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Distributed stabilisation of spatially invariant systems: positive polynomial approach

  • Petr Augusta
  • Zdeněk Hurák
Article

Abstract

The paper gives a computationally feasible characterisation of spatially distributed controllers stabilising a linear spatially invariant system, that is, a system described by linear partial differential equations with coefficients independent on time and location. With one spatial and one temporal variable such a system can be modelled by a 2-D transfer function. Stabilising distributed feedback controllers are then parametrised as a solution to the Diophantine equation ax + by = c for a given stable bi-variate polynomial c. The paper is built on the relationship between stability of a 2-D polynomial and positiveness of a related polynomial matrix on the unit circle. Such matrices are usually bilinear in the coefficients of the original polynomials. For low-order discrete-time systems it is shown that a linearising factorisation of the polynomial Schur-Cohn matrix exists. For higher order plants and/or controllers such factorisation is not possible as the solution set is non-convex and one has to resort to some relaxation. For continuous-time systems, an analogue factorisation of the polynomial Hermite-Fujiwara matrix is not known. However, for low-order systems and/or controller, positivity conditions on the closed-loop polynomial coefficients can be invoked. Then the computational framework of linear matrix inequalities can be used to describe the stability regions in the parameter space using a convex constraint.

Keywords

Multidimensional systems Algebraic approach Control design Positiveness Convex optimisation 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPraha 8Czech Republic
  2. 2.Faculty of Electrical EngineeringCzech Technical University in PraguePraha 6Czech Republic

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