A nonlinear system [ x 1 ˙ x 2 ˙ ] = [ x 2 3 + u u ] y = x 1 as the control objective to make y track y d , and the tracking error is e = y − y d .It is, y ˙ = x 1 ˙ and choosing the control law u = − x 2 3 − e + y d ˙ gives e ˙ + e = 0which is stable and converging to zero as the time t → ∞ (or in other words, the error equation has one pole in -1).The same control input u applies also to the second equation of the nonlinear system, representing the internal dynamics. With the choice of u as above, x 2 ˙ = u yields x 2 ˙ + x 2 3 = y d ˙ − e .With the choice of a bounded y d ˙ and e (bounded since e ˙ + e = 0), then it is | y d ˙ − e | ≤ D ,being D a positive constant.From now on my question starts, as I would like to know what are the math steps to derive the following conclusion:the example concludes that | x 2 | ≤ D 1 / 3 (i.e. x 2 is bounded too!), since x 2 ˙ &lt; 0 when x 2 &gt; D 1 / 3 , and x 2 ˙ &gt; 0 when x 2 &lt; D 1 / 3 . Can someone explain how to derive these inequalities?

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A nonlinear system      [                                                              x              1                        ˙                                                                                  x              2                        ˙                                ]    =      [                                                                x              2                        3                    +          u                                      u                      ]    y  =      x    1  as the control objective to make   y track       y    d  , and the tracking error is   e  =  y  −      y    d  .It is,       y    ˙    =            x      1        ˙   and choosing the control law  u  =  −              x      2        3    −  e  +            y      d        ˙  gives      e    ˙    +  e  =  0which is stable and converging to zero as the time   t  →  ∞ (or in other words, the error equation has one pole in -1).The same control input   u applies also to the second equation of the nonlinear system, representing the internal dynamics. With the choice of u as above,             x      2        ˙    =  u yields            x      2        ˙    +              x      2        3    =            y      d        ˙    −  e  .With the choice of a bounded             y      d        ˙   and e (bounded since       e    ˙    +  e  =  0), then it is  |            y      d        ˙    −  e  |  ≤  D  ,being   D a positive constant.From now on my question starts, as I would like to know what are the math steps to derive the following conclusion:the example concludes that   |      x    2    |  ≤      D          1          /      3      (i.e.       x    2   is bounded too!), since             x      2        ˙    &amp;lt;  0 when       x    2    &amp;gt;      D          1          /      3      , and             x      2        ˙    &amp;gt;  0 when       x    2    &amp;lt;      D          1          /      3      . Can someone explain how to derive these inequalities?

enfeinadag0 · Accepted Answer

Let us rewrite your equation as               x      ˙        2    =  −      x    2    3    +  d  (  t  ), where   |  d  (  t  )  |  ≤  D for all   t. If       x    2   is positive, then x˙2 is negative for all       x    2    &amp;gt;      D          1          /      3      . If       x    2   is negative, then               x      ˙        2   is positive for all       x    2    &amp;lt;  −      D          1          /      3      . It implies that the set   |      x    2    3    |  ≤  D is invariant and attractive: if the initial condition       x    2    (      t    0    ) belongs to the set, then the trajectory reamins in the set. If the initial condition       x    2    (      t    0    ) is outside the set, then it will converge to the set.Thus,       x    2    (  t  ) is bounded. However, the claim that   |      x    2    |  ≤      D          1          /      3       is valid only if you know for sure that the initial condition satisfies this inequality, or if you consider it as   t  →  ∞.

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