In control theory, the linear–quadratic–Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, having incomplete state information (i.e. not all the state variables are measured and available for feedback) and undergoing control subject to quadratic costs. Moreover, the solution is unique and constitutes a linear dynamic feedback control law that is easily computed and implemented. Finally the LQG controller is also fundamental to the optimal control of perturbed non-linear systems.
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The LQG controller is simply the combination of a Kalman filter, i.e. a linear–quadratic estimator (LQE), with a linear–quadratic regulator (LQR). The separation principle guarantees that these can be designed and computed independently. LQG control applies to both linear time-invariant systems as well as linear time-varying systems. The application to linear time-invariant systems is well known. The application to linear time-varying systems enables the design of linear feedback controllers for non-linear uncertain systems.
The LQG controller itself is a dynamic system like the system it controls. Both systems have the same state dimension. Therefore, implementing the LQG controller may be problematic if 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.
Continuous time
Consider the continuous-time linear dynamic system
where
where
The LQG controller that solves the LQG control problem is specified by the following equations:
The matrix
Given the solution
The matrix
Given the solution
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
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
The quadratic cost function to be minimized is
The discrete-time LQG controller is
The Kalman gain equals
where
The feedback gain matrix equals
where
If all the matrices in the problem formulation are time-invariant and if the horizon