linearization-procedure.md

Tangent-Line Linearization Procedure

Author: Percival Segui

Prepared as an independent technical reference.

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Purpose

Use this when you want to approximate a nonlinear differential equation with a linear one near a known operating point.

This is tangent-line linearization, not necesarily equilibrium linearization.

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General Form

You are given an equation of the form:

$$ \text{(linear terms)} + f(x) = 0 $$

Where:

• $f(x)$ is nonlinear (i.e., $\cos x$, $ln x$, $e^x$, etc.)
• You want to linearize near a point $x_0$

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Step-by-Step Procedure

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Step 1: Identify the Operating Point

Pick the known value around which you want to analyze the system:

$$ x_0 = \text{operating point} $$

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Step 2: Define the Deviation Variable

Let:

$$ \delta x(t) = x(t) - x_0 $$

This represents small changes around $x_0 $.

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Step 3: Replace the Nonlinear Term with the Tangent Line Approximation

Use the equation of the tangent line:

$$ f(x) \approx f(x_0) + f'(x_0) \cdot (x - x_0) = f(x_0) + f'(x_0) \cdot \delta x $$

This gives a linear approximation for $f(x)$ near $x_0$.

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Step 4: Plug into the Differential Equation

Substitute the approximation into the original equation:

$$ \text{(linear terms)} + f(x_0) + f'(x_0) \delta x = 0 $$

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Step 5: Subtract Off the Constant $f(x_0) $

Move the constant term to the other side:

$$ \text{(linear terms)} + f'(x_0) \delta x = -f(x_0) $$

Now your equation is fully linear in $\delta x$, and includes any necessary offset (forcing term) due to the original nonlinear shape.

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Clarifying Note on Operating Points and Forcing Terms

The appearance of the constant term $f(x_0)$ depends on how the operating point is chosen.

If $x_0$ is selected as an equilibrium point of the original nonlinear system, then it satisfies the steady-state condition:

$$ \text{(linear terms)} + f(x_0) = 0 $$

In this case, $f(x_0)=0$ when the system is expressed in deviation variables, and the linearized model describes small-signal dynamics about equilibrium. The resulting deviation equation is homogeneous.

If $x_0$ is not an equilibrium point, then $f(x_0)\neq 0$, and the linearized deviation equation includes a constant forcing term. This represents the fact that the original system would naturally accelerate away from that point even with zero deviation.

Both forms are mathematically valid:

• equilibrium linearization $\rightarrow$ small-signal dynamics
• non-equilibrium linearization $\rightarrow$ linear approximation with constant offset

The example shown below uses a non-equilibrium operating point to illustrate the general tangent-line procedure.

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Example

Original nonlinear equation:

$$ \frac{d^2 x}{dt^2} + 2 \frac{dx}{dt} + \cos x = 0 $$

Operating point: $x_0 = \frac{\pi}{4}$

Then:

• $f(x_0) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$
• $f'(x_0) = -\sin(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$

So:

$$ \cos x \approx \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot \delta x $$

Substitute and subtract constant:

$$ \frac{d^2 \delta x}{dt^2} + 2 \frac{d \delta x}{dt} - \frac{\sqrt{2}}{2}\delta x = -\frac{\sqrt{2}}{2} $$

This is the fully linearized version of the original equation.

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Summary Table

Step	Description
1	Choose operating point $x_0$
2	Define deviation: $\delta x = x - x_0$
3	Approximate: $f(x) \approx f(x_0) + f'(x_0) \delta x$
4	Plug into equation
5	Subtract constant $f(x_0)$ to isolate linear deviation

This gives a complete linear model, valid for small deviations around the point of interest.

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