Questions tagged [linearization]

On the many different ways to turn non-linear systems of equations into linear ones.

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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 ...
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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 ...
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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 $x^2$?
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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 $(x_1, x_2) = (1, 1)$ ...
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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: $\chi^{[0]}=\mu$ $\chi^{[i]}=\mu+\left(\sqrt{(n+\lambda)\Sigma}\right)_i\;\;\;i=1,...,n$ $\...
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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)...
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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(...
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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 ...
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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 ...
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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 $$\...
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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 $\vec{y}$? ...
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38 views

Linearizing a non-linear model

I have a non-linear programming model as below: $x_{ij}^2=k_{ij}(y_i-y_j), x_{ij}\in\mathbb{R}, y_i\in\mathbb{R^+}, y_j\in\mathbb{R^+}, \forall$ $i,j\in {[1,2,...,N]}$ $\sum_{j=1}^N a_{ij}.x_{ij}=b_i$ ...
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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 $K$ ...
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14 views

I need to linearize this Equation

I need to linearize this Equation: $F= K_F*i^2(t)$ can anyone help !?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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) = ...
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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 ...
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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
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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]^...
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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 ...
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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 $$x' = y$$ $$y'=-y+ax^2 + bxy.$$...
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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 ...
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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 ...
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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}
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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 ...
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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(...
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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} \...
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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) ...
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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. ...
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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'(...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 \...
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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}...
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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: ...
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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 ...
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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, & \...
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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 ...
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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 ...
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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}[...
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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
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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-...