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Observability and forward-backward observability of discrete-time nonlinear systems. (English) Zbl 1025.93007
The observability problem for single-input single-output discrete-time systems is considered. For the control system the observability property means distinguishability of any pair of states by suitable input and corresponding output sequences. The authors use differential geometric concepts of codistributions in the discrete time setting for the study of the observability property. The considered codistributions are generated by the set of functions obtained by iterative superposition of the right hand side of the system and its output. The functions derived in such a way may be used for state determination. The observability criteria are obtained in terms of dimensionality of appropriate codistributions. If the map generated by the system is a diffeomorphism for each input, then it is possible to obtain a motion in reverse time. For such invertible systems stronger observability criteria are derived. The alternate motion along straight and inverse trajectories generates a new class of forward-backward systems. The authors show that observability implies the forward-backward observability, but generally the converse is false. It is proven that under some regularity assumption the weaker notion of forward-backward observability is equivalent to the usual ones.
MSC:
93B07Observability
93C55Discrete-time control systems
93B29Differential-geometric methods in systems theory (MSC2000)
93C10Nonlinear control systems