Lyapunov Stability for a Nonlinear, Time-varying system
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I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems.
Say we are given a nonlinear system:
$$dot{x_1}(t)=-x_1(t) + x_2(t)[x_1(t)+g(t)]$$
$$dot{x_2}(t)= x_1(t)[x_1(t)+g(t)]$$
And we want to investigate the stability of the solution $x(t)=0$.
If we use a simple Lyapunov function
$$V(x) = 0.5x_{1}^{2} + 0.5x_{2}^{2}$$
I can find $dot{V}(x,t)$, but I am unsure of where to go from here. How do I prove some kind of stability/instability. Do I need Barbalat's Lemma?
control-theory nonlinear-system stability-theory non-linear-dynamics lyapunov-functions
asked Dec 8 '18 at 5:00
Chemical EngineerChemical Engineer
597
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add a comment |
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I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems.
Say we are given a nonlinear system:
$$dot{x_1}(t)=-x_1(t) + x_2(t)[x_1(t)+g(t)]$$
$$dot{x_2}(t)= x_1(t)[x_1(t)+g(t)]$$
And we want to investigate the stability of the solution $x(t)=0$.
If we use a simple Lyapunov function
$$V(x) = 0.5x_{1}^{2} + 0.5x_{2}^{2}$$
I can find $dot{V}(x,t)$, but I am unsure of where to go from here. How do I prove some kind of stability/instability. Do I need Barbalat's Lemma?
control-theory nonlinear-system stability-theory non-linear-dynamics lyapunov-functions
asked Dec 8 '18 at 5:00
Chemical EngineerChemical Engineer
597
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You could look at the case when $g(t)=0$ and if that is stable use vanishing perturbation.
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– Kwin van der Veen
Dec 10 '18 at 9:43
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I think you might have mistaken a $-$ sign with $+$
$endgroup$
– polfosol
Feb 4 at 21:27
add a comment |
$begingroup$
I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems.
Say we are given a nonlinear system:
$$dot{x_1}(t)=-x_1(t) + x_2(t)[x_1(t)+g(t)]$$
$$dot{x_2}(t)= x_1(t)[x_1(t)+g(t)]$$
And we want to investigate the stability of the solution $x(t)=0$.
If we use a simple Lyapunov function
$$V(x) = 0.5x_{1}^{2} + 0.5x_{2}^{2}$$
I can find $dot{V}(x,t)$, but I am unsure of where to go from here. How do I prove some kind of stability/instability. Do I need Barbalat's Lemma?
control-theory nonlinear-system stability-theory non-linear-dynamics lyapunov-functions
asked Dec 8 '18 at 5:00
Chemical EngineerChemical Engineer
597
$endgroup$
I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems.
Say we are given a nonlinear system:
$$dot{x_1}(t)=-x_1(t) + x_2(t)[x_1(t)+g(t)]$$
$$dot{x_2}(t)= x_1(t)[x_1(t)+g(t)]$$
And we want to investigate the stability of the solution $x(t)=0$.
If we use a simple Lyapunov function
$$V(x) = 0.5x_{1}^{2} + 0.5x_{2}^{2}$$
I can find $dot{V}(x,t)$, but I am unsure of where to go from here. How do I prove some kind of stability/instability. Do I need Barbalat's Lemma?
control-theory nonlinear-system stability-theory non-linear-dynamics lyapunov-functions
control-theory nonlinear-system stability-theory non-linear-dynamics lyapunov-functions
asked Dec 8 '18 at 5:00
Chemical EngineerChemical Engineer
597
asked Dec 8 '18 at 5:00
Chemical EngineerChemical Engineer
597
edited Dec 8 '18 at 7:19
Chemical Engineer
asked Dec 8 '18 at 5:00
Chemical EngineerChemical Engineer
597
asked Dec 8 '18 at 5:00
Chemical EngineerChemical Engineer
597
asked Dec 8 '18 at 5:00
Chemical EngineerChemical Engineer
597
597
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You could look at the case when $g(t)=0$ and if that is stable use vanishing perturbation.
$endgroup$
– Kwin van der Veen
Dec 10 '18 at 9:43
$begingroup$
I think you might have mistaken a $-$ sign with $+$
$endgroup$
– polfosol
Feb 4 at 21:27
add a comment |
$begingroup$
You could look at the case when $g(t)=0$ and if that is stable use vanishing perturbation.
$endgroup$
– Kwin van der Veen
Dec 10 '18 at 9:43
$begingroup$
I think you might have mistaken a $-$ sign with $+$
$endgroup$
– polfosol
Feb 4 at 21:27
$begingroup$
You could look at the case when $g(t)=0$ and if that is stable use vanishing perturbation.
$endgroup$
– Kwin van der Veen
Dec 10 '18 at 9:43
$begingroup$
You could look at the case when $g(t)=0$ and if that is stable use vanishing perturbation.
$endgroup$
– Kwin van der Veen
Dec 10 '18 at 9:43
$begingroup$
I think you might have mistaken a $-$ sign with $+$
$endgroup$
– polfosol
Feb 4 at 21:27
$begingroup$
I think you might have mistaken a $-$ sign with $+$
$endgroup$
– polfosol
Feb 4 at 21:27
add a comment |
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$begingroup$
You could look at the case when $g(t)=0$ and if that is stable use vanishing perturbation.
$endgroup$
– Kwin van der Veen
Dec 10 '18 at 9:43
$begingroup$
I think you might have mistaken a $-$ sign with $+$
$endgroup$
– polfosol
Feb 4 at 21:27