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Electrical Engineering QUESTION #2174
Question 1
In LQR optimal control, the cost $J = \int_0^\infty (x^TQx + u^TRu)\,dt$ is minimized by $u^* = -Kx$ where $K = R^{-1}B^TP$. The positive-definite matrix $P$ satisfies:
  • The Algebraic Riccati Equation: $PA + A^TP - PBR^{-1}B^TP + Q = 0$✔️
  • The Lyapunov equation: $PA + A^TP + Q = 0$
  • The Sylvester equation: $AP + PA^T = -Q$
  • The Hamiltonian eigenvalue problem only
Correct Answer Explanation
The steady-state matrix $P$ in LQR is the unique positive-definite solution of the Algebraic Riccati Equation (ARE): $PA + A^TP - PBR^{-1}B^TP + Q = 0$, from which the optimal gain $K = R^{-1}B^TP$ is derived.