This paper formulates the attitude filtering problem as a nonlinear stochastic filter problem evolved directly on the Special Orthogonal Group 3 (SO(3)). One of the traditional potential functions for nonlinear deterministic complimentary filters is studied and examined against angular velocity measurements corrupted with noise. This paper demonstrates that the careful selection of the attitude potential function allows to attenuate the noise associated with the angular velocity measurements and results into superior convergence properties of estimator and correction factor. The problem is formulated as a stochastic problem through mapping SO(3) to Rodriguez vector parameterization. Two nonlinear stochastic complimentary filters are developed on SO(3). The first stochastic filter is driven in the sense of Ito and the second one considers Stratonovich. The two proposed filters guarantee that errors in the Rodriguez vector and estimates are semi-globally uniformly ultimately bounded in mean square. Simulation results are presented to illustrate the effectiveness of the proposed filters considering high level of uncertainties in angular velocity as well as body-frame vector measurements.

Two novel nonlinear pose (i.e., attitude and position) filters developed directly on the Special Euclidean Group SE(3) able to guarantee prescribed characteristics of transient and steady-state performance are proposed. The position error and normalized Euclidean distance of attitude error are trapped to arbitrarily start within a given large set and converge systematically and asymptotically to the origin from almost any initial condition. The transient error is guaranteed not to exceed a prescribed value while the steady-state error is bounded by a predefined small value. The first pose filter operates based on a set of vectorial measurements coupled with a group of velocity vectors and requires preliminary pose reconstruction. The second filter, on the contrary, is able to perform its function using a set of vectorial measurements and a group of velocity vectors directly. Both proposed filters provide reasonable pose estimates with superior convergence properties while being able to use measurements obtained from low-cost inertial measurement, landmark measurement, and velocity measurement units. The simulation results demonstrate the effectiveness and robustness of the proposed filters considering large error in initialization and high level of uncertainties in velocity vectors as well as in the set of vector measurements.

This paper presents an application of adaptive internal model principle controller to an active noise cancellation problem. The algorithm is extended to handling the large phase delays that arise in active noise control problems because of the inherent transport delays. Internal model principle controllers, such as the integral action in a PI controller, require, as a necessary condition of stability, that gains are chosen to ensure negative feedback. Previously the algorithm had fixed gains which resulted in negative feedback, and stability, only for plants whose phase did not vary by more than 180 degrees. By adaptively tuning the two control gains in the feedback loop, this implicit phase requirement is eliminated. The new algorithm now requires, at a minimum, that no more than 100% uncertainty exists in the plant model. Simulations on Ben Amara's model of an acoustic duct show the effectiveness of the proposed approach.

An internal model based adaptive algorithm for canceling disturbances composed of sums of exponentially damped sinusoidal (EDS) signals is presented. The time varying state variables of an internal model principle controller in a feedback loop can provide estimates of the EDS signal parameters, the frequency and the damping factor. By using additional integral controllers, the estimation errors can be eliminated to achieve perfect cancellation. The convergence of the proposed adaptive algorithm and the stability of the feedback control system including the algorithm are justified using singular perturbation theory and averaging theory. Simulation results demonstrate the validity of the convergence and stability analysis.