minimum variance control and performance assessment of time-variant processes

Minimum variance control and performance assessment oftime-variant processes

Biao Huang*

Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G6

Received 21 October 2000; received in revised form 20 December 2000; accepted 4 April 2001

Abstract

This paper is concerned with (1) an explicit solution of a minimum variance control law for linear time-variant (LTV) processes

in the transfer function form, and (2) performance assessment of LTV processes using minimum variance control as the benchmark.It is shown that there exists a time-variant, absolute lower bound of process variance that is achievable under LTV minimum var-iance control and can be estimated from routine operating data. This lower bound can subsequently be used to assess the benefit ofimplementing LTV control such as adaptive control. The proposed methods are illustrated through simulated examples and an

industrial case study. # 2002 Elsevier Science Ltd. All rights reserved.

Keywords:Minimum variance control; Linear time-variant system; Performance monitoring; Non-stationary time series

1. Introduction

In this paper, we consider (1) an explicit solution ofthe minimum variance control law for linear time-vari-ant (LTV) processes in the transfer function form, and(2) performance assessment of LTV processes usingminimum variance control as the benchmark. Develop-ment of linear time-invariant (LTI) minimum variancecontrol can be attributed to Astrom [1]. The theory hasreceived considerable attention in the field of predictivecontrol [6,8,27,29,30], adaptive control [10–12,22], andself-tuning control [2,3,5,9,20]. Most recent interests inminimum variance control are in the emerging researcharea of control loop performance assessment [7,13,14,16,17,19,21,26,32,33] that uses minimum variance con-trol as a benchmark to evaluate control loop perfor-mance.Despite this development, design of minimum variance

control for LTV plants is very rare [23], and is onlyavailable in some very restrictive cases, namely one-step-ahead minimum variance control for LTV plants withcolored process noise, [4,23]. Li and Evans [23] recentlydeveloped a d-step-ahead minimum variance control

algorithm for LTV plants with colored process noise. Aset of pseudocommutation equations have to be solvedand the solutions are non-trivial. The method is alsorestricted to LTV autoregressive moving-average(ARMAX) models. Although the general problem ofLTV control may also be solved under the state spaceframework, the transfer function approach to minimumvariance control is of considerable interest particularlyin the studies of adaptive control [24] and control loopperformance assessment [13,19].The most distinguished feature of the minimum var-

iance control law in the transfer function form, perhaps,is its simplicity. It admits a closed-form solution withoutinvolving a numerical solution or solving the Ricattiequation. On the other hand, minimum variance controlis also, theoretically, the best possible control achievable.Its control error is simply a finite-order moving averagetime series. This fact has been used for diagnosis todetermine performance of control loops [1,13,19].Harris [13] has found that a ‘‘feedback control invari-

ant’’ or controller independent term of an LTI SISO pro-cess can be estimated from routine operating data, andthis term represents the process output under minimumvariance control. Desborough and Harris [7] furtherdeveloped this idea and suggested using autoregressivetime series modeling to fit the process output and con-sequently calculate the minimum variance term from

0959-1524/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

PI I : S0959-1524(01 )00026-9

Journal of Process Control 12 (2002) 707–719

www.elsevier.com/locate/jprocont

* Corresponding author. Tel.: +1-780-492-9016; fax: +1-780-492-

2881.

E-mail address: [email protected] (B. Huang).

linear regression. Since then, a number of extensionshave appeared in the literature. Readers are referred tothe review in Qin [28], Harris et al. [15] and Huang andShah [18] for details.However, just like studies on LTV minimum variance

control, almost all studies on control-loop performanceassessment have so far been limited to time series, thatare stationary or can be made stationary by some simpletransformation (e.g. differencing), or to time series thatcan be represented as a time-invariant linear function ofpresent and past values of an independent ‘‘white noise’’process. Non-stationary time series or time-variantdynamics are often observed in performance assessmentof control loops due to varying process dynamics, abruptchanges of disturbances, non-linearity of actuators andsensors, operator intervention, etc. The most intuitiveextension of performance assessment techniques fromLTI process to LTV process is thorough the recursiveestimation technique. Several recursive algorithms havebeen proposed to estimate control loop performance inthe presence of non-stationary characteristics in the data[7,18]. As one of the main objectives of this paper we willjustify these recursive approaches from a theoretical pointof view, and at the same time point out a potential diffi-culty in the handling of LTV operators and propose asolution strategy. The main contributions of this papertherefore include: (1) development of an algorithm fordesigning the LTV minimum variance control law in thetransfer function form; (2) theoretical rationalization ofthe existence of a time-variant lower bound of processvariance that is feedback control invariant and isachievable under LTV minimum variance control; (3)development of a methodology for performance assess-ment of LTV processes. Compared to the results of Liand Evans [23], the LTV minimum variance control lawdeveloped in this paper is directly applicable to the Box–Jenkins model structure which is more general than theARMAX model structure and the calculation of thecontrol law does not require solving a set of nontrivialpseudocommutation equations.The remainder of this paper is organized as follows.

In Section 2, an explicit solution of the transfer functionform of the LTV minimum variance control law isdeveloped. A methodology for performance assessmentof LTV processes using recursive algorithms is devel-oped in Section 3. Simulation examples are give inSection 4 and an industrial case study is discussed inSection 5, followed by concluding remarks in Section 6.To facilitate reading of this paper, most mathematicalproofs and derivations are presented in the Appen-dices.

2. Minimum variance control of time-variant processes

Consider the LTV SISO process shown in Fig. 1:

yt ¼ q�dT~ q�1; t� �

ut þN q�1; t� �

at ð1Þ

where d is the time-delay and is considered as a constant(the case where d is not a constant is considered inRemark (1)); T~ q�1; t

� �¼ A�1 q�1; t

� �B q�1; t� �

is thedelay-free LTV plant transfer function; N q�1; t

� �¼

D�1 q�1; t� �

C q�1; t� �

is the LTV disturbance transferfunction; where A q�1; t

� �; B q�1; t

� �; C q�1; t

� �and

D q�1; t� �

are polynomials in the backshift operator q�1,and D(q�1, t) and A(q�1, t) may or may not have com-mon factors; at is a white noise sequence with a zeromean and time-variant variance �2a tð Þ. Eq. (1) is alsoknown as the (time-varying) Box–Jenkins model [25].In this section, we derive a linear time-variant feed-

back control law ut=�Q(q�1, t)yt such that the follow-ing control objective function is minimized:

Jt ¼ E yt½ �2

We then show that the time-variant minimum var-iance or the lower bound of the objective functionmin[Jt]=minE[yt]

2 can be estimated from routine oper-ating data even though the process is not under LTVminimum variance control.Before proceeding, we need to introduce a basic prop-

erty of the backshift operator q�1 when it is multiplied byan LTV transfer function. This property will be usedthroughout this paper. That is, for any LTV transferfunction G(q�1,t), we have the following identities:

q�dG q�1; t� �

¼ G q�1; t� d� �

q�d

qdG q�1; t� �

¼ G q�1; tþ d� �

qd

The difficulty in manipulation of LTV transfer func-tions is the noncommutativity of the multiplication [24].For example, let A(q�1, t) and B(q�1, t) be two LTVpolynomials in the backshift operator:

A q�1; t� �

¼ a0 tð Þ þ a1 tð Þq�1 þ � � � þ an tð Þq

�n

B q�1; t� �

¼ b0 tð Þ þ b1 tð Þq�1 þ � � � þ bm tð Þq

�m

Fig. 1. Schematic of time-variant process.

708 B. Huang / Journal of Process Control 12 (2002) 707–719

The multiplication ofA(q�1, t) and B(q�1, t) is given by

A q�1; t� �

B q�1; t� �

¼Xni¼0

Xmj¼0

ai tð Þq�ibj tð Þq

�i

¼Xni¼0

Xmj¼0

ai tð Þbj t� ið Þq� iþjð Þ

while the multiplication of B(q�1, t) and A(q�1, t) isgiven by

B q�1; t� �

A q�1; t� �

¼Xmj¼0

Xni¼0

bj tð Þq�iai tð Þq

�i

¼Xni¼0

Xmj¼0

bj tð Þai t� jð Þq� iþjð Þ

Therefore, A(q�1, t)B(q�1, t)6¼B(q�1, t)A(q�1, t). Thisresult implies that the LTV minimum variance controllaw is not a trivial extension from the LTI minimumvariance control law.The following result presents the design of LTV

minimum variance control and performance assessmentof LTV processes using the minimum variance controlas the benchmark.

Result 1. For an LTV process shown in Eq. (1), thesolution of the LTV minimum variance control law andthe performance assessment problem is given by thefollowing steps:

(i) the LTV minimum variance control law is given by

Q q�1; t� �

¼ T~ �1 q�1; t� �

R q�1; tþ d� �

F�1 q�1; t� �

ð2Þ

where Fðq�1; tÞ and Rðq�1; tÞ are solved from a Dio-phantine identity:

N q�1; t� �

¼ f0 tð Þ þ f1 tð Þq�1 þ � � � þ fd�1 tð Þq

�dþ1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}F q�1;tð Þ

þ R q�1; t� �

q�d ð3Þ

(ii) where fiðtÞ ðfori ¼ 1; � � � ; d� 1Þ are time-variantimpulse response coefficients of the disturbancetransfer function, and Rðq�1; tÞ is the remainingrational, proper transfer function.The closed-loop response under minimum variancecontrol is given by

yt mv ¼ f0 tð Þ þ f1 tð Þq�1 þ � � � þ fd�1 tð Þq

�dþ1

at�� ð4Þ

(iii) If one models the closed-loop response, that isnot necessarily under minimum variance control, by amoving average model, then the model should havethe following form

yt ¼ f0 tð Þat þ f1 tð Þat�1 þ f2 tð Þat�2 þ � � � þ fd�1 tð Þat�dþ1

þ fd tð Þat�d þ � � �

i.e. the first d terms of this moving average modelconstitute the process output under LTV minimumvariance control as shown in Eq. (4).

The proof of this result is shown in the Appendices.

Remark 1. The minimum variance control law isdesigned according to the fact that time delay is one ofthe fundamental limitations in feedback control. Thestructure of the Diophantine equation [Eq. (3)] for thedesign of the minimum variance control law depends onthe time delay, which is assumed to be constant so far. Itis clear that changing of time delay will change the num-ber of terms in Fðq�1; tÞ, i.e. the polynomial Fðq�1; tÞvaries not only in its parameters but also in the numberof terms in the polynomial. The same argument appliesto the term Rðq�1; tÞq�d. Therefore, by assuming a timevarying d and then following the same derivation proce-dure as in Appendix A, the minimum variance controllaw can be derived as

Q q�1; t� �

¼ T~�1 q�1; t� �

R q�1; tþ d� �

F�1 q�1; t� �

which has the same form as Eq. (2). However, the twoterms in the control law, Rðq�1; tþ dÞ and F�1ðq�1; tÞ,vary not only in their parameters but also in theirstructures which can be determined from the Dio-phantine equation. The closed-loop response underLTV minimum variance control is given by

yt mv ¼ f0 tð Þ þ f1 tð Þq�1 þ � � � þ fd�1 tð Þq

�dþ1

at��

which has the same form as Eq. (4) but d is time vary-ing, i.e. the number of terms in this equation is timevarying.For performance assessment, one typically assumes

time delay is known a priori [7,13,18]. The rationale is thattime delay constitutes a fundamental limitation on theachievable control performance and must be known inorder to assess this fundamental performance limitation.Knowing time delay is much less demanding than know-ing a complete knowledge of the process. However, if thetime delay or its variation is unknown and if there is anexternal excitation to the control loop, then closed-loopidentification (recursive closed-loop identification for thetime varying system) [31] can be applied to on-line esti-mate the time delay. If there is no external excitationavailable, then the extended horizon performanceassessment (or a plot of the minimum variance termversus a range of time delay) can also be useful forassessment of controller performance by checking thetrajectory of the plot [15,18,33].

B. Huang / Journal of Process Control 12 (2002) 707–719 709

Remark 2. A minimum variance control law for theLTV ARMAX process has been discussed in Li andEvans [23]. Compared to Li and Evans [23], the LTVminimum variance control law shown in Eq. (2) is moregeneral and simpler in the following senses:The model used for the derivation of Eq. (2) is a gen-

eral Box–Jenkins model

yt ¼ q�dT~ q�1; t� �

ut þN q�1; t� �

at

while Li and Evans [23] uses an ARMAX model

A q�1; t� �

yt ¼ B q�1; t� �

ut þ C q�1; t� �

at

In order to solve the LTV minimum variance controlproblem, a set of pseudocommutation equations have tobe solved first in Li and Evans [23], which is non-trivial.In this paper, the solution of the LTV minimum var-iance control law does not require the pseudocommu-tation equations, and thus significantly simplifies thedesign procedure.In the Appendices, we will show that the LTV mini-

mum variance control law derived in this paper isequivalent to that derived by Li and Evans [23] if theBox–Jenkins model is reduced to an ARMAX model.

3. Performance assessment of LTV processes

One of the main objectives of this paper is to justifycontrol loop performance assessment for LTV processesusing LTV minimum variance control as the benchmarkthrough the recursive algorithm, and in the same time topoint out potential problem in the calculation of theminimum variance term and the solution strategy.As shown in Result 1, a stable LTV closed-loop

response can be written as an infinite-order moving-average (MA) process:

yt ¼ f0 tð Þat þ f1 tð Þat�1 þ f2 tð Þat�2 þ � � �

þ fd�1 tð Þat�dþ1 þ fd tð Þat�d þ � � � ð5Þ

The LTV minimum variance can be calculated fromthe first d-terms of the moving average model (5), i.e.

�2mv tð Þ ¼ f 20 tð Þ�2a tð Þ þ f

21 tð Þ�

2a t� 1ð Þ þ � � �

þ f 2d�1 tð Þ�2a t� dþ 1ð Þ ð6Þ

By comparing this minimum variance term with theactual process variance, one can determine how well thecurrent controller is operating. For LTV processes, thiscomparison indicates whether an LTV or adaptive con-trol should be used. To calculate the minimum varianceterm in practice, an LTV ARMA model may be esti-

mated first and then transferred into the moving aver-age form, Eq. (5). The following result shows that anystable closed-loop response of an (open-loop) LTV Box-Jenkins process under feedback control can be expres-sed by an LTV ARMA model.

Result 2. For the LTV process shown in Eq. (1), thestable closed-loop response under LTV feedback con-trol ut ¼ �Qðq�1; tÞyt ¼ �U�1ðq�1; tÞVðq�1; tÞyt can berepresented by an LTV ARMA time series model,where Uðq�1; tÞ and Vðq�1; tÞ are polynomials in thebackshift operator q�1.

Proof. Substituting ut=�Q(q�1, t)yt=�U�1(q�1, t)V(q�1, t)yt into Eq. (1) yields

yt ¼ 1þ q�dT~ q�1; t� �

Q q�1; t� �h i�1

N q�1; t� �

at

¼ 1þ q�dA�1 q�1; t� �

B q�1; t� �

U�1 q�1; t� �

V q�1; t� � �1

D�1 q�1; t� �

C q�1; t� �

at

¼ 1þ A�1 q�1; t� d� �

B q�1; t� d� �

U�1 q�1; t� d� �

V q�1; t� d� �

q�d�1D�1 q�1; t

� �C q�1; t� �

at ð7Þ

where A(q�1, t), B(q�1, t), C(q�1, t) and D(q�1, t) arepolynomials in the backshift operator q�1 as defined inEq. (1).It follows from Li and Evans [23] that there exists a

pseudocommutation equation such that

B q�1; t� d� �

U�1 q�1; t� d� �

¼ U~ �1 q�1; t� d� �

B~ q�1; t� d� �

ð8Þ

Substituting Eq. (8) into Eq. (7) yields

yt ¼ 1þ A�1 q�1; t� d� �

U~ �1 q�1; t� d� �

B~ q�1; t� d� �h

V q�1; t� d� �

q�d�1D�1 q�1; t

� �C q�1; t� �

at ð9Þ

Denote E�1(q�1, t�d)=A�1(q�1, t�d)U~ �1(q�1, t�d),and then Eq. (9) can be written as

D q�1; t� �

1þ E�1 q�1; t� d� �

B~ q�1; t� d� �h

V q�1; t� d� �

q�dyt ¼ C q�1; t

� �at

ð10Þ

Similarly the following pseudocommutation equationexists

D q�1; t� �

E�1 q�1; t� d� �

¼ E~�1 q�1; t� d� �

D~ q�1; t� �

ð11ÞSubstituting Eq. (11) into Eq. (10) yields

D q�1; t� �

þ E~�1 q�1; t� d� �

D~ q�1; t� �

B~ q�1; t� d� �h

V q�1; t� d� �

q�dyt ¼ C q�1; t

� �at ð12Þ


Multiplying Eq. (12) by E~ (q�1, t�d) yields

E~ q�1; t� d� �

D q�1; t� �

þD~ q�1; t� �

B~ q�1; t� d� �h

V q�1; t� d� �

q�dyt ¼ E~ q�1; t� d

� �C q�1; t� �

at

Denote this equation as

Acl q�1; t

� �yt ¼ Ccl q

�1; t� �

at ð13Þ

which has the form of the LTV ARMA model, whereAcl(q

�1, t) and Ccl(q�1, t) are polynomials in the back-

shift operator q�1.

Remark 3. Although in the proof of Result 2, we usepseudocommutation equations to derive the ARMA formof the closed-loop response, we do not use the pseudo-commutation equations to solve for Eq. (13) in practice.Instead, we directly estimate the coefficients of Eq. (13)from time series analysis of routine operating data. &

Therefore, any recursive time-series analysis algo-rithm can be used to estimate the LTV ARMA modelfor performance assessment of LTV processes [7,18].However, to calculate the minimum variance, care hasto be taken to the noncommutativity problem when wetransfer the LTV ARMA model to the LTV movingaverage (MA) model, which is a necessary step for thecalculation of the minimum variance term. Simple longdivision may not yield a correct solution if the plantmodel or the disturbance model varies sufficiently fast.This fact has so far been ignored in the literature. Forexample, for an LTV ARMA model

A q�1; t� �

yt ¼ C q�1; t� �

at

the LTV impulse response coefficients have to be solvedvia the following equation:

C q�1; t� �

¼ A q�1; t� �

f0 tð Þ þ f1 tð Þq�1 þ f2 tð Þq

�2 þ � � �

ð14Þ

Solving Eq. (14) by equating coefficients of the equa-tion is not obvious due to the noncommutativity pro-blem. Consider a simple LTV AR model:

1� l tð Þq�1

yt ¼ at ð15Þ

According to Eq. (14), the impulse response coeffi-cients can be solved via

1 ¼ 1� l tð Þq�1


�2 þ � � �

ð16Þ

This yields

1 ¼ f0 tð Þ þ f1 tð Þ � l tð Þf0 t� 1ð Þ½ �q�1

þ f2 tð Þ � l tð Þf1 t� 1ð Þ½ �q�2

þ f3 tð Þ � l tð Þf2 t� 1ð Þ½ �q�3 þ � � � ð17Þ

Equating coefficients of the both hand sides of Eq. (17)yields

f0 tð Þ ¼ 1f1 tð Þ ¼ l tð Þf2 tð Þ ¼ l tð Þl t� 1ð Þ

f3 tð Þ ¼ l tð Þl t� 1ð Þl t� 2ð Þ

..

.

8>>>>><>>>>>:That is

yt ¼ 1þ l tð Þq�1 þ l tð Þl t� 1ð Þq�2 þ l tð Þl t� 1ð Þl t� 2ð Þ

q�3 þ � � �at

However, direct long division of Eq. (15) yields

yt ¼ 1þ l tð Þq�1 þ l2 tð Þq�2 þ l3 tð Þq�3 þ � � �

at ð18Þ

which is not correct. In Li and Evans [23], the multi-plications as used in Eqs. (16) and (17) are called thenormal multiplication of LTV transfer functions; Themultiplication (or division) that is used to derive Eq.(18) is called pointwise multiplication (or division). Thenormal multiplication is non-commutative, while thepointwise multiplication (or division) is commutativebut not applicable to LTV transfer functions. Therefore,unless we can directly model the LTV time series by themoving average model, any attempt to transfer othermodels to the moving average model needs to take thenoncommutativity into account if the parameters varysufficiently fast. In any case, it would be recommendedto use normal multiplication/division rather than thepointwise multiplication/division when one deals withtime varying processes.

4. Simulations

Simulation 1. Consider a process transfer function givenby:

q�dT~ q�1� �

¼ q�4 1� 0:67q�1� ��1

Kp tð Þ ð19Þ

and the time-variant disturbance transfer function givenby

N q�1� �

¼ 1� l tð Þq�1� ��1

ð20Þ


In this simulation, the minimum variance control lawwill be calculated and closed-loop responses are simu-lated to demonstrate applicability of the proposed LTVminimum variance control law. The purpose of choos-ing this example is to test the proposed minimum vari-ance control law for several different scenarios, fromextremely rapid parameter change to relatively slowparameter change. The performance of traditionaladaptive control is also compared with the new LTVminimum variance control through this simulation.The Diophantine identity can be solved according to

Eq. (3) as:

1� l tð Þq�1� ��1

¼ f0 tð Þ þ f1 tð Þq�1 þ f2 tð Þq

�2 þ f3 tð Þq�3

þ 1� l tð Þq�1� ��1

g0q�4

or equivalently

1 ¼ 1� l tð Þq�1� �


�2 þ f3 tð Þq�3

þ g0q

�4 ð21Þ

Equating coefficients of both hand sides of Eq. (21)yields

f0 tð Þ ¼ 1f1 tð Þ ¼ l tð Þf2 tð Þ ¼ l tð Þl t� 1ð Þ

f3 tð Þ ¼ l tð Þl t� 1ð Þl t� 2ð Þ

g0 tð Þ ¼ l tð Þl t� 1ð Þl t� 2ð Þl t� 3ð Þ

8>>>><>>>>:That is

F q�1; t� �

¼ 1þ l tð Þq�1 þ l tð Þl t� 1ð Þq�2

þ l tð Þl t� 1ð Þl t� 2ð Þq�3 ð22Þ

R q�1; t� �

¼ 1� l tð Þq�1� ��1

l tð Þl t� 1ð Þl t� 2ð Þl t� 3ð Þ

ð23Þ

The LTV minimum variance control law can be cal-culated from Eq. (2):

Q q�1; t� �

¼ K�1p tð Þ 1� 0:67q�1

� �1� l tþ 4ð Þq�1� ��1

l tþ 4ð Þl tþ 3ð Þl tþ 2ð Þl tþ 1ð Þ 1þ l tð Þq�1 þ l tð Þ

l t� 1ð Þq�2 þ l tð Þl t� 1ð Þl t� 2ð Þq�3�1

ð24Þ

The time-variant minimum variance term can theoreti-cally be calculated from Eq. (4) as

ytjmv¼ 1þ l tð Þq�1 þ l tð Þl t� 1ð Þq�2 þ l tð Þl t� 1ð Þ

l t� 2ð Þq�3at

and the LTV minimum variance is given by

�2mv tð Þ ¼ 1þ l tð Þ2þ l tð Þl t� 1ð Þð Þ2

�þ l tð Þl t� 1ð Þl t� 2ð Þð Þ

2��2a

ð25Þ

In the following, we shall simulate closed-loopresponses of LTV minimum variance control for differ-ent sets of parameters.

Case 1. Consider l(t)=(�1)t and Kp(t)=0.33(�1)t

where t is integer. In this system the pole of the dis-turbance transfer function has a rapid change from 1 to�1 and vice versa, and the gain of the plant has a rapidchange from 0.33 to �0.33 and vice versa. The dis-turbance trajectory without control is shown in Fig. 2.The process output under LTV minimum variance con-trol is shown on the top graph of Fig. 3, and the con-troller output is shown on the bottom graph of Fig. 3.Clearly, the LTV minimum variance controller is able toregulate the disturbances even though the processdynamics may change arbitrarily fast.

Case 2. Consider the case when both plant and dis-turbance dynamics gradually change with time (in com-parison to the abrupt changes as discussed in theprevious case). This case would be naturally related toadaptive control. We shall compare the LTV minimum

Fig. 2. The trajectory of the disturbance with LTV dynamics.

Fig. 3. The trajectory of the output under LTV minimum variance

control.


variance control with the traditional adaptive minimumvariance control.Assume that the models can be exactly estimated, i.e. we

consider control performance of traditional adaptive con-trol without model–plant mismatch. This represents thebest possible performance of traditional adaptive mini-mum variance control. The traditional adaptive minimumvariance control, as discussed in Li and Evans [23], doesnot take into account the noncommutativity of the LTVtransfer functions. For this example, the traditionaladaptive minimum variance control law would be

Q q�1; t� �¼

1� 0:67q�1� �

l4 tð Þ

Kp tð Þ 1� l tð Þq�1ð Þ 1þ l tð Þq�1 þ l2 tð Þq�2 þ l3 tð Þq�3

where all multiplications (or divisions) are pointwise[23]. Pointwise multiplication (or division) does notcause any time delay in the coefficients of the poly-nomials and is commutative.Let us first consider that the model parameters change

continuously and relatively fast:

Kp ¼ 0:33þ 0:3sin 250tð Þ

l ¼ sin 250tð Þ

The simulation results are shown in Fig. 4. Theclosed-loop response under traditional adaptive mini-mum variance control clearly yields much poorer per-formance than the LTV minimum variance control. Thedifference of the performance is reduced when theparameter change slows down. Consider

Kp ¼ 0:33þ 0:3sin 100tð Þ

l ¼ sin 100tð Þ

Then the two minimum variance controllers yieldsimilar responses as shown in Fig. 5.

Simulation 2. Now consider the performance assessmentproblem. The process model is given by

yt ¼ q�40:33

1� 0:67q�1ut þ 1� l tð Þq�1

� ��11� � tð Þq�1� �

at

We consider that disturbance has three different timevariant dynamics:

Case 1.

� tð Þ ¼ 0:4� 0:4cos t=100ð Þ ð26Þ

l tð Þ ¼ 0:67þ 0:33cos t=100ð Þ ð27Þ

Case 2.

� tð Þ ¼ 0:4� 0:4cos t=10ð Þ ð28Þ

l tð Þ ¼ 0:67þ 0:33cos t=10ð Þ ð29Þ

Case 3.

� tð Þ ¼ 0:4� 0:4cos tð Þ ð30Þ

l tð Þ ¼ 0:67þ 0:33cos tð Þ ð31Þ

A Dahlin controller is used to control this process,which is given by

Q q�1� �

¼0:7� 0:47q�1

0:33� 0:10q�1 � 0:23q�4ð32Þ

In this simulation, we shall compare the differencebetween the normal multiplication and pointwise multi-plication for the calculation of the minimum varianceterm. This is an example where the disturbancedynamics varies with time in a relatively smooth way.The purpose of choosing this example is to compare theproposed performance assessment algorithm with thetraditional one for several different scenarios, i.e. fromrelatively slow parameter change to relatively fast para-meter change.The LTV minimum variance term can theoretically be

calculated via the Diophantine equation:

1� l tð Þq�1� ��1

1� � tð Þq�1� �

¼ 1þ f1 tð Þq�1 þ f2 tð Þq

�2

þ f3 tð Þq�3 þ 1� l tð Þq�1

� ��1r0 tð Þq

�4

or

Fig. 4. The trajectory of the outputs under traditional adaptive mini-

mum variance control and LTV minimum variance control.

Fig. 5. The trajectory of the outputs under traditional adaptive mini-

mum variance control and LTV minimum variance control.


1� � tð Þq�1 ¼ 1� l tð Þq�1� �

1þ f1 tð Þq�1 þ f2 tð Þq

�2�

þ f3 tð Þq�3�þ r0 tð Þq

�4 ð33Þ

It follows from Eq. (33) that

f1 tð Þ ¼ l tð Þ � � tð Þf2 tð Þ ¼ l tð Þf1 t� 1ð Þ

f3 tð Þ ¼ l tð Þf2 t� 1ð Þ

8<:It is interesting to compare the difference between the

normal multiplication and pointwise multiplication andhow this difference varies with the parameter changingrate. The comparisons are shown in Fig. 6. The top sub-figure shows the minimum variance term calculated usingthe normal multiplication (solid line) and pointwisemultiplication (dashed line) for the disturbance dynamics

given by case 1. The calculated minimum variance termsfor the disturbance dynamics given by case 2 are shownin the middle subfigure. The calculated minimum varianceterms for the disturbance dynamics given by case 3 areshown in the bottom subfigure. The three subfigures arearranged in the ascending order, from the top to the bot-tom, in terms of varying rate of the disturbance para-meters. It is clearly seen that the difference between thenormal multiplication and the pointwise multiplicationincreases with the increase of the parameter varying rate.

5. Case study on an industrial cascade control loop

The proposed performance assessment method isapplied to monitor control loop performance of anindustrial cascade control loop. The schematic of theprocess is shown in Fig. 7. The feed stocks are anhy-drous ammonia (NH3) and air. The ammonia goesthrough a two-stage heating process before entering thecatalytic reaction which contains a (gauze type) plati-num–rhodium catalyst. Process air at over 400�F and150 psig enters the reactor. The ammonia-air mixturereacts on the catalyst at over 1600�F and forms nitrogendioxide with other by-products (NOx). In order to max-imize the production of NO2 and minimize the by-prod-ucts which are harmful to the environment, the gauzetemperature is required to be kept as steady as possibleeven in the presence of disturbances in the ambient tem-perature air quality, ammonia feed temperature, andammonia flow rate. The present control configuration isthat the gauze temperature controller (outer loop) adjuststhe setpoint of the ammonia flow rate (inner loop).The time delay of the outer loop from a priori analysis is

known to be 15 s including the delay due to the zero-order-hold device. The sampling period is 5 s, and, there-fore, the time delay is three sampling periods, i.e. d=3.

Fig. 6. Comparison of time-variant minimum variance term between

normal multiplication and pointwise multiplication.

Fig. 7. Schematic diagram of the industrial cascade reactor control loop.


To compare the minimum variance term estimatedusing normal multiplication and pointwise multi-plication, the estimated time-variant minimum varianceof the outer loop is shown in Fig. 8. The difference canclearly be seen from the figure. Since the minimum var-iance term depends on disturbance dynamics only, thetime varying property of the minimum variance termactually reflects the time varying property of the dis-turbance dynamics. This case study once again showsimportance to use the normal multiplication rather thanthe pointwise multiplication for a process with timevarying characteristics.

Remark 4. This example demonstrates that it is importantto use normal multiplication in the estimation of theminimum variance term for an LTV process. The timevarying nature of process data is frequently observed inpractice, which is due to either varying process dynamicsor varying disturbance dynamics. In this example, it hasbeen suspected that the time varying nature is due to timevarying feed quality and/or ambient temperature.It is worthwhile, before concluding, to point out that

to calculate the minimum variance term using the pro-posed approach, one does not need to know process ordisturbance models, which is a very desired property inpractice for assessment of control loop performance.

6. Conclusion

An explicit transfer function form of the LTV mini-mum variance control law has been derived in thispaper. This result extends the recent contributions by Liand Evans [23] from an ARMAX model to the generalBox–Jenkins model. The solution presented here is sim-pler and no pseudocommutation equations need to besolved, which has greatly simplified the design of theLTV minimum variance control law. It has been shownthat the lower bound of process variance (minimum

variance) for an LTV process is feedback control invar-iant and can be estimated from routine operating data.The results have been verified by simulated examplesand an industrial case study.

Acknowledgements

This work was supported by the Natural Sciences andEngineering Research Council of Canada underResearch Grant 203057-98.

Appendix A. Proof of result 1

Proof. Let ut=�Q(q�1, t)yt be a time-variant controllaw. Substituting this into Eq. (1) yields

yt ¼ 1þ q�dT~ q�1; t� �

Q q�1; t� �� 1

N q�1; t� �

at ðA1Þ

Substituting Eq. (3) into Eq. (A1) yields

yt ¼ 1þ q�dT~ q�1; t� �

Q q�1; t� �� 1

F q�1; t� �

þ R q�1; t� �

q�d� �

at

ðA2Þ

Due to noncommutativity of the multiplication of LTVtransfer functions, we have to treat all manipulations ofthe LTV polynomials in the same way as the matrixmanipulation. Applying the matrix inversion lemma (seeAppendix C) to Eq. (A2) yields

yt ¼ 1� q�dT~ q�1; t� �

1þQ q�1; t� �

q�dT~ q�1; t� �� 1�

Q q�1; t� �

F q�1; t� �

þ R q�1; t� �

q�d� �

at ¼ F q�1; t� �

at

þR q�1; t� �

q�dat � q�dT~ q�1; t

� �1þQ q�1; t

� ��q�dT~ q�1; t

� ��1Q q�1; t� �

F q�1; t� �

at � q�dT~ q�1; t

� �1þQ q�1; t

� �q�dT~ q�1; t

� �� 1Q q�1; t� �

R q�1; t� �

q�dat

ðA3Þ

Denoting the third term on the right hand side of Eq.(A3) as rhs3, then

rhs3 ¼ q�dT~ q�1; t� �

1þQ q�1; t� �

q�dT~ q�1; t� �� 1

Q q�1; t� �

F q�1; t� �

at ¼ T~ q�1; t� d� �

qd þQ q�1; t� �

q�dT~ q�1; t� �

qd� ��1

Q q�1; t� �

F q�1; t� �

at

¼ T~ q�1; t� d� �

1þQ q�1; t� d� �

q�dT~ q�1; t� d� �� 1

Q q�1; t� d� �

F q�1; t� d� �

at�d ðA4ÞFig. 8. Estimated time-variant minimum variance.


Similarly, the fourth term on the right hand side of Eq.(A3) can be written as

rhs4 ¼ T~ q�1; t� d� �

1þQ q�1; t� d� �

q�dT~ q�1; t� d� �� 1

Q q�1; t� d� �

R q�1; t� d� �

q�dat�d

ðA5Þ

Substituting Eqs. (A4) and (A5) into Eq. (A3) yields

yt ¼ F q�1; t� �

at þ R q�1; t� �

at�d � T~ q�1; t� d

� �1þQ q�1; t� d

� �q�dT~ q�1; t� d

� �� 1Q q�1; t� d� �

F q�1; t� d� �

at�d � T~ q�1; t� d

� �1þQ q�1; t� d

� �q�dT~ q�1; t� d

� �� 1Q q�1; t� d� �

R q�1; t� d� �

q�dat�d ¼ F q�1; t� �

at þ L q�1; t

� �at�d ðA6Þ

where

L q�1; t� �

¼ R q�1; t� �

� T~ q�1; t� d� �

1þQ q�1; t� d� �

q�dT~ q�1; t� d� �� 1

Q q�1; t� d� �

F q�1; t� d� �

� T~ q�1; t� d� �

1þQ q�1; t� d� �

q�dT~ q�1; t� d� �� 1

Q q�1; t� d� �

R q�1; t� d� �

q�d ¼ R q�1; t� �

� T~ q�1; t� d� �

1þQ q�1; t� d� �

q�dT~ q�1; t� d� �� 1

Q q�1; t� d� �

F q�1; t� d� �

þ R q�1; t� d� �

q�d

¼ R q�1; t� �

� T~ q�1; t� d� �

1þQ q�1; t� d� �

q�dT~ q�1; t� d� �� 1

Q q�1; t� d� �

N q�1; t� d� �

ðA7Þ

which is a proper transfer function.Since F(q�1, t)at=f0(t)at+ � � �+fd�1(t)at�d+1 depends

on the white noise sequence from at�d+1 up to at, andL(q�1,t)at�d depends on the white noise sequence beforetime t�d+1, the two terms on the right hand side of Eq.(A6) are independent, and as a result,

Var ytð Þ ¼ Var F q�1; t� �

at� �

þ Var L q�1; t� �

at�d� �

Therefore

Var ytð Þ5Var F q�1; t� �

at� �

The equality holds [or the lower bound of Var(yt) isachieved] if and only if

L q�1; t� �

¼ 0 ðA8Þ

The LTV minimum variance control law can there-fore be solved from Eq. (A8). Using Eqs. (A7) and (A8),we have

R q�1; t� �

� T~ q�1; t� d� �

1þQ q�1; t� d� ��

q�dT~ q�1; t� d� ��1

Q q�1; t� d� �

N q�1; t� d� �

¼ 0

This can be simplified to

R q�1; t� �

� Q�1 q�1; t� d� �

T~�1 q�1; t� d� �

þ q�d� ��1

N q�1; t� d� �

¼ 0

or

Q�1 q�1; t� d� �

T~�1 q�1; t� d� �

þ q�d� ��1

¼ R q�1; t� �

N�1 q�1; t� d� �

Therefore, we have

Q q�1; t� d� �

¼ T~�1 q�1; t� d� �

½N q�1; t� d� �

R�1 q�1; t� �

� q�d��1 ¼ T~�1 q�1; t� d� �

F q�1; t� d� �

þ R q�1; t� d� �

q�d� �

R�1 q�1; t� �

� q�d �1¼ T~�1 q�1; t� d

� �F q�1; t� d� �

þ q�dR q�1; t� ��

R�1 q�1; t� �

� q�d�1

¼ T~�1 q�1; t� d� �

R q�1; t� �

F�1 q�1; t� d� �

ðA9Þ

or

Q q�1; t� �

¼ T~�1 q�1; t� �

R q�1; tþ d� �

F�1 q�1; t� �

ðA10Þ

The closed-loop response under LTV minimum vari-ance control can be solved by substituting Eq. (A10)into Eq. (A6). This yields

ytjmv¼ F q�1; t� �

at ¼

f0 tð Þ þ f1 tð Þq�1 þ � � � þ fd�1 tð Þq

�dþ1

atðA11Þ

In Eq. (A6), the controller transfer function Q(q�1, t)only affects the second term, L(q�1, t), the minimumvariance term F(q�1, t) is therefore feedback controllerinvariant. If we model routine operating data by amoving average model, then the model should have thefollowing form:

yt ¼ f0 tð Þat þ f1 tð Þat�1 þ f2 tð Þat�2 þ � � � þ fd�1 tð Þat�dþ1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}F q�1;tð Þat

þ fd tð Þat�d þ � � �|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}L q�1;tð Þat�d


i.e. the first d terms precisely constitute the process out-put under minimum variance control.

Appendix B. Equivalence of two LTV minimum

variance control laws

To verify the developed LTV minimum variance con-trol law, in the sequel we shall show that the LTVminimum variance control law derived in this paper isequivalent to that derived by Li and Evans [23] if theBox–Jenkins model is reduced to an ARMAX model.We will, in the following, introduce three lemmas beforethe proof of the equivalence of the two minimum var-iance control laws.

Lemma 5. For an LTV ARMAX model:

A q�1; t� �

ytþd ¼ B q�1; t� �

ut þ C q�1; t� �

atþd ðB1Þ

Eq. (B1) is I/O equivalent to the following LTV model:

0 D~ q�1; t� �

A~ q�1; t� �

�B~ q�1; t� �

" #xtþdzt

� �¼

utatþd

� �ðB2Þ

ytþd ¼ C~ q�1; t� �

0h i

xtþdzt

� �ðB3Þ

where A~ q�1; t� �

; B~ q�1; t� �

; C~ q�1; t� �

; and D~ q�1; t� �

are determined from the following pseudocommutationequations:

A q�1; t� �

C~ q�1; t� �

¼ C q�1; t� �

A~ q�1; t� �

B q�1; t� �

D~ q�1; t� �

¼ C q�1; t� �

B~ q�1; t� �

Proof. See Li and Evans [23].This lemma builds a bridge between the regular

ARMAX model and the model used to derive the LTVminimum variance control in Li and Evans [23]. Withthis new model the LTV minimum variance control lawcan be solved from the following lemma.

Lemma 6. The minimum variance control law for thenew I/O equivalent model in Lemma 5 is given by

C~ q�1; t� d� �

0

G q�1; t� �

E q�1; t� �

B~ q�1; t� �

" #xt tj

zt tj

� �¼

yt

y tþd

� �ðB4Þ

ut ¼ 0 D~ q�1; t� �h i

xt tjzt tj

� �ðB5Þ

where y t is the desired setpoint; E(q�1, t) and G(q�1, t)

are determined from the Diophantine identity

C~ q�1; t� �

¼ E q�1; t� �

A~ q�1; t� �

þ G q�1; t� �

q�d ðB6Þ

Proof. See Li and Evans [23].

Lemma 7. For the same notations as defined in Lemmas5 and 6, the following equality holds:

E�1 q�1; t� �

G q�1; t� �

C~ �1 q�1; t� d� �

¼ A~ q�1; t� �

C~ �1 q�1; t� �

G q�1; t� �

A~�1 q�1; t� d� �

E�1 q�1; t� d� �

ðB7Þ

Proof. Eq. (B7) is equivalent to:

C~ q�1; t� �

A~ �1 q�1; t� �

E�1 q�1; t� �

G q�1; t� �

¼ G q�1; t� �

A~ �1 q�1; t� d� �

E�1 q�1; t� d� �

C~ q�1; t� d� �

ðB8Þ

Substituting Eq. (B6) into the left hand side of Eq. (B8)yields

lhs ¼ E q�1; t� �

A~ q�1; t� �

þ G q�1; t� �

q�dh i

A~ �1 q�1; t� �

E�1 q�1; t� �

G q�1; t� �

¼ G q�1; t� �

þ G q�1; t� �

q�dA~ �1 q�1; t� �

E�1 q�1; t� �

G q�1; t� �

Substituting Eq. (B6) into the right hand side of Eq.(B8) yields

rhs ¼ G q�1; t� �

A~ �1 q�1; t� d� �

E�1 q�1; t� d� �

E q�1; t� d� �

A~ q�1; t� d� �

þ G q�1; t� d� �

q�dh i¼ G q�1; t

� �þ G q�1; t

� �q�dA~�1 q�1; t

� �E�1 q�1; t

� �G q�1; t� �

Thus lhs=rhs, i.e. Eq. (B8) holds and consequently Eq.(B7) is true.With these three lemmas, we are in the position to

show the equivalence of the LTV minimum variancecontrol law for an ARMAX model as derived in Li andEvans [23] and the LTV minimum variance control lawdeveloped in this paper when it is applied to an LTVARMAX model.

Corollary 8. For the LTV ARMAX model given by Eq.(B1), the LTV minimum variance control law shown inResult 1 is the same as the minimum variance control lawgiven in Lemma 6. However, the solution in Result 1 doesnot require solving pseudocommutation equations andtherefore simplifies the design procedure.


Proof. It follows from Eqs. (B2) and (B3) that the I/Oequivalent model of (B1) is given by

yt ¼ q�dC~ q�1; t� �

A~ �1 q�1; t� �

B~ q�1; t� �

D~ �1 q�1; t� �

ut

þ C~ q�1; t� d� �

A~ �1 q�1; t� d� �

at ðB9Þ

Comparing Eq. (B9) to Eq. (1) yields

T~ q�1; t� �

¼ C~ q�1; t� �

A~�1 q�1; t� �

B~ q�1; t� �

D~ �1 q�1; t� �

ðB10Þ

N q�1; t� �

¼ C~ q�1; t� d� �

A~ �1 q�1; t� d� �

ðB11Þ

Combining Eqs. (B6) and (B11) yields

N q�1; t� �

¼ E q�1; t� d� �

þ G q�1; t� d� �

A~ �1 q�1; t� 2d� �

q�d ðB12Þ

Comparing Eq. (B12) with (3) yields

F q�1; t� �

¼ E q�1; t� d� �

ðB13Þ

R q�1; t� �

¼ G q�1; t� d� �

A~�1 q�1; t� 2d� �

ðB14Þ

From Eqs. (B4) and (B5), (assuming a zero setpointfor simplicity) the minimum variance control law solvedby using pseudocommutation equations is given by

ut ¼ �D~ q�1; t� �

B~�1 q�1; t� �

E~�1 q�1; t� �

G q�1; t� �

C~ �1 q�1; t� d� �

yt ðB15Þ

Substituting Eq. (B7) into Eq. (B15) yields

ut ¼ �D~ q�1; t� �

B~�1 q�1; t� �

A~ q�1; t� �

C~ �1 q�1; t� �

G q�1; t� �

A~ �1 q�1; t� d� �

E�1 q�1; t� d� �

yt ðB16Þ

Using Eqs. (B10), (B13) and (B14), Eq. (B16) can bewritten as

ut ¼ �T~�1 q�1; t� �

R q�1; tþ d� �

F�1 q�1; t� �

yt

which is the same as the minimum variance control lawreported in Result 1.

Appendix C. Matrix inversion lemma

Given matrices A, B, C and D, then the followingidentity holds:

Aþ BCDð Þ�1¼ A�1 � A�1B C�1 þDA�1B

� ��1DA�1

where A and C are assumed to be invertible.

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