Towards Controller Synthesis for Systems with Negative Imaginary Frequency Response

• Published in: IEEE transactions on automatic control Vol. 55; no. 6; pp. 1506 - 1511
• Main Authors: Zhuoyue Song, Lanzon, Alexander, Patra, Sourav, Petersen, Ian R
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.06.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Actuators; Applied sciences; Bounded real; Computer science; control theory; systems; Control system synthesis; Control systems; Control theory. Systems; Exact sciences and technology; Feasibility; Force control; Force sensors; Frequency response; Fundamental areas of phenomenology (including applications); Gain; Image sensors; Invariants; Lighting control; lightly damped systems; Mechanical sensors; negative imaginary frequency response; Optimal control; Origins; passivity; Physics; positive real; Sensor systems; small-gain theorem; Solid mechanics; Structural and continuum mechanics; Studies; Synthesis; Transfer functions; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); {\cal H}_{\infty} control; Invariant; positive real; Force measurement; lightly damped systems; small-gain theorem; Spacecraft; H infinite control; Control synthesis; H; Real positive function; control; Passivity; Closed feedback; Transducer position; Optimal control; Small gain theorem; Bounded real; Position sensor; Position measurement; Force transducer; Vibration control; Feasibility; Frequency response; negative imaginary frequency response
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2010.2045700

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 H ∞ 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 H ∞ 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.