Rejection of sinusoids from nonlinearly perturbed uncertain regular linear systems

An application-motivated system class - nonlinearly perturbed regular linear systems (NPRLS) - is considered, and the response of the latter to periodic inputs is characterized. A recently proposed robust control scheme for tracking bandlimited periodic signals by uncertain exponentially stable regular linear systems (RLS) is then applied to an uncertain system belonging to the NPRLS class, whose linearization is an exponentially stable RLS, to reject internally generated sinusoids from the system output. Assuming the uncertain NPRLS to be unknown, but its gain at the frequency of interest known and bounded away from zero, and using the aforementioned characterization, the stability and disturbance rejection of the resulting topology are shown to be guaranteed for sufficiently small nonlinearity.

I. introduction

Internal model principle [1] has been used to track and reject periodic signals in finite dimensional linear [2]–[6] and nonlinear systems [7]–[9] when the plant model is known. When the plant model is unknown, this principle has been applied using only the transfer function (TF) gains at the frequencies of interest, to track bandlimited periodic signals in case of stable linear finite [10], linear infinite [11]–[13] and nonlinear finite [14] dimensional systems. The control schemes in [13] and [14] are similar and are motivated by a motion distortion problem in steel casting mold oscillation systems that include a servo-beam structure. This application can be characterized as a sinusoidally excited linear infinite dimensional structure with a small finite dimensional nonlinear perturbation causing the generation of unwanted harmonics which excite a structural resonance. The control objective is to eliminate the harmonic at the resonance frequency from the output. In the absence of the nonlinear perturbation, [13] addressed the tracking of bandlimited periodic signals by exponentially stable regular linear systems (RLS), a class which encompasses a large number of infinite-dimensional physical models. In the presence of nonlinear perturbation [14] addresses rejection of unwanted harmonics only in a stable finite dimensional linear system.