基于广义最小变差基准的多变量控制性能评估方法.pdf

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1、PROCESS SYSTEMS ENGINEERING Chinese Journal of Chemical Engineering,18(1)8694(2010)Multivariable Control Performance Assessment Based on Generalized Minimum Variance Benchmark*ZHAO Yu(赵宇),SU Hongye(苏宏业)*,CHU Jian(褚健)and GU Yong(古勇)State Key Laboratory of Industrial Control Technology,Institute of Cy

2、ber-Systems and Control,Zhejiang Univer-sity,Hangzhou 310027,China Abstract This paper is concerned with the control performance assessment based on the multivariable generalized minimum variance benchmark.An explicit expression for the feedback controller-invariant(the generalized mini-mum variance

3、)term of the multivariable control system is obtained,which is used as a standard benchmark for the assessment of the control performance for multi input multi output(MIMO)process.The proposed approach is based on the multivariable minimum variance benchmark.In comparison with the minimum variance b

4、enchmark,the developed method is more reasonable and practical for the control performance assessment of multivariable systems.The approach is illustrated by a simulation example and an industrial application.Keywords control performance assessment,generalized minimum variance,minimum variance contr

5、ol,multi-variable control system 1 INTRODUCTION In the modern large-scaled process industries,there are tremendous control loops in a plant.At the commission stage,all the loops run well.However,the performance of control loops will gradually deterio-rate,as influenced by many factors,such as the ch

6、ange of feedstock,malfunction of devices such as sensors and actuators,and some unmeasured disturbances.The excess variation throughout the process will re-duce the operability of equipment,increase operating costs,and lose the control for product quality.As re-ported,about 60%of the industrial cont

7、rollers have problems 1.Hence,prompt recognition and correc-tion of process-control malfunctions is needed for monitoring and evaluating the control loops.For solving the problems in the control loop monitoring and assessment,control performance as-sessment(CPA)has attracted growing interest both in

8、 academia and industries.The main objective of CPA is to provide an online automated procedure that delivers information to plant personnel for determining whether specified performance targets and response characteristics are being met by the controlled process variables and that evaluates the perf

9、ormance of the control system 2.Over the last decade,CPA has con-siderable achievements in industrial applications es-pecially with the univariate CPA.Many algorithms including commercial software are available.Several interesting reviews address related research achieve-ments in different stages 1-

10、5.After Harris first used the minimum variance controller(MVC)as the controller performance as-sessment benchmark 6,many researchers extended it to the feedforward/feedback(FF/FB)control loops 7-9.Ko and Edgar studied the cascade control loop performance assessment 10 and extended the MVC benchmark

11、to the constraint minimum variance control(CMVC)benchmark based on the model predictive control(MPC)algorithm 11.The MVC benchmark was further extended to the multivariable control sys-tems 12-15.In the performance assessment for mul-tivariable feedback systems,the so-called interactor matrix(or equ

12、ivalences)plays an important role.The interactor matrix is equal to the time delay in the uni-variate case,but its estimation needs more a priori in-formation of the system.For detailed information one can refer to 3,in which the performance assessment of time variant processes and non-minimum phase

13、 sys-tems based on the MVC benchmark is also considered.However,there are many stumbling blocks to use the MVC benchmark in practical applications.The main difficulty is that the minimum-variance control law often gives high gain,wide bandwidth and unre-alistically large control signal variations,re

14、ducing the value of the criterion 16.Consequently,more practi-cal benchmarks have been developed.Including the design specifications of the user(such as the rise time and settling time),Huang and Shah 3 has proposed the user defined benchmark.Similarly,in the design case benchmark,the value of objec

15、tive function is compared with the data generated by the designed model and the operating plant separately 17.The ap-propriate historical operation data are compared with the current normal closed-loop operation data and the relative performance index is determined,so the his-torical benchmark is de

16、veloped 18,19,in which the choice of the historical data series is critical.In con-sideration on the constraints of manipulated variables,the constrained minimum variance benchmark is brought out 11,which can be used to assess the Received 2009-03-23,accepted 2009-12-09.*Supported by the National Hi

17、gh Technology Research and Development Program of China(2008AA042902),the National Basic Research Program of China(2007CB714006)and the Graduate Creative Research Program of Zhejiang Province(YK2008024).*To whom correspondence should be addressed.E-mail: Chin.J.Chem.Eng.,Vol.18,No.1,February 2010 87

18、performance of MPC systems.In consideration of the control action penalization,three alternative advanced benchmarks have been pro-posed:linear quadratic Gaussian(LQG)benchmark 3,model predictive control(MPC)benchmark 20 and generalized minimum variance(GMV)control benchmark 16.By solving the LQG pr

19、oblem,which has the varying control action weighting,one can ob-tain a tradeoff curve.A linear controller can only op-erate in the region above the tradeoff curve,so the tradeoff curve can be used for the performance as-sessment purpose 3.The solution of LQG problem can also be obtained by solving a

20、 finite horizon gener-alized predictive control(GPC)problem 3,which is convenient to realize in the MPC toolbox of MAT-LAB.By defining the prediction horizon and the con-trol horizon as infinity,Julien et al.20 have proposed to use the optimal value of objective function as the benchmark,and showed

21、that“the MPC performance curve lies significantly above the LQG benchmark which illustrates the point that even a perfect MPC,designed in the absence of model-plant mismatch,will never fall on the LQG curve unless the actual distur-bance is a random walk”.The conclusion indicates that although the M

22、PC benchmark is similar with the LQG benchmark,the former is more practical.One important point in the MPC benchmark is that the tradeoff curve is easy to estimate.All the benchmarks mentioned above have the constant weighting,so that dynamic features of the loops may not be assessed.Grimble 16 prop

23、osed the generalized minimum variance(GMV)control bench-mark.He first defined a dynamic weighting objective function and then solved for its optimal solution to obtain an optimal controller.By substituting the opti-mal controller in the closed transfer function,the gen-eralized minimum variance(the

24、feedback invariant term)was obtained.With the GMV as benchmark,the performance assessment was realized.However,only the univariate case was considered 16.Later the work was extended to the multivariable case 21 based on the filtering and correlation(FCOR)algorithm 3.In this study,we extend the GMV t

25、o the multi-variable case with a novel and simple approach.The univariate GMV is extended to the multivariable GMV based on the multivariable MVC.The perform-ance assessment algorithm is based on the proposed benchmark.A simulation example and an industrial application are given to demonstrate the e

26、ssential fea-tures of the proposed method.2 THE MULTIVARIABLE MVC BENCHMARK A multivariable process can be modeled as tptdt=+YG UG a (1)where Gp and Gd are proper transfer function matrices for the plant and disturbance respectively,Yt is an output vector,and Ut the input vector.For stochastic syste

27、ms,at represents a white noise vector with zero mean and covariance matrix a.If Gp is a proper full rank transfer function matrix,a unitary interactor matrix D can be evaluated and pp=%DGG,where p%G is the delay-free transfer func-tion matrix of Gp 22,23.Eq.(1)can be expressed as 1tptdtptdt=+=+%YG U

28、G aD G UG a (2)Multiplying both sides of Eq.(2)by dqD,where q is the back shift operator and d is the maxi-mum delay order of interactor matrix D,we have tptdtdddqqq=+DYG UDG a (3)With ttdq=%YDY and dddq=GDG,Eq.(3)becomes tptdtdq=+%YG UG a (4)Since D is a unitary interactor matrix,the mini-mum varia

29、nce control law,which minimizes the ob-jective function of the interactor-filtered variable t%Y,()T1ttE=%JY Y,also minimizes the objective function of the original variable tY,()T2ttE=JY Y4,i.e.12=JJ,which means that()()TTttttEE=Y YY Y.It is further shown that if d%G is written as 4 11d011ddddqqqq+=

30、+%?GFFFRFR(5)the minimum variance can be calculated,()()()TtttminTT0011EtrCovtrdd=+%aaY YFaFFFF (6)If a time series transfer function Gcl is estimated from time series modeling of output data Yt only,by multiplying it with dqD and expanding cldqDG into the Markov parameter form,we have clddqq=+DGFL

31、Its first d terms summarized in F are the same as those expanded from d%G shown by Eq.(5).It is also known as control invariance property 12,15.Therefore,the so-called minimum variance term tFa can be esti-mated from routine operating data Yt.Harris et al.15 developed a parallel approach based on sp

32、ectral inter-actor and spectral factorization,which extracts the minimum variance term from the routine operating data.With the above conclusion,the objective function based performance measure(denoted as MIMO per-formance measure)can be calculated as()()()()TttmindTttTttmvminTttEminimum varianceact

33、ual varianceEEtrtrE=%?%YY YY YY YY Y (7)Chin.J.Chem.Eng.,Vol.18,No.1,February 2010 88 The performance indices of individual outputs are obtained from the diagonal elements of such a comparison:11mv,diagnyy=%Y where()diag=%YY.The individual output per-formance indices represent the performance of eac

34、h output with respect to the ideal output under multi-variable minimum variance control.Moreover,if an offset exists in the process output,the output variance,Y,should be replaced by the output mean square error in the above calculation for the performance in-dices 3.Minimum variance benchmark is a

35、popular benchmark for control performance assessment.It is the most frequently used benchmark owing to its theo-retical elegance and ability to provide the absolute lower bound of control variance.Many commercial performance assessment softwares use it as the pre-assessment strategy.However,a drawba

36、ck of this approach is resulted from the interactor matrix,which is conceptually difficult and computationally challeng-ing.In addition,the minimum variance control is an aggressive control and is rarely implemented in practice.Subsequently,many researchers developed alter-native approaches for the

37、performance assessment without the estimation of interactor matrix.Ko and Edgar 13 developed an algorithm by imbedding the calculation of the interactor matrix into the algebraic manipulation of the Markov parameters in the process model.Mcnabb and Qin 24 developed an alternative approach for minimu

38、m variance benchmark based on state space models estimated from subspace identifi-cation.Huang et al.25 estimated the bounds on minimum variance with the order of magnitude of the interactor matrix.These simpler methods require less a priori process knowledge but provide less diagnostic information.

39、There is a tradeoff between simplicity and diagnostic ability of the CPA methods.3 DEVELOPMENT OF THE MULTIVARIABLE GMV BENCHMARK Grimble 16 has pointed out that the results of univariate GMV benchmark may be easily extended to the multivariable case and the data analysis tools fol-low the procedure

40、s in Ref.12.However,there exist some obstacles:(1)The structure of the GMV objective function is similar with that of LQG,where control action pe-nalizetion is considered,while the MVC benchmark only considers the output error term.Thus the applica-tion of procedures in Ref.12 for the development of

41、 multivariable GMV is problematic.(2)The GMV uses the dynamic weighting term,while the MVC benchmark utilizes the constant weighting matrix.Such differences also make it diffi-cult to extend GMV to multivariable systems.Later Majecki and Grimble 21 developed the multivariable GMV benchmark based on

42、the FCOR algorithm in Ref.12.They formulated the general-ized open-loop and generalized disturbance transfer function matrix,which involves the dynamic matrices Pc and Fc.It needs to estimate the interactor matrix of the generalized open-loop plant.Then following the steps of the FCOR algorithm,the

43、feedback invariant term can be obtained and are used as the GMV benchmark.However,Pc and Fc should be selected carefully to ensure that the generalized plant is stable.Other approaches were used to estimate the mul-tivariable generalized minimum variance controller.Akira et al.26 proposed a new GMV

44、with new de-sign parameters by using coprime factorization ap-proach for MIMO case,in which only one Dio-phantine equation is to be solved by conventional method.Grimble 27 considered the design and im-plementation of GMV control laws for nonlinear mul-tivariable systems involving severe nonlinearit

45、ies,in which the time-delay matrix was assumed in the di-agonal form.It is not the general case,and is a special case of the general unity interactor matrix in Ref.3.Here we propose a method to avoid the above prob-lems and obtain the multivariable GMV benchmark based on the multivariable MVC in Ref

46、.3.Consider a general multivariable control system as shown in Fig.1,which can be modeled by Eq.(1).Figure 1 A simplified process control loop diagram For simplicity,we assume that the set-points r equal to zero.Substituting the controller C0 into Ut,t0t=UC Y (8)we obtain the closed loop transfer fu

47、nction,()1tp0dtclt1=+?YG CG aG a (9)where Gcl is obtained by the identification of closed-loop routine operation data set 3.By multiplying the interactor matrix filter dqD,the feedback invariant part based on the minimum variance benchmark can be estimated,clddqq=+DGFL (10)However,GMV benchmark must

48、 be considered for the multivariable control systems.The objective function is Chin.J.Chem.Eng.,Vol.18,No.1,February 2010 89 2c tctEm=+P eF U (11)where Pc is the dynamic cost-function weighting matrix represented by linear polynomial matrices,Fc stands for the control signal costing matrix,et and Ut

49、 are error vector and control vector,respectively.Typically Pc is low-pass and Fc is a high-pass transfer matrix 16.Substituting Eq.(8)and tt=eY (vector r equals to zero)into m,we have()()()22c tctcc0tT2tttEEEEm=+=+=?P eF UPF CYWYWYWY(12)The minimum variance of m is obtained from the following theor

50、em.Theorem If D is a unitary interactor matrix,a proper optimal control law that minimizes the linear quadratic(LQ)objective function of the interactor-filtered weighted output t%WY()()TttEm=%WYWY (13)also minimizes the LQ objective function of the original weighted output tWY,()()TttEm=WYWY i.e.mm=