Motion Control Design: Frequency Response Modeling of a Flexible Structure

by Mike Donnelly
5 November 2018

A versatile and effective modeling capability is available in our HyperLynx product family. It uses complex-pole fitting to extract simulation-ready models from measured frequency response data. This can be useful in a wide range of engineering design applications, from system level transfer function (signal flow) analysis, to modeling component interactions in a circuit simulation. The following example illustrates both of these aspects in a practical application: Design of a motion control loop that includes a flexible structure.

Figure 1: Compare Flexible Structure Physical and Complex-Pole Fitted Models

Figure 1 illustrates the ability to capture, in an executable model, the dynamic characteristics of a structure based solely on its frequency response data. It compares the frequency response of the "physical" model on the left (springs, masses, dampers), with its complex-pole fitted model on the right. The sensor displacement and the reaction force frequency response data from the physical model was exported to the fitting function in HyperLynx, and the resulting complex-pole parameters were simply copied and pasted into the model on the right. Of course this is not the “normal” use-case, wherein only the measured frequency response data would exist for a real structure, based on “shaker-table” or other types of vibration testing. That is, the “physical” spring-mass-damper model would not be required at all.

The comparative results for both sensor displacement (center-top plot) and reaction force (center-bottom plot) show that the transfer functions for these two models are nearly identical, with the only visible divergence coming at high frequency for the sensor displacement. This occurs when the transfer function gain is already "in the noise", at < -180 dB.

Figure 2:  Motion Control Compensation Design using Complex-Pole Fitted Model

In Figure 2, the fitted model of the flexible structure is used to represent the “plant” in a motion control loop driven by a Voice Coil Actuator (VCA). The top net in this schematic represents the output of an ideal position sensor, which provides the actual target displacement feedback. The loop compensation was designed using "AC" or frequency-domain analysis of the open-loop transfer function. The results are shown in the Magnitude and Phase difference plots (sensor_displacement - feedback), with the loop gain crossover at approximately 100 Hz and the phase margin is just over 43 degrees. The corresponding time-domain step response is shown in the left-most waveform plot.

It is important to note the two distinct uses made of the frequency response modeling method. The sensor displacement is a simple transfer function, from displacement of the bottom of the structure to displacement of the top (i.e. the sensor location). It assumes there is no mechanical loading or interaction force between the sensor and the rest of the control system, its only interaction is with the flexible mechanical structure itself. This was captured in the original frequency response data in Figure 1. On the other hand, the force frequency response is a reaction force. This force is also modeled as a transfer function, from the displacement of the bottom of the structure to the reaction force quantity value, which is then converted into an actual force and applied as a load on the voice coil. That is the purpose of the controlled force source on the far right of the schematic.

Figure 3: Motion Control System with VCA and PWM H-Bridge Drive

A more detailed implementation of this motion control system is shown in Figure 3. It illustrates using the complex-pole fitted model when making design decisions for the electronic drive circuit, such as choosing properly sized Power MOSFETs, selecting an appropriate PWM switching frequency, etc. The model supports better understand of the key electronic-mechanical dynamic interactions, as well as the overall system behavior.

We are just beginning to explore the value of this complex-pole fitting technology, to generate robust models for both time- and frequency-domain simulation from commonly available measured frequency response data. You can expect further blogs/articles on this topic from me, including other application examples as well as “How to use it” information. The latter may be evolving as we work to improve access within the SystemVision Cloud environment.