Feedback Control of Negative-Imaginary Systems: Large Flexible structures with colocated actuators and sensors

I. Highly resonant dynamics can severely degrade the performance of technological systems. Structural modes in machines and robots, ground and aerospace vehicles, and precision instrumentation, such as atomic force microscopes and optical systems, can limit the ability of control systems to achieve the desired performance. Consequently, control systems must be designed to suppress the effects of these dynamics, or at least avoid exciting them beyond open-loop levels. Open-loop techniques for highly resonant systems, such as input shaping [1], as well as closed-loop techniques, such as damping augmentation [2], [3], can be used for this purpose. Structural dynamics are often difficult to model with high precision due to sensitivity to boundary conditions as well as aging and environmental effects. Therefore, active damping augmentation to counteract the effects of external commands and disturbances must account for parametric uncertainty and unmodeled dynamics. This problem is simplified to some extent by using force actuators combined with colocated measurements of velocity, position, or acceleration, where colocated refers to the fact that the sensors and actuators have the same location and the same direction. Colocated control with velocity measurements, called negative-velocity feedback, can be used to directly increase the effective damping, thereby facilitating the design of controllers that guarantee closed-loop stability in the presence of plant parameter variations and unmodeled dynamics [1], [4]. This guaranteed stability property can be established by using results on passive systems [5], [6]. However, the theoretical properties of negative-velocity feedback are based on the idealized assumption of colocation and require the availability of velocity sensors, which may be expensive. Also, the choice of measured variable may depend on whether the desired objective is shape control or damping augmentation. An alternative approach to negative-velocity feedback is positive-position feedback, where position sensors are used in place of velocity sensors. Although position sensors can facilitate the objective of shape control, it is less obvious how they can be used for damping augmentation. Nevertheless, it is shown in [7], [8] that a positive-position feedback controller can be designed to increase the damping of the modes of a flexible structure. Furthermore, this controller is robust against uncertainty in the modal frequencies as well as unmodeled plant dynamics. As shown in [7]–[10], the robustness properties of positive-position feedback are similar to those of negative-velocity feedback. The present article investigates the robustness of positive-position feedback control of flexible structures with colocated force actuators and position sensors. In particular, …