24667-Homework 3: Linear Control Systems Solved

Exercise 1. Controllability and Observability (10 points)

Is the state equation

 

Controllable?(5 points) Observable?(5 points) Provide your derivation.


Exercise 2. Jordan form test (15 points)

Is the Jordan-form state equation controllable (7.5 points) and observable? (7.5 points)

2

0

0



x˙ = 0

 0



0

0
1

2

0

0

0

0

0
0

0

2

0

0

0

0
0

0

0

2

0

0

0
0

0

0

0

1

0

0
0

0

0

0

1

1

0
0

0

0 0x

0

0

1
 2

 2  1 

+  3 −1





 1

1
1

1

1

2

0

0

0
0

1

1 1u

1

1

0
 
2

1

1
1

1

1
3

2

1
−1

0

1
1

0

1
1
 
 
 
 

 

Exercise 3. Controllability (10 points)

Recall the Exercise 3 of Homework 2 from last week. Is that system controllable? (5 points) Why?

Now lets move the inlet pipe from tank 1 to tank 2, as shown in the figure. Is this system controllable now? (5 points) Why?

 

Figure 1: Revised Tank Problem

The system dynamics are

 

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

1.    Solve the following system of linear equations using Gauss Elimination Method

a)             x + y + z = 3 x + 2y + 3z = 0 x + 3y + 2z = 3

b)x + 2y − z = 1 2x + 5y − z = 3 x + 3y + 2z = 6

c) x1 + x2 − x3 + x4 = 1 2x1 + 3x2 + x3             = 4

3x1 + 5x2 + 3x3 −x4 = 5

2.    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,

 

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

2.    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.

1.    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)

2.    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.