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.
[1] 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).
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.
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 function.
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.
This problem is more difficult to solve because it is no longer separable.
Despite these facts numerical algorithms are available[4][5][6][7] to solve the associated optimal projection equations[8][9] which constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller.
[4] LQG optimality does not automatically ensure good robustness properties.
[10] [11] 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.
represents the vector of state variables of the system,
Both additive white Gaussian system noise
and additive white Gaussian measurement noise
Given this system the objective is to find the control input history
of the cost function becomes negligible and irrelevant to the problem.
These five matrices determine the Kalman gain through the following associated matrix Riccati differential equation: Given the solution
through the following associated matrix Riccati differential equation: Given the solution
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.
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).
tends to infinity the LQG controller becomes a time-invariant dynamic system.
Since the discrete-time LQG control problem is similar to the one in continuous-time, the description below focuses on the mathematical equations.
represent discrete-time Gaussian white noise processes with covariance matrices
The quadratic cost function to be minimized is The discrete-time LQG controller is and
is determined by the following matrix Riccati difference equation that runs forward in time: The feedback gain matrix equals where
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
tends to infinity the discrete-time LQG controller becomes time-invariant.