# American Institute of Mathematical Sciences

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 and Management Optimization, 2017, 13 (4) : 2067-2091. doi: 10.3934/jimo.2017032
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Convergence trend for GA-PD algorithm: 10 runs are represented
Convergence trend for PSO-PD algorithm: 10 runs are represented
Convergence trend for GA-LQR-PD algorithm: 10 runs are represented
Convergence trend for PSO-LQR-PD algorithm: 10 runs are represented
Root locus of the controlled systems
Comparison between altitude trend versus time in GAPD case (red line) and in PSO-PD case (black line)
Comparison between roll trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Comparison between pitch trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Comparison between yaw trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Comparison between thrust trend versus time in GAPD case (red line) and in PSO-PD case (black line)
Comparison between roll moment trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Comparison between pitch moment trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Comparison between yaw moment trend versus time in GA-PD case (red line) and in PSO-PD case (black line)
Comparison between altitude trend versus time in GA-LQR-PD case (red line) and in PSO-LQR-PD case (black line)
Comparison between roll trend versus time in GALQR-PD case (red line) and in PSO-LQR-PD case (black line)
Comparison between pitch trend versus time in GALQR-PD case (red line) and in PSO-LQR-PD case (black line)
Comparison between yaw trend versus time in GALQR-PD case (red line) and in PSO-LQR-PD case (black line)
Comparison between thrust trend versus time in GALQR-PD case (red line) and in PSO-LQR-PD case (black line)
Comparison between roll moment trend versus time in GA-LQR-PD case (red line) and in PSO-LQR-PD case (black line)
Comparison between pitch moment trend versus time in GA-LQR-PD case (red line) and in PSO-LQR-PD case (black line)
Comparison between yaw moment trend versus time in GA-LQR-PD case (red line) and in PSO-LQR-PD case (black line)
Air turbulence: altitude time history
Air turbulence: roll time history
Air turbulence: pitch time history
Air turbulence: yaw time history
Air turbulence: thrust time history
Air turbulence: roll moment time history
Air turbulence: pitch moment time history
Air turbulence: yaw moment time history
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 $K_{i, P}$ $K_{i, D}$ $K_{i, P}$ $K_{i, D}$ $z$ $38327$ $923.59$ $46879$ $1000$ $\phi$ $19776$ $632.13$ $20000$ $1000$ $\theta$ $16918$ $246.83$ $20000$ $1000$ $\psi$ $18142$ $162.16$ $20000$ $1000$
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 $K_{i, P}$ $K_{i, D}$ $K_{i, P}$ $K_{i, D}$ $z$ $32.89$ $33.013$ $707.11$ $207.48$ $\phi$ $447.21$ $35.386$ $447.21$ $42.964$ $\theta$ $252.13$ $32.336$ $447.21$ $42.964$ $\psi$ $310.79$ $41.579$ $447.21$ $50.254$
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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