Question

A system is "Controllable" if:

• A. All states can be reached from an arbitrary initial state using an input
• B. The output can be determined by observing the input
• C. The eigenvalues are all negative
• D. The transfer function has no zeros

Hint:

Kalman's Test for Controllability:
The system is controllable if and only if \( \det(Q_c) \neq 0 \) (for a square matrix).
Controllability is about guiding states with inputs; Observability is about estimating states from outputs.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the mathematical or physical definition of a "Controllable" system in state-space control theory.
Controllability is a fundamental property of a dynamic system that determines whether we can drive the system's internal states to any desired target using control inputs.

Step 2: Key Formula or Approach:
Mathematically, a system defined by the state equation \( \dot{x} = A x + B u \) is completely state controllable if its controllability matrix \( Q_c \):
\[ Q_c = \begin{bmatrix} B & AB & A^2B & \dots & A^{n-1}B \end{bmatrix} \] has a full rank of \( n \), where \( n \) is the number of state variables.
Conceptually, this algebraic condition guarantees that any state can be reached within a finite time using a suitable input vector \( u(t) \).

Step 3: Detailed Explanation:
Let us explore the physical meaning of controllability and compare it with the other options:

• State Controllability Definition:
- A system is completely state controllable if, starting from an arbitrary initial state \( x(t_0) \) at time \( t_0 \), there exists an unconstrained control input \( u(t) \) that can drive the system to any other target state \( x(t_f) \) in a finite time interval \( t_f - t_0 \ge 0 \).
- If any of the internal states are decoupled from the input signal, those states cannot be influenced, and the system is said to be uncontrollable.

• Comparison with other options:
- Observability (Option B) is the property where the internal states can be determined by observing the output and input over a finite time.
- Stability (Option C) is determined by having eigenvalues with negative real parts, which is independent of controllability.
- Zeros of the transfer function (Option D) relate to system transmission zeros and do not define state controllability.

Step 4: Final Answer:
A system is controllable if all states can be reached from an arbitrary initial state using an input signal.