Carleman linearization

In mathematics, Carleman linearization (or Carleman embedding) is a technique to transform a finite-dimensional nonlinear dynamical system into an infinite-dimensional linear system. It was introduced by the Swedish mathematician Torsten Carleman in 1932.[1] Carleman linearization is related to composition operator and has been widely used in the study of dynamical systems. It also been used in many applied fields, such as in control theory[2][3] and in quantum computing.[4][5]

Procedure

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Consider the following autonomous nonlinear system:

 

where   denotes the system state vector. Also,   and  's are known analytic vector functions, and   is the   element of an unknown disturbance to the system.

At the desired nominal point, the nonlinear functions in the above system can be approximated by Taylor expansion

 

where   is the   partial derivative of   with respect to   at   and   denotes the   Kronecker product.

Without loss of generality, we assume that   is at the origin.

Applying Taylor approximation to the system, we obtain

 

where   and  .

Consequently, the following linear system for higher orders of the original states are obtained:

 

where  , and similarly  .

Employing Kronecker product operator, the approximated system is presented in the following form

 

where  , and   and   matrices are defined in (Hashemian and Armaou 2015).[6]

See also

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References

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  1. ^ Carleman, Torsten (1932). "Application de la théorie des équations intégrales linéaires aux systèmes d'équations différentielles non linéaires". Acta Mathematica. 59: 63–87. doi:10.1007/BF02546499. ISSN 0001-5962. S2CID 120263424.
  2. ^ Salazar-Caceres, Fabian; Tellez-Castro, Duvan; Mojica-Nava, Eduardo (2017). "Consensus for multi-agent nonlinear systems: A Carleman approximation approach". 2017 IEEE 3rd Colombian Conference on Automatic Control (CCAC). Cartagena: IEEE. pp. 1–5. doi:10.1109/CCAC.2017.8276388. ISBN 978-1-5386-0398-7. S2CID 44019245.
  3. ^ Amini, Arash; Sun, Qiyu; Motee, Nader (2020). "Approximate Optimal Control Design for a Class of Nonlinear Systems by Lifting Hamilton-Jacobi-Bellman Equation". 2020 American Control Conference (ACC). Denver, CO, USA: IEEE. pp. 2717–2722. doi:10.23919/ACC45564.2020.9147576. ISBN 978-1-5386-8266-1. S2CID 220889153.
  4. ^ Liu, Jin-Peng; Kolden, Herman Øie; Krovi, Hari K.; Loureiro, Nuno F.; Trivisa, Konstantina; Childs, Andrew M. (2021-08-31). "Efficient quantum algorithm for dissipative nonlinear differential equations". Proceedings of the National Academy of Sciences. 118 (35): e2026805118. arXiv:2011.03185. Bibcode:2021PNAS..11826805L. doi:10.1073/pnas.2026805118. ISSN 0027-8424. PMC 8536387. PMID 34446548.
  5. ^ Levy, Max G. (January 5, 2021). "New Quantum Algorithms Finally Crack Nonlinear Equations". Quanta Magazine. Retrieved December 31, 2022.
  6. ^ Hashemian, N.; Armaou, A. (2015). "Fast Moving Horizon Estimation of nonlinear processes via Carleman linearization". 2015 American Control Conference (ACC). pp. 3379–3385. doi:10.1109/ACC.2015.7171854. ISBN 978-1-4799-8684-2. S2CID 13251259.
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