The problem is about a Single Input Single Output (SISO) system with a state-space representation. Here's the given information and required tasks:

1. System Representation: The state-space equations for the system are given by: $\dot{x} = Ax + Bu$ $y = Cx$ where:

   • $x$ is the state vector.
   • $u$ is the input.
   • $y$ is the output.
   • $A$, $B$, and $C$ are matrices defining the dynamics of the system.

2. Questions:

   • Question 1: Identify the names and dimensions of matrices $A$, $B$, and $C$.
   • Question 2: Determine the transfer function $G(p)$ of the system in terms of $A$, $B$, and $C$.

Solution:

1. Dimensions of Matrices $A$, $B$, and $C$:

• $A$ is the state matrix with dimensions $n \times n$, where $n$ is the order of the system (number of state variables).
• $B$ is the input matrix with dimensions $n \times 1$ because there is a single input.
• $C$ is the output matrix with dimensions $1 \times n$ because there is a single output.

2. Transfer Function $G(p)$:

The transfer function $G(p)$ of a state-space system can be derived using the formula: $G(p) = C(pI - A)^{-1}B$ where:

• $p$ is the Laplace variable (assuming $s$-domain representation).
• $I$ is the identity matrix of dimension $n \times n$.

This transfer function expresses the relationship between the input $u$ and output $y$ in the frequency domain.

Would you like further details on these steps or more explanations?

Related Questions for Further Understanding:

1. How do you interpret the role of the matrix $A$ in the dynamics of the system?
2. Why is $B$ called the input matrix, and how does it affect the system’s response?
3. What does the matrix $C$ signify in the context of output generation?
4. Can you explain how the inverse $(pI - A)^{-1}$ is computed and what it represents?
5. How would you find the poles of this transfer function $G(p)$?

Tip:

Understanding state-space representation is crucial for analyzing multi-variable systems, especially in control theory, as it provides a framework to describe system dynamics in a structured form.