survey of industrial optimized adaptive controlof adaptive systems became quite popular in the...

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2012; 26:881–918Published online 24 July 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/acs.2313

Survey of industrial optimized adaptive control

Juan M. Martín-Sánchez 1,*,†, João M. Lemos 2 and José Rodellar 3

1UNED/DIEEC, Juan del Rosal 12, 28040 Madrid, Spain2INESC-ID/IST(UTL), R. Alves Redol 9, 1000-029 Lisbon, Portugal

3UPC, Applied Mathematics III Department, Gran Capitán s/n, 08034 Barcelona, Spain

SUMMARY

This survey paper aims, to present in a systematic way, the ‘state-of-the-art’ of a class of so called optimizedadaptive control methodologies, where adaptive systems theory is complemented by optimal control the-ory. It is based on literature published since 1958 and emphasizes results with proven industrial relevance.After recalling previous adaptive systems, results that had prepared the ground for the new ideas, the anal-ysis departs from the work on adaptive predictive control in the mid 1970s, which provided the conceptualbasis for the development of this kind of technique. Well-known design methods, both from the stability andoptimization perspectives, along with their results, are interpreted in an intuitive way in order to understandhow control law and the adaptive mechanism complement each other to produce mature controllers thatcan reliably solve the industrial process dynamic stabilization problem. Surveys of industrial applicationsfrom both design perspectives are presented and industrial products based on optimized adaptive control arecited. Copyright © 2012 John Wiley & Sons, Ltd.

Received 2 February 2012; Revised 7 May 2012; Accepted 31 May 2012

KEY WORDS: optimized adaptive control; adaptive systems stability; predictive control; expert control;ADEX control; MRAS, model reference adaptive systems; self-tuning control; industrialapplications

1. INTRODUCTION

Producing a survey on adaptive control techniques would be an almost impossible task because ofthe large volume of works that have been published on this subject in the last half century. Althoughwe will refer briefly to this vast literature, our survey will be focused on a subset of adaptive controlmethods which uses optimal control theory and has provided reliable adaptive tools for industrialprocess control. We will also emphasize the relationship of these techniques to adaptive predictivecontrol (APC) as introduced by the first author in the mid 1970s.

This subset of adaptive control methods is based on a predictive model of the plant whoseparameters are adjusted in real time by an adaptive mechanism in such a way that the predictionerror converges towards zero. Additionally, this predictive model is used at every control instant toderive a desired future process output trajectory by minimizing an index that is chosen according toa performance criterion. Thus, adaptive systems theory is complemented by optimal control theoryto derive a class of methods that we will call ‘optimized adaptive control’.

The introduction of APC was significant in the development of the theory and practice of processcontrol. The dominating principle of negative feedback for the control law in the previous contextwas substituted by the new principle of predictive control. At the same time, the need for processidentification, always present in the previous state-of-the-art in control theory, was replaced by the

*Correspondence to: Juan M. Martín-Sánchez, UNED/DIEEC, Juan del Rosal 12, 28040 Madrid, Spain.†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

882 J. M. MARTÍN-SÁNCHEZ, J. M. LEMOS AND J. RODELLAR

requirement for ‘process identification with a view to control’, demanding only an orthogonalitycondition to be verified by the adaptive mechanism.

These new ideas arose in a changing technological context, where the introduction of digitalcomputers represented a revolution in the control field. Existing theories were translated toalgorithmic form and new developments for discrete time systems were formulated. However, thenew context offered the possibility for the implementation of new theoretical concepts suitable fordigital computation and capable of outperforming the previous ones.

Although the history of adaptive control is complex with many competing and parallel develop-ments, it is generally agreed that the first attempts to use adaptive control date back to 1958, whenKalman [1] proposed a self-tuning regulator (STR) and Whitaker, Yarmon, and Kezer presenteda ‘Design of a Model Reference Adaptive System (MRAS) for Aircraft’ (MIT, InstrumentationLaboratory Rept. R-164, September 1958).

A first generation of adaptive systems, named MRAS, tried to stabilize the dynamic characteristicsof a feedback control system in the presence of plant parameter changes. Early developments of thiskind of system, as previously cited, tried to implement autopilots aimed at high performance aircraft[2, 3]. Also a gain scheduling approach was used for this kind of application [4].

A complete survey of continuous and discrete time MRAS was presented by Landau in 1974[5]. In addition to local optimization and estimation methods, because of the nonlinear characterof MRAS, Lyapunov [6] and Hyperstability [7] methods were used to derive adaptation algorithmsin an attempt to ensure closed-loop stability. MRAS, in this survey and others reported later [8, 9],used the basic negative feedback control strategy, inherited from the theory developed for continuoussystems. This was a limiting stability feature which restricted their industrial application.

In the new field of digital systems, discrete time modeling of processes introduced, for the firsttime in control theory, the possibility of predicting the future evolution of process outputs in realtime. This capability allowed first the introduction of the minimum variance controller by Aström[10], derived in the context of optimal control theory. Then, from this controller formulation, STRsand controllers were derived [11, 12].

The minimum variance controller can be considered as the actual precursor of the methodologyof predictive control, which, for the first time, fully exploited the possibilities of real time predictionfor control purposes. This methodology was introduced by Martín-Sánchez already in the contextof APC [13–15].

This paper recalls the state-of-the-art in adaptive control prior to the introduction of APC andpresents the development and current state-of-the-art in optimized adaptive control methods, alreadyapplied in industry, from the basis established by the original formulation of APC.

Two different approaches in the development of optimized adaptive control can be identified.The first one is based on a stability perspective that had as background the stability approachused previously by MRASs. The second one is based on an optimization perspective that had asbackground the first STRs and controllers. These two kinds of optimized adaptive controls weredeveloped in a completely independent manner and we must consider both of them in our survey, aswell as other results on optimized adaptive control.

The paper is organized as follows:Section 2 recalls the background to optimized adaptive control, that is to say, the state-of-the-art

in adaptive control prior to the introduction of APC, which was represented by MRASs and STRsand controllers (STR and STC). Section 3 recalls the introduction of APC and analyses the driverblock concept, essential in predictive control, and how APC results from the simple combination ofa predictive controller and an adaptive system under a stability perspective, setting the basis for thedevelopment of optimized adaptive control methodologies.

Section 4 presents a survey of optimized adaptive control methodologies developed from thestability perspective. It recalls firstly the stability results that led historically to the develop-ment of the body of APC stability theory which supports these kinds of industrial optimizedadaptive control systems. Next, the section describes the synthesis of the driver block and theadaptive mechanism based on conditions for both which can guarantee the desired stabilityresult. An intuitive interpretation of the result obtained is also described. Finally, it recalls thesteps involved in the development and validation of standard systems for the optimized adaptive

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2012; 26:881–918DOI: 10.1002/acs

SURVEY OF INDUSTRIAL OPTIMIZED ADAPTIVE CONTROL 883

control of industrial plants and the applications carried out by a first generation of these kindsof systems.

Section 5 presents a survey of optimized adaptive control methodologies developed from theoptimization perspective. It recalls firstly the drawbacks related to the basic self-tuning designs anddescribes the optimal control theory and optimization concepts used to overcome these limitationsin the context of optimized adaptive control. Next, it describes the control laws used in the mostrelevant design alternatives and how adaptation is embedded in the corresponding control algo-rithms. Although in this case, rigorous global stability and convergence results are not available,the stability and convergence issues are analyzed. Finally, applications of these systems in a widevariety of industrial fields are reported.

Section 6 presents a comparative analysis of the results obtained using both design perspec-tives, and Section 7 presents new methodological results introduced to utilize available processoperating knowledge within the controller function itself. Thus, the new adaptive predictive expert(ADEX) controllers integrate various domains within their structure, enabling the configuration andapplication of APC and expert control as required, based on available process knowledge. Nonlinearpredictive controllers and their adaptive versions could also be integrated in this kind of structure.Applications of an ADEX-based standard product for the optimized control of industrial processesare also reported. The conclusions are presented in Section 8.

2. BACKGROUND TO OPTIMIZED ADAPTIVE CONTROL

2.1. Model reference adaptive systems

The problem of self adjusting the parameters of a controller in order to stabilize the dynamiccharacteristics of a feedback control system, when drift variations in the plant parameters occurred,was the main motivating factor for the development of adaptive control techniques. A first generationof adaptive systems became quite popular in the research community under the name of modelreference adaptive systems (MRAS). The state of the art of MRAS, prior to the introduction ofpredictive control, was presented in a famous survey paper in 1974 [5]. The basic scheme of anMRAS is presented in Figure 1.

The reference model gives the desired response of the adjustable system, and the task of theadaptation is to minimize a function of the difference between the outputs, or the states, ofthe adjustable system and those of the reference model. This was performed by the adaptationmechanism that modified the parameters of the adjustable system or generated an auxiliaryinput signal.

Figure 1. Basic configuration of a model reference adaptive system.



Model reference adaptive systems had a ‘dual’ character in the sense that they were used forapplications in the field of adaptive control systems and also for applications in the field of processidentification. When used for process identification, the process played the role of reference modeland the adjustable model was replaced by an adaptive model, whose parameters were adjusted bythe adaptation mechanism in order to obtain a dynamic characteristic similar to that of the process.

Three different configurations of the MRAS for process identification were possible depending onthe adaptive model structure: (1) a ‘parallel’ configuration, also called ‘output error method’; (2) a‘series-parallel’ configuration, also known as the ‘equation error method’ [16–18]; and (3) a ‘series’configuration called ‘the input error method’. A unified approach to the analysis and synthesis ofthese different structures was presented from the stability perspective in [19] using hyperstabilityconcepts [20]. The stability result guaranteed that, after a given time instant, the adaptationmechanism ensures that the absolute value of the error considered in Figure 1 is bounded by alimit that depends on the level of noise and perturbations acting on the process. If no noise orperturbations are involved, the error tends asymptotically towards zero.

Although the synthesis of stable parallel and series MRAS for process identification requiredcertain knowledge of the process parameters and the verification of a positivity condition [7,21], theseries-parallel structure was free of these conditions [22] and was later used on the APC formulation,as considered in the following of this paper.

When applied to adaptive control systems, MRAS did not consider a change in previous controlstrategies but only an adaptation of the controller parameters to prevent a decrease in the globalperformance of the control system in the presence of changes in the process dynamics. Thus,MRAS used traditional negative feedback methodology as the basic control strategy and adapted thecontroller parameters using different algorithms. This is one of the reasons for the lack of successfulindustrial control applications of these techniques. In the conclusions of [5], it is written that ‘Thegrowth of applications of MRAS will be determined in part by new improvements of the designmethods and the development of implementation techniques’.

2.2. Self-tuning regulators and controllers

The first attempt to design a self-tuner was described in [1]. The algorithm proposed combined theleast squares estimation of the parameters of a deterministic linear difference equation with a controllaw to be selected in accordance with the application considered in each case. Lack of adequatecomputer technology limited the impact of these ideas, although a dedicated analog computer wasbuilt at the time to test the controller.

The next step consisted in the inclusion of stochastic disturbances in the plant model and theformulation of the control problem as the minimization of the plant output variance at the earliestsample affected by the current manipulated variable move [11]. Specifically, the plant, assumed tobe linear, time-invariant and SISO, is represented by the autoregressive-moving-average model withexogenous inputs (ARMAX) model

A�q�1

�y .k/D B

�q�1

�u .k � d/CC

�q�1

�e .k/ (1)

where A, B , and C are polynomials in the backward shift operator q�1 [23], with A and C monicandC hurwitz, k is an integer index denoting discrete time, d > 1 is the input-output transport delay,y is the measured plant output, u is the manipulated variable, and e is a sequence of independent,identically distributed, zero mean random variables that are used to model the disturbances actingon the plant.

The control law is such that, at each discrete time k, the current value u.k/ of the manipulatedvariable minimizes the output variance d steps ahead

u.k/D arg minJMV (2)

where the minimum variance cost JMV is given by

JMV DEŒy2.kC d/jI k� (3)



Here, EŒ�jI k� stands for the mean conditioned on (the � -algebra induced by) the informationavailable up to time k, denoted I k .

A multivariable version of the aforementioned self-tuning has also been developed [24].The minimum variance control law has an important direct link with economic performance [10].

The application that motivated its use was paper production. The weight per unit area of the paper(process output) had to be above a given threshold with a specified probability. Because there arerandom disturbances, the set point has to be settled not at this specified point (in which case,assuming a gaussian distribution of the disturbances, half of the production would be below thespecification) but at a value high enough so that the area under the probability density functionof the paper weight and to the left of the specified threshold is the contracted one. By reducingthe output variance, it is possible to decrease the set point and save the raw material. In addition,fewer output fluctuations imply increased product homogeneity and hence increased quality. Tounderstand this, consider as an example that, if the paper is to be used in a photocopy machine,major changes in paper thickness might make transport of paper sheets throughout the machinedifficult. Another example is provided by thermal processes. If fluid temperature is to be as highas possible but without surpassing an upper threshold (e.g., for safety reasons), minimizing outputfluctuations will allow an increase in set point while maintaining the risk of surpassing the thresholdconstant. An example involving predictive adaptive control of superheated steam will be givenin Section 5.

In the STR of [11] adaptation is embedded by estimating the parameters of polynomials A andB with recursive least squares (RLS) [25]. In the cases in which the disturbance is not a whitenoise stochastic process (i.e., when C ¤ 1), a situation referred to as colored disturbance, twoissues arise: First, the design of the minimum variance controller also requires knowledge of poly-nomial C . Assuming that C D 1 when in fact it will not deviate the controller gains from theiroptimal values. Second, the estimates yielded by least squares have a bias, and hence the estimateswill not approach the parameters of A. The remarkable fact, known as the self-tuning property,is that these two effects compensate each other, and under some conditions, the gains will con-verge to the ones that minimize JMV . This fact is established using the so called ODE method [27]according to which a deterministic ODE is associated to the difference stochastic equation to bestudied. In rough terms, the solution of the ODE corresponds to the average behavior of thesolutions of the difference equation, the equilibrium points being the same. Applying this methodto the ‘standard’ self-tuner (RLS plus minimum variance) leads to the conclusion that, providedthat the closed loop is stable and under a positive real condition related to the C polynomial, thecontroller gains converge globally to the gains that minimize JMV [27].

Although providing significant conclusions, both with theoretical and practical impact, the ODEapproach makes the undesirably strong assumption that the system variables remain bounded. Theproblem was solved, including the multivariable case in [28], where it was proved that subject to aninverse stability condition and a positive real condition, the plant input and outputs are samplemean square bounded and the conditional mean square output tracking error converges to itsglobal minimum value with probability one. Therefore, the self-tuning control algorithm achievesthe same performance as could be achieved if the plant parameters were known. Stability andconvergence issues for the corresponding servo problem were addressed in [29, 30] assuming theinjection of dither signals and in [31] assuring persistency of excitation through conditions in thereference signal.

It should be remarked that, because the minimum variance control law yields a closed-loop thatcancels the zeros of the open-loop plant [23], if this is non-minimum-phase, there will be unstablemodes. Even when the closed loop is stable, minimum variance control is too reactive, yielding anexcessive control action.

Despite these limitations, the standard self-tuner was successfully used in a number of industrialapplications, including digester control [32], moisture control in a paper machine [33], continuouscement raw material mixing [34], a titanium dioxide kiln [35] and control of an ore crusher [36]. In[34], the results obtained are evaluated by plotting on the space of parameters the evolution of thegains computed by the self-tuners superimposed on the level curves of the minimum-variance gainobtained with a plant model obtained independently.



In order to overcome the limitations of the minimum variance control law, a new version of thebasic self-tuning was proposed [12, 37] in which the cost (3) is modified by the inclusion of a termthat penalizes the manipulated variable, resulting in

Jdetuned DEŒy2.kC d/C �u2.k/jI k� (4)

with � as a positive parameter (weight on the manipulated variable penalty). As in the standardself-tuner, a reference may easily be included (at least for simple problems) by replacing the plantoutput y by its deviations with respect to the reference.

Controller design based on the minimization of Jdetuned was an important modification becausethe extra term regularizes the control variable, providing the designer with a knob to change thetype of closed-loop response obtained. A faster response is obtained by decreasing �, whereas theincrease of � makes the closed loop more sluggish.

The resulting control law can be used to stabilize open-loop stable non-minimum phase systemsbut is not able to stabilize a plant that is both open-loop unstable and non-minimum phase. For thesetypes of plant, one should resort to a control law based on a cost function defined over an extendedhorizon, such as the ones provided by various forms of (adaptive) model predictive control that willbe thoroughly discussed in Section 5.

Making a parallel with linear quadratic optimal control [38], minimizing (4) corresponds toiterating backwards just one step of the difference Riccati equation. The resulting gains may thusbe quite different from the optimal ones and there is no guarantee that they stabilize the nominalclosed-loop system. Instead, the methods of Section 5 approximate the steady-state solution of theRicatti difference equation, yielding a stabilizing solution for the nominal parameters.

The algorithm based on the minimization of (4) received much attention, both to extend it andfor applications. In [39], a multivariable version was presented, and in [40], integral action wasincluded. In [41], the SISO version of the algorithm has been applied to control the temperaturein a catalytic reactor, including a detailed experimental study of the effect of parameter �. Otherapplication examples include the control of a binary distillation column [42], cement raw materialblending [43], pH neutralization [44], a phosphate drying process [45], domestic stored-energyheating systems [46], and bottom temperature control of a glass furnace [47]. Application totwo-dimensional processes, such as paper production, is addressed in [48, 49].

3. THE INTRODUCTION OF ADAPTIVE PREDICTIVE CONTROL

3.1. Predictive control: origin and basic concepts

The methodology of predictive control was introduced in 1974 in a doctoral thesis [13].Subsequently, the original basic principle was formally defined in a US patent [14] and presentedin [15]. This principle may be expressed in the following way: ‘On the basis of a model of theprocess, predictive control is the one that makes the predicted process dynamic output equal to adesired dynamic output conveniently predefined’. Defined in this way, predictive control is straightcommon sense and its objective of control has a clear physical meaning.

This predictive control strategy can be implemented through a predictive model and a driverblock, as presented in the block diagram of Figure 2.

Figure 2. Basic block diagram for predictive control.



As already mentioned, the predictive model generates, from the previous input and output (I/O)process variables, the control signal that makes the predicted process output equal to the desiredoutput. This will be generated by the driver block which is an essential concept in predictive control.

The original driver block design generated a desired output trajectory that, starting from the actualvalues of the process output, converged towards the set point following desired dynamics. Thistrajectory was produced at every control instant and was named projected desired trajectory (PDT).One simple way of generating it consists of using the output of a stable model with desired dynamics,having the set point as input and the actual process outputs as initial conditions.

From the driver block operation, described earlier, the concept of the driving desired trajectory(DDT) was derived. This is produced from the first values of each of the projected desired trajecto-ries that are redefined at the consecutive control instants. Then, the DDT is generated point-by-pointin real time, and from its values, the control action will be computed according to the predictivecontrol principle. Consequently, this trajectory is the one which has to guide the process outputto the set point in the desired way: rapidly, without oscillations and moreover, compatible with abounded control action.

The fact that the DDT was redefined, at each control instant, from the actual process output was akey factor to obtain satisfactory results in many practical applications, because it helped the DDT toapproach a physically realizable process output trajectory. The concept of physical realizability ofthe DDT is of great theoretical and practical relevance and may be defined as follows: ‘It is said thata DDT is physically realizable if, at any control instant, it can be realized through the application ofa control signal that always remains bounded’.

The driver block concept as defined in [14] was different from the previous MRAS referencemodel concept, where the reference model output was generated from previous model referenceoutputs without redefinition. Also, it distinguished predictive control from the minimum variancecontrol strategy of previous STRs and controllers [11, 12].

The original formulation of predictive control considered only a single step prediction for thecomputation of the control action and the driver block design did not use the information availablefrom the process dynamics in the predictive model. This way of applying predictive control is knownas the basic strategy of predictive control. Although successful in many practical applications in thecontext of adaptive control, it had important limitations that will be considered later in this paper.

3.2. The need for adaptation

It is not generally reasonable to expect excellent performance of predictive control in an industrialcontext, mainly because of the unknown and time-varying nature of industrial process dynamics.Obviously, when the predictions are not satisfactory because of ‘unadjusted’ model parameters, itwould be convenient to have an adaptation mechanism able to adjust the parameters of the predictivemodel in order to reduce the prediction error towards zero.

Accordingly, predictive control was already introduced in the framework of APC [13–15] bycombining the predictive control system with an adaptive system as represented by Figure 3. It canbe observed that the right half of this figure represents an adaptive system similar to those used forprocess identification within the MRAS context.

Because the knowledge gained in the adaptive system from the process dynamics should beimmediately used for prediction in the predictive scheme, in both cases, the same adaptive-predictive(AP) model should be used. Thus, an AP model with a series-parallel structure performs thefollowing two roles:

(i) Within the adaptive system, the AP model receives the same input signal as the process andgenerates the model output that, compared with the process output, allows the adaptationmechanism to adjust its parameters in order to obtain a stability result in the sense previouslyconsidered in MRAS.

(ii) Within the predictive part of the scheme, the AP model calculates the control signal from thedesired output generated by the driver block. This computation, carried out according to thepredictive control principle, renders the desired output equal to the predicted process output.



Figure 3. Adaptive predictive control system.

Consequently, when the adaptation mechanism makes the difference between the process and themodel outputs tend towards zero, the difference between the process output and the desired outputalso tends towards zero. In this way, global stability of the APC system can be achieved under anappropriate driver block design.

Thus, APC was introduced with a perspective of stability similar to that considered in the contextof MRAS but extended to the closed-loop APC system, unifying the global stability result with thedesired control system performance. In fact, at the same time that the methodology was formallypresented in [14], a first analysis of stability was published [15].

3.3. Basic implementation of APC

This section presents a general description of the original formulation of APC by assigning to thedifferent blocks in Figure 3 equations that conceptually rule their operation without going into thedetail of the practical implementation.

The dynamics of the generic process considered in Figure 3 can be described by MIMO differenceequations of the form

Y.k/D

hXiD1

Ai .k/Y.k � i � r/C

fXiD1

Bi .k/U.k � i � r/

C

gXiD1

Ci .k/W.k � i � r/C�.k/

(5)

where Y.k�i�r/, U.k�i�r/, andW.k�i�r/ are respectively, the increments at time k-i-r of themeasured output, input, and measurable disturbance vectors of the process with respect to its steadystate values. Ai .k/, Bi .k/, and Ci .k/ are time variant matrices of appropriate dimensions whichdetermine the most significant dynamics of the process. �.k/ is the perturbation vector, wheremeasurement noises, nonmeasurable perturbations, and other process dynamics are considered, andr represents the pure process time delay.

The AP model used in the adaptive system of Figure 3 calculates an a priori estimation of theprocess output Y.k/ as follows:

OY .kjk � 1/D

hXiD1

OAi .k � 1/Y.k � i � r/C

fXiD1

OBi .k � 1/U.k � i � r/

C

gXiD1

OCi .k � 1/W.k � i � r/

(6)



The adaptation mechanism uses the error of the a priori estimation, Y.k/� OY .kjk � 1/, to adjustthe AP model parameter matrices OAi , OBi , and OCi , at each control time k, according to the previouslyconsidered stability perspective.

It may be observed that the AP model of (6) is, in the context of the adaptive system, a series-parallel model [22] that produces the model output from the previous process outputs. A parallelmodel would have produced the model output from the previous model outputs [5]. This kind ofseries-parallel model was chosen in the formulation of APC for two reasons:

(i) The desired stability result for the adaptive system was obtained by means of a rather simpleadaptive mechanism that reduced the estimation error in the gradient direction [18]; and

(ii) It makes the computation of the predicted process output and of the predictive control signalstraightforward, as considered in the following.

The previously considered AP model is also used to predict, at time k, the process output for timekC r C 1 as follows:

OY .kC rC 1jk/D

hXiD1

OAi .k/Y.k� i C 1/C

fXiD1

OBi .k/U.k� i C 1/C

gXiD1

OCi .k/W.k� i C 1/ (7)

From the aforementioned prediction, the basic strategy of predictive control computed the controlvector that renders the predicted output OY .k C r C 1jk/ equal to the desired output Yd .k C r C 1/by means of

U.k/D OB1.k/�1Yd .kC r C 1/

� OB1.k/�1

24

hXiD1

OAi .k/ OY .k � i C 1/C

fXiD2

OBi .k/ OU .k � i C 1/C

gXiD1

OCi .k/W.k � i C 1/

35

(8)where the desired output Yd .kC r C 1/ is computed by the driver block at time k as follows:

Yd .kC r C 1/D

tXiD1

Fi Y.k � i C 1/C

sXiD1

Hi YSP .k � i C 1/ (9)

where YSP .k� i C 1/ represents the value of the set point vector at time k� i C 1, and matrices Fiand Hi are chosen to take into account the desired dynamics.

The basic implementation of APC presented in [14] and previously considered, was describedwith particular emphasis on its practical issues in [50], where a successful application to themultivariable control of a distillation column was presented.

3.4. Block diagram

The diagram in Figure 3 can be simplified to the one shown in Figure 4, which is the diagramgenerally used to represent the APC methodology. The functional description of the blocks in thisdiagram may be summarized as follows:

Figure 4. Block diagram of the adaptive predictive control system.



(i) The driver block. It generates the DDT that will guide the process output to the set point in an‘optimal’ way.

(ii) The predictive model. It calculates the control signal that ensures that the predicted processoutput follows the desired trajectory generated by the driver block.

(iii) The adaptation mechanism. It adjusts the predictive model parameters from the predictionerrors in order to make these errors tend towards zero efficiently. Likewise, it informs thedriver block of the deviation of the process output with respect to the desired trajectory. In thisway, the driver block can redefine the desired trajectory from the actual process output.

After the introduction of the basic APC concepts, considerable research effort was devoted to thedevelopment of the following:

(i) Adaptive mechanisms in different operating contexts for adjusting the AP model parametersin real time in such a way that the prediction error is reduced toward zero according to variouscriteria, and

(ii) Predictive control strategies to derive optimal desired future process output trajectories at everycontrol instant within a receding prediction horizon verifying various performance criteria.

This research effort gave rise to the industrial optimized adaptive control techniques that areconsidered in the next sections.

4. OPTIMIZED ADAPTIVE CONTROL FROM THE STABILITY PERSPECTIVE

This section recalls firstly the stability results that led historically to the development of the body ofAPC stability theory which supports optimized adaptive control systems designed from the stabilityperspective that has been applied in industry. Next, starting from conditions for the driver block andthe adaptive mechanism that can guarantee the desired stability result, it describes the syntheses ofthe driver block and adaptive mechanism and interprets the results obtained from an intuitive per-spective. Finally, it recalls the steps involved in the development and validation of standard systemsfor the optimized adaptive control of industrial plants and the applications carried out by a firstgeneration of these kinds of systems.

4.1. Stability results for adaptive control

The APC principles were key factors in the proof of the existence of simple, globally convergentadaptive control algorithms for discrete systems. The main results of the progress made towards thecreation of the body of APC stability theory that supports industrial application is considered in thefollowing, taking into account both the various subjects and a historical perspective.

Stable-inverse processes

The first analysis of stability for an APC algorithm [15] was based on the convergence propertiesof the a posteriori estimation error and the AP model parameters of a stable adaptive system,under the assumption of a bounded desired output. These conditions were sufficient to prove thestability of the APC closed loop in the case of a first-order process, but the extension of the stabilityresult to a more general order required a restrictive assumption on the stable-inverse nature of theprocess [51].

Because the convergence result on the prediction or control error in [15] was valid for a moregeneral order in spite of the unstable-inverse nature of the process, in order to overcome theconsidered restriction in the global proof of stability, it was suggested in [52] that a new driverblock design is capable of generating a desired output trajectory physically realizable by a boundedcontrol signal for the kind of unstable-inverse processes. The suggested driver block design wasformally introduced in a patent application [53] and defined the so called extended strategy ofpredictive control.



The restrictive assumption on the stable inverse nature of the process was introduced in [54],where the challenging proof of stability, for the same APC algorithm considered in [15], wasrigorously obtained for discrete deterministic systems. Analogous global stability results were alsoobtained for continuous systems [55–57] and hybrid systems [58].

Stochastic processes

Once the aforementioned results were known, interest in the analysis of stability focused onthe stochastic aspects of a system. Some stability results in the discrete stochastic case requiredthat the disturbance acting on the system verified the statistical properties of a moving average ofindependent zero mean random variables [28]. In practice, such conditions were generally veryrestrictive. Less restrictive assumptions led to the stability analysis in the presence of boundeddisturbances. Results derived for this case [59] suffer from certain restrictions on the a prioriestimate of the process parameters, and the proof of stability is technically very involved. Shorterproofs of analogous stability results were obtained for continuous [60] and discrete [61] systems,but these proofs also suffer from a condition on the a priori estimate of a process parameter. All ofthese results [59–61] used as a main argument in their proofs, the boundedness of the rate of growthin the system signal.

Global stability results for a general class of time-invariant stable-inverse in the presence ofbounded noises, and disturbances were presented in [62,63]. These results proved the minimizationof the control error and unlike earlier results, did not depend on any previous estimation of theprocess parameters nor use the argument of a limited divergence rate for the system signals. It isinteresting to note that the proof introduces a criterion for continuing or stopping parameter adap-tation and is derived directly from the same conditions considered in [15], plus the assumption ofinverse stability.

Unstable-inverse processes

All the previous stability results assumed the process to have a stable-inverse and to be time-invariant. Global convergence for deterministic continuous systems not requiring inverse stabilitywas reported in [64], but this is conditional on convergence of the estimated parameters to their truevalues. Local stability results for non-minimum-phase discrete systems [65] were based on adaptivepole assignment algorithms but required a priori knowledge of initial parameter estimates. Thestability of the extended strategy of predictive control, under some specific choices of driverblock design, was proved in [66], although the complete proof of stability, including the adaptivemechanism, was not considered. Global stability results, also based on adaptive pole assignment,were reported in [67] but they require a persistency-of-excitation condition.

Time-variant processes

The stability of algorithms designed for the time-invariant case [68] was analyzed in the time-variant case as a preliminary result of convergence. Other studies analyzed the robustness of adaptiveschemes where external exciting signals help to keep the parameter estimates close to the trueprocess parameters [69]. Also, the performance of previous algorithms [28], developed for the time-invariant case, were analyzed as applied to systems with converging martingale parameters [70] andexponentially convergent parameters [71].

Conditions for the design of globally stable APC in the multivariable stochastic time-variant casewere established in [72]. These conditions are quite general, in the sense that the only restrictiveassumption on the process equation is its stable-inverse nature and basically rely on assumedconvergence properties of the estimation system. Therefore, research into stability proofs in thetime-variant case may, using this approach, proceed to consider different models of the process, forwhich estimation systems that satisfy the required convergence properties exist or may be derived.It is clear that restrictive assumptions on the considered models of the process are necessary [73].



A methodology used to derive the required properties of the estimation system is also presentedin [72] and is based on a theorem closely related to hyperstability [74] and input–output passivitytheories [75].

Unmodeled plant dynamics

The stability problem related to the model-process order mismatch in the control schemewas illustrated in [76, 77], where it was demonstrated that straightforward application of stablealgorithms found in the literature may lead to instability when this mismatch is present. Stabilityresults reported in the presence of unmodeled dynamics, [78] and [79], are based on the followingassumptions: (1) the a priori knowledge or existence of a tuned set of controller parameters thatguarantee the stability result and (2) the modeling error is relatively bounded. Also, the results in[79] required of an additional projection in the adaptive law.

A generalization of previous stability analysis for APC is combined in [80] with a normalizedparameter estimation system to prove global stability for stable-inverse processes in the presence ofbounded disturbances and unmodeled dynamics. The generalization of the stability analysis states asingle condition for stability in terms of convergence properties of the a posteriori estimation errorand the AP model parameters. The normalized parameter estimation system permits a formal proofthat the modeling errors can be treated as a bounded disturbance. Thus, the on/off mechanismconsidered in [62, 63] is valid and the normalized estimation system verifies the convergenceproperties that guarantee global stability.

The results presented in [80] prove that the control system converges to a tuned set of parametersand guarantees stability without the need for any assumption on the knowledge or existence ofsuch a tuned set, neither the assumption of a relatively bounded modeling error nor the need ofan additional projection in the adaptive law. However, it shows that there is a stability limit in theprocess-model mismatch that depends on the process dynamics itself.

APC stability theory

The stability results presented in [81] encompass, unify, and generalize all the previous stabilityresults on APC and form a theoretical body of stability both in the context of predictive controland APC.

This body of stability theory considers three different classes of processes taking into accounttheir stability nature

(i) Processes of a linear and stable-inverse nature;(ii) Processes of a linear and stable nature; and

(iii) Unstable processes with an unstable inverse.

All of them are represented by autoregressive-moving-average (ARMA) type equations withtime-varying parameters, including a term for unmeasured perturbations and are characterized forverifying input/output properties previously derived for the same kind of linear processes in the caseof constant parameters. Process classes (1) and (2) can describe the dynamic behavior of almost theentire set of industrial processes. Class (3) was included for the sake of completion of the theoreticalanalysis.

For each of these process classes, a stability condition was established related to the boundednessand/or physical realizability of the DDT. In the first class, where the process inverse is stable, thecorresponding condition (DDT boundedness) may easily be satisfied using the basic or the extendedstrategy of predictive control in the driver block. In the second class, where the process inversemay be unstable, the satisfaction of the condition (DDT physical realizability) requires in practicethe use of the extended strategy. These conditions unified the analysis performed for each class ofprocesses and gave general validity to the stability results derived, irrespective of the specific driverblock design used.

The general global stability results obtained in this context rely only on convergence propertiesof the a posteriori estimation error and of the AP model parameters of a stable adaptive system,



similar to those already considered in [15]. These convergence properties can be reduced to a singleconvergence condition that has to be verified by the adaptive system. The analysis of the verificationof the convergence or stability condition by the adaptive system progresses within APC stability the-ory from the simplest formulation (no noise, perturbations, model-process mismatch, and constantprocess parameters of the ideal case) to a formulation defined by hypotheses that describe an indus-trial environment. Thus, in addition to the ideal case, three scenarios are considered: (1) the realcase with no difference in structure (no process-model mismatch and time-invariant process param-eters, but presence of bounded noise and unknown perturbations); (2) the real case with differencein structure (adds the process-model mismatch), and (3) the time-varying parameters case.

In the context of predictive control, where there is no adaptive system, the stability result dependson a measure of the modeling error that was mathematically formulated in relation to the processdynamics. This result was intuitive and could reasonably be expected. The basic motivation forAPC has been to overcome the stability problem because of the modeling error. In the ideal case,the expected result for APC was obtained, that is to say asymptotic stability guaranteeing that theprocess output converges towards the DDT with a bounded control signal.

As expected, asymptotic stability was not possible in the real case with no difference in structurebecause of the existence of the unknown and unpredictable perturbation vector. Nevertheless, theresults obtained proved stability for this case in terms of the boundedness of the control or trackingerror, approaching the corresponding boundary that would be obtained if the process parameterswere known and used in the predictive control law.

In the real case with difference in structure, it seems logical that stability may not always be guar-anteed and that there must be stability limits in terms of the model order reduction. The stabilityresults for this case derived the mathematical formulation of these limits and their relationshipwith the process dynamics. When the model order reduction is compatible with these limits, APCguarantees the stability result and the control error can be minimized by appropriate selection ofthe adaptation mechanism. These stability limits were also analyzed in [82] and their practicalconsequences were illustrated by means of simulation examples.

In the case of time-varying parameters, APC adaptation mechanisms guarantee tracking of theprocess parameters by the AP model parameters until the stability result is reached, which has beenrepeatedly confirmed by good practice in the implementation of the methodology.

The APC stability theory considered here has also been presented with minor changes in [83].It has determined the design of the driver block and the adaptive mechanism from the perspectiveof stability and provides theoretical support for the industrial application of APC methodology asa whole, that is, both for predictive control and for APC. Those areas of theoretical analysis whichare still an open subject for research were also indicated in [81], but it is important to emphasizethat the results arising from the APC stability theory already developed have clear implications in awidespread practical context.

In the following, the main concepts of the APC stability theory will be reviewed and interpreted ina simple but illustrative way, and their practical implications in their application to industrial plantswill be analyzed.

4.2. Stability conditions for the design of adaptive predictive controllers

As a starting argument for the development of APC stability theory presented in [81,83], a so called‘conjecture’ formulates conditions for the design of the driver block and the adaptive mechanismthat guarantees the global stability result under hypotheses that accord with an industrial environ-ment. In this section, we will consider these hypotheses, a description of the process and the APCsystem, and state and prove this conjecture using a SISO formulation and simplifying the notations.The extension of the results to the multivariable case is direct and easy.

Conjecture hypotheses

The conjecture departing hypotheses are the following:

(i) The process is described by linear equations with time varying parameters.



(ii) There exist measurement noise and unmeasurable disturbances randomly acting on theprocess.

(iii) The process and model equations may have different orders.

Process description

The process can be described by the following equation

y.k/D �.k/T �.k � d/C�.k/ (10)

It may be observed that (10) is the SISO case of (5), where y.k/ is the measured process output attime k, �.k/ is the process parameter vector, �.k�d/ is the regression vector that includes processinput, output, and measurable disturbances, d represents the process time delay that includes thediscretization delay plus the pure delay, that is, d D r C 1, and �.k/ is the perturbation signal thatrepresents the effect of the unmeasured perturbations and measurement noise acting on the process.It is interesting to note that (10) verifies the aforementioned conjecture hypotheses (a) and (b).

Adaptive system

In the adaptive system shown in Figure 3, the AP model can give an estimation of the processoutput at instant k using the AP model parameters also estimated at instant k, which will be denotedby O�r.k/, and the signals included in the regression vector. This estimation is expressed in the form

Oy.kjk/D O�r.k/T �r.k � d/ (11)

The dimensions of �r and O�r are usually less than or equal to the dimensions of � and � . �r containsa subset of the most recent process inputs and outputs included in �. These assumptions account forconjecture hypothesis (c).

The estimated parameter vector O�r.k/ is generated by the adaptation mechanism using theinformation available on the process inputs and outputs up to instant k. The following notationis adopted for the estimation error, the difference between the process and model outputs

e.kjk/D y.k/� Oy.kjk/D y.k/� O�r.k/T �r.k � d/ (12)

Oy.kjk/ and e.kjk/ are called respectively, the a posteriori process output estimation and thea posteriori estimation error because both of them are calculated after the model parametershave been adjusted at instant k. Another estimation error, extremely important in the analysis andsynthesis of adaptive systems, is the a priori estimation error, which is defined by the equation

e.kjk � 1/D y.k/� Oy.kjk � 1/D y.k/� O�r.k � 1/T �r.k � d/ (13)

where Oy.kjk � 1/ is the a priori estimation for the process output y.k/ calculated from the modelparameter vector adjusted at instant k � 1, that is, O�r.k � 1/.

Predictive controller

The predictive model calculates the control action u.k/ to make the predicted output at instantk C d equal to the DDT at the same instant. From (11), the predicted output can be expressed inthe form

Oy.kC d jk/D O�r.k/T �r.k/ (14)

Denoting now the DDT as yd .kC d/ and applying the principle of predictive control, we obtain

yd .kC d/D O�r.k/T �r.k/ (15)

This equation may also be written in the form

yd .kC d/D O�ro.k/T �ro.k/C O�1.k/

T u.k/ (16)



where O�1.k/ is the parameter included in vector O�r.k/ in equation (15) that corresponds to thecontrol signal u.k/ in the inner product. The predictive control law can be written from (16) inthe form

u.k/Dyd .kC d/� O�ro.k/

T �ro.k/

O�1.k/(17)

The difference between the process output and the DDT is defined as the control or tracking error

�.k/D y.k/� yd .k/ (18)

which plays an important role in characterizing the performance of APC systems.

Design from a stability perspective

Definition 1An adaptive predictive control system is said to be globally stable if the following conditionsare satisfied:

(i) j�.k/j6M <1 8k > kf > 0

(ii) k�.k/k6˝ <1 8k > kf > 0where k � k denotes the Euclidean norm.

The aforementioned definition corresponds to the stability result that will represent the desiredcontrol performance for APC in its application to industrial processes. It is implicitly expected thatthe limit M on the absolute value of the control or tracking error is reduced to a minimum valuethat depends on the level of noise and unmeasured perturbations acting on the process and theunmodeled dynamics. For the ideal case, the expected result corresponds to that of asymptoticstability, where the regression vector remains bounded and the tracking error converges to zero.

Stability conditions

The design principles for APC systems are stated by means of a conjecture in terms of specificconditions that, if satisfied by the driver block and the adaptation mechanism, would make the APCsystem globally stable.

Conjecture 1If the driver block verifies that the DDT yd .kC d/ is

(i) Physically realizable and

(ii) Bounded

and that, for certain values M and kf , the adaptive system satisfies the following conditions:

(a) �r.k/D �r.k � d/ 8k > kf > 0.

(b) je.kjk/j6M <1 8k > kf > 0.

Then, the APC system represented in Figure 3 will have the following properties:

(I) j�.k/j D jy.k/� yd .k/j6M <1 8k > kf > 0.

(II) jjX.k/jj6˝ <1 8k > kf > 0.

�

It may be arguable to use the term conjecture when, as proven in the following, it is a mathe-matical result. However, previous literature used this nomenclature because it established a sound



stability conclusion from premises that were not yet proven. Under this scheme, conditions (i), (ii),(a) and (b) are used as the guidelines for the design of the driver block and the adaptive mechanismin order to satisfy the stability objective.

Equation (15) can be written in the delayed form

yd .k/D O�r.k � d/T �r.k � d/ (19)

By comparing equations (11) and (19), it is clear that if condition (a) holds, then we may write

yd .k/D y.kjk/ (20)

From this result and condition (b), along with (12), property (i) of the conjecture immediatelyfollows. Property (ii) is derived solely from the boundedness and physical realizability of the DDT.

Once we have verified that the control objectives are achieved if the conditions of the conjecturehold, the problem to solve is that of the synthesis of the driver block and the adaptation mechanismthat are able to satisfy these conditions.

4.3. Driver block design

Basic strategy of predictive control

For stable-inverse processes, a bounded DDT is always physically realizable. Therefore, usingthe basic strategy of predictive control in the driver block design will make it satisfy, in this case,the conjecture stability conditions (i) and (ii). However, for unstable-inverse processes, to make theprocess output follow a bounded DDT may require an unbounded process input sequence, not appli-cable in practice. In this case, a driver block based on the basic strategy of predictive control mayviolate the DDT physical realizability stability condition. As previously mentioned, the need for anextended strategy of predictive control was first considered in [52] and later formalized method-ologically in [53].

Extended strategy

The extended strategy of predictive control uses the same basic principles of predictive control,but the driver block generates a desired output trajectory at each control instant that verifies a certainperformance criterion in a fictitious prediction horizon, taking into account the process dynamics asconsidered in the following.

The predictive model defines the available knowledge from the process dynamics and it can, ateach control instant, be used for one step ahead prediction of the process output (basic strategy) orto predict the effect on the process output of a sequence of control actions in a certain predictionhorizon. A predicted process output trajectory will correspond to each possible sequence of controlactions, and the convenience of applying each one of these sequences can be evaluated accordingto a certain performance criterion. The extended strategy of predictive control defines as PDT, thepredicted output trajectory that corresponds to the sequence of control actions which minimizes acertain performance index in a chosen prediction horizon. The following quadratic cost functionwas considered by the way of example in [53] as performance index

Jk D

rC�XjDrC1

Qj Œ Oy.kC j jk/� yr.kC j jk/�2C

��1XjD0

Rj� Ou.kC j jk/2 (21)

where Oy.k C j jk/ and � Ou.k C j jk/ are the predicted output and incremental input sequences inthe prediction horizon, respectively; r is the process pure time-delay, and yr.kCj jk/ is a referencetrajectory which can be generated in a form equivalent to the way that the PDT for the basic strategywas generated, irrespective of the dynamic nature of the process to be controlled. The cost function(21) imposes a compromise between the resultant PDT being as close to the reference trajectory as



possible and the required control action not being excessive.Qj andRj are weighting factors whichare chosen to weight the tracking of the reference trajectory and the magnitude of the incrementalcontrol action, respectively. Within this criterion, a predicted trajectory which requires an unlimitedcontrol sequence can obviously not be the resulting PDT. By selecting the reference trajectory, thedesigner can define, in a simple way and independently of the dynamics of the process, the timeresponse and damping that would be desirable for the PDT. The index of performance (21) was alsoused to describe the extended strategy of predictive control and some practical applications in [84]and analyzed as an optimal design for the driver block in [66].

The PDT determines the control action to be applied to the process, which is the first control actionof the corresponding predicted control sequence. A new PDT is defined at each control instant andthe corresponding sequence of PDT determines the sequence of control actions actually applied tothe process. The envelope of the first values of the sequence of PDT, which correspond to the controlactions being applied to the process is known as DDT.

A solution for industrial applications

The performance criterion of the extended strategy may include a condition to make the predictedcontrol sequence constant along the prediction horizon. A particular case of the performancecriterion, already considered in [53, 66, 84], adds to the previous condition that of making the PDTvalue at the end of the prediction horizon equal to the value at the same instant of a referencetrajectory that, starting from the actual process outputs, converges to the set point according todynamics chosen by the designer. This kind of performance criterion has been successfully usedwhen applying the extended strategy, in an adaptive predictive context, to a wide variety of industrialplants.

In order to explain the physical meaning for the APC operation of the choice of the predictionhorizon under the previously considered performance criterion, let us assume the ideal case in whichthere is no prediction error and the process initially at steady state. Then, let us analyze the followingthree alternatives to the prediction horizon choice in the predictive controller operation when a setpoint change is undertaken

(i) If the chosen prediction horizon is greater than the process time response and greater than thesettling time of the reference trajectory, that is, the reference value at the end of the predictionhorizon equals the set point value, the first PDT will reach the set point value following anatural process step response. All the subsequent PDT will follow the same path and the DDTand the process output trajectory will also follow this process of natural response.

(ii) If the prediction horizon chosen is equal to 1, the DDT and the process output will follow thepreviously considered reference trajectory, which in this case will also be equal to the PDT.

(iii) If the value of the prediction horizon lies between the two previous choices, the DDT andthe process output will follow a trajectory in between the process of natural response and thereference trajectory, that is, the dynamics of this trajectory will approach those of the naturalprocess step response as the prediction horizon grows and those of the reference trajectory asit gets closer to 1.

Physical realizability of the DDT

As explained earlier, the choice of the prediction horizon will determine the DDT, and therefore,the control law stability, because the physical realizability of the DDT will depend on the stabilitynature of the process as considered in the following.

Processes of a linear and stable-inverse nature have the bounded input-bounded output propertyand its inverse. Therefore, the control law stability is guaranteed for any choice of the predictionhorizon, with the only condition of a bounded reference trajectory.

Processes of a linear and stable nature do not have the bounded output-bounded input property,and in order to follow certain bounded output trajectories, these processes may require unboundedcontrol sequences. When the prediction horizon is ‘large enough’ and the DDT matches the naturalprocess step response, a bounded control sequence is guaranteed with the only condition of a



bounded reference trajectory. However, when the prediction horizon is equal to 1, the process outputto follow the DDT, which matches the reference trajectory, may require an unbounded controlsequence and the stability of the control law is not guaranteed.

Consequently, for processes of a linear and stable nature, the choice of a large prediction horizoncan guarantee stability, but the process output trajectory will approach the natural process response.On the other hand, a small prediction horizon can result in the process output trajectory approachingthe reference trajectory, but stability may not be guaranteed.

The aforementioned reasoning, which assumes the ideal case with no prediction error, is alsoconsistent in the industrial context. Industrial practice in the operation of APC has demonstratedthat, using the driver block design previously considered in combination with a stable adaptivesystem, it is easy to find a prediction horizon experimentally that makes the process output dynamicsapproach the reference trajectory dynamics at the same time that stability is guaranteed in adesired manner.

4.4. Adaptation mechanism design

In this section, the synthesis of an adaptive mechanism is first outlined in the ideal case, and the wayit approaches the conjecture stability conditions is described. Next, modifications on this adaptivemechanism are considered to meet the conjecture stability conditions under different hypothesesthat approach the operation in an industrial context.

Synthesis in the ideal case

In the ideal case, there is neither process-model order mismatch nor noises nor unmeasuredperturbations acting on the process, and therefore, the process and AP model can be described bythe following equations

y.k/D �T �.k � d/ (22)

Oy.kjk/D O�.k/T �.k � d/ (23)

and the a posteriori estimation error by

e.kjk/D Q�.k/T �.k � d/ (24)

where Q�.k/D � � O�.k/ is the parameter estimation error vector.The desired stability result is defined in the form

limk!1

e.kjk/D 0 (25)

The following proposition defines the design strategy for the adaptive system.

Proposition 1Property (25) holds if the following condition is satisfied:

s.kt / D

ktXkD1

e.kjk/2 6 ı2 <C1 8kt > 0 (26)

�

Proof 1Clearly, because it sums the squares of the error, s.kt / is a non-decreasing sequence that maybegin to grow from the instant at which the adaptation mechanism starts to operate, as illustrated inFigure 5. If condition (26) holds, this sequence is bounded by ı2 and thus its increments must tendto zero, that is, e.kjk/2! 0 as k!1.



Sampling instants0 1 2 3 4 5 6 7 8 9 10 11

0

)( tks

2 2

∑=

tk

kkke

1

2)|(

Figure 5. Graphical illustration of condition (26).

Therefore, an adaptive system satisfying (26) will be asymptotically stable.It is proven that condition (26) is satisfied if the AP model parameter vector is generated at instant

k by means of algorithms of the form

O�.k/D e.kjk/B�.k � d/C O�.k � 1/ (27)

where B is a positive definite matrix.Also, it is proven that the relation between the a posteriori estimation error and the a priori

estimation error, e.kjk � 1/D y.k/� O�.k � 1/T �.k � d/, is expressed by

e.kjk/De.kjk � 1/

1C �.k � d/TB�.k � d/(28)

and the adaptive algorithm can be written in the form

O�.k/DG.k/Œy.k/� O�.k � 1/T �.k � d/�C O�.k � 1/ (29)

where

G.k/DB�.k � d/

1C �.k � d/TB�.k � d/

With this formulation, (29) takes the form of a linear recursive filter with a variable gain: the newvector of the estimated parameters is obtained by adding to the previous one an increment equal tothe estimation error obtained using the preceding vector multiplied by the gain vector G.k/.

Other techniques of parametric estimation, such as those of the gradient [18], which minimize afunction of the square of the prediction error in the direction of the gradient (from which the nameis derived), or the estimation techniques based on optimization criteria [85], converge with smalldifferences towards the general expression that we have derived in this case from a perspectiveof stability.

Interpretation of the results

The interpretation of the results obtained in the previous synthesis of the adaptive mechanism issummarized in the following points:

(i) The upper limit ı2 of the non-decreasing sequence s.kt / is a quadratic function of theEuclidian norm of the initial parameter estimation error vector, Q�.0/.

(ii) If the initial value of the parameter estimation error vector was equal to zero, that is, O�.0/D � ,then ı2 would be zero and so would be for all kt , the non-decreasing sequence s.kt /, thatwould follow the X axis in Figure 5.



(iii) At any other time kt , the difference between the upper limit ı2 and the value of thenon-decreasing sequence s.kt / is a quadratic function of the Euclidian norm of the parameterestimation error vector at time kt , Q�.kt /. Therefore, when the sequence s.kt / increases, thelength of the parameter estimation error vector decreases. If the sequence s.kt / were to reachthe upper limit ı2, the parameter estimation error vector would be zero and the AP modelwould have identified the process equation, that is, O�.kt /D � .

(iv) When the a posteriori estimation error e.kjk/ is zero, the increment between k � 1 and kof the parameter estimation vector, O�.k/, is also zero, and therefore, the conjecture stabilityconditions are verified.

(v) The a posteriori estimation error e.kjk/ would be zero, see (24), only if the processparameters have been identified by the AP model parameters or if the parameter estimationerror vector, Q�.k/, and the regression vector, �.k � d/, are orthogonal.

(vi) In fact, when the non-increasing sequence s.kt / remains constant, the conjecture stabilityconditions are verified and the parameter estimation and regression vectors are orthogonal.

(vii) The parameter estimation algorithm (27) of the adaptive mechanism efficiently achieves theresult of stability, e.kjk/2! 0 as k!1, because the updating increments of the estimatedparameters look for the orthogonality condition between the parameter estimation errorvector and the regression vectors by reducing the square of the a posteriori estimation errorin the gradient direction.

(viii) Therefore, the parameter estimation performed by the adaptive mechanism produces anidentification of the process dynamics with a view to control. This concept introduced in[15] is of paramount importance in understanding the performance of stable APC systems, asoutlined in the following points.

(ix) The first goal of the identification with a view to control is to achieve the aforementionedorthogonality condition, which results in the verification of the conjecture stability conditionsand guarantees the desired control system performance. Looking for the orthogonalitycondition, the parameter estimation error vector is projected to the plane orthogonal to theregression vector, and in this way, the length of the parameter estimation error vector isreduced. At the same time, the non-increasing sequence s.kt / increases and approaches theupper limit ı2.

(x) Figure 6 illustrates the performance of the adaptive mechanism. When the regression vector,�.k � d/, changes from �1 to �2 and �3, the adaptive mechanism updates the AP modelparameters in such a way that the corresponding parameter estimation error vector, Q�.kt /,changes from Q�1 to Q�2 and Q�3, respectively, looking for the desired orthogonality.

1

θ1~

2

2

3

3

θ~

θ~

Figure 6. Tracking the orthogonality condition for identification with a view to control.



(xi) Practical application of APC demonstrates that the considered orthogonality condition ismuch easier to attain than process parameter identification. This is particularly relevantwhen considering changes in the process dynamics, where the control performance may notsignificantly deteriorate if this adaptive mechanism is applied.

(xii) Therefore, the identification with a view to control provides satisfactory APC performancewithout the need of identifying the process parameters.

The proof of stability for the adaptive mechanism, derived in the ideal case, implies the verifi-cation of the conjecture stability conditions when the regression vector is bounded. Obviously, theadaptive system does not produce the control signal and therefore, it cannot ensure the boundednessof the regression vector. However, the stability properties, derived for this and the following realcases, combined with the conjecture conditions on the DDT, allow APC stability theory to prove theboundedness of the regression vector, and as a result, to attain the desired global stability.

Synthesis in the real case with no difference in structure

In the real case with no difference in structure, the process and AP model can be described by thefollowing equations

y.k/D �T �.k � d/C�.k/ (30)

Oy.kjk � 1/D O�.k � 1/T �.k � d/ (31)

(31) gives the a priori process output estimation and (30) is equal to (10), but in this case, we assumethat there is a boundary �b such that

�b > j�.k/j C ı 8k, ı > 0 (32)

From equations (30) to (31), the a priori estimation error can be derived

e.kjk � 1/D y.k/� y.kjk � 1/D .� � O�.k � 1//T �.k � d/C�.k/ (33)

It can be observed in the right-hand side of (33) that the a priori estimation error is the sum of twoterms. The first one depends on the parameter estimation error vector, whereas the second is theperturbation signal.

In the ideal case, the only source contributing to the a priori estimation error is the parameterestimation error vector. Thus, it is logical to have a continuous adaptation in order to make theconvergence as fast as possible. The situation is different when noise and disturbances are present,because they are unpredictable and also contribute to the estimation error.

It seems logical that the information given by the a priori estimation error may or may not beuseful for adaptation depending on whether the estimation error is mainly due to the parameteridentification error or to the effect of noise and disturbances, respectively. When the estimationerror is mainly because of noise and disturbances, the information on which the parameter adap-tation is based will be misleading. On the other hand, if the estimation error is mainly due tothe parameter estimation error vector, this information is probably useful for adaptation. This factsuggests the convenience of performing the adaptation only when the second case occurs, instead ofdoing it continuously.

This idea, first presented for the real case with no difference in structure in [62, 63], is the coreof the approach followed by APC stability theory. It is essentially based on the introduction of acriterion that allows the adaptation of the model parameters only when it is certain that this adapta-tion will lead to a reduction in the norm of the parameter estimation error vector. This criterion is



based on some estimated knowledge about the size of the noise and disturbances that can influencethe process output unpredictably and can be described in the following simple terms:

(i) If the a priori estimation error is equal to or smaller than a function of the maximum level thatthe perturbation signal can reach (�b), it may be possible that such an error is due more to theperturbation signal than to the parameter estimation error vector. In such a case, the adaptationshould be stopped.

(ii) If the a priori estimation error is greater than such a function, such an estimation error ismainly due to the parameter estimation error vector and, consequently, the adaptation shouldbe performed.

The problem of synthesis was solved by using the same gradient parameter estimation algorithmsderived for the ideal case and determining and applying the aforementioned function to perform orstop adaptation [62, 63, 81, 83].

With the use of this adaptive mechanism, it was proved that

(i) When adaptation is stopped, the a priori and a posteriori estimation errors are bounded.(ii) When adaptation is performed, the norm of the parameter estimation error vector decreases.

(iii) The number of adaptations is bounded when the regression vector is bounded. Therefore, inthis case, the conjecture stability conditions are met.

(iv) When the bound�b approaches the maximum absolute value of the perturbation signal �.k/,the limit on the control error approaches the maximum control error obtained if the processparameters were known and used in the AP model.

At this point, it is important to note that the effect of the unmeasured disturbances in the processoutput is only totally unpredictable at an initial stage. However, because disturbances contribute tothe evolution of the process output, their effect can subsequently be predicted in part by the APmodel itself. Therefore, the contribution of unmeasurable disturbances to the maximum level of theperturbation signal is not usually as large as it might seem a priori. Thus, determining a reasonablevalue for �b is usually not difficult in practice.

The interpretation of the synthesis results obtained for this case can be summarized as follows:

(i) The adaptive mechanism looks permanently for an extended orthogonality condition in whichthe zero value for the ideal case is replaced by a dead band around zero.

(ii) When the a priori estimation error, which contains the inner product between the param-eter estimation error vector and the regression vector, enters the dead band, the extendedorthogonality condition is attained and adaptation is stopped.

(iii) The dead band is determined by the level of noise and unmeasured disturbances acting on theprocess.

(iv) The limit on the dead band can be chosen to make the control error to approach that obtainedif the process parameters were known and used in the predictive control law.

(v) The norm of the parameter estimation error vector becomes a non-increasing function thatdetermines stability.

Synthesis in the real case with difference in structure

The synthesis of adaptive systems able to handle not only the problem of noise and unmeasurabledisturbances but also the problem of difference in structure, that is, process-model order mismatchuses within APC stability theory the strategy introduced in [80] and outlined in the following:

(i) Definition of a normalized adaptive system, so that all the process input/output signals becomebounded, allowing dynamic terms of the process equation to be included in an extendedperturbation signal that remains bounded. In this way, the process order in the normalizedsystem is reduced to match the AP model order.

(ii) Similar to the case with no difference in structure, an adaptation mechanism is definedwhich guarantees that the norm of the reduced parameter estimation error vector is anon-increasing function.



(iii) From the result obtained in point (ii), the properties of convergence of the a posterioriestimation error and the AP model parameters are derived for the normalized adaptive system.

(iv) Finally, by the inverse process to normalization, the corresponding convergence properties arederived for the adaptive system without normalization.

With the use of the previously considered normalized adaptive mechanism, it is proven [81,83]

(i) When adaptation is stopped, the a priori and a posteriori estimation errors of the normalizedsystem are bounded, and the a posteriori estimation error of the adaptive system withoutnormalization is bounded by a function proportional to the norm of the regression vector,where the proportionality depends on the norm of the unmodeled parameter vector.

(ii) When adaptation is performed, the norm of the reduced parameter estimation error vectordecreases.

(iii) The number of adaptations is bounded, independently of the boundedness of the regressionvector.

Therefore in this case, the conjecture stability conditions for the adaptive system are also metwhen the regression vector is bounded. However, APC stability theory establishes limits on thestability result for the closed loop control system in the presence of unmodeled dynamics. Theexistence of these limits, which depend on the process dynamics and the norm of the unmodeledprocess dynamics included in the extended perturbation signal, was analyzed and illustrated in [82].

The stochastic and unmodeled dynamics hypotheses considered in this case for the synthesis ofthe adaptive system are closer to a real industrial context. In this sense, the results obtained approachthe desired control objectives for a realistic environment and encompass previous stability resultsobtained for the ideal case and the real case with no difference in structure. It can be interpreted inthis case that the adaptive mechanism looks permanently for an extended normalized orthogonalitycondition between the reduced regression vector and the reduced parameter estimation errorvector, and the norm of this vector becomes a non-increasing function that guarantees stability. Animportant practical advantage is that, although precise knowledge of the structure of the processequations was required before in order to define the AP model, a reduced-order AP model can nowbe chosen without requiring such knowledge.

Time-varying parameters

The dynamic nature of industrial processes is basically nonlinear, but it is known that ingeneral, their behavior can be described approximately by linear equations, such as those previouslyconsidered, with time varying parameters. These parameters usually undergo variations because ofthe effect of external perturbations acting on the process or changes in the point or conditions inwhich the process is operating; but these variations generally stop when the context of operationstabilizes and the process approaches steady state.

Adaptive predictive control stability theory shows that the adaptive mechanism, previouslyconsidered in the presence of unmodeled dynamics, when applied in the time-varying parametercase, maintains the property that the norm of the reduced parameter estimation error vector decreaseswhen adaptation is performed.

Therefore, although in this case, the proof of global stability requires logically some restrictiveassumptions on the process parameter changes [72], there is a guarantee that the adaptive mechanismwill adjust the AP model parameters in order to track the previously considered orthogonalitycondition and reduce the parameter estimation error vector. In the following section, we willconsider industrial applications of APC, where the excellent results achieved were due to thesatisfactory performance of the adaptive system in the presence of time-varying parameters.

4.5. Industrial application

The research steps involved in the development and application of standard systems for the opti-mized adaptive control of industrial plants from the stability perspective have been the following:



� Theoretical developments, such as those considered in the previous sections, in the stability,robustness, efficiency, and generality of the methodology as applied in an industrialenvironment.� Applications to simulations that reproduce complex processes and their challenging

control problems.� Applications to industrial pilot plants.� Experimental applications to critical control loops of operational industrial plants� Development, realization, and exhaustive testing of prototypes that have been designed to

satisfy the conversational and control requirements for an industrial control system.

This section will recall the main milestones of this research effort and the systematic industrialimplementation of this technology in the various industrial areas.

The application of an APC system to the design and evaluation of an automatic pilot for NASA’sF-8 digital fly-by-wire (DFBW) supersonic aircraft was the first important real time application ofthe new methodology. It was carried out in the spring of 1975 using the aircraft high-fidelity hybridsimulation of the Charles Stark Draper Laboratory, Cambridge, Massachusetts, USA. [81, 86].

In the first months of 1976, APC was successfully applied to single-input single-output andthe multivariable control of a pilot scale binary distillation column [14, 50] at the Department ofChemical Engineering at the University of Alberta, Canada. The multivariable control of a distilla-tion column was at that time a typical example of the difficulties found in the practical application ofmodern control theory [87]. This application and the previous one were carried out in the context ofa two-year research and development program (1974–1976) funded by the Juan March Foundationof Spain and, in both cases, the basic strategy of predictive control was used [88].

An APC system was applied for the first time to an industrial production process in the bleachplant of the pulp factory of CANFOR Ltd., in Port Mellon, British Columbia. This application[89, 90] used the extended strategy of predictive control. It was carried out in 1984 as a joint effortbetween the Paper and Pulp Research Institute of Canada and the University of Alberta, in thecontext of a 5-year research program supported by the Natural Sciences and Engineering ResearchCouncil of Canada.

A simple, systematic, and generalized industrial implementation of APC systems was the mainobjective of an intensive research and development effort initiated in 1986 in Spain and supportedby the Centro para el Desarrollo Tecnológico Industrial, the Comisión Interministerial de Cienciay Tecnología , the Dirección General de Nuevas Tecnologías del Ministerio de Industria, and thecompany SCAP Europa SA. The result was the development of several versions of the so calledSCAP optimization systems [81].

The SCAP optimization systems combined in their operation APC with a so called master system.This system was a rule-based system aimed at

(i) Identifying certain process operating conditions, which could deteriorate the APC systemperformance, and reacting to them in an efficient predefined manner.

(ii) Searching for optimal process performance under APC by evaluating the performance of theprocess operation periodically and moving the set point values for critical process variables toimprove performance. Because the optimum operating set points may also vary over time, thesearch conducted by the master system was permanent.

More than 150 industrial and commercial applications of SCAP optimization systems werecarried out until 1999 in industrial areas such as chemical [91], environment [92], food [93, 94],climatization [95–97], cement [98, 98–103], and energy [104–107].

It is worthwhile to mention that the last version of SCAP optimization systems was a distributedcontrol system (DCS) that used a multivariable adaptive predictive controller as a basic control toolfor continuous variables. It was integrated by means of an operator in the graphical programminglanguage of the DCS control logic. In 1993, this DCS was installed at the coal-fired thermal powerstation of Pasajes de San Juan (Spain) of the company Iberdrola and applied to the continuousprocesses of the plant. The results obtained were excellent, showing the suitability of APC for theoptimized adaptive control of thermal power stations and verifying almost completely the Electrical



Power Research Institute (EPRI) specifications for an ideal control system [108]. This SCAPoptimization system is still operating the plant with the same remarkable performance.

By the end of last century, a new generation of industrial optimized adaptive control from thestability perspective started to be applied in industry, as described later in Section 7.

5. OPTIMIZED ADAPTIVE CONTROL FROM THE OPTIMIZATION PERSPECTIVE

Although the self-tuning ideas discussed in Section 2.2 played a major role both in theory and appli-cations of optimized adaptive control from the optimization perspective considered in this section,the basic algorithms had a number of drawbacks that limited their scope. Significant limitations arethe fact that an unstable and non-minimum phase plant may not be stabilized in closed loop, theplant input–output transport delay must be known exactly and the important issue of operationalconstraints cannot be tackled in a satisfactory manner. These limitations motivated the considera-tion of another class of adaptive control algorithms with a tight relation to optimal control but onethat could easily and efficiently be implemented.

5.1. Optimal control theory and optimization concepts

Optimal control provides the backbone that underlies optimized adaptive control. Major tools aredynamic programming and linear filtering [38] that, together, are used to solve the linear quadraticstochastic (LQS) dynamic optimization problem in which the following cost is to be optimized:

JLQS .u/D limk!1

EŒy2.k/C �u2.k/� (34)

Here, u and y are, respectively, the manipulated input and output of a linear (discrete time) plantdisturbed by a sequence of independent, identically distributed random variables e.k/; EŒ�� is themean computed with the probability measure of e.k/. The plant, assumed to be linear, may bedescribed either by an ARMAX or by an equivalent state-space model.

When the plant is described by a state-space model, a separation principle applies and two sub-problems are to be solved in order to solve this so called LQS problem: The estimation of the stategiven the observations of input/output data, solved by a Kalman filter; and the control problemassuming the state is available, in which the solution of a Riccati equation plays a major role. Themanipulated variable to apply to the plant is then obtained by replacing, in this control law, the stateby its Kalman filter estimate.

When the plant is described by an input–output ARMAX model, the control is obtained by thesolution of a spectral factorization problem [109].

In all cases, under perfect knowledge of plant dynamics, the minimization of (34) yields astabilizing controller, even if the plant is simultaneously open-loop unstable and non-minimumphase and assuming controllability and observability assumptions. It should however be remarkedthat, although optimal in the sense of corresponding to the minimization of (34), the resulting perfor-mance may, for practical results, be poor, presenting an excess of overshoot in the step response. Toovercome this, a number of modifications of the basic problem are available, including the inclusionof dynamic weights, or the restriction of the closed-loop poles [110].

5.2. Control laws design

Some algorithms, such as [111–114], have been proposed by the direct interconnection (using thecertainty equivalence principle) of a parameter estimation algorithm and a design procedure such asthe ones briefly explained in Section 5.1. Another approach [115], with roots in [116] aimed at theexplicit minimization of JLQS . However, the most successful optimized adaptive control algorithmshave been based on the concept of model predictive control and rely on a multistep approximationto JLQS (34).

By comparing JLQS (34) with Jdetuned (4), a number of important differences arise. First,although the average operator that appears in (4) is a conditioned mean on (the � -algebra inducedby) the information available up to time k (typically input/output samples), the mean that appears in



(34) is unconditioned. Furthermore (but connected to this), JLQS refers to stochastic steady state,that is, to a situation in which enough time has passed such that transients have died out and the sta-tistical properties of the output are constant), whereas Jdetuned refers to a transient situation (justone step after current time). Thinking in terms of the Riccati equation, the controller gains that opti-mize JLQS correspond to the steady-state Riccati equation [38], whereas the controller gains thatoptimize Jdetuned correspond to just performing one iteration with the difference Riccati equation,a procedure that provides no assurance on stability.

Hence, the idea arose (e.g., [117–123]) of designing the control by minimizing an extendedhorizon cost that approximates the steady-state quadratic cost JLQS such as

JMPC DE

24

TXkDN1

y2.t C k/C

T�1XkDNu

��u2.t C k � 1/jI t

35 (35)

Here, �u denotes the increments of the manipulated variable and is used in order to force inte-gral action. Future samples of the output y are related with the samples of the manipulated variableusing linear predictive models that are linear combinations of past samples of y and u (or �u) thatare packed in a state x.t/, t being the current time. In this way, JMPC becomes a function of thesequence of future values of the manipulated variable, between t and t C T � 1. In order to applyfeedback control at all discrete times, according to a receding horizon strategy [124], only the firstvalue of this sequence is actually applied to the plant, and the whole procedure is repeated at thenext sampling time.

Predictive models are of the form

Oy.t C i/D

iXjD1

j�u.t C j � 1/C 0jx.t/ (36)

where Oy.tCi/ is the predictor of y.t/, i (scalar) and i (vector) are parameters and x.t/ is a vectorthat contains samples up to time t of the input, output, and accessible disturbances. In generalizedpredictive control (GPC) [122,123], it is assumed (Figure 7) that the increments of the manipulatedvariable vanish from tCNuC1 on, up to time tCT . ForNu D 1, the receding horizon minimizationof JJMLe7 is equivalent to iterating backwards T steps of the difference Riccati equation. Thus, forT large enough [38], a stabilizing controller results that is a good approximation of the optimumof (34).

Another possibility [121] consists in assuming that a constant feedback is acting on the plant atall future times (Figure 8). This allows simplified predictive models, reducing the dependency onfuture values of the manipulated variable to just u.t/. In addition, to simplify the computation of

x(t)

T-steps ahead predictor

2-steps ahead predictor

1-step ahead predictor

Constant feedbackassumed to act on the plant

t+1 t+2 t+N t+Tt

.

.

.

. . .Present time u(t)=?

. . .u

Figure 7. Extended horizon and predictors in generalized predictive control.



x(t)

T-steps ahead predictor

2-steps ahead predictor

1-step ahead predictor

Constant feedback assumed to act on the plant

t+Tt+2t+1t

.

.

.

. . .Present time u(t)=?

Figure 8. Extended horizon and predictors in MUSMAR.

the controller gains, it can be shown [121] that the minimization of 36 under this assumption is anapproximation to perform Kleinman’s iterations to solve the underlying Riccati equation. Therefore,a tighter approximation to the optimal LQS optimal gains is obtained with smaller values of T(and hence with a smaller computational load).

These algorithms have a number of interesting properties. In addition to being able to stabilizesystems that are simultaneously open-loop unstable and non-minimum phase, they are insensitiveto the uncertainty in plant input/output transport delay, a feature of paramount importance in manyprocess control applications.

An issue that raised a strong controversy at the end of the 1980s was the problem of finding avalue of the horizon T that ensures stability. This led to a number of modifications with varioustechniques [125–128].

Several modifications and extensions were also considered, including the multivariable case[50, 129–131], state-space modeling [132, 133] and soft (i.e., depending on the mean value of thevariables) constraints [134].

When hard constraints are included, the controller may no longer be expressed as a linear feed-back of a (possibly non-minimal) state and one has to resort to nonlinear programming numericalalgorithms to find the optimal value of the manipulated variables [135]. Recent formulations [136]allow constraint handling in an approximate, although systematic, way.

Another important issue is filtering and the inclusion of dynamic weights in one and two DOFcontrollers to enhance robustness. This is treated for GPC in [137] and for MUSMAR in [138].

5.3. Adaptive mechanism design

Adaptation is embedded in the algorithms described in Section 5.2 by estimating the parametersof the predictive models using RLS. In some cases, such as MUSMAR, this is justified, at least instochastic steady state, because, under the assumption that the control is given by a constant feed-back gain, an ARMAX plant admits, in closed loop, an ARX parametrization [139]. This propertygeneralizes for a class of predictive controllers the classical self-tuning property.

Figure 9 shows the structure of the adaptation mechanism for GPC [122]. The parameters of apredictive model that relate the input and the output for a horizon equal to the input/output transportdelay of the plant (referred as the ‘first’ predictor in Figure 9) are estimated using RLS (or a variant).The parameters of the remaining predictors are then computed from those by using an algorithm thatamounts to a polynomial division. From the estimates of the predictors, the controller gains are thencomputed in such a way as to optimize (35).

In MUSMAR (Figure 10), identification of predictive models takes advantage of the fact that theyshare a common regressor [121]. Thus, only one Riccati equation and one Kalman gain vector needsto be propagated in time to get the RLS estimates, a fact that greatly reduces the computational load.For each predictor, the corresponding parameter estimates are obtained by correcting the previous



Reference

Manipulatedvariable

Output

Adaptation mechanism

First predictor parametersestimation with RLS)

Controller gain redesign

All predictivemodel parameters

Controller Plant

Controller gains

First predictivemodel parameters

Polynomial division

Figure 9. Generalized predictive control adaptation mechanism structure.

Reference

Manipulatedvariable

Output

MultipleRLSidentifiers

Adaptation mechanism

Covariance andKalman Gain update

Common Kalman gain

Controller gain redesign

Predictive modelparameters

Predictor 1update

Predictor Tupdate

Controller Plant

Controller gains

Figure 10. MUSMAR adaptation mechanism structure.

estimate with a term given by the product of the Kalman gain (common to all predictors) by thecorresponding a priori prediction error. An important point consists in the fact that, because predic-tive models are separately estimated (using different prediction errors), this introduces a diversity(as opposed to GPC, where the predictors are obtained by extrapolating the first one) that increasesthe algorithm robustness with respect to plant/model mismatches and even mild nonlinearities.



5.4. Stability and convergence issues

A general result stating the global stability and convergence of optimized adaptive controlalgorithms is not available. This is because of the nonlinear character of the resulting equationsthat forces a number of simplifying assumptions to be made. A number of properties of this classof algorithms has nevertheless been established [38, 128, 133, 140, 141] although in many casesaddressing only the non-adaptive case.

In some algorithms, such as explicit criterion minimization [115] or MUSMAR [139], thecontroller gains F are updated in discrete time t in a way that is equivalent, up to second-orderterms, to

F.t/D F.t � 1/� gMrT JLQS .F.t � 1// (37)

where g > 0 is a parameter, M is a positive definite matrix, and rT JLQS .F.t � 1// denotes anapproximation of the gradient of the steady state cost (34) that becomes tighter when the horizonT increases. With the use of a simple Lyapunov function argument, it is thus concluded that, for Tlarge enough, the controller gains, if they converge, may only converge to local minima of JLQS ,constrained to the structure of the chosen controller regressor.

It is worth noting that a result of this type, connecting convergence points of the controller gainswith the local minima of the underlying cost, cannot be obtained for GPC because this algorithmdoes not exactly minimize a cost, because of the fact that the manipulated variable is assumed tobecome constant after Nu steps.

5.5. Industrial applications

Many applications are described in the literature for GPC and its variants. In [142], an applicationto batch-fed fermentation for penicillin production is presented, whereas in [143], modeling andestimation aspects in fermentation processes are examined. In [144], a multivariable GPC algorithmwith feedforward is applied to anesthesia. In [145], a variant of GPC with a reduced computationload is used to control the outlet temperature in a distributed collector field. In [146], several aspectsof GPC are examined in relation to two industrial control applications (a gas engine and high temper-ature plant). Reference [147] addresses penetration control in arc welding, a non-minimum-phaseplant with variable large orders and delays. The use of delta models and multivariable control inrelation to missile autopilots is addressed in [148]. In [149], applications to glass manufacturingare reported. In [150], a number of variants of GPC are considered in relation to conveyor control.Application to control of NOx emissions is made in [151]. An application to injection moldingis described in [152]. The design of a GPC-based PID controller for a weight-feeder is discussedin [153].

Although not as popular as GPC, MUSMAR has also been applied with success to a number ofindustrial systems. Examples include various forms of control of distributed collector solar fieldssuch as dual adaptive control [154], adaptive cascade control [155], and adaptive feedforwardfrom accessible disturbances [138], super-heated steam temperature control in large-scale boilers[156], trailing centerline rate of cooling control in arc welding [157], and level control in waterdelivery canals [158]. All these examples illustrate the ability of MUSMAR to tackle plantmodel mismatches.

The ability of MUSMAR to perform as an adaptive optimization controller is well-illustrated in[156]. As shown in Figure 11, MUSMAR adjusts the controller gains so as to minimize the extendedquadratic cost, achieving a performance that clearly surpasses the standard controller (a cascade ofPI controllers with several feedforward terms that has been carefully optimized by plant engineers).This also provides a link with economic performance: By reducing temperature fluctuations, the setpoint may be safely raised, resulting in an increase in turbine performance.

Another significant aspect to consider in industrial applications is smooth start-up. If the initialplant model parameter estimates are poor, a strong initial adaptation transient may result. Asexplained in [154], a way to avoid this consists in using a dual version of the algorithms that amountsto solving a bi-criteria optimization problem that takes into account both the control objective andthe parameter estimation criteria.



0 0.5 1 1.5530

531

532

533

534

535

536

537

538

Ste

am T

empe

ratu

re [o C

]

Adaptation starts ρ decreases

Time [hour]

Figure 11. An example of optimized adaptive control: Superheated steam temperature control in a largescale boiler controlled with MUSMAR.

6. COMPARATIVE ANALYSIS

6.1. Control laws

The surveys presented in Sections 4 and 5 of optimized adaptive control developments from thestability and the optimization perspectives, respectively, show how the underlying philosophies thatdetermine the control laws. On the stability perspective side, the driver block generates a desiredoutput trajectory (PDT) by projecting the future evolution of the process output according to selecteddesired dynamics. From the DDT value, the control law is derived through the application of thepredictive control principle. On the optimization perspective side, the concept of desired outputtrajectory is not considered in the implementation of the control law, which is determined by theminimization of a performance index that involves predicted process input/output signals.

The limitations found in practical applications from the initial solutions, that is, the basic strategyof predictive control in the stability approach, and the first self-tuning control law in the opti-mization approach, led to new developments on both sides that approached each other from animplementation point of view.

Thus, the extended strategy of predictive control was introduced to comply with the physicalrealizability of the DDT, imposed in the driver block design by the conjecture stability conditions.The extended strategy derives the PDT by minimizing a performance index involving processinput/output sequences in a prediction horizon, where the desired dynamics are taken into accountby means of a reference trajectory.

Similar control laws adopting performance indices defined in a future prediction horizon wereintroduced from the optimization perspective. This is the case of the control laws of the MUSMARand GPC approaches which have been extensively used combined with adaptive algorithms in opti-mized adaptive control, as considered in Section 5.3. Again, the tools used to solve the minimizationproblem are taken from optimal control theory, as in the extended strategy of predictive control.

Most industrial applications of optimized adaptive control, from the stability perspective, haveused a driver block design based on a particular solution of the extended strategy of predictive con-trol which considers a constant control sequence in the prediction horizon. This is a solution that inpractice allows the operator to choose the dynamics of a stable close-loop response in an intuitivemanner as described in Section 4.3.

Optimized adaptive control from the optimization perspective has also converged towards thiskind of solution by introducing a constant process input sequence in the formulation of the controllaw performance index in order to simplify the computational load and improve the solution



robustness, as described in Section 5.2 for the MUSMAR approach. Consequently, the convergenceof control law solutions in the development of both design approaches has been significant, butnot surprising.

6.2. Adaptive mechanisms

The adaptive mechanism of optimized adaptive control from the optimization perspective has beengenerally derived considering that the process description includes noise with statistical propertiessuitable for process parameter estimation algorithms that minimize an index of performance, such asthe RLS method. In this way, the resultant adaptation mechanisms have generally the form of a firstorder recursive filter described by (24), where the corresponding parameter estimates are obtainedby correcting the previous estimate with a term given by the product of a gain by the correspondinga priori prediction error.

However, the gain of the filter converges towards zero as the number of iterations grows and moredata is processed. This makes the adaptation mechanism progressively lose its capacity for adapta-tion. Under a time-invariant process description, this is not relevant, but if the process parametershave changes, as would be expected in a realistic context, this represents an important drawback forthe closed loop performance of the system.

Some alternatives have been proposed to tackle this problem. The most popular is the introduc-tion of a so called ‘forgetting factor’ in the adaptation mechanism. This forgetting factor tendsto diminish the weight in the estimation algorithm of the process input/output data as it becomes‘older’. The potential limitations of this alternative arise from the fact that valid data for theprocess parameter estimation may be disregarded, whereas poor data, including the effect of noiseand unmeasured perturbations, may receive an inadequate weight in the parameter estimationalgorithm, leading to ‘blow up’ phenomena in the parameter estimation process. Consequently, theselection of the forgetting factor becomes a critical issue for the industrial performance of this kindof systems.

From the stability perspective, the adaptation mechanism is synthesized to look for an orthogonal-ity condition between the parameter estimation error vector and the regression vector that verifies theconjecture stability conditions and therefore, guarantees the desired closed loop system performanceif the DDT is bounded and physically realizable.

In the ideal case, the adaptation mechanism guarantees that a non-decreasing function, integratingthe square of the estimation error, is bounded by an upper limit, and the orthogonality condition isreached when the estimation error is zero. The distance between the non-decreasing function and itsupper limit represents the norm of the parameter estimation error vector.

The adaptation mechanism also operates as a first-order recursive filter described by (24), but inthis case, the filter gain does not tend to zero with time. Thus, if the process parameters change,the distance from the non-decreasing function to its upper limit may increase or decrease, but theadaptation mechanism will not lose its adaptive capacity and will keep looking for the orthogonalitycondition, reducing the square of the estimation error in the gradient direction.

In real cases with and without structure differences, the only restrictive assumption on noise andunmeasured perturbations acting on the process is that their effect on the process output must bebounded, which is a realistic and general assumption. The limits on this boundary and the consid-eration of unmodeled dynamics due to the process-model order mismatch, determine an extendedorthogonality condition and an on/off criterion for parameter adaptation.

The adaptation mechanism inherits the gradient parameter algorithms from the ideal case butprocesses the data at every control instant before deciding if it should be used for adaptation. Whenadaptation is executed, the norm of the parameter estimation error vector is reduced, improvingknowledge of process dynamics and approaching the orthogonality condition. When adaptationstops, the orthogonality condition is reached.

Similar to the ideal case, the norm of the parameter estimation error vector is a non-increasingfunction with lower limit equal to zero. Again, if process parameters change, the norm of the param-eter estimation error vector may increase or decrease, but the adaptation mechanism will never loseits adaptive capability and will keep looking for the orthogonality condition until the estimation



Figure 12. Block diagram of an adaptive predictive expert controller.

error is bounded and the closed-loop system approaches the same performance which would havebeen obtained if the process parameters had been known and used in the predictive control law.

7. NEW RESULTS ON OPTIMIZED ADAPTIVE CONTROL, INDUSTRIAL APPLICATIONS,AND PERSPECTIVES

Experience in the industrial application of optimized adaptive control from the stability perspectivedescribed in Section 4.5, demonstrated the importance of using available knowledge on the processoperation in the overall control scheme. New results on optimized adaptive control and its industrialapplication involve the use of this knowledge within the controller itself and also in the developmentof control strategies required to optimize the process performance, as described in the following.

Adaptive predictive expert (ADEX‡) control [83, 159] has been introduced in order to integrateand generalize the use of the available process knowledge within the controller operation itself.Thus, ADEX controllers integrate different domains of operation, in which APC or expert controlis configured and applied, in a complementary and coherent manner, based on available processknowledge. In this way, the control role of the master system in previous SCAP optimization systemshas been advantageously integrated within the controller.

Figure 12 shows the block diagram of an ADEX controller, which is similar to that shown inFigure 4 for an AP controller but adds an expert block on top. This expert block identifies theprocess domain of operation in real time and accordingly determines if APC or expert control shouldbe applied.

In the first case, the control block in Figure 12 behaves as an AP model. In the second case, thecontrol block behaves as an expert system using rules to control the process in a domain wheregood manual control would provide a more robust and efficient control than APC. This situationcan arise, for instance, in a domain where there does not exist a cause-effect relationship betweenprocess input and output variables. Expert domains are usually defined in the extremes of normaloperating ranges of process variables or when abnormal operating conditions occur. Expert controlin these domains usually tries to drive the evolution of process variables towards AP domains inwhich optimized adaptive control can be applied.

In order to enable the design, execution, and application of control strategies that integrate ADEXcontrollers and can optimize the process performance, a software platform named ADEX COP(control and optimization platform) has been developed [83]. This platform is a windows-basedapplication, designed to operate in parallel with the local control system and linked via OPC orequivalent. The development and validation of optimized control strategies for a particular kind ofindustrial process usually represents a significant applied research effort, which is required to opti-mize the process performance. As a matter of fact, the use of optimized adaptive control in industryhas opened an extensive research area for the optimization of process performance.

Future developments in nonlinear predictive control [160–162] and their adaptive versions couldalso be integrated in an ADEX like structure.

‡ADEX is a trademark of adaptive predictive expert control ADEX, S.L.



Industrial and commercial applications of ADEX COP have been carried out with excellentresults in industrial areas such as environment [163–165], cement [166, 167], energy [168], andpetrochemical [169–172].

8. CONCLUSIONS

The design of industrial control systems aimed at optimizing process performance is necessarilybased on available knowledge of the operation process dynamic and the use of a controlmethodology. The optimization objective first of all requires stabilization of the process dynamic.This means that the critical process variables should remain under precise control around their setpoints with adequately bounded control signals, in spite of changes in process dynamics, context ofoperation and noise, and perturbations acting on the process.

The concept of an optimized process control system can be associated with the previouslymentioned concept of process dynamic stabilization plus the use of appropriate control strategiesto drive critical process variables in real time towards operating points where process performanceoptimization can be achieved.

Because of the time-varying nature of industrial processes, the desired objective of processdynamic stabilization is often difficult to achieve and maintain using fixed parameter controllers,such as those derived from control methodologies currently used in the industry.

This paper has presented a survey of industrial optimized adaptive control techniques, describingboth the stability and the optimization design perspectives and their evolution from their outset totheir current state of the art. It has also put special emphasis in the intuitive interpretation of theresults derived from both design perspectives, in order to understand how control law and theadaptive mechanism complement each other to produce mature optimized adaptive controllers thatcan reliably solve the industrial process dynamic stabilization problem.

The first reported application of an optimized adaptive control system to industrial plant datesback to 1984. Many other industrial applications have been reported since then with excellentresults and industrial products based on optimized adaptive control that have already appeared in themarket. These products are designed to allow the user to develop and apply optimized processcontrol systems that integrate optimized adaptive controllers within appropriate control andoptimization strategies. This kind of development and application has opened up a wide appliedresearch area for the optimized control of industrial plants.

ACKNOWLEDGEMENTS

The work of J. M. Lemos was supported by FCT (Portugal) under contract PTDC/EEA-CRO/102102/2008.The work of J. Rodellar was supported by the Ministry of Science and Innovation of Spain under projectDPI2011-28033-C03-01.The authors would like to gratefully acknowledge the comments and suggestionsof the reviewers of this paper.

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