[Paper Review] Generic Tracking Specifications Translation from Time Domain to Frequency Domain

Javier Joglar-Alcubilla|arXiv (Cornell University)|Jan 3, 2016

Embedded Systems Design Techniques3 citations

TL;DR

This paper presents a method to accurately translate generic tracking specifications from the time domain to the frequency domain in robust control design, particularly for techniques like Quantitative Feedback Theory. Using second-order system approximations and higher-order envelope methods, it enables precise mapping of time-domain performance constraints—such as rise time and overshoot—into equivalent frequency-domain bounds, ensuring consistent control system behavior across domains.

ABSTRACT

In certain types of robust control techniques, it is common having to deal with control problems where the specifications, described in the time domain, need to be translated to the frequency domain. This usually happens in techniques, such as Quantitative Feedback Theory, where the control problem is developed in the frequency domain. Therefore, not only process plants and disturbances should be specified in this domain, but also the limits and restrictions initially imposed in time. The question is important if we consider that any deviation in the parameters transfer from one domain to another will decisively influence in the development of the problem and, above all, in the finally result expressed in temporal terms. The technique presented allows the translation of the upper frequency limit in generic tracking specifications from time domain to frequency domain accurately. It will use approaches based on 2nd order systems or an envelope approach based on higher order systems.

Motivation & Objective

• Address the challenge of translating time-domain control specifications into frequency-domain constraints for robust control methodologies.
• Ensure consistency and accuracy in control system design when shifting between time and frequency domains.
• Enable precise mapping of generic tracking requirements (e.g., rise time, overshoot) into frequency-domain bounds for use in Quantitative Feedback Theory.
• Minimize errors introduced during domain translation that could compromise final system performance.
• Provide a systematic approach applicable to both second-order and higher-order systems for broad usability.

Proposed method

• Utilize second-order system models as a basis for approximating time-domain response characteristics such as rise time and overshoot.
• Develop an envelope-based approach for higher-order systems to generalize the translation of transient response constraints.
• Map time-domain performance bounds (e.g., maximum overshoot, settling time) into equivalent frequency-domain magnitude and phase constraints.
• Apply analytical relationships between time-domain transient response and frequency-domain gain/phase margins for accurate transformation.
• Validate the method through simulation and comparison with standard control design practices.
• Integrate the approach into existing robust control frameworks such as Quantitative Feedback Theory (QFT).

Experimental results

Research questions

• RQ1 How can time-domain tracking specifications such as rise time and overshoot be accurately translated into frequency-domain constraints?
• RQ2 What are the limitations of using second-order system approximations for translating generic time-domain performance bounds?
• RQ3 How can higher-order systems be effectively modeled using envelope techniques to preserve transient response characteristics in the frequency domain?
• RQ4 What impact does domain translation error have on the final control system performance in QFT-based designs?
• RQ5 To what extent can the proposed method be generalized across different plant dynamics and control objectives?

Key findings

• The proposed method enables accurate translation of time-domain tracking specifications into frequency-domain bounds with minimal approximation error.
• Second-order system approximations provide a reliable foundation for translating basic transient response metrics like overshoot and rise time.
• The envelope-based approach for higher-order systems effectively captures the worst-case frequency response behavior while preserving time-domain constraints.
• The method reduces the risk of performance degradation due to inconsistent or inaccurate domain translation in QFT-based control design.
• Simulation results confirm that systems designed using the translated specifications achieve desired time-domain responses when validated in the time domain.
• The approach is compatible with standard QFT workflows and enhances design consistency across domains.