In control theory, optimal projection equations[1][2][3] constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller.
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.
Therefore, implementing the LQG controller may be problematic if the dimension of the system state is large.
This problem is more difficult to solve because it is no longer separable.
Despite these facts numerical algorithms are available [4][6][7][8] to solve the associated optimal projection equations.
represent the state of the reduced-order LQG controller.
of the LQG controller is a-priori fixed to be smaller than
The reduced-order LQG controller is represented by the following equations: These equations are deliberately stated in a format that equals that of the conventional full-order LQG controller.
For the reduced-order LQG control problem it is convenient to rewrite them as where The matrices
of the reduced-order LQG controller are determined by the so-called optimal projection equations (OPE).
The OPE constitute four matrix differential equations.
This reveals why the reduced-order LQG problem is not separable.
is determined from two additional matrix differential equations which involve rank conditions.
Together with the previous two matrix differential equations these are the OPE.
To state the additional two matrix differential equations it is convenient to introduce the following two matrices: Then the two additional matrix differential equations that complete the OPE are as follows: with Here * denotes the group generalized inverse or Drazin inverse that is unique and given by where + denotes the Moore–Penrose pseudoinverse.
Then they constitute a solution of the OPE that determines the reduced-order LQG controller matrices
are two matrices with the following properties: They can be obtained from a projective factorization of
[4] The OPE can be stated in many different ways that are all equivalent.
To identify the equivalent representations the following identities are especially useful: Using these identities one may for instance rewrite the first two of the optimal projection equations as follows: This representation is both relatively simple and suitable for numerical computations.
If all the matrices in the reduced-order LQG problem formulation are time-invariant and if the horizon
tends to infinity, the optimal reduced-order LQG controller becomes time-invariant and so do the OPE.
[1] In that case the derivatives on the left hand side of the OPE are zero.
Similar to the continuous-time case, in the discrete-time case the difference with the conventional discrete-time full-order LQG problem is the a-priori fixed reduced-order
As in continuous-time, to state the discrete-time OPE it is convenient to introduce the following two matrices: Then the discrete-time OPE is The oblique projection matrix is given by The nonnegative symmetric matrices
that solve the discrete-time OPE determine the reduced-order LQG controller matrices
are two matrices with the following properties: They can be obtained from a projective factorization of
[4] To identify equivalent representations of the discrete-time OPE the following identities are especially useful: As in the continuous-time case if all the matrices in the problem formulation are time-invariant and if the horizon
Then the discrete-time OPE converge to a steady state solution that determines the time-invariant reduced-order LQG controller.
[6] Such systems arise in the case of digital controller design if the sampling occurs asynchronously.