On the existence of right-coprime factorization for functions meromorphic in a half-plane

Mossaheb, S.

doi:10.1109/tac.1980.1102353

Cited by 12 publications

(5 citation statements)

References 8 publications

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“…Hence,

\overset{F}{true} false(z false) : = {false(z + 1 false)}^{- 1} F false(ψ false(z false) false) = O false(z^{- 1} false)

as z → ∞ on

C^{+}

. Therefore, by the work of Mossaheb, we get a coprime factorization of

\overset{F}{true} false(z false)

over

boldH false(C^{+} false)

, which, in view of z +1≠0 on

C^{+}

, yields a coprime factorization of F ∘ ψ over

boldH false(C^{+} false)

and hence, via ψ −1 , a coprime factorization of F ( s ) over H (Ω ′ ). 5) Now, consider one such

Ω^{'} \subset normalΩ

and its coprime factorization F = M −1 N over H (Ω ′ ).…”

Section: Winding Number Nyquist Stabilitymentioning

confidence: 94%

“…Therefore, by the work of Mossaheb, 37 we get a coprime factorization ofF(z) over H(C + ), which, in view of z + 1 ≠ 0 on C + , yields a coprime factorization of F • over H(C + ) and hence, via −1 , a coprime factorization of F(s) over H(Ω ′ ).…”

Section: Winding Number Nyquist Stabilitymentioning

confidence: 96%

“…Proof By Mossaheb, the strictly proper G ( s )− G ( ∞ ) has coprime factorizations over

H^{\infty} false({double-struckC}_{α}^{+} false)

due to the sufficiently rapid decay O ( s − r ), and so has G ( s ). Since G is the frequency representation of an operator

𝒢 \in T I C_{σ} false(U, Y false)

, it follows from theorem 8.4.1(ii) in Staffans that G ( s ) has a jointly exponentially stabilizable and detectable well‐posed realization.…”

Section: Boundary and Distributed Pde Controlmentioning

confidence: 99%

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Structured H_∞‐control of infinite‐dimensional systems

Apkarian

Noll

2018

Intl J Robust & Nonlinear

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Summary We develop a novel frequency‐based H∞‐control method for a large class of infinite‐dimensional linear time‐invariant systems in transfer function form. A major benefit of our approach is that reduction or identification techniques are not needed, which avoids typical distortions. Our method allows to exploit both state‐space or transfer function models and input/output frequency response data when only such are available. We aim for the design of practically useful H∞‐controllers of any convenient structure and size. We use a nonsmooth trust‐region bundle method to compute arbitrarily structured locally optimal H∞‐controllers for a frequency‐sampled approximation of the underlying infinite‐dimensional H∞‐problem in such a way that (i) exponential stability in closed loop is guaranteed and that (ii) the optimal H∞‐value of the approximation differs from the true infinite‐dimensional value only by a prior user‐specified tolerance. We demonstrate the versatility and practicality of our method on a variety of infinite‐dimensional H∞‐synthesis problems, including distributed and boundary control of partial differential equations, control of dead‐time and delay systems, and using a rich testing set.

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“…Hence,

\overset{F}{true} false(z false) : = {false(z + 1 false)}^{- 1} F false(ψ false(z false) false) = O false(z^{- 1} false)

as z → ∞ on

C^{+}

. Therefore, by the work of Mossaheb, we get a coprime factorization of

\overset{F}{true} false(z false)

over

boldH false(C^{+} false)

, which, in view of z +1≠0 on

C^{+}

, yields a coprime factorization of F ∘ ψ over

boldH false(C^{+} false)

and hence, via ψ −1 , a coprime factorization of F ( s ) over H (Ω ′ ). 5) Now, consider one such

Ω^{'} \subset normalΩ

and its coprime factorization F = M −1 N over H (Ω ′ ).…”

Section: Winding Number Nyquist Stabilitymentioning

confidence: 94%

Section: Winding Number Nyquist Stabilitymentioning

confidence: 96%

“…Proof By Mossaheb, the strictly proper G ( s )− G ( ∞ ) has coprime factorizations over

H^{\infty} false({double-struckC}_{α}^{+} false)

due to the sufficiently rapid decay O ( s − r ), and so has G ( s ). Since G is the frequency representation of an operator

𝒢 \in T I C_{σ} false(U, Y false)

, it follows from theorem 8.4.1(ii) in Staffans that G ( s ) has a jointly exponentially stabilizable and detectable well‐posed realization.…”

Section: Boundary and Distributed Pde Controlmentioning

confidence: 99%

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Structured H_∞‐control of infinite‐dimensional systems

Apkarian

Noll

2018

Intl J Robust & Nonlinear

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show abstract

“…Hence F(z) := (z + 1) −1 F(ψ(z)) = O(z −1 ) as z → ∞ on C + . Therefore by Mossaheb [33] we get a coprime factorization of F(z) over H(C + ), which in view of z + 1 ̸ = 0 on C + yields a coprime factorization of F • ψ over H(C + ), and hence via ψ −1 , a coprime factorization of F(s) over H(Ω ′ ).…”

Section: πmentioning

confidence: 98%

“…Proof. By Mossaheb [33] the strictly proper G(s)−G(∞) has coprime factorizations over H ∞ (C + α ) due to the sufficiently rapid decay O(s −r ), and hence so has G(s). Since G is the frequency representation of an operator G ∈ T IC σ (U,Y ), it follows from [30,Thm.…”

Section: Remark 15mentioning

confidence: 99%

Structured H-infinity control of infinite dimensional systems

Apkarian¹,

Noll²

2017

Preprint

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We develop a novel frequency-based H ∞ -control method for a large class of infinite-dimensional Linear-Time-Invariant systems in transfer function form. Major benefits of our approach is that reduction or identification techniques are not needed thereby avoiding possible distortions. It can exploit either transfer function models or input/output frequency response data when available. It computes simple practical controllers of any size and structure. We use a non-smooth trust-region method to compute arbitrarily structured locally optimal H ∞ -controllers for a frequency sampled approximation of the underlying infinite-dimensional H ∞ -problem in such a way that exponential stability in closed-loop is guaranteed, and the optimal value of the approximation captures the value of the infinite-dimensional problem within a prior tolerance level. We demonstrate the versatility and practicality of our method on a variety of infinite-dimensional H ∞ -synthesis problems, including distributed and boundary control of PDEs, control of dead time and delay systems, and using a rich testing set.

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