# Tagged Questions

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### On the first Lyapunov method, when the linearization fails

I have been trying to apply the first Lyapunov method to decide about the stability of the origin for the following system \begin{equation*} \dot{x}=\sqrt[3]{-x}. \end{equation*} However, the ...
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### control engineering transfer function vibration

What does vibrational mode even mean? How do you tell it from the poles?
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### Prove that $(A,B)$ is uncontrollable $\Longleftrightarrow$ $\exists P$ $\in$ $\mathbb{R}^{nxn}$, $P \neq 0$: $PA - AP = 0$, $PB=0$

In my course advanced system Theory I had the following question: Prove the following equivalence for the pair $(A,B)$ $\in$ $\mathbb{R}^{nxn}$ x $\mathbb{R}^{nxm}$: $(A,B)$ is uncontrollable ...
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### Prove that A is marginally stable iff there exist a $P$ $\in$ $S^n$, $P \succ 0$ such that $A^T + PA \leq 0$

Asymptotic stability, which means that all eigenvalues of A are in the open left half plane is easily proven. See the scan in the attachment. However, in the book the proof for the second case where ...
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I got this system state representation: \begin{align} \frac{dx}{dt} &= ... 1answer 126 views ### Is this the correct way to get the state space representation of this system? In this exercise the state space representation of the imaged system is asked for.G_1(s) = \frac{s-1}{s+2} = 1 - \frac{3}{s+2} G_2(s)=\frac{1}{s-1}$$I can see that G_1(s) is "able to leap" ... 2answers 273 views ### State transform from one state space representation to another I have a state space representation, system S1, in the form of:$$ \frac{dx}{dt} = Ax + bu y = c^Tx$$This system is transformed with the state transform$$x=T z$$into the system S2:$$ ...
I have this 2nd order transfer function: $$G(s) = \frac{2}{s} + \frac{1}{s+2}$$ And I need to find a possible state space representation in the form of: $$\frac{dx}{dt} = Ax + bu$$ $$y = c^Tx$$ ...