Vector control systems with unpredictable actuation

Vector systems are fundamentally different from scalar systems since the dynamics of a vector system can both spin and stretch the system state. The magnitude and direction of control actions for multi-dimensional systems can need to be coordinated across time to achieve the desired behavior. Hence, planning for future actions is important. Consequently, system predictability, and the related ability to plan, can affect system stabilizability. This paper considers a simple d-dimensional state-space model for an unpredictable “spinning” control system. The spinning controller has a particular restriction: it may only apply a scalar control in a direction that is randomly chosen by nature at each time. This model is related to the classically studied models for intermittent Kalman filtering and intermittent control problems, but is simpler to analyze. The simplicity of the spinning model lets us use dynamic programming to characterize the decay rate of the system and thus understand the fundamental limitations imposed by the fact that future control directions are unpredictable. We can reduce the unpredictability of the model by providing non-causal side information about the future control directions, and further calculate how this affects the decay rate of the system. With k-steps of lookahead on the random control direction, the d-dimensional system decays as fast as a d-k dimensional system with no lookahead on the control action. This model is a first step to understand unpredictability in more general networked control systems.