Structural Stability of Polynomial Matrices Related to Linear Time-Invariant Singular Systems (original) (raw)
Related papers
Higher-Order Singular Systems and Polynomial Matrices
There is a one-to-one correspondence between the set of quadruples of matrices defining singular linear time-invariant dynamical systems and a subset of the set of polynomial matrices. This correspondence preserves the equivalence relations introduced in both sets (feedback-similarity and strict equivalence): two quadruples of matrices are feedback-equivalent if, and only if, the polynomial matrices associated to them are also strictly equivalent.
A criterion for structural stability of quadruples of matrices related to singular linear systems
Journal of Mathematical Sciences, 2007
The concept of structural stability, first introduced by A.A. Andronov and L.Pontryagin in 1937 in the qualitative theory of dynamical systems (structurally stable elements being those whose behaviour does not change when applying small perturbations) has been studied by many authors in Control Theory. We present in this work conditions for structurally stable linear time-invariant singular systems when considering different equivalence relations among them.
Systems & Control Letters
The concept of structural stability, first introduced by A.A. Andronov and L.Pontryagin in 1937 in the qualitative theory of dynamical systems (structurally stable elements being those whose behaviour does not change when applying small perturbations) has been studied by many authors in Control Theory. We present in this work conditions for structurally stable linear time-invariant singular systems when considering different equivalence relations among them.
Dimension of orbits of linear time-invariant singular systems under restricted system equivalence
Linear Algebra and its Applications, 2008
We consider the set of quadruples of matrices defining regular singular linear time-invariant dynamical systems under restricted system equivalence and derive miniversal deformations from a basis of the normal space to orbits under the Lie group action related to this equivalence relation. A lower bound and an upper bound for the dimension of the orbits are obtained. We conclude with examples and further comments about genericity. (J. Clotet), M.Dolors.Magret@upc.edu (M.D. Magret).
Robust stability of a special class of polynomial matrices with control applications
Computers & Electrical Engineering, 2003
The robust stability problem of uncertain continuous-time systems described by higher-order dynamic equations is considered in this paper. Previous results on robust stability of Metzlerian matrices are extended to matrix polynomials, with the coefficient matrices having exactly the same Metzlerian structure. After defining the structured uncertainty for this class of polynomial matrices, we provide an explicit expression for the real stability radius and derive simplified formulae for several special cases. We also report on alternative approaches for investigating robust Hurwitz stability and strong stability of polynomial matrices. Several illustrative examples throughout the paper support the theoretical development. Moreover, an application example is included to demonstrate uncertainty modeling and robust stability analysis used in control design.
Robust stability of linear systems described by higher-order dynamic equations
IEEE Transactions on Automatic Control, 2000
In this note we study the stability radius of higher order di erential and difference systems with respect to various classes of complex a ne perturbations of the coe cient matrices. Di erent perturbation norms are considered. The aim is to derive robustness criteria which are expressed directly in terms of the original data. Previous results on robust stability of Hurwitz and Schur polynomials 13] are extended to monic matrix polynomials. For disturbances acting via a uniform input matrix, computable formulae are obtained whereas for perturbations with multiple input matrices structured singular values are involved.
Approximate zero polynomials of polynomial matrices and linear systems
2011
This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials [1] and the exterior algebra [4] representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros [2], [4] of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials".
Stability bounds for higher order linear dynamical systems
2000
This paper derives analytic expressions for the real stability radius of polynomial matrices with respect to an arbitrary region in the complex plane. We are also discussing numerical issues for computing these radii for different perturbation structures, with application to robust stability of Hurwitz and Schur polynomial matrices.