LQRLINEARquadratic

LQRLINEARquadratic refers to the linear-quadratic regulator (LQR), a fundamental method in optimal control for designing a controller of linear dynamic systems that minimizes a quadratic cost. The approach provides a systematic way to compute a stabilizing state-feedback control law that balances control effort against a state error penalty.

In continuous-time form, the system is ẋ = A x + B u, and the objective is to minimize J = ∫_0^∞ (x^T Q x + u^T R u) dt, with Q ≥ 0 and R > 0. Finite-horizon problems use J = ∫_0^T (x^T Q x + u^T R u) dt, possibly with a terminal cost x^T(T) P_f x(T). The optimal policy is a linear state feedback u = -K x, where K = R^{-1} B^T P and P is the unique symmetric positive semidefinite solution to the continuous-time algebraic Riccati equation A^T P + P A - P B R^{-1} B^T P + Q = 0. The closed-loop system ẋ = (A - B K) x is then stabilized under mild controllability and detectability conditions.

In discrete time, x_{k+1} = A x_k + B u_k with J = ∑_k (x_k^T Q x_k + u_k^T R u_k). The solution uses the discrete-time Riccati equation P = A^T P A - A^T P B (R + B^T P B)^{-1} B^T P A + Q, and the optimal gain is K = (R + B^T P B)^{-1} B^T P A.

LQR assumes full state feedback and linear dynamics; it is widely used in engineering fields such as robotics and aerospace and forms the basis of extensions like LQR with integral action or the LQG framework when combined with state estimation.