On the design of discrete-time optimal dynamic controllers using the partial state observer

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Abstract
This paper considers the problem of designing optimal dynamic controllers, based on a partial state observer, for discrete-time linear time-invariant systems with inaccessible state. It is assumed that the initial system state is an unknown random vector with known mean and covariance. The performance index is taken to be the expectation, with respect to the initial state, of the standard quadratic one for the discrete-time regulator. Necessary and sufficient conditions for optimality are derived, provided that the subspace which ensures the existence of a partial state observer, is given. A design algorithm is also obtained which can be directly solved for the parameters of the optimal observer and requires the solution of a matrix Riccati equation of the observer order. As in the continuous-time case, the separation property in the restricted sense is seen to hold. Finally an example is presented to illustrate the procedure.