IEE Colloquium on Multidimensional Systems: Problems and Solutions
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 Location: London, UK
 Conference date: 14 Jan. 1998
 Conference number: 1998/225
 The following topics were dealt with: transfer functions; robust discretisation; filter banks and wavelets; 3D IIR filters; perfect combinatorial models; 2D polyphase component evaluation; process plant modelling; 1D root loci; 2D leastorder systems; a behavioural/algebraic approach; 1D models of 2D systems; and variablecoefficient LQR problems
12 items found

Transfer function models of multidimensional physical systems
 Author(s): R. Rabenstein
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Transfer functions models of linear systems are a well established tool for the design and analysis of onedimensional systems. This is not the case for multidimensional physical systems, given in terms of initial boundary value problems. Here, the only mathematical description of widespread use are partial differential equations. This contribution shows, how the onedimensional transfer function concept can be generalized to multidimensional systems with initial and boundary conditions. The idea is first introduced by a simple example and then extended to arbitrary spatial dimensions, general boundary conditions, space dependent coefficients and nonselfadjoint differential operators. (7 pages)

A behavioural/algebraic approach to multidimensional systems theory
 Author(s): J. Wood ; E. Rogers ; S. Benton ; P. Zaris ; D.H. Owens
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The field of multidimensional systems theory suffers from the lack of a framework in which its many diverse strands could be unified. We propose the behavioural approach as such a framework. In the study of autoregressive systems, the use of behavioural tools is particularly useful since it allows the application of Oberst's duality theory, in which every AR nD behaviour is identified with a unique finitely generated module over the polynomial (Laurent polynomial) ring in n indeterminates. We have illustrated the efficacy of this approach by considering a fundamental algebraic concept, the annihilator of a module. We have shown that this concept relates to the autonomy of a behaviour, the idea of poles of an nD system, and notions of primeness and Bezout identities. We believe that the combination of the behavioural approach with algebraic techniques is suitable for the unification and solution of many problems in the study of multidimensional linear shift invariant systems. (6 pages)

1D models for class of 2D linear systems
 Author(s): K. Galkowski ; E. Rogers ; D.H. Owens
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The basic unique control problem for repetitive processes arises from the explicit interaction between successive pass profiles. In particular, the output sequence of pass profiles can contain oscillations that increase in amplitude in the pass to pass direction. Such behaviour is easily generated in simulation studies and in experiments on scaled models of industrial examples such as longwall coal cutting. In longwall coal cutting this problem appears as severe undulations in the newly cut floor profile which means that cutting operations (i.e. productive work) must be suspended to enable their manual removal. This problem is one of the key factors behind the stop/start cutting pattern of a typical working cycle in a coal mine. In general, this problem cannot be removed by standard, i.e. 1D, control action. The basic reason for this is that such an approach essentially ignores their inherent 2D systems structure. Motivated by this key fact, Rogers and Owens (1992) have developed a stability theory for repetitive processes with linear dynamics and a constant pass length. This theory is based on an abstract model in a Banach space setting which includes all such processes as special cases. The results of applying this theory to a range of special cases are also known including discrete linear repetitive processes which are the subject of this paper. (4 pages)

Solution to the LQR problem of variable coefficients singular 2D systems using the wave advanced model
 Author(s): M. Shafiee and M. HajiRamazanali
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The optimal control problem of variable coefficients singular 2D systems is under study. To extend the results developed in Shafee (1995) to singular case, the singular wave advanced model (SWAM) is introduced. In particular, two dimensional LQR problems of fixed final stage, with free and fixed final states, are considered and their equivalent 1D settings are formulated using SWAM. Optimal control laws are obtained and numerical examples are illustrated for each case. (8 pages)

Robust discretization algorithms for the numerical integration of nonlinear PDEs with application to a generalized capacitor
 Author(s): F.N. Koumboulis ; M.G. Skarpetis ; B.G. Mertzios
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Numerical integration of nonlinear (NL) partial differential equations (PDEs) is studied via approximating the original continuousdomain physical system by a discrete multidimensional (MD) and passive system, using principles of wave digital filters. Resulting integration algorithms are highly robust, massively parallel, and imply only local interconnections. The numerically integrated system of NL PDEs is a special case of Maxwell's equations, namely the system of two conductor plates separated by a dielectric. Inductance, capacitance and resistance coefficients are appropriate nonlinear functions of the currents, voltage, space and time. (6 pages)

Multidimensional filter banks and wavelets  a system theoretic perspective
 Author(s): S. Basu
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We review the current status of multidimensional filter banks and wavelet design from the perspective of signal and system theory. The study of wavelets and perfect reconstruction filter banks are known to have roots in traditional filter design techniques. On the other hand, the field of multidimensional systems and signal processing has developed a set of tools intrinsic to itself, and has attained a certain level of maturity over the last two decades. We have noted a degree of synergy between the two fields of wavelets and multidimensional systems. This arises from the fact that many ideas crucial to the wavelet design are inherently system theoretic in nature. While there are many examples of this synergy manifested in previous publications, we provide a flavour of techniques germane to this development by considering a few specific problems in detail. The construction of orthogonal wavelets can be essentially viewed as a circuit and system theoretic problem of design of energy dissipative (passive) filters, the multidimensional version of which has very close ties with classic problem of lumpeddistributed passive network synthesis. Groebner basis techniques, matrix completion problems over rings of polynomials or rings of stable rational functions, i.e., QuillenSuslin type problems are still other examples, which feature in our discussion in an important manner. A number of open problems are also cited. (51 pages)

Implementation of digital 3D IIR filters for streamprocessing applications
 Author(s): G. Runze
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Presents an implementation structure for recursive threedimensional digital filters, based on the filter design presented in Runze and Steffen (1996), which yields either recursive or nonrecursive filters. The recursive design results have two interesting properties: the recurrence direction is oriented parallel to one coordinate axis (e.g. time axis in datastream processing), and the transfer function itself is composed of separable systems. These properties can be exploited to build up the whole threedimensional system, using only spatial shifts and timedirected onedimensional recursive filters. (5 pages)

Multidimensional systems based on perfect combinatorial models
 Author(s): V.V. Riznyk
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The paper is concerned with the innovative techniques of multidimensional systems design based on multidimensional combinatorial sequencing theory, namely the concept of gold ring bundles (GRB), which can be used for finding optimal solutions for wide classes of technological problems. Research into the underlying mathematical principles relating to the optimal placement of structural elements in spatially/temporally distributed multidimensional systems with nonuniform structure (e.g. twodimensional arrays of radio antennas) or to the optimal choice of parameters in multidimensional systems (e.g. vector data coding systems, vector distributed electrical circuits, vectorisation systems etc.). These design techniques make it possible to configure multidimensional systems with fewer components then at present, while maintaining or improving on resolving ability, highspeed operation, and the other significant operating characteristic of the system. In particular, these results have been developed for the synthesis of nonuniformly spaced thinned antenna arrays with low level of side lobes. The results, essentially, relate to the development of new algebraic constructions based on the idea of perfect combinatorial configurations, such as cyclic difference sets, due to the remarkable properties and structural perfection of GRBs provide an ability to reproduce the maximum number of combinatorial varieties in the system with a limited number of elements and bonds. (4 pages)

2D polyphase component evaluation via complex integration
 Author(s): D.C. McLernon
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This paper shows an alternative approach to the evaluation of the 2D complex integral used in the calculation of the ztransform of the subsampled sequence of a 2D signal, and in particular, the polyphase components of a 2D digital filter, H(z, w). (6 pages)

Statistical approaches to industrial process plant modelling
 Author(s): M. Hartnett and G.W. Irwin
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The objective of this contribution is to describe the role of the increasingly popular multivariate SPC (MSPC) methods in the context of process engineering. This involves a discussion of some of the techniques used, the application areas in which they are being implemented and some of their limitations which are being addressed in recent algorithm modifications. To illustrate some of the topics raised, a case study is briefly described in which a principal component analysis based approach is used for predictive modelling of an overheads condenser reflux drum system. (8 pages)

A possible two dimensional system equivalent to one dimensional rootloci
 Author(s): P. Seekings ; G.E. Taylor ; P. Taylor
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The root locus diagram for a one dimensional system shows graphically how the closed loop poles vary as the gain increases. It provides a powerful design tool, allowing the designer to see immediately how a particular choice of gain will affect closed loop stability and dynamic response. The situation for two dimensional systems is more complex in that singularities in the transfer function (the poles) take the form of surfaces in 4D space. To display the variation of such curves with gain hence requires, at the least, four dimensions and, because of the potential complexity, probably more. This paper demonstrates a graphical test of stability for open loop, two dimensional systems, and shows how this may be extended to investigate the stability in the closed loop case. (5 pages)

The invariants of 2D least order systems
 Author(s): A.C. Pugh ; S.J. McInerney ; G.E. Hayton
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System transformations for 2D linear systems are discussed and the invariants of such transformations are noted. This paper shows that it is zero coprime system equivalence which forms the basis of the generalisation of Rosenbrock's least order characterisation for the case of 2D systems. As such the invariants of 2D least order system matrices giving rise to the same transfer function can be given. (5 pages)