The class of stabilizing nonlinear plant controller pairs

Paice, Andrew; Schaft, A.J. van der

doi:10.1109/9.489201

Cited by 56 publications

(19 citation statements)

References 25 publications

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“…It is also known that in the case of nonlinear control systems, the stabilizing controllers can be characterized using kernel representation based on generalized left co-prime factorization [14][15][16] Here, the mapping expressed by Q represents a free parameter defining a stabilized nonlinear system, which determines the characteristics of the closed-loop system, while mapping expressed by K o represents a free parameter defining a stabilized nonlinear system, which determines the characteristics of the target value response. Representation by the above equations corresponds to a control system with two degrees of freedom and is an expansion of nonlinear parametrization of the previous system with one degree of freedom [14]. A block diagram of the control system based on this parametrization is shown in Fig.…”

Section: Controller Parametrizationmentioning

confidence: 99%

Design of discrete time polynomial nonlinear systems and its application to sequential control

Fujimoto

2004

Electrical Engineering Japan

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A systematic design methodology for discrete control systems is proposed, which is based on Galois field and Gröbner basis. The controller can be automatically designed using a model of the controlled plant. The controlled plant is represented on the Galois field. Then the inverse function of the plant is derived using the Gröbner basis so that the input variables are explicitly obtained as the function of the state variable and reference variable. An illustrative example of a material handling system is shown. The proposed method will realize cost reduction in software development and an improvement of reliability and safety of the PLC-based systems.

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Section: Controller Parametrizationmentioning

confidence: 99%

Design of discrete time polynomial nonlinear systems and its application to sequential control

Fujimoto

2004

Electrical Engineering Japan

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“…Anderson et al (1998), Sontag (1989) and Verma and Hunt (1993) discuss extensions of coprime factor robustness to cover non-linear systems. Paice and van der Schaft (1996) uses a Kernel representation to study the stability of plant-controller pairs and compares the results to those of coprime factorisation. The work on non-linear coprime factor robustness was then related to the non-linear gap in French (2005, 2003) and James et al (2005).…”

Section: Other Non-linear Generalisations Of the Gapmentioning

confidence: 99%

Robust stability for iterative learning control

Bradley

French

2009

2009 American Control Conference

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This thesis examines the notion of the long term robust stability of iterative learning control (ILC) systems engaged in trajectory tracking, using a robust stability theorem based on a biased version of the nonlinear gap metric. This is achieved through two main results:The first concerns the establishment of a nonlinear robust stability theorem, where signals are measured relative to a given trajectory. Although primarily motivated by ILC, the theorem provided is applicable to a wider range of problems. This is due to its development being made independently of any particular signal space, provided the space is furnished with a definition of causality. The theorem's formulation therefore permits its implementation on single-or multi-dimensional problems in a variety of different settings. Necessitated by an ILC constraint concerning reference signals, the trajectory that stability is measured relative to must often lie outside the signal space that is chosen. The robust stability theorem is therefore devised to address signals that lie in extended signal spaces throughout. Additionally, the theorem is applicable to nonlinear systems, and it is shown that the biased gap measure collapses to the standard nonlinear gap measure when the bias is set to zero. It is also shown to collapse to the classical linear gap when restricting the analysis to certain linear systems.The second result applies the robust stability theorem to ILC using a 2D signal space.Initially the subject of ILC is reviewed and some of the problems associated with controllers are described; in particular the issues of long-term stability and the criteria for convergence. ILC algorithms expressed in the 'lifted system' or 'supervector' formulation are discussed and then analysed using the biased robust stability theorem. Results are presented regarding the use of filtering in ILC algorithms to aid robustness, and also the robustness of inverse model-based techniques. The robust stability tool applied to ILC in this thesis is not restricted in its analysis to algorithms which are 'causal' (in an ILC sense); and is based on a general unstructured uncertainty model in contrast to the existing literature, whereby uncertainties are typically constrained to additive, multiplicative or parametric models.

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“…Definitions 2.3 and 2.4 are generalizations of the coprime factorisation and normalized coprime factorisation for linear operators (see [20]) to the nonlinear case, as considered previously by various authors. Definition 2.3 is given by Verma and Hunt in [19] (see also [12,18]) where the stability is in the sense of "bounded input implies bounded output" (resp. linear gain) between normed spaces.…”

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confidence: 99%

Graph Topologies, Gap Metrics, and Robust Stability for Nonlinear Systems

Bian¹,

French²

2005

SIAM J. Control Optim.

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Abstract. Graph topologies for nonlinear operators which admit coprime factorisations are defined w.r.t. a gain function notion of stability in a general normed signal space setting. Several metrics are also defined and their relationship to the graph topologies are examined. In particular, relationships between nonlinear generalisations of the gap and graph metrics, Georgiou-type formulae and the graph topologies are established. Closed loop robustness results are given w.r.t. the graph topology, where the role of a coercivity condition on the nominal plant is emphasised.Key words. gap metric, graph metric, graph topology, robust stability, nonlinear systems AMS subject classifications. 93D09, 93D25, 93C101. Introduction. The theory of coprime factorisations of linear signal operators is well known to be a significant tool in the study of robustness of stability for linear feedback systems and has been extensively studied (see [5,16,20]). Perturbations to normalized co-prime factors form a good description of physically realistic deviations from nominal models, since they allow a unified treatment of both low and high frequency uncertainties [8]. In the linear theory, it is well known that the graph topology is the appropriate topological description for studying robustness of stability and that co-prime factor perturbations can be used to induce the graph topology. Furthermore, the graph topology is metrizable, and both the gap metric [3,21] and the graph metric [20] provide suitable metrizations, the former being more suitable for calculations by standard H ∞ optimizations, (although both metrics are topologically equivalent) [5,16,21]. There is thus a rich set of equivalences between the notions of co-prime factorisations, gap/graph metrics and topologies and their attendant robust stability theorems. Moreover, this framework is a cornerstone of modern robust linear control theory.Given the richness and importance of this framework in the linear setting, it is natural to seek extensions to the nonlinear case, and to alternative signal spaces. Indeed, by adopting a notion of stability corresponding to the existence of a linear gain, (typically either in an L 2 or L ∞ setting), a number of authors have previously considered a nonlinear theory of co-prime factorisation. Here we highlight three contributions of particular relevance to the context of this paper. In [18], Verma defined a notion of co-prime factorisation for nonlinear mappings and, presented a stability result for a nonlinear system. In [2], Anderson, James and Limebeer generalised the linear theory of normalized co-prime factor robustness optimisation to the case of affine input nonlinear systems and presented a optimal robustness margin. In [10], a new definition of "normalized" was introduced for left representation for the graph of a nonlinear system and different gap metrics were studied. Many further pointers to a growing literature on nonlinear co-prime factorisation can be found in the monograph [14] and the references therein.On the other ...

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The class of stabilizing nonlinear plant controller pairs

Cited by 56 publications

References 25 publications

Design of discrete time polynomial nonlinear systems and its application to sequential control

Design of discrete time polynomial nonlinear systems and its application to sequential control

Robust stability for iterative learning control

Graph Topologies, Gap Metrics, and Robust Stability for Nonlinear Systems

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