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MPC tuning for systems with right half plane zerosWINSTON. GARCA-GABNEscuela de Ingeniera Elctrica

Universidad de Los AndesAv. Tulio Febres Cordero, Mrida 5101

VENEZUELA

EDUARDO.F. CAMACHODepartamento de Ingeniera de Sistemas y Automtica

Universidad de SevillaCamino de los Descubrimientos, Sevilla 41092

ESPAA

Abstract: - This paper demonstrates how instability problem will be present for a Model Predictive Control(MPC) in the case of it is applied to non-minimum phase systems. Instability appears when the control horizonhas the same value that the prediction horizon and the control weight is zero, because MPC achieves itsperformance by cancelling the plant zeros including the unstable zeros, which leads to a loss of internalstability of the feedback system. It is demonstrated that the instability problem can be solved with an adequatetuning of horizon values without using a big control weight as it is suggested in the literature. The results areillustrated with a non-linear example.

Key-Words: - Model predictive control, Non-minimum phase systems, stability, tuning controller.

1 IntroductionGeneralized Predictive Control was proposed byClarke, et al.[1] and has become one of the most

popular MPC methods both in industry andacademia. It has been successfully implemented inmany industrial applications, showing goodperformance. The basic idea of MPC is to calculatea sequence of future control signals in such a waythat it minimizes a multistage cost function definedover a prediction horizon. The index to be optimizedis the expectation of a quadratic function measuringthe distance between the predictive systems outputand some predictive reference sequence over thehorizon plus a quadratic function measuring controleffort. In order to implement a MPC, a model of the

plant is used to predict the future plant outputs. Thisprediction is based on past and current values of theinput and the output of the plant. The process modelplays, in consequence, a decisive role in thecontroller performance, and thus it is desirable tochoose a model quite similar to the plant in order topredict accurately.

A system is said to be a non-minimum phase processif it has Right Half Plane Zeros (RHPZ) or in thediscrete case if at least one of the zeros of thetransfer function is located outside the unit circle.These processes are common in industrialapplications and they are characterized by their

inverse response. The control engineers must bewarewith this kind of process, because they are animportant source of problems in practical

applications. It is well known that non-minimumphase systems present difficulty in applying controlstrategies, because they have an initial inverseresponse to step input in the opposite direction fromthe steady state, [2]. The presence of unstable zeroin a process transfer function is thus identified asbeing responsible for its difficult dynamic behavior;it is also the source of a considerable amount ofdifficulty in controller design. Another aspect ofcontrolling a process with unstable zero is theinstability problem, which arises in order to achievehigh performance when the controller contains an

inverse of the process model [3].

The instability problems applying MPC to SingleInput Single Output (SISO) systems with RHPZhave been reported in literature.[4,5,6,7,8,9] showedthat non-minimum phase systems produce instabilitywhen prediction horizon and control horizon areequal to one. Also an unstable behaviour in SISOMPC has been reported by [10,11] when it is appliedto a non minimum phase systems. This can besolved using a control weight parameter[1,6,7,12,13].

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This paper demonstrated how instability problemwill be present for whatever non-minimum phasesystems, when the control horizon has same valuethat the prediction horizon and control weight iszero, because MPC achieves its performance bycanceling the plant zeros, including the unstablezeros, which leads to a loss of internal stability ofthe feedback system. It is demonstrated that theinstability problem can be solved with an adequatetuning of horizon values without use a big controlweight as it is suggested in the literature.

This article is organized as follows: Section 2 givesa brief review of formulation of MPC, Section 3shows the procedure used to obtain MPC closedloop relationships. Section 4 shows the applicationof this controller for linear and non-linear non-

minimum phase systems. Finally, the conclusionsare presented.

2 Model Predictive ControlMost of the SISO plants, when consideringoperation around a particular set-point and afterlinearization, can be described by:

( ) ( ) ( ) tttzC

uzByzA

+=

1

111

(1)

Where:

ty : Output signal process

tu : Input signal process

t : Zero mean white noiseA, B, C are the following polynomials in thebackward shift operator z-1

A(z-1) = 1 + a1z-1 + + anz-naB(z-1) = b0 + b1z-1 + + bnz-nb

C(z-1) = 1 + c1z-1 + + cnz-nc

This model is known as the CARIMA Model(Controller Auto-Regressive Integrated Moving-Average). It has been argued that for many industrialapplications in which disturbances are non-stationary an integrated CARIMA model is moreappropriate [14].

The MPC algorithm consists of applying a controlsequence that minimizes a multistage cost function(2). This minimization produces u(t), u(t+1) ,,u(t+Nu), but only u(t) is applied. At time, t+1 anew minimization problem is solved. This

implementation is called the Receding Horizoncontroller.

( ) [ ]

[ ]

=+

=++

+=

uN

j

jt

N

Nj

jtjt

u

rytuJ

1

21

22

1

,

(2)

Subject to ut+j=0, j=Nu, ..., N2

Where:N1 : Minimum prediction horizonN2 : Maximum prediction horizonNu : Control horizon :Control weight

r : Reference trajectory

In order to solve the problem posed by theminimization of (2), the y(t+1) has been computed.The j-step ahead output forj = N1...N2 based on theinformation known at time tand the future values ofthe control increments. The following Diophantineequation is considered,

( ( ( ( 1111 += zFzzAzEzC jj

j(3)

The polynomialsEjandFj are uniquely defined withdegreesj-1 and na respectively.Combining the plant model (1), and Diophantineequation (3), the follow prediction output equationcan be obtained,

1 ++ += jtj

t

j

jt uC

BEy

C

Fy (4)

In this expression jty + is a function of a known

signal values at time t and of future control inputs,

which have not been computed yet. To distinguishpast and future control values a second Diophantineequation (5) has been used.

( ) ( ) ( ) ( )

( )1

1111

+=

zz

zCzGzBzE

j

j

jj

(5)

The following expression of the prediction wasobtained.

f

tj

f

tjjtjjt yFuuGy ++= ++ 11 (6)

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Wheref

tu andf

ty are

( )( ) t

f

t

t

f

t

yzCy

uzCu

11

11

=

= (7)

(8)

Finally, (6) can be rewritten as:

( )tjtjtjjt

yuzGy++

+ += 1

1 (9)

Where tjty

+

is the free response prediction of jty +

assuming that future control increments after time t-1 will be zero,

( ) ( ) ftjf

tjtjtyzFuzy 11

1

+ += (10)

Substituting Ej(q-1) of (3) into (5), this yields

( )

( ) 1

11

++=

CABFz

CzGAB

j

j

j

j

j

(11)

Define the vector f, composed of the free responsepredictions,

T

tNttttt

yyyf

21 2,,,

+++

= (12)

the vector of future control increments,

[ ] TNttt uuuuu

11 ,,,~

++ = (13)

and the vector of the predicted plant outputs,

[ ] TNttt yyyy

21 2,,, +++= (14)

From the prediction (10) the predicted input-outputrelationship of the plant can be written as the vectorequation,

fuGy += ~ (15)

Where the matrix G is composed of the stepresponse parametersgi of the plant model.

=

u

uu

NNNN

NN

ggg

ggg

gg

g

G

222 21

021

01

0

0

00

(16)

The quadratic minimization of (2) becomes a directproblem of linear algebra, assuming there are noconstraints on the control signal, which leads to:

)()(~ 1 frGIGGu TT += (17)

3 Closed loop relationshipsClosed loop relationships can be obtained for theMPC in order to show how the tuning parametersN1, N2, might affect the stability of the controlledplant. MPC is a receding horizon controller sotherefore only components i of the first row of thematrix equation (17) were considered. They can berewritten as,

2

2

2

22

1

1

1

1

Nt

N

i

iN

i

t

N

i

iiti

N

i

i

rzC

yFuzC

+=

+

=

=

+

=

+

(18)

With definitions for the polynomials R, S and T1,(18) is given by,

21 NtttrCTSyuR ++= (19)

Substituting (19) into CARIMA model (1), the close

loop relationship was obtained,

t

c

Nt

c

tCP

Rr

CP

BTy += + 2

1

(20)

Where, the characteristic polynomial can be definedas:

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( )

(

( )

c

N

i

iii

N

i

iii

CP

zGAB

AC

zBFA

CABSzRA

=

+=

+

+=+

=

=

2

2

1

1

1

1

1

(21)

When the control weight is zero, (17) is given by,

)()(~ 1 frGGGu TT = (22)

If the prediction horizon and the control horizon

have the same value Nu=N2=N, the first row of u~

isgiven by,

xNg

10

0...001

(23)

Therefore, the characteristic polynomial becomes,

( )0

0

0

1

g

BCgAB

gACP

C=

+= (24)

The characteristic polynomial has the numeratorpolynomial of the process model B. Consequently,close loop poles are the zeros of the plant model.

3.1 Solution using tuning parametersThe unstable behavior produced by RHPZ when

N2=Nu with =0 can be solved with an adequatedifference between prediction horizon and controlhorizon. This can be confirmed as follow,

Increasing the prediction horizon (N2 ) and

using an unitary control horizon (Nu = 1) the controlsignal during the prediction horizon are,

utu

Ntutututu

+=

+=+=+=

)1(

)1()2()1()( 2(25)

the control increment u is calculated to achievedthe final values of process output y(t+N2) must bethe reference r(t+N2) , hence using a big predictionhorizon,

)()()(1

ktwktuzGLimz

+=+

(26)

thus,

)()( 1 ktrGktu +=+ (27)

Then, the closed loop response in the predictionhorizon is obtained,

)()1()()( 1 ktrGzGkty +=+ (28)

Thus, the output dynamics is given by the open looppoles.

The above results can be summarized as follow,when MPC is applied to process with RHP zeros,the tuning parameters can be adjusted to avoid

Nu=N2 with =0 because this produce thecancellation of RHP zeros with RHP poles. Stable

behavior can be obtained using a prediction horizonbigger than control horizon. Using a big predictionhorizon and unitary control horizon the closed looppoles are given by the open loop poles.

4. Simulation examplesTwo examples have been used to illustrate theresults, a linear system to show the influence of thetuning parameters in the pole placement and a non-linear system to compare the tuning methods.

4.1 linear non-minimum phase system

The following linear non-minimum phase systemwas considered to obtain the pole placement of thecontroller in function of its tuning parameters.

1

11

9.01

5.11)(

+

=z

zzG (29)

Using the closed loop polynomial characteristic (21)the Biggest Pole Magnitude (BPM) is obtained foreach tuning parameters set. Figure 1, shows theBPM when the Prediction horizon (N2) was changed

from N2=1 to N2=100 with Nu=1 and =0. WhenN2=1 the controller has its pole in z = 1.5, it has thesame value of RHPZ of the plant model, thisproduces an unstable behavior. Subsequently, theprediction horizon was increased. When theN2 valueis close to Nu, the controller has an unstable pole, if

N2 is bigger than Nu the pole will be inside the unitcircle (the magnitude is smaller than one). Note thatthe closed loop pole tends towards the open looppole whenN2 is increased.

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Fig.1 BPM vs. Prediction horizon

The control weight parameter has the function of

including a penalty on the control signal of themultistage cost function (2). Fig. 2, shows the BPMobtained for the control weight change from =0 to=10, using the same prediction horizon and controlhorizon.

Fig.3. BPM vs Control weight (). N2=Nu=5.

Despite of the horizons have the same value, whenthe control weight parameter was increased the BPMof the controller was changed tending to inside the

unit circle. This is the solution proposed in theliterature to allow a stable behavior[1,6,7,12,13].

4.2 Non linear reactorIn order to compare the tuning methods a non-linear,non minimum phase process has been used. Theisothermal Van de Vussen reaction systems involveseries and parallel reactions. The equations thatgovern the systems are:

( )V

FCCCkCk

dt

dCainaaa

a += 231 (26)

VFCCkCk

dtdC bbab = 21 (27)

The desired output is the concentration ofB, Cb[mol/l], Ca and Ca in are the concentrations ofA[mol/l] in the reactor and in the feed respectively,the manipulate input,

Fis the dilution rate [l/min], Vis the volume [l], andthe rate constants are given by k1=5/6 [min-1], k2=5/3[min-1],k3=1/6 [mol/(litermin)] [15]After linearizing model (26,27) about the operatingpoint, the physical model gives the followingtransfer function (30). The discretization has beenmade with a sampling rate (Ts=0.2 min).

21

1

3951.02573.11

1745.00939.0)(

+

+=

zz

zzG (30)

The transfer function has a zero in 1.8584.Fig. 1 shows a step changes in the reference ofcontrolled variable when MPC is tuning withN2=8,

Nu=8, =0. The reactor has an unstable response,this behavior verified the cancellation of theunstable zero with an unstable pole that wasdemonstrated in Section 3.

Fig. 3. MPC tuning with N2=8, Nu=8, =0

To contrast the controller performance with twodifferent sets of tuning parameters a set-pointchange was produced. Fig. 4 shows the proposedtuning method against the tuning method proposedin the literature. The proposed method avoids theinstability using a prediction horizon bigger thancontrol horizon. The Fig. 4 illustrate that the raisedtime is a 28 % of the raised time required by themethod proposed in the literature, that avoids theinstability increasing the control weight. Thus, ascan be observed the MPC with a high value ofcontrol weight produces a soft control signal as wellas a soft output process. The proposed method

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allows adjusting a small control weight asconsequence the raised time is reduced considerably.

Fig. 4. Proposed tuning method (- -) vs method thatincrease the control weight (-)

4 ConclusionThis paper has presented how the presence of RHPzeros in the process model produces an unstabledynamics in MPC. The instability is produced by thecancellation RHP zeros with RHP poles. A tuningmethod is proposed to solve it. The method is basedin use a prediction horizon bigger than controlhorizon. It is showed that an appropriate selection ofthe difference between the prediction horizon andthe control horizon produce a stable performance. Inspite of the method proposed in the literatureproduces a stable behavior, a better raised time isobtained employing the proposed method. The badperformance is obtained because the control weightparameter produces soft moving of manipulatevariable as consequence a soft behavior in thecontrolled variable.In summary, small values of control weightparameter with a convenient difference between

prediction horizon and control horizon, it seems towork well for MPC applied to non-minimum phasesystems.

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[3] Bardley, R., and M. Morari Design ofResilient processing Plants. ChemicalEngineering Science, 40,1985, 59-74.[4] R. K. Mehra and R. Rouhani. Theoreticalconsiderations on model algorithmic control for

non minimum phase systems. Joint AutomaticControl Conference,1980.[5] Clarke, D., C. Mohtadi and P. Tuffs,Generalized predictive Control-Part II Extensionsand Interpretations, Automatica, 23,1987,149-160.[6] M.J. Grimble. Generalized PredictiveOptimal Control: an introduction to theadvantages and limitations. International Journalof Systems Science, 23, 1992,85-98.[7] W.S. Levine (Ed). The Control Handbook

CRC Press,1996.[8] J. M. Martn and J. Rodellar, Adaptive

predictive control: from the concepts to plantoptimisation,Prentice Hall,1996[9] J.M. Maciejowski, Predictive Control withConstraints, Prentice Hall,2001[10] D. Seborg and E. Thomas and D.Mellichamp, Process Dynamics and Control.John Wiley and Sons,1989[11] T. E. Marlin. Process Control: DesigningProcesses and Control Systems for DynamicPerformance, McGraw-Hill, 1995.[12] Bitmead, R., M. Gevers, and V. Wertz,Adaptative Optimal Control. Prentice Hall, 1990.[13] R. Soeterboek. Predictive Control: AUnified Approach, Prentice Hall,1991.[14] Camacho, E.F., and C. Bordons. ModelPredictive Control, Springer 1999.[15] B. Wayne Bequette, Process Dynamics:Modeling, Analysis, and Simulation, PrenticeHall,1998.