A negative imaginary lemma and state-feedback negative imaginary synthesis for commensurate fractional-order systems

Abstract

This article focuses on negative imaginariness of commensurate fractional-order linear time-invariant systems. Enlightened by the generalized Kalman–Yakubovic–Popov lemma, minimal state-space realization based necessary and sufficient conditions are developed to characterize negative imaginariness of commensurate fractional-order systems for both 0<α<1 and 1<α<2 cases. Meanwhile, the problem of negative-imaginary state-feedback controller synthesis for fractional-order systems is addressed. A fractional-order viscoelastic system, a fractional-order RLC circuit network and a numerical example are used as illustrative examples to demonstrate the main developed theory.