Stability analysis of 2-d linear discrete feedback control systems with state delays on the basis of lagrange solutions

Researches on the two dimensional (2d) systems back to 1950s, when themain concernwas the study of stability conditions for analog networked circuits (Levenstein, 1958; Ozaki & Kasami, 1960). Then, with the advent of new technologies and developments in the digital systems engineering as well as advances in the mathematical fields, this paradigm has evolved over the last decades into a major shift to discrete systems, addressing in addition to stability issues the control systems theory problems, which have ever since called the attention of mathematicians, digital signal processing community, control systems theorists and computer scientists among others. These investigations on the stability and control of 2d systems can be gathered into basically two approaches: the multidimensional z-transform framework (Bose, 1982; Lim, 1990) and the energy method (see for example (Du & Xie, 2002) and references therein). The z-transform formalism has contributed greatly to the stability analysis of systems expressed in terms of the transfer function representation by providing a variety of stability methods as the well known Shanks stability criteria. Due to the fact that these techniques are useful instruments to checking the bounded input bounded output (BIBO) stability of system (Lim, 1990), this philosophy has been applied to systems described by their state space model representations, and as a result many stability conditions have been established in terms of the characteristic equations and eigenvalues (Fornasini & Marchesini, 1978), which have provided helpful tools for people in the systems engineering to establish control systems design methodologies (Kaczorek, 1985). On the other hand, unlike the z-transform, the energy method consists essentially in finding a Lyapunov function that expresses the energy of the system, and then showing that this energy vanishes as the equations indices increase. Thus, since the success of this method relies fundamentally in one’s ability to formulate an adequate energy function, the role and the influence of the eigenvalues of the state space matrices are in many cases left uncovered. Incidentally, the discovering of a suitable function is also inherent in the stability and design procedures based on the linear matrix inequalities (LMI) approach, which is essentially a branch of the energy method (Boyd et al., 1994). Despite this point, LMI’s based 1