Stability of Control Systems with Multiple Sector-Bounded Nonlinearities for Inputs Having Bounded Magnitude and Bounded Slope

Abstract

This paper considers the input-output stability of a control system that is composed of a linear time-invariant multivariable system interconnecting with multiple decoupled time-invariant memoryless nonlinearities. The objectives of the paper are twofold. First and foremost, we prove (under certain assumptions) that if the multivariable Popov criterion is satisfied, then the system outputs and the nonlinearity inputs are bounded for any exogeneous input having bounded magnitude and bounded slope, and for all the nonlinearities lying in given sector bounds. As a consequence of using the convolution algebra, the obtained result is valid for rational and nonrational transfer functions. Second, for the case in which the transfer functions associated with the Popov criterion are rational functions, we develop a useful inequality for stabilizing the system by numerical methods. This is achieved by means of the positive real lemma and known results on linear matrix inequalities. To illustrate the usefulness of the inequality, a numerical example is provided.