Specific optimal control of the linear regulator using a dynamical controller based on the minimal-order Luenberger observer†

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Abstract
The problem is considered of designing the optimal, fixed configuration, dynamical feedback controller of order (n—m) required to control an nth-order linear system with output of order m(m the cost functional for the regulator being of the usual integral quadratic form. It is supposed that only the means and covariances of the system state initial conditions are known. The form of the dynamical controller is assumed to be that of the minimal-order Luenberger observer. Two problems are considered. First, a design algorithm is derived for obtaining the optimal dynamical controller (observer) on the assumption that its output will be fed into the regulator system through the feedback gain matrix normally used for the optimal state regulator. Secondly, the algorithm is extended to the case when the feedback gain matrix is treated as one of the design parameters of the dynamical controller. Although the problem is basically one of parameter optimization it can be considered from the point of view of designing the minimal-order observer required for use with the state regulator when some of the states are unavailable. However, it should be noted that the design parameters are chosen to minimize the expectation of the regulator cost functional and not the state estimation error.