Impossibility Results for Constrained Control of Stochastic Systems

Strictly unstable linear systems under additive and nonvanishing stochastic noise with unbounded supports are known to be impossible to stabilize by using deterministically constrained control inputs. In this article, similar impossibility results are obtained for the scenarios where the control input is probabilistically constrained and the support of the noise distribution is not necessarily unbounded. In particular, control policies that have bounded time-averaged second moments are considered. It is shown that for such control policies, there are critical average moment bounds, below which second moment stabilization of a linear stochastic system is not possible, and moreover, second moment of the state diverges regardless of the choice of control policy and the initial state distribution. Nonnegative-definite Hermitian matrices are exploited to extract sufficient instability conditions that can be assessed by using the eigenstructure of the system matrix and the distribution of the noise. The results indicate that in certain networked control system settings with noise, designing stabilizing constrained controllers is an impossible task, if the probability of successful transmissions of control commands over the network is known to be too small in average.