Controller for rejection of the first harmonic in a periodic signal with uncertain delay

TL;DR

This work addresses canceling only the first harmonic of a periodic disturbance with an unknown dead time between plant and controller. It designs a harmonics-based, linear time-invariant controller built from a bank of coupled harmonic oscillators (modified system of harmonic oscillators) using the internal model principle, ensuring rejection of the fundamental sine component while preserving all higher harmonics. Stability analysis with unknown delays reveals delay-dependent regions, including delay-induced stability, and shows that in the infinite-harmonic limit the stability condition becomes $\alpha>0$ and $\beta/\omega>-1/4$, with practical parameter choices providing robust operation up to a fraction of the period. The controller is motivated by rotational AFM tilt compensation and demonstrated via simulations, with extensions proposed to cancel additional harmonics, handle unknown frequencies, or integrate Smith predictors for longer delays, enabling larger scanning areas in high-speed AFM applications.

Abstract

The plant (the system to be controlled) is disturbed by a periodic external force with a broad spectrum of Fourier harmonics. The first Fourier harmonic (sine-type signal) is assumed to be undesirable and should be removed by a control force, whereas the other harmonics should be preserved without distortion. Because the measured plant data have an unknown amount of time delay (dead time) and the sensitivity of the plant to external force is unknown, therefore the amplitude and phase of a sine-type control force are also unknown. Based on the internal model principle, we developed a feedback controller described as a linear time-invariant system that can remove the first harmonic from the plant's output by constantly adjusting its control force parameters. Using this controller, we aimed to further extend the capabilities of a newly developed high-speed, large-area rotational scanning atomic force microscopy. In such a scanning technique, the sample is rotated, and the tilt angle between the normal of the sample surface and the axis of rotation produces a parasitic first Fourier harmonic, which significantly limits the scanning area.

Figures (10)

• Figure 1: Schematic illustration of the tilt problem occurring in the rotational scanning.
• Figure 2: The block scheme of the plant $P(s)=P_0(s)\mathrm{e}^{-\tau s}$ disturbed by the external periodic force $f(\omega t)$ and controlled by the feedback controller $C(s)$.
• Figure 3: The Nyquist plots of the open-loop system (\ref{['open_trf']}) for 8 topologically different cases when the frequency $\Omega$ increases from 0 to $+\infty$. The part of the Nyquist plot for $\Omega=-\infty$ to $\Omega=0$ can be obtained by reflecting the drawn part with respect to the $x$-axis. Each row corresponds to $\bar{\alpha}_1<0$ or $\bar{\alpha}_1>0$ while each column corresponds to four different cases, namely, $(K/\gamma)\bar{\beta}_1<-1$, $-1<(K/\gamma)\bar{\beta}_1<0$, $0<(K/\gamma)\bar{\beta}_1<1$ and $(K/\gamma)\bar{\beta}_1>1$. Asymptotes of the hyperbola are drawn by dashed grey lines. The red closed circle represents the point $(-1,0)$, while the black line depicts the unit circle. The critical frequencies $\Omega_{1}$ and $\Omega_{2}$ when the Nyquist curve goes outside and inside into the unit circle, $|G_0(\mathrm{i}\Omega_{1,2})|=1$, depicted by an open and closed circle, respectively.
• Figure 4: The closed-loop systems (\ref{['close_trf']}) stability region calculated numerically from Eqs. (\ref{['stab_bord1']}), (\ref{['stab_bord2']}), (\ref{['stab_bord3']}) with different values of the ratio $\tau/T$. The parameter $K/\gamma$ is set to $K/\gamma=1$.
• Figure 5: The results of the numerical simulation of closed-loop system (\ref{['main']}), (\ref{['one_harm']}) and (\ref{['out']}). Before the time moment $t=40$ the system was in a control-free regime ($K=0$). (a) -- the dynamics of the plant variable, (b) -- the same for the first controller variable $a_1(t)$. The parameters are following: $K/\gamma=1$, $\gamma=200$, $f_1=1$, $\omega=1$, $\alpha_1=-1$, $\beta_1=-0.5$ and the time delay $\tau/T=0.15$.