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The image you uploaded shows a problem related to **Linear-Quadratic Optimal Control** in a drilling machine system. Here’s a breakdown of the content in the image and the questions it poses:

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### Problem Overview:
- **Objective**: You are tasked with designing a controller that maintains the speed of a drilling machine constant.
- **Variables**:
  - \(\Delta n(t)\): Speed deviation.
  - \(\Delta T(t)\): Motor torque deviation.
  - \(T_d\): Load torque acting as a disturbance.

  The equation provided is:
  \[
  \Delta \dot{n}(t) = -a\Delta n(t) + b[\Delta T(t) - T_d(t)]
  \]
  
  **Controller**: 
  - An **LQR (Linear Quadratic Regulator)** control law is suggested to minimize the cost function \( J \), which considers the speed deviation and torque applied.

### Questions to Address:
a) **Bryson's Rule**: Determine \(r\) and \(q\) using Bryson's rule.
b) **LQR weight**: Determine the LQR weight \( r \) that ensures that \(\Delta T(t) = 10 Nm\).
c) **LQR design**: Using \(q = 1\), find the state-feedback control law.
d) **Validation**: Check whether the LQR law calculated satisfies the control requirements.
e) **LQR vs Non-optimal**: Compare the LQR solution with a manually selected controller.

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Let me walk through the problem-solving steps.

### (a) Bryson's Rule
Bryson's rule is used to find suitable values of \(r\) (control input weighting) and \(q\) (state error weighting). Typically, Bryson’s rule is given by:
\[
q = \frac{1}{\text{max allowable state deviation}^2}, \quad r = \frac{1}{\text{max allowable control input}^2}
\]

From the image, you can see:
- Maximum allowable speed deviation is on the order of \(2.5 \, \text{rpm}\).
- Maximum torque deviation is \(10 \, \text{Nm}\).

Using these values:
\[
q = \frac{1}{(2.5)^2} = \frac{1}{6.25}
\]
\[
r = \frac{1}{(10)^2} = \frac{1}{100}
\]

### (b) Determine the LQR weight for \(\Delta T(t) = 10 Nm\)
To ensure that the torque deviation is capped at \(10 \, \text{Nm}\), the control input weight \(r\) can be adjusted accordingly using the value obtained from Bryson's rule.

Thus:
\[
r = \frac{1}{10^2} = \frac{1}{100}
\]

### (c) State-Feedback Control Law
Using \(q = 1\) as given in the problem, the LQR state-feedback control law can be calculated. In general, for the system:
\[
u(t) = -Kx(t)
\]
where \( K \) is obtained by solving the **Riccati equation** using the values of \(q\) and \(r\).

### (d) Verify LQR Law
You’ll need to substitute the calculated state-feedback law into the system’s dynamics to verify that the requirements are met, particularly ensuring that the torque deviation remains within the given limits.

### (e) Comparison with Manually Selected Controller
The LQR controller is optimal because it minimizes a quadratic cost function, while a manually selected controller might not be as efficient. The performance difference between the two could be measured in terms of deviation minimization and control effort.

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Would you like further clarification on any of these steps, or should I proceed with solving the Riccati equation for part (c)?

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### 5 Related Questions:
1. What is the importance of Bryson's rule in designing LQR controllers?
2. How does the choice of \(q\) and \(r\) affect the performance of an LQR controller?
3. What is the Riccati equation, and how is it solved in optimal control problems?
4. How does an LQR controller differ from other control strategies, like PID?
5. Can you explain the concept of state-feedback control in simpler terms?

### Tip:
When solving LQR problems, ensure you understand how to balance the trade-off between state deviation (accuracy) and control effort (energy). The choice of \(q\) and \(r\) directly influences this balance.

A tool manufacturer has asked to design a controller that maintains the speed of a drilling machine constant. The speed deviation \(\Delta n(t)\) is described in the system dynamics \(\Delta \dot{n}(t) = -a\Delta n(t) + b[\Delta T(t) - T_d(t)]\). LQR control is suggested to minimize the cost function. Questions include determining values using Bryson's rule, designing LQR control, and comparing with a manually selected controller.

This problem involves designing an LQR controller to maintain constant speed in a drilling machine using control theory principles. It covers state-feedback control laws, Bryson's rule, and a comparison of LQR with manual control. Suitable for students in advanced control systems courses.

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