# Tagged Questions

1answer
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### Designing linear systems to respond to particular kinds of oscillations

Say that I have a linear system which is being perturbed by an oscillating signal of a single frequency, of the form $$\dot{\vec{x}}(t) = A\vec{x} + B \sin(\alpha t),$$ where $B$ is a vector of 1s ...
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### Is an Invariant set Connected?

Let an autonomous dynamical system is characterized by the state equation $$\dot x(t) = f(x(t)),\quad x(0)=x_0$$ with state $x(t)\in \mathbb R^n$. The definition of invariant set, as I came across, ...
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### Backstepping analysis of multi-input system

Suppose I have a simple system that's like following: $\dot{x}_1 = A x_2 + Bx_3 \\ \dot{x}_2 = u_1 \\ \dot{x_3} = u_2$ I am familiar with a standard method of backstepping if there was only one ...
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### Diagonalize Complex ODE

I'm trying to solve for the dynamics of one coordinate of a coupled system of linear differential equations with complex coefficients. Physically, a number of single-pole harmonic oscillators with ...
2answers
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### Rigorous mathematical treatments of engineering topics

I started out as an engineering student and got interested in mathematics. So after some point (Rigorous analysis and linear algebra, some real analysis, basic measure theory and topology etc.) I ...
1answer
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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 ...
1answer
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### Can I always put a system in modal form?

Given the transfer function of a system, can I always put the system in modal form? Are there exceptions?
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1answer
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### Local stability + global attractivity = global asymptotic stability?

I was wondering how could I prove such a property stated in [Angeli, 2004]. For instance, consider the system $\dot{x}=f(x)$, where $f:\mathbb{R}^n\to\mathbb{R}^n$ is Lipschitz continuous. Claim. ...
1answer
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### Stability of $\dot{x}=-A(t)x$

I already have an ODE of $A(t)$, that is $\dot{A}=-G(A(t)-A^*)$, where $G$ and $A^*$ are constant positive definite matrices. Thus I can deduce that $A(t)$ exponentially converge to $A^*$. Now I take ...
1answer
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### Lyapunov stabilty, elementary question

Let’s say I have a system 1/(T1s+1) or any other n-th order polynomial and a PI controller (KP and TI). I already know that the system is stable but for, let’s say, educational purposes (not ...
1answer
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### Prove $0$ is an exponentially stable equilibrium of the system $x'=f(x)+g(x)$ if $f(0)=g(0)=0$

Besides the conditions in the title, we have: $0$ is an exponential equilibrium of the system $y'=f(y)$ $|g(x)|\leq \mu|x|,\forall x \in \mathbb{R}^n$ $\mu$ is sufficiently small! What I have ...
1answer
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### Show that the system $\Sigma(SAS^{-1},SB,CS^{-1},D)$ is observable/controllable iff $\Sigma(A,B,C,D)$ is observable/controllable

I am given the two linear systems: \begin{eqnarray} \Sigma_1: \dot{x}&=&Ax+Bu\\ y&=&Cx+Du \end{eqnarray} and \begin{eqnarray} \Sigma_2: \dot{x}&=&\bar{A}x+\bar{B}u\\ ...
0answers
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### $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that $\exists p\in\Bbb R ^m$ s.t. $(A,Bp)$ is controllable iff $(A,B)$ is controllable

Let $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that there exists a vector $p\in \Bbb R ^m$ such that $(A,Bp)$ is controllable iff $(A,B)$ is controllable. Here when I say $(A,B)$ ...
1answer
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### Prove that $x \equiv 0$ of $\dot{x}(t)=a(t)x$ is Uniformly Asymptotically Stable

I have a problem: Consider the scalar equation: $$\dot{x}(t)=a(t)x \tag{I}$$ where $a(t) \in C(\mathbb{R}^+)$. Prove that $x \equiv 0$ of $(I)$ is Uniformly Asymptotically Stable iff ...
2answers
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### Show that System $(I)$ is stable iff $X(t)$ is bounded.

I have a theorem: For a linear homogeneous system: $$\dfrac{dx}{dt}=A(t)x \tag{I}$$ Where $A(t)=(a_{ij}(t))_{n \times n} \in C(\mathbb{R}^+,\mathbb{R}^{n \times n})$ Suppose that $X(t)$ be the ...
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### generalizing superfunctions of entire functions

Let $z$ be complex and $x$ real. Define $f(z,0) = f(z)$ where $f(z)$ is an entire function. Define $f(z,x)$ as the $x$ th superfunction of $f(z)$. We know that $f(z,x-1) = f(f^{-1}(z,x)+1)$ where ...
3answers
145 views

### Weird condition for a transfer function

I am reading a paper that presents the following system, represented by a second-order transfer function: $G(s) = \frac{K\times(1+0.036s)}{(1+0.0018s)(1+as)}$, where the gain $K$ is a known ...
1answer
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### Parametric uncertainty in conditional term of piecewise nonlinear dynamical system

Consider a Hammerstein nonlinear dynamical system of the form $\mathbf{\dot{x}} = \mathbf{Ax} + \mathbf{Bu}$, where the non-linearity is in the control term $\mathbf{u}$, and has a piecewise ...
1answer
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### When are attracting sets invariant?

Consider a control system of the form $\dot{x}(t) = f(x(t), u(t))$ where $u(t)$ is the control input, $x \in \mathbb{R}^{n}$, $u \in \mathbb{R}^{m}$. Assume $f$ is Lipschitz continuous so that the ...
1answer
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### Estimating the input to a system from a system state

[ Cross-posted to: http://dsp.stackexchange.com/questions/3098/estimating-the-input-to-a-system-from-a-system-state-using-ekf ] I have a system for which I have obtained a non-linear time-varying ...
2answers
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### Feedback characteristics of nonlinear dynamical systems

I am trying to understand the following article: A. Lahellec, S. Hallegatte, J.-Y. Grandpeix, P. Dumas and S. Blanco, Europhys. Lett. 81, 60001 (2008). In the beginning, it was quite easy to follow: ...