Robust Stability and Stable Member Problems for Multilinear Systems

In this paper, we consider robust stability and stable member problems for linear systems whose characteristic polynomials are nonmonic polynomials with multilinear uncertainty. For both problems, the results are given by using the reflection (box) coefficients and the extreme point property of multilinear functions defined on the box. Finding stable member in a polynomial family is one of the hard problems of linear control theory. This issue is considered by visualizing the cases n-l=2 and n-l=3. Necessary and sufficient conditions for robust stability and the existence of a stable member of the multilinear polynomial family using the reflection coefficients are obtained. Several examples are provided.

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[1] Bhattacharyya S.P., Chapellat H., Keel L., Robust control: The parametric approach, Prentice-Hall, New Jersey, (1995) 432-459.
[2] Nürges Ü., New Stability Conditions via Reflection Coefficients of Polynomials, IEEE Transactions on Automatic Control, 50 (9) (2005) 1354-1360.
[3] Anderson B.D.O., Kraus F., Mansour M., Dasgupta S., Easily Testable Sufficient Conditions for the Robust Stability of Systems with Multilinear Parameter Dependence, Automatica, 31 (1) (1995) 25-40.
[4] Tsing N.K., Tits A.L., When is the Multiaffine Image of a Cube a Convex Polygon?, Systems & Control Letters, 20 (6) (1993) 439-445.
[5] Akyar H., Büyükköroğlu T., Dzhafarov V., On Stability of Parametrized Families of Polynomials and Matrices, Abstract and Applied Analysis, Article ID 687951 (2010) 1-16.
[6] Dzhafarov V., Büyükköroğlu T., Akyar H., Stability Region for Discrete Time Systems and Its Boundary, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 27 (3) (2021) 246-255.
[7] Yılmaz Ş., Stable Polytopes for Discrete Systems by Using Box Coefficients, Circuits, Systems, and Signal Processing, 41 (2) (2022) 789-804.
[8] Barmish B.R., New tools for robustness of linear systems, Macmillan, New York, (1994) 237-256.
[9] Yılmaz Ş., Büyükköroğlu T., Dzhafarov V., Random Search of Stable Member in a Matrix Polytope, Journal of Computational and Applied Mathematics, 308 (2016) 59-68.
[10] Polyak B.T., Shcherbakov P.S., Hard Problems in Linear Control Theory: Possible Approaches to Solution, Automation and Remote Control, 66 (2005) 681-718.
[11] Fam A.T., Meditch J.S., A Canonical Parameter Space for Linear Systems Design, IEEE Transactions on Automatic Control, 23 (3) (1978) 454-458.
[12] Büyükköroğlu T., Çelebi G., Dzhafarov V., Stabilisation of Discrete-Time Systems via Schur Stability Region, International Journal of Control, 91 (7) (2018) 1620-1629.
[13] Nürges Ü., Avanessov S., Fixed-Order Stabilising Controller Design by a Mixed Randomized/Deterministic Method, International Journal of Control, 88 (2) (2015) 335-346.
[14] Petrikevich Y.I., Randomized Methods of Stabilization of the Discrete Linear Systems, Automation and Remote Control, 69 (11) (2008) 1911-1921.