Carleman linearization explained

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

Consider the following autonomous nonlinear system:

• x = f(x) + \sum ^{m} _{j =1} g _{j(x)d} _{j(t)}

where

x\in R ^{n} denotes the system state vector. Also, f and g _{i} 's are known analytic vector functions, and d _{j} is the j ^{th} 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

f(x)\simeq f(x _{0)+ \sum} ^{η} ^{k =1} (1)/(k!) \partial f _{[k]} \mid ^{x = x _{0}} (x - x ^{[k]} _{0 )}

where

\partial f _{[k]} \mid ^{x = x _{0}} is the k ^{th} partial derivative of f(x) with respect to x at x = x _{0} and x ^{[k]} denotes the k ^{th} Kronecker product.

Without loss of generality, we assume that

x _{0} is at the origin.

Applying Taylor approximation to the system, we obtain

• x\simeq \sum ^{η} ^{k =0} A _{k} x ^{[k]} + \sum ^{m} _{j =1} \sum ^{η} ^{k =0} B _{jk} x ^{[k]} d _{j}

where

A _{k = (1)/(k!)} \partial f _{[k]} \mid _{x=0} and B _{jk} = (1)/(k!) \partial g _{j[k]} \mid _{x=0} .

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

(d(x ^{[i]} ))/(dt) \simeq \sum ^{η- i +1} ^{k =0} A _{i,k} x ^{[k+i-1]} + \sum ^{m} _{j =1} \sum ^{η- i +1} ^{k =0} B _{j,i,k} x ^{[k+i-1]} d _{j}

where

A _{i,k} = \sum ^{i -1} ^{l =0} I ^{[l]} _{n} ⊗ A _{k} ⊗ I ^{[i -1- l]} _{n} , and similarly B _{j,i,\kappa} = \sum ^{i -1} ^{l =0} I ^{[l]} _{n} ⊗ B _{j,\kappa} ⊗ I ^{[i -1- l]} _{n} .

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

• x _{⊗} \simeq Ax _{⊗} + \sum ^{m} _{j =1} [B _{jx} _{⊗} d _{j+B} _{j0} d _{j]+A} _{r}

where

x _{⊗} = \begin{bmatrix} x ^{T} &x ^{{[2]} ^{T}} & ... & x ^{{[η]} ^{T}} \end{bmatrix} ^{T} , and A, B _{j} , A _{r} and B _{j,0} matrices are defined in (Hashemian and Armaou 2015).^[6]

See also

• Carleman matrix
• Composition operator

References

1. Carleman . Torsten . 1932 . Application de la théorie des équations intégrales linéaires aux systèmes d'équations différentielles non linéaires . Acta Mathematica . en . 59 . 63–87 . 10.1007/BF02546499 . 120263424 . 0001-5962. free .
2. Book: Salazar-Caceres . Fabian . Tellez-Castro . Duvan . Mojica-Nava . Eduardo . 2017 IEEE 3rd Colombian Conference on Automatic Control (CCAC) . Consensus for multi-agent nonlinear systems: A Carleman approximation approach . 2017 . https://ieeexplore.ieee.org/document/8276388 . Cartagena . IEEE . 1–5 . 10.1109/CCAC.2017.8276388 . 978-1-5386-0398-7. 44019245 .
3. Book: Amini . Arash . Sun . Qiyu . Motee . Nader . 2020 American Control Conference (ACC) . Approximate Optimal Control Design for a Class of Nonlinear Systems by Lifting Hamilton-Jacobi-Bellman Equation . 2020 . https://ieeexplore.ieee.org/document/9147576 . Denver, CO, USA . IEEE . 2717–2722 . 10.23919/ACC45564.2020.9147576 . 978-1-5386-8266-1. 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 . en . 118 . 35 . e2026805118 . 2011.03185 . 10.1073/pnas.2026805118 . 0027-8424 . 8536387 . 34446548. 2021PNAS..11826805L . free .
5. Web site: Levy . Max G. . January 5, 2021 . New Quantum Algorithms Finally Crack Nonlinear Equations . December 31, 2022 . Quanta Magazine.
6. Book: Hashemian . N. . Armaou . A. . 2015 American Control Conference (ACC) . Fast Moving Horizon Estimation of nonlinear processes via Carleman linearization . 2015 . 3379–3385 . 10.1109/ACC.2015.7171854 . 978-1-4799-8684-2 . 13251259.

External links

• A lecture about Carleman linearization by Igor Mezić