62 research outputs found

    Optimal control on the doubly infinite continuous time axis and coprime factorizations

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    Coprime factorization and optimal control on the doubly infinite discrete time axis

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    We study the problem of strongly coprime factorization over H-infinity of the unit disc. We give a necessary and sufficient condition for the existence of such a coprime factorization in terms of an optimal control problem over the doubly infinite discrete-time axis. In particular, we show that an equivalent condition for the existence of such a coprime factorization is that both the control and filter algebraic Riccati equation (of an arbitrary realization) have a solution (in general unbounded and even non densely defined) and that a coupling condition involving these solutions is satisfied

    Optimal Control on the Doubly Infinite Time Axis for Well-Posed Linear Systems

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    Linear Control Theory with an ℋ∞ Optimality Criterion

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    This expository paper sets out the principal results in ℋ∞ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods

    Analytic system problems and J-lossless coprime factorization for infinite-dimensional linear systems

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    AbstractThis paper extends the coprime factorization approach to the synthesis of internally stabilizing controllers satisfying an H∞-norm bound to a class of systems with irrational transfer matrices. Using the coprime factorization description, the H∞-control problem can be reduced to two stable analytic system problems. Such problems have solutions if and only if a certain J-lossless factorization exists. The full H∞-synthesis problem is shown to be equivalent to the solution of two nested J-lossless factorizations. If the irrational transfer matrix has a state-space realization, then the known state-space formulas for the H∞-control problem may be recovered using the relationship between J-lossless factorizations and solutions of Riccati equations. However, the results derived here are valid for a larger class of infinite-dimensional systems

    Normalized coprime representations for time-varying linear systems

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    Copyright © 2010 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.49th IEEE Conference on Decision and Control, Atlanta, USA, 15-17 December 2010By considering the behaviour of stabilizable and detectable, linear time-varying state-space models over doublyinfinite continuous time, we establish the existence of so-called normalized coprime representations for the system graphs; that is, stable and stably left (resp. right) invertible, image (resp. kernel) representations that are normalized with respect to the inner product on L²(−∞,∞); this is consistent with the notion of normalization used in the time-invariant setting. The approach is constructive, involving the solution of timevarying differential Riccati equations with single-point boundary conditions at either +∞ or −∞. The contribution lies in accommodating state-space models that may not define an exponential dichotomy

    Stabilizing solutions of the H∞ algebraic Riccati equation

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    AbstractThe algebraic Riccati equation studied in this paper is related to the suboptimal state feedback H∞ control problem. It is parametrized by the H∞-norm bound γ we want to achieve. The objective of this paper is to study the behavior of the solution to the Riccati equation as a function of γ. It turns out that a stabilizing solution exists for all but finitely many values of γ larger than some a priori determined bound γ−. On the other hand, for values smaller than γ− there does not exist a stabilizing solution. The finite number of exception points can be characterized as switching points where eigenvalues of the stabilizing (symmetric) solution can switch from negative to positive with increasing γ. After the final switching point the solution will be positive semidefinite. We obtain the following interpretation: The Riccati equation has a stabilizing solution with k negative eigenvalues if and only if there exists a static feedback such that the closed-loop transfer matrix has k unstable poles and an L∞ norm strictly less than γ
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