two degree-of-freedom of generalized predictive control based on polynoial approach using genetic...

7/31/2019 Two Degree-Of-Freedom of Generalized Predictive Control Based on Polynoial Approach Using Genetic Algorithm

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Two Degree-of-Freedom of Generalized Predictive Control Based on

Polynomial Approach Using a Genetic Algorithm

Akira YANOU

Abstract Generalized Predictive Control (GPC) achieves arobust tracking for step-type reference signal by including anintegrator in advance. Although author has proposed a designscheme of two degree-of-freedom GPC system which revealsan effect of integral compensation only if there exists modelingerror or disturbance, a gain for integral compensation mustbe selected by trial and error. In this paper, a new schemeof two degree-of-freedom of GPC system based on polynomialapproach is obtained by using a genetic algorithm for selectionof integral gain.

I. INTRODUCTION

Generalized Predictive Control (GPC) has been first pro-

posed by Clarke and others[1]. The controller is derived

by performance index, which includes parameters of pre-

diction horizon, control horizon and weighting factor. These

parameters are an interval of predicting a behavior of future

output based on a nominal model, an interval of calculating

optimal future inputs and a parameter on control input re-

spectively. The control signals are derived by minimizing the

performance index on future control inputs and re-calculated

receding from their horizons at each sampling time.

With these features, the control strategy follows the step-

type reference signal robustly by including an integrator in

the controller and it has been accepted by many of practical

engineers and many papers have been proposed[2], [3], [4],[5]. Whereas, if the controlled plant is modeled accurately

and there is no step-type disturbance, the controller can track

to the step-type reference signal without an integrator in it.

And the effect of an integral has the possibility of the change

for the worse of the characteristic of the transient response

or the increase of the control input. Therefore it is desirable

that an effect of an integral compensation appears only if

there exists a modeling error or a disturbance.

In this paper, this feature is defined as two degree-of-

freedom system because the characteristic of the output

response and the disturbance response can be designed

independently, that is, on one hand the characteristic of the

output response is designed by minimizing the performanceindex which includes a control input and a tracking error

between the reference signal and the output, on the other

hand the characteristic of the response by the disturbance is

designed by the gain of the integral compensator.

Although many papers have proposed two degree-of-

freedom optimal servo systems[6], [7] and the author has

proposed two degree-of-freedom of generalized predictive

A. YANOU is with School of Eng., Kinki University,1, takayaumenobe, Higashi-Hiroshima 739-2116, [email protected]

control based on polynomial approach for single-input

single-output systems[8], [9], [10], there is a problem for

a gain of integral compensation. That is, a gain for integral

compensation, which designs for the disturbance response,

must be selected by trial and error. Therefore this paper

newly proposes two degree-of-freedom of GPC by using

a genetic algorithm[11], [12], [13] for selection of integral

gain. By using a genetic algorithm, a gain for integral

compensation can be determined.

The proposed method is designed by the following steps.

First, although GPC strategy proposed by Clarke and others

has an integral action by including an integrator in theperformance index, it makes the strategy proposed in this

paper difficult, because in this paper the amount of the

integral action must be calculated analytically on condition

that there does not exist a modeling error and a disturbance.

Therefore the controller is first designed with no integral

action in the performance index. Second, a new controller

is designed by adding an integral action to the controller

in the first step. The effect of its integral action can be

designed by a gain introduced in the controller at once. This

controller always reveals the integral action and it may have

an extra integral compensation because the integral action

is merely added to the controller designed in the first step.

In the third step, two degree-of-freedom GPC is obtained

by calculating the amount of the integral for the controller

designed in the second step and subtracting its amount from

the control input designed in the second step. When there is

no modeling error and disturbance, its controller generates

the control input designed in the first step, which has no

integral compensation. And it reveals the effect of the integral

compensation in the case that there exists modeling error

or disturbance. Finally, a gain for integral compensation is

selected by using a genetic algorithm, that is, the proposed

controller is given.

When there is neither modelling error nor disturbance, the

proposed controller generates the same control input as onederived in the first design step with no integral compensation.

And the proposed controller shows an effect of integral

compensation in the case that there exists modelling error

or disturbance. This paper is consisted as the followings. In

section II, two degree-of-freedom of GPC based on poly-

nomial approach is shown. In section III, a search method

of a solution of integral gain is shown by using a genetic

algorithm. In section IV, numerical examples are shown

to verify the validity of the proposed method. Finally the

conclusion is given.

Proceedings ofthe 2009 IEEE International Conference onNetworking, Sensing and Control, Okayama, Japan, March 26-29, 2009

978-1-4244-3492-3/09/$25.00 2009 IEEE 492


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II. DESIGN SCHEME OF TWO DOF OF GPC

Two degree-of-freedom of GPC is designed by three steps.

In the first step GPC is designed without an integral action in

the performance index. The controller given by this step can

not achieve the control objective because it does not include

an integral action, that is, if there exists modeling error or a

step-type disturbance, a plant output by the controller in the

first step can not track to a reference signal. Therefore, inthe second step, the integral action is added to the controller

given by the first step. Then because of an integral action,

the output by its controller can track to reference signal even

if there exists modeling error or a step-type disturbance. But

the integral action always acts, there may be an extra integral

action, that is, the degradation of the transient response or the

increase of the control input may occur. Therefore in the third

step, by calculating the extra integral action and subtracting it

from the controller in the second step, two degree-of-freedom

of GPC controller is given. The controller in the third step

appears the effect of an integral compensation only when

there exists modeling error or a step-type disturbance.

A. PROBLEM STATEMENT

Consider a single-input single-output system,

A[z1]y(t) = zkmB[z1]u(t) (1)

where y(t) and u(t) denote the output and the input. A[z1]and B[z1] are the n-order and m-order polynomials respec-tively.

A[z1] = 1 + a1z1 + + anz

n

B[z1] = b0 + b1z1 + + bmz

m

The integrator w(t) is given by the following equation,

w(t) =1

e(t) (2)

where

= 1 z1

A tracking error is a signal between the reference signal r

and the output, and given as follows.

e(t) = r y(t)

z1 denotes backward shift operator: z1w(t) = w(t 1).The control objective is that the output y(t) tracks thereference signal r.

B. GPC WITHOUT INTEGRAL COMPENSATION

At first the prediction y(t) is given for the deviation systemof the plant (1). The steady state values y of y(t) and uof u(t) are derived as follows.

A[z1]y = zkmB[z1]u

From the previous equation of the steady state values, the

deviation system of the plant (1) is obtained,

A[z1]y(t) = zkmB[z1]u(t) (3)

where the deviations y(t) and u(t) are defined as followsrespectively.

y(t) = y(t) y

u(t) = u(t) u

The prediction for (3) can be derived by the Diophantine

equation,

1 = A[z1]Ej [z1] + zjFj [z

1]

Ej [z1]B[z1] = Rj [z

1] + zjSj [z1]

where

Ej [z1] = 1 + e1z

1 + + ej1z(j1)

Fj [z1] = f

j0 + f

j1z1 + + fjnz

n

Rj [z1] = r0 + r1z

1 + + rj1z(j1)

Sj [z1] = sj0 + s

j1z1 + + sjm1z

(m1)

In order to find j-ahead prediction y(t +j|t), j-ahead outputy(t +j) is derived.

y(t +j) = Rj [z1

]u(t +j km) + hj(t)where

hj(t) = Fj [z1]y(t) + zkmSj [z

1]u(t)

Because it is assumed that there exists no perturbation, j-

ahead prediction is derived as y(t + j|t) = y(t + j). Thenthe vector form of output prediction at each sampling time

and the series of future input, and the matrices R and H are

defined as follows.

Y = RU + H (4)

where

Y = [y(t + N1|t) y(t + N2|t)]T

U = [u(t) u(t + Nu 1)]T

H = [hN1(t) hN2(t)]T

R =

r0 0 0...

. . . 0rN21 r0

[N1, N2] and [1, Nu] denote the prediction horizon and thecontrol horizon respectively. To simplify, in this paper it is

assumed that N1 = km = 1 and N2 = Nu.The performance index for the deviation system (3) is

considered under the condition of y(t +j) = y(t +j|t).

J =

N2j=N1

y2(t +j) +

Nuj=1

u2(t +j 1)

And the performance index can be rewritten by the following

vector form.

J = (RU + H)T(RU + H) + UTU (5)

Minimizing the performance index J on U, the control law

is derived.

U = (RTR + I)1RTH

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Because this equation is about the deviation system of the

plant (1), the control law of the plant (1) is obtained as the

following equation.

u(t) = H0(z1)r F0(z

1)y(t) (6)

where

H0(z

1

) =

Fp[z1] + (1 + zkmSp[z

1])K

1 + zkmSp[z1]

F0(z1) =

Fp[z1]

1 + zkmSp[z1], K =

A[1]

B[1]

[pN1 , , pN2 ] = [1, 0, , 0](RTR + I)1RT

Fp[z1] =

N2j=N1

pjFj [z1]

Sp[z1] =

N2j=N1

pjSj [z1]

C. GPC WITH INTEGRAL COMPENSATION

When there exists modelling error or step-type distur-bance, the control law (6) can not achieve the control

objective and leaves steady state error between output and

reference signal because it does not have an integral com-

pensation.

Therefore the integral compensation G0w(t) is added tothe control law (6), where we denote an integral action as

w(t) and an integral gain as G0. Then GPC controller withan integral action is derived as follows.

u(t) = H0(z1)r F0(z

1)y(t) + G0w(t) (7)

The control law derived here can achieve the control objec-

tive, that is, the output can follow to the reference signal

even if there exists modelling error or step-type disturbance.

D. TWO DOF CONTROLLER

Because in the control law (7) the integral compensation

always acts, it may cause an extra input or a worse change

of the transient response. That is, the two degree-of-freedom

of GPC is desirable, which has the effect of the integral

compensation only if there exists modelling error or dis-

turbance. The two degree-of-freedom of GPC proposed in

this paper has the ability to adjust the characteristics of the

output response and the disturbance response independently.

In this subsection two degree-of-freedom of GPC controller

is designed. At first, the control law is expressed as follows

by including an integral action z(t) and a gain G0.

u(t) = H0(z1)r F0(z

1)y(t) + G0z(t) (8)

where H0(z1) and F0(z

1) is the same coefficients asin (6). Second, assuming that there is neither modelling

error nor disturbance, the tracking error e(t) = r y(t)is calculated. By this assumption, the control law expressed

here becomes the same controller of (6). Then the closed-

loop system by the control law (6) is obtained,

y(t) = T(z1)r

where T(z1) is the transfer function of the closed-loopsystem.

T(z1) =Tn[z

1]

Td[z1]

Tn[z1] = zkmB[z1]{Fp[z

1]

+(1 + zkmSp[z1])K}

Td[z

1] = A[z

1](1 + z

kmSp[z

1])+zkmB[z1]Fp[z

1]

If there exists no modeling error, then the tracking error e(t)is given by the following equation.

e(t) = (1 T(z1))r (9)

Now the integral compensation z(t) is derived by (2) and(9).

z(t) = w(t) 1

(1 T(z1))r

By using the previous z(t), if there exists no modeling erroror a step-type disturbance, z(t) is always equal to zero. Thatis, the effect of the integral compensation does not appear.

III. SEARCH BY A GENETIC ALGORITHM

In the previous section, the two degree-of-freedom of GPC

is given. This section considers to give a solution of integral

gain by a genetic algorithm[11], [12], [13].

In the first step, each genotype of individual G0 represents

in binary as strings of n bits. Each phenotype of individual

G0, that is, the integral gain G0 is calculated by converting

binary representation to decimal representation and corre-

sponding to a designed range.

In the next step, the fitness function to search the solution

of G0 is defined as integral of square error between the

reference signal and the output,

f(G0) =

Nt=1

{r yG0(t)}2 (10)

where yG0(t) is the plant output using G0 searched by agenetic algorithm. In this paper the solution of G0 is searched

by a genetic algorithm so that the fitness function f(G0)becomes small.

Therefore a disturbance response is given in the sense ofminimization of f(G0). And the genetic algorithm in thispaper is calculated as following steps and repeated until a

decided number of generations.

1) Initialization of population of genotype of G0. The

number of generation is set to be 1.

2) Evaluation of f(G0). It is repeated until a decidednumber of generations.

3) Creation of population of genotype of next generation

by selection, crossover and mutation.

4) The number of generation is added to 1. Go to step 2.

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IV. NUMERICAL EXAMPLE

In this section the numerical examples are shown to verify

the validity of the proposed method.

Consider the controlled plant,

y(t) =z1(1.2 0.1z1)

1 1.0z1 + 0.25z2u(t)

Simulation steps are 200, the initial values of the output and

the input are assumed to be zero . The disturbance as thewhite gaussian noise with the variance 2 = 0.03 is addedto the controlled plant output and each design parameter is

given by follows.

N1 = 1, N2 = 10, Nu = 10, = 1

The reference signal is a rectangular signal with amplitude

1 and the period of 50 steps. The parameters of the genetic

algorithm are given as follows. The number of individuals is

set to be 50. The integral gain G0 is represented in binary as

strings of 8 bits, and its phenotype is in the designed range.

The number of generations is set to be 50. The selection

is chosen to roulette rule and the crossover is one-point

crossover. The probability of mutation is set to be 0.1.The numerical example gives the minimal value of fitness

function to 17.532 and the integral gain to 0.1914.

Fig.1 shows the integral gain calculated by the genetic

algorithm and fitness value is shown in fig.2. Fig.3 and

fig.4 show the plant output and input by the integral gain

calculated by the genetic algorithm. It is shown that the

proposed method can track to the reference signal by G0calculated by the genetic algorithm.

Fig.5 and fig.6 show the comparison among the controllers

(6), (7) and (8). The controlled plant, simulation steps, the

initial values of the plant, design parameters and the integral

gain G0 are the same as the previous example. The reference

signal is a step-type signal with amplitude 1. The disturbance

as a step-type signal with amplitude 0.5 is added to the

controlled plant output after 100th step.

In fig.5 the solid line shows the output by the two degree-

of-freedom of GPC (8) proposed in this paper, the dotted line

shows the output by the GPC with integral compensation (7)

as z(t) = w(t), and the broken line shows the output bythe GPC without integral compensation (6). Fig.6 shows the

control input and each type of line is the same as Fig.5.

In Fig.5 the proposed method can track to the reference

signal and the overshoot in the transient response can be

controlled less than the controller (7) as z(t) = w(t). Inaddition the transient response proposed in this paper is the

same as the one by (6). It is shown that the effect of integral

compensation of the proposed method does not appear if

there is no disturbance. And it is also found that the effect

of integral compensation of it appears because there exists

disturbance after 100th step. That is, two degree-of-freedom

of GPC can be designed by the integral gain calculated by

the genetic algorithm.

V. CONCLUSION

In this paper, a selection method of integral gain for two

degree-of-freedom of GPC based on polynomial approach is

0.1

0.2

0.3

0.4

0.5

0.6

5 10 15 20 25 30 35 40 45 50

integral

gain

generation

Fig. 1. Integral gain

17.4

17.45

17.5

17.55

17.6

17.65

17.7

17.75

17.8

5 10 15 20 25 30 35 40 45 50

error

generation

Fig. 2. Fitness value

given by using a genetic algorithm. And numerical examplesare shown to verify the validity of the proposed method.

Although the gain for integral compensation must be selected

by trial and error in the conventional method, this paper gives

its selection method.

As future works, in the case that the proposed method is

applied to a general plant, it is necessary to consider aged

deterioration. That is, the integral gain must be re-calculated

periodically or online. There are also an extension to multi-

input multi-output systems of the proposed method and to

design a self-tuning controller.

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-1

-0.5

0

0.5

1

20 40 60 80 100 120 140 160 180 200

plantou

tput

step

Fig. 3. Plant output

-1.5

-1

-0.5

0

0.5

1

1.5

20 40 60 80 100 120 140 160 180 200

plantinput

step

Fig. 4. Plant input

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[1] D. W. Clarke, C. Mohtadi and P. S. Tuffs, Generalized PredictiveControl, Automatica, Vol. 23, No. 2, pp.137-160, 1987

[2] H. Demircioglu and P. J. Gawthrop, Continuoustime GeneralizedPredictive Control, Automatica, Vol. 27, No. 1, pp.55-74, 1991

[3] B. Kouvaritakis, J. A. Rossiter and A. O. T. Chang, Stable generalisedpredictive control:an algorithm with guaranteed stability, Proc. IEE,Vol. 139, No. 4, pp.349-362, 1992

[4] E. F. Camacho and C. Bordons, Model Predictive Control in theProcess Industry, Springer, 1995

[5] M.Deng, A.Inoue, A.Yanou and S.Okazaki, Stable anti-windupcontinuous-time generalised predictive control to a process controlexperimental system, The Journal of the Institute of Measurementand Control, Vol. 40, No. 4, pp.120-123, 2007

[6] Y. Fujisaki and M. Ikeda, Synthesis of Two-degree-of-freedom Opti-mal Servosystems, Transactions of SICE, Vol. 27, No. 8, pp.907-914,1991

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

20 40 60 80 100 120 140 160 180 200

plantou

tput

step

Fig. 5. Plant output

-0.2

0

0.2

0.4

0.6

0.8

1

20 40 60 80 100 120 140 160 180 200

plantinput

step

Fig. 6. Plant input

[7] T. Hagiwara, M. Ichiki, M. Kanaboshi, K. Fukumitsu and M. Araki,Digital Two-degree-of-freedom LQI Servo Systems -Design Methodand Its Application to Positioning Control of a Pneumatic Servo

Cylinder,Transactions of ISCIE

, Vol. 11, No. 2, pp.51-60, 1998[8] A. Yanou, S. Masuda and A. Inoue, A Design Scheme of Two-Degree-of-Freedom Generalized Predictive Control System in thePolynomial Approach, Preprints of 12th SICE Symposium inChugoku Branch(In Japanese), pp.98-99, 2003

[9] Akira YANOU, Shiro MASUDA and Akira INOUE, Two Degree-of-Freedom of Self-Tuning Generalized Predictive Control Based onPolynomial Approach, Proceedings of SICE Annual Conference 2005in Okayama, pp.1187-1190, 2005

[10] Akira YANOU, Shiro MASUDA and Akira INOUE, Two Degree-of-Freedom of Self-Tuning Generalized Predictive Control Based onPolynomial Approach with Computational Savings, Proceedings of2006 International Conference on Dynamics, Instrumentation andControl, 2006

[11] T. Yamamoto, Y. Mitsukura and M. Kaneda, A Design of PID

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Controllers Using a Genetic Algorithm, Transaction of SICE, Vol.35,No.4, pp.531-537, 1999

[12] H. Iba, The Basis of Genetic Algorithms(In Japanese), Ohmsha, 1994[13] R. Ishida, H. Murase and S. Koyama, The Basis and Application of

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