MIDA 2

Question 1

[Intro] What is the difference between White-Box and Black-Box modeling?

Answer 1

• White-box uses first-principle equations and requires knowledge of the system’s physical parameters.
• Black-box relies on data collected from experiments and doesn’t require knowledge of the underlying physical system

Question 2

[Intro] What is Software Sensing?

Answer 2

It’s an algorithmic approach to estimate un-measurable variables using indirect measurements.

Question 3

[Intro] What are some advantages and disadvantages to Software Sensing?

Answer 3

+ Measures variables otherwise unmeasurable
+ Reduces the need for physical sensors, saving costs.
+ Eliminates maintenance and fault risks of physical sensors.
- Development costs for designing and calibrating the algorithm
- Potential for higher variance in sensing errors due to indirect measurement.

Question 4

[Ch1] What are the three main mathematical representations of discrete-time linear systems?

Answer 4

1.- State Space (Internal representation)
2.- Transfer Function (external representation)
3.- Impulse Response (I.R.)

Question 5

[Ch1] What equations define the state-space representation?

Answer 5

• State Equation: x(t+1) = Fx(t) + Gu(t)
• Output Equation: y(t) = Hx(t) + Du(t)

Question 6

[Ch1] What do the matrices F, G, H, and D represent in state-space representation?

Answer 6

• F: State Matrix
• G: Input Matrix
• H: Output Matrix
• D: Direct Transmission Matrix (0 for strictly proper systems.

Question 7

[Ch1] What’s the formula to transform from State-Space to Transfer Function representation?

Answer 7

W(z) = H(z I - F)^-1 G

Question 8

[Ch1] What is the Z-transform? What does it mean?

Answer 8

The Z-transform converts discrete-time signals into the transform domain, simplifying the analysis of linear systems. It is crucial for moving between IR and TF representations..

Question 9

[Ch1] How is the impulse response representation defined?

Answer 9

It represents the output y(t) as the convolution of the input u(t) with the system’s impulse response:
y(t) = SUM_k=0^∞ w(k)u(t-k)
with w(k) being the impulse response

Question 10

[Ch1] Why is the impulse response representation rarely used in practice?

Answer 10

It requires all values of the impulse response to be noise-free and fully known, often impractical.

Question 11

[Ch1] How can the eigenvalues of the state matrix F determine system stabillity in state-space representation?

Answer 11

• If all eigenvalues of F lie within the unit circle in the complex plane, the system is asymptotically stable.
• Eigenvalues on the unit circle indicate simple stability.
• Eigenvalues outside the unit circle indicate instability.

Question 12

[Ch1] What’s the difference between asymptotical stability and simple stability?

Answer 12

Asy. stability guarantees that the system returns to it’s equilibrium after disturbances. Simple stability does not guarantee convergence, posing risk in dynamic systems.

Question 13

[Ch1] How does the concept of “Strictly Proper” systems relate to the impulse response?

Answer 13

Strictly proper systems ensure that the output doesn’t respond instantaneously to an input jump. This reflects physical system behavior without abrupt changes.

Question 14

[Ch1] What’s the formula to transfrom from a state-space representation to an Impulse Response?

Answer 14

y(t)= H F^(t-1) G for t>0

Question 15

[Ch1] Why is transformation from impulse response to state-space representation challenging (not recommended)?

Answer 15

It requires reconstructing state matrices from measured impulse responses, which are sensitive to noise and involves complex calculations.

Question 16

[Ch1] What is Observablity in the Context of a System?

Answer 16

A system is observable if the current state can be determined from the output history over time. Mathematically
O = [H ; HF ; HF^2; … ; HF^(n-1)]
MUST have a full rank. n is the order of the system.

Question 17

[Ch1] What is Controllability, How is it Tested?

Answer 17

Controllability means the system’s states can be fully controlled using the inputs. It is tested using the controllability matrix:
R= [ G FG F^2G … F^(n-1)G]
This must be Full rank.

Question 18

[Ch1] What are the implications of a system not being fully observable or controllable?

Answer 18

If a system is not fully observable, some states cannot be inferred from outputs. If not fully controllable, certain states cannot be influenced by inputs. They both limit differently the ability to design effective controls.

Question 19

[Ch1] What does 4SID mean?

Answer 19

Subspace-based State Space System Identification.

Question 20

[Ch1] What is the role of the Hankel matrix in the 4SID algorithm?

Answer 20

The Hankel matrix organizes impulse response data into a structured form. Its rank indicates the system’s order, and it can be factorized to estimate observability and controllability matrices.

Question 21

[Ch1] How is the system order determined using the Hankel matrix?

Answer 21

By increasing the matrix size and checking its rank. When the rank stops increasing with added rows or columns, the rank indicates the system order. (This happens when we find a Non-full-rank matrix)

Question 22

[Ch1] Why is 4SID considered non-parametric?

Answer 22

It does not assume specific model structure or use optimization. It directly estimates state-spacce matrices using impulse response data.

Question 23

[Ch1] What is SVD and why is it used in 4SID?

Answer 23

Singular Value Decomposition decomposes a matrix into three components (U,S,V):
H = USV^T
Separating the system dynamics from noise.
It is used when building a Hankel Matrix with all the dataset.

Question 24

[Ch1] What does the “knee” in the singular value curve represent in noisy data?

Answer 24

It represents the transition between system-related singular values and noise-related singular values, guiding the choice of system order.

Question 25

[Ch1] How does noise affect the Hankel Matrix in the 4SID algorithm?

Answer 25

Noise adds distortions, leading to overestimated ranks and inaccurate matrix factorization. SVD helps filter out noise by retaining only dominant singular values.

Question 26

[Ch1] What is the trade-off in choosing the size of the Hankel matrix?

Answer 26

A larger matrix provides better precision but increases computational complexity. The ideal size balances accuracy and feasibility.

Question 26

[Ch1] How can one verify observability and controllability using block schemes?

Answer 26

By visually analyzing the block scheme:
- Observabillity: Check if all state contributions can be traced to the output.
- Controllability: Ensure input paths can influence all states.

Question 27

[Ch1] What are some limitations of the 4SID algorithm?

Answer 27

• Highly sensitive to noise in the impulse response.
• Inefficient use of data

Question 28

[Ch1] How is the 4SID algorithm generalized for non-impulse inputs?

Answer 28

It adapts to handle general input signals by modifying the data organization and ID Steps, but increases complexity.

Question 29

[Ch1] Why is SVD critical for the 4SID algorithm?

Answer 29

It is used to decompose a noisy Hankel matrix into system-related singular values and noise-related singular values.

Question 30

[Ch1] How do singular values determine the system order in 4SID?

Answer 30

The number of singular values corresponds to the system order. In an ideal case, there is a sharp drop after the system-related singular values, making the order evident (Knee).

Question 31

[Ch1] What are the three matrices obtained from SVD, and what do they represent?

Answer 31

H = USV^T
where:
- U: Contains the left singular vectors related to observability.
- S: A diagonal matrix with singular values, indicating the system’s dynamics and noise levels.
- V: Contains the right singular vectors related to controllability.

Question 32

[Ch1] What are the key steps in applying 4SID algorithm using the full dataset?

Answer 32

1. • Build Hankel Matrix: using the measured dataset (W(1), W(2) … W(n)).
     2.- Perform SVD: Identify system/noise singular values.
     3.- Determine system order: Analyze the singular value curve for a “jump” or “knee”
2. • Reconstruct Clean Hankel Matrix: H_system = U_nS_nV_n^T
     5.- Factorize H_system decompose it into the extended observability and controllability matrices.
     6.- Estimate State-Space matrices

Question 33

[Ch1] How does the choice of q and d for the Hankel Matrix dimensions affect the use of SVD in 4SID?

Answer 33

• Larger q and d improve precision but is harder to compute.
• A balanced choice is recommended. q>d/2