Coprime Factorization and Dynamic Stabilization of Transfer Functions

Mikkola, Kalle

doi:10.1137/050641740

Cited by 9 publications

(7 citation statements)

References 32 publications

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“…A sixth equivalent condition is that P is dynamically stabilizable (i.e., I −Q −P I −1 ∈ H ∞ for some proper Q), as will be shown in [16] using Theorems 1.1 and 1. 3.…”

Section: Moreover a Realization A B C D Of P Is Output-stabilizable mentioning

confidence: 98%

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

Mikkola

2008

Math. Control Signals Syst.

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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.

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“…A sixth equivalent condition is that P is dynamically stabilizable (i.e., I −Q −P I −1 ∈ H ∞ for some proper Q), as will be shown in [16] using Theorems 1.1 and 1. 3.…”

Section: Moreover a Realization A B C D Of P Is Output-stabilizable mentioning

confidence: 98%

“…The connection between dynamic stabilization and (Bézout) coprime factorization has been established also for general nonrational functions in, e.g., [12,26,32,42] in the matrix-valued case, and in the operator-valued case in [5,6] and [16]; all these for transfer functions only. Fairly general state-space results are given in [45].…”

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confidence: 98%

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

Mikkola

2008

Math. Control Signals Syst.

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“…Even more is true: the right factorization can be chosen to be normalized and weakly coprime. See Smith [14] (for the case that U and Y are finite-dimensional) and Mikkola [10] (for the general case). Using the results in Mikkola [10] it is not too difficult to show that the input-output map of the system (5.1) with K = K f and E = S −1/2 f provides a normalized weakly right coprime factorization.…”

Section: Implies 3 If the Control Riccati Equation Of The Node [ A Bmentioning

confidence: 99%

Optimal State Feedback Input-Output Stabilization of Infinite-Dimensional Discrete Time-Invariant Linear Systems

Opmeer

Staffans

2007

Complex anal.oper. theory

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We study the optimal input-output stabilization of discrete timeinvariant linear systems in Hilbert spaces by state feedback. We show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a right factorization over H-infinity. A necessary and sufficient condition in terms of an (arbitrary) realization is that each state which can be reached in a finite time from the zero initial state has a finite cost. Another equivalent condition is that the control Riccati equation has a solution (in general unbounded and even non densely defined). The optimal state feedback input-output stabilization problem can then be solved explicitly in terms of the smallest solution of this control Riccati equation. We further show that after renorming the state space in terms of the solution of the control Riccati equation, the closed-loop system is not only input-output stable, but also strongly internally stable. Mathematics Subject Classification (2000). 47N70, 47A48, 47A20, 47A56, 47A62, 93B28, 93C55, 93D15.

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“…The transfer function G is said to have a right factorization if there exists a func- spaces; [11] for the general case of possibly infinite-dimensional input and output spaces).…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Coprime factorization and robust stabilization for discrete-time infinite-dimensional systems

Curtain

Opmeer

2011

Math. Control Signals Syst.

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We solve the problem of robust stabilization with respect to right-coprime factor perturbations for irrational discrete-time transfer functions. The key condition is that the associated dynamical system and its dual should satisfy a finite-cost condition so that two optimal cost operators exist. We obtain explicit state space formulas for a robustly stabilizing controller in terms of these optimal cost operators and the generating operators of the realization. Along the way we also obtain state space formulas for Bezout factors.

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Coprime Factorization and Dynamic Stabilization of Transfer Functions

Cited by 9 publications

References 32 publications

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

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

Optimal State Feedback Input-Output Stabilization of Infinite-Dimensional Discrete Time-Invariant Linear Systems

Coprime factorization and robust stabilization for discrete-time infinite-dimensional systems

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