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Research Papers

Parameter-Dependent Lyapunov Functions in the Robust Control of Discrete Linear Repetitive Processes Using Previous Pass-Windowed Information

[+] Author and Article Information
Błażej Cichy

Institute of Control and Computation Engineering, University of Zielona Góra, ul. Podgórna 50, 65-246 Zielona Góra, Polande-mail: b.cichy@issi.uz.zgora.pl

Krzysztof Galkowski

Institute of Control and Computation Engineering, University of Zielona Góra, ul. Podgórna 50, 65-246 Zielona Góra, Polande-mail: k.galkowski@issi.uz.zgora.pl

Eric Rogers

Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UKe-mail: etar@ecs.soton.ac.uk

Manuscript received August 23, 2014; final manuscript received February 24, 2015; published online April 20, 2015. Assoc. Editor: Athanasios Pantelous.

ASME J. Risk Uncertainty Part B 1(2), 021008 (Apr 20, 2015) (12 pages) Paper No: RISK-14-1040; doi: 10.1115/1.4029971 History: Received August 23, 2014; Accepted March 04, 2015; Online April 20, 2015

The variables in multidimensional systems are functions of more than one indeterminate, and such systems cannot be controlled by standard systems theory. This paper considers a subclass of these systems that operate over a subset of the upper-right quadrant of the two-dimensional (2D) plane in the discrete domain with a specified recursive structure known as repetitive processes. Physical examples of such processes are known and also their representations can be used in the analysis of other classes of systems, such as iterative learning control. This paper gives new results on the use of the parameter-dependent Lyapunov functions for stability analysis and controls law design of a subclass of repetitive processes that arise in application areas. These results aim to eliminate or reduce the effects of model parameter uncertainty.

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References

Rogers, E., Gałkowski, K., and Owens, D. H., 2007, Control Systems Theory and Applications for Linear Repetitive Processes (Lecture Notes in Control and Information Sciences, Vol. 349), Springer-Verlag, Berlin.
Ahn, H.-S., Chen, Y.-Q., and Moore, K. L., 2007, “Iterative Learning Control: Brief Survey and Categorization,” IEEE Trans. Syst. Man Cybern. Part C: Appl. Rev., 37(6), pp. 1099–1121. 10.1109/TSMCC.2007.905759
Bristow, D. A., Tharayil, M., and Alleyne, A. G., 2006, “A Survey of Iterative Learning Control: A Learning Based Method for High-Performance Tracking Control,” IEEE Control Syst. Mag., 26(3), pp. 96–114. 10.1109/MCS.2006.1636313
Hładowski, Ł., Gałkowski, K., Cai, Z., Rogers, E., Freeman, C. T., and Lewin, P. L., 2010, “Experimentally Supported 2D Systems Based Iterative Learning Control Law Design for Error Convergence and Performance,” Control Eng. Pract., 18(4), pp. 339–348. 10.1016/j.conengprac.2009.12.003
Paszke, W., Rogers, E., Gałkowski, K., and Cai, Z., 2013, “Robust Finite Frequency Range Iterative Learning Control Design and Experimental Verification,” Control Eng. Pract., 21(10), pp. 1310–1320. 10.1016/j.conengprac.2013.05.011
Roberts, P. D., 2000, “Numerical Investigations of a Stability Theorem Arising from 2-Dimensional Analysis of an Iterative Optimal Control Algorithm,” Multidimension. Syst. Signal Process., 11(1/2), pp. 109–124. 10.1023/A:1008490731182
Azevedo-Perdicoúlis, T. P., and Jank, G., 2012, “Disturbance Attenuation of Linear Quadratic OL-Nash Games on Repetitive Processes With Smoothing on the Gas Dynamics,” Multidimension. Syst. Signal Process., 23(1–2), pp. 131–153. 10.1007/s11045-010-0109-0
Cichy, B., Gałkowski, K., Rogers, E., and Kummert, A., 2011, “An Approach to Iterative Learning Control for Spatio-Temporal Dynamics Using nD Discrete Linear Systems Models,” Multidimension. Syst. Signal Process., 22(1–3), pp. 83–96. 10.1007/s11045-010-0108-1
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Gałkowski, K., Cichy, B., Rogers, E., and Lam, J., 2006, “Stabilization of a Class of Uncertain “Wave” Discrete Linear Repetitive Processes,” in Proceedings of the 45th IEEE Conference on Decision and Control, pp. 1435–1440.
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Cichy, B., Gałkowski, K., Rogers, E., and Kummert, A., 2013, “Control Law Design for Discrete Linear Repetitive Processes with Non-Local Updating Structures,” Multidimension. Syst. Signal Process., 24(4), pp. 707–726. 10.1007/s11045-012-0199-y
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Figures

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Fig. 1

Illustrating the updating structure of the current pass state vector in Eq. (2)

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Fig. 2

Illustrating the updating structure of the current pass profile vector in Eq. (2)

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Fig. 3

The transfer function of the gantry robot dynamics at the x-axis

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Fig. 9

Pass profiles and the input signal for model switching from vertex 1 to 2, then from 2 to 3, and back to 1—Scenario 3

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Fig. 7

Pass profiles of the controlled process—Scenario 2—vertices 1, 2, and 3

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Fig. 8

The controlled process inputs—Scenario 2—vertices 1, 2, and 3

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Fig. 4

Pass profile dynamics for uncontrolled process—Scenario 1 and vertex 1

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Fig. 5

Pass profiles for controlled process—Scenario 1—vertices 1, 2, and 3

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Fig. 6

The controlled process inputs—Scenario 1—vertices 1, 2, and 3

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