Control of time-delayed double integrator systems

The control of system with large time-delays is a very challenging problem. As Proportional-Integral-Derivative (PID) controllers and variants of these such as P, PI, or PD are widely used in industry, there has been extensive work to determine the range of their gains that will stabilize a linear time-invariant plant described by a rational transfer function. However, the extension of this work to systems with time-delays has been difficult. In this paper, the control of double integrators, a particular class of secondorder systems, with time-delays will be considered. Double integrator systems are a simple, but important, class of second order systems, as they model single-degree of freedom translational and rotational systems. The study of time-delayed double integrator systems in this paper is motivated by the formation flying challenge. In this paper, the ability for tight formation control in the presence of communication and measurement delays is investigated. The paper begins with a discussion of the cross-link requirements for formation control. A simple two degreeof-freedom model with communication delays is introduced for analysis. A typical approach for systems with small time-delays is to base the design on a nominal system model that does not contain the time-delays. The limitations of this approach on stability and closed-loop bandwidth are discussed. The time-delay is explicitly considered in the design of state-feedback (PD) controllers. The effects of the time-delay on stability, stability margins, and closed-loop bandwidth are investigated.