STABILITY TESTING OF TWO-DIMENSIONAL SYSTEMS

2D STABILITY testing of a Polynomial C(z1,z2). STABILITY testing of a 2D system given with a Roesser model.

STABILITY TESTING OF TWO-DIMENSIONAL SYSTEMS

1. StabTest_2D_Polynomial.m

STABILITY TESTING of a 2D Polynomial C(z1,z2) given by

C(z1,z2) = [z1^n,...z1^2,z1,1]*A*[z2^m,...,z2^2,z2,1]'

where A is a n x m coefficient matrix

C(z1,z2) is 2D structurally stable, iff C(z1,z2) has no zeros in the closed unit bidisc

U-bar^2={(z1,z2)| z1=<1, z2=<1}

The method for testing the 2D stability of C(z1,z2) is based on the following paper:

[1] Kraus F. and P. Agathoklis, "Stability Test of 2D Filters Based on Tschebyscheff Polynomials and Generalized Eigenvalues", Proceedings of the 2020 IEEE International Symposium on Circuits and Systems (ISCAS), May 17-20, 2020, Seville, Spain.

Extensions to 2D Continuous and mixed continuous /discrete system can be found in:

[2] Mohsenipour Reza and Panajotis Agathoklis, “Algebraic Necessary and Sufficient Conditions for Testing Stability of 2-D Linear Systems” IEEE Transactions on Automatic Control, Vol. 66, Issue 4, 2021.

The approach is based on using the conjugate of the original polynomial together with Tschebyscheff polynomials to transform the problem of stability analysis of a polynomial with coefficients depending on a complex variable to a problem with a polynomial with coefficients depending on a real variable. For the latter polynomial, the stability condition can be tested using a generalized eigenvalue problem.

2. Roesser_2D_CharPolynomial.m

Compute The Characteristic Polynomial of a 2D system given with a Roesser’s Model.

The approach is based on formulating the problem as a polynomial interpolation problem.

3. Lyapunov_2D.m

Compute Block Diagonal Positive Definite Matrix P and Positive Definite Matri Q satisfying the 2D Lyapunov Equation for a 2D system given with Roesser model.

The necessary and Sufficient conditions for the existence of such matrices and their relationship to the Stability of the 2D system have been presented in:

[3] Anderson, B.D.O., P. Agathoklis, E.I. Jury, and M. Mansour, "Stability and the matrix Lyapunov equation for discrete 2-dimensional systems", IEEE Trans. on Circuits and Systems, vol. CAS-33, pp. 261-267, March 1986.

The approach is based on formulating the problem as a variation of LMI optimization problems leading to 4 different pairs of (P,Q) for a 2D system given by a Roesser model. If (P,Q) are positive definite, the 2D system is stable. However, as shown in [3], there are 2D stable systems for which on positive definite (P,Q) exist.