Control of negative imaginary systems with rate-limited actuation using a dissipative framework

Rate limitation on the control input is ubiquitous in numerous mechanical actuators and demands special care in control systems design. Since several mechanical systems are known to satisfy negative imaginary (NI) system properties, this paper addresses the problem of finding suitable strictly negative imaginary (SNI) controllers for robustly stabilizing such linear time-invariant (LTI) NI plants, with no pole at origin, constrained by rate-limited actuation. The proposed design method relies on a dissipative characterization of NI systems and on a particular structure of a stabilizing SNI controller to effectively address the input rate constraint. It is shown that the stabilizing controller can be represented as an interconnection of an integrator preceded by strictly positive real (SPR) dynamics in the forward path and a positive definite gain matrix in negative feedback. This parameterization facilitates synthesis of SNI controllers guaranteeing rate-bounded control. In addition to satisfying the DC gain condition for closed-loop stability, a suitable design of the positive definite matrix, a part of the controller, ensures the magnitude shaping of the control input and impacts the closed-loop control performance. Moreover, by employing the existing technique of dynamical forward action to make a system NI, the proposed framework can be extended to design robust controllers for a larger family of non-NI systems with rate limits as well. The results are demonstrated through numerical examples.