Figures (10) and (11) show the experimental results obtained with disturbance. Figure

Figure 11. Fractional order PI-state feedback control experimental results with disturbance on the pendulum at and , using the second method.

Conclusion

Two new pole placement fractional PI-state feedback designs algorithms are presented in this book chapter. The first one consists in choosing poles of an integer polynomial corresponding to the closed-loop fractional characteristic polynomial. Matignon’s stability condition can thus be used to ensure the stability of the closed-loop. Nevertheless, the noninteger order must be rational. In addition, the poles of the integer polynomial are not arbitrary since they must verify constraints because some of the polynomial coefficients are zero. The second method is simple to implement, it consists in choosing poles of an integer polynomial and one pole of a fractional polynomial. A judicious choice of these poles provides a closed-loop particular response obtained when the Bode’s ideal transfer function is used, that is, an overshoot of the step response imposed by the fractional order and its dynamics by the pole of the fractional polynomial. These methods are then used to stabilize an inverted pendulum-cart system. The implementation on an experimental test-bed gave good results, especially in terms of stability, accuracy and robustness with respect to the variation of the cart mass as well as external disturbances applied on the pendulum.

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Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no Gr/34/5. The authors, therefore, acknowledge with thanks DSR technical and financial support.

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