DCDS 3 gyakorlat

Описание: This video is a practical exercise session for the "Theory of Discrete and Continuous Dynamic Systems" (DCDS) course, led by . It focuses on discrete-time systems, matrix exponentials, and state-space modeling.

Below is a summary of the key exercises and concepts covered:

1. Matrix Functions and Exponentials [00:47]
The instructor begins with elementary examples of calculating matrix exponentials.

Diagonal Matrices: He demonstrates that for a diagonal matrix, the exponential is simply the exponential of each diagonal element [03:17].

General Matrices: For non-diagonal matrices, he explains the use of diagonalization (Jordan form) where e
M
=Te
D
T
−1
[04:29].

Inverse Laplace Method: He shows how to find the matrix exponential e
Qt
using the inverse Laplace transform formula L
−1
{(sI−A)
−1
} [05:45].

2. Discretization of State-Space Models [08:27]
A significant portion of the video is dedicated to converting a continuous-time linear system into a discrete-time model with a sampling time h.

Discrete Matrices: He explains that while the C and D matrices remain the same, the state transition matrix Φ and input matrix Γ must be calculated [10:34].

Step-by-Step Calculation:

Φ is determined using the matrix exponential e
Ah
[11:24].

The process involves finding the inverse of (sI−A) and performing partial fraction decomposition to transform back to the time domain [16:53].

Γ is then calculated using the formula Γ=A
−1
(e
Ah
−I)B [22:14].

3. Pulse Transfer Operators and Canonical Forms [28:32]
The instructor moves on to higher-order difference equations.

Pulse Transfer Operator (H(q)): He demonstrates how to derive the operator by using the time-shift operator q [29:38].

State-Space Representations: He shows how to convert the transfer operator into specific forms:

Controller Canonical Form (Irányíthatósági normál forma) [34:43].

Observer Canonical Form (Megfigyelhetőségi normál forma) [38:04].

Non-Uniqueness: He highlights that state-space models are not unique, as infinitely many equivalent representations can describe the same input-output behavior [31:58].

4. Discrete-Time Impulse Response and Markov Parameters [39:13]
The final section covers the system's response to an impulse.

Sequence Calculation: He provides the formula for the impulse response sequence h(k)=CΦ
k−1
Γ for k≥1 [40:09].

Invariance Proof: The video concludes with a mathematical proof showing that Markov parameters are independent of the chosen state-space representation, meaning they remain the same regardless of similarity transformations [45:14].