The capacity of unstable dynamical systems-interaction of control and information transmission

Feedback capacity is extended beyond classical communication channels, to stochastic dynamical systems, which may correspond to unstable control systems or unstable communication channels, subject to average cost constraints of total power κ ∈ [0, ∞). It is shown that optimal conditional distributions or randomized strategies, have a dual role, to simultaneously control the output process and to encode information. The dual role is due to the interaction of control and information transmission; it states that encoders in communication channels operate as encoders-controllers, while controllers in control systems operate as controllers-encoders. The concepts are illustrated through the analysis of Gaussian control systems with randomized strategies, which are equivalent to Additive Gaussian Noise channels, Stable or Unstable, with arbitrary memory on past outputs, with an average constraint of quadratic form. It is shown that such unstable dynamical systems have Control-Coding Capacity which is operational, precisely as in Shannon's operational definition. However, the control-coding capacity is zero, unless the power κ allocated to the system, exceeds a threshold K
_min
, where K
_min
is the minimum cost of ensuring asymptotic stability and ergodicity. The excess power κ - K
_min
is turned into an achievable rate of information transmission over the dynamical system.