# need some help on the Linearization tool in MATLAB

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Randy Chen on 1 Nov 2021
Answered: Amey Waghmare on 18 Jan 2023
According to the documentation for linearize, a linear approximation of a nonlinear plant is returned at the specific operating point. I think the word operating point is somewhat confusing, and is it correct for me to interpret this as a snapshot of the behavior of the system at a specific time?
For example, if I linearize a plant for the inverted pendulum at a specific operating point and specified the input as the disturbance force and the output as the pendulum angle theta, then how is the linearized approxiamtion useful to me? Do I simply know what theta is given the input force at this current snapshot (operating point)?

Amey Waghmare on 18 Jan 2023
Hi,
An Operating Point of a dynamic system is the states and the inputs of the system at the specific time. It does not indicate the behaviour of the system at a specific time, rather it just indicates the states of the system at that time instant.While linearizing a plant, the operating point which we specify can be an ‘equilibrium point’ or ‘non-equilibrium point’.
• If we specify an equilibrium point, we can analyze and draw conclusions about the local stability of the system. If this linearized system is stable at that equilibrium, then there exists a neighbourhood around that point where the non linear system is also stable.
• If we specify non-equilibrium point, then we can’t draw the above conclusions as such.
If you linearize the plant for inverted pendulum at an ‘equilibrium point’, then you can perform stability analysis and controller design on the Linearized model. This analysis and design would also be valid on the non-linear plant around a small neighbourhood of that equilibrium point.
An equilibrium point is a solution to dynamical system where the states of the system do not change with time. It is a constant solution to a differential equation/ dynamical system.
Please refer to the following documentation for more details regarding the operating points;