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The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the LQ problem. 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.
The Kalman filter, the linear-quadratic regulator, and the linear–quadratic–Gaussian controller are solutions to what arguably are the most fundamental problems of control theory. In most applications, the internal state is much larger (has more degrees of freedom ) than the few "observable" parameters which are measured.
This control law which is known as the LQG controller, is unique and it is simply a combination of a Kalman filter (a linear–quadratic state estimator (LQE)) together with a linear–quadratic regulator (LQR). The separation principle states that the state estimator and the state feedback can be designed independently.
A set of differential equations forms a physics engine which maps the control input to the state space of the system. The forward model is able to simulate the given domain. For example, if the user pushes a cart to the left, a pendulum mounted on the cart will react with a motion. The exact force is determined by newton's laws of motion.
A particular form of the LQ problem that arises in many control system problems is that of the linear quadratic regulator (LQR) where all of the matrices (i.e., , , , and ) are constant, the initial time is arbitrarily set to zero, and the terminal time is taken in the limit (this last assumption is what is known as infinite horizon). The LQR ...
The algebraic Riccati equation determines the solution of the infinite-horizon time-invariant Linear-Quadratic Regulator problem (LQR) as well as that of the infinite horizon time-invariant Linear-Quadratic-Gaussian control problem (LQG). These are two of the most fundamental problems in control theory.
Two optimal control design methods have been widely used in industrial applications, as it has been shown they can guarantee closed-loop stability. These are Model Predictive Control (MPC) and linear-quadratic-Gaussian control (LQG). The first can more explicitly take into account constraints on the signals in the system, which is an important ...
where is the gain of the optimal linear-quadratic regulator obtained by taking = = and () deterministic, and where is the Kalman gain. There is also a non-Gaussian version of this problem (to be discussed below) where the Wiener process w {\displaystyle w} is replaced by a more general square-integrable martingale with possible jumps. [ 1 ]