Observer design for stochastic-parameter systems

Abstract

ENGIN YAZtConstant-gain linear observer design is utilized to reconstruct the state vector of stochastic systems with both multiplicative and additive noises. It is shown that if the system model is mean square stable, then it is very easy to design an effective estimation scheme. I. IntroductionDiscrete-time systems with observation equations containing multiplicative as well as additive noise have been important to communication engineers because such modelling corresponds to physical phenomena that may involve reflection of a transmitted signal at an ionospheric layer, amplitude modulation or random sampling of some kind. Optimal linear predictor design is considered by Nahi (1969) and Hadidi and Schwartz (1979) for scalar binary-valued multiplicative noise appearing in the measurement equation. A continuous-valued noise version of similar results was reported by Rajasekaran et al. (1971) and Tugnait (1981); the former paper derives the predictor, the latter proves uniform asymptotic stability in the large. Recently, De Koning (1984) assumed a more general model involving matrices of stochastic parameters and derived the optimal one-step predictor. He showed that mean square stability of the system is sufficient for the existence, uniqueness and stability of the time-invariant estimator.In this work we shall assume the more general stochastic model considered in De Koning (1984), but our approach is not from the optimization point of view; instead, it is shown that any constant-gain linear observer, whether optimal in some sense or not, will result in a stable observation scheme if the gain used does not destabilize the open-loop stable system.