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A derivation of an output feedback solution to the statistical control problem which minimizes a finite linear combination of first k cost cumulants of the finite-horizon integral quadratic cost associated with a linear stochastic system, when the controller measures the noisy states is first presented. Of course for the k = 1 case, there has been the celebrated Kalman result. Also, there are already some results for infinite-horizon output-feedback risk sensitive control which minimizes a certain linear combination of denumerable cost cumulants. The contribution of this paper is to fill in the other values of k ≥ 2 here, in hoping to complete the picture of linear k-cost-cumulant (kCC) control theory. The phase I benchmark problem for response control of smart base-isolated buildings then follows, in which the realistic 8-story, steel braced building representing mid rise buildings in the city of Los Angeles, California is protected by a linear elastometric isolation system under seven different earthquakes with fault normal and parallel components acting in two perpendicular directions. A group of nine non-dimensionalized evaluation criteria is used to assess the performance of both base isolation system and benchmark structure. Simulation results indicate that the active control of the augmenting devices at the isolation layer using output feedback statistical control paradigm offers broad improvement in structural performance over the baseline LQG design. Therefore, statistical control is well applicable to the protection of civil structures.