New Results on Robust Stability Analysis and Synthesis for MIMO Uncertain Systems

This paper provides theoretical preliminary results and develops a first methodology, that allows one to efficiently estimate the maximum time constant of a dynamic matrix of an uncertain system with rational multi-affine structure with respect to parameters, and a second one to find a majorant system of a multi-input multi-output (MIMO) system with rational multi-affine structure with respect to parameters. The developed methodologies are used to estimate the evolution of an uncertain LTI system with additional bounded nonlinearities and/or additional bounded input signals. Moreover, the above results are also used to design a robust controller for an uncertain MIMO system with unmeasurable states and subject to a rate-bounded disturbance in order to track a rate-bounded reference signal. The obtained theoretical results are illustrated by three examples. In the first example, some estimates of the maximum time constant of the dynamic matrix of an uncertain LTI system with additional bounded nonlinearities and additional bounded input signals and some majorant systems are found. In the second example some majorant systems are computed in order to estimate the output and the control signal of a control system in the presence of a not bounded additional nonlinearity and of a bounded disturbance; for this system a new control law with saturation is also designed. Finally, in the last example, a robust controller for an uncertain electro-mechanical system with unmeasurable state is designed to track a rate-bounded reference signal in the presence of a disturbance with bounded derivative.