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Performance assessment and online input design for closed-loop identification of machine directional… Yousefi, Mahdi 2014

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Performance Assessment and Online Input Design forClosed-Loop Identification of Machine DirectionalProperties on Paper MachinesbyMahdi YousefiB.Sc. in Electrical Engineering, Amirkabir University of Technology(Tehran Polytechnic), 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Electrical and Computer Engineering)The University Of British Columbia(Vancouver)August 2014© Mahdi Yousefi, 2014AbstractModel-based controllers based on incorrect estimates of the true plant behaviourcan be expected to perform badly. Due to the fact that machine directional proper-ties in paper machines can be controlled by model predictive control, it is importantfor us to use a valid model of the process in the controller to keep controller per-formance high. Performance is measured to detect model-plant mismatch using aminimum variance index and a closely related user-specified criterion. In this the-sis, we define a sensitivity measure that relates system performance to model-plantmismatch, and use it to explore this sensitivity for three realistic types of paramet-ric modelling errors. This analysis shows the power of the indices to detect modelplant mismatch. In addition, the effect of model-plant mismatch on the closed loopbehaviour is discussed.To compensate controller performance in the case of model-plant mismatch,the process needs to be re-identified to update the process model. This thesispresents a new approach to input design for closed loop identification. The ideais to maximize the trace of the Fisher information matrix associated with the plantmodel in a moving horizon framework, while enforcing explicit constraints on bothinputs and outputs. The result is the richest possible excitation signal for which theoperation of a running closed-loop system remains within acceptable bounds. Themethod can be combined with a fixed model variable regressor technique to esti-mate time delays.The suggested technique is implemented and used to monitor performance ofmachine-directional processes in an industrial paper machine and identify the pro-cess if any degradation in controller performance because of model-plant mismatchis detected.iiPrefaceThe current thesis proposes a technique to measure performance of controllerswhich control machine directional properties of paper in paper machines to detectmodel-plant mismatch. It also proposes a novel approach to identify the processand retune the controllers to compensate for the lost performance if model-plantmismatch is detected.Chapter 2 and 3 introduce minimum variance as well as user-specified bench-marking for both single-input/single-output and multi-input/multi-output systems.Using the above benchmarking techniques to estimate performance of feedfor-ward/feedback control systems is the contribution of the author in these two chap-ters.In chapter 4, the effect of different types of parametric model-plant mismatch arestudied for both single-input/single-output and multi-input/multi-output systems.This analysis is originally carried out by the author. The result generated in thischapter is generated by the author using Honeywell Industrial Paper Machine Sim-ulator. Some parts of this chapter with analysis in chapters 2 and 3 were publishedin American Control Conference 2014 and Multi-Conference on Systems and Con-trol 2014,• M. Yousefi, MG. Forbes, RB. Gopaluni, PD. Loewen, GA. Dumont, and J. Back-strom. Sensitivity of MIMO controller performance to model-plant mismatch, withapplications to paper machine control. In Multi-Conferences on Systems and Con-trols (MSC), Antibes/Nice, 2014,• M. Yousefi, MG. Forbes, RB. Gopaluni, GA. Dumont, J. Backstrom, and A. Mal-hotra. Sensitivity of controller performance indices to model-plant mismatch: Aniiiapplication to paper machine control. In American Control Conference (ACC), Port-land, 2014.In chapter 5, the author proposes a novel approach to distinguish the effect ofmodel-plant mismatch from the effect of changes in disturbance characteristicson system performance. The results presented in this chapter are generated by theauthor using Honeywell Industrial Paper Machine Simulator.In chapter 6, the author is suggested a technique to design excitation signals forclosed-identification purposes.The idea originally came from the paper by R. Pat-wardhan and B. Gopaluni [36]. However, the suggested technique is different fromthe one suggested by them. The result generated in this chapter is generated by theauthor using Honeywell Industrial Paper Machine Simulator.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Performance Assessment of Single-Input/Single-Output Systems . . 52.1 Minimum Variance Benchmarking . . . . . . . . . . . . . . . . . 62.1.1 FCOR Algorithm . . . . . . . . . . . . . . . . . . . . . . 82.1.2 The Effect of Measured Disturbances . . . . . . . . . . . 92.2 User-Specified Performance Index . . . . . . . . . . . . . . . . . 112.2.1 The Effect of Measured Disturbances . . . . . . . . . . . 132.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Performance Assessment of Multi-Input/Multi-Output Systems . . . 153.1 Minimum Variance Benchmarking . . . . . . . . . . . . . . . . . 15v3.2 User-specified Benchmarking . . . . . . . . . . . . . . . . . . . . 183.2.1 Performance Assessment of Feedforward/Feedback Con-trol Systems . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 The Impact of Model-Plant Mismatch on the Performance Indices . 214.1 Effects of MPM on the Sensitivity Function . . . . . . . . . . . . 234.2 The Effect of MPM in MIMO Systems . . . . . . . . . . . . . . . 294.2.1 Minimum Variance Control . . . . . . . . . . . . . . . . 294.2.2 User-specified Control . . . . . . . . . . . . . . . . . . . 314.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3.1 Sensitivity of The Performance Indices to MPM . . . . . 344.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . 354.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Detection of MPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.1 Prediction Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . 475.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 535.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 Moving-Horizon Predictive Input Design for Closed-Loop Identifi-cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . 586.3 Input Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.3.1 Moving Horizon Framework . . . . . . . . . . . . . . . . 646.4 Time Delay Estimation . . . . . . . . . . . . . . . . . . . . . . . 656.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 666.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 727.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73viBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74viiList of TablesTable 4.1 Nominal plant model, P0 (continuous time) . . . . . . . . . . . 36viiiList of FiguresFigure 2.1 The block diagram of a feedback control system . . . . . . . 6Figure 2.2 The block diagram of a FFFB control system . . . . . . . . . . 10Figure 4.1 Nyquist Diagram . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 4.2 The effect of gain mismatch on the sensitivity function . . . . 27Figure 4.3 The effect of delay mismatch on the sensitivity function . . . 27Figure 4.4 The effect of time constant mismatch on the sensitivity function 28Figure 4.5 The effect of a poor performance output on other outputs . . . 37Figure 4.6 The effect of different types of mismatch in G21 on the perfor-mance index . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 4.7 User-specified performance index versus gain mismatch . . . 39Figure 4.8 Sensitivity of the user-specified performance index to gain mis-match . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 4.9 User-specified performance index versus time delay mismatch 41Figure 4.10 Sensitivity of the user-specified performance index to time de-lay mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 4.11 User-specified performance index versus time constant mismatch 43Figure 4.12 Sensitivity of the user-specified performance index to time con-stant mismatch . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 5.1 Correlation of the filtered prediction error and the filtered pro-cess input, dc = 2, d = 4. . . . . . . . . . . . . . . . . . . . . 54Figure 6.1 The closed loop system during the identification experiment . 62Figure 6.2 Illustration of moving horizon approach in input design . . . . 64ixFigure 6.3 The closed loop system during the identification experiment . 64Figure 6.4 The designed excitation signal . . . . . . . . . . . . . . . . . 68Figure 6.5 The closed loop response of the system . . . . . . . . . . . . 68Figure 6.6 The estimated time delays . . . . . . . . . . . . . . . . . . . 69Figure 6.7 The estimated parameters . . . . . . . . . . . . . . . . . . . . 70Figure 6.8 Dry weight signal perturbed with a step disturbance . . . . . . 70xGlossaryARIX Autoregressive Integrated With Exogeneous InputARX Autoregressive With Exogeneous InputFCOR Filtering and CorrelationFFFB Feedforward/FeedbackFMVRE Fixed Model Variable Regressor EstimationIAE Integral Absolute ErrorMD Machine DirectionalMIMO Multi Input-Multi OutputMHP Moving Horizon PredictiveMPC Model Predictive ControlMPM Model Plant MismatchMVC Minimum Variance ControlMVI Minimum Variance Performance IndexSISO Single Input-Single OutputRSI Relative Sensitivity IndexUSI User-Specified Performance IndexxiAcknowledgementsI would like to express my deepest appreciation to my academic supervisors, Prof.G.A. Dumont, Prof. R.B. Gopaluni. Prof. P.D. Loewen and Prof. M.S. Davies fortheir supportive guidance through this project; their discussions, ideas, and feed-back have been absolutely invaluable. They continually and convincingly conveyedto me the spirit of adventure associated with conducting research.I would like to thank my supervisors at Honeywell Process Solutions, NorthVancouver, namely, Dr. Michael G. Forbes and Mr. Johan Backstrom, who backedme up through the project with technical hints which could not normally be foundin academia. Furthermore, I would like to thank Mr. Ashish Malhotra, Dr. DanleiChu, Mr. Qiugang Lu, Mr. Lee Rippon and my other friends and collogues atHoneywell Process Solutions and UBC Pulp and Paper Center for their great helpand support during throughout this research.Finally, I would like to thank and dedicate this thesis to my sweetheart, Nazaninand my parents for the love, support, and constant encouragement I have gottenover the years.xiiChapter 1IntroductionThe main objective of this work is to monitor performance of controllers associ-ated with Machine Directional (MD) processes in paper machines to detect poorperformance situations and compensate the lost performance. Model PredictiveControl (MPC) is often used to control MD processes in paper machines and thecontrol objective is to keep output perturbations as low as possible. Other objectiveis to track set points. Any mismatch between a process model used in controller de-sign and the process will affect performance of such controllers and consequently,increase output variabilities. Thus, a purpose of monitoring performance of MDprocesses is to detect Model Plant Mismatch (MPM) in a related control loop. Mini-mum Variance Control (MVC) based benchmarking techniques are used to estimatesystem performance and detect model plant mismatch. In this type of benchmark-ing, the output variance is compared with some desired variances. Due to thefact that the objective of the controller is to keep the output variance low, MVCbenchmarking is an appropriate tool for our objective. Harris et al. [19], Huangand Shah [20] and Qin [27] present comprehensive surveys of performance assess-ment techniques for univariate and multivariable control systems. The MinimumVariance Performance Index (MVI) is defined as the ratio between the minimumtheoretically achievable variance in the output variable of a process and the ac-tual achieved variance [17][1]. However, in practice MVI has often proved to bean extremely stringent requirement on the variance. It thus leaves engineers withcontrollers that are in theory performing poorly but have limited scope for improve-1ment. In order to circumvent this discrepancy between theory and practice, [20]introduced user-specified minimum variance benchmarking where the variance iscompared with the user-specified variance. Lynch and Dumont [7], Desboroughand Harris [9], Jofriet [28] and Owen [35] have applied the above mentioned per-formance assessment technique to pulp and paper processes and show how it canbe used in industry.In this work, we show how MPM affects a closed-loop system and changesthe output variance. In fact, we discuss how the shape of the sensitivity functionchanges as a result of MPM. Moreover, we study sensitivity of the above mentionedperformance indices to different types of model parametric mismatch. The indicesare shown not to be sensitive to some types of mismatch, such as time constant mis-match. To remedy this problem, we suggest a technique to measure performanceof closed loop systems in servo mode, when there is transient behaviour in systemoutputs. It is shown that this technique increases the sensitivity of the performanceindices to MPM which helps us to detect MPM better.MVC benchmarking is very sensitive to disturbance characteristics. Indeed,any changes in disturbance characteristics may cause the indices to drop. Thus,such a benchmarking technique is not a robust tool to detect MPM. However, adistinct advantages if MVC method is that it does not require external excitation.Other techniques suggested to detect MPM in which there must be some sorts ofexcitations. See [44], [34], [30], [14], [29], [3], [2] and [6]. In this thesis, wepropose a technique to determine whether the performance index drop is due toMPM or it is because of changes in disturbance characteristics. We show that inthe presence of MPM in a closed-loop system, correlation between prediction errorand process input is dominant at some specific lags. Hence, by comparing thiscorrelation in poor and high performance situations, we can determine if MPM ispresent.If MPM is present in a control system and leads to degradation of system perfor-mance, the process model needs to be re-identified using closed loop identificationtechniques, e.g., direct method, projection method, etc. To do so, the process needsto be excited using a proper excitation signal. The accuracy of an estimated modeldepends on the richness of the excitation signals used to probe the system. Theidentifiability condition [40] quantifies the degree of excitation required. Identifi-2cation in industry often proceeds by adding a PRBS1 to the plant input. Of course,any perturbation of the process inputs also affects outputs, and hence the quality ofthe output products. Thus, identification experiments are not free: the need to ex-tract sufficient information about the process at an acceptable cost has driven manyresearchers to study optimal input design for closed loop identification. Typically,one tries to find an input with minimum variance that optimizes some function ofthe covariance matrix of estimated parameters and guarantees the identifiablity ofa related system. Jansson and Hjalmarsson [23], [24] propose a general frameworkfor such problems. They also [25] suggest a method to solve the input design prob-lem by parameterization of the covariance matrix. Gopaluni et al. [16] use a par-ticle filtering approach to solve the input design problem for nonlinear stochasticdynamic systems. Valenzuela et al. [41] use results from graph theory to decreasethe computational difficulties in optimal input design. Bombois et al. [5] specify acondition for costless identification which is not met in most practical cases. Yan etal. [47] show how a process with an MPC controller can be sufficiently excited bychanging the controller parameters instead of modifying the input signal directly.In this work, the Moving Horizon Predictive (MHP) input design techniqueis introduced for closed loop identification experiment. This method is suggestedbased on the fact that the trace of the Fisher information matrix is a convex quadraticfunction of the optimization parameters for systems modelled by AutoregressiveWith Exogeneous Input (ARX) model. For such functions, constrained minimiza-tion is a well-understood and efficient computational problem. Here, however, thegoal is to maximize the trace of the information matrix subject to given constraints.This is considerably more challenging, especially when the vector of parametersto be chosen is high-dimensional. Special characteristics of typical control sys-tems make it possible to suggest a practical method that decreases the dimensionof the problem by splitting it into a sequence of lower-dimensional problems. Thistechnique is implemented in a moving horizon framework. The moving horizonimplementation brings several advantages. For instance, it guarantees that the pro-cess outputs do not go beyond the output constraints.Once the system is excited, recursive least square estimation is implemented to1Pseudo random binary sequence3identify the process model in direct framework. Finally, the model in the MPC con-troller is updated with the estimated model and consequently, the lost performanceis compensated.Recapitulating, this work has made the following basic contributions:• A new user-specified benchmark to measure performance of FFFB controlsystems is developed,• The effect of MPM on the new performance indices is analysed,• Sensitivity of the performance indices to different types of parametric mis-match is quantified,• Sensitivity of the performance indices to MPM is increased,• An algorithm to distinguish the effect of MPM on the performance indicesfrom the effect of disturbance is introduced,• TheMoving Horizon Predictive input design technique to generate excitationsignals for closed-loop identification purposes is proposed,and the following papers have been published out of the context of this work:• M. Yousefi, MG. Forbes, RB. Gopaluni, PD. Loewen, GA. Dumont, and J. Back-strom. Sensitivity of MIMO controller performance to model-plant mismatch, withapplications to paper machine control. In Multi-Conferences on Systems and Con-trols (MSC), Antibes/Nice, 2014,• M. Yousefi, MG. Forbes, RB. Gopaluni, GA. Dumont, J. Backstrom, and A. Mal-hotra. Sensitivity of controller performance indices to model-plant mismatch: Anapplication to paper machine control. In American Control Conference (ACC), Port-land, 2014.4Chapter 2Performance Assessment ofSingle-Input/Single-OutputSystemsIn this chapter, minimum variance control is introduced as a benchmark for mea-suring performance of control systems. This method shows how close the outputvariance is to the minimum theoretically achievable variance of a control system.In our application of interest, the control objective is to minimize output perturba-tions. Thus, MVC benchmarking is an appropriate tool to measure performance ofthe system. Due to the fact that the minimum variance is a rigorous benchmarkin industrial applications, user-specified benchmarking is proposed to compare theoutput variance with some benchmark defined by the user. The new benchmark foroutput variance is defined based on the user’s other desired control objectives.The contribution made by this work, is to use user-specified benchmarkingto assess performance of Feedforward/Feedback (FFFB) control systems. In thegeneral case, the machine directional processes in paper machines can be modelledas a lower triangular transfer function matrix. The special structure of this modelallows us to assume the processes as several FFFB systems, rather than a singleMIMO system. Thus, using FFFB user-specified analysis instead of MIMO analysiswill reduce the computational loads for such systems substantially.52.1 Minimum Variance BenchmarkingIn MVC benchmarking, output variance is Harris [17] has developed an efficienttool for measuring the performance of control systems by comparing the minimumtheoretically achievable variance of a control system with its actual output vari-ance. In [17] he shows that for a system with time delay d, a portion of the outputvariance is feedback-invariant, so it can be estimated by time series analysis ofroutine operating data. This portion of the output variance, which is actually theminimum variance of the system, can be achieved by applying minimum variancecontrol (MVC) to the process. Using MVC as a benchmark does not mean that onehas to implement such a controller on the actual process. Rather, the performanceof MVC is used to compare performance of a given controller with.The general feedback control system shown in fig.2.1, is used to review theideas behind the minimum variance index for SISO systems. In this figure,C(q1),P˜(q1) and N(q1) denote the controller, delay-free process and disturbance trans-fer functions, respectively. d is the time delay in the process. yt and ut are theprocess input and output, respectively. Moreover, et denotes the unmeasured dis-turbance assumed to be uniformly distributed white noise with zero mean and vari-ance s2e . q1 denotes the back-shift operator. For the sake of simplicity, the depen-dence on q1 is not shown explicitly for most transfer functions in this thesis. Wecan calculate the output of the above system under regulatory control as follows:yt =N1+qdP˜Cet . (2.1)Using the Diophantine identity, N can be written as:C(q1) qdP˜(q1)N(q1)utetytFigure 2.1: The block diagram of a feedback control system6N = f0 + f1q1 + · · ·+ fd1qd+1| {z }F+Rqd (2.2)where fi (i= 0, . . . ,d1) are constant coefficient and R is a rational proper transferfunction. Substituting (2.2) into (2.1) yieldsyt =F +qdR1+qdP˜Cet=F +RFP˜C1+qdP˜Cet= Fet +Letd (2.3)where L = RFP˜C1+qdP˜C. In the above equation, Fet is the feedback invariant term ofthe output, since it cannot be affected by the controller. Based on the statisticalindependence of the white noise sequence, et , we haveVar(yt) =Var(Fet)+Var(Letd). (2.4)ThereforeVar(yt)Var(Fet). (2.5)Equality holds when L = 0 and leads to minimum variance control law:Cmv =RP˜F. (2.6)Therefore, applying minimum variance control to the system results in the follow-ing dynamics:yt|mv =⇣f0 + f1q1 + · · ·+ fd1qd+1⌘et (2.7)The minimum achievable variance of the process can be calculated as follows:s2mv =Var(Fet)=f 20 + f21 + · · ·+ f2d1s2e . (2.8)7Consequently, MVI can be defined to compare the performance of the actual outputvariance, s2y =Var(yt) with the minimum variance performance.MVI , s2mvs2y. (2.9)2.1.1 FCOR AlgorithmHuang and Shah [20] suggest an algorithm to calculate MVI, based on the cross-correlation between output and noise sequences, namely, the Filtering and Correla-tion (FCOR) algorithm. A stable closed-loop process can be written as an infinite-order time series:yt =⇣f0 + f1q1 + · · ·+ fd1qd+1 + fdqd + . . .⌘et . (2.10)Calculating the cross-correlation of the above-mentioned series with et from lag 0to d1 yieldsrye(0) = E[ytet ] = f0s2erye(1) = E[ytet1] = f1s2e...rye(d1) = E[ytetd+1] = fd1s2e(2.11)Therefore, the minimum variance of the process iss2mv =f 20 + f21 + · · ·+ f2d1s2e="✓rye(0)s2e◆2+✓rye(1)s2e◆2+ · · ·+✓rye(d1)s2e◆2#s2e=⇥r2ye(0)+ r2ye(1)+ · · ·+ r2ye(d1)⇤/s2e . (2.12)8Thus, according to (2.9), MVI can be written as:MVI =⇥r2ye(0)+ r2ye(1)+ · · ·+ r2ye(d1)⇤/s2y s2e= r2ye(0)+r2ye(1)+ · · ·+r2ye(d1) (2.13)whererye(k) =rye(k)qs2y s2e. (2.14)Throughout the above calculation, it is assumed that the noise sequence, et isknown. Therefore, it is important to estimate the noise. To do so, the observedoutput signal can be whitened by filtering yt =N1+qdP˜Cet ; the estimated noise iseˆt = (N1+qdP˜C)1yt . (2.15)2.1.2 The Effect of Measured DisturbancesDesborough and Harris [8] extendMVI to univariate FFFB control systems and thusaccount for the reduction in output variance due to measured disturbances. In suchsystems, we can use the extra information provided by measured disturbances, todecrease output variations more. Fig. 2.2 depicts the structure of FFFB controlsystems. The output of such systems can be expressed as:yt =11+qdP˜C"mÂi=1(Ni +qdP˜Ci)et,i +Net#. (2.16)In the above equation, et,i is the ith measured disturbance with zero mean, Ni isthe corresponding transfer function, and m is the number of measured disturbances.Using the Diophantine identity, shown in (2.2), the closed-loop equation can be9C qdP˜Ni NCiuetet,iytFigure 2.2: The block diagram of a FFFB control systemwritten as:yt =F +RFP˜C1+qdP˜Cqdet +mÂi=1Fi +P˜Ci +RiFiP˜C1+qdP˜Cqdet,i (2.17)where Fi and Ri are the solutions of the Diophantine identity with respect to Ni.Huang and Shah in [20] show that the output of FFFB control systems under MVCcan be written in the following form:yt|mv = Fet +mÂi=1Fiet,i (2.18)where yt|mv is the output of the FFFB system under MVC. In the above equation, thefirst term on the right hand side is the feedback control invariant term of the outputdue to the unmeasured disturbance and the second term is the feedforward controlinvariant term due to the measured disturbances. The corresponding minimumvariance feedforward and feedback controllers are given by the following transferfunctions:Cmv =RP˜F, (2.19)Ci,mv =RFiRiFP˜F. (2.20)Therefore, the output variance of a system controlled by FFFB minimum variance10algorithm iss2mv =Var Fet +mÂi=1Fiet,i!(2.21)and, just as for feedback control systems, the minimum variance index for FFFBcontrol systems is defined by the ratio in (2.9).2.2 User-Specified Performance IndexIn practice, the minimum variance benchmark has been found to be an extremelystringent requirement on the performance of a controller. Moreover, a minimumvariance controller is rarely used in practice due to the physical limitations. Forinstance, non-minimum phase zeros cannot be cancelled by stable controllers andthis affects the closed-loop dynamics. Moreover, the minimum variance controllerprovides very aggressive control moves, potentially damaging actuators and alsoit is not a very robust controller. Hence, we may get a low MVI for a controlsystem whose performance is otherwise considered satisfactory. Therefore, min-imum variance benchmarking is modified to user-specified assessment techniquefor feedback control systems by Huang and Shah [20]. The idea behind their ap-proach is to compare the output variance of the system with a variance defined by auser based on historical data and performance of the respective control loops, theseeffects appear as additional terms in the desired closed loop dynamics, which nowtake the form:yt|usr =⇣f0 + f1 + · · ·+ fd1 +qdGR⌘et , (2.22)where GR is a stable and proper transfer function. There are many ways to specifyGR based on desired closed-loop specifications, e.g., closed loop time constant, de-sired variance, frequency domain characteristics, robust performance, etc. In fact,it is totally arbitrary how to specify GR. Huang and Shah [20] suggest defining:GR = (1GF)R, (2.23)11where GF is a proper and stable transfer function and R is defined based on theDiophantine identity (2.2). Choosing GF = 1 makes GR = 0 and reduces (2.22)to (2.7). Thus, the user-specified approach includes minimum-variance methodsas a special case. In general, GF can be specified according to the closed loopdynamics. So, if we consider it as a desired closed loop transfer function, 1GFis going to be a desired sensitivity function. GF is defined as:GF =1a1aq1 . (2.24)a is specified asa = exp( Tstdes) (2.25)where Ts is a sampling interval and tdes is the desired time constant of the closedloop system. GR can be also selected as a desired closed loop transfer function fromthe unmeasured disturbance to the output. In this case, there is no need to estimateand use the disturbance transfer function, N. The only information required toperform user-specified benchmarking is process time delay and desired responseof the closed loop system to the disturbance.Consider the term L in (2.3). Under user-specified control,L =RFP˜C1+qdP˜C= GR. (2.26)Using (2.23), we haveRFP˜C1+qdP˜C= (1GF)R. (2.27)Consequently, we can calculate a user-specified controller as follows:Cusr =GFRFP˜. (2.28)(In the above equation, if GF = 1, we would have the minimum variance controller12in (2.20).) Accordingly, user-specified variance of the system iss2usr =Var ((F +GR)et) . (2.29)Finally, the User-Specified Performance Index (USI) can be defined by comparingthe actual output variance with user-specified variance.USI =s2usrs2y. (2.30)2.2.1 The Effect of Measured DisturbancesIn this thesis, the concept of user-specified benchmarking is generalized and ap-plied to FFFB control systems. In such control systems, the user can specify dy-namics based on which measured disturbances are rejected. So, we can take thisinformation into account to have a better estimation of the system’s performance.The effect of user-specified dynamics on feedforward terms can be considered byadding extra terms to the control invariant part of the output associated with themeasured disturbances:yt|usr = (F +qdGR)et +mÂi=1(Fi +qdGR,i)et,i (2.31)where GR,i is a stable and proper transfer function. In equation (2.31), settingGR,i = 0 denotes that user-specified control is used for feedback control and mini-mum variance control is applied to reject measured disturbances. Similar to user-specified benchmarking for feedback control systems, there are many ways to spec-ify GR,i. For instance, it can be defined as a desired transfer function from eachmeasured disturbance to the output or we can define it by analogy with (2.23).Therefore, the variance of the output under user-specified FFFB control is:s2usr =Var (F +qdGR)et +mÂi=1(Fi +qdGR,i)et,i!. (2.32)The user-specified performance index for FFFB control systems is then calculatedas in (2.30). In the above mentioned equation, we can specify GR,i as we do for GR13in (2.23):GR,i = (1GF,i)Ri (2.33)where GF,i is a proper and stable transfer function that can be defined as in (2.24),with Ri calculated by solving the Diophantine identity associated with Ni. There-fore, the feedback and feedforward controllers which result in (2.31), areCusr =GFRP˜F, (2.34)Ci,usr =RGFNiGF,iRiFqdGF,iRiP˜F. (2.35)Setting GF,i to 1 in the above equations results in applying user-specified feedbackcontrol and minimum variance feedforward control.2.3 ConclusionsIn this section, we introduced minimum variance as well as user-specified bench-marking. In these techniques, a performance index is defined which shows howclose performance of a control system is to the performance of minimum vari-ance or user-specified control. We also showed that how MVC benchmarking canbe applied to FFFB control systems. The contribution made to the work in thissection was to apply user-specified benchmarking to FFFB control performancemonitoring. Indeed, the application of interest in this work, i.e., MD control is aFFFB control system. So, it is of utmost important for us to apply user-specifiedbenchmarking to such systems due to the advantages of this technique over MVCbenchmarking.14Chapter 3Performance Assessment ofMulti-Input/Multi-OutputSystemsIn this chapter, SISO MVC as well as user-specified benchmarking are extendedto assess performance of MIMO systems. Moreover, it is shown that similar toSISO analysis, we can use user-specified benchmarking to estimate performance ofMIMO FFFB control systems. In the application of interest in this thesis, i.e. MDcontrol, we do not deal with such systems; however, the analysis presented in thischapter can be used for other similar applications, e.g., cross directional control inpaper machines.3.1 Minimum Variance BenchmarkingHuang et al. [21] and Harris et al. [18] use the minimum variance approach tomeasure performance of MIMO systems. For a multivariable process the outputvector Yt , of dimension n, satisfiesYt = PUt +Net (3.1)15Ut is the m-dimensional input, et is a noise vector in Rn with zero mean andVar(et) = Se, P is a n⇥m transfer matrix representing the process model, N isa disturbance transfer matrix (assumed diagonal, n⇥n). We decompose the matrixP in (3.1) asP = D1P˜, (3.2)where the “interactor matrix ” D1 is a diagonal transfer matrix consisting of thetime delays in the diagonal terms of P, and P˜ is the resulting filtered transfer matrix.P˜ = DP (3.3)The interactor matrix is introduced by Wolovich and Elliott [45], Wolovich andFalb [46] as well as Goodwin and Sin [15] for MVC and other purposes. Huang andShah [20] present a comprehensive survey on the characteristics of the interactormatrix.The minimum variance control law is obtained by choosing the transfer matrixC such that the controllerUt =CYt minimizes the following objective function:J = E⇥(Yt E[Yt ])T (Yt E[Yt ])⇤. (3.4)Proposition. For any linear time invariant system with the transfer functionshown in (3.1), regardless of a controller type, the minimum value of J is:Jmin = tr(Var(Fet)) (3.5)where F is a matrix with polynomial entries such thatN = F +D1R (3.6)and R is a matrix whose entries are proper rational transfer functions. Since thematrix N is diagonal, R and F are diagonal matrices too.16Proof. With a set-point of zero, the closed loop system obeysYt =I+D1P˜C1Net . (3.7)Substituting (3.6) into 3.7 givesYt =I+D1P˜C1 F +D1Ret=⇣ID1P˜I+CD1P˜1C⌘F +D1Ret= Fet +D1⇣R P˜I+CD1P˜1CN⌘et, Fet +D1Let (3.8)whereL = R P˜(I+CD1P˜)1CN. (3.9)In (3.8), the term Fet is control-invariant. Hence the minimum variance controllaw is obtained by setting L = 0, i.e., by definingCmv = P˜1RF1. (3.10)The system output under minimum variance control isYmv = Fet , (3.11)and using this in (3.4) reveals the minimum value shown in (3.5).Accordingly, a vector-valued performance index for a MIMO system can bedefined as follows.[hy1mv, . . . ,hynmv] = diagSmvS˜1Y (3.12)where, Smv is a covariance matrix of the system output under minimum variancecontrol, S˜Y = diag(SY ) and SY = Var(Yt). Furthermore, a single index can be17defined as follows to show the performance of the MIMO system at once,hmv =1nnÂi=1hyimv. (3.13)To calculate the above indices, the minimum variance of outputs must be calcu-lated. The system output under minimum variance control, defined in (3.11), canbe written in the following form.Ymv =⇣F0 +F1q1 + · · ·+Fd1qd+1⌘et (3.14)where d is the maximum time delay in the Interactor matrix, D1. Accordingly,the covariance matrix of the output signal under minimum variance control isSmv = E[YmvYTmv] =d1Âi=0FiSeFTi (3.15)where Set is a noise covariance matrix. Due to the fact that the noise sequence isstatistically independent, the minimum variance of the ith output can be calculatedass2yi,mv =d1Ân=0(Fiin )2Siiet . (3.16)3.2 User-specified BenchmarkingAs discussed in previous sections, minimum variance control provides very aggres-sive control moves and due to the existence of some constraints on plant inputs aswell as stability and robustness concerns, it is rarely utilized in practice. Hence,the performance of control loops is always far from the theoretical performanceunder minimum variance control and the minimum variance performance index al-ways looks poor. As for SISO systems, minimum variance benchmarking can beextended to user-specified benchmarking for MIMO systems. In this approach, theoutput covariance matrix is compared with a covariance matrix specified by a user.To do so, an extra dynamic term is added to the output under user specified control18shown in (3.11), resulting inYusr =F +D1GRet (3.17)where GR is a proper transfer function matrix which is assumed to be diagonal inthis work. A control law which results in the above output isCusr = P˜1(RGR)(F +D1GR)1. (3.18)Accordingly, the user-specified performance index is defined as follows.[hy1usr, . . . ,hynusr] = diagSusrS˜1Y , (3.19)husr =1nnÂi=1hyiusr, (3.20)where Susr = Var(Yusr). To calculate Susr, first of all, eq.(3.17) should be writtenas a moving average process,Yusr =⇣F0 +F1q1 + · · ·+Fd1qd+1⌘et+D1GR,0 +GR,1q1 + . . .et . (3.21)Then, we haveSusr = E[YusrYTusr] =d1Ân=0FnSet FTn +•Ân=0GR,nSetGTR,n . (3.22)Thus, the variance of the ith output under user specified control iss2yi,usr =d1Ân=0(Fiin )2Siiet +•Ân=0(GiiR,n)2Siiet . (3.23)193.2.1 Performance Assessment of Feedforward/Feedback ControlSystemsAs in SISO case, information coming from measured disturbances in MIMO FFFBcontrol systems can be used to decrease output variations. Following a similar ap-proach which was discussed in the previous section, Huang et al. [22] apply mini-mum variance benchmarking to the performance assessment of feedforward/feed-back MIMO control systems. Moreover, user-specified benchmarking for SISOFFFB control systems which introduced in the section 2.2.1, can be extended touser-specified performance assessment of MIMO FFFB control systems.3.3 ConclusionsIn this section, we showed that how MVC and user-specified benchmarking canbe applied to MIMO systems. Using user-specified benchmarking for FFFB controlsystems, which was discussed in the previous section, can be extended and appliedto MIMO FFFB control systems. It can have an application in performance moni-toring of applications other than MD control, e.g., cross-direction control in papermachines.20Chapter 4The Impact of Model-PlantMismatch on the PerformanceIndicesThe performance indices in (2.30) and (3.20) refers to controllers (2.28) and (3.18)that depend on the plant, P˜. In practice the true plant P0 is not known exactlyand the controller is designed using some model P. We refer to the discrepancybetween P and P0 as model-plant mismatch, MPM. There is scant literature onunderstanding the effect of MPM on minimum variance and user-specified perfor-mance indices. For example, Yousefi et al. [48] [49] analyse the sensitivity of thementioned performance indices to different types of parametric MPM. They showthat mismatch in different parameters influences the indices differently. However,there is a lack of explanation for their observations.There are also few articles on MPM detection in control systems using otherapproaches. For instance, in [43], Wang et al. analyse the influence of model-plant mismatch on control loop behaviour. They introduce the Integral AbsoluteError (IAE) index to measure performance of control systems to detect MPM. TheIAE index is defined by integrating the absolute value of the output error. Theyclaim that the smaller the index the better the performance. Indeed, they show thatMPM increases IAE index. However, they don’t define any benchmark to comparethe defined index with, nor do they indicate what the index should be in the normal21operating condition.Badwe et al. [3] define a Relative Sensitivity Index (RSI) by comparing an ac-tual sensitivity function with a designed sensitivity function to quantify the impactof model plant mismatch. This index is defined as follows:RSI =SaSd•(4.1)where Sa is an actual sensitivity function and Sd is a designed sensitivity function.To calculate the above index, we only need to estimate Sa. Badwe et al. [3] showthat if MPM happens in a closed loop system, the index becomes greater that 0 dB.They use the closed loop information to estimate the actual sensitivity function.To do so, there must be set-point changes in the closed loop system. However, inindustrial applications, set-point changes do not happen frequently and most con-trol systems work in regulatory mode. So, RSI cannot be implemented to monitorclosed loop performance in most cases.In this section, we investigate how MPM affects the performance indices definedin the previous chapters. We consider the effect of different types of parametricmismatch, namely, gain mismatch, time constant mismatch and delay mismatch,on the output variance in the closed loop system. This analysis will answer theobservations presented by Yousefi et al. [48] [49]. Their observations show thatgain mismatch influences the performance indices more than either time constantor delay mismatch. In addition, it is shown that sluggish mismatch affects the tran-sient response of systems more and due to the fact that the introduced performanceindices are used to estimate the performance of closed loop systems in regulatorymode, the effect of such mismatch doesn’t appear in the indices. Therefore, amethod is proposed to estimate the performance of closed loop systems in servocontrol mode to remedy this problem.In principle, the performance of each output in a MIMO system can be affectedby modelling errors in every single element of the transfer function matrix rep-resenting the process model. In this chapter, we outline some reasonable circum-stances where, if there is no mismatch in transfer functions associated with a certainoutput in a multivariate process, the performance of that output does not deteriorateeven if there is mismatch in other transfer functions. In such cases, the performance22of each output can be analysed independently.4.1 Effects of MPM on the Sensitivity FunctionIn this section we show how MPM affects a closed-loop sensitivity function. Thecomplementary sensitivity function T for a SISO system, is defined as follows:T =CP1+CP, (4.2)where C and P denote the controller and the process model, respectively. Theprocess model can be written as follows:P = P0+D (4.3)where, P0 is a nominal plant model, which is used in controller design and D is adifference between the nominal model and the plant model.Stability MarginFor a closed loop system, the stability margin Sm is defined as the shortest distancefrom the Nyquist curve to the critical point on the complex plane, which is (1,0).This distance is illustrated in fig. 4.1 .It can be shown that the stability margin is proportional to the inverse of sen-sitivity peak (kSclk•). The closed loop sensitivity function can be demonstrated asfollows:Scl =11+L(4.4)where L is the open loop transfer function,L =CP. (4.5)For a particular frequency w > 0 , the distance dw of the corresponding open loop23−1 −0.5 0 0.5 1 1.5−0.8−0.6−0.4− AxisImaginary AxisSmFigure 4.1: Nyquist DiagramNyquist plot point L( jw) from the critical point (1,0) in the complex plane isdw = |L( jw) (1)|= |L( jw)+1|= 1/|S( jw)| (4.6)while the minimum distance, dmin of the whole Nyquist plot of L(s) from the criti-cal point (1,0) is well known to bedmin = infwdw= infw(1/|S( jw)|)= 1/supw(|S( jw)|)= 1/Ms (4.7)Here, Ms stands for the sensitivity function peak value or equivalently, the H•24norm:Ms = supw(|S( jw)|) = kS(s)k•. (4.8)According to the above equation, dmin denotes the stability margin Sm. So, we canwriteMs =1Sm. (4.9)Based on the above equation, if the stability margin decreases or in other words,if the open loop gain increases, the sensitivity function peak grows. Among dif-ferent types of MPM, namely, gain mismatch, time constant mismatch, and delaymismatch, positive gain mismatch increases the open loop gain and equivalently,increase the sensitivity peak more.The existence of a peak in the Bode diagram of the sensitivity function impliesthat the closed loop system not only fails to reject disturbances around that fre-quency region, but also it amplifies them. Indeed, in a case of mismatch that thesensitivity peak increases, the closed loop system amplifies disturbances within aspecific range of frequencies more and consequently, the output variance increases.This is the reason why the performance indices drop when mismatch happens inclosed loop systems. Fig. 4.2, 4.3 and 4.4 compare the sensitivity peak with dif-ferent values of MPM for different types of MPM. These figures are associated witha closed-loop system which controls a first order process with following transferfunction:P(s) =kts+1etd (4.10)where k, t and td denote process gain, time constant and time delay, respectively.25Percentages of mismatch are define as follows:Gain Mismatch =k k0k(4.11)Time Constant Mismatch =t t 0t (4.12)Time Delay Mismatch =td  t 0dtd(4.13)where k0, t 0 and t 0d denote nominal gain, time constant and time delay, respectivelywhich are used in control design.Comparing these figures shows that positive gain and delay mismatch increasethe sensitivity function peak more than time constant mismatch. This is becausethese types of mismatch change the open loop transfer function and decrease thestability margin more. The small changes in the sensitivity peak due to the changesin time constant mismatch indicates that the output variance doesn’t change whentime constant mismatch happens in a control system. It implies that the perfor-mance indices are not sensitive to time constant mismatch. In addition, the negativegain mismatch makes the open loop gain smaller and stability margin bigger andconsequently, decreases the sensitivity peak. Thus, this type of mismatch causesno increase in the output variance, indeed it decreases it and improves the perfor-mance of the system. However, in this case and in other cases where the sensitivitypeak doesn’t change very much, the bandwidth of the closed loop system changes.This change in the closed loop system affects the performance of the system inservo mode in which there are some transient behaviours. To strengthen the effectof model-plant mismatch on performance of control systems, we can take tran-sient responses into account by measuring the performance of the following signalinstead of the actual output:Yt,new = Yt Yt,des=hI+D1P˜C1D1P˜CTdesirt +I+D1P˜C1Net (4.14)where P˜ and D1 denote delay free transfer function matrix and interactor matrix,respectively which were defined in previous sections. Yt,des and Tdes are a desired2610−3 10−2 10−1−30−25−20−15−10−505101520Magnitude (dB)  Frequency  (rad/s)No mismatch−50% gain mismatch+50% gain mismatch+100% gain mismatchFigure 4.2: The effect of gain mismatch on the sensitivity function10−3 10−2 10−1−20−15−10−5051015Magnitude (dB)  Frequency  (rad/s)No mismatch−50% delay mismatch+50% delay mismatch+100% delay mismatchFigure 4.3: The effect of delay mismatch on the sensitivity function2710−3 10−2 10−1−20−15−10−50510Magnitude (dB)  Frequency  (rad/s)No mismatch−50% time constant mismatch+50% time constant mismatch+100% time constant mismatchFigure 4.4: The effect of time constant mismatch on the sensitivity functionoutput signal and closed-loop transfer matrix (Complementary Sensitivity func-tion), respectively, and rt is a vector of set-points. The above definition makes itpossible to measure the performance of a system in servo mode. Clearly, if there isno MPM andI+D1P˜C1D1P˜C = Tdes,the performance of Yt,new will be the same as the performance of Yt . On the otherhand, if mismatch is present, the difference between the desired response and thesystem’s actual response will help us to detect the mismatch better by decreasingthe performance indices more. Several advantages will be observed: a) Mismatchwith no effect on the system’s steady state performance can be detected. b) Per-formance can be assessed continuously. (Note that current assessment techniquescannot be applied during transients or set-point changes.) c) A comparison ofthe system performance in transients with the performance in the steady state willprovide extra information about the type of mismatch. These advantages will bediscussed further in the next section.284.2 The Effect of MPM in MIMO SystemsIn principle, the performance of each output in a MIMO system can be affected bymodelling errors in every single element of the transfer function matrix represent-ing the process model. In this section, we outline some reasonable circumstanceswhere, if there is no mismatch in the transfer functions associated with a certainoutput in a multivariate process, the performance of that output does not deteriorateeven if there is mismatch in other transfer functions. In such cases, the performanceof each output can be analysed independently.We now investigate how a system’s output variance is affected by mismatchbetween the true plant, P˜, and the model used in its controller, P˜0. The mismatch isthe transfer matrixD= P˜ P˜0. (4.15)We will show that for output components whose transfer functions are modelledexactly, mismatch in other components has no effect on the variance. This allowsfor a modest decoupling of the full sensitivity problem. Our analysis relies onthe assumptions that both the disturbance transfer matrix N(q1) and the filter GRin (3.17) are diagonal. Similar methods treat both minimum-variance and user-specified control strategies.4.2.1 Minimum Variance ControlUsing the plant model P˜0 in (3.10) produces the mismatched controllerCmv = (P˜D)1RF1. (4.16)29Using this in (3.9) gives, after simplification,L = R P˜I+CmvD1P˜1CmvN= RC1mv P˜1 +D11N= RFR1(P˜D)P˜1 +D11N= RFR1FR1DP˜1 +D11N= RRNFR1DP˜1R1N= RRIN1FR1DP˜1R1. (4.17)Thus, by the matrix inversion lemma,L = RR⇣I+N1FR1DI P˜1RN1FR1D1P˜1R⌘=RN1FR1DI P˜1RN1FR1D1P˜1R, KDM, (4.18)whereK =RN1FR1, (4.19)M =I P˜1RN1FR1D1P˜1R. (4.20)The matrix K is diagonal, being a product of diagonal matrices. Thus the elementsof matrix L are simplyLi j =nÂp=1DipKiiMp j. (4.21)We can extract the static gain of Dip and write the above equation as follows:Li j =nÂp=1d ipD˜ipKiiMp j=nÂp=1d ipLip j (4.22)30where d ip is the static gain of Dip, Dip = d ipD˜ip and Lip j = D˜ipKiiMp j. When theoutput equation (3.8) is expressed as a moving average process,Yt =⇣F0 +F1q1 + · · ·+Fd1qd+1⌘et+D1L0 +L1q1 + . . .et , (4.23)the coefficient matrices Ln come from (4.21):Li jn =nÂp=1d ipLip jn . (4.24)Here Lip jn is the n th infinite impulse response coefficient of Lip j. Finally, the vari-ance of output component i is given bys2yi =d1Ân=0(Fiin )2Siie+•Ân=0"nÂl=1"nÂq=1" nÂp=1d ipLipqn!Sqle nÂp=1d ipLipln!###. (4.25)The first term on the right in (4.25) is the minimum variance of the ith output.Model-plant mismatch influences the second term only, entering through d ip, thetransfer-function discrepancy from the pth input to the ith output. If d ip = 0 foreach p, the second term vanishes. That is, if there is no mismatch in the transferfunctions associated with the ith output, then the predicted variance of the ith outputwill be identical to the value predicted using an exact model. Mismatch in transferfunctions related to other outputs cannot affect the calculated performance indexfor component i.4.2.2 User-specified ControlIn this part, the same analysis is carried out for user-specified control. The user-specified control law under mismatch conditions can be written asCusr = (P˜D)1(RGR)(F +D1GR)1. (4.26)31The substitution of the above equation into (3.9) results inL =R P˜I+CusrD1P˜1CusrN=RC1usr P˜1 +D11N=R(F +D1GR)(RGR)1(P˜D)P˜1 +D11N=R (RGR)N (F +D1GR)(RGR)1DP˜1(RGR)1N=R (RGR)IN1(F +D1GR)(RGR)1DP˜1(RGR)1. (4.27)Using the matrix inverse lemma, we haveL =R (RGR)⇥I+N1(F +D1GR)(RGR)1DI P˜1(RGR)N1(F +D1GR)(RGR)1D1P˜1(RGR)⇤=GR (RGR)N1(F +D1GR)(RGR)1DI P˜1(RGR)N1(F +D1GR)(RGR)1D1P˜1(RGR), GR +KDM, (4.28)whereK = (RGR)N1(F +D1GR)(RGR)1, (4.29)M =I P˜1(RGR)N1(F +D1GR)(RGR)1D1P˜1(RGR). (4.30)It is obvious that the matrix N is diagonal, because all of the matrices in (4.29) arediagonal. Accordingly, the elements of matrix L can be calculated as follows.Li j =8><>:Ânp=1DipKiiMp j i 6= jGiiR +Ânp=1DipKiiMpi i = j=8><>:Ânp=1 d ipLip j i 6= jGiiR +Ânp=1 d ipLipi i = j(4.31)32where d ip is the static gain of Dip and Lip j = D˜ipKiiMp j. Based on the aboveequation, the elements of the coefficient matrix Ln in (4.23) are calculated basedon the following equation:Li jn =8><>:Ânp=1 d ipLip jn i 6= jGiiR,n +Ânp=1 d ipLipin i = j.(4.32)In the above equation, Lip jn is the n th infinite impulse response coefficient of therelated transfer function, Lip j. Hence, the variance of the ith output under user-specified control, when there is mismatch between the model used in the controllerand the plant, can be calculated as follows:s2yi =d1Ân=0(Fiin )2Siie+•Ân=0264nÂl=1l 6=i264nÂq=1q6=i" nÂp=1d ipLipqn!Sqle nÂp=1d ipLipln!#375+nÂl=1l 6=i" GiiR,n +nÂp=1d ipLipin!Sqle nÂp=1d ipLipln!#+nÂq=1q6=i" nÂp=1d ipLipqn!Sqle GiiR,n +nÂp=1d ipLipin!#+ GiiR,n +nÂp=1d ipLipin!Siie GiiR,n +nÂp=1d ipLipin!#. (4.33)In the above equation, d ip is associated with mismatch in the transfer function fromthe pth input to the ith output. Accordingly, if d ip is zero, the variance of the ithoutput is the same as the user-specified variance (s2yi,usr), shown in (3.23). In otherwords, no matter how much mismatch there is in transfer functions related to otheroutputs, the performance of the ith output would be high if there is no mismatch intransfer functions associated with this output.Based on the above analyses, the performance of each output in MIMO system33is independent from other output performance. Hence, the sensitivity of the ithoutput performance to the model-plant mismatch can be studied regardless of otheroutputs’ performance. As mentioned before, this analysis is valid under the basicassumption that both a) The disturbance transfer function matrix N and b) the filterused in user-specified control GR, are diagonal.4.3 Case Study4.3.1 Sensitivity of The Performance Indices to MPMThe process model is a key ingredient in defining performance criteria for bothminimum-variance and user-specified benchmarking. (4.25) and (4.33) show howmodel-plant discrepancy degrades performance. Sensitivity analysis will revealhow model-plant mismatch affects the performance indices, and help focus re-identification efforts aimed at restoring plant performance.In the most general view, a performance index is a function that maps a model,plant, and controller to a scalar value. Expressing the true plant P˜ in terms ofdeviations D from a nominal plant P˜0 suggests defining the general performancefunctions2yi(t,Dip) : R+⇥S ! R+, (4.34)whereS is a set of transfer functions capturing all possible uncertainties Dip. (Thecontrollers considered here are implicitly determined by the plant model, so thereis no need to show explicit controller-dependence in the functions above.) For boththe minimum variance and user-specified performance indices, the performancecriterion is the variance. Thus we take hyi = s2yi above, and express the relativesensitivity of the performance index to model uncertainty asS(t,Dip) =∂s2yi(t,Dip)∂DipDips2yi(t,Dip). (4.35)where Dip is the uncertainty in the transfer function from the pth input to the ithoutput.34Discussions about the performance of control systems in the literature usuallyrefer to the performance in steady state, also known as regulatory mode. In ad-dition, in all equations the set-points are always assumed to be zero. In the nextpart, it is shown that the performance indices are insensitive to mismatch in timeconstants. Indeed, for a stable control system, variations in time constants of thesystem do not affect the indices.4.3.2 Simulation ResultsIn this part, first of all, a paper machine’s control loop which controls the machinedirectional (MD) properties of paper, such as dry weight, is introduced. Then,using the mentioned system, it is shown that a poorly performing output cannotinfluence the performance of other outputs as proved in section 6.4. In the next step,the sensitivity is calculated for a particular output with respect to three types ofparametric mismatch (gain, time constant and delay) in two different control modes(regulatory and servo control) to see how sensitive the performance indices are tothe model mismatch. The aim of comparing the sensitivity of the performanceindices in two modes is to check if such mismatch does not affect the performance.Since the related analyses for minimum variance and user specified benchmarksare similar, only the sensitivity of the user-specified control performance indexis discussed, as the system is not controlled by a minimum variance controller.In fact, user-specified benchmarking can be easily reduced to minimum variancebenchmarking by setting GR = 0.Three important product characteristics in paper machines are dry weight (y1),size press moisture (y2) and reel moisture (y3). The dominant manipulated vari-ables which influence these properties are stock flow (u1) and two different dryerpressures (u2 and u3). The following equation shows the relation between the men-tioned variables.264y1y2y3375| {z }Yt=264P11 0 0P21 P22 0P31 P32 P33375| {z }P264u1u2u3375| {z }Ut+264N11 0 00 N22 00 0 N33375| {z }N264e1e2e3375| {z }et(4.36)35Input \ Output Stock Flow 1st Dryer Pressure 2nd Dryer PressureDry Weight1.3366.6s+1e72s  Size Press Moisture0.29646s+1e90s0.1490s+1e114s Reel Moisture0.553154.8s+1e42s0.238211.2s+1e30s0.055524.84s+1e90sTable 4.1: Nominal plant model, P0 (continuous time)For such a system, each matrix entry Pi j is a first order transfer function withdead time (Table 4.1 ). These transfer functions are used as the plant model todesign a controller. A special type of MPC controller is implemented to controlthe above properties. In [33], it is asserted that the controller is very robust tomodel uncertainties. In fact, the performance indices are not expected to be verysensitive to model mismatch. Our simulations shown in this section will quantifyhow sensitive the controller and performance indices are to the model mismatchesin two different operating modes, the regulatory and servo control modes. For thegiven control system, the desired transfer function matrix, Tdes is specified as adiagonal matrix shown as follows:Tdes =2666664e72s99.9s+10 00e114s135s+100 0e90s37.26s+13777775. (4.37)Assuming that GR = Tdes, all of the conditions used in the analyses above aresatisfied. Accordingly, in the next step, an effect of a poor performance output onother outputs is illustrated based on the simulation results.Fig. 4.5 shows the performance of 3 outputs of the system in 3 different cases:• Case 1: No mismatch,• Case 2: 100% gain mismatch in G11,36• Case 3: 100% gain mismatch in G22.Case 1 Case 2 Case 300. Indices  Dry WeightSize Press MoistureReel MoistureFigure 4.5: The effect of a poor performance output on other outputsIn the first case, there is no mismatch between the plant and the model used in thecontroller. Thus, we get good performance for all of the outputs. In the secondcase, the only difference between true plant P and the model P0 is 100% mismatchadded to the gain of P11 (4.11). In the last case, the nominal model matches theplant exactly, except for 100% gain mismatch added to P22. As we expected, incase 1, all of the outputs show good performance since the model used in the con-troller truly represents of the plant’s dynamics. But, when there is mismatch in thetransfer function from the first input to the first output, the performance of the firstoutput deteriorates. However, as proven in the previous sections, it doesn’t affectthe performance of two other outputs. Similarly, as a result of gain mismatch inP22, the performance index for the size press moisture decreases, while it doesn’tchange the indices for the other outputs. Thus, the sensitivity of the performanceindices of each output can be analysed individually. The size press moisture isselected to analyse the sensitivity of its performance to model-plant mismatch. Inthe next step, the sensitivity of the user-specified control performance index to 3different types of parametric mismatch, namely time constant, time delay and gain37No Mismatch Gain Mismatch Time Constant MismatchTime Delay Mismatch00. Index  200% Mismatch400% MismatchN0 MismatchFigure 4.6: The effect of different types of mismatch in G21 on the perfor-mance indexmismatch, is calculated, and illustrated. Fig. 4.6 shows that even a big mismatchin the parameters of P21 does not have a significant effect on the performance indexfor the size press moisture. This is because the dominant control variable whichaffects the size press moisture is u2 which is the result of our assumption that thedesired transfer function is diagonal. Therefore, in the rest of this section, weanalyse sensitivity of the performance index to the model mismatch in P22, assum-ing that the mismatch in P21 does not change the performance of the size pressmoisture. Thus, the sensitivity of the performance index to the different types ofmismatch only in P22 is analysed. Furthermore, the performance of the size pressmoisture in 2 different modes, servo control and regulatory modes, is calculatedand compared.Gain MismatchNow suppose the model used in the controller is kept untouched, but the gain of thetransfer function in the plant is changed. Then, the performance index of the sizepress moisture is calculated for each gain sample and accordingly the sensitivity ofthe performance indices to the gain mismatch is calculated base on (4.35). In this38case, the transfer function P22 is specified as follows.P22 =knom(1+Dk)tnoms+1etdnoms (4.38)where knom, tnom and tdnom are nominal gain, time constant and time delay used inthe controller, respectively, and Dk is the mismatch between the gain of G22 in theplant and the model used in the controller. The following figures show the user-specified performance index and the sensitivity of the index for different valuesof gain mismatch. The performance index and the sensitivity of the index arecalculated for the response of the system in two different modes, regulatory andservo control, and subsequently compared. Fig. 4.7 shows that the performanceindex falls dramatically as the mismatch increases in both modes. Indeed, thesmall mismatch in the gain of the transfer function causes a significant drop in theperformance index. Fig. 4.8 shows sensitivity of the index to the gain mismatch.It illustrates that the index is very sensitive to the mismatch in the gain and thesensitivity increases as the mismatch grows. Furthermore, the sensitivity of theindex is almost the same for both the regulatory and the servo control mode.0 0.5 1∆bPerformance Index   Regulatory ModeServo Control ModeFigure 4.7: User-specified performance index versus gain mismatch390 0.2 0.4 0.6 0.8 1 1.2 1.400.511.522.5∆bSensitivity   Regulatory ModeServo Control ModeFigure 4.8: Sensitivity of the user-specified performance index to gain mis-matchDelay MismatchNext, the time delay in P22 is changed and the performance index and the sensitivityof the performance to the delay mismatch Dtd are calculated and plotted. In thiscase, P22 can be represented asP22 =knomtnoms+1etdnom (1+Dtd )s. (4.39)Figs. 4.9 and 4.10 depict the index and the sensitivity of the index to the delay mis-match for the different values of the delay mismatch, respectively. As illustrated infig. 4.9, the growth in the delay mismatch degrades the performance of the system.In addition, it can be noticed that the performance of the transient response of thesystem is affected more than the performance of the steady state response. Thisfact is depicted in fig. 4.10 as well. This figure shows that the index in the servocontrol mode is more sensitive when there is small mismatch in the time delay.In fact the small delay mismatch affects the performance of the transient responsemore rather than the performance of the steady state response.400 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8∆dPerformance Index  Regulatory ModeServo Control ModeFigure 4.9: User-specified performance index versus time delay mismatch0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8∆τdSensitivity  Regulatory ModeServo Control ModeFigure 4.10: Sensitivity of the user-specified performance index to time delaymismatch41Time Constant MismatchIn this part, the effect of the time constant mismatch is studied on the performanceof the system in servo control and regulatory modes. In the case of time constantmismatch, P22 can be represented as follows.P22 =knomtnom(1+Dt)s+1etdnoms (4.40)where Dt is the amount of mismatch in the time constant. The following figuresshows the effect of the time constant mismatch on the user-specified performanceindex as well as the sensitivity of the index to the mismatch.According to fig. 4.11, the mismatch in the time constant has no significant influ-ence on the performance of the steady state response of the system. Nonetheless, itaffects performance of the system in the servo control mode more. Fig. 4.12 con-firms this. This figure shows that the transient response performance of the systemis more sensitive to time constant mismatch than the performance of the system inthe regulatory mode.For this type of system, we can conclude that if the system shows good per-formance in the regulatory mode and poor performance in the servo control mode,the problem is likely due to mismatch in the time constant. This information canbe used in experiment design for system identification, which is the next step toimprove the performance.4.4 ConclusionsIn this chapter, we demonstrated how different types of parametric mismatch affectthe sensitivity function and change the output variance and subsequently, how theyaffect the performance indices. Furthermore, we claimed that in MIMO systemsunder some reasonable circumstances, if there is no MPM in transfer functions as-sociated with a specific output, its performance cannot be affected by MPM in othertransfer functions. Moreover, we showed that the performance indices are not sen-sitive to some sorts of parametric mismatch, e.g. time constant mismatch. Thus, weproposed a technique to increase sensitivity of the indices to MPM, making thosetypes of mismatch that cannot be detected using MVC benchmarking, detectable.420 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8∆τPerformance Index  Regulatory ModeServo Control ModeFigure 4.11: User-specified performance index versus time constant mis-match0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8∆τSensitivity  Regulatory ModeServo Control ModeFigure 4.12: Sensitivity of the user-specified performance index to time con-stant mismatch43Chapter 5Detection of MPMOne of the disadvantages of using MVC benchmarking techniques to detect MPMis that these techniques are very sensitive to changes in disturbance characteristics.Moreover, any deterministic high frequency disturbance increases output variancesand consequently, system performance deteriorates. Therefore, MVC benchmark-ing techniques are not a very robust method to detect MPM. Nevertheless, MVCbenchmarking, is passive over closed-loop systems, and thus is considered as oneof its major advantages. In almost all other techniques in literature to detect MPMin closed-loop systems, some sort of excitation signal is applied. See [44], [34],[30], [14], [29], [3], [2] and [6].In this section, we propose a technique that, without introducing any excita-tion, enables us to determine which of these factors results in deterioration in theperformance indices: MPM or changes in disturbance characteristics. We show thatif MPM happens in a closed loop system, it will change the correlation between theprediction error and the process input at some specific lags. Indeed, by compar-ing the correlation of the mentioned signals in a case when the performance indexdrops with the correlation in normal operation, one can detect MPM. Some of theabove mentioned articles also consider the correlation between some closed-loopsignals; however, all of them require excitation to be introduced to the system.445.1 Prediction ErrorAs discussed in the previous sections, the output of a closed-loop system can bespecified as:yt = Put +Net . (5.1)The above equation can be re-written as follows:yt = (P0+D)ut +dt (5.2)where P0 denotes the nominal process model and D is the variation of the processmodel from the nominal model. dt is the output disturbance which is specified by:dt = Net . (5.3)The process output can be predicted using the nominal model as follows:yˆt = P0ut . (5.4)Accordingly, we can define the prediction error et by:et = yt  yˆt= Dut +dt . (5.5)If there is no mismatch between the process model and the nominal model, we haveD= 0, so the prediction error equals the output disturbance:et = dt . (5.6)In closed loop systems, the relation between output disturbance and processinput is:ut =C1+PCdt (5.7)45The above equation can be represented as:dt = Hut (5.8)whereH =1+PCC=✓1C+P◆(5.9)Substitution of (5.8) into (5.5) yields:et = yt  yˆt= (D+H)ut . (5.10)(5.7) can be shown as:ut =C˜qdc1+PCdt (5.11)where if dc( 0) is a lag which the controller introduces, C˜ is a delay free transferfunction of the controller. Also, (5.9) can also be shown as:H =(C+P) (5.12)where C= 1C. If P is stable andC is a minimum phase controller, we can concludethat H is a stable transfer function. Therefore, when we express H in the form ofinfinite impulse response series,H = h0qdc +h1qdc1 + · · ·+hdc +hdc+1q1 + · · ·+hdc+dqd +hdc+d+1qd1 + . . . ,(5.13)where d is the process time delay. In the above equation, the impulse response46coefficients (hi) are assumed to satisfy:|h0| > |h1| > · · · > |hdc | > |hdc+1| > .. . (5.14)and since H has a negative static gain, h0 < 0. Corresponding to (5.13) is theinfinite impulse response for the plant,P = qd(p0 + p1q1 + . . .). (5.15)5.2 Correlation AnalysisWe now show how to detect MPM by analysing correlation of the prediction errorand the process. Indeed, if performance of a closed-loop system deteriorates, wewould be able to say whether MPM causes the problem or it is because of changesin disturbance characteristics. We analyse the correlation of the prediction errorand the process input in two different cases: a) when there is mismatch and b)when there is no mismatch. Finally, we propose an algorithm to detect MPM whichis not sensitive to disturbance.Case 1: No MPM (D= 0)When there is no MPM in the closed-loop system, we have:et = Hut . (5.16)It is known that every stationary signal can be represented as filtered white noise,ut = Get (5.17)where G is a proper stable filter and et is a white noise. Assuming that G is aminimum phase transfer function, we havee¯t = Het (5.18)47wheree¯t = G1et . (5.19)Using (5.13) in (5.18) yields:e¯t =h0et+dC +h1et+dC1 + · · ·+hdCet +hdc+1et1 + . . .+hdc+ded +hdc+d+1ed1 + . . .=•Âj=0h jet+dc j. (5.20)Based on (5.9), the impulse response coefficients shown in (5.13) can be specifiedas follows:h0 =l0h1 =l1...hdc =ldc...hdc+d =ldc+d  p0hdc+d+1 =ldc+d+1 p1... (5.21)where li and pi are the impulse response coefficients associated with C and P,respectively.Proposition. If H is stable, the correlation between e¯t and G1ut (et) is mini-mized at t = dc.mintCorre¯t ,G1ut+t =Corre¯t ,G1ut+dc. (5.22)Proof. Recall that the correlation between generic signals ft and gt is defined by48fixing some N > 0 and settingCorr( ft ,gt) = E NÂi=0ftigti!. (5.23)Thus, for any shift t ,Corr(e¯,G1ut) =Corr(e¯,et)= E NÂi=0e¯tiet+ti!= E NÂi=0•Âj=0h jet+dc jiet+ti!= E •Âj=0NÂi=0h jet+dc jiet+ti!=•Âj=0h jCorret+dc jet+t. (5.24)In particular, we have for any t and t thatCorr(et ,et+t) =(0, i f t 6= 0,N +1, i f t = 0.(5.25)Therefore,Corr(e¯,G1ut) = (N +1)hdct . (5.26)Based on (5.14) and the assumption that the his are negative, we havemintCorr(e¯,G1ut) =mint{hdct}=h0 (5.27)andargmintCorr(e¯,G1ut) = dc. (5.28)49Therefore,mintCorre¯t ,G1ut+t =Corre¯t ,G1ut+dc. (5.29)Case 2: System with MPM (D 6= 0)In a system with MPM, (5.10) can be simplified as follows:e 0t = (D+H)ut=✓PP0 ✓1C+P◆◆ut=C+P0ut= H 0ut (5.30)whereH 0 =C+P0. (5.31)Filtering both sides of (5.30) with G1 yieldse¯ 0t = H 0et (5.32)wheree¯ 0t = G1e 0t . (5.33)Writing the filtered prediction error e¯ 0t in the form of an infinite impulse responseyields:e¯ 0t = h00et+dc +h01et+dc1 + · · ·+h0dcet +h0dc+1et1 + . . . (5.34)+h0dc+detd +h0dc+d+1etd1 + . . . . (5.35)50h0i in the above equation can be specified as:h00 =l0h01 =l1...h0dc =ldc...h0dc+d0 =ldc+d0  p00h0dc+d0+1 =ldc+d0+1 p01... (5.36)where the p0i denote infinite impulse response coefficients associated with P0 andd0 is the nominal time delay. Just as in the previous case, it can be shown that thet-shifted correlation between the filtered prediction error and the filtered processinput is minimized at t = dc, (5.22). For the sake of simplicity, in this part weonly consider positive gain mismatch. In addition, we assume that the process gainis positive. However, the technique proposed here can easily be extended to othertypes of mismatch. So, we have:d0 = d,limjw!0|P0( jw)| > limjw!0|P( jw)|,p0i > pi.51Comparing the impulse response coefficients of H and H 0 yields (H 0 H):h00h0 = 0h01h1 = 0...h0dc hdc = 0...h0dc+d hdc+d = p0 p00 > 0h0dc+d0+1hdc+d0+1 = p1 p01 > 0... (5.37)The above relations show that the infinite impulse response coefficients of H 0 arebigger than the ones associated with H. Therefore, we can easily show that fort d, Corre¯ 0t ,G1ut+t>Corre¯t ,G1utt. Thus, by comparing correlationbetween the prediction error and the process input at some lags around the timedelay when the performance of a closed-loop system deteriorates with the casewhen performance is high, we can say if MPM causes the problem. This correlationis not sensitive to disturbance characteristics. Accordingly, to distinguish the effectof positive gain mismatch on system performance from the effect of disturbance,the following procedure is suggested:1. Keep track of maximum correlation between the prediction error and theprocess input at some lags around the process time delay when performanceis good.2. When performance degrades, compare the maximum correlation for the sameinterval with the correlation value produced during high performance condi-tions.3. If the difference is greater than a user-specified constant (l ), conclude thatthe problem is because of MPM.l denotes the level of conservativeness that the algorithm has. For a process withnegative gain, maximum correlation must be replaced with minimum correlation in52the above-mentioned steps. The analysis mentioned in this section can be extendedto other types of MPM, e.g., delay mismatch.5.3 Simulation ResultsExistence of high frequency deterministic disturbances in closed-loop systems isone of the situations in which the indices show poor performance. In this case,because output variance increases and our benchmark doesn’t change, the perfor-mance indices drop. However, there is no mismatch between the model used incontroller design, so the process and the controller cannot do a better job.To show the efficiency of the proposed technique, a MPC control system isimplemented to control a SISO process with the following transfer function,P =0.0510.6q1q4. (5.38)The process output is perturbed with a filtered white noise, which can be consid-ered as an unmeasured disturbance. Fig. 5.1 compares the normalized correlationbetween the prediction error and the process input in 4 different situations:1. No MPM (USI= 0.92),2. No MPM with high frequency deterministic disturbance(Sine wave) (USI=0.34),3. 150% positive gain mismatch (USI= 0.22),4. 150% positive gain mismatch with high frequency deterministic disturbance(Sinewave) (USI= 0.45) .As mentioned above, USI is low when there is a high frequency disturbance inthe system. Fig. 5.1 depicts that the minimum correlation in all cases happens att = dc = 2. On the other hand, it can be seen that when there is MPM in the closed-loop system, the correlation is higher at lags around the time delay. Moreover, thisfigure shows that we can detect MPM in the presence of high frequency disturbancestoo. In fact, this correlation analysis is insensitive to disturbances. Accordingly,53when the performance index drops, by comparing correlation between the predic-tion error and the process input at lags around the time delay with correlation inhigh performance operating conditions, we can distinguish the effect of MPM fromthe effect of disturbances on system performance.−10 −5 0 5−1−0.8−0.6−0.4−τcorr(ut+τ,ε t)  No MismatchNo Mismatch + High Freq. Deterministic Dist.150% Gain Mismatch150% Gain Mismatch + High Freq. Deterministic Dist.Figure 5.1: Correlation of the filtered prediction error and the filtered processinput, dc = 2, d = 4.There is a situation in which we cannot use the suggested technique to detectMPM. High frequency deterministic disturbance causes the correlation betweenthe prediction error and the process input to oscillate with the same frequency asthe frequency of the disturbance. Thus, if process time delay is comparable withthe period of the deterministic disturbance, there is an overlap between a peakcaused by MPM and a peak caused by the periodic behaviour of the correlation.Therefore, the following relation must hold between the period of the deterministicdisturbance (T ) and the process time delay:d <T4. (5.39)54Accordingly, detectability of MPM increases for smaller time delay in comparisonwith the period of the deterministic disturbance.5.4 ConclusionsIn this chapter, we showed that by analysing the correlation between the predictionerror and the process input, one can say whether MPM or changes in disturbancecharacteristics causes the performance indices to drop. In this thesis, we have sofar used only MVC and user-specified benchmarking only to detect MPM. Now, toaddress the fact that MVC benchmarking is very sensitive to disturbance charac-teristics, we can use correlation analysis to decrease a number of false-positivesassertions of MPM based only on MVC conditions.55Chapter 6Moving-Horizon Predictive InputDesign for Closed-LoopIdentification6.1 IntroductionThe accuracy of an estimated model depends on the richness of the excitation sig-nals used to probe the system. The identifiability condition [40] quantifies thedegree of excitation required. Identification in industry often proceeds by addinga PRBS1 to the plant input. Of course, any perturbation of the process inputs alsoaffects the outputs, and hence the quality of the output products. Thus, identifica-tion experiments are not free: the need to extract sufficient information about theprocess at an acceptable cost has driven many researchers to study optimal inputdesign for closed loop identification. For instance, Wahlberg et al. [42] calculatea minimum power input sequence using which the estimated model satisfies thecontrol performance. See also, [37], [12], [4], [26], [39], [13], [38]. Typically,one tries to find an input with minimum variance that optimizes some function ofthe covariance matrix of estimated parameters and guarantees the identifiablity ofa related system. Jansson and Hjalmarsson [23], [24] propose a general framework1Pseudo random binary sequence56to deal with such problems. They also [25] suggest a method to solve the inputdesign problem by parameterization of the covariance matrix. Gopaluni et al. [16]use a particle filtering approach to solve the input design problem for nonlinearstochastic dynamic systems. Valenzuela et al. [41] use results from graph theoryto decrease the computational difficulties in optimal input design. Bombois et al.[5] specify a condition for costless identification which is not applicable in mostpractical cases. Yan et al. [47] show how a process with an MPC controller canbe sufficiently excited by changing the controller parameters instead of modify-ing the input signal directly. Most of the above mentioned input design methodsrely on knowledge of true systems. In addition, they are very complex and com-plicated to use in practice. Patwardhana and Gopaluni [36] suggest a completelydifferent technique. They show that for a process controlled by a MPC controller,the acsMPC objective function is equal to the trace of the Fisher information ma-trix, assuming the process is compatible with an ARIX model. Therefore, if weswitch the MPC optimization problem from minimizing the objective function tomaximizing the objective function, the process will be sufficiently excited for theidentification purposes. Due to the fact that we keep all MPC constraints, the pro-cess inputs and outputs never diverge. However, the main characteristics of thistechnique that may concerned users is that the controller does not work in a nor-mal operating mode. Technically, in this situation, the process does not operateunder closed-loop control and some concerns about the stability and other controlperformance criteria show up.The method proposed in this chapter needs only an initial approximation for theprocess model and an estimate of the maximum perturbation from the true modelto ensure robustness. This method is implemented in a moving horizon framework.At each sample time, by estimating the process model recursively, using a directmethod, the model used in the input design algorithm gets updated. Over time, itconverges to the true model. The approach is compatible with output constraintsalong with amplitude and frequency constraints on the inputs. Thus, it complieswith requirements expressing the safety and financial concerns of industrial prac-tice.In the rest of this chapter, we show that the trace of the Fisher informationmatrix is a convex quadratic function of the optimization parameters. For such57functions, constrained minimization is a well-understood problem, for which effi-cient computational solutions exists. Here, however, the goal is to maximize thetrace of the information matrix subject to given constraints. This is considerablymore challenging, especially when the vector of parameters to be chosen is high-dimensional. Special characteristics of typical control systems make it possible tosuggest a practical method that decreases the dimension of the problem by split-ting it into a sequence of lower-dimensional problems. The suggested techniqueis practical, easy to implement, and applicable to closed loop systems built aroundany type of controller.6.2 Parameter EstimationWe will consider a linear MIMO system in discrete time, described by the followingARX model:A(q,q)yt = B(q,q)ut + et (6.1)Here yt is a (column) vector of n output components, ut is a vector of m inputs, etis an n-component vector of white noise with invertible covariance matrix Se. Thesystem matrices A(q,q) and B(q,q) have dimensions n⇥n and n⇥m, and dependon the delay operator q1 as follows:A(q,q) = IA1q1 · · ·AnAqnA (6.2)B(q,q) = B1qd + · · ·+BnBqdnB (6.3)The positive integers nA and nB define the orders of the polynomial entries in ma-trices A and B, and d  1 is the minimum time delay in matrix B. The coefficientmatrices in (6.2)–(6.3) constitute the parameters of the system model, and can begathered into the block matrix q of shape n⇥ (nnA +mnB) defined below:q =hA1 . . . AnA B1 . . . BnBi. (6.4)58With these definitions, the output vector at time t satisfiesyt = qyt , (6.5)where yt is the (nnA +mnB)⇥1 vector defined byyt =hy0t1 . . . y0tnA u0td . . . u0tdnBi0. (6.6)It is assumed that the true system has the form described above for some (unknown)value q = q0. The one-step ahead predictor for the model (6.1) isyˆt = (IA(q,q))yt +B(q,q)ut (6.7)and the vector of prediction errors ise(t,q) = yt  yˆt . (6.8)The parameters can be estimated by minimizing the mean squared prediction er-ror [32]. Given N data values, this produces the estimateqˆN = argminq E⇥e(t,q)0e(t,q)⇤. (6.9)Well-known methods [32] lead to the explicit formulaqˆN = YY0(YY0)1 (6.10)whereY =hyt . . . ytN+1i, (6.11)Y=hyt . . . ytN+1i. (6.12)The above process can be implemented recursively (recursive least squares) to up-date parameter matrix qˆN at each sample time. The accuracy of the estimate de-pends on the covariance of the estimated parameters. According to the mean squareconsistency theorem [31], if the process is stable and the error et is normally dis-59tributed with mean 0 and cov(et) = s2, then the least square estimate is meansquare consistent iflimN!•s2tracecov(qN)1= 0 (6.13)where cov(q) = (YY0)1. Accordingly, smaller values for cov(q) correspond tobetter estimates. Meanwhile, equations (6.6), (6.12) show that the covariance of theestimated parameters depends on the inputs. Thus, the design of the input signalhas a direct effect on the quality of the resulting estimate.6.3 Input DesignAs noted in (6.13), the quality of an estimate depends on the covariance of the es-timated parameters. The covariance, in turn, depends on the input signal—recall(6.6), (6.12), (6.13). Thus, input design strategies for identification purposes areoften formulated to minimize some scalar associated with the size of the covari-ance matrix. For instance, one can minimize the trace, the largest eigenvalue, orthe determinant of the covariance matrix. In the large literature on optimal inputdesign, a typical goal is to find the minimum-power input sequence that guaranteesprocess identifiability [40]. This is a challenging and complex problem.In this work, we do not minimize the trace of the covariance matrix. Instead, wemaximize the trace of its inverse—i.e., the trace of the Fisher information matrixF = cov(qN)1 =YY0. (6.14)Clearly trace(F) is an unbounded function of the inputs, so suitable constraintsmust be introduced to produce a well-posed maximization problem.In detail, Ru = trace(F) is given byRu =nÂp=1nAÂj=1NÂi= j⇣y(p)ti⌘2+mÂp=1nBÂj=1NÂi= j⇣u(p)td+1i⌘2(6.15)where y(p)t and u(p)t denote the pth output and input in output and input vectors,respectively. We are assuming that the true process model has the form described60in Section 6.2. It follows that the process outputs for times t N through t  1satisfyYt = cUtd (6.16)where Yt = [y0t , . . . ,y0tN+1]0, Utd = [u0td , . . . ,u0tdN+1]0, and c is an Nn⇥Nmmatrix whose elements are generated by the true system parameters. Using (6.16)and assuming that the input before time t dN is zero, the Ru = trace(F) canbe written as follows:Ru =nAÂi=1Y 0tiYti +nBÂi=1U 0td+1iUtd+1i=nAÂi=1U 0tdc 0iciUtd +nBÂi=1U 0tdi 0i iiUtd . (6.17)Here ci and ii are block matrices of shapes Nn⇥Nm and Nm⇥Nmm, respectively,defined byci ="Oin⇥NmhI(Ni)n⇥(Ni)n O(Ni)n⇥ini⇥c#, (6.18)ii ="O(i1)m⇥NmO(Ni+1)m⇥(i1)m I(Ni+1)m⇥(Ni+1)m#. (6.19)Defining the symmetric, positive-semidefinite matrixG =nAÂi=1c 0ici +nBÂi=1i 0i ii (6.20)leads to a succinct presentation of (6.17):Ru =U 0tdGUtd . (6.21)This is a convex quadratic function of the vector variableUtd .Since Ru is a convex quadratic function, any critical point will be a globalminimizer. However, we seek a maximizer. Existence of a solution is guaranteedwhenever the set of competing inputs to be closed and bounded. Thus the mathe-61C(q1)H(q1)P(q1)IDutr(t)u?tu˜?t ytetuctFigure 6.1: The closed loop system during the identification experimentmatical formulation corresponds to the need to explicitly recognize the limitationson permissible inputs for the process under study. In closed loop identification, thedesigned input signal will be added to the control signal. To keep output variationssmall and to control the power of the input, we impose simple box constraints onthe inputs and outputs. These inequalities between vectors, to be interpreted com-ponentwise, express the limits of acceptable variation in the signals and the signalto noise ratio:yL  yt  yH , (6.22)uL  ut  uH . (6.23)In addition to the above constraints, the bandwidth of the excitation can be adjustedby filtering. As shown in Fig. 6.1, the controller’s output uct is augmented by u˜⇤t ,whereu˜⇤t = H(q1)u⇤t (6.24)for some filter H(q1) chosen to satisfy the frequency constraints. Here H(q1) isa low-pass filter with cut-off frequency located at fb. By analogy with (6.16), thefiltered input can be written asU˜t = JUt , (6.25)where Ut = [u0t , . . . ,u0tN+1]0 and J is an Nm⇥Nm matrix which is built based on62H(q1). Accordingly, the matrix G, first defined in (6.20), is modified as follows,G =nAÂi=1c 0ici +nBÂi=1J 0iJi, (6.26)whereJi ="O(i1)m⇥NmhI(Ni+1)m⇥(Ni+1)m O(Ni+1)m⇥(i1)mi⇥J#. (6.27)Further, the matrices ci in (6.26) have the same block structure as in (6.18), butnow the matrix c used to build them must be made from the best current estimateof the system parameters. (The true parameter values are unknown.)In summary, the input design problem is the following non-standard quadraticoptimization problem with convex constraints:maximizeutRusubject to yL  yt  yH ,uL  ut  uH . (6.28)A maximizer in this problem provides an input sequence that gives maximum in-formation about the process subject to the given input/output constraints. Thisinformation can be used to estimate the best possible model of the process underthese conditions.Fig. 6.1 shows the closed loop system during the modelling experiment. In thisblock diagram, P(q1) and C(q1) denote the process and the controller, respec-tively. Once the raw signal u⇤t is designed, it is filtered and added to the controlsignal uct . Their sum becomes the process input. Subsequently, by measuring theinput and output signals, the process can be identified using the direct method re-cursively. When the estimated parameters appear to settle on values we can use,we can stop exciting the process and use the estimated model for other purposes.63timekk+1k+2 k+Nk+N+1k+N+2max Rumax Rumax RuFigure 6.2: Illustration of moving horizon approach in input designC(q1)H(q1)P(q1)IDSignal Generatorutr(t)u˜?t ytetuctu?tFigure 6.3: The closed loop system during the identification experiment6.3.1 Moving Horizon FrameworkThe length of the excitation signal we design for identification purposes can becalled a prediction horizon. In the suggested input design technique, we build theFisher information matrix by predicting the future outputs. The output predictionover the prediction horizon is carried out based on the initial approximation for theprocess model. Thus, when the system is excited with the designed signal, there isno guarantee that the outputs do not go beyond constraints, due to the differencesbetween the true model and the model used in the input design formulation as wellas existence of output disturbance. To remedy the problem, we can implement the64algorithm in moving horizon fashion. Using this framework, instead of solvingthe input design problem once, it can be solved at each sample time. Then, thefirst element of the designed input sequence is sent through the process to providethe excitation and the remaining terms are discarded. In the next sample time, theprocess model is updated using the recursive estimation technique and the updatedmodel is used in the input design problem to generate a new excitation signal. Fig.6.2 depicts the moving horizon approach to input design over time. Fiq. 6.3 illus-trates the structure of the closed loop system during the closed loop identificationexperiment while the moving horizon framework is used.6.4 Time Delay EstimationThe time delay plays an important role in the input design formulation discussed inthe previous sections, where it was assumed to be known. Thus, it is important tohave an acceptable estimate of the time delay before starting to design an excitationsignal. A number of methods [10] [11] have been suggested for online estimationof the time delay d in (6.1)–(6.3). Elnaggar [10] proposes Fixed Model VariableRegressor Estimation (FMVRE) for this purpose in industrial processes. Lynch andDumont [7] demonstrate the effectiveness of FMVRE for finding the time delayin the performance monitoring of industrial processes. On the strength of thisproven success, ease of implementation, rapid convergence, and independence ofthe estimation approaches for other model parameters, we will use FMVRE toestimate the time delay in (6.1).When the delay d is known only to lie in an interval [dmin,dmax], an estimate dˆcan be obtained by solving the following minimization problem:dˆ = argmind0E⇥e(t,d0)0e(t,d0)⇤. (6.29)Here e(t,d0) is the prediction error from (6.8). Elnaggar [10] shows that for anauxiliary model of first order and a process with a positive gain, the minimization65in (6.29) is equivalent to the maximization ofE1(d0) = ruy(d0) ruy(d0 1)= E⇥(yt  yt1)0utd0⇤, (6.30)where ruy(d0) is the cross-correlation between input and output signals at lag d0 (Ifthe process gain is negative, the function to be minimized is E1). To solve thisestimation problem recursively, introduce the time-varying quantityE1(t,d0) = lE1(t1,d0)+⇥(yt  yt1)0utd0⇤,where l is a tuning parameter which affects the rate of convergence. Then, in-stead of the one-shot maximization of E1 in (6.30), generate a sequence of delayestimates as follows:dˆ(t) = argmaxd0E1(t,d0) |d0 2 [dmin,dmax] . (6.31)Elanggar [10] shows that to obtain a good estimate of the time delay, the input mustbe sufficiently rich.As discussed above, one of the advantages of FMVRE method for estimatingtime delay is the rapid convergence of the algorithm. Therefore, we can design anexcitation signal based on the current model of a process. Using the designed sig-nal, we can estimate the time delay of the process. Based on the updated time delay,the excitation signal can be redesigned and used to estimate the other parametersof the process.6.5 Simulation ResultsIn this section, we use the technique described above to identify machine direc-tional processes in a paper machine operating in closed loop. The system has 2inputs and 2 outputs, namely,u(1) stock flow, y(1)dry weight,u(2)dryer pressure, y(2) size press moisture.66As in many processes, each input/output relation can be captured by a first ordermodel with dead time. We postulate264y(1)ty(2)t375 =266664b11qd111a11q10b21qd211a21q1b22qd221a22q1377775264u(1)tu(2)t375 . (6.32)This equation can be rearranged to produce the deterministic elements of the stan-dard ARX model (6.1). The constraints on the designed excitation signals and theoutput signals are"125# yt "125#(6.33)"1010# ut "1010#(6.34)fb  0.159rads(6.35)where fb denotes the bandwidth of the designed signal. Now MHP is used to gen-erate the excitation signal at each sample time and the designed signal is addedto the control signal to provide the excitation needed for identification. Fig. 6.4illustrates a sequence of the calculated excitation signal over time.Figure 6.5 shows the outputs of the process under the closed loop condition.It can be seen that the outputs sometimes cross the upper and lower limits. Thishappens because we add the excitation signal to the control signal, and the feedbackcontroller acts to to reject disturbances. Thus, these two signals together cause anextra perturbation in the outputs. Indeed, we are trying to keep the outputs withinthe tight limits. Therefore, if we tighten the output constraints, we will make surethat the output values will stay inside the specified bounds.Once the optimal excitation signals are determined and added to the processinputs, we start using the measured inputs and outputs of the process to estimatethe model parameters recursively. See Fig. 6.1. In the first step, we estimate thetime delay from each input to each outputs using FMVRE. Figure 6.6 shows theresults. As discussed in section 6.4, the estimated values converge to the true values670 10 20 30 40 50 60 70 80 90 100−15−10−5051015Dryer Presure, u(2)Samples0 10 20 30 40 50 60 70 80 90 100−15−10−5051015Stock Flow, u(1)Samples  Actual signalUpper and lower boundsFigure 6.4: The designed excitation signal0 10 20 30 40 50 60 70 80 90 100160170180190200Dry Weight, y(1)Samples  0 10 20 30 40 50 60 70 80 90 100024681012Size Press Moisture, y(2)SamplesActual signalSet pointUpper and lower boundsFigure 6.5: The closed loop response of the system68very quickly. Since the estimated time delays are the same as the given model, thereis no need to update the time delay in the optimization problem equations. Given0 1 2 3 4 5 6 7 8 9 100123Samplesd 11  0 1 2 3 4 5 6 7 8 9 100123d 21Samples0 1 2 3 4 5 6 7 8 9 1001234d 22SamplesEstimated valueTrue ValueFigure 6.6: The estimated time delaysthe estimated time delays, we estimate the parameters of the model using recursiveleast squares. Fig. 6.7 shows how the estimated parameters converge to the truevalues.As mentioned earlier, the suggested approach to input design is independent ofthe type of controller used in the closed loop system. In addition, since the signal isadded to the control signal, it does not disturb the controller to reject disturbancesappearing in the outputs. Fig. 6.8 shows that if a big step disturbance happensduring the modelling experiment, although it causes the output to cross the upperlimits, the controller brings it back within the desired bounds. In this figure, a stepdisturbance of size 10 happens at sample time 18.6.6 ConclusionsIn this chapter a Moving Horizon Predictive technique was proposed for generatingsequences of input perturbations for the purpose of closed-loop process identifica-tion. The method maximizes the trace of the Fisher information matrix, a quadratic690 10 20 30 40 50−1−0.500.51a 11Samples 0 10 20 30 40 50−1−0.500.51b 11Samples  0 10 20 30 40 50−1−0.500.51Samplesa 210 10 20 30 40 50−1−0.500.51Samplesa 220 10 20 30 40 5000. 210 10 20 30 40 50−0.2− 22Estimated ValueTrue ValueFigure 6.7: The estimated parameters0 10 20 30 40 50 60 70 80 90 100160165170175180185190195200SamplesDry Weight  Actual signalSet pointUpper and lower limitsFigure 6.8: Dry weight signal perturbed with a step disturbance70function of the system inputs, subject to constraints on process inputs and outputsover the horizon of the planned experiment. It can be implemented in parallel withany type of controller, and allows for adjustment of the constraints to ensure ro-bustness to the unavoidable mismatch between the existing process model and thetrue plant characteristics.71Chapter 7Conclusions and Future Work7.1 ConclusionsIn this thesis, we used user-specified benchmarking to estimate the performance ofmachine directional (FFFB) control systems in paper machines to detect MPM. Wedefined a sensitivity measure that relates system performance to MPM, and used itto explore this sensitivity for three realistic types of parametric modelling errors.This analysis demonstrated the power of the indices to detect MPM. In addition,the effect of MPM on the closed loop behaviour was discussed.Once performance degrades and it is confirmed that it is because of MPM, theprocess model needs to be re-identified to compensate for the lost performance.This thesis presented a new approach to input design for closed loop identifica-tion. The idea is to maximize the trace of the Fisher information matrix associatedwith the plant model in a moving horizon framework, while enforcing explicitconstraints on both inputs and outputs. The result is the richest possible excitationsignal for which the operation of a running closed-loop system remains within ac-ceptable bounds. The function to be maximized is a convex quadratic. The methodcan be combined with a fixed model variable regressor technique to estimate timedelays.The suggested technique was implemented and used to monitor performanceof machine-directional processes in an industrial paper machine and to identify theprocess if degradation in controller performance due to model-plant mismatch is72detected.7.2 Future WorkThe suggested performance monitoring algorithm and the related analyses were ap-plied to machine directional control in paper machines. This work can be extendedand applied to cross directional processes too. Since it is a more complex systemsome modification and adjustment needs to be done to make this work applica-ble to cross directional processes. Moreover, it is worth investigating the effect ofparametric mismatch in a model of cross directional processes to have insight intohow different types of parametric MPM influence performance of cross directionalcontrol systems.Another possibility for future work is to find a method to solve the constrainedquadratic problem defined in chapter 6 in the proposed input design technique.This is a high dimensional and non-convex problem to solve. Indeed, to generateN input samples for a process whose input vector u has dimension m, the number ofscalar choice variables is Nm. So it is a very difficult problem to solve for excitingquadratic solvers that can handle non-convex function, e.g., fmincon function inMATLAB, and there is no guarantee to get a global optimum answer. The functionRu is convex, so its maximum over any compact convex domain will be achievedat an extreme point. The constraints in (6.28) determine a polyhedron in the spaceof possible inputs and the extreme points of this set are the vertices of the poly-hedron. It is easy to construct examples in which a given vertex provides a localmaximum but not a global one and hence situations in which standard optimizationsolvers fail to identify a true maximizer. The number of vertices in (6.28) growsexponentially with the dimension of the problem, so any strategy based on simplyvisiting each one is out of the question. Therefore, it helps a lot if one can suggesta solver which reduces the computational burden of solving the high-dimensionalnon-convex optimization problem and guarantees the global optimum answer.73Bibliography[1] K. Astro¨m. Introduction to stochastic control theory (mathematics in scienceand engineering vol 70)(new york: Academic). 1970. ! pages 1[2] A. Badwe, R. Gudi, R. 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