## 2 Transforming the Original Dynamics to Quadratic or to Bilinear Format
### Bilinear Systems
#### 2.1.1 Carleman's Linearization
For example, the following formulas hold true:  
\[\mathbf{F}_{2,k}=\mathbf{F}_{k}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes \mathbf{F}_{k},\ \ \mathbf{F}_{3,k}=\mathbf{F}_{k}\otimes\mathbf{I}_{n}\otimes\mathbf{I}_{n}+ \mathbf{I}_{n}\otimes\mathbf{F}_{k}\otimes\mathbf{I}_{n}+\mathbf{I}_{n} \otimes\mathbf{I}_{n}\otimes\mathbf{F}_{k}. \tag{8}\]  
In a similar manner, define \(\mathbf{G}_{j,k}\) for \(j\geqslant 1,\ k\geqslant 0\). It follows that the time derivative of \(\mathbf{x}^{(j)}(t)\) can indeed be written in terms of the newly defined matrices as:  
\[\dot{\mathbf{x}}^{(j)}(t)=\sum_{k=1}^{N-j+1}\mathbf{F}_{j,k}\mathbf{x}^{(k+1)} (t)+\sum_{k=0}^{N-j}\mathbf{G}_{j,k}\mathbf{x}^{(k+1)}(t)\mathbf{u}(t),\ \ j\geqslant 1.\]  
Increase the dimensions of the original state vector from \(\mathbf{n}\) to \(\mathcal{N}=n+n^{2}+\ldots+n^{N}=\frac{n^{N}-n}{n-1}\), whereas \(\mathbf{n}\) is the dimension of the original state variable \(\mathbf{x}(t)\). In doing so, we also introduce a new state variable \(\mathbf{x}^{\otimes}(t)\) defined as:  
\[\mathbf{x}^{\otimes}(t)=\Big{[}\ \ \mathbf{x}(t)\ \ \ \mathbf{x}^{(2)}(t)\ \ \ \ldots\ \ \ \mathbf{x}^{(N)}(t)\ \ \Big{]}^{T}\in\mathbb{R}^{n^{(N)}}. \tag{9}\]  
By substituting the newly introduced notation in (9), we obtain a bilinear system with the following realization \[ \begin{cases}\dot{\mathbf{x}}^{\otimes}(t)=\mathbf{A}^{\otimes}\mathbf{x}^{ \otimes}(t)+\mathbf{N}^{\otimes}\mathbf{x}^{\otimes}(t)u(t)+\mathbf{B}^{ \otimes}\mathbf{u}(t),\\ \mathbf{y}=\mathbf{C}^{\otimes}\mathbf{x}^{\otimes}(t),\end{cases}  \tag{10}\]  
where \(\mathbf{x}^{\otimes}(0)=\mathbf{0}\) and the matrices \(\mathbf{A}^{\otimes},\mathbf{N}^{\otimes}\in\mathbb{R}^{N\times\mathcal{N}}, \mathbf{B}^{\otimes},\left(\mathbf{C}^{\otimes}\right)^{T}\in\mathbb{R}^{ \mathcal{N}}\) are as below:  
\[\mathbf{A}^{\otimes} = \left[ \begin{array}{cccc}\mathbf{F}_{1,1}&\mathbf{F}_{1,2}& \ldots&\mathbf{F}_{1,N}\\ \mathbf{0}&\mathbf{F}_{2,1}&\ldots&\mathbf{F}_{2,N-1}\\ \mathbf{0}&\mathbf{0}&\ddots&\mathbf{F}_{3,N-2}\\ \vdots&\vdots&\vdots&\vdots\\ \mathbf{0}&\mathbf{0}&\ldots&\mathbf{F}_{N,1}\end{array} \right],\mathbf{N}^ {\otimes}=\left[ \begin{array}{cccc}\mathbf{G}_{1,1}&\mathbf{G}_{1,2}& \ldots&\mathbf{G}_{1,N-1}&\mathbf{0}\\ \mathbf{G}_{2,0}&\mathbf{G}_{2,1}&\ldots&\mathbf{G}_{2,N-2}&\mathbf{0}\\ \mathbf{0}&\mathbf{G}_{3,0}&\ddots&\mathbf{G}_{3,N-3}&\mathbf{0}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ \mathbf{0}&\mathbf{0}&\ldots&\mathbf{G}_{N,0}&\mathbf{0}\end{array} \right],\] \[\mathbf{B}^{\otimes} = \left[ \begin{array}{cccc}\mathbf{G}_{10}&\mathbf{0}&\cdots& \mathbf{0}\end{array} \right],\mathbf{C}^{\otimes}=\left[ \begin{array}{cccc} \mathbf{C}&\mathbf{0}&\cdots&\mathbf{0}\end{array} \right].\]  
This procedure involves approximation techniques (based on Taylor series expansion and truncation) to yield a BTI system that approximates the original system with generic nonlinearities. The main challenge with this method is that the dimensions of the derived bilinear system could be considerably higher than those of the original system. Indeed, by increasing the truncation index \(N\) will result in a better approximation. However, the increase in dimension of the bilinear system's dimension \(\mathcal{N}\) might yield serious memory storage problems (the storage of very large matrices). For many practical applications, one would restrict to low values of \(N\), such as \(N=2\) or \(N=3\). **Example 1**. : _Consider a non-linear scalar system characterized by the equations:_  
\[\mathbf{\Sigma}_{\mathbf{N}0}: \begin{cases}\dot{x}(t)=-3x(t)+x(t)\cos\left(x(t )\right)+u(t),\\ y(t)=2x(t).\end{cases} \]  
_Next, we use the Taylor series around the origin for the function \(\cos(x(t))\)_  
\[x\cos\left(x(t)\right)=\sum_{k=1}^{\infty}c_{k}x^{k}(t)=x(t)-\frac{x^{3}(t)}{2!}+\frac{x^{5}(t)}{4! }-\ldots,\ \ c_{k}= \begin{cases}0,\ k\ even\\ \frac{(-1)^{(k-1)/2}}{(k-1)!},\ k\ odd\end{cases}