Kalman meets Shannon

We consider the problem of communicating the state of a dynamical system via a Shannon Gaussian channel. The receiver, which acts as both a decoder and estimator, observes the noisy measurement of the channel output and makes an optimal estimate of the state of the dynamical system in the minimum mean square sense. The transmitter observes a possibly noisy measurement of the state of the dynamical system. These measurements are then used to encode the message to be transmitted over a noisy Gaussian channel, where a per sample power constraint is imposed on the transmitted message. Thus, we get a mixed problem of Shannon's source-channel coding problem and a sort of Kalman filtering problem. We first consider the problem of communication with full state measurements at the transmitter and show that optimal linear encoders don't need to have memory and the optimal linear decoders have an order of at most that of the state dimension. We also give explicitly the structure of the optimal linear filters. For the case where the transmitter has access to noisy measurements of the state, we derive a separation principle for the optimal communication scheme, where the transmitter needs a filter with an order of at most the dimension of the state of the dynamical system. The results are derived for first order linear dynamical systems, but may be extended to MIMO systems with arbitrary order

Analysis of Interconnected Systems

In this paper analysis of interconnected dynamical systems are considered. A solid framework for the analysis of the stability of interconnection is given. The results from Fax and Murray (2002) that studies the SISO-case for a constant interconnection matrix are generalized to the MIMO-case where arbitrary interconnection is allowed. The analysis show existence of a separation principle that is very useful in the sense of the simplicity for stability analysis. Stability could be checked graphically using a Nyquist-like criterion. The problem with time-delays and interconnection variation and robustness appear to be natural special cases of the general framework