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.

References

References
1.
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
.
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
Roesser
,
R. P.
,
1975
, “
A Discrete State-Space Model for Linear Image Processing
,”
IEEE Trans. Autom. Control
,
20
(
1
), pp. 
1
10
.10.1109/TAC.1975.1100844
10.
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
.
11.
Fornasini
,
E.
, and
Marchesini
,
G.
,
1978
, “
Doubly Indexed Dynamical Systems: State-Space Models and Structural Properties
,”
Theory Comput. Syst.
,
12
(
1
), pp. 
59
72
.10.1007/BF01776566
12.
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
13.
Daafouz
,
J.
, and
Bernussou
,
J.
,
2001
, “
Parameter Dependent Lyapunov Functions for Discrete Time Systems With Time Varying Parametric Uncertainties
,”
Syst. Control Lett.
,
43
(
5
), pp. 
355
359
.10.1016/S0167-6911(01)00118-9
14.
Geometric Bounding Toolbox
, http://www.sysbrain.com/gbt/gbt/index.htm.
This content is only available via PDF.
You do not currently have access to this content.