Ayhan Sebastian Kammer

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.

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.

Sign inverting and Delay Scheduling Control concepts with multiple rationally independent delays

In this paper we introduce a novel combination of Sign Inverting Control (SIC) and Delay Scheduling (DS) concepts for systems with multiple rationally independent and large delays. The combined scheme functions with a single premise that, the union of a proper choice of control and its sign inverted version (SIC) would provide a larger stable operating region in the domain of the delays. The outstanding question to properly perform this novel control strategy is the crisp knowledge of the stable operating regions for each of the two control schemes. This question is uniquely answered using the CTCR (Cluster Treatment of Characteristic Roots) paradigm. CTCR declares the stable regions very efficiently and exhaustively with no conservatism. The resulting switched operating mode renders more robust control performance against much larger delay variations vis-à-vis each of the schemes. For unstable delay compositions, a paradoxical scheme called the Delay Scheduling Control (DS) is deployed by increasing the present delays in order to regain stability. The crucial question of “how much increase in delays?” can also be answered by the CTCR without conservatism. The combination of Sign Inverting Control (SIC) and Delay Scheduling Control (DS) schemes provides previously-unexplored tools to control designers. Experimental validations of these methodologies are provided on a simple setup with a single-axis manipulator.

Stability and Control of a Thermoacoustic Device: The Rijke’s Tube

This study presents an intersection of two seemingly separate areas of research frontiers, “prediction and control of thermoacoustic instability” and “stability analysis of neutral-class linear-time-invariant (LTI) and time-delayed systems (TDS)”. The former is a coveted capability which has been elusive to the scientific community over 1½ centuries. Analytical capabilities have been limited due to the complex physics invoking the “combustion” phenomenon. Most available results rely on accumulated empirical knowledge. In this paper we consider a benchmark combustion test platform, which is known as Rijke’s tube. Its representation is simplified to an LTI neutral TDS, stability of which is assessed using a recent mathematical paradigm called the Cluster Treatment of Characteristic Roots (CTCR). CTCR provides a unique non-conservative and exhaustive stability declaration for a Rijke’s tube within the space of its parameters, naturally, under some simplifying assumptions. For those operating conditions which induce instability, we also propose a conventional and simple control strategy which can recover stability. This method is also analyzed using the CTCR paradigm for the first time.Copyright

Non-conservative stability assessment of LTI dynamics with distributed delay using CTCR paradigm

A general class of LTI feedback control systems with a distributed delay in the feedback line is taken into account. We start with a theorem which states the equivalence of a general class of distributed delay system to a discrete delay system with multiple independent delays. While this connection has been recognized earlier, what is interesting and novel to this study is the deployment of a paradigm called The Cluster Treatment of Characteristic Roots (CTCR) for this class of systems. CTCR exhaustively declares the stability outlook of the system in the space of the delays. This capability forms a very important basis over which we can build further research connected to systems with distributed delays.