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IMPROVING GPC TUNING FOR NON-MINIMUM PHASE SYSTEMS

W. Garca-Gabn, E.F. Camacho, D. Zambrano

Escuela de Ingeniera Elctrica

Universidad de Los Andes

e-mail: [email protected]

{winston, darine}@cartuja.us.es

Escuela Superior de Ingenieros,

Universidad de Sevilla

Camino de los Descubrimientos s/n, 41092-Sevilla (Spain)Phone:+34954487347, Fax:+34954487340,

e-mail: [email protected]

Abstract: This paper presents how an appropriate selection of tuning parameters of a

Generalized Predicted Controller for non-minimum phase systems avoids instability

problems produced by cancellations of unstable zeros with unstable pole. Some ideas

about tuning are presented and controller performance is judged using a non-minimum

phase system. Copyright CONTROLO 2000

Keywords: Predictive control, Non-minimum phase systems, stability, tuning,controller.

1. INTRODUCTION

A discrete system is said to be a non-minimum phase

process if at least one of the zeros of the transfer

function is located outside the unit circle. This kind

of processes is common in industrial applications and

they are characterized by their inverse response. It is

well known that non-minimum phase systems presentdifficulty in applying control strategies, because they

have an initial inverse response to step input in the

opposite direction from the steady state, (Ogunnaike,

1994). The presence of unstable zero in a process

transfer function is thus identified as being

responsible for its difficult dynamic behavior; it is

also the source of a considerable amount of difficulty

in controller design. Another aspect of controlling a

process with unstable zero is the instability problem,

which arises in order to achieve high performance

when the controller contains an inverse of the process

model. The unstable zero is cancelled by an unstable

pole, (Bradley and Morari, 1985).

Generalized Predictive Control (GPC) was proposed

by Clarke, et al., (1987a) and has become one of the

most popular Model Predictive Control methods both

in industry and academia. It has been successfully

implemented in many industrial applications,

showing good performance. The basic idea of GPC is

to calculate a sequence of future control signals in

such a way that it minimizes a multistage cos tfunction defined over a prediction horizon. The index

to be optimized is the expectation of a quadratic

function measuring the distance between the

predictive systems output and some predictive

reference sequence over the horizon plus a quadratic

function measuring control effort. In order to

implement a GPC, a model of the plant is used to

predict the future plant outputs. This prediction is

based on past and current values of the input and the

output of the plant. The process model plays, in

consequence, a decisive role in the controller

performance, and thus it is desirable to choose a

model quite similar to the plant in order to predictaccurately.

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As is well known, reports in predictive control

literature (Clarke, et al., 1987b), (Bitmead, et al.,

1990), show that GPC has stability problems

because, for non-minimum phase plants, this control

law suffers from requiring excessive control input inorder to effect the optimal output variance. The

controller achieves its performance by canceling of

the plant zeros, including the unstable zeros, which

leads to a loss of internal stability of the feedback

system.

Previous works by Clarke, et al., (1987b), showed

that non-minimum phase systems produce instability

when Nu=N2=1, while Bitmead, et al., (1990),

proposed that it can be solved using a control weight

parameter. This paper shows how instability problem

can be present for whatever non-minimum phase

systems, when the control horizon has same valuethat the prediction horizon and it can be solved

without use the control weight.

This article is organized as follows: Section 2 gives a

brief review of formulation of GPC, Section 3 shows

the procedure used to obtain GPC closed loop

relationships. Section 4 shows the application of this

controller for a system with non-minimum phase.

Finally, the conclusions are presented.

2. GENERAL PREDICTIVE CONTROL

Most Single-Input Single-Output (SISO) plants,

when considering operation around a particular set-

point and after linearization, can be described by:

( ) ( ) ( ) tttqC

uqByqA

+=

1

1

11 (1)

Where:

ty : Output signal process

tu : Input signal process

t : Zero mean white noise

A, B, C are the following polynomials in the

backward shif operator q-1

A(q-1

) = 1 + a1q-1

+ + anaq-na

B(q-1

) = b0 + b1q-1

+ + bnbq-nb

C(q-1

) = 1 + c1q-1

+ + cncq-nc

This model is known as the CARIMA Model

(Controller Auto-Regressive Integrated Moving-

Average). It has been argued that for many industrial

applications in which disturbances are non-stationary

an integrated CARIMA model is more appropriate

(Camacho and Bordons, 1999).

The General Predictive Control algorithm consists of

applying a control sequence that minimizes amultistage cost function (2). This minimization

produces ut,ut+1 ,, ut+Nu, but only utis actuallyapplied. At time t+1 a new minimization problem is

solved. This implementation is called the Receding

Horizon controller.

( ) [ ]

[ ]

=+

=++

+=

uN

j

jt

N

Nj

jtjt

u

rytuJ

1

2

1

22

1

,

(2)

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

Where:

N1:Minimum prediction horizon

N2: Maximum prediction horizon

Nu: Control horizon

:Control weightr: Reference trajectory

In order to solve the problem posed by the

minimization of (2), yt+1 has been computed the j-

step ahead output for j = N1...N2 based on theinformation known at time tand on the future values

of the control increments. The following Diophantine

equation is considered,

( ) ( ) ( ) ( )1111 += qFqqAqEqC jj

j (3)

The polynomials Ej and Fj are uniquely defined with

degreesj-1 and na respectively.

Combining the plant model (1), and Diophantine

equation (3), the follow prediction output equation

can 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 also of future control

inputs which have not been computed yet. Using a

second Diophantine equation (5) to distinguish past

and future control values.


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( ) ( ) ( ) ( )

( )1

1111

+=

qq

qCqGqBqE

j

j

jj

(5)

The following expression of the prediction was

obtained.

f

tj

f

tjjtjjt yFuuGy ++= ++ 11 (6)

Withf

tu andf

ty being filtered versions of

tu and ty ,

( )

( )t

f

t

t

f

t

yqCy

uqCu

11

11

=

=

(7)

(8)

Finally, (6) can be rewritten as:

( )tjtjtjjt

yuqGy ++

+ += 11

(9)

Wheretjt

y + is the free response prediction of jty +

assuming that future control increments after time t-1

will be zero,

( ) ( )f

tj

f

tjtjt yqFuqy1

1

1

+ += (10)

SubstitutingEj(q-1

) of (3) into (5), this yields

( )

( ) 1

11

++=

CABFq

CqGAB

jj

jj

j

(11)

Define the vector f, composed of the free response

predictions,

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-output

relationship of the plant can be written as the vector

equation,

fuGy += ~ (15)

Where the matrix G is composed of the step response

parameters g i 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 direct

problem of linear algebra, assuming there are no

constraints on the control signal, which leads to:

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

Where, ris the reference trajectory vector.

3. CLOSED LOOP RELATIONSHIPS

Closed loop relationships can be obtained for the

GPC in order to show how the tuning parameters N1,

N2, might affect the stability of the controlled plant.GPC is a Receding Horizon controller so therefore

only components i of the first row of the matrixequation (17) were considered. They can be rewritten

as,

2

2

2

22

1

1

1

1

Nt

N

i

iN

i

t

N

i

iiti

N

i

i

rqC

yFuqC

+=

+

=

=

+

=

+

(18)

With definitions for the polynomials R, Sand T1, (18)

is given by,

21 NtttrCTSyuR ++= (19)

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

loop relationship was obtained,


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t

c

Nt

c

tCP

Rr

CP

BTy += + 2

1 (20)

Where, the characteristic polynomial can be defined

as :

( )

(

( )

c

N

i

i

ii

N

i

iii

CP

qGAB

AC

qBFA

CABSqRA

=

+=

+

+=+

=

=

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-N1+1=N, the first row of u~

is

given by,

xNg 100...00

1

(23)

Therefore, the characteristic polynomial becomes,

( )0

0

0

1

g

BCgAB

gAC =

+ (24)

Consequently, close loop poles are the zeros of the

plant model.

4. SIMULATION EXAMPLE

The following non-minimum phase system was

considered to show the controller behaviour.

1

1

1

9.01

5.11)(

+=

q

qzG (25)

Figure 1, shows the close loop pole when the

Prediction horizon (N2) was changed from N2=1 to

N2=100 with Nu=1 and the control weight =0.When N2=1 the controller has its pole in q = 1.5 , it

has the same value as unstable zero of the plant

model, with an unstable behaviour. Consequently, to

improve the performance, the prediction horizon was

increased. For anN2 value close toNu, the close loop

has an unstable pole, ifN2 is much longer thanNu the

pole will be inside the unit circle. Note that theclosed loop pole tend towards the open loop pole as

N2 is increased.

Fig.1 Pole placements vs. Prediction horizon

Fig.2 Pole placements vs. Control horizon

In order to show how the pole placement was

influenced by changes in the control horizon, it was

changed from Nu=1 to Nu=N2=10 with =0,Figure 2,When Nu was increased the close loop pole that was

initially inside the unit circle changed it placement.

For control horizon near the prediction horizon the

pole increases, producing an unstable pole when Nu

has the same value asN2.

The control weight parameter has the function of

including a penalty on the control signal of the

multistage cost function. Figure 3, shows the

maximum closed loop pole magnitude obtained for

the control weight change from =0 to =10 , using


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the same prediction horizon and control horizon; the

line of crosses represents the behaviour with an

horizon N2=Nu=5 and the dashed line an horizon

N2=Nu=6. Despite the horizons having the same

value, when the control weight parameter wasincreased the maximum pole placement of the

controller was changed tending to inside the unit

circle. With an horizon value N2=Nu=6the maximum

pole is more stable than N2=Nu=5. The Figure 3,

shows how more dynamic information produces

better stability in the system.

Fig.3. Pole Placement vs Control weight ().(x)N2=Nu=5, and (..)N2=Nu=6.

To observe the controller performance with twodifferent sets of tuning parameters a unitary set-point

change was produced. Figure 4, and Figure 5, show

the results; as can be observed the GPC with a high

value of control weight produces a soft control signal

as well as a soft output process.

Fig.4. Output process with two different tuning

Parameters. (x)N2=Nu=5, =10 . and(-)N2=Nu=6, =1.

Fig.5. Control signal with two different tuning

Parameters. (x)N2=Nu=5, =10 . and(-)N2=Nu=6, =1.

Figure 6, and Figure 7, present the output process and

the control signal for comparing the controller

performance when different prediction and control

horizons are used for tuning the controller, compared

to dynamic behavior using the same prediction and

control horizons. Figure 6, shows how a suitable

difference between the prediction horizon and the

control horizon with a small control weight

parameter can produce fast overdamped behaviour.

The Control signal in Figure 7, shows that fast

dynamics in the output process needs fast movementin the control signal. Special requirements in the

control elements can need a slow movement in the

control signal, this may be accomplished by

increasing the control weight parameter.

Fig. 6. Output process with different prediction and

control horizons strategy. (- -)Nu=2,N2=10,

=0.8, (-)N2= Nu =6,=1


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Fig. 7. Control signal with different prediction and

control horizons strategy. (- -)Nu=2;N2=10,

=0.8, (-)N2= Nu =6,=1

5. CONCLUSIONS

This paper gives some ideas about tuning a General

Predictive Controller for non-minimum phase

systems. The way in which the It was presented how

presence of unstable zeros in the process model can

produce unstable dynamics in GPC has been present.

Appropriate selection of the difference between the

control horizon and the predictive horizon produces a

stable performance. In spite of that the controller

having stable behaviour, the control weightparameter produces soft moving of manipulate

variable.

In summary, small values of control weight

parameter with a convenient difference between

control prediction and control horizon, it seems to

work well for non-minimum phase systems.

REFERENCES

Bardley, R., and M. Morari (1985). Design of

Resilient processing Plants. Chemical

Engineering Science,40, 59-74.

Bitmead, R., M. Gevers, and V. Wertz (1990).

Adaptative Optimal Control. Prentice Hall,

Australia.

Camacho, E.F., and C. Bordons (1999). Model

Predictive Control. Springer.

Clarke, D., C. Mohtadi and P. Tuffs (1987a).

Generalized predictive Control-Part I. Basic

Algorithm .Automatica,23 , 137-148.

Clarke, D., C. Mohtadi and P. Tuffs (1987b).

Generalized predictive Control-Part II

Extensions and Interpretations. Automatica, 23,

149-160.

Ogunaike, B., W. Harmon (1994). Process

Dynamics. modeling, and Control. Oxford

University Press, New York.


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