Minimum controller substructure for generic arbitrary pole placement

multicommodity flow and TSP based formulations

Atreya Kotoky, Ashutosh Mahajan, Ashwin Arulselvan, Madhu N. Belur, Rachel K. Kalaimani

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

Abstract

This paper deals with finding a 'least interaction' controller that generically achieves pole placement, given a actuator-sensor interaction possibility for the controller. The structure of the plant and the controller are modeled as an undirected and bipartite graph. Assuming that the plant/controller structures are specified, we find a minimum controller substructure within the specified controller structure such that the controller substructure allows generic eigenvalue assignability. The minimum is in the sense that the bipartite graph consisting of the proposed controller has the minimum number of edges. The complexity of a brute-force algorithm to identify a minimum controller substructure would be exponential and hence we propose two formulations for solving this problem, using recent results about equivalence of generic pole-assignability and covering of plant edges using cycles. The first uses multi-commodity flow networks to include all plant edges in some cycle using the least number of controller edges. We show that an integer solution to this formulation gives a minimum controller substructure for arbitrary pole placement problem. Since the problem of finding a feasible integer flow in multicommodity networks is NP-complete, there is ample reason that identifying a minimum controller substructure is NP-hard. The second formulation uses the framework of travelling salesman with profits (TSP with profits) to cover all vertices of the bipartite graph by cycles using the least number of controller edges. The TSP-with-profits problem too belongs to the class of NP-hard problems. We show that our formulation is equivalent to the so-called Generalized Travelling Salesman Problem (GTSP) thus allowing branch-cut algorithms developed for GTSP problems.
Original languageEnglish
Title of host publication2016 European Control Conference (ECC)
Place of PublicationPiscataway, NJ
PublisherIEEE
Pages849-854
Number of pages6
ISBN (Electronic)9781509025916
DOIs
Publication statusPublished - 9 Jan 2017
Event2016 European Control Conference - Aalborg, Denmark
Duration: 29 Jun 20161 Jul 2017

Conference

Conference2016 European Control Conference
Abbreviated titleECC16
CountryDenmark
CityAalborg
Period29/06/161/07/17

Fingerprint

Poles
Controllers
Profitability
Traveling salesman problem
Computational complexity
Actuators
Sensors

Keywords

  • bipartite graph
  • mathematical model
  • electronic mail
  • differential equations
  • NP-hard problem
  • sensors

Cite this

Kotoky, A., Mahajan, A., Arulselvan, A., Belur, M. N., & Kalaimani, R. K. (2017). Minimum controller substructure for generic arbitrary pole placement: multicommodity flow and TSP based formulations. In 2016 European Control Conference (ECC) (pp. 849-854). Piscataway, NJ: IEEE. https://doi.org/10.1109/ECC.2016.7810395
Kotoky, Atreya ; Mahajan, Ashutosh ; Arulselvan, Ashwin ; Belur, Madhu N. ; Kalaimani, Rachel K. / Minimum controller substructure for generic arbitrary pole placement : multicommodity flow and TSP based formulations. 2016 European Control Conference (ECC). Piscataway, NJ : IEEE, 2017. pp. 849-854
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Kotoky, A, Mahajan, A, Arulselvan, A, Belur, MN & Kalaimani, RK 2017, Minimum controller substructure for generic arbitrary pole placement: multicommodity flow and TSP based formulations. in 2016 European Control Conference (ECC). IEEE, Piscataway, NJ, pp. 849-854, 2016 European Control Conference, Aalborg, Denmark, 29/06/16. https://doi.org/10.1109/ECC.2016.7810395

Minimum controller substructure for generic arbitrary pole placement : multicommodity flow and TSP based formulations. / Kotoky, Atreya; Mahajan, Ashutosh; Arulselvan, Ashwin; Belur, Madhu N.; Kalaimani, Rachel K.

2016 European Control Conference (ECC). Piscataway, NJ : IEEE, 2017. p. 849-854.

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

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AB - This paper deals with finding a 'least interaction' controller that generically achieves pole placement, given a actuator-sensor interaction possibility for the controller. The structure of the plant and the controller are modeled as an undirected and bipartite graph. Assuming that the plant/controller structures are specified, we find a minimum controller substructure within the specified controller structure such that the controller substructure allows generic eigenvalue assignability. The minimum is in the sense that the bipartite graph consisting of the proposed controller has the minimum number of edges. The complexity of a brute-force algorithm to identify a minimum controller substructure would be exponential and hence we propose two formulations for solving this problem, using recent results about equivalence of generic pole-assignability and covering of plant edges using cycles. The first uses multi-commodity flow networks to include all plant edges in some cycle using the least number of controller edges. We show that an integer solution to this formulation gives a minimum controller substructure for arbitrary pole placement problem. Since the problem of finding a feasible integer flow in multicommodity networks is NP-complete, there is ample reason that identifying a minimum controller substructure is NP-hard. The second formulation uses the framework of travelling salesman with profits (TSP with profits) to cover all vertices of the bipartite graph by cycles using the least number of controller edges. The TSP-with-profits problem too belongs to the class of NP-hard problems. We show that our formulation is equivalent to the so-called Generalized Travelling Salesman Problem (GTSP) thus allowing branch-cut algorithms developed for GTSP problems.

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Kotoky A, Mahajan A, Arulselvan A, Belur MN, Kalaimani RK. Minimum controller substructure for generic arbitrary pole placement: multicommodity flow and TSP based formulations. In 2016 European Control Conference (ECC). Piscataway, NJ: IEEE. 2017. p. 849-854 https://doi.org/10.1109/ECC.2016.7810395