An Explicit Method for Stability Analysis of 2D Systems Described by Transfer Function

Stability analysis of 2‐D discrete and continuous state‐space systems

“…is stable 2.3: The matrix Lyapunov equation with complex elements G (e j 𝜔 ) − P (e j 𝜔 ) t G (e j 𝜔 )P (e j 𝜔 ) = W (e j 𝜔 ) (17) has a positive definite and Hermitian solution G (e j 𝜔 ) for any given (n + m) × (n + m) positive definite and Hermitian matrix W (e j 𝜔 ) for 𝜔 ∈ [0, 2𝜋] and P (e j 𝜔 ) is given by ( 16) 2.4: In statement 2.2, the 2-D stability problem has been reduced in a 1-D stability problem with a matrix P (e j 𝜔 ) with entries which are functions of e j 𝜔 for 𝜔 ∈ [0, 2𝜋].…”

“…For a given positive definite and Hermitian matrix W (e j 𝜔 ), one may solve the Lyapunov matrix equation (17) to obtain a matrix solution G (e j 𝜔 ) which is a Hermitian matrix. Denoting the 𝜆th-order principal minor of G (e j 𝜔 ) by g 𝜆 (𝜔), we note that the principal minors g 𝜆 (𝜔), 1 ≤ 𝜆 ≤ (m + n) are real rational functions of one real variable 𝜔 over the closed interval [0, 2𝜋].…”

“…where Δ 𝜆 (e j 𝜔 ), 𝜆 = 1, 2, … , n + m are the quantities of the Mansour matrix associated with the matrix P (e j 𝜔 ), 𝜔 ∈ [0, 2𝜋]. Consider the matrix Lyapunov equation with complex elements given in (17). Using the arguments presented in [65] we choose…”

“…The first studies of the stability analysis of 2-D discrete systems were carried out in the frequency domain leading to many well known tests [9], [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. The introduction of statespace models for 2-D systems allowed the investigation of stability using the state-space approach and several stability criteria have been formulated in state-space using either the Roesser model [30] or the Fornasini-Marchesini model 73].…”