A Probabilistic Algorithm to Test Local Algebraic Observability in Polynomial Time

The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input–output behaviour of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of thelocal algebraic observability problem which is concerned with the existence of a non-trivial Lie subalgebra of model’s symmetries letting the inputs and the outputsbe invariant. We present a probabilistic seminumerical algorithm that proposes a solution to this problem in polynomial time. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the complexity of evaluation of the model and in the number of variables. Furthermore, we show that the size of the integers involved in the computations is polynomial in the number of variables and in the degree of the system. Last, we estimate the probability of success of our algorithm.