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The orbit of a satellite that repeats in the Earth fixed coordinates is determined by combining GPS tracking data from multiple arcs. The satellite dynamics are modeled with the epoch state and a set of parameters, called the bin parameters, that account for the effect of the local gravitational field on the satellite current state. The epoch state is specific to each arc, and the bin parameters are common to all repeat arcs. The estimation algorithm is based on the Square Root Information Filter. It involves partitioning of the measurement matrix and use of the Householder transformation to combine multiple arc data and solve for the epoch states and the bin parameters. The bin parameters can then be converted into the Earth's gravitational field with a modest amount of computation.

f AZIMUTH 2 ] bsa2 BIAS I-----^AZIMUTH DRIVE 2 V c2 6pa2 -.XELERROR Fig. 2 Rate loop control system of the DSS-13 antenna.Space Network antenna model. 2 As a result, the 90-state model has been reduced to a 27-state model, which preserves the full-model properties. 30 35 5 10 15 20 25 COMPONENT NUMBERFig. 3 Singular values of the balanced antigrammians of the rate loop model. FREQUENCY, Hz Fig. 4 Magnitude of the elevation rate transfer function (rad/s) for the full and the reduced rate loop model. Consider the NASA Deep Space Network DSS-13 antenna rate loop model (see Fig. 2; for details of the model see Ref.2). It consists of the antenna structural model and models of elevation and azimuth drives. The 35-state rate loop model has three poles at zero and is reduced while using balanced antigrammians; their singular values are shown in Fig. 3. By deleting the states with the largest singular values, the model has shrunk from 90 states to 27 states. The reduced model preserves the full-model properties, as illustrated by the frequency-response plots in Fig. 4. ConclusionsAntigrammians, similarly to grammians, reflect controllability and observability properties of a system. Unlike grammians, antigrammians do exist for systems with integrators, hence making their reduction possible. In this Note the reduction algorithm for systems with integrators is derived. The algorithm is based on balancing the antigrammians and has the same computational effectiveness as the regular balancing procedure. It was applied to the reduction of the NASA Deep