Simultaneous input and state smoothing for linear discrete-time stochastic systems with unknown inputs

This paper considers the problem of simultaneously estimating the states and unknown inputs of linear discrete-time systems in the presence of additive Gaussian noise based on observations from the entire time interval. A fixed-interval input and state smoothing algorithm is proposed for this problem and the input and state estimates are shown to be unbiased and to achieve minimum mean squared error and maximum likelihood. A numerical example is included to demonstrate the performance of the smoother.