Signalling and Control in Nonlinear Partially Observable Stochastic Systems: An Information State Approach and LQG Applications

We formulate and analyze the operational definition, called control-coding capacity, C<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FB</inf> in bits/second, of discrete-time nonlinear partially observable stochastic systems. C<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FB</inf> is the maximum rate of encoding signals or messages into randomized controller-encoder strategies, and reproducing the messages at the output of the system using a decoder, with arbitrary small asymptotic error probability.In the first part of the paper, we characterize C<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FB</inf> by an information theoretic optimization problem over randomized strategies, using posterior distributions of nonlinear filtering theory and information states.In the second part of the paper, we derive the formula of C<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FB</inf>, for linear-quadratic Gaussian partially observable stochastic systems (LQG-POSSs). We express C<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FB</inf> in terms of solutions of two Riccati equations of Kalman-filtering and a control Riccati equation. To derive C<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FB</inf> we prove, (1) optimal randomized strategies consist of signalling, control, and estimation, and (2) a decentralized semi-separation principle holds between the control problem and the signalling and estimation problems.