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Open AccessFeature PaperArticle

A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework

Department of Electrical, Control and Computer Engineering, Opole University of Technology, Prószkowska 76 Street, 45-758 Opole, Poland
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Symmetry 2019, 11(10), 1322; https://doi.org/10.3390/sym11101322
Received: 4 October 2019 / Revised: 17 October 2019 / Accepted: 18 October 2019 / Published: 22 October 2019
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
The new general theory dedicated to the stability for LTI MIMO, in particular nonsquare, fractional-order systems described by the Grünwald–Letnikov discrete-time state–space domain is presented in this paper. Such systems under inverse model control, principally MV/perfect control, represent a real research challenge due to an infinite number of solutions to the underlying inverse problem for nonsquare matrices. Therefore, the paper presents a new algorithm for fractional-order perfect control with corresponding stability formula involving recently given H- and σ -inverse of nonsquare matrices, up to now applied solely to the integer-order plants. On such foundation a new set of stability-related tools is introduced, among them the key role played by so-called control zeros. Control zeros constitute an extension of transmission zeros for nonsquare fractional-order LTI MIMO systems under inverse model control. Based on the sets of stable control zeros a minimum-phase behavior is specified because of the stability of newly defined perfect control law described in the non-integer-order framework. The whole theory is complemented by pole-free fractional-order perfect control paradigm, a special case of fractional-order perfect control strategy. A significant number of simulation examples confirm the correctness and research potential proposed in the paper methodology. View Full-Text
Keywords: stability criteria; feedback control methods; zero sets; pole zero assignment; minimum-phase systems; robust control; matrix inversion; state–space models; MIMO stability criteria; feedback control methods; zero sets; pole zero assignment; minimum-phase systems; robust control; matrix inversion; state–space models; MIMO
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Hunek, W.P.; Wach, Ł. A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework. Symmetry 2019, 11, 1322.

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