A linear-quadratic-regulator-based (LQR) controller originates from a homogeneous set of state-space equations, and consists of a matrix of constant feedback gains. If the state equations are made nonhomogeneous by adding a vector of deterministic forcing terms, the standard LQR solution is no longer optimal. The present paper develops a matrix solution to this augmented (nonhomogeneous) LQR problem. The solution form consists of constant-gain feedback of the full-state vector, summed with a matrix preview (Duhamel integral) term. A practical and usable approximation is presented for the optimal preview term, having the form of a constant preview gain matrix. An example shows the improvement obtainable in controller performance with the use of this preview gain matrix, for exponentially decaying disturbances with a range of time constants.
A Practical Solution to the Deterministic Nonhomogeneous LQR Problem
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Hampton, R. D., Knospe, C. R., and Townsend, M. A. (June 1, 1996). "A Practical Solution to the Deterministic Nonhomogeneous LQR Problem." ASME. J. Dyn. Sys., Meas., Control. June 1996; 118(2): 354–359. https://doi.org/10.1115/1.2802329
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