Linear–quadratic–Gaussian control
In control theory, the linear–quadratic–Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns linear systems driven by additive white Gaussian noise. The problem is to determine an output feedback law that is optimal in the sense of minimizing the expected value of a quadratic cost criterion. Output measurements are assumed to be corrupted by Gaussian noise and the initial state, likewise, is assumed to be a Gaussian random vector. Under these assumptions an optimal control scheme within the class of linear control laws can be derived by a  completion-of-squares argument. 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. LQG control applies to both linear time-invariant systems as well as linear time-varying systems, and constitutes a linear dynamic feedback control law that is easily computed and implemented: the LQG controller itself is a dynamic system like the system it controls. Both systems  have the same state dimension. A deeper statement of the separation principle is that the LQG controller is still optimal in a wider class of possibly nonlinear controllers. That is, utilizing a nonlinear control scheme will not improve the expected value of the cost functional. This version of the separation principle is a special case of the separation principle of stochastic control which states that even when the process and output noise sources are possibly non-Gaussian martingales, as long as the system dynamics are linear, the optimal control separates into an optimal state estimator (which may no longer be  a Kalman filter) and an LQR regulator. In the classical LQG setting, implementation of the LQG controller may be problematic when the dimension of the system state is large. The reduced-order LQG problem (fixed-order LQG problem) overcomes this by fixing "a priori" the number of states of the LQG controller. This problem is more difficult to solve because it is no longer separable. Also, the solution is no longer unique. Despite these facts numerical algorithms are available to solve the associated optimal projection equations which constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller . LQG optimality does not automatically ensure good robustness properties. The robust stability of the closed loop system must be checked separately after the LQG controller has been designed. To promote robustness some of the system parameters may be assumed stochastic instead of deterministic. The associated more difficult control problem leads to a similar optimal controller of which only the controller parameters are different. It is possible to compute the expected value of the cost function for the optimal gains, as well as any other set of stable gains. Finally, the LQG controller is also used to control perturbed non-linear  systems. Mathematical description of the problem and solution. Continuous time. Consider the continuous-time linear dynamic system where formula_3 represents the vector of state variables of the system, formula_4 the vector of control inputs and formula_5 the vector of measured outputs available for feedback. Both additive white Gaussian system noise formula_6 and additive white Gaussian measurement noise formula_7 affect the system. Given this system the objective is to find the control input history formula_8 which at every time formula_9 may depend linearly only on the past measurements formula_10 such that the following cost function is minimized: where formula_13 denotes the  expected value. The final time (horizon) formula_14 may be either finite or infinite. If the horizon tends to infinity the first term formula_15 of the cost function becomes negligible and irrelevant to the problem. Also to keep the costs finite the cost function has to be taken to be formula_16. The LQG controller that solves the LQG control problem is specified by the following equations: The matrix formula_19 is called the Kalman gain of the associated Kalman filter represented by the first equation. At each time formula_9 this filter generates estimates formula_21 of the state formula_22 using the past measurements  and inputs. The Kalman gain formula_19 is computed from the matrices formula_24, the two intensity matrices formula_25 associated to the white Gaussian noises formula_6 and formula_7 and finally formula_28. These five matrices determine the Kalman gain through the following associated matrix Riccati differential equation: Given the solution formula_31 the Kalman gain equals The matrix formula_33 is called the feedback gain matrix. This matrix is determined by the matrices formula_34 and formula_35 through the following associated matrix Riccati differential equation: Given the solution formula_38 the feedback gain equals Observe the similarity of the two matrix Riccati differential equations, the first one  running forward in time, the second one running backward in time. This similarity is called duality. The first matrix Riccati differential equation solves the linear–quadratic estimation problem (LQE). The second matrix Riccati differential equation solves the linear–quadratic regulator problem (LQR). These problems are dual and together they solve the linear–quadratic–Gaussian control problem (LQG). So the LQG problem separates into the LQE and LQR problem that can be solved independently. Therefore, the LQG problem is called separable. When formula_40 and the noise intensity matrices formula_41, formula_42 do not depend on formula_9 and when formula_14 tends to infinity  the LQG controller becomes a time-invariant dynamic system. In that case the second matrix Riccati differential equation may be replaced by the associated algebraic Riccati equation. Discrete time. Since the discrete-time LQG control problem is similar to the one in continuous-time, the description below focuses on the mathematical equations. The discrete-time linear system equations are Here formula_47 represents the discrete time index and formula_48 represent discrete-time Gaussian white noise processes with covariance matrices formula_49, respectively, and are independent of each other. The quadratic cost function to be minimized is The discrete-time LQG controller is  and formula_54 corresponds to the predictive estimate formula_55. The Kalman gain equals where formula_57 is determined by the following matrix Riccati difference equation that runs forward in time: The feedback gain matrix equals where formula_60 is determined by the following matrix Riccati difference equation that runs backward in time: If all the matrices in the problem formulation are time-invariant and if the horizon formula_62 tends to infinity the discrete-time LQG controller becomes time-invariant. In that case the matrix Riccati difference equations may be replaced by their associated discrete-time algebraic Riccati equations. These determine the time-invariant  linear–quadratic estimator and the time-invariant linear–quadratic regulator in discrete-time. To keep the costs finite instead of formula_63 one has to consider formula_64 in this case.