**Exercise 1. ***Controllability and Observability *

Is the state equation

Controllable?Observable?** **Provide your derivation.

**Exercise 2. ***Jordan form test **(15 points)*

Is the Jordan-form state equation controllable and observable?

2_{}00x˙ = ^{}_{}0 0_{}00 |
1200000 | 0020000 | 0002000 | 0000100 | 0000110 | 00_{}0 0_{}x0_{}0_{}1 |
2_{ }2 1 + _{ }3 ^{}−1_{ }11 |
1112000 | 01_{}1_{ }1_{}u1_{}1_{}0 |

211 | 111 | 321 | −101 | 101 | 1 | ||||

**Exercise 3. ***Controllability *

Recall the Exercise 3 of Homework 2 from last week. Is that system controllable? ** **Why?

Now lets move the inlet pipe from tank 1 to tank 2, as shown in the figure. Is this system controllable now? ** **Why?

Figure 1: Revised Tank Problem

The system dynamics are

**Exercise 4. ***Gauss Elimination and LU Decomposition **(20 points)*

- Solve the following system of linear equations using Gauss Elimination Method
*x*+*y*+*z*= 3*x*+ 2*y*+ 3*z*= 0*x*+ 3*y*+ 2*z*= 3

*x*+ 2*y*−*z*= 1 2*x*+ 5*y*−*z*= 3*x*+ 3*y*+ 2*z*= 6

*x*_{1 }+*x*_{2 }−*x*_{3 }+*x*_{4 }= 1 2*x*_{1 }+ 3*x*_{2 }+*x*_{3 }= 4

3*x*_{1 }+ 5*x*_{2 }+ 3*x*_{3 }−*x*_{4 }= 5

- Solve the following system of linear equations using LU Decomposition Method

*x*_{1 }+ 2*x*_{2 }+ 4*x*_{3}= 3

3*x*_{1 }+ 8*x*_{2 }+ 14*x*_{3}=13

2*x*_{1 }+ 6*x*_{2 }+ 13*x*_{3}= 4

Provide your derivation.

**Exercise 5. ***SVD **(15 points)*

Use SVD to compress the following image to 50%, 10%, and 5% of the original file size. You will find the image in the Canvas homework folder. For this problem you need to upload code and attached the corresponding compressed images.

Figure 2: CMU Grayscale.png

**Exercise 6. ***Design for Controllability and Observability **(20 points)*

Given the following Linear Time Invariant (LTI) system with a tunable parameter *γ*,

- What values of
*γ*makes the system controllable but not observable?**(10 points)** - What values of
*γ*makes the system observable but not controllable?**(10 points)**

**Exercise 7. ***State Space Representation, Controllability **(10 points)*

We have an LED strip with 5 red LEDs whose brightnesses we want to set. These LEDs are addressed as a queue: at each time step, we can push a new brightness command between 0 and 255 to the left-most LED. Each of the following LEDs will then take on the brightness previously displayed by the LED immediately to its left.

- Model the system as a discrete system with input
*u*(*t*) as the brightness command to the left-most LED. The state to be the brightness of the five LEDs. Output equals to the state. Write out the state equations in matrix form.**(5 points)** - Check the system’s controllability. Explain intuitively what the controllability means in this system.
**(5 points)**

Note: you do NOT need to consider the 0-255 constraints on the input.

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