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This study investigates the observability problem and the observer design of partially observed finite automata via a matrix approach. Using semi-tensor product of matrices, finite automata are modelled in the form of discrete-time bilinear systems. Matrix-form necessary and sufficient conditions for both the initial and current state observability, either with or without input information, are first proposed. Based on that, a constructive method for the observer design is provided.

Polarization aberration (PA) is a serious issue that affects imaging quality for optical systems with high numerical aperture. Numerous studies have focused on the distribution rule of PA on the pupil, but the field remains poorly studied. We previously developed an orthonormal set of polynomials to reveal the pupil and field dependences of PA in rotationally symmetric optical systems. However, factors, such as intrinsic birefringence of cubic crystalline material in deep ultraviolet optics and tolerance, break the rotational symmetry of PA. In this paper, we extend the polynomials from rotationally symmetric to M-fold to describe the PA of M-fold optical systems. Two examples are presented to verify the polynomials.

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