The document contains a practical session guide for Model Reference Adaptive Control (MRAC) using MATLAB. It is designed for third-year Bachelor students in Electrical Engineering, focusing on Adaptive Control. Below is a summary of key parts:

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Objective:

The goal is to control a system (plant) so that its behavior matches a desired reference model by dynamically adjusting the controller's parameters in real-time. Specifically, students will:

• Simulate the step responses of both the plant and the reference model.
• Apply adaptive control to minimize the error between them.
• Analyze system performance using MATLAB.

Theoretical Concepts:

1. Adaptive Control: Adaptive controllers adjust their parameters in real-time to ensure optimal performance under system uncertainties, unlike fixed-gain controllers.

2. Model Reference Adaptive Control (MRAC): In MRAC, the plant's behavior is adjusted to follow a reference model. The controller parameters are continuously updated based on the error between the plant output and the reference model output.

3. Control Law: The control signal $u(t)$ is given by: $u(t) = \theta(t) \cdot u_{ref}(t)$ where $\theta(t)$ is the adaptive gain, and $u_{ref}(t)$ is the reference input.

4. Adaptation Law: The adaptive gain $\theta(t)$ is updated as: $\theta(t+1) = \theta(t) + \gamma \cdot e(t) \cdot u_{ref}(t)$ where $\gamma$ is the adaptation gain, and $e(t)$ is the error between the reference model output and the plant output.

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MATLAB Implementation Steps:

1. Defining the Reference Model and Plant: The reference model can be represented by a second-order system: $G_{ref}(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$ where $\omega_n$ is the natural frequency and $\zeta$ is the damping ratio.

2. Implementing the Adaptive Control Law:

   • Use a loop to dynamically adjust the controller gain based on the tracking error.
   • Choose a parameter adaptation algorithm, such as the MIT rule or the Lyapunov method.

3. Plotting Results:

   • Plot the output of the reference model against the plant output.
   • Analyze and discuss the results based on tracking performance.

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Questions for Discussion:

1. Plot the initial step response of the plant and reference model, and discuss differences.
2. After applying MRAC, plot the resulting output and describe how well the plant follows the reference model.
3. Explain the control process: How does the adaptive law help minimize the error between the plant and reference outputs?
4. Adaptation gain impact: How does changing $\gamma$ affect system performance?
5. Choose an adaptation method: Compare the MIT rule and the Lyapunov method.

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Would you like a detailed MATLAB code example for this MRAC implementation?

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Tip: Start by defining simple plant and model dynamics to validate your MRAC design before experimenting with complex systems.