Instructions: Complete this exercise using Matlab and Simulink software. You may use the student version (with the Control Systems Toolbox added on)

This practical will be assessed as part of the Class Participation. You can complete each task at your own pace and submit at the end of the course for assessment.

The exercise requires the student to make use of Matlab tools to evaluate a MIMO system and to try Statistical Process Control techniques.

The objective of Part 1 of this lab exercise is to gain experience with the design of an LQR MIMO control system for an industrial process.

The objective of Part 2 is to gain experience with the use of Statistical Process Control techniques.

In an industrial process, we have two sensors and four actuators. The transfer functions were obtained by experimentation and they are as follows: From input 1 to output... -10 1: -------------- s^2 + 72 s + 9

1

2: -------------- s^2 + 72 s + 9

From input 2 to output... 1 1: -------------- s^2 + 72 s + 9

-12 2: -------------- s^2 + 72 s + 9

From input 3 to output...

0.2 1: -------------- s^2 + 72 s + 9

0.1 2: -------------- s^2 + 72 s + 9

From input 4 to output...

0.1 1: ----- s + 1

0.2 2: ------

s + 1

Question 1: (4 marks) Provide the DC gain matrix of the given system.

Question 2: (4 marks) Apply Singular Value Decomposition on the DC gain matrix and provide the obtained matrices.

Question 3: (4 marks) You are required to use two actuators only. Which ones would you choose? Update the transfer matrix to consider the selected actuators. The obtained transfer matrix should be a 2x2 matrix.

Question 4: (4 marks) Provide the Relative Gain Array of the new system (2x2).

Question 5: (4 marks) Provide the State-Space model of the new system (2x2).

Question 6: (4 marks) Design a Linear Quadratic Regulator for the new system (2x2) and provide the optimal gain matrix and a screenshot of the step response of the corresponding closed-loop system.