The Vinnicombe metric for nonlinear operators

Anderson, Brian D. O.; Brinsmead, Thomas S.; Bruyne, F. De

doi:10.1109/tac.2002.802767

Cited by 32 publications

(32 citation statements)

References 24 publications

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“…The challenges include especially providing a sound basis for picking representative models, which is likely to require some form of ν-gap metric suitable for nonlinear systems. While such an entity has been defined, [34], the calculations would appear to be overpoweringly complicated.…”

Section: Discussionmentioning

confidence: 99%

Failures of adaptive control theory and their resolution

Anderson

2005

Communications in Information and Systems

159

108

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Abstract. Adaptive control is a very appealing technology, at least in principle. Yet its use has been conditioned by an attitude of distrustfulness on the part of some practitioners. In this paper, we explain why such distrustfulness is warranted, by reviewing a number of adaptive control approaches which have proved deficient for some reason that has not been immediately apparent. The explanation of the deficiencies, which normally were reflected in unexpected instabilities, is our main concern. Such explanations, coupled with remedies for avoiding the deficiencies, are necessary to engender confidence in the technology. These include the unpredictable failure of the MIT rule; the bursting phenomenon, and how to prevent it; the Rohrs' counterexample, which attempted to disqualify all adaptive control algorithms; the notion that identification of a plant is only valid conceptually for a restricted range of controllers (with the implication that in adaptive control, certain controller changes suggested by adaptive control algorithms may introduce instability); and the concept of multiple model adaptive control.

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Section: Discussionmentioning

confidence: 99%

Failures of adaptive control theory and their resolution

Anderson

2005

Communications in Information and Systems

159

108

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“…We now state the following lemma (in the z-domain) which asserts that the winding number condition (3) cannot be violated over a homotopy without passing through a point in the homotopy where there is a violation of the scalar version of the determinant condition (2). This is the discrete time analogue of Lemma 4.2 in [1] which applies to continuous time systems.…”

Section: Implications For Scalar Transfer Function Homotopiesmentioning

confidence: 98%

“…For linear systems, it is known that the graph metric (also known as the gap metric [8]) and the ν-gap (Nu-gap or Vinnicombe gap) metric [14,17] induce the weakest operator topologies under which both stability and performance are robust properties [6] 1 . In addition, various extensions of such metrics to nonlinear operator spaces have also been proposed [2,3,11,12,15].…”

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confidence: 99%

Properties of the graph topology for scalar transfer functions

Brinsmead

Anderson

2002

Communications in Information and Systems

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Abstract. Necessary and sufficient conditions for the existence of a scalar homotopy in the Vinnicombe metric between two given transfer functions are presented. Previous results show that for multivariable transfer functions, the existence of a homotopy is related to a winding number condition on the homotopy end-points. Here, it is shown that for scalar transfer functions the existence of a homotopy is also related to a Cauchy index condition. Extensive use is made of the relationship, previously noted by Brockett, between the Cauchy index of a real function and the argument of a particular related complex function.

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“…The corresponding issue of existence of normalized coprime factorizations for time-varying linear state-space models has been considered in several papers over the last two decades [15], [13], [1]. In [15], systems which are uniformly completely controllable and observable, and which define an exponential dichotomy, are considered.…”

Section: Introductionmentioning

confidence: 99%

Normalized coprime representations for time-varying linear systems

Mueller

Cantoni

2010

49th IEEE Conference on Decision and Control (CDC)

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Abstract-By 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 L 2 (−∞, ∞); 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.

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The Vinnicombe metric for nonlinear operators

Cited by 32 publications

References 24 publications

Failures of adaptive control theory and their resolution

Failures of adaptive control theory and their resolution

Properties of the graph topology for scalar transfer functions

Normalized coprime representations for time-varying linear systems

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