Towards Controller Synthesis for Systems with Negative Imaginary Frequency Response

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This technical note provides a reformulation of closed-loop systems that have negative imaginary frequency response into closed-loop systems that have bounded gain, so that theory and results from <formula formulatype="inline"> <tex Notation="TeX">${\cal H}_{\infty}$</tex></formula> control can be borrowed to enable controller synthesis for the former class of systems. Systems with negative imaginary frequency response (negative-imaginary systems for short) arise for example in structures with collocated position sensors and force actuators, and finding a systematic controller synthesis treatment for such systems has important applications in, for example, lightly damped large space structure problems. The key result in this technical note assists by reformulating such systems into a bounded-real framework. This technical note also addresses a controller synthesis problem in an <formula formulatype="inline"><tex Notation="TeX">${\cal H}_{\infty}$</tex></formula> optimal control framework for a generalized plant with an invariant zero at the origin in its (1,2) element which is due to the reformulation of the closed-loop system from a negative-imaginary system to a bounded-real framework. An example demonstrates the feasibility of the reformulation given herein. </para>