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PDF Chain-scattering approach to h[infinity] control

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Thus the chain-scattering representation, the J-lossless factorization, and the J-lossless conjugation are the three key notions that provide the thread of development in this book. The book is completely self contained and requires little mathematical background other than some familiarity with linear algebra. Kimura's textbook is a useful source of information for everybody who wants to learn this part of modern control theory in a thorough manner.

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The book provides a quite complete picture of the chain-scattering techniques for the solution of conjugation, factorization and control problems. It somehow complements other books on similar arguments. The advent of H -control was a truly remarkable innovation in multivariable theory. It captures an essential feature of the control systems design, reducing it to a J-lossless factorization, which leads naturally to the idea of H-infinity-control.

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It will be useful to applied mathematicians and practicing engineers in control system design and as a text for a graduate course in H -control and its applications. The J-lossless conjugation, an essentially new notion in linear system theory, then provides a powerful tool for computing this factorization. Thus the chain-scattering representation, the J-lossless factorization, and the J-lossless conjugation are the three key notions that provide the thread of development in this book.

The book is completely self contained and requires little mathematical background other than some familiarity with linear algebra. Kimura's textbook is a useful source of information for everybody who wants to learn this part of modern control theory in a thorough manner.

The book provides a quite complete picture of the chain-scattering techniques for the solution of conjugation, factorization and control problems. It somehow complements other books on similar arguments.

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Glover, P. Khargonekar, and B. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control. Upper Saddle River. A result is given in [ 11 M. However, the Riccati equation approach has the advantage that it is numerically less demanding than the LMI approach. The results will be stated in terms of the following pair of complex algebraic Riccati equations:. If the above Riccati equations have suitable solutions, a quantum controller of the form 58 is constructed as follows:. The following Theorem is presented in [ 14 A.

Theorem 8 Necessity: Consider a quantum plant 57 satisfying the above assumptions. If the controller 58 is such that the matrices F c , G c , H c are as defined in 62 , then the resulting closed-loop system will satisfy the conditions 55 , Note that this theorem does not guarantee that a controller defined by 58 , 62 will be physically realizable. However, if the matrices defined in 62 are such that. However, a number of other recent results on aspects of quantum linear systems theory have not been covered in this paper. These include results on coherent quantum LQG control see [ 11 M.

Furthermore, in order to apply synthesis results on coherent quantum feedback controller synthesis, it is necessary to realize a synthesized feedback controller transfer function using physical optical components such as optical cavities, beam-splitters, optical amplifiers, and phase shifters. In a recent paper [ 22 H. An alternative approach to this problem is addressed in [ 19 I. For this class of quantum systems, an algorithm is given to realize a physically realizable controller transfer function in terms of a cascade connection of optical cavities and phase shifters.

An important application of both classical and coherent feedback control of quantum systems is in enhancing the property of entanglement for linear quantum systems. Entanglement is an intrinsically quantum mechanical notion which has many applications in the area of quantum computing and quantum communications.

To conclude, we have surveyed some of the important advances in the area of linear quantum control theory. However, many important problems in this area remain open and the area provides a great scope for future research. This is exactly what Open Access Journals provide and this is the reason why I support this endeavor. Open Access publishing is therefore of utmost importance for wider dissemination of information, and will help serving the best interest of the scientific community.

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On weight adjustments in H∞ control design

Abstract This paper surveys some recent results on the theory of quantum linear systems and presents them within a unified framework. Article Information. View Abstract. Download PDF. Download ePub. Quantum Harmonic Oscillators We begin by considering a collection of n independent quantum harmonic oscillators which are defined on a Hilbert space e.

A Generalized Framework of Linear Multivariable Control

We also have the commutation relations 3 For a general vector of operators on H , we use the notation to denote the corresponding vector of adjoint operators. Also, we specify the coupling operator for the quantum system to be an operator of the form where and. Also, we write In addition, we define a scattering matrix which is a unitary matrix. Lemma 1 A matrix satisfies if and only if where We now consider the case when the initial condition in the QSDE 5 is no longer the vector of annihilation operators 1 but rather a vector of linear combinations of annihilation operators and creation operators defined by where is non-singular.

Then, it follows from 4 that where 10 The relationship 11 is referred to as a generalized commutation relation [ 14 A.

Annihilation Operator Linear Quantum Systems An important special case of the linear quantum systems 12 , 13 , 16 corresponds to the case in which the Hamiltonian operator H and coupling operator L depend only of the vector of annihilation operators a and not on the vector of creation operators a. In this case, the linear quantum system can be modelled by the QSDEs 19 where It follows from 22 that 23 and hence 24 Rather than applying the transformations 21 to the quantum linear system 7 which satisfies the canonical commutation relations 4 , corresponding transformations can be applied to the quantum linear system 12 which satisfies the generalized commutation relations These transformations are as follows: 25 When these transformations are applied to the quantum linear system 12 , this leads to the following real quantum linear system: 26 where 27 These matrices are all real.

Also, it follows from 10 that 28 where 29 which is a Hermitian matrix. Now, we can re-write the operators H and L defining the above collection of quantum harmonic oscillators in terms of the variables and as where 30 Here 31 where the matrix R is real but the matrix V may be complex. However, using 30 and 16 , we can write where is real as in Also, we have 36 using 32 and 17 ; i.

Physical Realizability for General Linear Quantum Systems The formal definition of physically realizable QSDEs requires that they can be realized as a system of quantum harmonic oscillators.