A Perfect Explicit Model following Control Solution to Imperfect Model following Control Problems

For cases in which perfect model following is not possible for a particular desired model, a class of candidate models is defined that can be followed perfectly by the given plant. A candidate model that most closely matches the dynamics of the desired model is then determined through constrained parameter optimization. The result is perfect model following of a model that has an eigenstructure that resembles that of the desired model. In the development of this method, a new variation on perfect model following control law development is shown. This method explicitly displays the feedforward and feedback gains that determine the system error dynamics, which may be arbitrarily selected by conventional pole placement methods if the plant is completely controllable. The method is applied to a problem involving the linearized lateral-directional equations of motion of the B-26 airplane. The results show that a candidate model can be found that has virtually the same dynamic behavior as the desired model, and that it can be followed perfectly by the original plant with arbitrarily assigned error dynamics. NOMENCLATURE A System Matrix, nxn B Control Input Matrix, nxm β Sideslip Angle ∆ Matrix of differences between A matrices and B matrices δa Aileron Input δr Rudder Input e Error Vector, nx1 φ Bank Angle φ̇ Roll Rate G Matrix relating rows of a matrix to a basis for its row space Ke Error Dynamics Gain Matrix, mxn M Modal Matrix r Yaw Rate T Transformation Matrix u Control Vector, mx1 W A weighting matrix. x State Vector, nx1 Superscripts * Conjugate operation (matrix conjugate transpose) + Pseudo inverse -1 Matrix inverse T Matrix transpose w Left generalized inverse *Graduate Student, Aerospace and Ocean Engineering Department **Professor, Aerospace and Ocean Engineering Department. Associate Fellow AIAA