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?
|200x˙ = 0 000||1200000||0020000||0002000||0000100||0000110||000 0x001|| 2 2 1 + 3 −1 11||1112000||011 1u110|
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 + 2y + 3z = 0 x + 3y + 2z = 3
- x + 2y − z = 1 2x + 5y − z = 3 x + 3y + 2z = 6
- x1 + x2 − x3 + x4 = 1 2x1 + 3x2 + x3 = 4
3x1 + 5x2 + 3x3 −x4 = 5
- Solve the following system of linear equations using LU Decomposition Method
x1 + 2x2 + 4x3= 3
3x1 + 8x2 + 14x3=13
2x1 + 6x2 + 13x3= 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.