Linearity and nonlinearity of systems

Question

My teacher of Control Systems did some exercises at the seminar and I don't get it why he said that this system is not linear:

• $x_1'= x_1 + 2x_2 + 3x_2u_1$
• $x_2'= x_2 + 3u_2$;
• $y_1 = x_1$

Variables are $x_1$ and $x_2$ (both aren't squared)

The system has the following properties:

Not linear

Order: $2$ ($x_1'$ and $x_2'$ state variables)

Outputs: $1(y_1)$

Inputs: $2(u_1,u_2)$

Answer 1

There shouldn't be products of the terms either, like $x_2 u_1$. Consider this simple example:

$$x'=x u$$

Let $u=1$, then the solution is $x(t) = x(0) e^t$. Now let $u=2$, then the solution is $x(t) = x(0) e^{2t} \neq 2 x(0) e^t$ which violates the linearity conditions.