Robustified control design by an approximate min-max technique

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A design method for controllers that utilizes a Linear Quadratic Gaussian (LQG) cost function with approximated worst-case real parameter uncertainty is developed. The method aims to reduce the sensitivity of closed-loop performance to uncertainty of real open-loop parameters. Given a performance criterion, the best controller of a given structure is designed by finding the design whose worstcase open-loop parameters yield a cost that is lower than any other controller's worst-case cost. The parameters are assumed to be constant but unknown and bounded by a hyper-ellipse. The cost function is the LQG cost function expanded in a Taylor series with respect to the uncertain parameters out to the quadratic terms. An important property of this cost function is that its global maximum with respect to parameter variations can be calculated analytically for a given controller design. A second cost function, one that depends only on controller design parameters, is defined as the maximum of the original cost function with respect to the uncertain open-loop parameters. Controller design is carried out by using nonlinear programming techniques to minimize the latter cost function with respect to the controller design parameters. This cost function can have a discontinuous first derivative when two equal global maxima occur during the worst-case parameter calculations, but this situation can be handled by appropriate techniques from the field of non-smooth optimization. The new design technique is applied to two example problems.