Arun K. Tangirala

Quantitative analysis of directional strengths in jointly stationary linear multivariate processes

Identification and analysis of directed influences in multivariate systems is an important problem in many scientific areas. Recent studies in neuroscience have provided measures to determine the network structure of the process and to quantify the total effect in terms of energy transfer. These measures are based on joint stationary representations of a multivariate process using vector auto-regressive (VAR) models. A few important issues remain unaddressed though. The primary outcomes of this study are (i) a theoretical proof that the total coupling strength consists of three components, namely, the direct, indirect, and the interference produced by the direct and indirect effects, (ii) expressions to estimate/calculate these effects, and (iii) a result which shows that the well-known directed measure for linear systems, partial directed coherence (PDC) only aids in structure determination but does not provide a normalized measure of the direct energy transfer. Simulation case studies are shown to illustrate the theoretical results.

Quantification of interaction in multiloop control systems using directed spectral decomposition

Interactions among control loops are a critical and challenging issue in the control of multivariable systems. The focus of this work is on the analysis and quantification of interactions in multiloop or decentralized control systems. Existing interaction measures suffer from one or more of the following limitations: (i) the lack of a direct connection to a performance metric, (ii) assumption of the availability of process models and (iii) the approximate and/or a heuristic nature of the approach to their development, resulting only in approximate indicators of interaction. This work presents an exact quantifier of interaction that arises out of a directional decomposition of the loop variance using methods of causality (directional) analysis in frequency-domain. The main result is that the spectrum of the filtered output can be decomposed into (i) an interaction-and-feedback invariant term and (ii) an interaction-dependent term. The associated filter can be derived from the closed-loop data and is related to the diagonal element of the multiloop sensitivity function. The invariant term for each output is the spectral density of that output when the corresponding loop is under open-loop conditions. It is further shown to be solely a function of the control pairing. Variance measures corresponding to the invariant and interaction terms are introduced. The utility of the measure is that it can be computed from closed loop data as well as from the process model. Applications to simulated systems and a real time distillation process are presented to demonstrate the theoretical ideas.

Source separation in systems with correlated sources using NMF

Non-negative Matrix Factorization (NMF) has been used for source separation in various fields. However, the existing methods have ignored the presence of interactions among sources/measurements which leads to incorrect results. Interactions are common in a multivariate process where the variables are physically related/correlated with one another (for example: pressure-temperature dependency in an industrial process). In this work, conventional methods are extended to take into account the interactions. The contributions of this work are as follows: (i) an augmented NMF method to correctly determine the number of sources in the presence of multiple interactions; (ii) an algorithm to identify the correct signatures of the physical sources. The conventional method of NMF is shown to be a special case of the proposed method. Simulation studies are presented to demonstrate the practicality and utility of the proposed method.

Reconstructing Plant Connectivity Using Directed Spectral Decomposition

Abstract Process connectivity is a key information that is sought in a diverse set of applications ranging from design to fault diagnosis of engineering and biological processes. The present work develops a methodology for reconstruction of plant connectivity from dynamic data using directional spectral analysis, a novel adaptation of ideas from neurosciences and econometrics. The method is based on the concept of Granger causality while the procedure rests on the directional decomposition of power spectrum into direct and indirect energy transfers. The quantification of effective connectivity is obtained using a structural vector auto-regressive (SVAR) representation of the process. Results from simulation studies demonstrate the potential of the proposed method.

Wavelets Applications in Modeling and Control

Abstract Wavelets have been on the forefront for more than three decades now. Wavelet transforms have had tremendous impact on the fields of signal processing, signal coding, estimation, pattern recognition, applied sciences, process systems engineering, econometrics, and medicine. Built on these transforms are powerful frameworks and novel techniques for solving a large class of theoretical and industrial problems. Wavelet transforms facilitate a multiscale framework for signal and system analysis. In a multiscale framework, the analyst can decompose signals into components at different resolutions followed by the application of the standard single-scale techniques to each of these components. In the area of process systems engineering, wavelets have become the de facto tool for signal compression, estimation, filtering, and identification. The field of wavelets is ever-growing with invaluable and innovative contributions from researchers worldwide. The purpose of this chapter is threefold: (i) to provide a semi-formal introduction to wavelet transforms for engineers; (ii) to present an overview of their applications in process systems engineering, with specific attention to controller loop performance monitoring and empirical modeling; and (iii) to introduce the ideas of consistent prediction-based multiscale identification. Case studies and examples are used to demonstrate the concepts and developments in this work.

Spline Wavelets for System Identification

Abstract The paper introduces spline wavelets as a modelling tool for system identification and proposes the technique of consistent output prediction using wavelets for estimating system parameters. It suggests that direct weighted summation of projections in approximation space could be used for deriving consistent output prediction in case model structure is built with spline wavelets. This can be viewed as identification using prefiltered input and output. The prefiltering is motivated to decorrelate samples such that local fit can be considered as a possible solution. An iterative algorithm, alternately projecting the solution in time and wavelet domain for penalized minimization of local error in wavelet coefficients could be designed for estimating system parameters. The algorithm is computationally efficient and exhibits excellent performance in cross validation. As a case study, the paper addresses the problem of modelling Liquid Zone Control System (LZCS) in a large Pressurized Heavy Water Reactor (PHWR). In this work, an identification scheme of a single input single output (SISO) linear time invariant (LTI) model of the LZCS system is studied. Excellent approximation is achieved by modelling with Biorthogonal spline wavelets used for deriving consistent output prediction of the LZCS process.

Interaction assessment in multivariable control systems through causality analysis

Abstract Multivariable control systems are continually challenged by the presence of interactions among control loops. The issue of interactions is even more pronounced in systems involving multiloop (decentralized) control schemes. This paper presents a novel method to assess the level of interactions in multivariable control systems. The proposed approach is based on the total directed energy transfer between a pair of variables. The merit of the method lies in its ease of interpretation and versatility in accommodating different controller structures and operating conditions. Further, the method can be applied for both theoretical (when a process model is known) as well as data-driven analysis (when operating data is available). Implementation on synthetic processes are illustrated to demonstrate the usefulness of the proposed approach.

An Adaptive Basis Estimation Method for Compressed Sensing with Applications to Missing Data Reconstruction

Abstract The subject of compressed sensing, especially, the related concept of sparse representation has been growing into an exciting area with a diverse set of applications in the fields of image sensing and analysis, signal compression, network reconstruction, etc. The efficacy of the associated techniques depends on the ability to discover a suitable basis for a sparse representation of the underlying signal. This paper presents a method for discovering this basis adaptively from the data. Specifically, the method estimates the dictionary of basis functions that maps the sub-sampled signal to the sparse representation of the signal. We present an application of this technique to the reconstruction of missing data, which is an important problem in all data-driven methods. Two case studies, namely, the reconstruction of missing data in a liquid level system and missing pixels of a 2-D signal (image) are presented. Results show that the proposed algorithm outperforms the existing KSVD algorithm in terms of both accuracy and speed of the reconstruction.

SYSID: An open-source library for system identification

Through this paper we introduce an exclusive library sysid for system identification in the R® platform. This open-source library, the first of its kind on this platform, is designed primarily for classroom training and academic purposes. The library contains routines for input design, simulation and standard estimation methods for understanding the subject of and developing data-driven models for linear-time invariant (LTI) systems. In the present version of the library, the class of models are restricted to non-parametric (impulse, step and frequency-response descriptions) and parametric input-output LTI models (the prediction-error family), while the methods include both the prediction-error minimization and the instrument-variable approaches. Methods for recursive estimation are also included. We demonstrate the various functionalities on two simulation case-studies.

A systems perspective on non-normality in low-order thermoacoustic models: Full norms, semi-norms and transient growth

Non-normal transient growth of energy is a feature encountered in many physical systems. Its observation is intimately related to the norm used to describe the system dynamics. For a multi-physics problem such as thermoacoustics, where a heat source is in feedback with acoustic waves and a flow field, the appropriate metric is an ongoing matter of debate. Adopting a systemic perspective, it is argued in the present paper that an energy norm is, in principle, a matter of choice, but one that is critically tied to the dynamics described by the system model. To illustrate our arguments, it is shown that different norms exhibit the non-normal dynamics of thermoacoustic systems differently, but that this difference is fully explicable by the energy flux and source terms related to the formulation of the model. The non-normal dynamics as such is unaffected by the choice of norm, and transient growth merely results from a maximization of the flux and source terms governing the energy balance associated with the specific model formulation. Investigating transient growth for arbitrary energy norms requires the capability to handle semi-norm optimization problems. In the present study, we propose an approach to do so using the singular value decomposition. Non-normal transient growth around a stable fix point is then investigated for a low-order model of a simple thermoacoustic configuration of a premixed flame enclosed in a duct with non-zero mean temperature jump and bulk mean flow. The corresponding optimal mode shapes and pertinent parameters leading to transient growth are identified and discussed. For transient growth resulting from the interaction of the flame with the acoustic field, it is found that heat sources with a fast response lead to more transient growth than slow heat sources, because the system can bear a larger source term before becoming linearly unstable. Furthermore, the amount of transient energy growth does not increase monotonically with the amplitude of the initial perturbation of the flame.