On the convergence of Lion's identification method with random inputs

An interesting identification scheme for scalar-input-scalar-output linear systems, proposed by Lion in [1], is investigated for a wide class of random inputs. The inputs include the class of functions
u(t) = \Sigma k_{i}\bar{u}_{i}(t)
, where
\{\bar{u}_{1}(t),..,\bar{u}_{k}(t)\}
is a Markov process which is asymptotically stationary. An invariant set theorem for random systems [2] is used to prove convergence (in probability) of the identification algorithm proposed in [1]. Indeed, in view of the power of the deterministic invariant set theorem, it is of great interest to study methods of useful application of the stochastic analogy. One of the main contributions is the illustration of its potential power, via the vehicle of the identification problem.