Linear Matrix Inequalities for Digital Redesign Under Delay Suitable for PI Controllers with Application to PMSMs

  • Thieli S. Gabbi
  • Gustavo G. KochEmail author
  • Rodrigo P. Vieira
  • Vinícius F. Montagner
  • Eduardo S. Tognetti
  • Renato A. Borges
  • Luiz A. MaccariJr.
  • Ricardo C. L. F. Oliveira


This paper investigates the digital redesign of proportional integral (PI) controllers implemented under a delay of one sampling period. The usual digital redesign based on state matching cannot be directly applied here. The digital control gains are obtained by means of linear matrix inequalities that minimize the \({\mathcal {H}}_{\infty }\) norm associated with an error signal defined as the difference between the closed-loop system with continuous-time controller and the closed-loop system with the discrete-time controller implemented under the delay. Three synthesis conditions are given, exploring extra matrix variables and suitable matrix structures to provide more efficient redesigns that can deal even with large control delays in the implementation of the redesigned controller. As a contribution, the proposed conditions provide a theoretical guarantee that digital PIs, operating under delay, will perform very close to originally designed continuous-time PIs. A numerical example illustrates the efficiency of the proposed conditions, and a practical application for permanent magnet synchronous motor speed regulation shows good experimental results, illustrating the viability of the proposal.


Digital redesign Linear matrix inequalities Permanent magnet synchronous motor 



This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Ní vel Superior—Brasil (CAPES/PROEX)—Finance Code 001. The authors would also like to thank the INCT-GD and the finance agencies (CNPq 465640/2014-1, CAPES 23038.000776/2017-54 e FAPERGS 17/2551-0000517-1).


Funding was provided by Fundação de Apoio à Pesquisa do Distrito Federal and Conselho Nacional de Desenvolvimento Científico e Tecnológico (BR) (Grant Nos. 477487/2013-0, 480415/2013-7, 306197/2015-4).


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Copyright information

© Brazilian Society for Automatics--SBA 2019

Authors and Affiliations

  • Thieli S. Gabbi
    • 1
  • Gustavo G. Koch
    • 1
    Email author
  • Rodrigo P. Vieira
    • 1
  • Vinícius F. Montagner
    • 1
  • Eduardo S. Tognetti
    • 2
  • Renato A. Borges
    • 2
  • Luiz A. MaccariJr.
    • 3
  • Ricardo C. L. F. Oliveira
    • 4
  1. 1.Federal University of Santa Maria – UFSMSanta MariaBrazil
  2. 2.University of Brasilia – UnBBrasíliaBrazil
  3. 3.Federal University of Santa Catarina – UFSCBlumenauBrazil
  4. 4.University of Campinas – UNICAMPCampinasBrazil

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