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.

Procedure
Consider the following autonomous nonlinear system:



\dot{x}=f(x)+\sum_{j=1}^m 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 _{k=1}^\eta \frac{1}{k!}\partial f_{[k]}\mid _{x=x_0}(x-x_0)^{[k]} $$

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



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

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



\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 $$A_{i,k}=\sum _{l=0}^{i-1}I^{[l]}_n \otimes A_k \otimes I^{[i-1-l]}_n$$, and similarly $$B_{j,i,\kappa}=\sum _{l=0}^{i-1}I^{[l]}_n \otimes B_{j,\kappa} \otimes I^{[i-1-l]}_n$$.

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



\dot x_{\otimes}\simeq Ax_{\otimes} +\sum_{j=1}^m [B_jx_{\otimes}d_j+B_{j0}d_j]+A_r $$

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