In the state-space form x_dot = A x + B u; y = C x + D u, what does the matrix B represent?

• A. The input mapping to state derivatives
• B. The state-transition matrix
• C. The output mapping to the state
• D. The feedthrough from input to output

Answer: (A)

B represents how the input vector u enters the dynamics to influence the rate of change of the state. In x_dot = A x + B u, the term B u adds to x_dot based on the current input, so each column of B shows how a particular input drives each state's derivative. The A matrix governs how states evolve without input, the C matrix maps states to outputs, and the D matrix captures any direct feedthrough from input to output. For example, if you have n states and m inputs, B is an n-by-m matrix whose entries tell you how much a unit input affects each state's rate of change.