3. This differential equation describes an LTI system with input u and output y: ÿ+ay+by = cù+u → a, b, c € R (a) Determine a state-space representation for this system (b) Determine the transfer function for the system from the state-space representation from (a)
3. This differential equation describes an LTI system with input u and output y: ÿ+ay+by = cù+u → a, b, c € R (a) Determine a state-space representation for this system (b) Determine the transfer function for the system from the state-space representation from (a)
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![### Problem 3: Differential Equation and LTI System Analysis
This differential equation describes a Linear Time-Invariant (LTI) system with input \( u \) and output \( y \):
\[
\ddot{y} + a\dot{y} + by = c\dot{u} + u \quad \rightarrow \quad a, b, c \in \mathbb{R}
\]
#### Tasks:
(a) **Determine a state-space representation for this system**
(b) **Determine the transfer function for the system from the state-space representation found in (a)**
---
#### Explanation:
- **Differential Equation Structure:**
- \(\ddot{y}\) is the second derivative of \( y \) with respect to time.
- \(\dot{y}\) is the first derivative of \( y \) with respect to time.
- \( \dot{u} \) is the derivative of the input \( u \) with respect to time.
- **State-Space Representation:**
- This involves expressing the higher-order differential equation as a system of first-order differential equations.
- The state-space representation includes matrices that describe the dynamics of the system.
- **Transfer Function:**
- The transfer function is derived from the state-space representation.
- It relates the input and output of the system in the Laplace domain, providing insight into the system's frequency response.
---
This problem is aimed at understanding the conversion of differential equations describing LTI systems into their equivalent state-space and transfer function forms, fundamental in system dynamics and control theory.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf671330-1dc4-433a-bf19-1991be3b60ec%2F02c600d8-4699-4921-a337-8be768e86e45%2Fp3oxx9n_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 3: Differential Equation and LTI System Analysis
This differential equation describes a Linear Time-Invariant (LTI) system with input \( u \) and output \( y \):
\[
\ddot{y} + a\dot{y} + by = c\dot{u} + u \quad \rightarrow \quad a, b, c \in \mathbb{R}
\]
#### Tasks:
(a) **Determine a state-space representation for this system**
(b) **Determine the transfer function for the system from the state-space representation found in (a)**
---
#### Explanation:
- **Differential Equation Structure:**
- \(\ddot{y}\) is the second derivative of \( y \) with respect to time.
- \(\dot{y}\) is the first derivative of \( y \) with respect to time.
- \( \dot{u} \) is the derivative of the input \( u \) with respect to time.
- **State-Space Representation:**
- This involves expressing the higher-order differential equation as a system of first-order differential equations.
- The state-space representation includes matrices that describe the dynamics of the system.
- **Transfer Function:**
- The transfer function is derived from the state-space representation.
- It relates the input and output of the system in the Laplace domain, providing insight into the system's frequency response.
---
This problem is aimed at understanding the conversion of differential equations describing LTI systems into their equivalent state-space and transfer function forms, fundamental in system dynamics and control theory.
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