4. Simulation and analysis

AT

dV dt ¼ �<sup>e</sup>

Accordingly, the derivative to time of function V is

Thus, we obtain the following parameter adaptation laws

Figure 5. MRAC control of quadrotor: block diagram.

dθ<sup>1</sup>

dθ<sup>2</sup>

matrix.

asymptotically stable.

110 Adaptive Robust Control Systems

Therefore, if we choose a stable Am matrix, we will get always a P and Q positive definite

Where the function V is a Lyapunov function negative semi-definite ensures the output error between the real system and the reference model will tend to be zero, and the system is

<sup>m</sup>P þ PAm ¼ �Q (32)

dt <sup>¼</sup> <sup>p</sup>11e1x<sup>1</sup> <sup>þ</sup> <sup>p</sup>12e2x<sup>1</sup> <sup>þ</sup> <sup>p</sup>13e3x<sup>1</sup> (34)

dt <sup>¼</sup> <sup>p</sup>11e1x<sup>2</sup> <sup>þ</sup> <sup>p</sup>12e2x<sup>2</sup> <sup>þ</sup> <sup>p</sup>13e3x<sup>2</sup> (35)

TQe (33)

This section presents several simulations test made to prove the performance of MRAC controller to stabilize a mini-UAV quadrotor. As mentioned before, only orientation dynamic (angle position, angular velocity and acceleration) are considered. Analyzing Eq. (18), it is easy to see that roll, pitch, and yaw dynamics are very similar; for this reason, only roll moment is used as example in simulation.

The test begins considering controller parameters are unknown and by using an online adaptive mechanism to determine the values that permit the convergence of plant response to reference model response. It is important to note that the MRAC approach seeks to keep the tracking error (x � xm) equal to zero by an adjustment of the controller parameters and do not to seek to identify the real parameters of the plant.

Figures 6–8 shows a comparative between the states response of the plant and the state response of the model reference, where can be observer than all states of the plant converge asymptotical to the reference model states. To verify this, the Figures 9–11 are presented; these figures show the tracking error of states goes to zero. Additionally, in Figure 12, the reference input is compared to the plant and model reference output.

Figure 6. Comparison between x<sup>1</sup> and xm1.

Figure 7. Comparison between x<sup>2</sup> and xm2.

Figure 11. Error between x<sup>3</sup> to xm3.

Figure 12. Roll response.

Figure 10. Error between x<sup>2</sup> to xm2.

Attitude Control of a Quadcopter Using Adaptive Control Technique

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Figure 8. Comparison between x<sup>3</sup> and xm3.

Figure 9. Error between x<sup>1</sup> to xm1.

Figure 10. Error between x<sup>2</sup> to xm2.

Figure 11. Error between x<sup>3</sup> to xm3.

Figure 8. Comparison between x<sup>3</sup> and xm3.

Figure 9. Error between x<sup>1</sup> to xm1.

Figure 7. Comparison between x<sup>2</sup> and xm2.

112 Adaptive Robust Control Systems

Figure 12. Roll response.

Figure 13 shows the variations of the controller parameters during the adjustment process. This mechanism started many times is necessary to assure the perfect tracking of the plant to the desire response. Finally, in Figure 14, the simulation diagram is presented.

5. Conclusions

Author details

References

the 2011 IEEE. 2011

started only when it is needed.

Silvia Florida Melo and David Lara Alabazares

\*Address all correspondence to: rm\_ibarra@live.com

This work presents an adaptive control technique to stabilize the attitude of a quadrotor UAV using MRAC schema, which requires no information of the plant model. The asymptotic stability was demonstrated using the well-known Lyapunov's theory, obtaining in this way the adaptation law of the controller parameters. Simulations results demonstrate that the adaptive control approach proposed have a good performance to perform the asymptotic tracking of model reference output. It is important to note that the adaptive mechanism is

Attitude Control of a Quadcopter Using Adaptive Control Technique

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Ramiro Ibarra Pérez\*, Gerardo Romero Galvan, Aldo Jonathan Muñoz Vázquez,

UAM Reynosa Rodhe, Autonomous University of Tamaulipas, Tamaulipas, México

flying [dissertation]. École Polytechnique Fédérale de Lausanne; 2007

Electrical, Electronics, and Optimization Techniques (ICEEOT)—2016

Conference. 2016; DOI: 10.1109/ChiCC.2016.7555074

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Figure 13. Tuning of controller parameters.

Figure 14. Simulink diagram.
