Optimal projection equations

In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller.

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, incomplete state information (i.e. not all the state variables are measured and available for feedback) also disturbed by additive white Gaussian noise and 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 perturbation control of non-linear 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.

The reduced-order LQG control problem is almost identical to the conventional full-order LQG control problem. Let ${\displaystyle {\hat {\mathbf {x} }}_{r}(t)}$ represent the state of the reduced-order LQG controller. Then the only difference is that the state dimension ${\displaystyle n_{r}=dim({\hat {\mathbf {x} }}_{r}(t))}$ of the LQG controller is a-priori fixed to be smaller than ${\displaystyle n=dim({\mathbf {x} }(t))}$, the state dimension of the controlled system.

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