S. PackianatherR. I. Grosvenor and E. E. Eldhukri, The application of IWO in LQR controller design for the Robogymnast, SAI Intelligent Systems Conference (IntelliSys). IEEE, (2015), 274-279.   Google Scholar

[18]

R. A. Krohling and P. R. Joost, Design of optimal disturbance rejection PID controllers using genetic algorithms, Evolutionary Computation, IEEE Transactions, 5 (2001), 78-82.   Google Scholar

[19]

A. A. Mian and W. Daoboo, Modeling and backstepping-based nonlinear control strategy for a 6 DOF quadrotor helicopter, Chinese Journal of Aeronautics, 21 (2008), 261-268.   Google Scholar

[20]

A. MoharamM. A. El-Hosseini and H. A. Ali, Design of optimal PID controller using hybrid differential evolution and particle swarm optimization with an aging leader and challengers, Applied Soft Computing, 38 (2016), 727-737.   Google Scholar

[21]

S. Nesmachnow, An overview of metaheuristics: Accurate and efficient methods for optimisation, International Journal of Metaheuristics, 3 (2014), 320-347.   Google Scholar

[22]

D. NodlandA. GhoshH. Zargarzadeh and S. Jagannathan, Neuro-optimal control of an unmanned helicopter, Journal of Defense Modeling and Simulation: Applications, Methodology, Technology, 11 (2014), 5-18.   Google Scholar

[23]

T. PenchevaK. Atanassov and A. Shannon, Modelling of a stochastic universal sampling selection operator in genetic algorithms using generalized nets, Proceedings of the Tenth International Workshop on Generalized Nets, Sofia, (2009), 1-7.   Google Scholar

[24]

G. J. J. Ruijgrok, Elements of Airplane Performance, Delft University Press, 1990. Google Scholar

[25]

A. L. S. SalihM. MoghavvemiH. A. F. Mohamed and K. S. Gaeid, Flight PID controller design for a UAV quadrotor, Scientific Research and Essays, 5 (2010), 3660-3667.   Google Scholar

[26]

R. F. Stengel, Flight Dynamics, Princeton University Press, 2004. Google Scholar

[27]

R. Umarani and V. Selvi, Particle swarm optimization evolution, overview and applications, International Journal of Engineering Science and Technology, 2 (2010), 2802-2806.   Google Scholar

[28]

S. WangB. Li and O. Geng, Research of RBF neural network PID control algorithm for longitudinal channel control of small UAV, Control and Automation (ICCA), 2013 10th IEEE International Conference on, (2013), 1824-1827.   Google Scholar

[29]

Z. XieY. Xia and M. Fu, Robust trajectory-tracking method for UAV using nonlinear dynamic inversion, IEEE 5th International Conference on Cybernetics and Intelligent Systems (CIS), (2011), 93-98.   Google Scholar

[30]

R. ZhangQ. Quan and K. Y. Cai, Attitude control of quadrotor aircraft subject to a class of time-varying disturbances, IET Control Theory and Applications, 5 (2011), 1140-1146.  doi: 10.1049/iet-cta.2010.0273.  Google Scholar

[31]

J. G. Ziegler and N. B. Nichols, Optimum settings for automatic controllers, Journal of Dynamic Systems, Measurement, and Control, 115 (1993), 220-222.   Google Scholar

show all references

References:
[1]

Y. M. Al-YounesM. A. Al-Jarrah and A. A. Jhemi, Linear vs. nonlinear control techniques for a quadrotor vehicle, Mechatronics and its Applications (ISMA), 2010 7th International Symposium on. IEEE, (2010), 1-10.   Google Scholar

[2]

K. H. AngG. Chong and Y. Li, PID control system analysis, design, and technology, IEEE Transactions on Control Systems Technology, 13 (2005), 559-576.   Google Scholar

[3]

V. Artale, C. L. Milazzo, C. Orlando and A. Ricciardello, Genetic algorithm applied to the stabilization control of a hexarotor, AIP Conference Proceedings, 1648 (2015), 780003. Google Scholar

[4]

R. AruneshwaranS. SureshJ. Wang and T. K. Venugopalan, Neural adaptive fight controller for ducted-fan UAV performing nonlinear maneuver, IEEE Symposium on Computational Intelligence for Security and Defense Applications (CISDA), (2013), 51-56.   Google Scholar

[5]

S. BouabdallahP. Murrieri and R. Siegwart, Design and control of an indoor micro quadrotor, Proceedings of IEEE International Conference on Robotics and Automation, 5 (2004), 4939-4398.   Google Scholar

[6]

S. BouabdallahA. Noth and R. Siegwart, PID vs LQ control techniques applied to an indoor micro quadrotor, proceedings of IEEE International Conference on Intelligent Robots and Systems IROS, (2004), 2451-2456.   Google Scholar

[7]

H. BoubertakhS. Bencharef and S. Labiod, PSO-based PID control design for the stabilization of a quadrotor, Systems and Control (ICSC), 2013 3rd International Conference on, (2013), 514-517.   Google Scholar

[8]

P. CastilloR. Lozano and A. Dzul, Stabilization of a mini rotorcraft with four rotors, Control Systems, IEEE, 25 (2005), 45-55.  doi: 10.1109/MCS.2005.1550152.  Google Scholar

[9]

F. G. CastellanosN. MarchandS. Lesecq and J. Delamare, Bounded attitude stabilization: Real-time application on four-rotor mini-helicopter, Proceedings of the 17th World Congress The International Federation of Automatic Control, (2008), 3167-3173.   Google Scholar

[10]

H. ChenK. Chang and C. S. Agate, UAV path planning with tangent-plus-lyapunov vector field guidance and obstacle avoidance, IEEE Transactions on Aerospace and Electronic Systems, 49 (2013), 840-856.   Google Scholar

[11]

M. CutlerN. K. UreB. Michini and J. P. How, Comparison of fixed and variable pitch actuators for agile quadrotors, AIAA Conf. on Guidance, Navigation and Control, Portland, OR, (2011), 2011-6406.   Google Scholar

[12]

M. Cutler and J. P. How, Actuator constrained trajectory generation and control for variable-pitch quadrotors, AIAA Guidance, Navigation, and Control Conference (GNC), (2012), 2012-4777.   Google Scholar

[13]

M. A. Duarte-Mermoud and R. A. Prieto, Performance index for quality response of dynamical systems, ISA Transactions, 43 (2004), 133-151.   Google Scholar

[14]

G. A. Garcia and S. Keshmiri, Nonlinear Model Predictive Controller for Navigation, Guidance and Control of a Fixed-Wing UAV, AIAA Guidance, Navigation, and Control Conference, 2011. Google Scholar

[15]

D. GreerP. McKerrow and J. Abrantes, Robots in urban search and rescue operation, Proceedings of Australasian Conference on Robotics and Automation ACRA, (2002), 25-30.   Google Scholar

[16]

J. M. Herrero, X. Blasco, M. Martínez and J. V. Salcedo, Optimal PID Tuning With Genetic Algorithms for Non Linear Process Models, IFAC 15th Triennial World Congress, 2002. Google Scholar

[17]

H. A. IsmailM. S. PackianatherR. I. Grosvenor and E. E. Eldhukri, The application of IWO in LQR controller design for the Robogymnast, SAI Intelligent Systems Conference (IntelliSys). IEEE, (2015), 274-279.   Google Scholar

[18]

R. A. Krohling and P. R. Joost, Design of optimal disturbance rejection PID controllers using genetic algorithms, Evolutionary Computation, IEEE Transactions, 5 (2001), 78-82.   Google Scholar

[19]

A. A. Mian and W. Daoboo, Modeling and backstepping-based nonlinear control strategy for a 6 DOF quadrotor helicopter, Chinese Journal of Aeronautics, 21 (2008), 261-268.   Google Scholar

[20]

A. MoharamM. A. El-Hosseini and H. A. Ali, Design of optimal PID controller using hybrid differential evolution and particle swarm optimization with an aging leader and challengers, Applied Soft Computing, 38 (2016), 727-737.   Google Scholar

[21]

S. Nesmachnow, An overview of metaheuristics: Accurate and efficient methods for optimisation, International Journal of Metaheuristics, 3 (2014), 320-347.   Google Scholar

[22]

D. NodlandA. GhoshH. Zargarzadeh and S. Jagannathan, Neuro-optimal control of an unmanned helicopter, Journal of Defense Modeling and Simulation: Applications, Methodology, Technology, 11 (2014), 5-18.   Google Scholar

[23]

T. PenchevaK. Atanassov and A. Shannon, Modelling of a stochastic universal sampling selection operator in genetic algorithms using generalized nets, Proceedings of the Tenth International Workshop on Generalized Nets, Sofia, (2009), 1-7.   Google Scholar

[24]

G. J. J. Ruijgrok, Elements of Airplane Performance, Delft University Press, 1990. Google Scholar

[25]

A. L. S. SalihM. MoghavvemiH. A. F. Mohamed and K. S. Gaeid, Flight PID controller design for a UAV quadrotor, Scientific Research and Essays, 5 (2010), 3660-3667.   Google Scholar

[26]

R. F. Stengel, Flight Dynamics, Princeton University Press, 2004. Google Scholar

[27]

R. Umarani and V. Selvi, Particle swarm optimization evolution, overview and applications, International Journal of Engineering Science and Technology, 2 (2010), 2802-2806.   Google Scholar

[28]

S. WangB. Li and O. Geng, Research of RBF neural network PID control algorithm for longitudinal channel control of small UAV, Control and Automation (ICCA), 2013 10th IEEE International Conference on, (2013), 1824-1827.   Google Scholar

[29]

Z. XieY. Xia and M. Fu, Robust trajectory-tracking method for UAV using nonlinear dynamic inversion, IEEE 5th International Conference on Cybernetics and Intelligent Systems (CIS), (2011), 93-98.   Google Scholar

[30]

R. ZhangQ. Quan and K. Y. Cai, Attitude control of quadrotor aircraft subject to a class of time-varying disturbances, IET Control Theory and Applications, 5 (2011), 1140-1146.  doi: 10.1049/iet-cta.2010.0273.  Google Scholar

[31]

J. G. Ziegler and N. B. Nichols, Optimum settings for automatic controllers, Journal of Dynamic Systems, Measurement, and Control, 115 (1993), 220-222.   Google Scholar

Figure 1.  Convergence trend for GA-PD algorithm: 10 runs are represented
Figure 2.  Convergence trend for PSO-PD algorithm: 10 runs are represented
Figure 3.  Convergence trend for GA-LQR-PD algorithm: 10 runs are represented
Figure 4.  Convergence trend for PSO-LQR-PD algorithm: 10 runs are represented
Figure 5.  Root locus of the controlled systems
Figure 6.  Comparison between altitude trend versus time in GAPD case (red line) and in PSO-PD case (black line)
Figure 7.  Comparison between roll trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Figure 8.  Comparison between pitch trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Figure 9.  Comparison between yaw trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Figure 10.  Comparison between thrust trend versus time in GAPD case (red line) and in PSO-PD case (black line)
Figure 11.  Comparison between roll moment trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Figure 12.  Comparison between pitch moment trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Figure 13.  Comparison between yaw moment trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Figure 14.  Comparison between altitude trend versus time in GA-LQR-PD case (red line) and in PSO-LQR-PD case (black line)
Figure 15.  Comparison between roll trend versus time in GALQR-PD case (red line) and in PSO-LQR-PD case (black line)
Figure 16.  Comparison between pitch trend versus time in GALQR-PD case (red line) and in PSO-LQR-PD case (black line)
Figure 17.  Comparison between yaw trend versus time in GALQR-PD case (red line) and in PSO-LQR-PD case (black line)
Figure 18.  Comparison between thrust trend versus time in GALQR-PD case (red line) and in PSO-LQR-PD case (black line)
Figure 19.  Comparison between roll moment trend versus time in GA-LQR-PD case (red line) and in PSO-LQR-PD case (black line)
Figure 20.  Comparison between pitch moment trend versus time in GA-LQR-PD case (red line) and in PSO-LQR-PD case (black line)
Figure 21.  Comparison between yaw moment trend versus time in GA-LQR-PD case (red line) and in PSO-LQR-PD case (black line)
Figure 22.  Air turbulence: altitude time history
Figure 23.  Air turbulence: roll time history
Figure 24.  Air turbulence: pitch time history
Figure 25.  Air turbulence: yaw time history
Figure 26.  Air turbulence: thrust time history
Figure 27.  Air turbulence: roll moment time history
Figure 28.  Air turbulence: pitch moment time history
Figure 29.  Air turbulence: yaw moment time history
Table 1.  Proportional and Derivative coefficients gained by GA-PD and PSO-PD algorithms
GA-PD PSO-PD
i $K_{i, P}$ $K_{i, D}$ $K_{i, P}$ $K_{i, D}$
$z$ $38327$ $923.59$ $46879$ $ 1000$
$\phi$ $19776$ $632.13$ $20000$ $1000$
$\theta$ $16918$ $246.83$ $20000$ $1000$
$\psi$ $18142$ $162.16$ $20000$ $1000$
GA-PD PSO-PD
i $K_{i, P}$ $K_{i, D}$ $K_{i, P}$ $K_{i, D}$
$z$ $38327$ $923.59$ $46879$ $ 1000$
$\phi$ $19776$ $632.13$ $20000$ $1000$
$\theta$ $16918$ $246.83$ $20000$ $1000$
$\psi$ $18142$ $162.16$ $20000$ $1000$
Table 2.  Proportional and Derivative coefficients gained by GA-LQR-PD and PSO-LQR-PD algorithm
GA-LQR-PD PSO-LQR-PD
i $K_{i, P}$ $K_{i, D}$ $K_{i, P}$ $K_{i, D}$
$z$ $32.89$ $33.013$ $707.11$ $207.48$
$\phi$ $447.21$ $35.386$ $ 447.21 $ $42.964$
$\theta$ $252.13$ $32.336$ $ 447.21$ $42.964$
$\psi$ $310.79$ $41.579$ $ 447.21$ $50.254$
GA-LQR-PD PSO-LQR-PD
i $K_{i, P}$ $K_{i, D}$ $K_{i, P}$ $K_{i, D}$
$z$ $32.89$ $33.013$ $707.11$ $207.48$
$\phi$ $447.21$ $35.386$ $ 447.21 $ $42.964$
$\theta$ $252.13$ $32.336$ $ 447.21$ $42.964$
$\psi$ $310.79$ $41.579$ $ 447.21$ $50.254$
Table 3.  Comparison of objective function values at the final iteration step
GA-PD PSO-PD GA-LQR-PD PSO-LQR-PD
min 1.5442 1.5152 51.846 30.978
max 1.6271 1.5159 957.32 49.788
mean 1.5799 1.5156 457.47 37.234
std 0.02871 0.00036 348.55 7.3161
ISE 0.00358 0.00356 0.05474 0.00950
IAE 0.04889 0.05031 0.33813 0.13091
GA-PD PSO-PD GA-LQR-PD PSO-LQR-PD
min 1.5442 1.5152 51.846 30.978
max 1.6271 1.5159 957.32 49.788
mean 1.5799 1.5156 457.47 37.234
std 0.02871 0.00036 348.55 7.3161
ISE 0.00358 0.00356 0.05474 0.00950
IAE 0.04889 0.05031 0.33813 0.13091
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