Minimum-time optimal output transitions using pre- and post-actuated inputs: Impact of zeros on the structure of the optimal control profile
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A great deal of focus is now placed on reliable approaches to feedforward control for precise positioning of flexible structures, such as hard disk drives, gantry cranes, wafer scanners, large-scale manipulators, and flexible spacecraft. This includes shaping the reference input for a stable system, i.e., feedback stabilized system or an open-loop control such as time-optimal control which provides the nominal trajectories which are followed by a perturbation feedback controller. Methods are now available for robust control to both desensitize the reference shaper to model parameter uncertainties and minimize excitation of unmodeled modes through the introduction/inclusion of control constraints to minimize jerk or smooth out trajectories. Precise positioning is achieved through minimization of residual vibrations with the use of input shapers/time-delay filters. Until recently, the majority of the these control problems have be posed in the framework of a state-to-state transition (SST) problem, with application to rest-to-rest maneuvers where a stable or asymptotically stable system transitions from one set of equilibrium points to another. Optimal trajectories have been determined without direct consideration of the system zeros in frequency domain derivation of the minimum-time control. In systems whose transfer functions are characterized by zeros, the output is a function of multiple states, and one can pose the control design as an output-to-output transition problem. To address remaining gaps in derivation of the time-optimal control, a new approach is proposed, which allows the system zeros to impact the optimal control profile in the formulation of a minimum-time Optimal Output Transition (OOT) problem using pre- and post-actuated inputs. The result greatly simplifies the derivation of the control, provides a clearer interpretation about implementation to the true physical system, provides a means of deriving solutions robust to inherent system uncertainties, In addition, results clearly demonstrate the significant decrease transition time when compared to the traditional SST solution. Presented techniques derive a frequency domain (input-shaper/ time-delay filter) to time-optimal control using pre- and post-actuated inputs for stable or asymptotically stable systems. Derivation of the control begins in the time-domain where optimality is proven and closed-form solutions to the system inputs are determined, thus allowing for accurate parameterization in the frequency domain. It is shown that time-optimal output transitions for a system with minimum phase zeros (left-half plane zeros) is achieved through the use of post-actuation and that the optimal post-actuated control is equal to placement of a pole of the input-shaper at the location of the minimum phase zeros. Additionally, for systems with nonminimum phase zeros (right-half plane zeros) the time-optimal pre-actuation control is achieved by canceling the nonminimum phase system zeros with the poles of the input-shaper. Robustness can be achieved by placing multiple zeros at the nominal location of the system poles and, when knowledge of the support of the uncertain poles are known, a minimax optimization problem can be formulated to minimize the worst performance of the controller over the domain of uncertainty. The generalized approach to the minimum-time OOT problem with post-actuation is then derived as a Linear Programming (LP) problem for straight-forward solution in a convex framework and robustness is added through the inclusion of sensitivity states of the system (i.e. placement of multiple zeros at the nominal location of the system poles). Finally, a convex minimax problem is derived, which minimizes the maximum of the L 2 norm in tracking error during post-actuation, and determines the robust mimimum-time OOT solution over a domain of uncertain system models.