A lattice approach to analysis and synthesis problems

Quadrat, Alban

doi:10.1007/s00498-005-0159-2

Cited by 29 publications

(57 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(in our terms); the rest of the theorem is fairly straight-forward. We are grateful to one of the referees for pointing out to us the existence of [27] and [28] and that one obtains an alternative proof of the existence of a w.r.c.f. (and of Theorem 6.14) in the scalarvalued case from [27,Proposition 8 & Corollary 5], whose proofs can be applied to the matrix-valued case too when combined with [28, Theorem 1].…”

Section: Notesmentioning

confidence: 97%

See 1 more Smart Citation

Weakly coprime factorization and state-feedback stabilization of discrete-time systems

Mikkola

2008

Math. Control Signals Syst.

View full text Add to dashboard Cite

The LQ-optimal state feedback of a finite-dimensional linear timeinvariant system determines a coprime factorization NM −1 of the transfer function. We show that the same is true also for infinite-dimensional systems over arbitrary Hilbert spaces, in the sense that the factorization is weakly coprime, i.e., N f, M f ∈ H 2 ⇒ f ∈ H 2 for every function f . The factorization need not be Bézout coprime. We prove that every proper quotient of two bounded holomorphic operator-valued functions can be presented as the quotient of two bounded holomorphic weakly coprime functions. This result was already known for matrix-valued functions with the classical definition gcd(N , M) = I , which we prove equivalent to our definition. We give necessary and sufficient conditions and further results for weak coprimeness and for Bézout coprimeness. We then establish a variant of the inner-outer factorization with the inner factor being "weakly left-invertible". Most of our results hold also for continuous-time systems and many are new also in the scalar-valued case.

show abstract

Section: Notesmentioning

confidence: 97%

“…6 and a rich algebraic description of w.r.c.f. 's in the matrix-valued case is given in [24] and in [28].…”

mentioning

confidence: 99%

Weakly coprime factorization and state-feedback stabilization of discrete-time systems

Mikkola

2008

Math. Control Signals Syst.

View full text Add to dashboard Cite

show abstract

“…Otherwise one can only discuss the algebraic or module theoretic part of the systems as it has been done, for instance, in Quadrat's paper [20] and its predecessors. Theorem 2 of the present paper shows that the action or scalar multiplication…”

Section: Corollary 4 the Equationmentioning

confidence: 98%

Multidimensional BIBO stability and Jury’s conjecture

Scheicher

Oberst

2008

Math. Control Signals Syst.

View full text Add to dashboard Cite

Twenty years ago E. I. Jury conjectured by analogy to the case of digital filters that a two-dimensional analog filter is BIBO stable if its transfer function has the form H = 1/P where P is a very strict Hurwitz polynomial (VSHP). In more detail he conjectured that the impulse response of the filter is an absolutely integrable function. However, he did not specify the exact equations of these filters and did not prove the existence of the impulse response. In the present paper we generalise Jury's conjecture to arbitrary proper transfer functions H = Q/P where P is a bivariate VSHP and prove this generalisation. In particular, we show the existence of a suitable impulse response or fundamental solution for any multivariate proper rational function. However, this impulse response is a measure and not a function. We have not succeeded to prove an analogue of Jury's conjecture in higher dimensions than two yet, but we propose a new conjecture in context with the robustly stable multivariate polynomials investigated by Kharitonov et. al. For the discrete case we prove that the structurally stable rational functions after Bose, Lin et al. coincide with the stable rational functions discussed in context with the stabilisation of discrete input/output systems. These rational functions are BIBO stable, but the converse is not true as established by several authors.Keywords Multidimensional input/output system · Impulse response · Fundamental solution · BIBO stability · Jury's conjecture Financial support of M. Scheicher through the Austrian FWF-project P18974 is gratefully acknowledged.M. Scheicher (B) · U. Oberst

show abstract

“…The set of fractions of elements of A (σ ) = e −σt A is called the Callier-Desoer class and is a key concept to describe robust stabilization methods for a large class of distributed systems. The matter continues to generate interesting results, see for instance A. Quadrat [20], or P. Lakkonen [13] for a recent survey.…”

Section: Convolution Kernelsmentioning

confidence: 99%

“…The set A is sometimes called Wiener algebra (see, e.g. [20]). Many properties of the set A are exposed in [10], and its use in control theory was gradually introduced by various authors, among them C. A. Desoer [5,6,1], F. Callier [5,1] and M. Vidyasagar [6].…”

Section: Convolution Kernelsmentioning

confidence: 99%

Polyhedral Invariance for Convolution Systems over the Callier-Desoer Class

Loiseau

2017

Positive Systems

View full text Add to dashboard Cite

A lattice approach to analysis and synthesis problems

Cited by 29 publications

References 37 publications

Weakly coprime factorization and state-feedback stabilization of discrete-time systems

Weakly coprime factorization and state-feedback stabilization of discrete-time systems

Multidimensional BIBO stability and Jury’s conjecture

Polyhedral Invariance for Convolution Systems over the Callier-Desoer Class

Contact Info

Product

Resources

About