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. 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 and in quantum computing.

Consider the following autonomous nonlinear system:

${\displaystyle {\dot {x}}=f(x)+\sum _{j=1}^{m}g_{j}(x)d_{j}(t)}$

where ${\displaystyle x\in R^{n}}$denotes the system state vector. Also, ${\displaystyle f}$and ${\displaystyle g_{i}}$'s are known analytic vector functions, and ${\displaystyle d_{j}}$is the ${\displaystyle j^{th}}$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

${\displaystyle f(x)\simeq f(x_{0})+\sum _{k=1}^{\eta }{\frac {1}{k!}}\partial f_{[k]}\mid _{x=x_{0}}(x-x_{0})^{[k]}}$

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

Without loss of generality, we assume that ${\displaystyle x_{0}}$is at the origin.

Applying Taylor approximation to the system, we obtain

${\displaystyle {\dot {x}}\simeq \sum _{k=0}^{\eta }A_{k}x^{[k]}+\sum _{j=1}^{m}\sum _{k=0}^{\eta }B_{jk}x^{[k]}d_{j}}$

where ${\displaystyle A_{k}={\frac {1}{k!}}\partial f_{[k]}\mid _{x=0}}$and ${\displaystyle B_{jk}={\frac {1}{k!}}\partial g_{j[k]}\mid _{x=0}}$.

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

${\displaystyle {\frac {d(x^{[i]})}{dt}}\simeq \sum _{k=0}^{\eta -i+1}A_{i,k}x^{[k+i-1]}+\sum _{j=1}^{m}\sum _{k=0}^{\eta -i+1}B_{j,i,k}x^{[k+i-1]}d_{j}}$

where ${\displaystyle A_{i,k}=\sum _{l=0}^{i-1}I_{n}^{[l]}\otimes A_{k}\otimes I_{n}^{[i-1-l]}}$, and similarly ${\displaystyle B_{j,i,\kappa }=\sum _{l=0}^{i-1}I_{n}^{[l]}\otimes B_{j,\kappa }\otimes I_{n}^{[i-1-l]}}$.

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

${\displaystyle {\dot {x}}_{\otimes }\simeq Ax_{\otimes }+\sum _{j=1}^{m}[B_{j}x_{\otimes }d_{j}+B_{j0}d_{j}]+A_{r}}$

where ${\displaystyle x_{\otimes }={\begin{bmatrix}x^{T}&x^{{[2]}^{T}}&...&x^{{[\eta ]}^{T}}\end{bmatrix}}^{T}}$, and ${\displaystyle A,B_{j},A_{r}}$and ${\displaystyle B_{j,0}}$matrices are defined in (Hashemian and Armaou 2015).

• Carleman matrix
• Composition operator

• A lecture about Carleman linearization by Igor Mezić