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This paper deals with the design of quadratic and higher order normal forms for the left invertibility problem. The linearly observable case and one-dimensional linearly unobservable case are investigated. The interest of such a study in the design of a delayed discrete-time observer is examined. The example of the Burgers map with unknown input is treated and a delayed discrete-time observer is designed. Finally, some simulated results are commented.

This note deals with quadratic observability normal form for nonlinear discrete-time single-input-single-output (SISO) system. First of all, the main concept of quadratic equivalence with respect to the observability property, is introduced for discrete-time systems. Subsequently, normal form structure for discrete time system is developed for system with unobservable linear approximation in one direction. Finally, the effect of the so-called resonant terms on the observer design and synchronization of chaotic systems is pointed out in an illustrative example.Index Terms-Chaotic systems synchronization, observability normal form, quadratic equivalence, reduced observer design.

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