Synthesis of Strictly Negative Imaginary Controllers Using a H∞ Performance Index

Negative imaginary (NI) systems are those characterized by a negative imaginary frequency response. A NI system connected in a positive feedback interconnection with a strictly negative imaginary (SNI) controller is internally stable if and only if a DC gain condition is satisfied. This can be interpreted as a robust stability result in situations where plant uncertainty does not destroy the NI nature of the plant nor the DC gain condition. Motivated by a desire to realize improved closed-loop performance, this paper considers the design of H
_∞
-optimal SNI controllers. The proposed synthesis method makes use of convex optimization and linear matrix inequality (LMI) tools. Another contribution of this paper is pointing out how to realize tip position control, rather than joint position control, of a flexible manipulator within a NI framework. Numerical simulation results are included.