Questions tagged [linearization]
On the many different ways to turn non-linear systems of equations into linear ones.
1
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0answers
15 views
Linear approximation of $3$ variable function and its maximum error?
I've got a problem which asks me to find linear approximation of multi-variable function and its maximum error.
Here's the problem :
By about how much will
$$g(x,y,z)=x+x\cos(z)-y\sin(z)+y$$
...
0
votes
1answer
21 views
Linearization of a function: can someone explain me this last step?
I'm studying a linearization of a differential equation. $x(t)$ and $r(t)$ are really small signals and G, K, B and M are constants. I understand everything until I reach
$$ \frac{d^2x(t)}{dt^2}=G+\...
-2
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1answer
19 views
linearization of constraints
I am dealing with an optimization model where my binary variables xi have to follow this type of constraint (in the attached link):
...
0
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1answer
13 views
Linearization of Autonomous Equations
Consider $y'=y(y-50)(y-100)$
(a) Let $y_e$ be the stable equilibrium solution and let $u=y-y_e$, rewrite the equation as a differential equation for $u$.
$y_e=50, u=y-50, u'=(u-50)(u)(u+50)$
(b) ...
0
votes
0answers
8 views
linearizing dynamics about non fixed point for LQR implementation.
I am trying to implement LQR control for the cart pole system. I am curious if I can maintain a constant non-zero pole angle. So, I need to linearize my dynamics about my goal state. I know we can use ...
0
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0answers
19 views
Linearization between two ranges
We are currently trying to solve a problem where the input is range of variables x and y.
y $\in$ {min, max}.
x $\in$ {0, 255}
min & max are integers of range $\{-2^{32}, 2^{32} \}$
Need to ...
1
vote
0answers
32 views
Determining the original, unapproximated form of a rotation vector
a bit of an odd one here:
I'm reading a paper that supposedly rotates a basis in a symmetric way, using two scalar rotation quantities, $\theta_1$ and $\theta_2$. The idea, I believe, is for each ...
0
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0answers
31 views
Carleman matrix of multivariable functions - Carleman tensor?
Recently I learned about a matrix called
Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying.
Carleman linearization is a technique used to embed a finite
...
0
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1answer
67 views
How to solve binary nonlinear programming problems?
I have written binary nonlinear programming problem:
Now I want to solve this problem. My decision variables are $x_{i,j}, y_{i,j}$ and $z_{i,j}$. The other terms are constants. N=30 and K=4.
I ...
3
votes
1answer
59 views
How do I linearise the rational function to analyze the critical points?
For the system $$\frac{dx}{dt}=\frac{3xy}{1+x^2+y^2}-\frac{1+x^2}{1+y^2}\\\frac{dy}{dt}=x^2-y^2,$$ the point $\begin{pmatrix}1\\1\end{pmatrix}$ is
A. an unstable node
B. a stable node
C....
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votes
1answer
41 views
Linearizing two variable function
I have another linearization question similar to the one in here. This time, I have got two variables in my equation and I am in search of an "$A+B\rho$" or possibly "$A+B\rho+C\theta$" approximation. ...
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0answers
37 views
how I can linearize or simplify these complicated and non-linear constraints?
I have some constraints which are in the form $$ \dfrac{x_{1}-x_{2}+x_{3}+\cdots+x_{n}}{(x_{i}-x_{j})^{2}+\cdots+(x_{l}-x_{k})^{2}}+\cdots+\dfrac{x_{1}+x_{2}-x_{3}+\cdots+x_{n}}{(x_{j}-x_{i})^{2}+\...
1
vote
3answers
85 views
Linear Approximation of x/ (1-x)
I am trying to linearize the following function, but, having difficulties.
Let,
$x = \frac{l}{m},$
where $l,m \in R^+$ and $l<m$
Assume $l$ is a variable, while, $m$ is a constant (parameter), ...
1
vote
1answer
45 views
Linearized Pitot system
I am trying to create a linearized model of a compressible pitot tube system with altitude $h$ as the input and velocity as the output.
When I take the derivative and try and linearize around a point ...
3
votes
1answer
108 views
Linearization of differential system of equation
I would like to ask if I understand correctly the process of linearization for analyzing critical points.
I was given differential equation:
$\dot x = xy+1$
$\dot y = x+xy$
And my task was to ...
0
votes
0answers
49 views
Linearization of nonlinear Schrodinger equation
Picture below is from the 56th page of
Lyapunov stability of ground states of nonlinear dispersive evolution equations. The nonlinear Schrodinger equation (NLS) is
$$
i\phi_t(x,t)+\Delta \phi(x,t)+ ...
1
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0answers
68 views
About linearization around an equilibrium point
Consider the following non-linear differential equation
$$\label{star}
\dot{x}(t) = f(x(t))+g(x(t),u(t)), \quad t\ge 0, \ x(0)=x_0\in\mathbb{R},\tag{$\star$}
$$
where $f(\cdot)$ and $g(\cdot)$ are ...
2
votes
1answer
94 views
LQR Robotic Arm
I have a robotic arm model in Simulink and I'd like to control the position of the end-effector such that it follows a given trajectory. This is done by inputting joint angles and comparing the output ...
2
votes
0answers
27 views
(Checking) derivatives of determinant, cofactor, trace
I'm considering an $n\times n$ matrix $T_\epsilon$ such that there exists a unique inverse for $\epsilon \in (-\delta,\delta)$ for some $\delta > 0$. I'm trying to check whether I've correctly ...
0
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1answer
18 views
Properties of matrix $T_\epsilon$ such that there exists a unique inverse for $\epsilon \in (-\delta,\delta)$
I'm considering an $n\times n$ matrix $T_\epsilon$ such that there exists a unique inverse for $\epsilon \in (-\delta,\delta)$ for some $\delta > 0$. I'm trying to determine several quantities of ...
0
votes
4answers
139 views
Newtons method for finding reciprocal
Define a function 1 which is $f_1(x)=a-1/x$
and function 2 which is $f_2(x)=1-ax $
If I set both to zero I am looking for when $x=1/a$ as the root using Newtons method.
When I do this I get two ...
0
votes
0answers
37 views
How to linearize a system of ordinary differential equations, ODEs, around a periodic solution (using numerical methods)?
I am trying to linearize the thermal analysis of a spacecraft model with N nodes over M working points in one orbit.
The non-linear equation is given by the following system of first order ODEs, In ...
1
vote
0answers
19 views
How to linearize this function around a working point?
How can I linearize the following function, where my variables are $\phi_d$, $\dot{\phi_d}$ and $\dot{L}_1$:
$\dot{L}_1=\frac{1}{2\sqrt{(l_d\cos{\phi_d}-x_Q)^2+(l_d\sin{\phi_d}-x_Q)^2-R_2^2}}\small(2(...
1
vote
0answers
23 views
If $ \|G(x+ty)\|<\|G(x)\| $, is then $ \|G(x) + tG'(x)[y]\| <\|G(x)\| $?
It seems intuitive that given a $ C^1 $ function $ f : \mathbb R \to \mathbb R $, where
for some $ t_0 > 0 $ we have $ |f(x+t)| < |f(x)| $ for all $ t \in (0,t_0) $, and
$ f'(x) \neq 0 $,
...
1
vote
0answers
22 views
Similarity in the temperature coefficient of resistivity and coefficient of linear (length) expansion
This was posted on physics stackexchange, https://physics.stackexchange.com/questions/390785/why-change-in-resistivity-is-proportional-to-the-original-resistivity? , but got downvotes and close votes. ...
2
votes
1answer
100 views
Stability of Euler's Method for non-linear ODE
Consider the ODE
$$y'(t) = \lambda y(t), \quad y(t_0) =y_0.$$
Euler's method $y_{i+1}=y_i+h\lambda y_i $ is stable (meaning that the solution decays or stays constant as $ i \to \infty$) provided that ...
0
votes
1answer
695 views
Linearizing Logarithmic Function
I have a given set of data points (y,x) with uncertainties.
When I plot those points on a graph, the trendline appears to follow the equation y = c + a*ln(x).
I want to be able to find the ...
0
votes
1answer
50 views
Finding contstant in remainder term for linearization of $exp(x)$
I've managed to find the remainder term for the linearization of $\exp(x)$ about $x=0$ in Lagrange form:
$$
R_1(x)=\exp(θ_Lx)\frac 12x^2 \text{ ,where } θ_L∈[0,1].
$$
My question is how would I ...
0
votes
1answer
124 views
Product and Quotient Rule proof using linearisation
So I've recently been introduced to the concept of linearization and now I'm beginning to apply this concept to prove certain differenation rules. I've managed to prove the chain rule so far, but I ...
3
votes
1answer
101 views
What is the significance of the linearization of a non-linear PDE?
This may be too general a question so please let me know if I need to make it more specific.
I am a first year graduate student in PDEs, and as such have not had much exposure to non-linear PDEs. I ...
1
vote
2answers
52 views
Taylor approximation with 2 variables, not sure how to solve this
This is the problem I'm trying to solve:
Let $f(x,y)=ay+sin(bx)+c$. Evaluate the Taylor polynomial at $P(0,0)$ and find the values for $a$, $b$ and $c$ if $P(x,y)=-1+2x-y$
I do know the ...
0
votes
1answer
710 views
How to linearize a constraint including product of two binary variables in summation with different indexes?
I am trying to linearize the following two expressions:
$\sum_{k=1}^K \sum_{t=1}^T\sum_{h=1}^W x_{ijkt} a_{hjt} =\sum_{k=1}^K \sum_{t=1}^T x_{ijkt} k , i\in N, j \in M$
$\sum_{k=1}^K \sum_{t=p_{ijk}...
0
votes
1answer
61 views
Show that $L(x,y)=ax^2+by^2$ is a Lyapunov function for the equilibrium at the origin
I am looking at past paper questions and I'm a little stuck on this one.
I have the following system of ODEs:
$\dot{x}=(\epsilon x+2y)(x+1)$
$\dot{y}=(-x+\epsilon y)(x+1)$
where $\...
3
votes
0answers
36 views
A function that turns a power to a product/sum?
I just recently started learning about the logarithm functions, and its concept is quite amazing (f(x.y)=f(x)+f(y)).
Now I'm asking for a similar function but instead of x.y ; we use ...
0
votes
1answer
50 views
What were the steps in the linearization of the following equation?
I'm verifying a linear form of an equation, i.e.
I can't find how the equation 6 leads to the equation 7
3
votes
1answer
240 views
Stationary points and linearisation of non-linear system
So, the problem is:
Find and discuss the behavior of the stationary points of the system :
$$ x'=-y+x\cdot (x^2+y^2)\cdot \sin\sqrt{x^2+y^2} =f(x,y)$$
$$ y'=x+y\cdot (x^2+y^2)\cdot \sin\...
0
votes
0answers
37 views
Ricatti Substitution and Linearization
Recently, I was watching a lecture on asymptotic methods by Dr. Carl Bender. At this point in the lecture, he introduces the Ricatti equation as it arises in the factorization of a differential ...
2
votes
2answers
132 views
Linearization of Differential Equation
Find a linearization of the differential equation for $x$ near $0$.
$$x''(t) + x(t) e^{0.05x} = 0$$
Not sure what to do here. My book isn't any help either..
Any help would be appreciated :)
1
vote
1answer
98 views
Linearization of a Differential Equation
Can someone please help me to linearize this system, which is given by the differential equation shown in the picture below.
All variables are expressed as deviations from initial values
(0, for ...
0
votes
1answer
12 views
Equation of tangent plane at the solution of an ODE
I have the following differential equation:
$$P_0-P_n = \int_0^L f(x,P(x),Q)\,dx$$
In practice, $P_0$ is the pressure at the start of a pipe $(x=0)$, $P_n$ is the pressure at the end $(x=L)$, and $Q$...
0
votes
0answers
49 views
Linearization of a State Space
I have a state space model $x=(x_1, x_2, x_3,x_4,x_5,x_6)$ like this:
$$x_1=\phi, x_2=\dot{\phi}, x_3=\theta, x_4=\dot{\theta}, x_5=\psi, x_6=\dot{\psi}$$
and my function is:
$$f(x,u)= (x_2, ...
-2
votes
1answer
283 views
Exact discretization of non-linear dynamical system which was linearized at a non-equilibrium point. [closed]
0. Question
tl;dr: just see section 2. c)
I'd like to know how a non-linear system of first order ODEs (non-linear dynamical system) which was linearized at a point which is not the equilibrium of ...
0
votes
1answer
99 views
Linearization/Directional Derivative relationship?
I'm a little confused by the following question:
You know that a function f(x, y, z) satisfies f(0, 0, 0) = 33 and
$D_{<1,1,1>/
√
3}$
f(0, 0, 0) = 4/√3 and $D_{<1,1,0>/
√2}$
f(0, ...
1
vote
2answers
226 views
Limit of 4x + 1 as x approaches 0
Okay, so I was approached with this question in my math class and I can answer the first two parts correctly but the third is throwing me off...
${f(x) = 4x + 1}$
a) Table of Values
${x = -0.0001 ...
5
votes
1answer
75 views
Understanding a proof about Riemannian metrics in three dimensions always being diagonalizable
I've recently been working through Deturck's and Yang's Existence of elastic deformations with prescribed principal strains. First and formost, I'm interested in it's proof that Riemannian metrics can ...
0
votes
0answers
39 views
Linear representation of prime factors. What could it be used for?
Imagine we have a vector ${\bf d} \in \mathbb N^n$. Now imagine we have another vector $${\bf w} = [\log(2),\log(3),\log(5),\log(7),\cdots,\log(p_n)]^T$$
$\exp({\bf w^Td})$ will then be the integer ...
2
votes
0answers
53 views
Analysing a dynamical system [closed]
I am developing a model and my model equations are
${dA\over dt}=r_1A(1-{A+B\over k})-a_1 AE-m_1{AN\over (xN+A)}-m_4A$
${dB\over dt}=r_2B(1-{A+B\over k})+a_1 AE-m_2{BN\over (xN+B)}$
${dC\...
0
votes
0answers
32 views
Linearizations for matrix polynomials?
Having tried a bit on my own with limited success here, I'm wondering what other methods there exist to try and linearize this type of polynomial matrix equations:
$$P({\bf X}) = \sum_{k=0}^N {\bf ...
0
votes
0answers
43 views
How do I linearize this?
I'm given the following and I 'm asked to linearize around $x_1=0,x_2=0$$$x_1=x_2^2 \\ x_2=e^{x_1}+u$$
Only thing I know how to do is find the value for $u$ which is $-1$. The only problems I've faced ...
0
votes
1answer
129 views
How to find out the maximum value of absolute error?
Given $$f(x,y) = x^2-3xy+5,$$ where $$R : |x-2| \leq 0.1, \ |y-1| \leq 0.1$$
Now, this function is approximated by a polynomial $$L(x,y) = x-6y+7$$ at the point $(2,1)$.
How to find out the maximum ...
