All Questions
Tagged with nonlinear-dynamics lyapunov-functions
17
questions
4
votes
1
answer
171
views
Is a system, that is globally asymptotically stable for any constant input also input-to-state stable? [closed]
I am referring to the ISS definition by Sontag of
${\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).}$
I understand that 0-GAS is a necessary condition for ISS. But is GAS for all ...
- 51
1
vote
1
answer
109
views
Non vanishing gradient condition in control barrier funcions.
I am reading about barrier functions in control engineering/dynamical systems. These tools are used to prove that the system is forward invariant with respect to a set $\mathcal{C}$ (i.e., starting in ...
- 87
4
votes
1
answer
146
views
Exponential Stability and Lasalle's Invariance Theorem
It is well known that a system $\dot{x}=f(x)$ with $x \in \mathbb{R}^n$ is exponentially stable if there exists a Lyapunov function $V(x)$ which satisfies
\begin{align}
k_1\Vert x \Vert \leq V(x) &...
- 380
0
votes
0
answers
94
views
What should I prove to show the states lie within a compact set?
I'm trying to prove the local stability of a nonlinear system and got the following inequality.
$
\|x(t)\|\leq c_1\|x(t_0)\|\exp(-c_2(t-t_0))+c_3\epsilon_m\cdots
$(i)
where $c_1, c_2, c_3$ are ...
- 165
0
votes
0
answers
47
views
Parameter Estimation of Dynamical System when Model is known
Im working on a nonlinear control based on Lyapunov theory and its working really well. I am able to implement it on a dynamical model of the system in simulink. However I think it has a really big ...
- 79
2
votes
1
answer
143
views
Any theorems for Input-output or input-state stability for non-asymptotically stable nonlinear systems?
Update for clarification:
Assume $\dot{x_1}=f(x_1 , x_2)+ au$
where $x_1$ is asymptotically stable for all bounded values of $x_2$. If $x_2$ is kept bounded, will input-output stability theorem apply ...
- 21
6
votes
1
answer
205
views
How does this expression follow algebraically from the last one?
I was reading this paper:
Global stability for an HIV/AIDS epidemic model with different latent stages and treatment
Everything is understood apart from on page 7 of the pdf (page 1486 in the document)...
user644376
1
vote
0
answers
31
views
Lyapunov dimension
I have a nonlinear differential equation system composed of 4 equations. I calculated Lyapunov's dimension of each of the states to be a little bit over 3 (say 3.11, 3.1, 3.13, 3.14). How can I ...
- 21
5
votes
1
answer
812
views
Does Asymptotic Stability Imply the Existence of a Lyapunov Function for a Nonlinear System?
For a linear time-invariant system $\dot x = Ax,$ the inverse Lyapunov theorem asserts that if the origin is asymptotically stable, then a Lyapunov function in the form $V(x) = x^\top P x$ for some ...
- 394
1
vote
0
answers
91
views
What is the difference between Lyapunov Exponent vs time and Lyapunov Exponent vs parameter?
What is the difference between Lyapunov Exponent vs time and Lyapunov Exponent vs parameter?
I noticed that some plot graph of Lyapunov Exponent but are not usually the same x-axis. Some plot graph ...
- 466
2
votes
0
answers
105
views
Convergence proof of Spring Mass Damper example in Sastry's text
I have a question regarding Shankar Sastry's text "Nonlinear Systems, Analysis, Stability and Control", Page 199, Spring-mass system with damper
The equations that describe this system is,
$\...
- 652
4
votes
3
answers
668
views
How to find a Lyapunov function
I have the following system: $$\dot{x_1} = x_2(x_3-2) \\ \dot{x_2} = x_1(x_3-2) \\ \dot{x_3}=-x_3^3 $$ and I want to determine its equilibrium points together with their stability. To find the ...
- 2,807
3
votes
1
answer
223
views
Problem with lyapunov function
I have a question about a differential equation I tried to analyse:
$$
\begin{align}
\dfrac {dx}{dt} &= v \\
\dfrac {dv}{dt} &= -x+x^3-v^3 \\
\end{align}
$$
I plotted this differential ...
- 133
1
vote
1
answer
103
views
Finding Liapunov Function
System
$$\dot{x}=-x+2y^3-2y^4$$
$$\dot{y}=-x-y+xy$$
The Liapunov function of this system could be something like $V=x^m+ay^n$. I am trying to figure out the appropriate values for $m$, $n$, and $a$. ...
- 833
0
votes
2
answers
83
views
How to show the state equation that $x_1=x_2$ can only happen at the origin
Take a look at this system:
$$
\begin{align}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= -\frac{x^2_1}{x_2} - x_2 + x_1
\end{align}
$$
Take a Lyapunov function as
$$
V(x_1,x_2) = x^2_1 + x^2_2
$$
Its ...
- 1,246