Question

In State-Space representation, the "State Vector" consists of:

• A. Only the input variables
• B. The minimum set of variables that describe the system's future behavior
• C. The output variables only
• D. The coefficients of the transfer function

Hint:

The dimension of the state vector (number of state variables) is equal to the order of the system's differential equation and the number of independent energy-storage elements in the circuit.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the definition and physical significance of the "State Vector" in state-space control system modeling.
State-space representation is a modern time-domain modeling technique that describes physical systems using a set of first-order differential equations.

Step 2: Key Formula or Approach:
The state equations for a linear time-invariant system are written as:
\[ \dot{x}(t) = A x(t) + B u(t) \] \[ y(t) = C x(t) + D u(t) \] Where:
\( x(t) \) is the state vector of dimension \( n \times 1 \).
\( u(t) \) is the input vector.
\( y(t) \) is the output vector.

Step 3: Detailed Explanation:
Let us define the core concepts of the state-space model:

• State Variables:
- The state of a dynamic system is defined as the smallest, minimum set of variables (called state variables, \( x_1, x_2, \dots, x_n \)) such that knowledge of these variables at an initial time \( t = t_0 \), together with the knowledge of the input signal \( u(t) \) for \( t \ge t_0 \), completely determines the behavior and state of the system for any future time \( t \ge t_0 \).

• State Vector:
- If a system requires \( n \) state variables to be completely described, these variables are organized into an \( n \)-dimensional column vector:
\[ x(t) = \begin{bmatrix} x_1(t) \\ x_2(t) \\ \vdots x_n(t) \end{bmatrix} \] - This vector is called the state vector. It represents a point in the \( n \)-dimensional state space.

• Comparison with other options:
- The state vector is not restricted to input variables (Option A) or output variables (Option C).
- It is a physical internal representation of the system's energy-storing elements (like capacitors, inductors, or masses), distinct from the coefficients of the transfer function (Option D).

Step 4: Final Answer:
The state vector consists of the minimum set of variables that completely describe the system's future behavior.