H(infinity) optimized wave-absorbing control - Analytical and experimental results

Abstract

The wave-absorbing control is a control concept to absorb waves traveling in a flexible structure at actuator positions. This paper presents an approach to design a broadband compensator by applying the //«, control theory to the wave-absorbing control method. This approach aims to minimize effects of the incoming waves on the outgoing waves at the actuator positions in the sense that the //«, norm of the closed-loop scattering matrix is minimum. Vibration suppression control for a flexible beam is studied analytically and demonstrated experimentally to exemplify the controller design approach. Compensators are designed for a collocated torque actuator and angle sensor and also for a noncollocated torque actuator and bending moment sensor. Performance of the compensators is analyzed in the frequency domain, and measured open- and closed-loop transfer functions are obtained from random excitation tests. The designed compensators are shown to attain good broadband damping, and results of the experiments are shown to agree well for the range of frequency below 50 Hz with those of the numerical simulations. I. Introduction A CTIVE control of vibrations in large flexible structures has received considerable attention in recent years. The modal model is a powerful technique both for the dynamic analysis and for the control design. However, limitations on the applicability of the structural modal analysis exist1 when the requirements for vibration suppression and pointing accuracy for flexible structures become stringent. The flexible mode frequencies and shapes are extremely sensitive to inevitable modeling errors, and modal analysis cannot provide a sufficiently accurate design model over a modally rich frequency range. One alternative is the traveling wave approach. This approach is based on the property that the response of a flexible structure to a typical locally applied force can be viewed in terms of traveling elastic disturbances. Mathematically, traveling waves belong to homogeneous solutions of partial differential equations describing the vibration of continua. At controller positions, relations between incoming and outgoing wave vectors and control inputs are derived in a matrix form by representing boundary conditions in terms of the traveling wave vectors. Outgoing waves are produced by the reflection of the incoming waves and are generated by control inputs. Transfer functions from the incoming wave and control input vectors to the outgoing wave vector are called scattering and generating matrices, respectively. Control inputs are set to be in the output-feedback form. This leads to the closed-loop relations between outgoing and incoming waves. Compensators are selected so that the effects of the incoming waves on the outgoing waves are reduced in some sense by adequately selecting elements of the closed-loop scattering matrix. Characteristic elements of the wave-propagation model, such as a scattering matrix, are smooth functions with respect to frequency and are more insensitive to model uncertainties than mode frequencies and shapes. The approach can provide a sufficiently accurate model for a controller design over a modally dense frequency region, and considerable research has been done on the wave control methods.18 However these methods also have drawbacks, such as 1) the designed compensator is not guaranteed to be a causal and real function with