HIGH ORDER OPTIMAL TRACKING CONTROL SENSITIVITY CALCULATIONS USING COMPUTATIONAL DIFFERENTIATION
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An optimal tracking control is developed where the optimal control is calculated by optimizing a universal quadratic penalty. The optimal tracking problem formulation is generalized by modeling the control gains as a Taylor series in the parameter uncertainty. The generalized control formulation is computed as an off-line calculation for the sensitivity gains. The goal of the generalized control formulation is to eliminate the need for gain scheduling for handling model parameter variations. An estimator is assumed to be available for predicting the model parameter changes. Higher-Order control sensitivity calculations are applied on the full nonlinear model using computational differentiation tool. Several attitude error representations are presented for describing the tracking orientation error kinematics. Compact forms of attitude error equation are derived for each case. The attitude error is initially defined as the quaternion (rotation) error between the current and the reference orientation. Transformation equations are presented that enable the development of nonlinear kinematic models that are valid for arbitrarily large relative rotations and rotation rates. The nonlinear error dynamics for kinematics and the equation of motion is retained, yielding a tensor-based series solution for the Co-State as a function of error dynamics. Control sensitivity calculations are performed to handle model and parameter uncertainty in the real system. The OCEA (Object Oriented Coordinate Embedding) computational differentiation toolbox is used for automatically generating the first- through fourth-order partial derivatives required for the generalized control sensitivity differential equation. Several numerical examples are presented that demonstrate the effectiveness of the proposed approach. The methods presented are expected to be broadly useful for control applications in science and engineering. Copyright 2013 by The Math Works, Inc.
