# Questions tagged [linearization]

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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: ...