Data-based automated diagnosis and iterative retuningof proportional-integral (PI) controllers

Tim Spinner a, Babji Srinivasan b, Raghunathan Rengaswamy a,c,n

a Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, United Statesb Department of Chemical Engineering IIT Gandhinagar, Ahmedabad 382525, Indiac Department of Chemical Engineering, IIT Madras, Chennai 600036, India

a r t i c l e i n f o

Article history:Received 1 June 2013Accepted 11 March 2014

Keywords:PI controllersControl loop performance assessmentPerformance monitoringController retuning

a b s t r a c t

This work presents a new look at the existing data-based and non-intrusive PI (proportional-integral)controller tuning assessment methods for SISO (single-input single-output) systems under regulatorycontrol. Poorly tuned controllers are a major contributor to performance deterioration in processindustries both directly and indirectly, as in the case of actuator cycling and eventual failure due toaggressive tuning. In this paper, an extensive review and classification of performance assessment andautomated retuning algorithms, both classical and recent is provided. A subset of more recent algorithmsthat rely upon classification of poor tuning into the general categories of sluggish tuning and aggressivetuning are compared by their diagnostic performance. The Hurst exponent is introduced as a method fordiagnosis of sluggish and aggressive control loop tuning. Also, a framework for more rigorous definitionsthan previously available of the terms “sluggish tuning” and “aggressive tuning” are provided herein. Theperformance of several tuning diagnosis methods are compared, and new algorithms for using thesetuning diagnosis methods for iterative retuning of PI controllers are proposed and investigated usingsimulation studies. The results of these latter studies highlight the possible problem of loop instabilitywhen retuning based upon the diagnoses provided by data-based measures.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The typical process control engineer is responsible for severalhundred or more loops. They must split their time betweenimplementing new assets and maintaining existing controllers(Desborough & Miller, 2001). Perhaps as a result, surveys reportthat more than 60% of controllers provide less than acceptablelevels of performance, leading to poor product quality and loss ofproduction (Desborough & Miller, 2001; Bialkowski, 1993; Ender,1993). These industrial surveys indicate that at least 30% of controlloops were increasing the variability of the process variablecompared to the use of manual control, and another 36% ofprocesses were in open loop. Among the major causes exists poorcontroller tuning, with sometimes one quarter of all loops neveradjusted from the default controller parameters (Ender, 1993).Lack of manpower and lack of tuning knowledge, combined withthe time varying nature of process and disturbance behaviors(Chia & Irving, 2003) make the disappointing results unsurprising.The field of closed-loop performance monitoring and diagnosis(CLPM&D) seeks to provide tools to aid plant personnel in

identifying poorly performing loops and suggesting remedialaction, which may include controller retuning.

CLPM&D is a maturing area with several excellent articles(Harris, Seppala, & Desborough, 1999; Jelali, 2006; Qin, 1998; JoeQin & Yu, 2007; Shardt et al. 2012) and books (Huang & Shah,1999; Jelali, 2013; Ordys, Uduehi, & Johnson, 2007) providing ageneral overview, with additional monographs available on morespecialized topics such as valve stiction detection and diagnosis(Choudhury, Shah, Thornhill, & Shook , 2006; Jelali & Huang, 2010).Some of the major causes for poor control loop performance thatCLPM&D techniques attempt to identify include oscillatory dis-turbances (Babji, Nallasivam, & Rengaswamy, 2012), sensor oractuator faults (such as in the case of valve stiction), or poorcontroller tuning. Interest in the field of CLPM&D has increaseddramatically following the appearance of Harris's (1989) paper(Harris, 1989) on the minimum variance benchmarking of loopperformance. Comparing against the theoretical minimum var-iance of the process variable provides control engineers a way toquantitatively assess the current performance of each loop. Sincethe original minimum variance benchmark only considered theperformance limiting effects of time delay, subsequent workssought to include other limitations on loop performance such asinteractions in multivariate systems (Huang, Shah, & Kwok, 1995;Harris, Boudreau, & MacGregor, 1996), right half plane zeros (Tyler

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Control Engineering Practice

http://dx.doi.org/10.1016/j.conengprac.2014.03.0050967-0661/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ91 806 742 1765; fax: þ91 806 742 3552.E-mail address: raghu.rengasamy@ttu.edu (R. Rengaswamy).

Control Engineering Practice 29 (2014) 23–41

& Morari, 1995), but especially the effects of restricted controllerstructure (Grimble, 2002). Notably, several methods exist whereinprocess and disturbance model information for a given system isused in an attempt to define an upper bound on performanceachievable by PI or PID control (Ko & Edgar, 1998, 2004; Sendjaja &Kariwala, 2009).

PID is the dominant controller algorithm within the processindustries, with one survey reporting that 97% of controllers wereof this type (Desborough & Miller, 2001), while other referencesreporting that the actual implementation of these controllers isusually in PI form (Visioli, 2005; Åström, 1995). Although theseminal work of Ziegler and Nichols occurred over 70 years ago(Ziegler & Nichols, 1942), research in the field of PID control is stillexperiencing a rapid growth in the number of publications (Wang,Ye, Cai, & Hang, 2008), and a volume by O’Dwyer contains over400 tuning correlations for PID controllers (O’Dwyer, 2006). Still,proper tuning of these controllers for optimal performance is notalways a priority that the responsible engineer has time for. Oftenthese PI controllers are retuned only in the case that oscillationshave been detected and thus remain sluggishly tuned otherwisedue to lack of manpower or expertise (Hägglund, 1999). Tuning intimes of minimal disturbance can result in loops unable toproperly attenuate common disturbances properly. Retuning isalso necessary due to process changes or changes in operatingregimes. Instrument wear and equipment fouling (e.g. Matsumura,Ogata, Fujii, & Shioya, 1998) cause the process dynamics to driftand time delays to increase. When applying PI or PID control tononlinear processes, a change in operating regime should be animpetus for parameter retuning. Controller or process parameterchanges within interacting loops will also necessitate retuning. Allof these scenarios indicate a need for CLPM&D techniques toidentify problematic loops in need of retuning.

Long before the formal advent of the CLPM&D field, there hasexisted a wide assortment of automated methods to assess andcorrect poor controller tuning, these belonging to the fields ofautomatic tuning and adaptive control (Åström & Wittenmark,1994; Isermann, Matko, & Lachmann, 1992). In fact, it is likely thatmany of the techniques discussed throughout this work werepresent in industry long before being documented in the litera-ture. It is proposed that these existing methods should be able toachieve improved outcomes if combined with other CLPM&Dtechniques. For example, many adaptive tuning methods wouldidentify excessive oscillations as being associated with aggressivecontroller tuning (Seem, 2006), when in fact they could bepresent for a variety of reasons such as valve or sensor degradationother sources of hysteresis within the loop, or externally induced,oscillations beyond the capablity of the loop to compensate. This iswhy it is recommended (Jelali, 2006) to attempt detection of otherspecific types of malfunctions, for example by applying nonlinearityindices (Choudhury, Shah, & Thornhill, 2004) or Hammersteinstiction detection methods (Srinivasan, Rengaswamy, Narasimhan,& Miller, 2005), before assigning poor loop performance to con-troller tuning. In this way, controller tuning will not be inadver-tently worsened when other problems are afflicting the loop.

Closed-loop performance monitoring and diagnosis based PI/PID tuning assessment techniques and the previously existingdata-based automated tuning methods can be classified in asimilar way. In one category, a model of the open loop processand possibly a model of the disturbance filter are used to calculatean upper bound on performance, referred to as PI or PID achiev-able performance, and then the current performance is judgedagainst this benchmark. As a result of the parameter optimizationused to predict the best achievable performance, the optimalcontroller tunings are also acquired. However, the requirementof model information by these assessment techniques is not easilyachieved. Process models are available for only a small minority of

control loops (Desborough & Miller, 2001). Therefore, use ofmodel-based techniques produces the need for either identifica-tion experiments or else the existence of specific conditions(excitation by set point changes) that may not be present in mostloops. Thus the model information dependent techniques may beseverely limited in applicability. Other types of PI/PID controllerassessment and retuning require only operating data that includesresponses to either step-type or stochastic disturbances, in orderto diagnose and/or correct tuning problems. The comparativedisadvantage of this type of assessment technique is that theycannot produce knowledge of the distance from the optimal set oftuning parameters, so that retuning with these techniquesrequires an iterative approach towards acceptable performance.However, the largest advantage of these data-based methods isthat they can be implemented without the expense and intrusionof plant identification experiments.

1.1. Contributions of this work

Data-based techniques for controller tuning assessment andcorrection have an important role to play in increasing processplant performance. This work presents review and discussion ofseveral aspects concerning data-based diagnosis and retuning andnew ideas and improvements to existing techniques are proposed.It should be noted that Jelali (2013) gives a comprehensive over-view of a wide range of topics within the CLPM&D framework andespecially that chap. 14 of Jelali (2013) includes the basis of severaltechniques improved by this work. In the following, we present

(1) A new categorization of the multitude of available loop tuningassessment and retuning techniques is proposed (Section 2).

(2) A subset of the tuning assessment techniques is selected,concentrating on several diagnosis measures which categorizepoor controller tuning as either sluggish or aggressive. First adescription of each of the selected diagnosis measures isprovided (Section 3).

(3) A novel use of the Hurst exponent as an additional tuningdiagnosis measure will also be explored. Section 4 providesdetails of this schema.

(4) Several definitions of the classifications “sluggish” and“aggressive” are examined, and a rigorous definition of theseterms is proposed (Section 5).

(5) Comparison studies rate the selected diagnosis measuresbased upon correct classification of PI controller parameterssets into the newly defined sluggish and aggressive categories.The diagnosis measures are calculated upon the responses ofsystems subjected to a step in load disturbance magnitude(Section 6).

(6) Finally, use of the selected diagnosis measures within aniterative retuning algorithm is explored (Section 7).

Throughout, issues with and limitations of the use of data-based techniques are highlighted. The problem of stability duringretuning and the insufficiency of current and proposed techniquesin this regard are stressed.

2. Classification of tuning assessment and retuning techniques

The idea of using automated performance monitoring andadaptive tuning of PID controller parameters has existed for manyyears. By 1950, Caldwell (1950) had proposed an intricate mechan-ical design for adjusting the tuning knobs of a PID controller inorder to reach Ziegler–Nichols (Ziegler & Nichols, 1942) typetuning. In Caldwell's design, the integral and derivative gains wereadjusted to be a set proportion and inverse proportion,

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–4124

respectively, of the closed-loop cycling period, while the propor-tional gain was further adjusted to prevent either continuouscycling or long aperiodic deviations. The act of continuouslyvarying a controller's parameters may seem quite different fromthe task of monitoring control loop operating data and alertingoperators when a poor condition is detected as a control looptuning assessment tool might do. However, techniques for com-pleting either duty can be similarly classified based upon theinformation they require from the loop under consideration alongwith the way in which they process this information.

Among automated tuning techniques, generally referred to asautotuning or adaptive algorithms, a first level of demarcation canbe made based upon the method used for initiation of tuning(Åström & Wittenmark, 1994; Leva, Cox, & Ruano, 2002); autotun-ing is defined as the case where the human user commands thesystem to undergo a single automated determination of newcontroller parameters; whereas adaptive controllers continuouslymonitor the process and initiate tuning based upon their internalprogrammed logic. Among other identifiable differences, autotun-ing almost certainly involves some plant experiment to obtain therequired information for tuning to occur, while adaptive controlmay or may not require plant experiments. Leva et al. (2002)highlight a hybrid case, in which adaptive tuning is not performedcontinuously, but only upon fulfillment of some requirement, e.g.the control error exceeding some threshold. This is how mostcommercial applications of adaptive tuning behave in the processindustries; Leva et al. and Jelali (2013) refer to this type ofimplementation as autotuning, while Åström and Wittenmark(1994) refer to it as adaptive control. Here, we take the notationof Åström and Wittenmark, so that controllers that even sporadi-cally retune without user intervention are called adaptive. Adap-tive controllers that retune upon detection of specific criteria usesimilar methods to CLPM&D tools that continuously monitor theprocess and then perform diagnostic functions when some certainconditions are detected.

For the remainder of this section, we will group PID adaptationmethods together with PID controller assessment methods, andsegregate this combined group of methods based upon someessential characteristics. The reasoning behind this is that severalproposed CLPM&D methods incorporate a retuning function (Jelali,2013; Goradia, Lakshminarayanan, & Rangaiah, 2005), and severalothers provide estimated optimal controller parameters as abyproduct of their controller assessment procedure (Ko & Edgar,1998, 2004). Likewise many adaptive controllers incorporatemonitoring functions to detect when retuning is necessary anduse diagnostic techniques to decide upon new controller para-meters (Seem, 2006, 1998). Fig. 1 shows the categorization oftechniques for automated PID performance assessment and retun-ing. Techniques in Category I are classical adaptation techniques,with theory and practical aspects discussed in depth in Åströmand Wittenmark (1994) and Isermann et al. (1992) among othersources. These adaptive control algorithms can require intrusivemethods such as relay feedback tests or the injection of pseudo-random binary sequences (PBRS) (Qin, 1998; Jelali, 2013), or moresimply the switch to P-only control (Shamsuzzoha &Skogestad,2010). These techniques are distinguished from techniques inCategory II, which do not require a priori models, change infeedback configurations, or artificial excitation of the loop in orderto perform, with the possible exception of necessitating a singleset-point change. This practical way of categorization of methodsis based on the demands of many industrial users that there willbe no plant tests or upsets to the process under considerationduring performance assessment and correction procedures (Jelali,2006). Whether set-point response is a feature contained innormal operating data-sets varies, since in many loops of a processplant, set-point might never be changed (Srinivasan, Nallasivam, &

Rengaswamy, 2011); however, there exist some applications wherethis data is readily available, and in other cases some of theexisting methods relying on set-point responses may be modifiedto be made applicable to load disturbance responses, as inVeronesi and Visioli (2012). Whether a technique is meant forperformance assessment or parameter retuning, it usually containssome diagnostic capability to determine the state of the loop.

It is shown in Fig. 1 that techniques in Category II are furtherdivided into 3 subcategories based upon the way in which theirfunctions are performed. Tables 1, 2 and 3 provide a survey oftechniques from categories IIA, IIB, and IIC, respectively. Withincategory IIA, features of the closed-loop response to a step changein either set-point or load disturbance magnitude are observed,and these features are combined with knowledge of the currentcontroller settings to identify the parameters within simplifiedFOPTD (first order plus time delay) or SOPTD (second order plustime delay) open-loop model structures that are assumed for theplant under control. From the open-loop models, one of the manyexisting methods for PI/PID controller synthesis is used to generatea new set of tuning parameters according to the specified goal ofthe control loop. Notable in Category IIA is the work of Veronesiand Visioli, who applied a model simplification and parameterestimation technique to develop controller assessment and retun-ing techniques, first for self-regulatory loops with set-pointchanges (Veronesi & Visioli, 2009a), then later extended thismethod to the case of distributed-lag processes (Veronesi &Visioli, 2009b), processes with integrating plants (Veronesi &Visioli, 2010b), as well as cascade control loops (Veronesi &Visioli, 2011). Most recently they adapted their method forassessment and retuning of processes mostly subjected to loaddisturbances (Veronesi & Visioli, 2012) and also presented meth-ods for tuning of set-point filters when present.

Each of Tables 1–3 contains a column describing the informa-tion required for performance assessment or retuning. This infor-mation requirement is of three types: response to a step change inset-point (sp), response to a step change in load disturbance (ld),and response to stochastic disturbances (stoch). In Table 2, meth-ods for adaptive tuning of PI and PID controllers are presentedfrom Class IIB. These rely on identification of certain features of theclosed-loop response (rise time, overshoot, period), and then

PID performance assessment and adaptation mechanisms

information requirements

(I)Require initial model or

artificial excitation

(II)Require features available fromroutine operating data such as

the response to a load disturbance step (ld), stochastic disturbance (stoch), or a single

step in set-point (sp)

tuning diagnosis categories

(IIA)Closed loop

features provide estimate of open-

loop model

(IIB)PI parameter

updates have direct relation with closed

loop features

(IIC)Closed loop

features used for pattern

classification

Fig. 1. Classification of techniques.

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–41 25

setting the controller parameters as a prescribed function of theidentified feature values. Several authors state that this procedureis similar to carrying out Ziegler–Nichols tuning, but in reality, itdiffers in several respects (i) the features are identified on normaltransient responses and not responses inflicted on the loop byexperiments, and (ii) the responses may be decaying oscillatory innature, rather than continuously cycling. Of the adaptive techni-ques of Category IIB, at least two have reached commercialsuccess, these being the Foxborro EXACT controller which wasinstalled on thousands of loops (Åström & Wittenmark, 1994) andthe Pattern Recognition Adaptive Controller (PRAC) which hasbeen implemented in 500,000 digital controllers (Seem, 2006).

Table 3 describes members of Class IIC, methods which firstperform categorization of the control loop tuning state beforesuggesting retuning action. The remainder of this work concen-trates on this category of methods. From Table 3, it is clear thatseveral of the methods involve classifying loop tuning into thecategories “sluggish” and “aggressive”. Rigorous definitions ofthese sets are absent from the literature, so Section 5 discusses aproposed classification scheme. The methods of classes IIB and IICare the ideological descendants of the pattern recognition adaptivecontrol methods pioneered by Bristol (1970, 1977). He cham-pioned these pattern recognition based approaches due to troublewith identification based adaptive controllers, which would

sometimes cycle continuously due to poor model-plant agree-ment. This type of pattern recognition adaptation was also thebasis for the aforementioned EXACT controller (Bristol & Kraus,1984). In Section 7, we present results that suggest that instabilityis also a concern for data-based retuning based on patternclassification.

Also notable within Tables 1–3 are several contributions (Seem,1998, 2006; Hensel, Ploennigs, Vasyutynskyy, & Kabitzsch, 2012;Hensel, Vasyutynskyy, Ploennigs, & Kabitzsch, 2012; Salsbury,1999, 2005) from researchers in the field of heating ventilationair-conditioning (HVAC) controls. Like the process industries, thisfield is also cost sensitive, with insufficient manpower devoted tocontroller testing and tuning, and it relies heavily upon PIregulators (Seem, 1998). HVAC systems are also nonlinear andsubject to time varying system, loop interaction, and disturbancecharacteristics (Seem, 1998), complications which suggest adap-tive tuning as necessary to maintain good control. The relevance ofperformance assessment and retuning to the field was indicatedby a large survey of research into buildings controls systemsissues, examining 40 studies which incorporated 450 controlsystems problems reported across more than 70 buildings, whichsuggested that software problems were responsible for about 1/3of reported control system issues, and that regulator tuning was asignificant contributor to this category (Barwig, House, Klaassen,

Table 1Techniques for PID controller assessment and/or tuning from Category IIA.

Reference Type of technique Informationrequirement

Model estimation Goal

Morilla et al. (2000) Adaptive PIDretuning

sp Estimates SOPTD (second-order plus time delay) closed-loop model,then uses knowledge of controller parameters to construct a FOPTD(first-order plus time delay) open-loop model

Achieve specified closed-looprelative damping

Bai and Zhang (2006)Bai and Zhang (2007)

Adaptive PI retuning sp/ld Estimates discrete time FOPTD model using recursive least squares λ- tuning

Qu and Zaheeruddin(2004)

Adaptive PI retuning sp Estimates discrete time FOPTD model using recursive least squares.Needs user specification of time delay for accuracy.

Robust set-point tracking

Sung and Lee (1999) Adaptive PIDretuning

sp Use least-squares to identify high order open loop transfer functionand then perform model reduction to obtain SOPTD model

Minimum ITAE

Veronesi and Visioli(2012, 2009a, 2009b,2010a, 2010b, 2011)

Assessment andadaptive retuning ofPID controllers

sp/ld Identifies FOPTD or SOPTD plant models using current controllerparameters and integrals of loop signals

Benchmarks loop IAE against IMCcontrol

Yu, Wang, Huang, & Bi(2011)

Assessment andadaptive retuning ofPID controllers

sp Identifies FOPTD or SOPTD plant models using current controllerparameters and integrals of loop signals

Benchmark loop IAE against IMCcontrol for various reference signals

Yu & Wang (2012) Assessment andadaptive retuning ofPID controllers

ld Identifies FOPTD plant models using measured disturbance, currentcontroller parameters and integrals of loop signals

Benchmark loop IAE, robustness,control effort, against DS-d control(Chen and Seborg 2002)

Table 2Techniques for PID controller assessment and/or tuning from Category IIB.

Reference Type of technique Informationrequirement

Parameter tuning Goal

Hensel et al. (2012a,2012b)

Adaptive PIretuning

sp Steady state gain, overshoot For remote sensors, utilizing level crossing sampling,minimize the objective function J ¼Nsc� ISE where Nsc isthe number of samples transmitted

da Silva et al. (1988) Adaptive PIDretuning usingexpert system

sp/ld Observed damping, overshoot, and periodof decaying oscillatory response

Reach desired damping, overshoot, and period of decayingoscillatory response

Litt (1991) Adaptive PIDretuning usingexpert system

sp Observed damping, overshoot andovershoot ratio, rise time, and period ofdecaying oscillatory response

Reach desired damping, overshoot and overshoot ratio, risetime, and period of decaying oscillatory response

Seem (1998, 2006) Adaptive PIretuning

sp/ld Extrema and slopes within responses tostep changes in load or set-point

Minimize IAE

Bristol & Kraus, (1984)Kraus & Myron June(1984) (Foxboro EXACT)

Adaptive PIDretuning

sp/ld Observed damping, overshoot, and periodof transient response

Desired damping, overshoot, and period

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–4126

Ardehali, & Smith, 2002; Ardehali & Smith, 2001). Moreover, theauthors of the prior references indicated their suspicion that manysystem problems reported across other categories had originallybeen due to poor controller tuning, such as in the case of actuatorlinkages being continuously cycled until mechanical failure.Within the presented diagnostic and retuning algorithms, thecontributions from the HVAC field have carefully considered thehandling of noise, the proper detection of transience, and theidentification of problems beyond the controller's capability tocorrect (Seem, 2006, 1998; Salsbury, 1999, 2005).

3. Review of diagnosis measures for controller tuningassessment

This section reviews diagnosis measures available in the literaturefor assessing controller tuning. The usefulness of such techniques isby no means restricted to PI controllers; however, the remainder ofthis work concentrates upon this simple control structure bothbecause it provides for tractable analysis and because it holds adominant place in industrial applications (Visioli, 2005; Åström,1995). The techniques considered were developed for linear pro-cesses with relatively simple disturbances. Nonlinearities may affectinterpretations of these measures by inflicting additional hysteresisin the response curves, necessitating the use of additional monitoringtools to detect the presence of sources of these confounding effects.

3.1. Controller tuning diagnosis measures which require stepchanges in load disturbance

The methods which fall into this category give best resultswhen the required controller output and/or process variable datasatisfies the following conditions: (i) the data contains the sys-tem's response to a large step change in load disturbance so thatthe data represents a closed-loop step response, (ii) when

containing multiple step changes in load disturbance, the responseto each are separable in time, and (iii) is generally free of otherdisturbance types and noise. To identify when conditions (i) and(ii) are met from the recorded process measurement and con-troller output signals, an abrupt load detection procedure such asin Veronesi and Visioli (2008) or Hägglund and Åström (2000) willbe required. Suitable filtering is specific to each technique anddiscussed in the following subsections.

3.1.1. Idle indexHägglund (1999) noted that often during a load disturbance

response, sluggish tuning will yield positive correlation betweenthe signs of the slopes in the controller output (OP) and processvariable (PV) signals, and he proposed the idle index to detect thissituation. Upon an isolated step response to a changing loaddisturbance, tpos is defined as the total time for which theincrements of the process variable and controller output havethe same sign and tneg as the total time for which the incrementshave opposite signs. Then the Idle index (II) is defined as¼ ðtpos�tneg=tposþtnegÞ. Extensive prefiltering of the data may benecessary in many practical situations. The range �0.4o IIo0.4was originally proposed (Hägglund, 1999) to indicate well-tunedloops, with values above 0.4 indicating sluggish tuning, and valuesbelow �0.4 providing no diagnosis as these could result from eitherwell-tuned or aggressive systems. Later, Visioli (2005), among otherrequirements, specified that the idle index had to take on a value ofless than �0.6 in order for the loop to be diagnosed as well-tuned.Advantages: This index was introduced solely for detection of sluggishsystems, which makes it an attractive partner for one or moreoscillation detection and diagnosis techniques. Notes: As a measurecomputed upon signal derivatives, this index is sensitive to the effectsof noise, and its calculation requires isolated and denoised featuresfrom the closed loop load disturbance response in OP and PV to beprovided. This can require a multistep procedure of load change

Table 3Techniques for PID controller assessment and/or tuning from Category IIC.

Reference Type of Technique InformationRequirement

Categories Categorization basedupon

Goal

Zhou & Liu (1998) Adaptive PIretuning

sp Long settling time with oscillations /acceptable settling time / long settlingtime with no oscillations

Spectral distribution ofsquared error

Low settling time andISE

Hong et al. (1992) Adaptive PIDretuning

sp Five zones on the K – Ti plane Overshoot, decay ratio,settling time, and timedelay

Minimum settlingtime

Seif (1992) Adaptive PIretuning

sp Combinations of the 8 elementarypatterns

Qualitative shape analysisof the process variable'sstep response

Not specified

Chia & Irving (2003) (ControlsoftINTUNE)

Adaptive PIDretuning

sp Slow / medium / fast Decay ratio, overshoot,settling time, oscillationsymmetry

User-specifiedresponse speed

Salsbury (1999) Assessment of PIcontroller

sp/ld Sluggish / no fault / oscillatory Reversal index andsettling time

Acceptableperformance

Hägglund (1999) Sluggish loopdiagnosis

ld Sluggish / not sluggish Idle index Fast response with noovershoot

Goradia et al. (2005) Assessment andadaptive retuning ofPI controllers

stoch Sluggish / Optimal / Aggressive Pattern matching of theestimated closed loopimpulse response

Minimum meansquared error

Visioli (2005) Assessment of PItuning

ld For each of K and T: high / near optimal /low

Area, idle, output indices Minimum IAE

Salsbury (2005) Assessment of PItuning

ld Sluggish / acceptable / aggressive R-index Critical damping

Howard & Cooper (2009, 2010);Howard (2009); Howard &Cooper(2010); Howard & Cooper, (2009)

Assessment andadaptive retuning ofPI controllers

stoch Sluggish / well-tuned / aggressive Relative damping index User-defineddamping factor

This paper (Section 4) Assessment of PItuning

stoch Sluggish / aggressive Hurst exponent Minimum IAE

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–41 27

detection, denoising, and exclusion of steady state data. These andvarious other implementation aspects that are discussed in Hägglund(2005) and Kuehl and Horch (2005).

3.1.2. Area indexVisioli (2005) found that aggressiveness or sluggishness of the

loop's load disturbance response is characterized by the manner inwhich oscillation decays within the controller output (OP) signal.For an isolated response to a step change in load disturbance,define A1 as the area between the first two crossings of the finalsteady-state uss of the response of the OP and Atot as the total areain the response of the OP after the first crossing of the final steadystate value. Then, the index is calculated by AI¼ ðA1=AtotÞ ¼A1ð∑1

i ¼ 1AiÞ�1. Fig. 2a illustrates the definition of these quantities.If the response of the controller output signal does not present anycrossing of its final steady state value, the index is simply set tounity. Calculation of the index requires isolated features from theclosed loop load disturbance response within the OP signal to beprovided, and additionally, it requires the load change response tobe sufficiently abrupt in nature. This will require load changedetection and abruptness quantification. Separating the effects ofthe load change from the effects of noise within the OP data alsorequires filtering or noise thresholding (Visioli, 2005). The range0.3oAIo0.7, among other requirements, was originally proposed(Visioli, 2005) to indicate well-tuned loops, with lower valuesindicating aggressive tuning and higher values indicating slug-gishness. Later, Jelali (2010) proposed to extend the method tostochastic systems by applying the index to the estimated impulseresponse of the process output. Advantages: As a measure calcu-lated upon areas, it is relatively robust to noise.

3.1.3. Output indexVisioli (2005) also reported upon how overly high reset time

could be diagnosed in oscillating loops by asymmetry of theoscillations about zero. This asymmetry occurs because thereexists a dominant pole on the real axis within the closed looptransfer function. For an isolated response within the processvariable (PV) data to a step change in load disturbance, define An

as the integral of negative areas with respect to the final steadystate value, and Atot as the sum of all areas after the first zerocrossing of the response. Then the output index is simplyOI¼ An=Atot where An ¼ Σ1

i ¼ 1Ani and Atot ¼ Σ1i ¼ 1ApiþΣ1

i ¼ 1Ani withthese quantities illustrated in Fig. 2b. The index is only calculatedin the case of an aggressive response being detected by othermethods, because it requires the presence of a decaying oscillatorytrend. Calculation of the index requires isolated features from theclosed loop load disturbance response in PV to be provided. Thebeginning and ending of the response within the data need to belocated with a load detection procedure and steady state detec-tion, respectively. For loops already detected as being aggressive,Visioli (2005) proposed that a value of OIo0.35 indicates that thereset time is too high relative to its optimal setting. Advantages: Asa measure calculated upon areas, it is relatively robust to noise.Note: This measure has only been proposed in conjunction withother diagnosis measures and possesses little utility on its own.

3.1.4. R-IndexAccording to Salsbury (2005), aggressiveness or sluggishness of

the loop's load disturbance response can be characterized byshape of the initial response peak in the process output signal.For an isolated response within the PV to a step change in loaddisturbance magnitude, Ar is defined as the area between the startof the disturbance response and its peak value, and Ad as the areabetween this peak value and the first zero crossing of theresponse. These quantities are illustrated in Fig. 2c. Then, the

R-index is calculated as RI¼ Ar=Ad. Calculation of the indexrequires isolated features from the closed loop load disturbanceresponse in PV to be provided. Diagnosis criteria were not offeredin the original work, but in the several examples presented, thevalues 0.7 and 0.85 were obtained from loops considered sluggish,values of 0.48 and 0.6 were obtained from systems consideredwell-tuned, and a value of 0.36 was obtained from data for anaggressively tuned system. Advantages: As with other measurescalculated using areas, it is relatively robust to noise.

3.1.5. PI controller assessment using multiple diagnosis measuresBy combining several of the above diagnosis measures into one

tuning assessment technique, Visioli (2005) was able to give amore specific retuning recommendation for PI controllers than a

Fig. 2. Calculation of the (a) the area index, (b) the output index, and (c) theR-index.

Table 4Visioli's proposed combination (Visioli, 2005) of the area index, output index, andidle index to diagnose where current PI controller parameters are located relativeto the set providing minimal IAE.

IIo�0.6 (low) �0.6r IIr0 (med) II40 (high)

AIo0.35 (low) K higha T high T high0.35rAI r0.7 (med) Ok K low, T low K low, T highb

AI40.7 (high) K low K low, T low K low, T high

a If OIo0.35, then T is also too high.b This combination is not provided for in Visioli’s original paper and was given

by Veronesi & Visioli (2008).

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–4128

simple sluggish or aggressive diagnosis could provide. Table 4displays the detection criteria used for that purpose. This tech-nique attempts to solve a very classical pattern recognition/classification problem: allow identification of systems from severalcategories by using the measurement of several features from eachsystem (in this case, these are features of the closed loop response)(Bishop, 2006). Advantages: The combination of several featuresprovides a more thorough diagnosis of incorrect controller tuningthen the simple indication of sluggish/aggressive.

3.2. Controller tuning diagnosis measures that can handle stochasticsystems

For the proper application of the techniques of Section 3.1 fortuning assessment, requires either: (i) the existence of a set of veryspecific conditions (i.e. a step change in load disturbance, theresponse to which can be isolated from any other inputs to theclosed-loop system), and/or (ii) copious data pre-processing inorder to remove noise and steady state data. Many loops in theprocess industries may not regularly satisfy (i), and the use of thefiltering procedures of (ii) may require tuning of noise thresholdsand other parameters that can vary loop to loop and preclude easeof large scale automation. This subsection introduces techniqueswhich are more robust to noise and whose performance may infact increase with noise exciting the closed loop response. The firsttechnique makes use of a closed loop model fit upon the processvariable (PV), while the second relies upon the autocorrelationfunction (ACF) of process variable.

3.2.1. Impulse response curve methodGoradia et al. (2005) suggested making sluggish and aggressive

tuning diagnoses based upon pattern matching the system'sestimated closed loop impulse response under disturbance to aset of 9 characteristic impulse response curves. The system'simpulse response is estimated with an AR(20) model fit to theavailable process output data, from which a finite impulseresponse (FIR) model is obtained. Since the method relies on theestimated impulse response, data from routine operation (exclud-ing actuator saturation, manual control efforts, or set-pointchanges) should be acceptable. It was originally proposed thatthe FIR coefficients would be visually matched to the characteristicresponse curves (Goradia et al., 2005), though the use of neuralnetworks for this purpose has also been demonstrated (Jelali,2010). Characteristics of sluggish impulse response curves high-lighted by the original authors included presence of offset and lackof zero crossings. For characterizing aggressive tuning, importantquantities were the magnitude of overshoot and the number ofdiscernible cycles of oscillation due to complex conjugate poles.Advantages: The use of the estimated impulse–response instead ofraw OP or PV signals may allow for many of the diagnosismeasures reviewed in Section 3.1 to have their use extended topurely stochastic systems. Note: The method as first presentedrequires a human to perform the pattern matching. One canimagine many ways to automate this method, and it is an openproblem to determine which choice of features of the impulseresponse can be used to achieve the most successful classification.

3.2.2. Relative damping indexIt has been known for many years that the autocorrelation

function is a valuable tool for control loop performance assess-ment (Bialkowski, 1993; Ziegler & Nichols, 1942; Caldwell, 1950;Hägglund, 2005; Srinivasan, Spinner, & Rengaswamy, 2012).Howard and Cooper (2010) used the fact that if the closed-loopsystem can be properly described by linear model, then theautocorrelation function of the process variable data should

theoretically have the same poles as the applicable linear model.The authors demonstrated this relation in a practical situationusing boiler level data. A set of load disturbance response data wasshown to be similar in shape to the ACF of data from a separatetime period containing lower magnitude stochastic disturbances.Based on this useful property of the ACF, the authors furtherdemonstrated sluggish/aggressive diagnoses based upon thedamping coefficient, ζ, of a second order model fit to the estimatedautocorrelation function of the system's closed loop output. Therelative damping index (RDI) is a rescaling of the dampingcoefficient with user defined bounds on aggressive and sluggishvalues, denoted by ζagg and ζslug , respectively. After the estimatedACF is obtained, the second order model is iteratively fit, wherethe number of ACF coefficients regressed upon varies betweensuccessive iterations. We found that it was preferential to fit adiscrete time model to the ACF first, which was then converted tothe continuous time equivalent in order to obtain the dampingcoefficient. The discrete time model can better capture thebehavior of the ACF since both can display a magnitude of one atlag zero, which is untrue of a continuous time model. An alter-native solution was presented in Jelali (2010), wherein a transfor-mation of the ACF curve was used to make it take the appearanceof an open loop step response, upon which the second ordercontinuous time model could be properly fit. The dampingcoefficient was used to define the relative damping index, whichis given by

RDI¼ ζ�ζaggζslug�ζ

Since the method index is calculated upon the estimated ACF,any PV data from routine operation (excluding actuator saturation,manual control efforts, or set-point changes) should be acceptable.The method has been demonstrated upon data containing bothlarge load disturbance responses and stochastic noise. For self-regulating plants, the recommended criteria were that ζo0.6indicates aggressive tuning ðζagg ¼ 0:6Þand ζ40.8 indicates slug-gishness ðζslug ¼ 0:8Þ. Similar to the characteristic closed-loopimpulse response patterns provided by Goradia et al. (2005), theauthors provided a set of autocorrelation curves for visual deter-mination of poor tuning. Advantages: By using the ACF of theprocess variable, this technique can be applied to systems afflictedby mixtures of deterministic and stochastic disturbances. Notes: Asuggested range of acceptable damping ratios is provided, how-ever, it is unclear what condition the “well-tuned” specificationindicates, whether it corresponds to minimization of a perfor-mance metric, acceptable bounds on some closed-loop responsefeature, or something else. Also, whether the diagnosis measure inthis case (RDI) behaves as expected depends on how well a secondorder model can fit the ACF.

3.3. Remarks on tuning assessment techniques

To assess the state of loop tuning, each of the above techniquesidentified some characteristic of a PV and/or OP signal, wherethese signals were from (i) an isolated load disturbance response,(ii) an estimated closed-loop impulse response, or (iii) an esti-mated autocorrelation function. For diagnosis measures definedupon one type of signal, it may be possible to extend their use tothe other signal types (with a proper adjustment of detectioncriteria). One example (Jelali, 2010) has already appeared in whichthe area index, originally designed for use on an isolated loaddisturbance response in the OP signal, was extended for use onstochastic data by calculating it for the estimated closed loopimpulse of the PV signal instead. In Section 6, the performance ofseveral of the tuning diagnosis measures reviewed here is exam-ined. The assessment method relying upon pattern matching of

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–41 29

the estimated closed-loop impulse response (Goradia et al., 2005)is not considered further because of the variety of different waysthis technique could be automated.

3.4. Applications of tuning diagnosis techniques

Kuehl and Horch (2005) demonstrated the use of the idle indexupon industrial flow loop and pressure data. It has also beenreported that the idle index has been implemented in both ABBIndustrial IT system (Hägglund, 2005) as well as a performancemonitoring system at Eastman Chemical Company for 14,000 PIDloops in 40 plants at nine sites worldwide (Ordys et al., 2007).Salsbury (2005) gave results from several examples of using the R-index applied to monitor HVAC control systems. In some priorworks, (Goradia et al., 2005; Jelali, 2010) using the tuning assess-ment methods for iterative loop retuning towards an optimalvalue was proposed. Thus far, no documentation on the applica-tion of these techniques for automated retuning in an experi-mental or industrial context has appeared.

4. The Hurst exponent method for loop tuning assessment

The previously introduced diagnosis measures rely upon thetype of features present in the OP and PV signals, with themethods based upon responses to isolated step changes in loadmagnitude more suited to this type of disturbance and thoserelying upon the estimated impulse response or autocorrelationfunctions more suited to processes continuously excited by noise.Here we introduce a technique that always begins by processingthe raw PV (process variable) data. To characterize performance ofcontrol loops under PI control, we propose the use of the Hurstexponent, calculated by means of detrended fluctuation analysis(DFA) (Bialkowski, 1993), to detect and quantify loop sluggishnessor aggressiveness. In our previous work (Srinivasan et al., 2012), aspecific scaling of this quantity, referred to as the Hurst Index, wasused to characterize overall loop performance for a wider class of

systems. Therein, it was demonstrated that the Hurst Index couldbe used to estimate performance without any process knowledge,providing an index value that replicated trends in the widely usedminimum variance index. Now, it is proposed to apply theunscaled Hurst exponent to diagnose specific types of poorperformance in PI controlled loops, namely sluggish or aggressiveloop tuning.

In the context of analyzing process signals, we rely on the Hurstexponent's ability to quantify the persistence of correlations oranti-correlations (such as present in an oscillatory signal) existingin a signal. More generally, the Hurst exponent is a measureapplied to a time series to determine the self-similarity of thesignal. A time series yt is self-similar (contains sub-units whichresemble the whole structure) if it holds that ytðkÞ � aαyðk=aÞ,where a represents the scaling factor along the x-axis (time axis),aα the scaling factor along the y-axis, and exponent α is the selfsimilarity parameter. Letting the scaling factors along the x and yaxes be Mx and My, respectively, then the exponent is given byα¼ lnMy= ln Mx, or statistically, α¼ ln sy= ln sx. The scalingexponent α is referred to as the (generalized) Hurst exponent,which can be obtained for a stationary signal by mapping the timeseries to a self similar process through use of numerical integra-tion. The method used here to calculate the Hurst exponent isdetrended fluctuation analysis (Peng et al. 1994), which isregarded to be more reliable than several alternative methods(Taqqu, Teverovsky, & Willinger, 1995). A multitude of applicationsof the DFA procedure have been found in finance, earth science,and medicine (a large list of applications is collected in theintroductions of Chen et al. (2005) and Chen, Ivanov, Hu, andStanley (2002)). For example, the Hurst exponent calculated byDFA exhibits different values between the heart-beat signals ofhealthy individuals and sick patients (Peng, Havlin, Stanley, &Goldberger, 1995).

The DFA procedure is presented in Srinivasan et al. (2012), Hu,Ivanov, Chen, Carpena, and Stanley (2001) and can be performedon noise afflicted process loops without modification since theHurst exponent calculation requires only routine operating data.

Fig. 3. Illustration of the steps of Hurst exponent calculation.

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–4130

Data containing set-point responses should be excluded from thecomputation though as these cause additional nonstationarity inthe PV which will be misinterpreted if the exponent is computedwithout modification. Fig. 3 summarizes the Hurst exponentcalculation through DFA. Briefly, the PV data, of Panel (a) inFig. 3 is first numerically integrated resulting in a nonstationarysignal as in Panel (b). This signal is then piecewise detrended bylines regressed over a data window of length L as seen in Panel (c).The root-mean-squared (rms) value of the resultant detrendeddata is termed the fluctuation, F(L), which is a function of thewindow length over which linear trends were regressed. Thefluctuation calculation is repeated over a range of window lengthsL. Finally, the Hurst exponent is calculated by fitting a linear trendbetween the logarithms of window size and fluctuation magnitudeas shown in Panel (d) of Fig. 3.

Mathematically, N samples of stationary PV signal yt , whichvaries around its setpoint, are mapped to the detrended andintegrated signal Yt ¼∑N

k ¼ 1ðyt�yÞ, where y is the mean value.The integrated series is then segmented into W windows of lengthL. On each window j¼ 1…W , a linear trend is regressed according

in order to the objective function minaj ;bjϵ2j ðLÞ ¼minaj ;bjΣ

jLi ¼ ðj�1ÞLþ1

ðYi�aji�bjÞ2 and then this trend is subtracted. After this proce-dure is repeated for every window of length L, the detrendedfluctuation on window length L can be calculated as

FðLÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1WLΣ

Wj ¼ 1ϵ

2j ðLÞ

q. The final scaling exponent reflects the

growth in detrended fluctuations as the window size increases.This can be repeating the calculation over a range of L values, andobtaining the slope from a linear fit between L and FðLÞ. To avoidspurious artifacts from occurring when the ratio of the windowsize and sample size becomes too large, it is recommended to keepLrN=4 (Hu et al., 2001).

For the purposes of sluggish and aggressive tuning diagnosis inthis work, the window length parameter L of the DFA techniquewas varied from 100 to 400 samples at increments of 10. Based onthe guideline of maximum window size of N=4, this translates to atotal requirement of 1600 or more samples in a single record (withno set-point changes or other irregularities contained in therecord). Our Matlab implementation of the Hurst exponent calcu-lation takes less than 1 s to run on a personal computer. Theproposed use of the Hurst exponent as a tuning diagnosis measureis to combine it with other loop monitoring tools. Specifically, foran isolated control loop without interactions, these other diagnosistools might include a nonlinearity or stiction detection algorithmto monitor equipment problems and a performance monitor suchas the minimum variance index or Hurst index to indicate overallloop performance. If bad performance is indicated, and nonlinear-ity is not suspected to be the cause, the Hurst exponent can beused to diagnose a possible problem due to aggressive or sluggish(although the possibility of disturbances beyond the capability of

the loop to correct would still remain in that scenario and falsediagnoses would be possible). Table 5 shows the theoretical Hurstexponent values of several basic signal types and the informationthat similar behavior within the PV signal of a control loop canyield about controller tuning. A signal appearing indistinguishablefrom white noise with no oscillations or extended deviations fromzero is generally preferable to one that contains those features.Long drifts of the mean PV value indicate sluggish tuning andtypically lead to high Hurst exponent values, while oscillationssuch as due to aggressive tuning will lead to lower Hurst exponentvalues. Over a broad range of loop types, the value of 0.5,indicative of white noise behavior is a useful standard to compareindividual loops' Hurst exponent values against. Higher valuesmay indicate sluggish behavior, while lower values will indicateoscillatory behavior (which may be due to aggressive tuning).Within Section 6 that follows, we make a comparison among theseveral diagnosis measures discussed so far.

5. Framework for the evaluation of controller tuning

To proceed with the discussion of data-based controller tuningdiagnosis methods, a framework fromwithin which to understandthe different existing diagnosis techniques is first presented. Themethods for diagnosing poor controller tuning in the literaturegenerally contain at least 2 components, these being

(i) a control loop performance metric, the optimization of whichis the goal of controller retuning, and

(ii) a data-based diagnosis measure to indicate in which directionto move the controller parameters in order to obtain a bettervalue of the performance metric.

Following these, a retuning technique to implement the sug-gestions of the diagnosis method is sometimes also considered.Items (i) and (ii) are discussed in Sections 5.1 and 5.2, respectively.

5.1. Control loop performance metrics

In (i) above, the performance metric is often chosen to be eitherthe integral of absolute error (IAE), integral of squared error (ISE),or the integral of time-weighted absolute error (ITAE), as reflectedin Tables 1–3 of Section 2. Tuning to minimize ISE punishes largedeviations from the set-point, but also produces aggressive action(Seborg, Mellichamp, Edgar, & Doyle, 2010). Shinskey (2002) hassuggested that the integral of absolute error has the closestrelationship to economic considerations. Tuning to produce mini-mal ITAE gives the most conservative set of controller parameters,corresponding to the slowest response (Seborg et al., 2010). In theCLPM&D field, the metric chosen is often Harris's minimumvariance index (MVI) (Harris, 1989), which is defined as

MVI¼s2y=s2

MV

where s2y is the variance of the process output measurement and

s2MV is the theoretical minimum variance achievable for the given

process with any linear controller. This index has a firm relationwith ISE, one being the integrated square error, and s2

y being theexpected value of the square error when the process fluctuatesabout its set-point. However, the open-loop plant time delayneeded to compute the minimum variance index is likely to beunknown information and is not essential for the diagnosis andretuning algorithms. Therefore in this work, ISE is considered itselfwith the knowledge that the MVI could be recovered with theappropriate information, and that optimizing one metric shouldoptimize the other too.

Table 5Theoretical Hurst exponent values of several signal types and what their appear-ance in process output data suggests about the state of control loop tuning.

Signal type Hurst exponent(Hu et al., 2001;Palma, 2007)

Loop tuning interpretation

Pure sinusoidal oscillation 0 Aggressive controller iscausing oscillations

White noise 0.5 Fast decay of loop response todisturbances indicates well-tuned loop

Brownian noise 1.5 Sluggish controller is failingto reject drifting disturbances

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–41 31

In the following, simulation results generated with MatlabSimulink are used to study the definitions of sluggish andaggressive tuning. Considering now only the case of systems underPI control, we introduce the symbols K and T to represent,respectively, controller gain and reset time, so that for continuoustime, the PI controller Q has the form Q ðsÞ ¼ Kðsþð1=TÞ s�1. Forillustration, a representative system with open-loop plant modelðe�5s =ð10sþ1ÞÞ is chosen, and the closed-loop is subjected to aunit magnitude step disturbance at the plant input under varioussets of PI controller parameters (K,T). Simulations are run for2000 s at each parameter set. The following results seek toillustrate qualitative behavior of systems under PI control, and itis noted that the exact parameter values where certain featuresoccur will vary from system to system. Fig. 4a shows a plot of theK,T-plane with contours of IAE for the chosen system, along withthe set of controller parameters providing global optimal IAE. InFig. 4b an additional boundary is defined to separate systemsresponding to the stepwise change in disturbance with overshootin the PV signal from systems responding to the disturbancewithout overshoot. Lack of overshoot in the step response isconsidered necessary by some authors in order for the controlloop to be considered well-tuned (Harris et al., 1999), thoughopinions on this matter vary. In this case, reaching a constrainedglobal optimum becomes the goal for retuning. In Fig. 4b, it can benoted that for the chosen system, the constraint causes a minimaleffect on the location and value of the optimum.

While most previous studies have pursued diagnoses of slug-gish and aggressive tuning, Visioli proposed a method of detectingwhere the current parameters lie relative to the global optimum,as discussed in Section 3.1.5 and in reference Visioli (2005). Thecombination of diagnosis measures applied was used in anattempt to segment the K,T-plane into the categories shown inFig. 5. A small area of the plane surrounding the global optimumpoint is considered to be the region of acceptable tuning. Thenumerous classifications of Fig. 5 make unnecessary the additionalcategorization of the loop behavior as sluggish or aggressive. Theefficacy of this method is reviewed along with the other proposedmethods within Section 6.

5.2. Proposed definitions for sluggish and aggressive

Now we consider the bulk of the literature on data-basedtuning diagnosis, which concerns itself with the detection ofsluggish and aggressive controller tuning. Within the literature,the definitions of what constitutes sluggish and aggressive tuningare almost as numerous as the techniques to detect these

conditions. One way sluggish and aggressive tuning could bedefined for PI controllers is according to the corrective actionsuggested when each of these conditions are diagnosed. Forsluggish tuning, this corrective action is to increase the propor-tional gain and/or decrease the reset time, while for the case ofaggressive tuning, it is suggested to decrease proportional gainand/or increase the reset time. Goradia et al. (2005) stated that theperformance metric, which was in their case the minimumvariance index, also referred to as the closed-loop performanceindex (CLPI), behaved as follows: “…the CLPI, not surprisingly, takesa unimodal locus. Once the proper direction to improve the controllerperformance is determined, (i.e., to make the controller aggressive ordetuned) one can proceed iteratively in that direction as long as CLPIcontinues increasing. After reaching the peak, CLPI starts to decreaseeven though we are moving in the same direction. The peak value ofCLPI is the PI achievable performance.” Fig. 6a and b below showshow the sluggish and aggressive regions are defined according tothis idea, for the case where retuning of each tuning parameter isconsidered separately. As shown in each (a) and (b), adjusting achosen controller parameter to move closer to its correspondingoptimal curve will improve the value of the performance metric.

However, as shown in Fig. 7a, when both parameters areconsidered together, there exist regions where the sluggish diag-noses for one parameter overlap with the aggressive diagnoses ofthe other parameter and vice versa. This behavior occurs becausethe curves ToptðKÞ and KoptðTÞ coincide at only a single point, thisbeing the global optimum. As an example of the resulting contra-dictory behavior, at a given parameter set in Region IV, the IAE canbe decreased either by decreasing K or by decreasing T. So, if we

Fig. 4. (a) Contours of IAE for a load disturbance response over the space of controller parameters. (b) An additional line (dashed) separates systems responding withovershoot from those which do not, and presents the constrained global optimum (x).

Fig. 5. Classification of the current parameter values relative to the global optimalset (x).

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–4132

follow the statements of Goradia et al., parameter sets in thisregion would be considered aggressive when tuning gain andsluggish if we were tuning reset time. Obviously this definitionof sluggish and aggressive is troublesome because our next stepis to find a data-based diagnosis method to generally indicatesluggish or aggressive tuning, so we need well-defined regionsto judge our measure's performance against. To resolve this, it isseen that even when leaving Regions III and IV in Fig. 7aundefined, we can still classify most of the relevant portion ofthe K,T-plane into the categories of sluggish and aggressive asshown in Fig. 7b. Here sluggish tuning corresponds to a condi-tion where the performance metric could be improved byincreasing gain and/or reducing reset time, while aggressivetuning is defined as a parameter set within the region where theperformance metric can be improved through reduction of gainand/or increasing the reset time. At this point, the usefulness ofa given data-based diagnosis measure can be judged accordingto whether it takes upon contrasting values between thesluggish and aggressive regions.

The definition of sluggish and aggressive against the oneparameter optimal curves ToptðKÞ and KoptðTÞ captures a muchlarger portion of the K,T-plane compared to if we defined thesetwo tuning classifications by judging the controller parametersrelative to their global optimal values. That the classificationssluggish and aggressive would occupy a much reduced area in thatcase is demonstrated in Fig. 8, which shows their definitions uponthe stable region of parameter space for a single system.

5.2.1. Definition of sluggish and aggressive with consideration ofovershoot

Finally, let us revisit the case of a constraint placed uponovershoot within the load disturbance response. In Fig. 4b aboundary line was plotted separating systems with and withoutovershoot in their disturbance responses. We could define sluggishtuning as parameter sets having lower gain and higher reset timethan points upon this boundary and aggressive tuning as para-meter sets having higher gain and lower reset time than pointslying on the boundary line. Adding these definitions upon theoptimal curves of Fig. 7b, we can note from the resulting Fig. 9 that

Fig. 6. For System 1, definitions sluggish of aggressive: (a) Kopt ðTÞ, the optimal gain as a function of reset time and (b) Topt ðKÞ, the optimal reset time as a function of gain.

Fig. 7. (a) Definitions of sluggish and aggressive resulting from overlapping Fig. 4a and b. (b) Proposed segregation of the tuning plane into regions of sluggish, aggressive,and undefined controller parameters.

Fig. 8. Sluggish and aggressive tuning definitions based upon parameter valuesrelative to their values at the global optimum (x).

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–41 33

(i)Controller parameters in the sluggish region provide noovershoot in their disturbance step response but have worseperformance than the constrained global optimum.

(ii) Controller parameters in the aggressive region contain over-shoot and may or may not have a better value of the objectivefunction than the constrained global optimum.

(iii) For the region of the K,T-plane shown, the constrained versionsof the functions ToptðKÞ and KoptðTÞ have converged to a singlecurve (the overshoot constraint line which separates sluggishand aggressive systems). The purpose of retuning sluggishsystems is to approach the boundary line in order to increaseperformance. The purpose of retuning aggressive systemstowards the boundary is to meet the constraint on overshoot.

5.2.2. Sluggish and aggressive definitions in the literatureWe now consider sluggish and aggressive definitions that have

been used with data-based diagnosis measures in the literature.Jelali (2010) largely followed the definitions of Goradia et al.(2005) that were considered above in Fig. 7. Hägglund (1999)mentions that well-tuned loops have a fast response with noovershoot. This idea fits into the framework shown in Fig. 9, whichincludes the presence of constraints on overshoot. Salsbury (2005)proposed critical damping of the closed loop system as the optimalcondition to aim for. Many closed loop systems will have too manypoles for this goal to be achieved with the PI controller structure.However, the use of critical damping criteria does indicate thatovershoot in the disturbance response is not considered accepta-ble to the author, and again we can view this detection method inthe framework of Fig. 9. Howard and Cooper (2010) have definedsluggish, aggressive, and acceptable tuning based upon the valueof their own diagnosis measure with user specified thresholds.Finally, as mentioned previously, Visioli (2005) largely side-stepped the issue of sluggish and aggressive categorization by

the use of a more thorough classification of controller tuning likethat shown in Fig. 5.

5.2.3. Use of the proposed aggressive and sluggish classificationframework

In the following sections, we consider the case of classifyingtuning versus the optimal IAE of the process variable's loaddisturbance response (with no constraint on the amount of over-shoot). This corresponds to the definitions of sluggish and aggres-sive of Fig. 7b, discussed in Section 5.2.2.

6. Performance of tuning diagnosis methods

In each of the works in which data-based controller tuningdiagnosis methods were introduced or discussed, results for onlyseveral examples (at most, a few sets of tuning parameters for anysingle system) were shown. However, we find that the perfor-mance of the data-based diagnosis methods can vary considerablyacross different regions of the controller parameter space (the K,T-plane). By looking at several systems across a large portion ofparameter sets likely to be encountered in practice, we hope toprovide new results on these diagnosis methods' performance.

6.1. Evaluation of several methods for sluggish and aggressivedetection:

Four diagnosis measures from the literature for detectingsluggish or aggressive tuning, as well as the Hurst exponent baseddiagnosis technique introduced in Section 4, are evaluated on anexample set of systems. The criteria for sluggish and aggressiveclassification by each method are provided in Table 6. These werechosen by a combination of guidelines from the literature and ourexperience using these methods on a set of training systems.

In order to obtain a training set of controller tuning, for each ofthe five continuous time systems in Table 7, the classifications ofFig. 7b are produced for the case of a unit step change to the inputof the corresponding disturbance filter. The systems in Table 7were obtained from several different references on PI controllertuning or performance assessment (Visioli, 2005; Morilla,Gonzáles, & Duro, 2000; Howard & Cooper, 2010; Wang & Shao,2000). Notably, this set of systems contains both a variety of timedelays and number of lags. This corresponds to a large range ofvalues of the normalized dead-time parameter (Åström, Hang,Persson, & Ho, 1992), η¼ ðdead timeÞ=ðtime constantÞ, also referredto as the process's degree of difficulty (Hensel et al., 2012). Theloop performance goal is taken to be minimum IAE of thedisturbance step response, and the curves ToptðKÞ and KoptðTÞ wereconstructed accordingly. In this way, the classifications of thetuning parameter sets (K,T) for each of the five systems inTable 7 were created according to Fig. 7b. For each of the five

Fig. 9. The proposed definitions of sluggish and aggressive regions of controllerparameters under the presence of a constraint upon overshoot (solid line). Alsoshown are Topt ðKÞ and Kopt ðTÞ (dashed and dash–dot lines, respectively), theconstrained global optimum (x) and the stability boundary (dotted line).

Table 6Detection criteria.

Diagnosis method Sluggish condition Aggressive condition

Idle index (II) II4�0.6 IIr�0.6R-index (RI) RIo0.7 RIZ0.7Damping coefficient (ζ) ζ41 ζr1Hurst Exponent (HE) HE40.5 HEr0.5Area index (AI) AI40.35 AIr0.35

Table 7Systems used in simulation studies.

Open loop plant Disturbance filter Normalized dead-time

e�5s

10sþ11

10sþ10.5

1ðsþ1Þ4

1ðsþ1Þ4

0

e�3s

ð15sþ1Þð5sþ1Þð2sþ1Þ1

ð17sþ1Þð4sþ1Þðsþ1Þ0.2

e�5s

ðsþ1Þ31

ðsþ1Þ35

e�10s

ðsþ1Þðsþ2Þðsþ3Þ1

ðsþ1Þðsþ2Þðsþ3Þ10

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systems, 50 sets of tuning parameters were randomly chosen fromwithin both the sluggish and aggressive classified portions of the(K,T)-plane. An additional 21 points were chosen from the regionsclassified as “undefined” in Fig. 7b. At each set of tuning para-meters, a closed loop simulation was run for a unit step change tothe input of the disturbance filter. Then each of the five diagnosismethods of Table 6 was applied to the resulting sampled processoutput and controller signals recorded once a second for 2000samples. Simulations were conducted using Matlab Simulink.

The results for each diagnosis method are shown in Table 8,and when compared, it is seen that under the conditions studiedthe best choice for diagnosing aggressive and sluggish tuningversus the framework of Fig. 7b is either the R-index or thedamping coefficient ζ, since they outperform the other methodsin detecting both sluggish and aggressive systems. For eachdiagnosis measure, an approximately equal percentage of sluggishand aggressive systems are correctly detected (within a fewpercent). This indicates that changing the values of the detectioncriteria would have little positive effect on the overall perfor-mance. To see this fact, it is useful to consider each diagnosismeasure acting as a simple classifier, with a threshold valueseparating sluggish and aggressive diagnoses. Changing thethreshold can increase the correct diagnoses of one type of system(either sluggish or aggressive), but only at the expense of decreas-ing correct diagnoses of the other system type. Also, these resultsare for the case of a single isolated load disturbance. As discussedin Sections 3 and 4, the ability of each method to handle signalswith other types of disturbances varies markedly. Notably, theHurst exponent is expected to perform better in the case ofstochastic disturbances.

It is notable that the training parameter sets were chosen fromone of three categories (sluggish, aggressive, and undefined),while the diagnosis methods only classified tuning into the twocategories “sluggish” and “aggressive”; however, the reason forthis are two-fold: (i) most of the tuning parameter sets in the“undefined” category are by no means well-tuned, and by puttingthese parameter sets into one of the categories “sluggish” or“aggressive”, the retuning algorithms introduced in the nextsection can seek to achieve better tuning for these systems;(ii) Fig. 10 Parts b–d show that there is no range of values forthe considered diagnosis measures that can define a localized“well-tuned” region of the (K,T)-plane. That is because values ofthese diagnosis measures present open-contours, which contrastto the closed contours of the performance metric IAE (Part (a) ofFig. 10). Therefore, we suggest, that the diagnosis measures whoseplots are in Fig. 10(b–d), when used alone, are better suited forstrictly making sluggish or aggressive diagnoses, since a categoryof “well-tuned” would inevitably contain systems far away fromoptimal performance when using these diagnosis measures.

For the area index, however, the use of a range of this diagnosismeasure's values to define a well-tuned region on the K,T-plane

makes more sense relative to the other methods. Fig. 11 shows aplot of AI contours (solid lines) overlaid upon the contours of IAE(dotted lines) for System 3. At the higher values of T in this plot,the contours have almost completely converged and the combinedline separates systems having an AI value of unity from thosehaving AI values of close to zero. At lower values of T, the contoursof AI have opened up and now take on intermediate values. Theseintermediate values occur near the inner contour of IAE valueswhich makes this diagnosis measure more attractive than theothers to detect close to optimal tuning. In previous works, valuesof the AI between 0.35 and 0.7 were used to detect close tooptimal tuning (Visioli, 2005; Jelali, 2010). Despite the advanta-geous behavior of the area index at higher T values, from Fig. 11, itis also seen that when moving from the point of optimal tuningtowards the origin (low K and T values), the IAE increases in value,yet the contours of AI do not close. This means the index will giveincorrect detections in this region if a “well-tuned” classification isdefined. From the foregoing analysis, it is recommended to use thetuning diagnosis measures only to provide general diagnoses ofsluggishness or aggressiveness, and not try to indicate well-tunedvalues, since these methods will often give this indication at farfrom optimal values of the performance objective. When used inthis way, these methods will less satisfy the needs of continuouscontroller monitoring (because they will constantly indicate tun-ing is either sluggish or aggressive), but are instead more applic-able for the iterative controller retuning methods introduced inSection 7.

6.2. Evaluation of alternative method for tuning diagnosis

As previously mentioned, Visioli (2005) introduced a tuningdiagnosis technique which combined three indices to qualitativelyestimate the position of the controller's current parameter setversus that of the global optimal tuning set. Since the results ofthree separate indices are combined to make each diagnosis, aspreviously illustrated in Fig. 5, many possible categories ofdiagnoses can result from the method. Fig. 12 shows the resultsof this method applied to a grid of points on the K,T-plane. To aidin visualization of this method's performance, the suggesteddirection of retuning implied by the diagnosis at each point isdepicted with an arrow. Diagonal arrows indicate that both tuningparameters have been diagnosed as away from their optimalvalues, while vertical and horizontal arrows indicate that the needfor retuning only one of the parameters has been detected.The plots show large regions of the K,T-plane where the methodworks correctly for either one or both parameters. Nevertheless,a problem arises for both systems near a portion of the stabilityboundary (in the bottom center of each plot), where the methodrecommends decreasing T. The result occurs because even thoughthe response is decaying oscillatory in this region, the idle index isin the medium range according to the criteria of Table 4. Depend-ing on the magnitude of the step taken, following the diagnosismethod's advice in this region will likely bring the control loop toan unstable setting, which of course is highly undesirable. Thenext section will compare the performance of autotuning algo-rithms which utilize either: (1) one of the five diagnosis methodsevaluated for sluggish and aggressive tuning detection in Section6.1, or (2) the tuning classification method of Visioli.

7. Data-based retuning of PI controllers

Previous works in the field of CLPM&D have also introducedalgorithms for automated retuning based upon model-free assess-ment of controller tuning (Jelali, 2013, 2010; Goradia et al., 2005);the algorithms presented here contain additional logic to deal with

Table 8Results of classification of 600 training sets of parameter pair (K,T) into categories“sluggish” and “aggressive”.

Diagnosismeasures

Sluggishsystems

Aggressivesystems

Systems in undefined regions

% Correctlyidentified

% Correctlyidentified

% Identified assluggish

% Identified asaggressive

HE 85.6 81.2 19.0 81.0II 80.4 75.2 61.0 39.1AI 66.8 71.2 28.8 71.2RI 93.6 97.2 14.3 85.7ζ 92.4 95.6 24.8 75.2

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–41 35

cases where the diagnosis measures are incorrectly indicating theproper direction for adjustment of controller parameters. It isstrongly noted that the tuning maps shown in the previoussections were for illustration only and are unavailable in practiceunless the process and disturbance models are known, which isusually not the case (hence the use of data-based diagnosismeasures). Therefore, in this section, we assume that the onlyinformation available is routine operating data which can be usedto calculate the values of the tuning diagnosis measures. Thesimulation studies of this section are for the case of a single unitstep in the magnitude of the disturbance variable, and it is

assumed that the process output, controller output, and set-point data are available. The set-point data is solely used toexclude portions of the process variable (PV) and controller output(OP) recordings where set-point changes have an effect. Thesuitability of each detection method for other signal types (e.g.those driven by stochastic disturbances) was discussed in detail inSections 3 and 4.

To retune controllers based upon the diagnosis methods dis-cussed in Sections 3–6, two new algorithms were developed,whose purpose is to obtain better performance with regards to ametric such as IAE or ISE. One algorithm is for use with the user'schoice of a single data-based diagnosis measure, and the secondalgorithm relies upon the diagnosis technique of Visioli (2005)where the AI, II, and OI are used in complementary fashion. Thefirst algorithm (an improvement of one proposed in Jelali (2010)),shown in Fig. 14, utilizes a user choice of one of the diagnosismeasures appearing in Section 6. Considering the case where IAEis the performance metric, then for whichever diagnosis measureis chosen, the detection criteria of Table 6 are used to determine adiagnosis of either sluggish or aggressive tuning at the initialtuning parameter set (if a different performance metric is chosenthen the associated detection criteria should be developed).Depending on the diagnosis at the original controller setting, a20% change in the controller gain K is made. If the result is adecrease in IAE, a new diagnosis is made. If a switch in diagnosisoccurs relative to the first one, the diagnosis measure is indicatingthat a minimum of IAE with respect to K was just crossed. Thispossible minimum is explored with smaller 10% steps in controllergain K. If the first change in gain led to an increase in IAE (a “badmove” occurred), the algorithm returns the system to the originalparameter set.

Fig. 10. For System 5 subjected to the case of a unit load disturbance change, plots of integral of absolute error (IAE) and three different controller tuning diagnosis measures(Hurst exponent, R-index, ζ). In each plot the dotted line represents the stability boundary and the (x) is point where minimum IAE is obtained.

Fig. 11. Contours of the area index overlaid upon contours of the integral ofabsolute error.

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–4136

The algorithm tunes one parameter K or T at a time, switchingparameters after either an optimum point is detected or an IAEincrease occured. The proposed method is supervised by theperformance metric (in this case IAE), meaning that if thisperformance objective is worsened by moving in one direction,the algorithmwill not continue in the same direction, regardless ofthe advice of the diagnosis measure. As previously suggested(Goradia et al., 2005; Jelali, 2010), a basic step of 20% in the variedparameter was used in each iteration. Smaller steps of 10% wereused to explore both the points providing minimum IAE in thevaried parameter and also the boundaries between sluggish andaggressive diagnoses. To obtain faster convergence, larger stepsizes should be considered in the case that the changes are carriedout manually under human supervision.

The performance of the algorithm was explored with each ofthe diagnosis methods of Section 6 (Hurst exponent, idle index,area index, R-index, and damping coefficient ζ). Results wereobtained for each choice of diagnosis method at each of the 250starting sets of parameters. Two cases were attempted: (i) tuningboth gain K and reset time T, and (ii) only retuning gain. Theretuning was applied to noise free simulations of the systemresponse to a unit step at the input of the disturbance filter. OPand PV signals were sampled at a rate of once per second duringsimulations of 2000 s. A set of 50 starting points for PI controllertuning on the K,T-plane were randomly generated for each of the5 systems of Table 7. The starting points (K,T) were chosen tosatisfy IAEðK; TÞo4� IAEmin, where IAEmin is the PI-achievable IAEfor each system for the disturbance described, in order to providemoderately (but not extremely) poor initial tuning. All simulationswere performed using Matlab Simulink.

Table 9 contains one example of the use of the Hurst exponentalong with the retuning algorithm of Fig. 14 to search for better

controller parameters. When a switch in the diagnosis occurs inIteration 1, the algorithm neither continues in the original direc-tion to check if a local minimum has been traversed, nor makes afull 20% step in the reverse direction as now indicated by thediagnosis measure. Instead the boundary between diagnosisclassifications is explored with a 10% step. Even though a bettervalue of the performance metric was reached in the first iteration,the retuning algorithm will not continue in the original directionto search for local minima when it is against the advice of thediagnosis measure, out of concern for stability. In this case thechoice to reverse direction in fact provides us with the localminimum to within 10%. After reversing directions, the algorithmswitches the tuning parameter to reset time T. In this case, whenincreasing the reset time according to the aggressive diagnosis, boththe initial 20% move and a reduced 10% step in T cause an increase inIAE. Since no further improvement in IAE is occurring when makingmoves in either K or T according to the recommendations of thediagnosis measure, the algorithm terminates. The effects upon PV ofthe iterative retuning process detailed in Table 9 are illustrated inFig. 13. Although the results of the table indicate substantialimprovement in performance, from the plot of PV it appears thatthe response is still oscillatory and more improvement should bepossible. This is a characteristic of the data-based retuning methodspresented here, in that the significant improvement is usuallyachieved, but the algorithm still terminates far from the optimalcontroller parameter set due to the desires to limit the number ofiterations and to make parameter changes only on the basis ofsupport from the diagnosis measure and performance metric.

A second retuning algorithm is presented in Fig. 15, whereinthe determined direction of retuning relies upon the diagnosiscriteria of Visioli (2005). During the initial step, either a 20% in asingle controller parameter or 10% in each controller parameters is

Fig. 12. For (a) System 3 and (b) System 5 each subjected to the case of a unit load disturbance change, diagnosis of controller tuning using the method of Visioli (2005).

Table 9Demonstration of the retuning algorithm of Fig. 14 for the case of isolated load disturbance changes afflicting System 2 with the Hurst exponent chosen as the method fordiagnosis of sluggish or aggressive tuning.

Iteration K T IAE HE Comments

0 3.53 16.2 9.8 0.486 HE indicates aggressive turning, decrease gain 20%.1 2.83 16.2 5.7 0.514 Diagnosis has changed, explore boundary at 10%.2 3.18 16.2 5.3 0.499 Local minimum found within 10%. Now switch and increase T by 20% according to the aggressive diagnosis.3 3.18 19.5 6.1 0.604 IAE has increased, try 10% step instead.4 3.18 17.8 5.6 0.552 IAE is still increased, return to best parameter set.5 3.18 16.2 5.3 0.499 Neither changing K nor T can give decreased IAE in indicated direction of retuning. Terminate.

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–41 37

taken in accordance with the diagnosis criteria of Table 4. If thismove resulted in an increase in IAE, the parameters were returnedhalf way towards the original set. If at this smaller step, IAE wasstill increased compared to the original set, the search is termi-nated and the controller settings are returned to the optimalparameter set encountered thus far, against the advice of thediagnosis method (which is providing incorrect results). Thus thistuning method is supervised by IAE. The algorithm can alsoterminate due to a diagnosis of near optimal tuning according tothe criteria of Table 6. One example application of algorithm ofFig. 15 is presented in Table 10. It is noted that tuning according tothe diagnosis criteria without considering changes in the perfor-mance metric, the parameters tend to cycle back and forthbetween differently classified regions leading to a large numberof iterations and often times instability. Instead, as shown inIteration 2 of Table 10, when the proposed algorithm with IAE

supervision encounters an indication that parameter changes needto be reversed, the controller is taken to the best parameter setencountered so far, and the algorithm is terminated. In this way,cycling between closely located sets of parameters does not occur.

Full results of all of the algorithms applied to the entire set of250 retuning trials are displayed in Table 11. The results in theupper portion of the table refer to the algorithm of Fig. 14 whichwas applied to each of the possible choices of diagnosis measures.One notable observation is that retuning of reset time in this firstalgorithm added marginal benefit; most of the improvement inperformance was due to the initial retuning of gain alone. Thealgorithm terminated in a similar number of iterations for all fiveof the diagnosis measures tested, so no advantage of any technique

Fig. 13. PV of System 2 corresponding to the results in Table 9.

Fig. 14. Proposed retuning algorithm using one of the sluggish/aggressive con-troller diagnosis methods (DM). Fig. 15. Proposed retuning algorithm using the controller diagnosis method of

Visioli (2005).

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is seen in this respect. Most notable from Table 11, it is seen thatinstability could be a major problemwhen retuning with several ofthese diagnosis measures. Specifically, the Hurst exponent,R-index, and damping factor ζ all bring the systems to unstableparameter sets numerous times. While the area and idle indicesdid not have this problem in the trials studied, there is noguarantee that these methods will keep the loop at stable para-meter sets for all systems. But due to the extreme problemsencountered by the three other methods, it is recommended toonly attempt this type of automated retuning with one of themore stable methods, either the AI or II. It is noted that thestability issue has not previously been discussed in the literatureon these types of data-based controller diagnosis and retuningmethods, and the issue needs more exploration. It is noted that theinstability problem occurs when a set of parameters is diagnosedas sluggish (either correctly or incorrectly) and the resultantincrease in gain or decrease in reset time moves the parameterset across the stability boundary. This implies that the problemcould be mitigated by adjustment of the sluggish and aggressivedetection criteria to shift the boundary of these classificationsaway from the stability line. However, even this solution could notguarantee to eliminate the problem, and it would be at theexpense of algorithm performance.

Also, the lower section of Table 11 contains results from thesecond retuning algorithm of Fig. 15, the IAE supervised algorithmwhich makes retuning decisions based on the criteria of Table 4.For contrast, the original version of this algorithm, proposed inJelali (2010), also for use along with the diagnosis method ofVisioli (2005), was implemented without IAE supervision. Lack ofvirtual supervision is ill-advised, since retuning can occur repeat-edly in a given direction even though the performance metric tobe minimized becomes worse at each step. As shown in Table 11,this leads to instability: 35 out of 250 trials were brought tounstable parameter sets for the algorithm without supervision,compared to only 2 trials in total for the IAE supervised method.Also, lack of supervision by the performance metric led to morethan double the number of average iterations, for a small improve-ment in final IAE reached compared to the supervised algorithm.

Using Table 11 to compare between the algorithms of Figs. 14 and15, it is concluded that tuning with the algorithm using thediagnosis criteria of Visioli provides the most benefit comparedto the other algorithm regardless of the choice of diagnosismeasure. This result indicates the utility of combining the infor-mation available from several diagnosis measures. In the future,other combinations of diagnosis measures should be considered tosee if further improvements with respect to performance andstability can be gained. Redundant information between differentdiagnosis measures could be investigated by use of singular valueddecomposition, and support vector machines or other learningalgorithms could then be applied to synthesize a final diagnosisbased on the values of the set of diagnosis measures. Future workmay also find that certain diagnosis techniques have betterperformance when dealing with data having certain disturbancetypes not considered here. For example, the Hurst exponent and ζare both diagnosis measures designed for use with stochasticdisturbances, as the underlying calculation of the detrendedvariance and the autocorrelation function, respectively, are statis-tical techniques for use with random data. Therefore, efficacy ofthese methods should improve for processes which are respond-ing to disturbances that occur more frequently (having overlap intheir response) and varied in direction and magnitude than thesingle step in load magnitude used in this study.

Above all, the results of this section raise concerns with thestability of automated data-based techniques and more generally,retuning based upon heuristic measures of process behavior. It iswell known that traditional adaptive systems require persistentlyexciting input in order to guarantee global exponential stability,and it has been clearly demonstrated that the lack of excitation canlead the adaptive controller to instability ((Åström & Wittenmark,1994) for example). The methods reviewed here, relying solelyupon closed-loop features, are considered advantageous for indus-trial implementation because of their low requirements for experi-mentation and excitation. However, very little is known about thesetechniques with respect to stability. We have shown some veryspecific examples of how instability can be reached, but morerigorous mathematical foundations explaining stability of these

Table 10Demonstration of the retuning method proposed for use with the diagnosis method of (Visioli, 2005) for the case of isolated load disturbance changes afflicting System 5.

Iteration K T IAE AI II OI Comments

0 3.45 18.2 5.3 1.00 0.380 Increase gain and lower reset time according to AI, II.1 3.80 16.4 4.3 1.00 �0.349 IAE has decreased, so continue retuning according to AI, II based diagnosis.2 4.18 18.0 4.3 0.01 �0.787 0.002 Diagnosis method is indicating to reverse last move. Instead terminate at best parameter set reached.3 3.80 16.4 4.3 1.00 �0.349 AI, II indicate to change turning, but the suggested move has previously been attempted. Terminate.

Table 11Summary of retuning results for 5 systems retuned from 50 starting points.

methods used Retuning using both parameters K adjustment only

Mean % change inIAE

Meaniterations

# of trials instabilityreached

Mean % change inIAE

Meaniterations

# of trials instabilityreached

I. retuning based upon a single diagnosismeasure

�20.5 7.2 38 �18.3 5.4 38

HE �23.3 6.8 0 �23.2 5.8 0II �25.9 7.2 24 �24.4 5.6 24RI �18.1 6.7 0 �17.8 5.6 0AI �27.8 8.4 17 �25.0 5.6 17Zeta

II. turning based on Visioli's diagnosismethod

�42.0 14.5 35

unsupervised �37.8 6.3 2with IAE supervision

III. theoretically achievable �47.5

T. Spinner et al. / Control Engineering Practice 29 (2014) 23–41 39

methods are lacking both here and in the literature. As a practicalmatter, as proposed above, perhaps the combination of severaldiagnosis measures can also yield a better prediction of whetherretuning in a considered direction will result in instability. It hasbeen demonstrated here that the use of supervision by a perfor-mance metric is another way to mitigate the effects of erroneousdiagnoses when performing heuristic data-based retuning.

8. Conclusions and future work

This work has examined several previously proposed data-based methods for PI controller tuning diagnosis and retuning.Most of the existing techniques for controller diagnosis classifypoor tuning as sluggish or aggressive, but a rigorous definition ofthe meaning of these terms was previously absent. Here, weintroduced a definition of the sluggish and aggressive categoriesupon the controller's parameter space that can be used to evaluatethe performance of data-based controller tuning diagnosis tech-niques. The Hurst exponent as a diagnosis measure for sluggishand aggressive loop tuning was introduced, and its performancewas deemed comparable to the other available diagnosis mea-sures. By examining behavior of the diagnosis measures across therelevant part of the K,T-plane, we gained insight into how thesemethods will perform under different tuning conditions. Finally,new adaptive algorithms for PI controller retuning were intro-duced that take advantage of the data-based diagnosis measures,and it was found that the stability is a major concern that couldlimit the applicability of such techniques. Among the proposedautomated retuning techniques based on data-based diagnosismeasures, it was found that diagnosis based on the combinedinformation from several measures provided superior perfor-mance to any single measure considered alone. This indicatesthat, in the spirit of using the values of multiple feature types forpattern recognition problems, future work in the area of data-based tuning diagnosis should concentrate on developing methodswhich synthesize information from several diagnostic measures.

Acknowledgments

The authors express appreciation for the assistance of thereviewers in improving this work. Funding for this work wasobtained through National Science Foundation GOALI Award#0934348.

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