Question: write the code in Octave software and provide the observations in each case.

Consider a temperature control system where a heater is used to regulate the temperature of a process. Design and implement a PID controller in Octave for this temperature control system.

System Description:

The temperature control system is modeled as a first$-$order system with a time constant $($tau$)$ of $10$ seconds.

The heater has a variable power input $(u)$ that influences the temperature of the process.

Controller Design:

Implement a PID controller to regulate the temperature of the system to a desired setpoint. Choose appropriate values for the PID gains $($Kp$,$ Ki$,$ Kd$).$

Experiment with different PID gain values to observe their impact on the system's response. Analyze the trade$-$offs between faster response, overshoot, and steadystate error.

Disturbance Handling:

Introduce a disturbance in the system after a certain time $($e$.$g$.,30$ seconds$).$ Analyze how well the PID controller can handle the disturbance and bring the temperature back to the setpoint.

Performance Metrics:

Implement performance metrics such as rise time, settling time, overshoot, and steady$-$state error. Evaluate the performance of the PID controller using these metrics.

Optimization:

Explore methods to optimize the PID gains automatically for optimal performance. Consider the use of optimization algorithms or tuning rules.

Real$-$world Considerations:

Discuss real$-$world considerations and challenges in implementing a PID control system for temperature regulation. Consider factors such as sensor noise, actuator constraints, and system non$-$linearities.