One of the main results in the theory is that the solution is provided by the linear–quadratic regulator (LQR), a feedback controller whose equations are given below.

Most of these involve variants on the case of linear dynamics and convex (e.g. positive quadratic) cost. The simplest case, called the linear quadratic regulator (LQR), is formulated as …

One of these [Kalman and Bertram 1960], presented the vital work of Lyapunov in the time-domain control of nonlinear systems. The next [Kalman 1960a] discussed the optimal control …

[K,S,P] = lqr(A,B,Q,R,N) calculates the optimal gain matrix K, the solution S of the associated algebraic Riccati equation and the closed-loop poles P using the continuous-time state-space …

Jun 10, 2025 · The LQR problem is a fundamental concept in optimal control theory. It involves finding a control law that minimizes a quadratic cost function subject to a linear dynamic system.

Sep 16, 2019 · Since the plant is linear and the PI is quadratic, the problem of determining the SVFB K to minimize J is called the Linear Quadratic Regulator (LQR). The word 'regulator' …

Jul 2, 2025 · LQR is highly significant due to its ability to design controllers that can manage the state of a system optimally. It is extensively used in various fields such as aerospace, robotics, …

LQR finds applications in aerospace, robotics, and process control. It determines the best control inputs to minimize a quadratic cost function, balancing system state regulation and control …

Before proceeding we need to learn how to solve the above Lyapunov equation in X and K. This is not always possible. In this case, because R ≻ 0, we can complete the squares, rewriting …

Pt can be found from a differential equation running backward in time from t = T the LQR optimal u is easily expressed in terms of Pt we start with x(t) = z