This paper concentrates on the stability of discrete-time systems. The stability of the system is deduced from the roots of the determinant of a matrix polynomial of nth order. Nell' and less restrictive conditions than those published previously are derived to ensure that roots of the determinant of the matrix polynomial are located inside the unit circle. The conditions are given in terms of the spectral radius of matrix constructed from the coefficient matrices. Examples are given for illustration.
The leading coefficient is the identity matrixWhen A" = I"" we have the following result.Theorem I: When An = 1 m , the matrix polynomial (2) is asymptotically stable if the following condition is satisfled:It is obvious that det [P(z)] and det IP I (z)] have the same roots. Consequently the stability of (2) may be deduced from that of (5). By contradiction, assume that det [PI (z)] =°yields [z]~I. Then one has 1~[z] = I·X;(-A n -I -A n _ 2 z-1 _ ..• A _-n+2 A -"+')1 -I~-OZ . satisfy [z] < I. Several results have been published on this topic (Hermite 1854, Parks 1964, Kalman 1965, Ahn 1982, Jury 1982, Behouari et al. 1992, where the criteria are established in terms of the rational functions of the coefficients only. So, for control system synthesis, simpler conditions should be of great interest. In this paper, new and less restrictive sufficient stability conditions for discrete-time systems are derived. The results may be easily extended to cover the uncertainties in the matrix polynomial. (5) (4) Proof: The matrix polynomial can be written as P(z) = zn-I PI (z), PI (z) = zl; + An_I + A n_2z-1 + ... + A IZ-n + 2 + Aoz-n + l • where Nomenclature [z] modulus of the complex number z jAI absolute matrix, that is IAI = (Iaul) det (A) determinant of the matrix A p(A) spectral radius of A IIA lie matrix norm of A (c = 1,2,00)t; identity matrix of R m " "
