## Description

## Problem 1: Pole Placement (Hand Calculations) (6 points)

Consider the state-space model from the previous assignment:

�(� + 1) = (

0.1 ∗ � −0.4

0.1 ∗ � 0.252 �(�) + 3

1

0

4 �(�)

where p and q are the last two digits of your roll number. Thus, if your roll number is CH01B234,

then p = 3, q = 4 and the matrix A=[0.3,-0.4;0.4,0.25];.

1. Compute the pole-placement design that gives deadbeat controller. Please return the controller

gain as a matrix L_db.

2. Compute the pole-placement design that gives a controller with one pole at 0.25 and another at

0.4. Please return the controller gain as a matrix L_pp.

## Problem 2 Demo Example: Simulations of closed-loop system (0 points)

This is now a demo problem and it will not be graded: Consider regulating the system from

�! = [1 0.5]

“. We will do this for five discrete steps, to obtain �(1), �(2), �(3), �(4), �(5).

Prefer to do this with hand calculations, though MATLAB is also permitted.

• Using dead-beat controller, compute �(1) to �(5). Please return the results in a 2 × 5 matrix Xdb

• Repeat for the pole placement controller. Please return the results in a 2 × 5 matrix Xpp

## Problem 2: Steady State LQR (6 points)

3. Using the algebraic Ricatti equation or discrete Lyapunov equations in MATLAB, compute the

controller gain if Q = I and R = 1. Please report the results as a matrix L.

4. What are the controller poles for this controller? Please return them in a 2 × 1 vector LQRpoles.

Problem 3: Effect of Q and R (8 points)

5. Aggressive control: Repeat Problem 3 with new weights: � = 100�, �# = 1. Compute the poles

6. Sluggish control: Repeat again with � = �, �# = 100. Compute the poles

7. Very aggressive control: Repeat again with � = 10$�, �# = 1. Compute the poles

8. Very sluggish control: Repeat again with � = �, �# = 10$. Compute the poles