Bounded‐error state and parameter estimation of tip‐tilt disturbances in adaptive optics systems

Abstract

The main objective of this paper was to estimate tip-tilt disturbances in adaptive optics systems. In a bounded-error context, set inversion methods based on interval analysis are used to guarantee both state and parameter estimation of tip-tilt disturbances. Consequently, two methods are performed. The first method is based on contraction-bisection, and the second one is based only on contraction. Both methods are thus compared, and results are discussed according to computational time and pessimism introduced on each estimated parameter. Deformable mirror (DM): This component features a reflective surface that can be deformed in real time using an array of piezo-stack or voice-coil actuators. The control input is the vector of actuation signals (usually voltages). Wavefront sensor (WFS): This component measures the distorted wavefront. Several kinds of WFS exist. However, one of the most used in AO systems is the Shack-Hartmann sensor, because of its simple and robust technology [1]. Real-time computer: It estimates disturbance parameters, and accordingly, the DM is then deformed to correct the disturbance effect on the wavefront.Deformed wavefront High resolution camera Corrected wavefront Beam splitter Figure 1. Scheme of adaptive optics system.Hence, the WFS measures the residual wavefront or (in the case of so-called open-loop AO systems) directly the disturbance wavefront. WFS measurements are fed into the real-time computer, which computes the DM controls in order to minimize the residual distortion seen by the imaging camera.Tip-tilt disturbances, according to the Zernike representation [5], are the first two terms of the atmospheric disturbance modal expansion. As their name suggests, they are generated by simply tilting a flat wavefront along the x or y axis. Thus, tip tilt is induced not only by atmospheric turbulence but also by displacements of the optical beam relatively to the optical system. This includes displacements of the optical system itself generated by various sources, such as wind shake (wind-induced telescope movements) and mechanical vibrations of the telescope's structure and/or instruments. One can refer to [6-10] and the references therein for more details.Another particularity of the tip and tilt modes is that they can be compensated in a decoupled manner, using two dedicated SISO controllers. In order to decouple these specialized loops from the rest of the AO system, on the actuation side, the separation can be effected by constraining the control vector into the appropriate 2D subspace, by adding to the AO system a specialized flat tiptilt mirror, or by mounting the DM itself on a tip-tilt platform. On the sensing side, adequate tip-tilt measurements are provided by computing the so-called x and y mean slopes, that is, the averages of WFS's slopes in the x and y directions. In an ideal case (Shack-Hartmann perfectly linear), and by neglecting both noise and the effect of higher-order modes, the mean slopes correspond respectively to tip and tilt modes with nearly multiplicative ...

State Estimation Using Dependent Evidence Fusion: Application to Acoustic Resonance-Based Liquid Level Measurement

“…For nonlinear systems, the corresponding studies are relatively rare. Khemane et al and Jaulin proposed bounded-error state and parameter estimations for nonlinear systems [ 13 , 14 ]. Gning proposed a relatively simple and fast bounded-error method based on interval analysis and constraint propagation, which was successfully applied to dynamic vehicle localization [ 12 ], but when the noise bounds cannot be precisely determined, its robustness will unavoidably decline [ 7 ].…”