GMVC and Pole Placement

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Proceedings of the 2004 IEEEIntemational Conference on Control ApplicationsTaipei, Taiwan, September 24 ,2 004

Generalized Minimum Variance with Pc

.

Assignment Controller Modified r Practical Applications

Abstract - mong self-toningeontrollcrs, the explicit typt is themost commonly ose in industrial app l ica t ions ue to its simplestrnctore; on the other hand the implicit shllctores are lessa t t r ac t i v e because of their largest shpchlr~lcomplexity. Thispaper presents a modification to the generalized minimumvar iance with pole assignment controller. The modiPr*tionpretends to simplify the origi0.1 algorithm and to make easier itsimplementation io practical projects, conserving most o f theoriginal attributes.PRINCIPAL SYMBOLSA . M Polynomialsof order n mb and n corresponding to,respectively, system output, control input, anddisturbance nput; e+=co=l.Ed) Expectation operation,Wt, Generalized output function.p.e,AEFGHT Pole assigmnent polynomialU(,),CO) Unconelatedz e r o - m m random sequence.k System time delay.I

I . INTRODUCTIONThe self-tuning controllers have been studied during several

decades. Some developments have been successfully adoptedin industrial controllers [1,2]; being the most popular amonginstruments manufacturers, the explicit self-tuning controllertype [3,4] which synthesize the control law using the processestimated parameters. On the other hand, the implicit self-tuning controllers have not achieved the impact of the explicitcontrollers due to the complexity of their practicalimplementation. However the former try to achieve an optimalperformance of the overall system: this is a characteristic thatcould be used in industrial applications in a broader way

Cos polynomialsofordern,, nQ nd na actingon,respectively,system output, input and reference.General polynomial acting on the reference.Generalpolynomial acting on he perturbationGeneral polynomialacting on he output.Generalpolynomial acting on he input.r(l) System output, input and set point, respectively.

Time measured in sample instants.

In this document, the generalized minimum variancecontroller (GMVC) [7,81 and the generalized minivariance controller with pole assignment (GWCPA) [9] are

presented and compared with the self-hming controller usingpole assiplent (STCPA) [5,6] . From this comparison amodification of the Allidina controller is also presented,incorporating the best characteristics of a STC and the GMVC.This modification simplifies the control algorithm andfacilitates its implementation in a practical way when theprocess parameters are unknown at the controller s-Another advantage of this modification is that the parameterselection variability is reduced, producing a more reliablecontroller.2. AUTOTUNING CONTROL WITH POLE ASSIGNMENT

The main objective of a pole assignment control is to forcethe system to achieve a set of previously-defined closed looppoles. The self-tuning cnnh-oller with pole assignment(STCPA) is briefly reviewed in the following paragraphs.

Consider the system described by the CARMA modelAy / )= z - 'Bu ( / ) + c&r) 1 )

A , R and C are polynomials in z , where the polynomial A ismonic and coprime in relation to K, k represents the systemdelay in samples of time, while I) is an unconelated zeromean random sequence. To achieve that the plant output y 0 ,follow to certain reference rp) , inside a pole assignmentoutline (Fig.l), it is necessary to design the contmlpolynomials H,G and E.

In Fig. 1we can be observe that the controller structure isgiven by

H N I )= E r ( / ) G, t) (2 )The closed loop poles (represented by the rmts of T) are

assigned in the desired locations hy the appropriate selectionofHand G according to the diophantine equation[lo].

HA +:IBG = TC (3)The self-tuning control with pole assignment (STCPA) can

he synthesized in the following algorithm.

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I L p gCo*oller I-___-___-_-__ ________I

Fig.1Closed Imp control s m m e2.1. Algorithm: Self-tuning control with pole arsignment

Given T polynomialStep 1 Use recursive least squares to estimate A , B , C from (1).

Step 2 Synthesize he controller polynomials from

A A A

A A A A AA H + :-I B G = C TA 76)"E = - C

Step 3 Generate the control output u t ) fromA A AH u ( t )= Rr l)-Gy Ij

Step 4 Wait for sample time interval to elapse and then returnto Step 1

This controller has several advantages; the most evident isthe simplicity. This controller also offers robustness [ 6 ] andthe ability to control n o n - m i n i phase systems. Because ofits capacities. the STCPA surpasses the elementag minimumvariance controllers.3. GENERALIZEDMINIMUMVARLANCE CONTROL

In this section, the G M V C [7,8] is briefly reviewed asfollows. Let us defme a generalized output or pseudo output as

((t+@ =Py(t+k)+Qu(?)-Rr(t) (4)where P , Q, R (denominated cost polynomials) are used to

modify the control performance (Fig.2).

The cost function'to be minimized is the mathematicalexpectation of& + k ) .

J=E, ( (tCv) ( 5 )The control polynomials are computed according to the

control law

H U ( ~ ) + G Y ( ~ ) +r ( r ) = o ( 6 )and G is calculated from

P C = F A + ~ - ~ G (7)where:H = B F + Q C y E = - C R 8)

The generalized minimum variance control can hesynthesized in the FolloWiog algorithm.

Fig. 2 Generalized output

3.1 Algorithm: GMVCGiven P. Q and R,Step 1 Form the generalized output I )

((1, =Py(1, +Q u(t-k)-Rr(t-k)step 2 Estimate , ,E and i from

Kri- & u ( t - k ) + & y ( t - k ) + k r ( t - ) + i ( t )Step 3 Apply control up) using

k u t ) =&(Ij- (l)

Step 4 Wait for the sample time interval to elapse and retum toStep I .

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Convergence analysis related to this type of controllers canhe found in [4, 141.

The GMV algorithm has some disadvantages:1. The algorithm is steady state one. the system performancedepends directly on the parameters convergence [4].2. A situation that complicates the convergence is thecontroller predictive character, which depends on a sub-dimensioned estimation of the control parameters from a time-displaced ).4. GENERALIZED MlNlMUM VARIANCE CONTROLWITH POLE ASSIGNMENT.

Allidina, and Hughes [9] presented a modification, whichconsisted on self-tune P and Q in such a way that a desiredclosed loop poles is achieved, represented by a Tpolynomial.

4. I Algorithm: GMVCPAGiven P, Q,R initials and T,Step 1 Form the generalized output

&t)=Pv(i)+Qu(t-k)-Rr(t-k)Step 2 EstimateH G and E from

Step 3 Apply control U( , using. r i . (1 )= - (1)-kr(1)

Step4 RecomputeP and Q from:PH - q - k ~ ~b m

Step 5 Wait for the sample time interval to elapse and return toStep 1The term h 8 cts as a polynomial observer where ho is the

fust element of H . This structure pretends to eliminate the offline adjustment of P and Q; however this algorithm presentsthe following disadvantages:1. P and Q convergence is subordinated to the processparameters knowledge, because this iuformation is used todesign the generalized output function. A well designedfunction forces the parameter convergence.2. There is not convergence guarantee with arbitraryinitialization of P and Q [4].

3. If it was necessruy to estimate the process when the plant isunknown, then it is not possible that P,Q and R converge,because the generalized output function cannot be formedappropriately on-line.

It can be deduced that the AUidina algorithm is onlyfeasible when the process is previouslyh o w . n recent yearsnumerous papers have been published about GMV controllers,some examples are [11-14]; However most of thesedevelopments consider that the process is known beforebeginning the control, this situation avoids implementingGMCVPA as a commercial controller.5 . GENERALIZED MIMlMUM VARIANCE CONTROLWITH POLE ASSIGNMENT MODIFIED FOR PRACTICALAPPLICATIONS.

The first practical application possibility for the GMVCPAis to introduce between the step 1 and the step 2 a processestimation step (see the section 4.1). But this modification isinefficient because requires double estimate. However a biggerproblem exists, if the plant parameters are unknown, anappropriate generalized output function canuot be designed,and the convergence (in the case of existing) only would he acasual event. Such problems in the practical implementationmotivate the modification of the existent algorithm; thereforewe propose modifications that i z e the causedinconveniences by insufficient previous information. To makepossible the convergence in the modification (Generalizedminimum variance control with pole assignment modified forpractical applications) is assumed that C t) it is equivalent toC(t-1) (canceling the predictive mode on the controller). andwe also assume that the coefficients estimation of the generalpolynomials tend to the m e ones (with this supposition theestimate of the general polynomials can be omitted). Instead ofproposing P and Q arbitrarily, these are calculated on lineusing the equation

A AB P + A Q = T (9)

maintaining the relationship between the optimization and thepole assignment. Next the modified algorithm is list ed5. Algorithm: Modified GMVCPAStep 1 Using recursive least squares estimate A , B , CStep 2 Calcnlate P and Q from

A A A

f i nR P +AQ=TStep 3 With P and Q calculate directly the control parameters.G,6 and from

PC = F A + ?GH = B F + Q CE = -CR

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Step 4 Apply control u(t) using

STCPA GMVCT O nitializethe recursive - To initialue the recursive eninmion..The processparameters areesIimati0 .Imkno\w.

. elect valuesP.Q and o fom the(,)= P.d,)+Qri(t - k ) - R r [ t - )

Enlmate H , G , E from # ( f ) h e r e t h egeneralized o q l n s

generalized o q d

Usingreclnnive least squares A A AA A Al oea lm a te A,B,C

( I ) = U({ k +6yii- ) + i r ( t k ) + ; ( I )

f U /) 1. / )-2 r 1)

GMVCPA- To initialire the recursive eaimation.- Select n i t i a l va l w s P.Q and o fom~ [ ~ ) = P y [ l ) + Q ~ [ l - k ) - ~ ( r - k )

Estimate H , G , E from f) wherethegeneralized o q u s

the generalized o q u t

A A

( I ) = . ,(I - +;. (I - k ) + &(I k ) + ; I)

Step 5 Wait for the sample time interval to elapse and return toStep 1.

In Table I, we present a comparison of the algorithmsreviewed.6. CONTROLLERRESULTS

The GMVCPA modified was tested with a set of linearprocesses defined in the literature [2]. One of these processesis

whose discrete equivalent. using a sampling period of 0.1seconds and considering an perturbation, is

y :) = 1,81?y(:)- 0.819;~ y(.-)+ O.O0468;~*y(;)+0.00438~- '+0.05e( . - )

For this example. we select the following pole assignmentT(z)=: -1 .879~+0 .8884

The initial conditions used for the simulation are: forgettingfactor h=0.98, initial estimates t3=(0.5,0.5,0.5,0.5,0.5) and theinitial covariance matrix was selected as P I O O I , where I isthe identity matrix whose size is Q x Q. (The initial conditionsdescribed in this section will be effective in all the testspresented in this document).

polynomial:

TABLE 1. ALGORITHMS COMPARlSON

Calculate H and G solvlngand E: homA H + Z- B G = c TT 1 ) -l i= C

&)T apply U l) from

A h

A A A ASolve P H - ;? CQ = h,FT to obtanP and Q effective for thened s m p hpenod

modified GMVCPA-To nitialize the recursiveealmation

o m .- The prOEeSS parmeters areusing recursive least squares toestimate A,B,Cn ~ l

For a pole assignment T, olve theequationB P + A Q = T

With the estimated plynomialsA , B, C. , . and to solve

PC = FA + :-'GH = m + Q CE=-CR

to obtain H, G and ETo apply U( )om

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certain pole assignment ( 9 ) , the success of the suboptimalcontroller will depend on the perfect emulation of the dynamicbehavior represented by T. Therefore if we choose T qual tothe open loop process denominator, a successful poleassignment would cause a wutrol output with the same shapeas the reference input.

In Fig. 3, the controller and the evolution of the estimatedparameters are presented.

1 Y 5To illustrate this situation, we use the plant described in

section 6 and Ti s assigned to the pole placement of the openloop process denominator. In Fig. 4 we present the referenceand the control input; note that U is equivalent to the r shape

*o m 80 I O ,m m 1 4 ?el 1Mtime (S)

0 m 4c m m rm 120 4 18 * l ot u n e S)Fig 3 Sundahoo of Lhe mothtied GMVCPA

7. COMPUTATTONS EFFICIENCYWhen considering applying a controller in a practical

application. computing efficiency becomes an essential featureof an algorithm. In this paper, we compare the above reviewedcontrollers regarding execution time and memoryrequirements. All tests were performed in a PC Pentium 4,Matlab was used as programming platform and themeasurements were done using Matlab utilities. Table 2summarizes these results

A Execution d m e (seconds).B Static memoly d-g the execution (byies).C Memory occupied by the code @yies).E: hoportiooal increase to the Smallest ststic memory Jlocatim.Propadonal increase 10 the d e s 1 xecvfion une.

From Table 2, we can state that STCPA is the fastestcontroller, followed by the modified GMVCPA; however, thelatter performs much faster that GMVC and GMVCPA.

Regarding static memory space, STCPA and modifiedGMVCPA need almost the same space: whereas GMVC andGMVCPA require a lot more. In terms of code length, STCPAgenerates the smallest code, fallowed closely by the modifiedGMVCPA.

Fig. 5 Generalizedoutputs and performance indicators.To show the optimization process developed by the

modified GMVCPA algorithm, the generalized output function(Fig. 5a) was reconstructed (because in the modified controlleris implicit). The next equation

shows the difference that exists between the Ki,l and eo).Theoretically, the cost function should be minimized toachieve 0. without considering the pmrbation (Fig. 5b).Findly, if the pole assignment was successful, for this

8. SUBOPTDMALITYWhen we include a cost function in the design of adaptive

controller. the control law is optimized when we also have astochastic characteristic, such control systems must be particular case the relationshipconsidered suboptimal, If the cost function is defmed by a

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11 ACKNOWLEDGEMENTSThe authors wish to th nk to CENIDET and Tecnologico de

Monterrey Campus Cindad de M6xico for the support providedto develop this project.1 1. REFERENCES[I] Hagglund T., Astrom, K.J. Industrial adaptive controllers

based on ffeauencv remonse techniques, Automatica,

must have a tendency towards 0. In this test, the average m o rfor a sampling period was 0.0181, which is equivalent to apercentual error of 0.181 with FIO (Fig. Sc).

The cost function in the modified GMVCPA algorithm isdetermined by (9). If an algebraic manipulation is carried out,the control law structure of the modified GMVPA dli beequivalent to the control law used by the STCPA. However,the modified algorithm capacities are not restricted to thedevelopment of pole assignment, because there is thepossibility of modifying the cost function.9. PRACTICAL IMPLEMENTATION

To cany out tests, a laboratory apparatus (CEl10 ServoTrainer, Tecquipment) was selected, which is a D.C. motoraffected by inertial dynamics. The process is perturbed with aload change in the second 77; in the Fig. 6, we can see how theprocess response overcomes the load change. In this figure itcan also be appreciated that the generalized output functionamplitude. is notably small (standard deviation of 3 . 0 ~ 1 0 ~ nd- 3 . 0 ~ 1 0 . ~ean), which shows the capacity of optimization ofthe algorithm in practice.

IO CONCLUSIONSWe can say that the modification proposed for the

GMVCPA gives verygood results With the cases studied sofar, since it considerably reduces the number of parametersthat need to he initialized, its implementation complexity iscomparable with explicit control structures, and fmally, itallows the controller t o can y out optimization tasks in areliable way.

2 7 , 1 9 9 1 , ~99-609. .121 Lundh. M.. AstrBm. K.J. Automatic Initialization of a_robust self tuning controller, Automatica, 30, 1994,

pp1640-1662.[3] Astrom, K.J., Wittenmark B. Adaptive control (2ded.).Addisan-Wesley 1995.

[4] Wellstead, P.E., Zarrop M.B. Self-tuning sysfsms. JohnWiley & Sous, England 1991.[SI Wellstead P.E., Edmunds J.M., Prager D., % d e r P.Self-tuning polelzero assignment regulators, Int. J.Control, 30, 1979, pp 1-26.[6] Wellstead P.E., Prager D., Zanker P. Pole assignmentself-tuning regulator, IEE Proc., 126, 1979, pp 781-787.[7] Clarke, D.W., Gawthrop B.A. Self-tuning controller,IEE Proc., 122,1975,pp 929-934.[SI Clarke, D.W., Gawtluop B.A. Self-tuning control, IEEProc., 126, 1979, pp 633-640.[9] Allidma, A.Y., Hughes F.M. Generalized self-tuningcontroller with pole assignment_ EE Proc., 127, 1980,

pp 13-18.[ IO] KnEera V. Diophantine equations in control-a surveyAutomatica. 29, 1993, pp1361-1375.[ l l ] a n a k a H., Fujikawa, H.. Yamada S. A design method

of generalized minimum variance for tracking referencewith ramp input, Proceedings of the InternationalConference on Industrial Electronics, Coiitrol andInstrumentation, 1993, pp. 2212-2217.[I21 Yamamoto T., Fujii K., Kaneda M. Self-TuningTemperature Control of a Polymerizing Reactor,Proceeding of the 1998 IEEE International Conference onControl Applications, Trieste, 1998, pp. 1110-1114.

[I31 Yaacoh S. , Mohamed F.A. On-line self-tuning controllerfor induction motor based on generalized minimumvariance method, S CE _1998, pp. 875-878.[I41 Masayoshi D._ Yasuchika Mori, A study on RobustAsymptotic Tracking Property for Generalized MinimumVariance Control, Proceedings of the Americm ControlConference, Anchorage, 2002, pp. 1472-1477

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