Global linearization method

The idea behind the control design through global linearization is to find a suitable feedback that linearizes a non-linear system and thus simplifies control . Usually the output is fed back for this purpose, which is why the method is also known as linearization through output feedback.

Global linearization is mainly used in control engineering, which is why we will now consider such an example.

Global linearization in control engineering

In control technology, a controlled system is the physical variable to be controlled, such as temperature.

A non-linear controlled system can be expressed in the state space representation as follows:

${\displaystyle {\begin{aligned}{\dot {x}}_{1}=x_{2}&&{\dot {x}}_{2}=x_{3}&&...&&{\dot {x}}_{n}=f(x_{1},\cdots ,x_{n})+b(x_{1},\cdots ,x_{n})u,\end{aligned}}}$

which follows from the general state space representation for single-variable systems:

${\displaystyle {\begin{bmatrix}{\dot {x}}_{1}(t)\\{\dot {x}}_{2}(t)\\\vdots \\{\dot {x}}_{n}(t)\\\end{bmatrix}}=\underbrace {\begin{bmatrix}a_{11}&\dots &a_{1n}\\a_{21}&\dots &a_{2n}\\\vdots &\ddots &\vdots \\a_{n1}&\dots &a_{nn}\\\end{bmatrix}} _{\text{System-Matrix}}\ \cdot \underbrace {\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\\vdots \\x_{n}(t)\\\end{bmatrix}} _{\text{Zustd.-Vektor}}+\underbrace {\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\\\end{bmatrix}} _{\text{Eingangs-Vektor}}\cdot \underbrace {\begin{bmatrix}u(t)\\\end{bmatrix}} _{\text{Eing.-Variable}}.}$

This non-linear control system can be controlled by the feedback

${\displaystyle u={\frac {1}{b(x_{1},\cdots ,x_{n})}}(v-f(x_{1},\cdots ,x_{n}))}$

linearize.

Will the state feedback

${\displaystyle v=-k_{1}x_{1}-k_{2}x_{2}-\cdots -k_{n}x_{n}\;}$

selected as controller, is the linearized controlled system

${\displaystyle {\begin{aligned}{\dot {x}}_{1}=x_{2}&&{\dot {x}}_{2}=x_{3}&&...&&{\dot {x}}_{n}=-k_{1}x_{1}-k_{2}x_{2}-\cdots -k_{n}x_{n}.\end{aligned}}}$

The controlled system is asymptotically stable if all eigenvalues ​​of the system matrix have a negative real part .

Example: Van der Pol system

A van der Pol system (named after the Dutch physicist Balthasar van der Pol , who published it in 1927) is described by the following differential equation:

${\displaystyle {\ddot {x}}+\epsilon (x^{2}-1){\dot {x}}+x=u.}$

After rewriting in the canonical controllability normal form with , and one obtains ${\displaystyle x_{1}=x\;}$ ${\displaystyle {\dot {x}}_{1}={\dot {x}}=x_{2}}$ ${\displaystyle {\dot {x}}_{2}={\ddot {x}}}$

${\displaystyle {\begin{aligned}{\dot {x}}_{1}&=x_{2}\\{\dot {x}}_{2}&=\epsilon (1-x_{1}^{2})x_{2}-x_{1}+u=f(x_{1},x_{2})+b(x_{1},x_{2})u.\end{aligned}}}$

So is

${\displaystyle {\begin{aligned}f(x_{1},x_{2})&=\epsilon (1-x_{1}^{2})x_{2}-x_{1}\\b(x_{1},x_{2})&=1\end{aligned}}}$

and thus the repatriation

${\displaystyle u=-\epsilon (1-x_{1}^{2})x_{2}+x_{1}+v.}$

The linearized state space representation is thus

${\displaystyle {\begin{aligned}{\dot {x}}_{1}&=x_{2}\\{\dot {x}}_{2}&=-k_{1}x_{1}-k_{2}x_{2}.\end{aligned}}}$

The corresponding homogeneous, linear differential equation reads:

${\displaystyle {\ddot {x}}+k_{2}{\dot {x}}+k_{1}x=0.}$

Individual evidence

1. ↑ Lutz, Wendt: Pocket book of control engineering , chapter: Control by state feedback .
2. ↑ Van der Pol, B. and Van der Mark, J., “Frequency demultiplication”, Nature , 120 , 363-364, (1927).
3. ↑ Kaplan, D. and Glass, L., Understanding Nonlinear Dynamics , Springer, 240-244, (1995).