Questions tagged [linearization]
On the many different ways to turn non-linear systems of equations into linear ones.
172
questions
0
votes
0answers
19 views
Linearization for Extended Kalman filter measurement equation for IMU
I have to implement Extended Kalman Filter for an IMU.
I have solved the equations for the prediction of the next state using the Jacobian.
I am having trouble calculating the Jacobian for the ...
0
votes
1answer
25 views
Help me understand this equality? (assessing stabilty of critical points of a dynamic system)
I'm reading through the solution to a question I became stuck on, and I'm struggling to understand why the following is true:
$$ -kx_{\pm}-k_{1}x^{3}_{\pm}-k\xi-3k_{1}x^{2}_{\pm}\xi = 2k\xi $$
Any ...
0
votes
1answer
26 views
How to linearize the square of a positive continuous variable
Suppose we have a positive continuous variables $0 \le x \le UB$ where $UB$ is a known upper bound.
How can we linearize the term $x^2$?
2
votes
1answer
96 views
Linearized system for $ \begin{cases} \frac{d}{dt} x_1 = -x_1 + x_2 \\ \frac{d}{dt} x_2 = x_1 - x_2^3 \end{cases} $ is not resting at rest point?
Assume there is the dynamical system
$$
\begin{align}
\frac{d}{dt} x_1 &= -x_1 + x_2 \\
\frac{d}{dt} x_2 &= x_1 - x_2^3
\end{align}
$$
The system is at rest at the point $(x_1, x_2) = (1, 1)$ ...
0
votes
0answers
19 views
Redundant sigma points in Unscented Kalman Filter?
According to the Unscented Transform equations in an Unscented Kalman Filter, sigma points are chosen via:
$\chi^{[0]}=\mu$
$\chi^{[i]}=\mu+\left(\sqrt{(n+\lambda)\Sigma}\right)_i\;\;\;i=1,...,n$
$\...
3
votes
3answers
83 views
Use linearisation of a certain function to approximate $\sqrt[3]{30}$
Background
Find the linearisation of the function
$$f(x)=\sqrt[3]{{{x^2}}}$$
at
$$a = 27.$$
Then, use the linearisation to find
$$\sqrt[3]{30}$$
My work so far
Applying the formula
$${f\left( x \right)...
0
votes
1answer
14 views
Series expansion of the absolute value of complex variables
Consider the function $f(u)=|u|^{2k}$ where $u$ is a complex number, and $k$ is an integer. I need to 'linearize' this equation* around the value $\phi$. In other words I need a series expansion of $f(...
0
votes
0answers
57 views
minimizing a concave function
After decoupling a big optimization, the inner one is as follow:
$min_X \quad log_2 \left(det(A+BX\right))$
$s.t. norm(X)\leq \gamma,$
where $A\in \mathbb{C}^{n\times n}$ and $B\in \mathbb{C}^{n\times ...
0
votes
1answer
42 views
Derivative of Input in nonlinear State Space representation
I am dealing with obtaining an space state representation of a nonlinear differential equation that arises from an inverted pendulum. It includes some terms that reflect the fact pendulum is ...
0
votes
0answers
31 views
Linearising system of ODEs
Suppose we have a system
$$\dot{x}=2x+y^2-1,$$
$$\dot{y}=6x-y^2+1.$$
The system has two equilibria at $(0,1)$ and $(0,-1)$. Now, when we linearise around these equilibria, we find the Jacobian
$$\...
0
votes
0answers
49 views
How to linearize $\mid \lambda \vec{e_{1}} - \vec{y}\mid$?
Given a scalar constant $\lambda \in {\rm I\!R}$ and two vectors $\vec{e_{1}}, \vec{y} \in {\rm I\!R^{3}}$, how would one linearize the quantity $\mid \lambda \vec{e_{1}} - \vec{y}\mid$ in $\vec{y}$? ...
0
votes
0answers
38 views
Linearizing a non-linear model
I have a non-linear programming model as below:
$x_{ij}^2=k_{ij}(y_i-y_j), x_{ij}\in\mathbb{R}, y_i\in\mathbb{R^+}, y_j\in\mathbb{R^+}, \forall$
$i,j\in {[1,2,...,N]}$
$\sum_{j=1}^N a_{ij}.x_{ij}=b_i$ ...
0
votes
1answer
29 views
How do I linearise this differential equation?
Consider a system of two genes whose dynamics are described by the following two equations:
$$\dot{X}_1=\frac{K}{1+X_2^n}-X_1, \quad \dot{X}_2=\frac{K}{1+X_1^n}-X_2. $$
Find the critical value of $K$ ...
0
votes
0answers
14 views
I need to linearize this Equation
I need to linearize this Equation:
$F= K_F*i^2(t)$
can anyone help !?
1
vote
1answer
32 views
Rigourous argument for linearization of systems of difference equations
As the title says I'm interested in rigourously justifying the process of linearization of systems of non
-linear difference equations. I recently started reading a book about dynamical systems and ...
0
votes
1answer
32 views
linear programming, discard objective function results below threshold, set them to 0.
i want to create a fantasy football lineup that maximizes the number of weeks it scores over a threshold.
i have every score for every player for every week.
i currently use a solver to maximize my ...
1
vote
1answer
69 views
Linearization of a three tank system resulting in singularity
This question is very closely related to another question previously asked here on Math Stackexchange, where linearization of a three-tank-system is discussed.
After linearization you would get at ...
0
votes
0answers
25 views
approximate floor function with a linear one
How to linearize floor function in form of transfer function?
suppose you have input and out put to the floor function as r and y.
y = floor(r),
I want to ...
0
votes
0answers
16 views
Introducing time to calculate relative percentage error using differentials
I solved a problem asking for the maximum corresponding percentage error in computing a cube's (a) surface area and (b) volume assuming the edge $x$ is measured with an error of at most $0.5\%$ using ...
-1
votes
2answers
34 views
Calculus Problem Regarding Voltage, Differentials, and Linearizations
I don't know how to begin solving this problem. If anyone could help me get the equations that I need and small hints, I would hugely appreciate it!
The voltage across a resistor is given by $V(t) = ...
1
vote
0answers
22 views
Linear transformation for regression
Assuming we have some function with y =
f(a, b, x, e), this relationship should be transformed into a simple linear regression function with y = A + Bx + e, with A only depending on a, B only ...
0
votes
0answers
30 views
Linearization of a Jacobian
I have to take just to linear order (maybe second), the following Jacobian:
$J = \left(1+\frac{v_z}{z}\right)^{-2}\left(1+\frac{d \, v_z}{d \, z}\right)^{-1}$
Can someone help?
Thank you
0
votes
1answer
15 views
Linearization of control system
How can i linearize the following system ?
State equations:
$ x'_1 = (x_1-2)x_2 - 2x_2 $
$ x'_2 =-(x_1-2)^2 + x_2 + u -1 $
Output equation:
$ y = x_1 $
assume the eq. point : $ \tilde x=[2 ,0]^...
2
votes
1answer
128 views
Consider the system $x'=-y-x^2$, $y'=x.$ Find a reversor $S$, identify Fix$(S)$ and show that the origin is a nonlinear center. [closed]
Can I please have help solving the following? I am not quite sure how to approach this problem and was wondering if someone could please help. Thanks!
Consider the system $$x'=-y-x^2$$ $$y'=x.$$ Find ...
2
votes
2answers
101 views
Computing the first 2 terms of the Taylor series exp. for the center manifold & find the reduced equation on the center manifold.
I am having difficulty solving the problem below. It is from Meiss Dynamics book. Can I please receive help solving the following system? Thank you
Consider the system $$x' = y$$ $$y'=-y+ax^2 + bxy.$$...
1
vote
2answers
41 views
How to linearize $\min\{\max\{0,x\},y\}$ as constraint in MILP?
I am formulating a MILP and one of the constraints is
$\min\{\max\{0,y-x+a\},b\} \leq c$.
with decision variables $x, y \geq 0$ and $a,b,c$ as constants.
How would I ideally introduce auxiliary ...
0
votes
0answers
17 views
Argument variable vs. indexed variable in Fourier transform.
Why is the transformed variable in some fourier transforms written as an index whereas sometimes it is written as an argument ? For example,
$$G_{\bf k}({\bf{v}},t)= \frac{1}{(2\pi)^3}\int g({\bf ...
0
votes
0answers
9 views
How do I make a linearization at an equilibrium point
How do I make a linearization of this function?
h(y)/f(x), where
\begin{array}{cc}
h(y)=&py^2+qy+s \\
f(x)=& ax^2+bx+c
\end{array}
3
votes
1answer
54 views
How to linearize system of equations with partial derivatives?
I have a system of equations that I wish to linearize around an equilibrium point. The system is as follows:
We have two functions $h(x,t)$ and $u(x,t)$ and constants $d$ and $g$. We then have the ...
0
votes
0answers
19 views
Linearization of a parametrized function
I have trouble understanding the notation in the following exercise:
Find linearization of function $ f(x_1(p),x_2(p))$ for small changes in parameters $p$ i.e. $p_0+\delta$ using the Jacobian:
$$ f(...
0
votes
0answers
20 views
Recursion relation for the expansion coefficients of the product of two Jacobi polynomials in terms of one Jacobi polynomial
I need the recursion relation satisfied by the expansion coefficients $\left\{ {{c_k}} \right\}$ in the following series:
$P_n^{(\mu ,\nu )}(x)P_m^{(\mu ,\nu )}(x) = \sum\limits_{k = \left| {n - m} \...
2
votes
3answers
164 views
Solving y=ax^b with logarithms
[y = 7 when x = 2 and y = 8 when x = 3]
[a,b > 0 and are fixed real numbers]
I understand that this equation can be linearized using logarithms:
$$y = ax^b$$
$$log(y) = log(ax^b)$$
$$log(y) = log(a) ...
2
votes
2answers
63 views
Linearization of $C^r$ ODE system
Consider the following ODE system
\begin{align*}
\dot{x} &= x^5 + y^3 = f(x\,,y) \\
\dot{y} &= x^3 - y^5 = g(x\,,y)
\end{align*}
Clearly, $(x\,,y) = (0\,,0)$ is the only fixed point. ...
0
votes
1answer
19 views
Find linearized equation at equilibrium
The differential equation below has an equilibrium at $x=1$:
$\frac{dx}{dt} = (e^{x}-1)(x-1)$
The problem asks to find the linearized equation but in terms of $h$, where $h=x-a$.
$\frac{dh}{dt}=f'(...
0
votes
0answers
27 views
Why is linearization in least-squares regression less accurate than nonlinear regression when the datapoints resemble a curve?
Say that you have a set of 10 datapoints which could be modelled by a logarithmic curve. What is the concrete reason why linearizing and then using systems of linear equations to obtain the formula of ...
0
votes
0answers
25 views
Optimal order for the Extended Kalman Filter
For a set of observations and a nonlinear relation to the hidden state, the extended Kalman filter works by linearizing the relation in each step around the previous prediction. This can lead to poor ...
0
votes
0answers
12 views
Linearization of S-shaped fonction in logaritmic domain.
Given $R = \frac{I^n}{I^n+\sigma^n}$,
How to show that $R\sim a+b\log(I)$ around $\sigma$?
It apear in some book, I tried many things but couldn't show it.
0
votes
0answers
39 views
Hartman-Grobman theorem and linearization about non-equilibrium point for linear control
I recently came across an interesting quadcopter control paper (http://rpg.ifi.uzh.ch/docs/ICRA18_Foehn.pdf), that also seemed very strange with respect to the linearization scheme used on the ...
-1
votes
1answer
12 views
Volume and Radius Linearization Practice
Equations to help:
$V = \frac 43\pi r^3$, $p = m/v$ , $s = ut + \frac12 at^2$ , $v^2 = u^2 + 2as$
In each this situation you are told which data is collected. Please determine using the collected ...
0
votes
1answer
22 views
Reformulation logical AND in integer programming maximization problem
Suppose we have variables $x_1,x_2,y \in \{0,1\}$ such that $y=1$ if and only if $x_1 = x_2 = 1$ and we want maximize the value of $y$. I know that this reduces to the following Integer programming ...
-1
votes
1answer
37 views
Combination of AND OR in Linear Programming
I have three binary variables: $x,y,z$.
I want to define $U$ as follows:
$$U = x \wedge (y \vee z)$$
Following this, I have already tried defining
$$yz = y \vee z$$
and then, doing
$$U = x \...
0
votes
0answers
55 views
How can I reduce this system (differential equations below) to a space state (matrix A, B, C, D)?
I have the system below
\begin{align}
\ddot{\varphi } &=\frac{ \tau_e -m_P.R.\ddot{\alpha}.l.\cos\alpha+m_P.R.\dot{\alpha}^2.l.\sin\alpha-C_D.\pi.R^5.\rho.\dot{\varphi}^2}{J_B.R^2}\\
\ddot{\alpha}...
0
votes
1answer
23 views
How can I linearize the drag force without any constants terms?
How can I linearize the drag force at $V=V_{medium}$:
$F_d = \frac{1}{2}C_d.A.\rho .V(t)^2$
without any constant term? in the form:
$F_d=K_1V(t)+K_2$
K2 is the constant term and should be 0.
EDIT:
...
0
votes
0answers
38 views
determine if the origin is a spiral sink, a spiral source, or node, center, or saddle?
Solve the linear system and decide whether the critical point (0, 0) is stable or unstable. Sketch
the direction field and use it to decide whether (0, 0) is a node, a center, a saddle or a spiral
...
0
votes
1answer
102 views
Maximize a piecewise function
I'm trying to linearize the problem:
$\max f(x)\\\text{s.t.}\\g(x)\geq 0$
Where $g(x)$ are already linear functions, but $f(x)$ is the following piecewise function:
$f(x)=\begin{cases}
bx, & \...
0
votes
1answer
79 views
Linearizing quadratic 0-1 function
I have the following as part of my objective function
z=$y_{wb} \sum_{w' \neq w} p(w') y_{w'b}$
s.t. $\sum_{w' \neq w} y_{w'b}=1$
where $y\in \{0,1\}$, and $p(w')$ is the probability of $w'$.
The ...
-1
votes
1answer
10 views
Diameter increase approximation (Linearization)
The diameter of a tree was 10 in. During the following year, the
circumference increased 2 in. About how much did the tree’s
diameter increase? The tree’s cross-section area?
we are using ...
1
vote
1answer
62 views
Linearizing a nonlinear dynamical system with Sigmoid function
We have the following dynamical system to linearize in order to find the critical points:
$$\dot{y_0}(t) = y_3(t) \\
\dot{y_1}(t) = y_4(t) \\
\dot{y_2}(t) = y_5(t) \\
\dot{y_3}(t) = Aa\mathrm{Sigm}[...
0
votes
0answers
20 views
Aizermann (linearization), and Krasovski methods
Could you please give a simple summary and example shows the difference between Aizermann (linearization), and Krasovski methods, to find Lyapunov function for nonlinear system
1
vote
0answers
53 views
Lyapunov-Function with the Method of Aizermann/Krasovski
Consider the following autonomous system:
$$
\begin{align}
\dot{x}_1 &= -2 x_1 \\
\dot{x}_2 &= -2 x_2 + 2 x_1 x_2^2
\end{align}
$$
Derive for the equlibrium $x_1 = x_2 = 0$ a Lyapunov-...
