Carleman linearization

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).^[6]

• Carleman matrix
• Composition operator