I am trying to understand a detail from an example, from the control textbook Slotine and Li (1991) "Applied Nonlinear Control", Prentice-Hall, Example 6.4, pag. 220. A linear system is given:

$$\eqalign{ \left[ {\matrix{ {{{\dot x}_1}} \cr {{{\dot x}_2}} \cr } } \right] =& \left[ {\matrix{ {{x_2} + u} \cr u \cr } } \right] \cr\\ y =& {x_1} \cr} $$

where the output \$y\$ is desired to track \$y_d\$. Differentiating the output, an explicit relation between \$y\$ -output- and \$u\$ -control input- is obtained:

$$\dot y = {x_2} + u$$

Until here it is clear. Now, the authors choose a control law:

$$u = - {x_2} + {{\dot y}_d} - (y - {y_d})$$

and say that this yield the tracking error equation:

$$\dot e - e = 0$$

with the \$e\$ being the tracking error, defined as \$e=y-y_d\$ and the internal dynamics: $${{\dot x}_2} + x = {y_d} - e$$ ... and the problem continues.

My question is: how does one define the control law \$u = - {x_2} + {{\dot y}_d} - (y - {y_d})\$ ?? And how does this relate to the two following equations (\$\dot e - e = 0\$ and \${{\dot x}_2} + x = {y_d} - e\$)?

Any clarifying answer is much appreciated.



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