Nonlinear Adaptive Control of an Electroslag Remelting Process

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NONLINEAR ADAPTIVE CONTROL OF AN ELECTROSLAG REMEL TING PROCESS B. Castillo and Ja. Alvarez IJ l'jillllillll l' lI/O

r/I' /lI p,n l infll F!h / rim. .)·('( ci li ll r/f' COII/ro! A II/o///(ilic(). Ci 11 ; '1'.\ / 111'-/ PS, AY. /-1 -7-1 0 . 07()()O

Abstract. Automatic control of an electroslag remelting process allows to improve its productivity under certain operating conditions. The complexity of the model due to nonlinearities and time varying parameters ma kes difficult the design of control algorithms. A nonlinear control technique is proposed and in order to take into account parametric changes, this scheme is adaptive. The simulation results show the adequate performance of this scheme.

Keywords Nonlinear control systems, adaptive control, feedback,metallurgical industries, electroslag remelting process.

INTRODUCTION The electroslag remelting process (ESR) is one of the most important processes which have been developed for the production of special steels and alloys. In order to assure product quality and to optimize production rate, the ESR process must be closely controlled . The control of the melting rate is crucial for maximization of the furnace life and also to obtain ingots with optimum structures. On the other hand a change in the electrode, immersion in the slag pool, due to current changes, is undesirable since it usually results in poor ingot surface. The up and down motion of the electrode must be replaced by a control scheme where a change in current will result in a change in voltage but no change in the relative position of the electrode in the slag pool. That is, a control of the electrode - ingot distance is needed. One of the main problems to achieve suitable control schemes with the ESR process is the complexity of its mathematical model. In particular, nonlinearities and time varying parameters make difficult the design of control algorithms. There are some papers dealing with the design and implementation of control schemes for controlling the melting rate and the electrode - ingot distance. [1] The control techniques used correspond to classical and optimal approaches. These aproaches assume a time invariant linear model. However, some problems arise with this assumption and a nonlinear control technique could be much more adequate. The main objective of this work is to present a recently developed adaptive nonlinear control technique that improves the results obtained with the ones mentioned above. The control algorithm that we present is able to take into account parametric changes in the process as well as process nonlinearities.

,\U\IC(),

D.F.

MODEL OF TH E PROCESS For control purposes the model utilized must have a very simple mathematical structure. On the other hand, that model must still describe the main process variables, with enough precision. A common practice in a process control development is to have first a very complete model of the process and then to obtain a simplified model for control purposes. This simplified model must allow to bu ild a control law that attains the objective of getting a high quality product, by controlling the most influent process variables. A complete model of this process can be found in [2]. In this paper we will present the reduced model based on the mechanical-electrical, thermal and metallurgical stages. This model assumes the following hypothesis are valid. H 1. The input flows to the jacket are considered constant, as well as the temperatures of the water flowing out the jacket H2. The ingot and electrode temperatures are kept constant. We select the states, inputs and outputs of the system in the following way: Xl

U1

= Du

;

X2

= Dei

= Lea

Y1

= X2

Y2 =

X3

where : De, : electrode inmersion in the slag pool Dei : electrode-ingot distance

T, : slag temperature

T/ melting temperature in the slag Le, : distance from the furnace base to the top of the electrode. V : voltage applied to the electrode.

and the reduced model in state variable representation is written as:

X3

+ b 1U1 = a2x3 + b2U ( )() aaZlz,+a. = a 3 X3 + a7 + aSX3 xl + x2 + a6 + z.[alo-al1(z.+a,)!U2

Y1

= x2

Xl X2

.

= a1x3

222

B. Caslillo and J a. Al\"arez (6)

Table 1 shows the values for the nominal process parameters.

10- 7

/18

= 0.12397 X 10- 5

/19

=

/11

= -2 .9:1:16281

/12

= 2.185:1298 x 10-7

/13

=

/14

= 1809

= 1.3704 x 10-3 /111 = 2.61 x 10-3

/15

= -0.2516771

/112

/16

= -7.1889544

/17

=

X

-0.0414147:15

yields the closed loop decoupled linear system

2.7079917

/110

=

[1] ~ []

2.03 x 154

bl = 1.0 b2 = -1.3333

-369.3006605

table 1

The feedback law (6) is referred to as a static-state feedback Iinearizing control law.

BASIC THEORY

If A(x) is singular, linearization may still be achieved using dynamic state feedback. (3)

Exact IInearlzatlon technlgues Among the entire class of non-linear systems, a large class can be made to have linear input-output behavior via a non-linear state feedback control law. A brief review of the recent theory is given here. Consider the m-input m-output nonlinear system. m

X = f(x) +

L gj(x)Uj

(1)

j=l

I"

= h,(x) i

(7)

(2)

= 1, ... , m

Minimum phase Non-LInear Systems

In this section we review some definitions due to Isidori (4) Consider the system given by equations ( 1), ( 2). If A(x) is nonsingular and rt + r2 + ... + rm = r < n, then it can be shown that at cach xOElR n there exists a neighborhood Uo of XO such that a news set of (partial)coordinates in the state space can be chosen as

with ulR n ; UElR m and f,g"h, smooth. We differentiate the output 11. with respect to time, to get.

this set may be completed with functions '11, . . . , '1n-r defined as Z2 = ['Il. .. . , '1n - r] where Lp& stands for the Lie derivative of s with respect to p. If Lgjh, = 0 for each i we derivate the output until at least one of the inputs appear in 11;;, i .e. m

L (Lr;-lh') 11,r;-Lr;h - f • +" ~ gj f \ Uj

. 1 \ = , .. . , m

(4)

j=l

such that the mapping ~ = [Z1. Z2]T : UO diffeomorfism onto its image. (2)

-+

!Ji'n is a

In the new coordinates Z = [Z1. Z2]T ,the system (1 )-(2) may be written in the normal form as the system

~l,l = ~12

with at least one of the Lgj(L'j,-lh;) # o. Equation (4) can be written as.

~l,r-l = ~l,rl m

~l,rl = LJ' hl + L Lgj(LT

l

htJUj

;=1 (8) (5)

where the matrix on the right side of equation (5), which we note as A(x) is called the decoupling matrix. If A(x) is nonsingular, (r1. ... , rm) is defined as the (vector) relative degree of the system (1 )-(2). It is clear from equation (5) that the control law

~m, l = ~m,2 m

~m,r_ = L'j-h m + LLgj(L'j--lhm)u;

;=1

(9)

COlllrol of all Elenroslag Rcmdlillg Process x = 0 is an equilibrium point of the undriven system ) (2). i.e. !(O) = 0 and hi(O) = 0 i = 1, ... ,m, then the ynamics

If we define?

le

=

=0 -

°

as the parameter error vector

(e', eg;)£!Rnf+En.;I, then after some calculation,

(15) yields

Z2 = W(0,Z2)

(16)

; defined as the zero-dynamics of system (1 )-(2). Je say that the nonlinear system (1 )-(2) is "minimum hase" if its zero dynamics are asymptotically stable, 1en the control law given by (6) results in bounded acking provided that v given by

; stable, i .e., the polynomial lurwitz.

.r;

+ a,1. r ;-1 + ... + a,r is

his is, Z1 is bounded and le reference IIm,'

11,

asymptotically tracks

where

and

•

Idaptlve control of lInearlzable systems

'he control law (6) yields an exact linearization only • there exists exact cancellation of nonlinear terms. jowever, if there is an uncertainty in the knowledge If the parameters of f and g, the cancellation is not !xact and the resulting input-output behaviour will lot be longer linear. To solve this problem, a paraleter adaptive control scheme to attain asymptotic :ancellation is proposed. The following explanation ntends to make the methodology clear. . The Relative degree one case ri - 1 ::;onsider the system given by (1 )-(2) with Lg;hi(x) f. 0 or some j and for each i = 1, .. . , m (Relative degree " = 1) and let !(x) and g;(x) have the form.

The control law for tracking is v = IIm

+ a(lIm -

11)

(17)

which leads to the following error equation e=wT~+ae

(18)

where e = 11 - IIm The next theorem gives the results about asymptotic parameter adaptation.

Theorem [1]

nf

!(x) = LO,d;(x)

(11)

,=1

n" g;(x)

= LIJg;,g;;(x)

j= 1, ... ,m

(12)

Consider a minimum phase nonlinear system with f and g as (11)- (12) and the control law given by (15), (17). Asume A(x) nonsingular. Then, if IIm; is bounded, the parameter update law

,=1

(19)

with 0" , Og" , ... , Og_, unknown parameters and I;(x),g;,(x) known functions. If we consider 0" and Og;, the estimates of the parameters, then the estimates of !(x) and g;(x) are given by: nf

!(x) = LO,di(x)

(13)

yields bounded lI(t) asymptotically converging to IIm(t). Furthermore all state variables x(t) are bounded. The proof of this theorem is given in [1] so that we will omit the details. Also, the extension to higher order relative degree systems is straightforward, though involves more complicated calculations.

i=l

n,;

g;(x) = L Og;;g;,(x)

APPLICATION TO AN ELECTROSLAGj = 1, ... , m

(14)

,=1

REMELTING

PROCESS

Also, the control law takes the form:

(15)

In the following, we apply the results presented above to the process described before. After some easy calculations, we get.

B. Castillo and .Ia ..-\harl'1

224

=>rt=l

Then, after some calculations, the control law ( 15) takes the form:

0 ] A(x) = [ 0 b1 a.%'%2+a. => A(x) %2(a,o all(%.+a.))

is nonsingular. By choosing Zl

= "'1 = Xz

;

= "'z

Zz

= X3

the system takes the form:

= azzz + bZ U1 %Z = a3zZ + ~(a7 + aszz)(bz z 1 + bdzz - Z3)) + as

%1

These equations, together with (16), (17), (18) and (15) provides the nonlinear adaptive scheme.

Simulation results %3 =

Y1

= Zl; yz = zz

Thus, the zero dynamics is described by %3 = zz(O, Z3) = 0, which is asymptotically stable. As it was shown, the model is not linear with respe~t to some parameters, so we propose a reparametrlzation to get a model which is linear with respect to the all set of parameters. We have chosen

so that

I, gl

A set of simulation results was carried out with the model given and the equations described above, which form the nonlinear adaptive scheme. Due to the fact that there are twelve parameters, we have simulated the variations of the parameter which explicity affect the outputs behaviour, namely, 9f" i = 3, ... ,8, and 912, 9g z1 , 9g 22. In figure 1 we have introduced variations in the parameters according to table 1

gZ have the form:

and

8

I(x) =

L 9f.l;(x)

•=1

We have omitted variations an 9fs because in some previous experiments, we have seen that the process is extremely sensitive to variatios on this parameter . In figure 1.a and 1.b we show the outputs due to this set of changes, and figures 1.c, 1.d show the inputs to the process. As we can easily see, the process has a large sensitivy to variations in the parameters, but the outputs tends to maintain their set value.

with

In figure 2 we show the case in which only one parameter (9/3) varies 10% about its nominal value. It can be seen that the outputs are well regulated after a certain number of iterations.

It =

fs =

[~l ;fz = [Zi 1;h = [!] ;14 = [;] ;

[~] . Is = [ +~ zz ]; h = [~] ;/8 = [ ~ ] Zz Z3(Zl + zz) l'

Zl

z

gd x ) =

L 99, .gli (x) ;; .=1

z

gz(x) =

L8gz .gz.(x) :

.=1 9g21

= a8al1;

9gzz

= -a9al1

Conclusions From the result obtained by using the nonlinear adaptive scheme, we can concluded that this scheme, provides a good performance when we treat the case of changes in few parameters. This is due to the relative complexity of the model, which is nonlinear with respect to some parameters. Even though, when one considers changes in the entire set of parameters, it is still possible to make a reparametrization of the model to get a model linear with respect to the parameters. We have shown that doing this, it is possible to get an adequate behavior of the system when all the parameters vary.

225

Control of an Electruslag Re melting Process REFERENCES R.J. Roberts. "A System for the Automatic Meaement and control of Melt Rate During ESR". Proc. :he Fifth International Symposium on ESR and otSpecial Technologies, 1974. p. 425. J. de Leon. "Control and Estimation of an ESR cess" (in Spanish.) M. Sc. thesis. Research and lanced studies Center. Mexico, 1987. S.S. Sastry, A. Isidori; "Adaptive Control of LiHizable Systems". Memorandum No. VCB/ERL :7/53. University of California, Berkeley, March 38. A. Isidori (1985). Nonlinear Control System: an -eduction. Lecture Notes in Control and Informal Sciences. vol 72 Springer Verlag.

parameter r of i terations ntage of change

0.03 0.02

-0. 02~ ________- r __________~ ________- ' ___

o

5

1000

1500N

o

(}f3 (}/4 (}/5 (}/1 (}/8 (}g12 (}g21 (}g22

50 60 70 80 90 100 1 10 120 2

2

2

2

2

2

2

2

(%)

-3000

table 1 -4000

o

1000

Figure 1. System Response for variations in the parameters given on table 1.

Y, (m)

Y,

O.03X)4

(m)

O.OnJ2

0 .03

0029!XJ.

10'00

•

1500 002996 0

/. h) .3

lOO

200

300

400

N

Y2 (kg/s)

0.11

>2 (Kg/s) O. lce

o

0 .106

·3

o

500

10 0

1500 N

0 .104 L-----r-----r-----,r------:~

o

700

200

300

Figure 2. System Outputs for variations of 10% on (}/3

~