The operating point is a state and input pair around which the system is linearized.

1. Understand Physical Significance

• Steady-State: Often, $(x_0, u_0)$ is chosen to be a steady-state operating point where the derivatives of the state variables are zero $(\dot{x}_0 = 0)$.
• Equilibrium: In the context of an AUV, an equilibrium or trim condition might be a situation where the AUV maintains a constant depth and orientation without accelerating.

2. Analyze Desired Operation

• Mission Profile: Consider the typical mission profile of your AUV. Is it often cruising at a particular speed or depth?
• Common Scenarios: Identify scenarios where the AUV spends a significant amount of time or where control is crucial.

3. Solve for Equilibrium

• Equations of Motion: Use the nonlinear equations of motion and set the derivatives of the state variables to zero to find equilibrium points. $f(x_0, u_0) = 0$
• Solve: Solve the above equation for $x_0$ and $u_0$. This might involve finding the control inputs $u_0$ that balance out the forces and moments for a particular state $x_0$.

4. Stability Considerations

• Stability: Ensure that the operating point chosen is stable. Small perturbations from this point should not lead to unbounded state growth.
• Control: Ensure that the AUV can be controlled effectively around the chosen operating point.

5. Analyze Linearization Validity

• Validity: Ensure that the linearized model is a valid approximation of the nonlinear system in the neighborhood of $(x_0, u_0)$.
• Comparison: Compare the responses of the nonlinear and linearized systems to ensure they are similar for expected disturbances and control inputs.

6. Control Design Relevance

• Relevance to Control Objective: Ensure that $(x_0, u_0)$ is relevant to your control objectives and the tasks the AUV will perform.

7. Iterative Refinement

• If the performance is not satisfactory, refine the choice of $(x_0, u_0)$ and re-linearize the system.