LQRpohjaiset

LQRpohjaiset refers to controllers designed using the Linear-Quadratic Regulator (LQR) framework. They are used to achieve optimal state feedback control for linear dynamical systems by minimizing a quadratic cost function that penalizes deviations in state and control effort. The approach is common in engineering fields such as robotics, automotive, aerospace, and process control, where a systematic method is needed to balance performance and energy use.

In the standard continuous-time formulation, the system is modeled as ẋ = A x + B u, and the objective is to minimize J = ∫0^∞ (x^T Q x + u^T R u) dt. Here Q is symmetric positive semidefinite and R is symmetric positive definite. The solution involves the continuous-time algebraic Riccati equation (CARE): A^T P + P A - P B R^{-1} B^T P + Q = 0, from which the state feedback gain is K = R^{-1} B^T P and the control law is u = -K x. In discrete time, with x_{k+1} = A x_k + B u_k and J = Σ (x_k^T Q x_k + u_k^T R u_k), the discrete-time algebraic Riccati equation (DARE) is solved for P, yielding K = (R + B^T P B)^{-1} B^T P A and the law u_k = -K x_k.

Design typically involves linearizing a nonlinear system around an operating point and verifying controllability and detectability. If full state measurement is unavailable, the LQR framework is combined with an observer (Kalman filter) in an LQG setup.

Advantages include a principled trade-off between performance and control effort and guaranteed stability for appropriate Q and R. Limitations include reliance on an accurate linear model, sensitivity to model errors, and the need for state measurements or estimators; real-world constraints may require extensions such as model predictive control or robust/constraint-aware methods.