Question

Which of the following is not a valid state-space representation?

• A. \(\dot{x}=Ax+Bu,\; y=Cx+Du\)
• B. \(\dot{x}=Ax,\; y=Cx+Du\)
• C. \(\dot{x}=Ax+Bu,\; y=Cx\)
• D. \(\dot{x}=Ax+Bu,\; y=Du\)

Hint:

The standard state-space model is \[ \boxed{ \begin{aligned} \dot{x}&=Ax+Bu, y&=Cx+Du. \end{aligned} } \] where \(A,B,C,\) and \(D\) are the system matrices.

Answer 1

The Correct Option is B

Step 1: Recall the general state-space model. The standard state-space equations are \[ \boxed{ \begin{aligned} \dot{x}&=Ax+Bu, y&=Cx+Du. \end{aligned} } \]

Step 2: Identify the invalid representation. A valid state equation must include the input term \(Bu\) when the system has an external input. The option \[ \dot{x}=Ax,\qquad y=Cx+Du \] omits the input from the state equation while retaining it in the output equation, making it inconsistent for a general controlled system. Hence, \[ \boxed{(B)} \] is the correct answer.