Applying negative imaginary systems theory to non-square systems with polytopic uncertainty

Abstract

This paper aims to apply Negative Imaginary (NI) systems theory to non-square LTI systems, for example, flexible structure systems with redundant sensors and actuators. The paper proposes static pre- and post-compensation schemes to transform stable fat (i.e. no. of inputs more than the no. of outputs) and tall (i.e. no. of outputs more than the no. of inputs) LTI plants into the class of Strongly Strict Negative Imaginary (SSNI) systems, a subset of the Strictly Negative Imaginary (SNI) systems. The proposed compensators can also stabilise a non-square or a non-NI square plant in a positive feedback loop. Moreover, the compensators can be utilised to design a simple constant reference tracking scheme exploiting the integral controllability propoerty of SSNI systems. Subsequently, observer-based pre- and post-compensators are developed to handle the cases when some of the system states are not available for direct measurement. Then, the compensation schemes are extended to stable, non-square or non-NI square systems with polytopic uncertainty. Finally, the post-compensation technique is applied to stabilise a class of tall/square uncertain LTI plants preceded by a slope-restricted nonlinearity.