Closed-loop transfer function

Abstract: This paper compares four current control structures for selective harmonic compensation in active power filters. All controllers under scrutiny perform the harmonic compensation by using arrays of resonant controllers, one for the fundamental and one for each harmonic of interest, in order to achieve zero phase shift and unity gain in the closed-loop transfer function for selected harmonics. The complete current controller is the superposition of all individual harmonic controllers and may be implemented in various reference frames. The analysis is focused on the comparison of harmonic and total closed-loop transfer functions for each controller. Analytical similarities and differences between schemes in terms of frequency response characteristics are emphasized. It is concluded that three of them have identical harmonic behavior despite the fact that their implementation is significantly different. It emerges that the fourth one has superior behavior and robustness and can stably work at higher frequencies than the others. Theoretical findings and analysis are supported by comparative experimental results on a 7-kVA laboratory setup. The highest harmonic frequency that can be stably compensated with each control method has been determined, indicating significant differences in the control performance.

Abstract: Statistical parameters of the phase-error behavior of a phase-locked loop tracking a constant frequency signal in the presence of additive, stationary, Gaussian noise are obtained by treating the problem as a continuous random walk with a sinusoidal restoring force. The Fokker-Planck or diffusion equation is obtained for a general loop and for the case of frequency-modulated received signals. An exact expression for the steady-state phase-error distribution is available only for the first-order loop, but approximate and asymptotic expressions are derived for the second-order loop. Results are obtained also for the expected time to loss of lock and for the frequency of skipping cycles. Some of the results are extended to tracking loops with nonsinusoidal error functions. Validity thresholds of widely accepted approximate models of the phase-locked loop are obtained by comparison with the exact results available for the first-order loop.

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.1"8 However these methods also have drawbacks, such as 1) the designed compensator is not guaranteed to be a causal and real function with