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18
Operator Theoretic Approach to the Optimal TwoDisk Problem
 IEEE Transactions on Automatic Control
, 2004
"... Abstract—The nonstandard twodisk problem plays a fundamental role in robust feedback optimization. Here, it is shown via Banach space duality theory that its solutions satisfy an extremal identity, and may be viewed as a dual extremal kernel of a particular 1optimization problem. A novel operator ..."
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Cited by 9 (9 self)
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Abstract—The nonstandard twodisk problem plays a fundamental role in robust feedback optimization. Here, it is shown via Banach space duality theory that its solutions satisfy an extremal identity, and may be viewed as a dual extremal kernel of a particular 1optimization problem. A novel operator theoretic framework to characterize explicitly its solutions is developed, in particular, the twodisk optimization is shown to be equal to the induced norm of a specific operator defined on a projective tensor product space involving a nonHilbert version of a vector valued 2 space. Moreover, this operator is shown to be a combination of multiplication and Toeplitz operators. Under certain conditions, existence of maximal vectors is established leading to an explicit formula for the optimal controller. An “infinite matrix ” representation with respect to a canonical basis is derived, together with an algorithm to compute it. The norm of the relevant operator is approximated by special finite dimensional optimizations whose solutions lead to solving semidefinite programming problems involving the computation of a matrix projective tensor norm. Index Terms—Feedback systems, functional analysis, optimal control, optimization methods, robustness, uncertainty. NOTATION stands for the field of complex numbers. denotes either the inner or duality product depending on the context. denotes the identity map. If is a Banach space, then denotes its dual space. For anvector, where denotes thedimensional complex space, is the Euclidean norm. is the space of matrices, where is the largest singular value of., and denote the complex Banach space ofvectors, , and with, respectively, the norms and (1) Clearly, is the dual space of and vise versa.,, , and denote the complex Banach space of matrices, , with, respectively, the following norms:
Disturbance rejection and robustness for LTV Systems
 Proceedings of the 2006 American Control Conference
"... Abstract — In this paper, we consider the optimal disturbance rejection problem for (possibly infinite dimensional) linear timevarying (LTV) systems using a framework based on operator algebras of classes of bounded linear operators. In particular, after reducing the problem to a shortest distance ..."
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Cited by 7 (7 self)
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Abstract — In this paper, we consider the optimal disturbance rejection problem for (possibly infinite dimensional) linear timevarying (LTV) systems using a framework based on operator algebras of classes of bounded linear operators. In particular, after reducing the problem to a shortest distance minimization in a space of bounded linear operators, duality theory is applied to show existence of optimal solutions, which satisfy a “timevarying ” allpass or flatness condition. With the use of the theory of Mideals of operators, it is shown that the computation of timevarying (TV) controllers reduces to a search over compact TV Youla parameters. This involves the norm of a TV compact Hankel operator and its maximal vectors. Moreover, an operator identity to compute the optimal TV Youla parameter is also derived. The results are generalized to the mixed sensitivity problem for TV systems as well, where it is shown that the optimum is equal to the operator induced of a TV mixed HankelToeplitz operator generalizing analogous results known to hold in the LTI case. Finally, a numerical algorithm to compute optimal TV controllers is proposed. DEFINITIONS AND NOTATION B(E, F) denotes the space of bounded linear operators from a Banach space E to a Banach space F, endowed with the operator norm. ℓ2 denotes the usual Hilbert space of square summable sequences with the standard norm Pk the usual truncation operator for some integer k, which sets all outputs after time k to zero. An operator A ∈ B(E, F) is said to be causal if it satisfies the operator equation PkAPk = PkA, ∀k positive integers The subscript “c” denotes the restriction of a subspace of operators to its intersection with causal (see [7] for the definition) operators. “⊗ ” denotes for the tensor product. “ ⋆ ” stands for the adjoint of an operator or the dual space of a Banach space depending on the context [5], [6]. I.
MIMO Disturbance and Plant Uncertainty Attenuation by Feedback
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2004
"... This paper investigates the ability of feedback to reduce plant and disturbance uncertainties in the multipleinput–multipleoutput (MIMO) case, by solving two fundamental problems posed by Zames in the late seventies. The first problem is termed the MIMO extension of the optimal robust disturbance ..."
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Cited by 6 (6 self)
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This paper investigates the ability of feedback to reduce plant and disturbance uncertainties in the multipleinput–multipleoutput (MIMO) case, by solving two fundamental problems posed by Zames in the late seventies. The first problem is termed the MIMO extension of the optimal robust disturbance attenuation problem, the second is a filtering of plant uncertainty twodegree of freedom feedback problem. The optimal solutions of these problems are characterized using Banach space duality theory and shown to satisfy an “allpass ” condition and an extremal identity. Moreover, the duality description leads to a dual pair of optimizations and the introduction of two nonstandard matrix norms. In particular, the primaldual optimization reduces naturally to approximate finite dimensional convex optimizations within desired tolerance. Whereas the computation of the nonstandard matrix norms are shown to be equivalent to specific semidefinite programming problems, and a numerical solution based on convex programming is provided. It is also shown using Douglas ’ range inclusion theorem that performance is a monotonic function of uncertainty, and some qualitative implications for feedback are derived.
STRUCTURED DOUBLING ALGORITHMS FOR WEAK HERMITIAN SOLUTIONS OF ALGEBRAIC RICCATI EQUATIONS
"... Abstract. In this paper, we propose structured doubling algorithms for the computation of weak Hermitian solutions of continuous/discretetime algebraic Riccati equations. Under the assumptions that partial multiplicities of purely imaginary and unimodular eigenvalues (if any) of the associated Hami ..."
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Cited by 1 (1 self)
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Abstract. In this paper, we propose structured doubling algorithms for the computation of weak Hermitian solutions of continuous/discretetime algebraic Riccati equations. Under the assumptions that partial multiplicities of purely imaginary and unimodular eigenvalues (if any) of the associated Hamiltonian and symplectic pencil, respectively, are all even, we prove that the developed structured doubling algorithms converge to the desired Hermitian solutions globally and linearly. Numerical experiments show that structured doubling algorithms perform efficiently and reliably. 1.
Robustness in the gap metric and coprime factor . . .
"... In this paper, we study the problem of robust stabilization for linear timevarying (LTV) systems subject to timevarying normalized coprime factor uncertainty. Operator theoretic results which generalize similar results known to hold for linear timeinvariant (infinitedimensional) systems are deve ..."
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In this paper, we study the problem of robust stabilization for linear timevarying (LTV) systems subject to timevarying normalized coprime factor uncertainty. Operator theoretic results which generalize similar results known to hold for linear timeinvariant (infinitedimensional) systems are developed. In particular, we compute a tight upper bound for the maximal achievable stability margin under TV normalized coprime factor uncertainty in terms of the norm of an operator with a timevarying Hankel structure. We point to a necessary and sufficient condition which guarantees compactness of the TV Hankel operator, and in which case singular values and vectors can be used to compute the timevarying stability margin and TV controller.
Flight Control Division FOR THE COMMANDER
, 1990
"... When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely Governmentrelated procurement, the United States Government incurs no responsibility or any obligation whatsoever. The fact that the Government may have formulated or in any w ..."
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When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely Governmentrelated procurement, the United States Government incurs no responsibility or any obligation whatsoever. The fact that the Government may have formulated or in any way supplied the said drawings, specifications, or other data, is not to be regarded by implication, or otherwise in any manner construed, as licensing the holder, or any other person or corporation; or as conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto. This report is releasable to the National Technical Information Service (NTIS). At NTIS, it will be available to the general public, including foreign nations. This technical report has been reviewed and is approved for publication.
NAVAL AIR DEVELOPMENT CENTER
"... REPORT NUMBERING SYSTEM The numbering of technical project reports issued by the Naval Air Development Center is arranged for specific identification purposes. Each number consists of the Center acronym, the calendar year in which the number was assigned, the sequence number of the report within th ..."
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REPORT NUMBERING SYSTEM The numbering of technical project reports issued by the Naval Air Development Center is arranged for specific identification purposes. Each number consists of the Center acronym, the calendar year in which the number was assigned, the sequence number of the report within the specific calendar year, and the official 2digit correspondence code of the Command Officer or the Functional Department responsible for the report. For example: Report No. NADC8802060 indicates the twpnti',th Center report for the year 1988 and prepared by the Air Vehicle and Crew Systems Technology Department. The numerical codes are as follows: