An observer design method for stochastic and deterministic robustness is developed so that the observer is less sensitive to uncertainties in transient and in steady-state observer performance. The uncertainties include not only deterministic factors such as unknown initial estimation error, round-off error, modeling error and sensing bias, but also stochastic factors such as disturbance and sensor noise. From a stochastic perspective, a small value in estimation error variance represents robustness to the stochastic uncertainties. It is shown that the upper bound of the error variance can be minimized by reducing the observer gain and by increasing the decay rate of the observer. From a deterministic perspective, a small value in the L2 norm condition number of the observer eigenvector matrix guarantees robust estimation performance to the deterministic uncertainties. An optimization problem constrained by a linear matrix inequality condition is formulated for both the deterministic and the stochastic robustness. The observer gain can be selected as a trade-off solution and the estimation performance to stochastic and deterministic uncertainties is demonstrated on simulation examples.

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