Observer design for nonlinear systems with Markov chain

We present a novel observer design for a class of single-output nonlinear systems with Markov jumps. The Markov jump process interferes with a deterministic nonlinear dynamics at random times and retains its state for a certain amount of time (dwell time). The estimation process is reset at these random times, depending on the reset values of the state process, and then evolves as a deterministic estimate of the state process itself. The novelty is given by the reset mechanism adopted for the estimation process itself, depending on the reset values of the slate process. We prove that, as long as the mathematical expectation of the dwell times has a positive lower bound and the transition rate of the jump process at the first exit time out of any point is bounded, the state estimation error of the switching dynamics asymptotically converges to zero with probability one. The state estimate over each dwell time is designed using the novel technique of 'output immersion': each single-output nonlinear dynamics is 'immersed' into a system with as many outputs as its states. The immersed dynamics can be split into one-dimensional dynamics each one with its state and output. By estimating the state of each one-dimensional dynamics we determine a state estimator for the dynamics before immersion. Copyright (c) 2008 John Wiley & Sons, Ltd.