In the state-space equations, which matrix maps the input vector to the state derivatives?

• A. The input-to-state derivative mapping
• B. The state-to-output mapping
• C. The system dynamics matrix A
• D. The input-to-output feedthrough D

Answer: (A)

In state-space form, the rate of change of the state is written as ẋ = A x + B u. This shows that the input directly influences the state derivatives through the term that multiplies u. That term is the input-to-state mapping, which captures how external signals drive the evolution of the state. So the correct choice is the input-to-state derivative mapping.

To connect the others to their roles: the system dynamics matrix A determines how the current state x affects its own derivative, independent of the input. The state-to-output mapping (often denoted C) links the state to what you measure as the output. The input-to-output feedthrough (often denoted D) maps the input directly to the output, bypassing the state.