Paper:

# Polynomial Linear Quadratic Gaussian and Sliding Mode Observer for a Quadrotor Unmanned Aerial Vehicle

## Abdellah Mokhtari^{*}, Abdelaziz Benallegue^{**},

and Abdelkader Belaidi^{***}

^{*}University of Science and Technology Oran, BP 1505 El M’naouer, 31000 Oran, Algeria

^{**}Laboratoire de Robotique de Versailles, 10-12 av de l’Europe, 78140 Velizy, France

^{***}E.N.S.E.T Oran, Algérie

A polynomial Linear quadratic Gaussian method is used to compute a controller which is mixed with feedback linearization to control altogether a non linear Quadrotor UAV. A dual criterion involving the minimization of the error and control signal variances is analyzed. The introduction of the feedback linearization is seen to be useful to transform the MIMO system into a non interacting one. This will facilitate the computation of spectral factorization where the behavior is like SISO system. An algorithm is developed in this context. A sliding mode observer is added to feedback loop to estimate the unmeasured states necessary to the inner loop controller. The convergence of output state vector is obtained despite the non-robustness exact linearization when uncertainties on system parameters and disturbances occur. However the sensitivity and complementary sensitivity are presented to confirm the performance and robustness theory and validate the efficiency of results. The weighting functions choice is analyzed through frequency domain. The whole observer-estimator-control constitutes an original suggestion of control system with minimum sensors used. The robustness study has been realized on simulation taking into account uncertainties, disturbances, with a corrupted measured state by noise, and the actuator saturation. The results obtained show the convergence in finite time of estimated values and a satisfying tracking error of desired trajectories.

*J. Robot. Mechatron.*, Vol.17, No.4, pp. 483-495, 2005.

- [1] C. Gökçek, P. T. Kabamba, and S. M. Meerkov, “An LQR/LQG Theory for Systems With Saturating Actuators,” Trans. Autom. Control, Vol.46, No.10, p. 1529, 2001.
- [2] V. Kucera, “Discrete Linear Control,” Wiley: Chichester, 1979.
- [3] V. Kucera, “Stochastic multivariable control: A polynomial equation approach,” IEEE Trans. Autom. Control, Vol.AC-25, No.5, pp. 913-919, 1980.
- [4] J. J. E. Slotine, “Sliding controller design for nonlinear systems,” International Journal of Control, Vol.38, pp. 465-492, Feb. 1984.
- [5] J.-J. E. Slotine, and J. K. Hedrick, “Robust Input-Output Feedback Linearization,” International Journal of Control, Vol.57, pp. 1133-1139, 1993.
- [6] W. T. Baumann, and W. J. Rugh, “Feedback control of nonlinear systems by extended linearization,” IEEE Transactions on Automatic Control, Vol.AC-31, No.1, pp. 40-46, Jan. 1986.
- [7] K. Youcef-Toumi, and O. Ito, “Time Delay Controller of Systems with Unknown Dynamics,” ASME Journal of Dynamic Systems, Measurement and Control, 112(1), pp. 133-142, Mar. 1990.
- [8] D. Luenberger, “Observers for multivariable systems,” IEEE Trans. Autom. Control, Vol.11, pp. 190-197, 1966.
- [9] W. Wang, and Z. Gao, “A Comparison Study of Advanced State Observer Design Techniques,” American Control Conference, at Denver, CO. June 4-6, 2003.
- [10] V. Mistler, A. Benallegue, and N. K. M’Sirdi, “Exact Linearization and Non-interacting Control of a 4 Rotors Helicopter via Dynamic Feedback,” 10th IEEE Int. Workshop on Robot-Human Interactive Communication (Roman’2001), Paris, Sep. 2001.
- [11] H. K. Khalil, “Nonlinear Systems,” McMillan Publishing Company, New York, 1992.
- [12] M. J. Grimble, “Controller for LQG self tuning application with coloured measurement noise and dynamic costing,” IEE Proc., Vol.133, Pt D, No.1, 1983.
- [13] M. J. Grimble, “LQG design of discrete systems using a dual criterion,” IEE Proc., Vol.192, No.2, pp. 66-68, 1985.
- [14] M. J. Grimble, “Dual criterion stochastic optimal control problem for robustness improvement,” IEEE Trans. Autom. Control, Vol.AC-31, No.2, pp. 181-185, 1986.
- [15] E. William, and S. Levine, “The Control Handbook,” CRC Press/IEEE Press, 1996.
- [16] F. M. Callier, and J. Winkin, “The spectral factorization problem for SISO distributed systems,” in Modelling, robustness and sensitivity reduction in control systems (R.F. Curtain, ed.), NATO ASI Series, Vol.F34, Springer-Verlag, Berlin Heidelberg, pp. 463-489, 1987.
- [17] F. M. Callier, and J. Winkin, “Spectral factorization and LQ-optimal regulation for multivariable distributed ystems,” Int. J. of Control, Vol.52, No.1, pp. 55-75, 1990.
- [18] A. H. Sayed, and T. Kailath, “A Survey of Spectral Factorization Methods,” Journal of Numerical Linear Algebra With Applications, Vol.8, pp. 467-496, 2001.
- [19] C. M. Dorling, and A. S. I. Zinober, “A comparison study of the sensitivity of observers,” Proceeding First IASTED Symposium on Applied Control and Identification, Copenhagen, pp. 6.32-6.37, 1983.
- [20] J. Slotine, J. K. Hedrick, and E. A. Misawa, “On sliding observers for nonlinear systems,” Journal of Dynamic Systems, Measurement, and Control, Vol.109, p. 245, 1987.
- [21] J. Slotine, and S. Sastry, “Tracking Control of Non-linear Systems Using Sliding Surfaces, with Application to Robot Manipulators,” Int. J. Control, Vol.38, pp. 465-492, 1983.
- [22] V. I. Utkin, “Sliding Modes in Control and Optimization,” Springer-Verlag, Berlin, 1992.
- [23] S. Drakunov, “Sliding mode observer based on equivalent control method,” Proceeding of the 34th IEEE CDC92, pp. 2368-2369, 1992.
- [24] S. Drakunov, and V. Utkin, “Sliding mode observer: Tutorial,” Proceeding of the 34th IEEE CDC95, 1995.

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