Linearization

How do we approximate a nonlinear system by a linear one?

Assume that a time-invariant system is described by the state space model:

$$\begin{array}{rl}\dot{\mathbf{x}}(t)& =\mathbf{f}(\mathbf{x}(t),u(t))\\ y(t)& =g[\mathbf{x}(t),u(t)]\end{array}$$

where $\mathbf{f}$ and $g$ are continuously differentiable functions.

The operating point of the system is a triple of constant vectors $(u_0,{\mathbf{x}}_0,y_0)\in \mathbb{R}\times {\mathbb{R}}^n\times \mathbb{R}$ if

$$\begin{array}{rl}0& =\mathbf{f}({\mathbf{x}}_0,u_0)\\ y_0& =g({\mathbf{x}}_0,u_0)\end{array}$$

Physical meaning: If the system has initial condition ${\mathbf{x}}_0$ and a constant input $u_0$ is applied, then the state and output will stay at constant values ${\mathbf{x}}_0$ and $y_0$, respectively, for all time, i.e.,

$$\{\begin{array}{l}u(t)=u_0\\ \mathbf{x}(0)={\mathbf{x}}_0\end{array}⟹\{\begin{array}{l}\mathbf{x}(t)={\mathbf{x}}_0\\ y(t)=y_0\end{array}$$

Since $\mathbf{f}$ and $g$ are differentiable (sufficiently smooth), we can say that

$$\{\begin{array}{l}u(t)-u_0\\ \mathbf{x}(0)-{\mathbf{x}}_0\end{array}\text{are small}⟹\{\begin{array}{l}\mathbf{x}(\mathbf{t})-{\mathbf{x}}_0\\ y(t)-y_0\end{array}\text{are small}$$

Denote

$$\begin{array}{rl}\tilde{u}(t)& =u(t)-u_0\\ \tilde{\mathbf{x}}(t)& =\mathbf{x}(t)-{\mathbf{x}}_0\\ \tilde{y}(t)& =y(t)-y_0\end{array}$$

Replace $\mathbf{f}$ and $g$ by their differentials:

where we have:

Since $\tilde{u}(t),\tilde{\mathbf{x}}(t),\tilde{y}(t)$ are small, we can neglect the higher-order terms and approximate the original system by the following linear system:

$$\begin{array}{rl}\dot{\tilde{\mathbf{x}}}(t)& =\mathbf{A}\tilde{\mathbf{x}}(t)+\mathbf{b}\tilde{u}(t)\\ \tilde{y}(t)& =\mathbf{c}\tilde{\mathbf{x}}(t)+d\tilde{u}(t)\end{array}$$

where

$$\begin{array}{rl}\mathbf{A}& =\frac{\partial f}{\partial x}{|}_{x=x_0,u=u_0}\\ \mathbf{b}& =\frac{\partial f}{\partial u}{|}_{x=x_0,u=u_0}\\ \mathbf{c}& =\frac{\partial g}{\partial x}{|}_{x-x_0,u=u_0}\\ d& =\frac{\partial g}{\partial u}{|}_{x-x_0,u=u_0}\end{array}$$