# Iterative learning control for over-determined under-determined, and ill-conditioned systems

Konstantin Avrachenkov; Richard Longman

International Journal of Applied Mathematics and Computer Science (2003)

- Volume: 13, Issue: 1, page 113-122
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topAvrachenkov, Konstantin, and Longman, Richard. "Iterative learning control for over-determined under-determined, and ill-conditioned systems." International Journal of Applied Mathematics and Computer Science 13.1 (2003): 113-122. <http://eudml.org/doc/207619>.

@article{Avrachenkov2003,

abstract = {This paper studies iterative learning control (ILC) for under-determined and over-determined systems, i.e., systems for which the control action to produce the desired output is not unique, or for which exact tracking of the desired trajectory is not feasible. For both cases we recommend the use of the pseudoinverse or its approximation as a learning operator. The Tikhonov regularization technique is discussed for computing the pseudoinverse to handle numerical instability. It is shown that for over-determined systems, the minimum error is never reached by a repetition invariant learning controller unless one knows the system exactly. For discrete time uniquely determined systems it is indicated that the inverse is usually ill-conditioned, and hence an approximate inverse based on a pseudoinverse is appropriate, treating the system as over-determined. Using the structure of the system matrix, an enhanced Tikhonov regularization technique is developed which converges to zero tracking error. It is shown that the Tikhonov regularization is a form of linear quadratic ILC, and that the regularization approach solves the important practical problem of how to intelligently pick the weighting matrices in the quadratic cost. It is also shown how to use a modification of the Tikhonov-based quadratic cost in order to produce a frequency cutoff. This robustifies good learning transients, by reformulating the problem as an over-determined system.},

author = {Avrachenkov, Konstantin, Longman, Richard},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {pseudoinverse; Iterative Learning Control; over-determined; under-determined; ill-conditioned systems; Iterative learning control; over-determined systems; under-determined systems; regularization},

language = {eng},

number = {1},

pages = {113-122},

title = {Iterative learning control for over-determined under-determined, and ill-conditioned systems},

url = {http://eudml.org/doc/207619},

volume = {13},

year = {2003},

}

TY - JOUR

AU - Avrachenkov, Konstantin

AU - Longman, Richard

TI - Iterative learning control for over-determined under-determined, and ill-conditioned systems

JO - International Journal of Applied Mathematics and Computer Science

PY - 2003

VL - 13

IS - 1

SP - 113

EP - 122

AB - This paper studies iterative learning control (ILC) for under-determined and over-determined systems, i.e., systems for which the control action to produce the desired output is not unique, or for which exact tracking of the desired trajectory is not feasible. For both cases we recommend the use of the pseudoinverse or its approximation as a learning operator. The Tikhonov regularization technique is discussed for computing the pseudoinverse to handle numerical instability. It is shown that for over-determined systems, the minimum error is never reached by a repetition invariant learning controller unless one knows the system exactly. For discrete time uniquely determined systems it is indicated that the inverse is usually ill-conditioned, and hence an approximate inverse based on a pseudoinverse is appropriate, treating the system as over-determined. Using the structure of the system matrix, an enhanced Tikhonov regularization technique is developed which converges to zero tracking error. It is shown that the Tikhonov regularization is a form of linear quadratic ILC, and that the regularization approach solves the important practical problem of how to intelligently pick the weighting matrices in the quadratic cost. It is also shown how to use a modification of the Tikhonov-based quadratic cost in order to produce a frequency cutoff. This robustifies good learning transients, by reformulating the problem as an over-determined system.

LA - eng

KW - pseudoinverse; Iterative Learning Control; over-determined; under-determined; ill-conditioned systems; Iterative learning control; over-determined systems; under-determined systems; regularization

UR - http://eudml.org/doc/207619

ER -

## References

top- Arimoto S., Kawamura S. and Miyazaki F. (1984): Bettering operation of robots by learning. - J.Robot. Syst., Vol. 1, No. 2, pp. 123-140.
- Astrom K., Hagander P. and Strenby J. (1980): Zeros of sampled systems. - Proc. IEEE CDC'80, Albuquerque, pp. 1077-1081.
- Avrachenkov K.E. (1998): Iterative learning control based on quasi-Newton methods. - Proc. IEEE CDC'98, Tampa, (on CD-ROM).
- Avrachenkov K.E. and Pervozvansky A.A. (1998a): Regularization and robustness of learning-based control algorithms. - J.Comput. Syst. Sci., Vol. 37, No. 2, pp. 338-340. Zbl1092.93612
- Avrachenkov K.E. and Pervozvansky A.A. (1998b): Iterative learning control for singularly perturbed systems. - Proc. ILC Workshop, IEEE CDC'98, Tampa, pp. 71-73.
- Avrachenkov K.E., Beigi H.S.M. and Longman R.W. (1999): Updating procedures for iterative learning control in Hilbert space. - Proc. IEEE CDC'99, Phoenix, pp. 276-280.
- Beigi H.S.M. (1997): New adaptive and learning-adaptive control techniques based on an extension of the generalized secant method. - J. Intell. Automat. Soft Comp., Vol. 3, No. 2, pp. 171-184.
- Beklemishev D.V. (1983): Additional Chapters of Linear Algebra. -Moscow: Nauka, (in Russian). Zbl0532.15002
- Campbell S.L. and Meyer C.D. (1979): Generalized Inverses of Linear Transformation. - London: Pitman. Zbl0417.15002
- Dennis J.E. Jr. and Schnabel R.B. (1983): Numerical Methods for Unconstrained Optimization and Nonlinear Equations. -Englewood Cliffs: Prentice-Hall. Zbl0579.65058
- Ding J. and Huang L.J. (1996): Perturbation of generalized inverses of linear operators in Hilbert spaces. - J. Math. Anal. Appl., Vol. 198, No. 2, pp. 506-515. Zbl0867.47003
- Frueh J.A. and Phan M.Q. (2003): Linear quadratic optimal learning control (LQL). - Int. J. Contr., Special Issue on Iterative Learning Control, (in print). Zbl0995.49018
- Jang H.S. and Longman R.W. (1994): A new learning control law with monotonic decay of the tracking error norm. - Proc. 32-nd Ann. Allerton Conf. Communication, Control, and Computing, Monticello, Illinois, pp. 314-323.
- Jang H.S. and Longman R.W. (1996): Design of digital learning controllers using a partial isometry. - Adv. Astronaut. Sci., Vol. 93, pp. 137-152.
- Juang J.-N., Phan M., Horta L.G. and Longman R.W. (1993): Identification of observer Kalman filter Markov parameters: Theory and experiments. - J. Guid. Contr. Dynam., Vol. 16, No. 2, pp. 320-329. Zbl0775.93259
- Longman R.W. (1998): Designing Iterative Learning and Repetitive Controllers, In: Iterative Learning Control: Analysis, Design, Integration and Applications (Z. Bien and J.-X. Xu, Eds.). - Boston: Kluwer Academic Publishers, pp. 107-146.
- Longman R.W. (2000): Iterative learning control and repetitive control for engineering practice. - Int. J. Contr., Special Issue on Iterative Learning Control, Vol. 73, No. 10, pp. 930-954. Zbl1006.93598
- Longman R.W. and Chang C.-K. (1990): Learning control for minimizing a quadratic cost during repetitions of a task. - Proc. AIAA/AAS Astrodynamics Conf., A Collection of Technical Papers, Part 2, Portland, Oregon, pp. 530-536.
- Longman R.W. and Huang Y.-C. (2003): The phenomenon of apparent convergence followed by divergence in learning and repetitive control. - Intell. Automat. Soft Comput., Special Issue on Learning and Repetitive Control, Vol. 8, No. 2, (to appear).
- Longman R.W., Beigi H.S.M. and Li C.J. (1989): Learning control by numerical optimization methods. - Proc. Conf. Modeling and Simulation, Instrument Soc. of America, Vol. 20, Part 5, pp. 1877-1882.
- Moore K.L. (1993): Iterative learning control for deterministic systems. - London, U.K.: Springer-Verlag. Zbl0773.93002
- Moore K.L. (1997): Iterative learning control- An expository overview.- Tech. Rep., No. 9798 002, (to appear in Appl. Comput. Contr. Signal Process. Circ.). Zbl0955.93500
- Oh S.J., Longman R.W. and Phan M.Q. (1997): Use of decoupling basis functions in learning control for local learning and improved transients. - Adv. Astronaut. Sci., Vol. 95, pp. 651-670.
- Owens D.H., Amann N. and Rogers E. (1995): Iterative learning control-An overview of recent algorithms. - Appl. Math. Comput. Sci., Vol. 5, No. 3, pp. 425-438. Zbl0850.93003
- Pervozvansky A.A. (1995a): Learning control and its applications. Part 1. Elements of general theory. - Avtomatika i Telemekhanika, No. 11, Engl. Transl. in Automation and Remote Control.
- Pervozvansky A.A. (1995b): Learning control and its applications. Part 2. Frobenious systems and learning controlfor robot-manipulators. - Avtomatika i Telemekhanika, No. 12, Engl. Transl. in Automation and Remote Control.
- Pervozvansky A.A. and Avrachenkov K.E. (1997): Learning control algorithms: convergence and robustness. - Proc. Australian Control Conf., Sydney, pp. 366-371.
- Plotnik A.M. and Longman R.W. (1999): Subtleties in the use of zero-phase low-pass filtering and cliff filtering in learning control. - Adv. Astronaut. Sci., Vol. 103, pp. 673-692.
- Rogers E. and Owens D.H. (1992): Stability Analysis for Linear Repetitive Processes. - Berlin: Springer-Verlag. Zbl0772.93072
- Tikhonov A.N. and Arsenin V.Ya. (1974): Solution Methods for Ill-Posed Problems. - Moscow: Nauka, (in Russian).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.