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Assume I have a dynamical system where I am only interested in the stability of some of the states, probably because the "unimportant" states are not stable.

An example could be a system where time is defined as a state:

$$ \begin{align} \dot{x}_1 &= f(x_1, x_2) \\ \dot{x}_2 &= 1 \,. \end{align} $$

Clearly, the state $x_2$ is unstable, as it represents time: $x_2\rightarrow \infty$ as $t \rightarrow \infty$. However, I don't care about this state, I am only interested in the stability of $x_1$.

Is there a (general) way how to deal with this situation? I am especially interested in the case when an unstable state does not represent time, but something else that can safely "blow up"?

EDIT:

To give a different example than time:

$$ \begin{align} \dot{z}_1 &= -2 z_1 + \tanh(z_2^2) - 1 \\ \dot{z}_2 &= z_2 \,. \end{align} $$

Clearly, $z_2$ will diverge. However, as $z_2 \rightarrow \pm \infty$, the first equation $\dot{z}_1 = -2 z_1$, which is stable. So, although $z_2$ blows up, $z_1$ is still stable. If I now "dont care" about $z_2$, I am done.

But what can I do if things are not as easy as in this simple example? Is there a way to formally prove stability of the part of the system I am interested in?

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  • $\begingroup$ $x_2$ is not a "state", it is a variable. $\endgroup$ Commented Feb 23, 2018 at 21:17
  • $\begingroup$ Why a variable? It is part of the state space representation of this system, so it should be a state... or how is it different to $x_1$, which qualifies as a state then? $\endgroup$ Commented Feb 23, 2018 at 21:44
  • $\begingroup$ $x_2$ is a state. (By the way a state, an input, an output are all variables, everything else is a parameter) $\endgroup$ Commented Feb 23, 2018 at 23:33
  • $\begingroup$ You may consider the concept of input to state stability and stability of interconnected systems. in the 2nd example you could consider $z_2$ as an input to the first system. $\endgroup$ Commented Feb 23, 2018 at 23:41
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    $\begingroup$ The $h$ was thought to generalize input functions as $tanh$, $sign$, $sat$,... Just an idea from myself how to may be approach this problem... if the input is saturated it doesn’t matter how large it its $\endgroup$ Commented Feb 25, 2018 at 10:43

1 Answer 1

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As the second equation results in

$$x_2(t)=t+c,$$

we can rewrite the first equation as

$$\dot{x}_1(t)=f(x_1,t+c).$$

Now you could try to study the behaviour of a scalar first order time-variant system.

Edit: An alternative method would be to use differential inequalities. Let us consider

$\dot{x}_1=f(x_1,x_2)$

in which the state $x_2$ is unstabel. Assume we can find bounds for $f(x_1,x_2)$ given by the following double inequality:

$g(x_1)\leq f(x_1,x_2) \leq h(x_1)$.

Then it is possible to bound the derivative of $x_1$ by

$g(x_1)\leq \dot{x}_1 \leq h(x_1)$.

Then use the theorem of Petrovitsch. By this procedure, you could at least bound the solution trajectories of $x_1$.

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  • $\begingroup$ Thanks, but what can I do if the "problematic" state isn't time? I have edited my original question and posted a different example. $\endgroup$ Commented Feb 23, 2018 at 22:13
  • $\begingroup$ Editing your question should not change the initial question. Next time you should be more carefull when you are formulating your questions. $\endgroup$ Commented Feb 23, 2018 at 22:58
  • $\begingroup$ Well it didn't really change the original question tbh, as I wrote that time is just one possible example. But the wording was maybe not precise enough, so sorry for any inconvenience... I didn't know Petrovitschs theoren. This looks definitly interesting, thanks. $\endgroup$ Commented Feb 24, 2018 at 13:19

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