Stability Robustness of a Feedback Interconnection of Systems With Negative Imaginary Frequency Response

A necessary and sufficient condition, expressed simply as the dc loop gain (i.e., the loop gain at zero frequency) being less than unity, is given in this note to guarantee the internal stability of a feedback interconnection of linear time-invariant (LTI) multiple-input multiple-output systems with negative imaginary frequency response. Systems with negative imaginary frequency response arise, for example, when considering transfer functions from force actuators to colocated position sensors, and are commonly important in, for example, lightly damped structures. The key result presented here has similar application to the small-gain theorem, which refers to the stability of feedback interconnections of contractive gain systems, and the passivity theorem, which refers to the stability of feedback interconnections of positive real (or passive) systems. A complete state-space characterization of systems with negative imaginary frequency response is also given in this note and also an example that demonstrates the application of the key result is provided.