Existence Conditions of Asymptotically Stable 2-D Feedback Control
Systems on the Basis of Block Matrix Diagonalization
Giido Izuta
Yonezawa Women’s Junior College, Yonezawa, Yamagata, Japan
Keywords:
2-D Systems, Asymptotic Stability, Feedback Control, Block Matrix Diagonalization.
Abstract:
This work is concerned with the existence of asymptotically stable 2-D (2-dimensional) systems by means of a
feedback control model represented by the system of partial difference equations and their Lagrange solutions.
Thus, the goal is to establish a controller thatprovides a feedback control system with state variables depending
solely on its Lagrange solution in the sense that the solution to the variable state is not a linear combination
of other Lagrange solutions. Roughly speaking, the results showed that, to achieve such a control system,
the controller has to diagonalize the block matrices of the matrices composing the system description model.
Finally, a numerical example is presented to show how the controller is designed in order to generate a stable
feedback control with given Lagrange solutions.
1 INTRODUCTION
Discrete 2-D control systems have been a subject
of research for over half a century now; and many
approaches have been proposed to study the stabil-
ity and design of feedback control systems. As far
as the mathematical model is concerned with, either
the Roesser system description (Givone and Roesser,
1972) or the FM model (Attasi, 1973; Fornasini and
Marchesini, 1978; Fornasini and Marchesini, 1980),
which are essentially equivalent to one another in the
sense that it is possible to transform the partial dif-
ference equations describing one model into the other
(Kaczorek, 1985), is taken as the system representa-
tion to handle this type of control systems. In addi-
tion, the formalism adopted has spread over a large
range of mathematical ﬁelds. Among those, the z−
transform showed to be a very effective strategy to in-
vestigate systems with single input and single output
corresponding to what is called the class of ‘delayless
systems’ in the ordinary 1-D system counterpart the-
ory; and a great deal of stability criteria and controller
design procedures were accomplished for 2-D con-
trol systems (Anderson and Jury, 1974; Bose, 1982;
Tzafestas, 1986; Lim, 1990).
During the time period around the turning of the
century, a remarkable shift in the paradigm from han-
dling those problems on the grounds of analytical
mathematics into the incorporation of computational
methods, which in control system theory is closely as-
sociated to the energy method framework, took place.
In fact, this approach relies basically on the Lyapunov
stability theory as well as the optimization algorithms,
and has undergone tremendous developments hand in
hand with the advances in computational techniques;
so that the formalization of the control system de-
sign based on the linear matrix inequalities gained
widespread popularity and established its status as a
standard procedure to handle not only the ordinary
1-D but also multi-input multi-output n-D systems
(Piekarski, 1977; Du and Xie, 2002; Pazke et al.,
2004; Izuta, 2007; Rogers et al., 2007). However,
this procedure has been pointed out to yield conserva-
tive controllers in the control theory sense. Also it is
worth noting that many other techniques have ﬂour-
ished and contributed to the broadening of the ﬁeld
(Jerri, 1996; Zerz, 2000; Cheng, 2003; Elaydi, 2005;
Izuta, 2012; Izuta, 2014a).
Unlike the theoretical accounts aforementioned,
this paper examines the existence conditions of
asymptotically stable 2-D feedback control systems
from the Lagrange method frame of reference, which
has shed light on the stability analysis problem (Izuta,
2010; Izuta, 2014b). In these reports, the Lagrange
method was used in conjunction with Jordan matrix
transformation in order to transform the original sys-
tem of partial difference equations into a system com-
posed by only diagonal matrices, which allow the
transformed system be solved analytically.
Motivated by these results, this paper aims to ﬁg-