Question

For an \(n^{\text{th}}\) order system to be state controllable, which of the following is not a correct statement?

• A. The controllability matrix must be of rank n
• B. The controllability matrix must contain n linearly independent column vectors
• C. The controllability matrix must contain n linearly dependent column vectors
• D. The determinant of controllability matrix is not singular

Hint:

For controllability, think: Rank = \(n\) = Non-singular = Linearly Independent. Any option mentioning "linearly dependent vectors" or "zero determinant" indicates an uncontrollable system.

Answer 1

The Correct Option is C

Concept:& nbsp;
In state-space control theory, an \(n^{\text{th}}\)-order continuous-time linear time-invariant (LTI) system is described by the state equations: \[ \dot{x}(t)=Ax(t)+Bu(t) \] where \(A\) is an \(n\times n\) system matrix and \(B\) is an \(n\times m\) input matrix. According to Kalman's Controllability Criterion, the system is fully state controllable if and only if the composite controllability matrix \(Q_c\), defined as \[ Q_c= \begin{bmatrix} B & AB & A^2B & \cdots & A^{n-1}B \end{bmatrix}, \] possesses full row rank. For an \(n\)-dimensional state vector, the rank of \(Q_c\) must be exactly equal to \(n\).& nbsp;

Step 1: Evaluate the mathematical properties of a full-rank matrix.
If a matrix \(Q_c\) of size \(n\times n\) (assuming a single-input system for standard dimension matching) has full rank equal to \(n\):

1. The matrix must contain exactly \(n\) linearly independent column vectors. Linearly independent columns mean that no column can be expressed as a linear combination of the remaining columns.
2. The determinant of the matrix must be non-zero, i.e., \[ |Q_c| \neq 0. \] A matrix with a non-zero determinant is called a non-singular matrix.

Step 2: Compare these properties against the given options to find the false statement.
& nbsp;

• Option (A): “The controllability matrix must be of rank \(n\).”
  This is the fundamental requirement of Kalman's controllability test, so it is correct.
• Option (B): “The controllability matrix must contain \(n\) linearly independent column vectors.”
  This is necessary for a matrix to have rank \(n\), so it is correct.
• Option (C): “The controllability matrix must contain \(n\) linearly dependent column vectors.”
  If the columns are linearly dependent, then \[ \operatorname{rank}(Q_c) < n, \] making the system uncontrollable. Therefore, this statement is false.
• Option (D): “The determinant of the controllability matrix is non-singular.”
  This means \[ |Q_c| \neq 0, \] which is required for full controllability. Hence, this statement is correct.

Conclusion:
Since the question asks for the statement that is not correct, the answer is: \[ \boxed{\text{Option (C)}} \]