To incorporate sensors with noise for measuring $\theta$ (angle) and $\dot{\theta}$ (angular velocity) in the cart-pole system, we can add blocks representing noisy measurements in the control diagram. Here's a step-by-step approach to modify this control system:

Steps to Add Sensors with Noise

1. Sensor Blocks for $\theta$ and $\dot{\theta}$:

   • Add a sensor block for each measurement ($\theta$ and $\dot{\theta}$) right after the outputs $\theta$ and $\dot{\theta}$ from the integrators.
   • The sensor blocks will represent the measurements obtained from physical sensors.

2. Noise Addition:

   • Introduce noise sources (like Gaussian noise) in the sensor measurement paths.
   • Connect a noise block to each sensor to simulate real-world imperfections. This can be represented as an additional summation block where the noise signal is added to the actual $\theta$ and $\dot{\theta}$ values.

3. Feedback Path Adjustment:

   • Route the outputs of these noisy sensor blocks as the inputs to the Feedback Linearized Controller instead of the direct $\theta$ and $\dot{\theta}$ outputs from the system.
   • This setup ensures that the controller uses the noisy measurements for feedback, mimicking real-world sensor limitations.

4. Display Noisy Signals:

   • If desired, display both the true and noisy signals in separate blocks or labels to compare the true system behavior versus what the controller "perceives" via the noisy sensors.

Would you like a more detailed step-by-step explanation on how to implement this in a simulation environment (e.g., MATLAB Simulink), or would you prefer additional guidance on sensor noise modeling?

Here are five related questions to deepen your understanding:

1. What are common types of noise used to model sensor imperfections in control systems?
2. How does sensor noise affect the stability and performance of a feedback control system?
3. What are potential methods to filter or mitigate sensor noise in control applications?
4. What kind of sensors are typically used to measure angle and angular velocity in physical cart-pole systems?
5. How does the choice of noise parameters (e.g., mean, variance) impact the system's control accuracy?

Tip: When modeling sensor noise, Gaussian (white) noise is often a realistic choice, but always tailor the noise characteristics to the physical context for the most accurate simulation.