# From PID to Extended Learning Control

### Cristiano Maria Verrelli

#### Abstract

It has been recently shown in Marino, Tomei, and Verrelli (2011) that the output error feedback regulation problem with (unknown) periodic reference and/or disturbance signals of known common period can be effectively solved for the class of single-input, single-output, minimum phase, nonlinear, time-invariant systems in output feedback form (of known relative degree one or two) which are affected by unknown parameters and unknown output-dependent nonlinearities. The resulting nonlinear control, which relies on advanced learning control techniques, can be interpreted as a generalization of the classical PID control which solves the problem when both reference and disturbance signals are constant. In this paper, we present sophisticated analytical arguments which prove that the learning control designed in Marino, Tomei, and Verrelli (2011) can be endowed with a period identifier when the output reference signal is periodic of uncertain period but available at each time instant. The generalized resulting control preserves the achievement of the closed loop properties obtained in Marino, Tomei, and Verrelli (2011) while maintaining an overall simple structure. The application of the presented control techniques to the position synchronization problem for current-fed permanent magnet step motors with non-sinusoidal flux distribution and uncertain position-dependent load torque allows us to provide a solution to a yet unsolved problem.

#### References

- H.-S. Ahn, Y. Chen, and K. L. Moore (2007). Iterative learning control: brief survey and categorization. IEEE Transactions on Systems, Man, and Cybernetics, C, 37, 1099-1121.
- P. Andry, P. Gaussier, S. Moga, J. P. Banquet, and J. Nadel (2001). Learning and communication via imitation: an autonomous robot perspective. IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans 31, 431-442.
- S. Bifaretti, V. Iacovone, A. Rocchi, P. Tomei, and C. M. Verrelli (2011). Global learning position controls for permanent magnet step motors. IEEE Transactions on Industrial Electronics, 58, 4654-4663.
- S. Bifaretti, P. Tomei, and C. M. Verrelli (2011). A global robust iterative learning position control for currentfed permanent magnet step motors. Automatica, 47, 227-234.
- D. A. Bristow, M. Tharayil, and A. G. Alleyne (2006). A D. M. Dawson, J. Hu, and T. C. Burg (1998). Nonlinear Control of Electric Machinery, Marcel Dekker.
- N. Kalouptsidis (1997). Signal Processing Systems, John Wiley & Sons.
- P. Krishnamurthy and F. Khorrami (2003). Robust adaptive voltage-fed permanent magnet step motor control without current measurements. IEEE Transactions on Control Systems Technology, 11, 415-425.
- R. Marino, P. Tomei, and C. M. Verrelli (2012). Robust adaptive learning control for nonlinear systems with extended matching unstructured uncertainties. International Journal of Robust and Nonlinear Control, 22, 645-675.
- R. Marino, P. Tomei, and C. M. Verrelli (2011). Generalized PID learning control. 18th IFAC World Congress, 18, 28 August-2 September.
- N. Minorsky (1922). Directional stability of automatically steered bodies. American Society of Naval Engineers.
- Automatic steering. Transactions of E. Sperry (1922). ASME.
- I. Tyukin (2011). Adaptation in Dynamical Systems, Cambridge University Press.
- C. M. Verrelli (2011a). Fourier series expansion for synchronization of permanent magnet electric motors. Applied Mathematics and Computation, 217, 4502- 4515.
- C. M. Verrelli (2012). Synchronization of permanent magnet electric motors: new nonlinear advanced results. Nonlinear Analysis: Real World Applications, 13, 395-409.
- C. M. Verrelli (2011b). Adaptive learning control for robotic manipulators driven by permanent magnet synchronous motors. International Journal of Control, 84, 1024-1030.

#### Paper Citation

#### in Harvard Style

Maria Verrelli C. (2012). **From PID to Extended Learning Control** . In *Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,* ISBN 978-989-8565-21-1, pages 464-469. DOI: 10.5220/0004029104640469

#### in Bibtex Style

@conference{icinco12,

author={Cristiano Maria Verrelli},

title={From PID to Extended Learning Control},

booktitle={Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},

year={2012},

pages={464-469},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004029104640469},

isbn={978-989-8565-21-1},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,

TI - From PID to Extended Learning Control

SN - 978-989-8565-21-1

AU - Maria Verrelli C.

PY - 2012

SP - 464

EP - 469

DO - 10.5220/0004029104640469