Extensions of the first and second complex-step derivative approximations

AbstractA general framework for the first and second complex-step derivative approximation to compute numerical derivatives is presented. For first derivatives the complex-step approach does not suffer roundoff errors as in standard numerical finite-difference approaches. Therefore, since an arbitrarily small step size can be chosen, the complex-step approach can achieve near analytical accuracy. However, for second derivatives straight implementation of the complex-step approach does suffer from roundoff errors. Therefore, an arbitrarily small step size cannot be chosen. In this paper the standard complex-step approach is expanded by using general complex-step sizes to provide a wider range of accuracy for both the first- and second-derivative approximations. Even higher accuracy formulations are obtained by repetitively applying Richardson extrapolations. The new extensions can allow the use of one step size to provide optimal accuracy for both derivative approximations

Robust Control of Nonlinear Systems Using Model-Error Control Synthesis

A new approach for the robust control of nonlinear systems is presented. This approach employs an optimal real-time nonlinear estimator to determine model error corrections to the control input using a one time-step ahead technique. Control compensation is achieved by using the estimated model error as a signal synthesis adaptive correction to the nominal control input so that maximum performance is achieved in the face of extreme model uncertainty and disturbance inputs. A significant advantage of this approach over other adaptive methods is that model parameters need not be updated online. Instead, the effects of these errors are used to update the actual control signal, which leads to a simple design strategy. The model-error control synthesis approach is used in conjunction with a variable structure controller to suppress the wing-rock motion of a slender delta wing. Simulation results indicate that the model-error control synthesis approach is extremely effective in providing clos..