Convergence analysis of estimation algorithms for dual-rate stochastic systems

Identification is to estimate the unknown parameters of systems by using the measured input–output data {u(t), y(t)}. Most existing identification approaches assume that the input–output {u(t), y(t)} is available at each sampling instant t. This paper focuses on a class of dual-rate sampled-data systems in which all inputs u(t) are available, but only scarce outputs {y(qt)} are available (q > 1 being an integer). We derive a mathematical model for such dual-rate systems by using a polynomial transformation technique, and present new algorithms for parameter identification and intersample output estimation using directly the dual-rate input–output data {u(t), y(qt)}, and study in detail convergence properties of the algorithms in the stochastic framework by using the stochastic process theory and stochastic martingale theory. We show that (1) the parameter estimation error consistently converges to zero under the persistent excitation condition; (2) the intersample output estimation error is uniformly bounded. Finally, we illustrate and test the proposed algorithms with example systems, including an experimental water-level system.