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Doctoral Thesis
DOI
10.11606/T.3.2010.tde-13082010-163547
Document
Author
Full name
Hector Bessa Silveira
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2010
Supervisor
Committee
Silva, Paulo Sérgio Pereira da (President)
Barros, Carlos José Braga
Corrêa Filho, Carlos
San Martin, Luiz Antonio Barrera
Tonelli, Pedro Aladar
Title in Portuguese
Formas triangulares para sistemas não-lineares com duas entradas e controle de sistemas sem arrasto em SU(n) com aplicações em mecânica quântica.
Keywords in Portuguese
Controle de sistemas quânticos
Controle não-linear
Estabilidade de Lyapunov
Formas triangulares
Grupo especial unitário
Platitude diferencial
Porta lógica quântica C-NOT
Sistemas diferenciais exteriores
Sistemas não-lineares
Teoria de controle geométrica
Abstract in Portuguese
A presente tese aborda dois problemas distintos e independentes: triangularização de sistemas não-lineares com duas entradas e controle de sistemas sem arrasto que evoluem no grupo especial unitário SU(n). Em relação ao primeiro, estabeleceu-se, através da generalização de resultados bem conhecidos, condições geométricas para que um sistema com duas entradas seja descrito por uma forma triangular específica após uma mudança de coordenadas e uma realimentação de estado estática regular. Para o segundo problema, desenvolveu-se uma estratégia de controle que força o estado do sistema a rastrear assintoticamente uma trajetória de referência periódica que passa por um estado objetivo arbitrário. O método de controle proposto utiliza os resultados de convergência de tipo- Lyapunov que foram estabelecidos pela presente pesquisa e que tiveram como inspiração uma versão periódica do princípio da invariância de LaSalle. Apresentou-se, ainda, os resultados de simulação obtidos com a aplicação da técnica de controle desenvolvida a um sistema quântico consistindo de duas partículas de spin-1/2, com o objetivo de gerar a porta lógica quântica C-NOT.
Title in English
Triangular forms for nonlinear systems with two inputs and control of driftless systems on SU(n) with applications in quantum mechanics.
Keywords in English
C-NOT quantum logic gate
Control of quantum systems
Differential flatness
Exterior differential systems
Geometric control theory
Lyapunov stability
Nonlinear control
Nonlinear systems
Special unitary group
Triangular forms
Abstract in English
This thesis treats two distinct and independent problems: triangularization of nonlinear systems with two inputs and control of driftless systems which evolve on the special unitary group SU(n). Concerning the first, one has established, by means of the generalization of well-known results, geometric conditions for a system with two inputs to be described by a specific triangular form after a change of coordinates and a regular static state feedback. For the second problem, one has developed a control strategy that forces the state of the system to track in an asymptotic manner a periodic reference trajectory which passes by an arbitrary goal state. The proposed control method uses Lyapunovlike convergence results that were established in this research and which were inspired in a periodic version of LaSalles invariance principle. Furthermore, one has shown the simulation results obtained from the application of the developed control technique to a quantum system consisting of two spin-1/2 particles, with the aim of generating the C-NOT quantum logic gate.
 
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Publishing Date
2010-09-20
 
WARNING: The material described below relates to works resulting from this thesis or dissertation. The contents of these works are the author's responsibility.
  • SILVEIRA, Hector Bessa, et al. A Computer Control Method for a General Class of Nonlinear Systems with Low Sampling Rates. In XIX Congresso Brasileiro de Automática (CBA), Campina Grande, 2012. CDROM., 2012. Available from: http://https://sites.google.com/a/dee.ufcg.edu.br/cba-2012/.
  • SILVEIRA, Hector Bessa, e SILVA, P. S. P. A Triangular Form for Flat Nonlinear Systems with Two Controls and Five or Less States. In 3rd IFAC Symposium on System, Structure and Control (SSSC), Foz do Iguaçu, 2007. CDROM IFAC SSSC07 Preprints., 2007.
  • SILVEIRA, Hector Bessa, PAGANO, D., e SILVA, P. S. P. Low-Rate Sampling Computer Control of Nonlinear Systems. In XVI Congresso Brasileiro de Automática (CBA), Salvador, 2007. CDROM XVI Congresso Brasileiro de Automática (CBA)., 2006.
  • SILVEIRA, Hector Bessa, Pereira da Silva, Paulo Sergio, and ROUCHON, Pierre. A Time-Periodic Lyapunov Approach for Motion Planning of Controllable Driftless Systems on SU(n). In IEEE Conference on Decision and Control CDC'09, Xangai, 2009. CDROM IEEE CDC'09., 2009.
  • SILVEIRA, Hector Bessa, Pereira da Silva, Paulo Sérgio, e ROUCHON, Pierre. Explicit Control Laws for the Periodic Motion Planning of Controllable Driftless Systems on SU(n). In 51st IEEE Conference on Decision and Control (CDC'2012), Maui, Hawai, 2012. CDROM 51st IEEE Conference on Decision and Control (CDC'2012)., 2012. Dispon?vel em: http://control.disp.uniroma2.it/cdc2012/.
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