Umut Zalluhoglu

Investigation of Local Stability Transitions in the Spectral Delay Space and Delay Space

The stability boundaries of LTI time-delayed systems with respect to the delays are studied in two different domains: (i) delay space (DS) and (ii) spectral delay space (SDS), which contains pointwise frequency information as well as the delay. SDS is the preferred domain due to its advantageous boundedness properties and simple construct of stability transition boundaries. These transitions at the mentioned boundaries, however, present some conceptual challenges in SDS. This transition property enables us to extract the corresponding local stability variation properties in the DS, while it does not have any implication in the preferred SDS. The novel aspect of the investigation is to introduce a comparison mechanism between these two domains, DS and SDS, from the stability transition perspective. Interestingly, we are able to prove their equivalency, which provides complementary insight to the parametric stability variations. [DOI: 10.1115/1.4027171]

Predicting Thermoacoustic Instability: A Novel Analytical Approach and Its Experimental Validation

The birthplace of modern combustors, the Rijke tube, is considered in this paper. The findings of earlier investigations are used as the departure point, and a complementary analytical approach is introduced for predicting its thermoacoustic instability with experimental validations. In the core of the novel analytical work resides a unique mathematical tool that evolved only within the last decade. It is called the cluster treatment of characteristic roots paradigm, which uniquely reveals the stable operating regions for thermoacoustic instability in the space of the operational parameters, such as geometric dimensions or thermal parameters. For linearizable systems, these stability regions are declared exhaustively and nonconservatively. The experimental validation of this analytical approach is the primary contribution of the present paper. The main intention of this paper is to supplement the combustion science literature with the analytical capabilities of cluster treatment of characteristic roots.

Combination of sign inverting and delay scheduling control concepts for multiple-delay dynamics

Abstract In this paper we treat a novel combination of two controversial control concepts, Sign Inverting Control (SIC) and Delay Scheduling (DS), for systems with multiple independent and large delays. SIC suggests the inversion of the control polarity and DS prolongs the existing delays. The combined scheme functions with a single requirement that, the union of the control schemes provides a larger stable operating region than each of its components does, in the domain of the delays. The critical knowledge that is needed to execute such a unified control strategy is the crisp description of these stable regions for each time-delayed control scheme. This need can be fulfilled using the recent Cluster Treatment of Characteristic Roots (CTCR) paradigm, which establishes the stable regions exhaustively and non-conservatively. The resulting options in selecting operating modes render more robust control performance against much larger delay variations than each of the schemes. We also investigate the disturbance rejection speeds within these enlarged stable regions in order to improve the control performance even further. Such multi-faceted paradoxical combinations provide previously-unexplored tools to control designers. Experimental validations of these novel concepts are presented on a simple setup with a single-axis manipulator.

A New Perspective in Designing Delayed Feedback Control for Thermo-Acoustic Instabilities (TAI)

This article suggests the deployment of a unique mathematical tool for assessing the thermo-acoustic instability (TAI) in a Rijke tube and proposes an analytical design strategy for its feedback control. A widely accepted characteristic of TAI is its time-delayed dynamics, which originate from the regenerative acoustic coupling terms. Linear systems theory has also evolved on similar classes of problems especially in recent years. This document offers a bridge between the two veins of research. We first review the analytical model of the TAI phenomenon, which renders a set of delayed differential equations. Then, we apply a new mathematical tool called the cluster treatment of characteristic roots (CTCR) paradigm on this dynamics. CTCR provides non-conservative and exhaustive stability predictions for this class of systems. This capability is employed for both uncontrolled and feedback-controlled Rijke tube structures. The findings are unique from two angles: (i) stability declarations are made in the parametric space of the system, such as geometric dimensions (much differently from the peer studies that are at best point-wise evaluations), and (ii) these declared sets of stable operating parameters are exhaustive (i.e., for a given system no other parametric selection can provide stability). These capabilities become crucial when designing thermoacoustically stable combustors as well as determining their operating conditions. As a highlight contribution in this article, for those operating conditions that induce instability, we offer a methodology to synthesize a feedback control law that can recover stability, again utilizing the CTCR paradigm. Example case studies and analytical justifications of these novelties are provided.

Critical Effects of the Polarity Change in Delayed States Within an LTI Dynamics With Multiple Delays

This note concerns the polarity inversion of the delayed states in linear time-invariant multiple time delay systems (LTI-MTDS). To start with, for such systems the assessment of asymptotic stability is complicated due to the infinite dimensionality introduced by the delays. It is shown that the mentioned polarity inversion influences the respective delay-dependent stability maps in interesting ways. There are some intriguing invariant characteristics and correspondences between the original and inverted systems, which form the primary contributions of this note. These findings are shown to be the enabling features to relate the stability maps of the two systems. The Cluster Treatment of Characteristic Roots (CTCR) paradigm is utilized to identify these features. An example case study is provided to illustrate the claims.

A study of Helmholtz resonators to stabilize thermoacoustically driven pressure oscillations

This paper studies passive control of thermoacoustic instabilities from an unconventional mathematical perspective. These instabilities are notoriously known to result from the complex dynamic exchange between the unsteady heat release and the acoustic waves within a finite volume such as a combustor. One possible passive control strategy is to utilize Helmholtz resonators. Under certain simplifications, the ensemble combustion dynamics including the resonators reduces to a linear-time invariant-multiple time-delayed system (LTI-MTDS). As the main contribution of the paper, an exact analytical procedure is proposed to determine the placement of the resonators to avoid instabilities. A unique mathematical paradigm, called the cluster treatment of characteristic roots, is used to accomplish this task. It declares exactly the necessary and sufficient stability conditions for an LTI-MTDS in the space of the system parameters. This concept paper is written with the mindset that this analytical tool can invite yet unexplored design capabilities for similar noise control applications where acoustic dampers are used.

Deployment of Time-Delayed Integral Control for Suppressing Thermoacoustic Instabilities

In this paper, active control of thermoacoustic instabilities is studied using a particular feedback control law and a mathematically novel perspective. The instabilities result from the complex dynamic exchange between the unsteady heat release and the pressure oscillations in a finite volume. A laboratory-scale Rijke tube, which is a simple thermoacoustic device, is considered in the paper. When linearized, its dynamics exhibits the neutral characteristics with multiple delays. To suppress the instabilities in this setting, a time-delayed integral control law is synthesized, and its performance is studied in detail. A recent mathematical paradigm, called the cluster treatment of characteristic roots is used to accomplish the stabilization of the time-delayed system. The paradigm declares exact stability conditions in the space of the system parameters. The analytical findings in this document are also validated with experimental results.

Parametric investigation of thermoacoustic instability (TAI) in a rijke tube: a time-delay perspective

This paper presents a novel deployment of a recent mathematical paradigm for predicting the thermo-acoustic instability (TAI) of a Rijke tube in the relevant parametric space. This benchmark problem in combustion science has been studied for over 1½ centuries with phenomenal achievements both in theoretical and practical fronts. The new paradigm is called the Cluster Treatment of Characteristic Roots (CTCR), which is originally developed to assess the asymptotic stability of Linear Time Invariant (LTI) Time-delayed Systems (TDS). A notorious subcategory within LTI-TDS is called “Neutral TDS”, which matches the characteristics of the linearized dynamics of thermo-acoustic instability. The CTCR is shown to reveal a non-conservative and exhaustive linear stability map of the Rijke tube within the space of its geometric and operational parameters. We present a review of this paradigm as well as several case studies to demonstrate its capabilities and some encouraging comparison with the earlier literature. This paper is a concept document and it is prepared with the intent of providing a breeding ground for studies beyond its present coverage.

Delayed Feedback Control Laws for Rijke Tube Thermoacoustic Instability, Synthesis, and Experimental Validation

We approach the thermoacoustic instability problem in Rijke tubes from a mathematically novel perspective. In this benchmark experimental setting, the complex dynamic exchange between the unsteady heat release and the acoustic pressure variations creates the instability. When linearized, this behavior leads to a neutral class time-delayed dynamics with multiple independent delays, where our major contribution arrives. We aim to synthesize a series of stabilizing feedback control strategies for this dynamics. A recent mathematical paradigm, called the cluster treatment of characteristic roots (CTCR), is used to facilitate this objective. This paradigm, in essence, declares the necessary and sufficient stability conditions in the space of the system and controller parameters. The main contributions of this brief are in the first-time deployment of CTCR in the syntheses of various feedback control laws and validating experiments.

Some critical properties of sign inverting control for LTI systems with multiple delays

A novel control logic, sign inverting control (SIC) is proposed for linear time-invariant multiple time delay systems (LTI-MTDS). SIC simply inverts the sign of a nominal control logic which is formulated for non-delayed dynamics using any known technique. The main objective of the inversion is to increase the delay robustness of the system. Three necessary and sufficient conditions are provided for SIC to be a viable option. This inversion method gains its strength from a recent paradigm called the Cluster Treatment of Characteristic Roots (CTCR). CTCR provides the necessary exact and exhaustive declaration of the stable regions in the domain of the delays for such systems. As the highlight of the paper, the spectral and root tendency invariance properties between the nominal and the SIC applied systems are presented with detailed proofs. Finally, we demonstrate an unexpected feature by an example, that SIC may also be applied to systems which are closed-loop non-delayed unstable for both control schemes.