Metrics for Interaction Assessment in Multivariable Control Systems

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Metrics for Interaction Assessment in Multivariable Control Systems using Directional Analysis Abhinav Garg, and Arunkumar Tangirala Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03671 • Publication Date (Web): 21 Dec 2017 Downloaded from http://pubs.acs.org on December 26, 2017

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Industrial & Engineering Chemistry Research

Metrics for Interaction Assessment in Multivariable Control Systems using Directional Analysis Abhinav Garg and Arun K. Tangirala∗ Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai-600036, India E-mail: [email protected] Abstract This work presents a novel method to assess and quantify the level of interactions in multivariable control systems in a hierarchical manner. The proposed metrics are based on the total directed (causal) power transfer between a pair of variables, constructed from a jointly linear stationary representation of the process. Three prime features of these metrics, specifically, (i) the ease of interpretation and versatility in accommodating different controller structures and operating conditions (ii) the ability to use both first-principles linearized and data-driven (empirical) models and (iii) the means for quantifying and evaluating the suitability of a decentralized versus a centralized control scheme, make them highly practical and valuable in the design and assessment of multivariable control schemes. An important outcome of this work is also an operator-friendly visual tool for inspection of interactions in various loops that can be generated for different tuning methods and controller configurations. Simulation studies on four different benchmark processes are presented to demonstrate the efficacy of the proposed method.

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Introduction

Industrial processes manifest themselves as complex multivariable systems with numerous interconnections and feedback loops. The performance of these processes can be significantly affected by interactions, propagated faults, sensor failures, equipment degradation, etc. The most significant among these factors is the interactions between loops/process units particularly with the increasing integration of various process operations. Interactions, as is widely known, present unique challenges in the design of multivariable feedback systems as it results in propagation of disturbances in one output to another, raising two prime concerns: (i) loss of control-loop performance and (ii) higher risk of instabilities. Multivariable controller design schemes can broadly be categorized into (i) decentralized and (ii) centralized control. The former class of schemes are widely used in industries as they are simpler to design and operate than their multivariable counterparts. The tolerance of decentralized schemes to loop failures also makes them a preferred choice over the latter. However, a major challenge encountered in the design and monitoring of the associated controllers is the limited degrees of freedom available to the designer in mitigating interactions. An additional challenge lies in selecting the right choice of control pairs (manipulated and controlled variables), which can have a significant impact on the extent of interactions. Multivariable controllers on the other hand, offer more tuning knobs to the designer in reducing interactions due to the presence of additional controllers in the off-diagonal as against the diagonal structure of a decentralized controller. The trade-off involved, however, is the complexity of the design problem and loss of robustness to subsystem failures. The aforementioned aspects of multivariable controller design have been studied extensively over the last four to five decades, resulting in a number of interaction measures, naturally for decentralized control schemes, e.g., Relative Gain Array (RGA), 1 µ interaction measure, 2 Structured Singular Value (SSV) 3 and their variants. 4,5 To overcome the limitations of the static RGA, dynamic RGA (DRGA) was proposed 6 which takes into account process dynamics. This is achieved by utilizing transfer function model of the plant instead 2


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of its steady state gain matrix in the definition of RGA. Some variants of DRGA also involve detailed controller design such as. 7–9 Another approach to define the DRGA can be found in. 10 However, as in the case of RGA, DRGA assumes perfect control at all frequencies. These measures are primarily aimed at aiding the designer in selecting appropriate control pairs. A major shortcoming of these measures is that they are not generally suited for assessment of interactions during a continual operation of the process, primarily because they require the knowledge of transfer functions and/or the controllers in place. These two critical pieces of information, especially process models, are seldom available for several multivariable processes. Therefore, there is a strong case for developing metrics of interaction that can work with both transfer functions and process data. Performance assessment of control loops has been widely studied over the last two decades, 11 where most of the efforts have been aimed at quantifying the performance of either the individual loops or the overall multivariable system. On a relative scale, very few methods 12,13 have been formulated to assess the loop-loop interaction from process data. Among these few methods that exist, a majority operate in time-domain, 14,15 whereas in a control system analysis it is highly desirable to have measures that can offer insights into the frequency-domain characteristics of these interactions. In the work by Seppala et al., 14 the authors proposed use of multivariate impulse response function of a joint stationary representation of the process for its dynamic analysis. This approach was largely qualitative in nature and input disturbances were not accounted for, which would result in confounding. The first significant contribution in interaction quantification of decentralized control systems was proposed by Gigi and Tangirala. 16 They presented an interaction measure for multiloop (decentralized) control systems based on a directional decomposition of the output variance in the spectral (frequency) domain. The approach consists of decomposing the filtered output spectrum into an interaction-and-feedback invariant and an interaction dependent term. The invariant term is only dependent on the control pairing, whereas the interaction-dependent term is affected by both controller parameters

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and the interconnections. While that method offers an exact quantification of interaction with respect to the variance metric, it is limited to the analysis of decentralized control systems. Moreover, the measure quantifies the interactions under loop-open conditions, i.e., it opens up the loop under study, through the use of sensitivity-like functions. This is also the case with several existing theoretical measures such as RGA. In practice, what would be desirable is to measure interactions while keeping all loops in feedback. However, then an exact decomposition of the output spectrum into an interaction-and-feedback invariant and interaction dependent term is, at least for the present, elusive. As an extension of these ideas, the preliminary results for quantification of interactions in a multivariable control system, such that the formulation was independent of the controller structure, was presented in Garg and Tangirala. 17 Present work builds on this and provides a detailed theoretical extension to it. More recently, Naghoosi and Huang 15 used the idea of total transfer function decomposition 16 to quantify interactions through a time-domain measure based on impulse response. The development in this work stands on the perspective that interaction is a causal phenomenon i.e., it is characterized by both magnitude and direction. The concept of causality is relatively new, particularly in engineering systems. A majority of the applications of causality analysis is witnessed in the fields of econometrics, social sciences and neuroscience owing to the origins of these methods. 18–21 In process engineering, analyzing processes using concepts of causality brings in a few significant advantages over that using the traditional input-output framework, especially because processes acquire a network representation where variables can play dual roles of cause and effect. Such a setting is natural for multivariable feedback control systems. Importantly, the strength of a directional (cause-effect) pathway can be quantified. To-date traditional approaches have been proposed in the literature for quantification of interactions. In this work, a frequency domain causality analysis based approach is proposed for assessment and quantification of interactions that offer merit over available methods. This paper serves as a valuable contribution to the emerging field of

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causality analysis in the domain of engineering systems. With this motivation, the objective of this work is to develop a quantitative measure of interaction such that it is, • Applicable to both data-based and model-based environments: Traditionally, methods for quantification of interactions have been tied to availability of process model. However, with the advent of big data revolution in process industries, it is imperative to have a metric that can be used in both the cases. • Invariant to controller structure / configuration: Most of the metrics available in the literature are applicable only to a decentralized controller and a clear definition of interactions in a centralized controller is not explicitly defined. Ideally, the metric should be able to compare different controller structures based on interactions. • Offer zoom-in and zoom-out features: The interaction metric should be able to provide a holistic view of the system as well as the frequency breakdown of interactions for a loop-by-loop assessment. Inline with the above mentioned objectives, this paper presents a measure and a visualization tool that provides insights into the frequency-domain characteristics of loop-to-loop interactions thereby assessing the disturbance and measurement error propagation in a multivariable control system. The measure is based on the total power transfer along a directed pathway between two variables of the multivariable system. These power transfers also lead to a visualization tool that offers an aerial picture of the interactions experienced in the control system. Theoretical support and the necessary computations for the measures are provided by a joint stationary representation of the closed-loop process. In summary, the key highlights of this work are: (a) a metric for interaction assessment that can be computed both from data and process model, (b) a setup for quantitative comparison of performance using different controller structures for a given multivariable process, (c) development of a novel equivalence between centralized and decentralized controllers, and 5



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finally (e) three metrics for assessment and quantification of interactions with zoom-in and zoom-out features. The scope of this work is restricted to stationary linear time-invariant processes. The article is organized as follows. Section 2 lays down the necessary theoretical preliminaries for the developments in this work. The proposed method for interaction assessment is discussed in Section 3, where the power map is also introduced. Applications of the method are illustrated through simulated case-studies in Section 4. The paper concludes with a few remarks in Section 5.

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Foundations

The basis for the proposed measure as well as the general development in this work is a joint stationary representation of the multivariable process, which in turn stems from the celebrated spectral factorization theorem. 22 Note that joint input and output modeling is reminiscent of a class of closed-loop identification methods; however, in this work, the closed loop model is not the object of interest and hence not identified. This section first reviews the spectral methods and the related concepts that are applied in this work.

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Spectral Factorization Theorem

The essence of the spectral factorization theorem is that the cross power spectral density T (matrix) Φzz (ω) of a multivariate jointly stationary process z = y u , can be factored as

Φzz (ω) = H(ω)Σe H∗ (ω)

(1)

where H(ω) is the frequency response function of the stationary process. It is also related to the vector autoregressive (VAR) and vector moving average (VMA) representations of the process as discussed in the following sections. 6


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2.2

Vector Moving Average (VMA) Representation

The VMA representation of order Q for a process z[k] , yT uT

where y ∈

Metrics for Interaction Assessment in Multivariable Control Systems

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