I am trying to understand an example from [1]. In detail I do not understand how the equation for the dynamic of the tracking error is chosen. I am not a mathematician so please forgive me if I may not use the most proper jargon. The example below, from [p. 216, 1], refers to a third order system. The nonlinear state equations are:

$\dot{x\_1}= sin (x\_{2})+ (x\_2 + 1) x\_3$

$\dot{x\_2}={x\_1}^5 + x\_3$

$\dot{x\_3}={x\_1}^2 + u$

$y=x\_1$

where $x\_1$, $x\_2$, $x\_3$ are the states, $u$ is the control input and $y$ the output. Now, the objective of the example is to find a direct equation linking the output to the control input $u$.

Differentiating $y$ yields

$\dot{y} = sin (x\_{2}) + (x\_2 + 1) x\_3 $, 

and differentiating the above gives:

$\ddot{y}=(x\_2 + 1)u + f(x)$

showing a direct relationship between the output $y$ and the control $u$, also 

$f(x)=({x\_{1}}^5 + x\_3) x3 + cos(x\_2) + (x\_2+1){x\_1}^2$. 

If the control input is chosen as 

$u=\frac{1}{x\_2 + 1}(v-f(x\_1))$

with $v$ a new 'auxiliary' control input, then the nonlinearity in the equation for $\ddot{y}$ is canceled, and the simple linear relation $\ddot{y}=v$ is obtained. Now, [p. 217, 1] says that designing a *tracking* controller for this double integrator is simple. Letting the *tracking error* be $e=y-y\_d$, where $y\_d$ is the reference, and choosing the new input as 

$v=\ddot{y\_d} - k\_{1} e - k\_{2} \dot{e}$

with $k\_1, k\_2 >0$, the *tracking error dynamic* is:

$\ddot{e}+ k\_{1} e + k\_{2} \dot{e} =0$

which represents a stable error dynamic. 

**My question is** 

1) how does one come up with the choice $v=\ddot{y} - k\_{1} e - k\_{2} \dot{e}$? Is this done in order to have $\ddot{e}+ k\_{1} e + k\_{2} \dot{e} =0$. 

2) Why is the choice $\ddot{e}+ k\_{1} e + k\_{2} \dot{e} =0$ made? Rather than for instance simply choosing $k\_{1} e + k\_{2} \dot{e} =0$?

Any (simple) explanation would help a lot!

Thank you

References:
[1] Slotine and Li, Applied Nonlinear Control, Prentice-Hall, New Jersey, 1991