Tagged Questions
3
votes
0answers
75 views
Root Locus Diagrams - “Breakaway Point”
Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...
1
vote
3answers
69 views
conditions under which real-matrix exponential are equivalent
Consider $M_{1}$, $M_{2}\in\mathbb{R}^{2\times2}$, $k\in\mathbb{R}$, $M_{1}\neq M_{2}$. Under what conditions is $e^{M_{1}}=e^{kM_{2}}$?
Thanks!
0
votes
0answers
58 views
Long term behavior & marginal stability of a discrete non-homogeneous dynamical system?
A simple self-made example about the long term behavior of a discrete non-homogeneous dynamical system in the form of $x(k+1) = Ax(k) + b$:
$$
A = \begin{bmatrix}\frac 12&3\\0&1\end{bmatrix}, ...
1
vote
1answer
98 views
Show that $\ker W$ is $A$-invariant?
Consider the system.:
$$\dot x = Ax$$
$$y= Cx $$
The observability matrix then will be
$$ W= \left(\begin{matrix}
C \\
CA \\
CA^2\\
\vdots \\
CA^{n-1}
...
2
votes
3answers
133 views
How to solve simple systems of differential equations
Say we are given a system of differential equations
$$
\left[ \begin{array}{c} x' \\ y' \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix}
$$
Where $A$ is a $2\times 2$ matrix.
...
1
vote
1answer
77 views
Kalman filter and data extrapolation
Context of the situation:
I have a system set up that can give me the position of a person in a room. I also have a light that shines on this position. However, the light are lagging behind by 0.300 ...
0
votes
1answer
62 views
How to model a system for tracking a person using kalman filter?
I need to model a system for human motion. The following link shows for to build a system for a plane.
I am currently reading the documentation for a kalman filter library ...
0
votes
0answers
54 views
Studying a nonlinear system under constraints in linear fashion
Suppose that $a = aBc$ where $a$ and $c$ are vectors and $B$ is some matrix that changes as time "continuously" goes on - making this system dynamical system. But suppose that at any time, if $B$ is ...
2
votes
3answers
358 views
Poles and Zeros of Linear Systems
This period I follow a course in System and Control Theory. This is all about linear systems
$$\frac{dx}{dt}= Ax + Bu $$ $$y = Cx + Du $$ where A,B,C,D are matrices, and x, u and y are vectors.
...
4
votes
1answer
262 views
Why are hyperbolic toral automorphisms (e.g. Arnold's cat map) ergodic?
Let $\varphi : \mathbb{T}^2 \to \mathbb{T}^2$ be a hyperbolic automorphism of the torus, induced by a linear map $A : \mathbb{R}^2 \to \mathbb{R}^2$ of determinant $\pm 1$ with no eigenvalues of ...
0
votes
1answer
65 views
Meaning of index in matrices
Question is, what does "index" mean?
For systems of order greater than the number of characteristic roots of $C$ of index one
Also, can anyone explain why is $u_1 + u_2 + n -1 =0$ and what ...
1
vote
2answers
66 views
The system of $x(t+1) = Ax$ growing and retaining stability possible?
This is about general equilibrium:
Suppose that $x(t)$ represents outputs of all sectors and parts of the
whole economy - represented as matrix. How outputs evolve to $x(t+1)$
...
2
votes
1answer
82 views
Stability of a system that has (Jacobian-like) matrix with eigenvalue of less than 1 that has $x$ as non-eigenvector
This is about general equilibrium:
Suppose that $x(t)$ represents outputs of all sectors and parts of the
whole economy - represented as matrix. How outputs evolve to $x(t+1)$
is ...
6
votes
2answers
135 views
Prove that Anosov Automorphisms are chaotic
Let me add some detail first. An Anosov automorphism on $R^2$ is a mapping from the unit square $S$ onto $S$ of the form
$\begin{bmatrix}x \\y\end{bmatrix}\rightarrow\begin{bmatrix}a && b \\c ...
2
votes
1answer
64 views
Dynamical changing of an eigenvector
Consider a matrix $A\in\mathbb{R}^{n\times n}$. One of the eigenvalues of $A$ is zero and all the others are positive. Suppose $w\in\mathbb{R}^n$ is an eigenvector with the zero eigenvalue, i.e,
...
2
votes
0answers
56 views
A question about conjugacy between two systems
We've been doing some examples of showing when two specific systems are conjugate in class. I came across this more general question and I'm having trouble finding such a conjugacy.
Suppose there ...
0
votes
1answer
182 views
Solving Linear Systems with Singular Matrices
Good morning! For (say, homogenous) linear systems of the form
$$x_{n+1} = A x_n,$$
where $A$ is a nonsingular matrix, each initial value problem can be solved by the method of finding a general ...
2
votes
1answer
205 views
State-space to transfer function
I’m looking into MATLAB’s state-space functionality, and I found a peculiar relation that I don’t believe I’ve seen before, and I’m curious how one might obtain it. According to this documentation ...
3
votes
2answers
849 views
Linear Algebra: Finding a steady state matrix
Here is the problem:
And here is what I tried to do:
I tried doing it in my calculator and got that the answer is 40 and 160 as you keep multiplying PX. Can anyone point out what I'm doing wrong ...
2
votes
1answer
54 views
solving coupled discrete systems
Suppose you have a discrete system, whose evolution is governed by the following equations:
$\mathbf{x}[k+1] = f_1(\mathbf{F}[k], \mathbf{x}[k])$
$\mathbf{F}[k+1] = f_2(\mathbf{F}[k], ...
0
votes
1answer
143 views
convergence rate of matrix product
Suppose you have a linear system like this:
$$\mathbf{x}[k+1] = \mathbf{D} \mathbf{x}[k]$$
where matrix $\mathbf{D}$ is diagonal. Assume its diagonal entries are real, greater than zero and less than ...
0
votes
1answer
130 views
does the following dynamic system converge to a steady state?
This is an economics problem, but I'm pretty sure this kind of thing comes up elsewhere. I've used dynamic programming to find the optimal path of a system (law of motion), which is:
...
2
votes
1answer
695 views
Zero State, Stable Equilibrium, Dynamic System
Could someone please help?
The question reads:
For which real numbers $k$ is the zero state a stable equilibrium of the dynamic system
$x_{t+1} = Ax_t$?
$A = \begin{bmatrix} 0.1 &k \\ 0.3 & ...
