# Linearization Help: y" (t)+ y'(t)+y(t)=u2(t)-1

• ktoobi
In summary, the conversation is about finding linearization for a system with given conditions, and whether it is necessary to have a linear system around a specific point or linearize the input function. It is suggested that both methods, obtaining a state-space representation and linearizing the input function, can be used to solve the system. The input function is also given as an example, f(x) = x^2.
ktoobi
urgent help for Linearization

Dear All,

y" (t)+ y'(t)+y(t)=u2(t)-1

Linearize the system about y(t)=0, u(t)=1, for all t>= 0

can we say that this equation is already linear at the given point

which will be y" (t)+ y'(t)+0=1-1 => y" (t) + y'(t)= 0

and no need for linearization.

No. What you do is only valid for that point (0,1). You need to have a linear system around this point mimicking almost the nonlinear system in a neighborhood hence your input function must be linearized.

thank you alot, so how to start solving this system?

do i have to get the state-space representation of this system first? or what?

any hint on that?

Last edited:

$f(x) = x^2$ is your function to be linearized.

Thank you alot

## 1. What is linearization and why is it important?

Linearization is the process of approximating a nonlinear system with a linear one. It is important because many real-world systems exhibit nonlinear behavior, but linear systems are easier to analyze and control mathematically.

## 2. How do you linearize a differential equation?

To linearize a differential equation, you need to find the linear approximation of the nonlinear terms. This is typically done by taking the first derivative of the nonlinear term and evaluating it at a specific operating point.

## 3. What is the purpose of the "y(t)+y'(t)+y(t)" term in the given equation?

This term represents the dynamics of the system, or how the output variable y(t) changes over time. It is necessary to include this term in the equation in order to accurately model the behavior of the system.

## 4. Can you explain the significance of the "u2(t)-1" term in the equation?

This term represents the input to the system, u2(t), and a constant offset, -1. It is important because it affects the output of the system, y(t), and can be controlled to manipulate the behavior of the system.

## 5. How can linearization help in solving differential equations?

Linearization can help by transforming a complex nonlinear differential equation into a simpler linear one, which is easier to solve. It also allows for the use of well-established mathematical techniques and tools for linear systems, such as Laplace transforms and transfer functions.

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