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Input estimation is a signal processing technique associated withdeconvolution of measured signals after filtering through a known dynamicsystem. Kitanidis and others extended this to the simultaneous estimation ofthe input signal and the state of the intervening system. This is normallyposed as a special least-squares estimation problem with unbiasedness. Theapproach has application in signal analysis and in control. Despite theconnection to optimal estimation, the standard algorithms are not necessarilystable, leading to a number of recent papers which present sufficientconditions for stability. In this paper we complete these stability results intwo ways in the time-invariant case: for the square case, where the number ofmeasurements equals the number of unknown inputs, we establish exactly thelocation of the algorithm poles; for the non-square case, we show that the bestsufficient conditions are also necessary. We then draw on our previous resultsinterpreting these algorithms, when stable, as singular Kalman filters toadvocate a direct, guaranteed stable implementation via Kalman filtering. Thishas the advantage of clarity and flexibility in addition to stability. Enroute, we decipher the existing algorithms in terms of system inversion andsuccessive singular filtering. The stability results are extended to thetime-varying case directly to recover the earlier sufficient conditions forstability via the Riccati difference equation.

Simultaneous input & state estimation, singular filtering and stability