GMA Welding Process Regulation via Hierarchical GPC with Assured Stability

Copyright © IFAC Algorithms and Architectures for Real-Time Control, Vilarnoura. Ponugal, 1997

GMA WELDING PROCESS REGULATION VIA HIERARCmCAL GPC WITH ASSURED STABILITY

S. G. Tzafestas and E. J. Kyrianuakis

Intelligent Robotics and Automation Laboratory National Technical University ofAthens Department ofElectrical and Computer Engineering 15773, Zografou Campus, Athens, Greece E-mail: [email protected]

Abstract: This paper deals with the regulation of the thennal characteristics of Gas Metal Arc (GMA) welding. A previously developed model is used for the open loop predictions. At the first level of the hierarchy, a parameterized Generalized Predictive Control (GPC) algorithm is selected among other control techniques, due to its robustness against modeling errors and parameter variations. A coordinator, at the second level of the hierarchy, specifies a set of reliable values for the parameters of GPC, so that stability is assured. A representative set of simulation results is included, along an evaluation of the advantages and limitations of the proposed hierarchical GPC technique. Keywords: Predictive control, Hierarchical structures, Regulation, Stabilization, Arc welding.

of a non-linear and non-stationary process which is very difficult to be modeled (Doumanidis, 1994). Since the welding process contains uncertainties and disturbances that appear as model parameter variations, predictive adaptive control techniques (Doumanidis and Hardt, 1991 ; Nishar et al., 1994 ; Henderson et al., 1991) seem to be among the most appropriate techniques for facing the welding control problem.

1. INTRODUCTION

Arc Welding constitutes one of the most important applications of intelligent robotics. Automated welding solves the problems of low manual productivity, dangerous working conditions for people, and constant acceptable quality of seams. The control of welding processes requires simultaneous regulation of the following physical characteristics: - weld bead location relative to the desired seam and bead cross section geometry, thermally induced stresses and strains, metallurgical microstructure and final material properties. Because of the complexity of the overall physical process and the lack of comprehensive specifications, only a few control systems have been developed so far to deal with more than one variables within the previous groups. Particularly, the thermal properties were shown to be the outcome

Generalized Predictive Control (GPC) is a digital control algorithm, the basic principles of which were developed in the industrial process control field in the mid-seventies (Richalet, 1990 ; Keyser, 1990). It is somewhat similar to optimal control but has many attractive features suitable for industrial systems such as: (i) it allows the existence of model uncertainties, (ii) it updates the outputs of the model by closed-loop corrections, (iii) it optimizes the control law on a moving horizon and (iv) it is very fast, allowing for real-time control (Clarke, 1990).

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2. TIffiRMAL MODELING OF GMA

The response of the bead cross section area NS to step inputs (either QI or V) can be approximated by an overdamped second-order behavior, in which one pole clearly dominates over the other, so that NS may be adequately modeled by a first order transfer function, with respect to either QI or V. Consequently, the proposed transfer functions are of the form:

The final microstructure and material properties of the joint are described here by the following welding outputs as suggested by Doumanidis and Hardt: (a) The weld nugget cross section (NS) defined by the solidus isotherm Tm. This is adopted as a collective measure of the extent of the solidification defects, such as porosity, inclusions, incomplete fusion, micro-cracks, micro-segregation and nucleation of undesirable phases. It also characterizes the dilution of the base material with filter material, when a consumable electrode is used. (b) The heat affected zone (HZ) defined by an enveloping isotherm Tb. This may indicate the extent of weak zones, such as the recovery, recrystallization and grain growth areas, or the width of the zone in which some undesirable, thermodynamically favored phase is formed, such as the sensitization zone in stainless steel. It may also characterize the extent of the contamination zone in cases of increased reactivity of the material with its environment during welding. (c) The centerline cooling rate (CR) defined at the critical temperature Tc:

The response of the heat affected zone HZ to steps in either QI or V has a nonminimum phase form. Accordingly, it can be closely approximated by a nonminimum phase second-order behavior of the form:

where the two modes can be associated with the dynamics of the isotherms Tb and Tlib the width difference of which defines HZ. The sensitivity of HZ to the third input 02 is almost insignificant and can be approximated by an almost zero, first order, transfer function.

CR:;::~T=T. This may determine the crystallization of undesirable, kinetically favored phases, or supply a measure of the cracking tendency of the weldment caused by thermal stresses.

HZ (s) :;:: Kb ('tb S + 1) , Q\ ('tlS + I)('t 2s + 1) HZ K' ('t' ·s+I) -(s):;:: , b b , V ('t 1 ·s+I)('t 2 '5+1)

Analytic steady-state modeling of these temperature field characteristics have demonstrated these outputs on three process variables which were proposed as the welding inputs: (a) The heat input QI of the primary torch, (b) The heat input Q2 of the secondary torch trailing at a fixed distance (required to separately modulate HZandCR), (c) The common velocity V of the two torches (see Fig. I for the double torch welding configuration).

A To

m

~~ T !

HZ K" -Cs) = " b ~0 Q2 't2 · s + 1

(lb)

Finally, the response of the centerline cooling rate CR to steps in either QI or V may be approximately described by an over-damped second order behavior owing to the existence of two dominating real poles. The dependence of CR on the third input Q2 is also of second order, but with the one pole dominating over the other, so that it can be reduced to a first order transfer function:

..... . . . .. . .... .....• •. . . . ..•..T.m ... •. ... \ .

... . . "r . .... ... .. j:

::

~ ~.~.

C'

:

I

. .

· ····· r ·~~·

CR (s):;:: Kc , Q\ ('t.s+ l)('tbs+ 1) CR (s):;::, K~ . V ('t a ' 5+I)('tb '5+1)

Fig. 1. The double torch welding configuration

CR (s) = • K:

Using the step-input identification technique, a linear low-order model of the welding dynamics was produced. The dynamic model can be linearized in the neighborhood of the nominal conditions for limited ranges of variation of the inputs, and described in terms of low-order transfer functions.

Q2

(lc)

't a ·s+ 1

The values of the gains of all the proposed transfer functions were easily determined from the steadystate values of the corresponding outputs and the

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The predictions can then be calculated as:

magnitudes of the steps imposed to the corresponding inputs. The time constants were calculated by a weighted average of those values that make the linearized responses match the experimental data at the corresponding sequence of time instants. The calculated values of the gains and the time constants of the linearized model in the neighborhood of the nominal conditions QI"=2.5KW. V"=5mm1s.
LN

Ym(t+k/t) =LH(J)u(Hk-jlt) j=1 i-I

= LH(k-J)u(Hjlt) j=i-41 !l(i)

= LP(k-J)u(Hjlt) j=i-41

k=I.2 ..-.L y

(4)

where ~(i)=min(k.W-l. H(k - J) P(k-J)= {

(2)

k < Lu or j < Lu -1

~H(k-l) k~Lu andj=Lu-1 I=L.-I

and q(t+klt) is the closed-loop correction vector

based on the available information set at time t. A

3. MODEL BASED PREDICTIVE CONfROL AND TIIE GPC ALGORITHM

recommended form for q(t+klt) is the following: q(t + kIt) = yet) - Ym(t)

GPC is a control algorithm which uses a model for open-loop predictions. optimizing the control inputs on a moving horizon and updating the outputs of the model by closed loop predictions. Especially. at each time 1. the output y(t+klt) is predicted over a future period of time k=1.2 •...•Ly where Ly is the prediction horizon. The predictions are symbolized by Yp(t+klt) and are determined by means of a model. for example state-space model or impulse response matrix (Richalet. 1990). Both models are also convenient for the MIMO case. Here the impulse response matrix model will be used since it is the available model for the thermal process. The predictions Yp(t+kIt). k=1.2 •...•Ly depend on future control values u(t+klt) • k=O.l •....Lu-l. where Lu is the control horizon (Lu~). In the control horizon: u(t+Lu +k/t) = u(t+Lu -1).

where y(t) is the measured value of the output vector at time t. A reference trajectory r(t+klt). k=1.2 •...•Ly is defined over the prediction horizon, which describes how one wants to guide the output vector y(t) to its setpoint w(t). i.e. r(t+k/t)=w(t+k/t)+u(t+k/t) (6) where u(t+k) is a correction vector based on the previous error information set:

{w(t)-y(t). w(t-l)-Yi(t-l) •...• w(l)-y(I)} A simple form which gives good results is the following (Clarke. 1990):

(7)

\>(1+ k/t)=a le [w(t)-y(t)]

k~O

where 0<351 is a tuning parameter that specifies the desired closed-loop dynamic (a-+O: fast control; a~ 1:slow control). The reference trajectory is initiated at the current measured output i.e. r(tlt)=y(t). It is noted that if the future setpoint values w(t+klt). k=1.2 •. ..•LY are unknown at time 1. one can assume that:

The output predictions can be calculated as: Yp (t + kIt) = Ym(t + kIt) + q(t + kIt)

(5)

(3)

where LN

Ym(t+k) = LH(J)u(t+k- j) j=1

w(t+k/t)=w(t). k=L2 •...•L y

is the finite convolution model. {H(j). j=I.2 •.. .•LN} is the set of impulse response matrices of the process. LN is the modeling horizon which by definition implies H(jp.<) when j>LN. and Ym(t+k) is the model output vector. It is worth noting that the previous model is an approximate representation of the linear model:

All the above concepts are illustrated in Fig. 2. The cost function has the form: 1

L,

minJ (t)=-::f 2

L

Ie=L.

2

Ilr(Hklt)-Y p (t+klt)ll~

1Q(1e)

L -I

OD

+

Ymet + k) = LH(J)u(t +k - J)

t

1:=0

j=1

223

Ilu(Hklt)ll!(Ie)

(8)

PAST

coincideDce horizon

r"f-.

•

l

t+Lo l+Lu

KZ, Cll.) T

t+Ly

Fig.3. The 2-level control strategy

Fig.2. The reference trajectory, the set-point trajectory, the prediction horizon and the coincidence horizon

4.1. The where Q(k)~ for k=Lo, ... ,Ly and R(k)>O for k=O, ...,Lu-1. Since J(t) varies with time t and has a moving optimization horizon, only the first term in the optimal solution is implemented to control the process. The optimization parameter Lo determines, together with Ly, the "coincidence horizon" during which the predicted output is to follow the reference trajectory over the time interval [t+Lo, ...,t+Ly] . It is of significant importance to mention the theorem of Clarke and Mohtadi (Bitrnead et al. 1990) concerning the stability properties of GPC: Theorem. The GPC controller results in a stable deadbeat system if : • L~ Lu=I..y-Lo+ I~ where n is the order of the process to be controlled., and • the control signal weights (i.e. the elements of the matrix R) are small enough compared to the weights of the matrix Q • The above properties will be embedded to the coordinator at the second level, along with some appropriate search rules for determining a set of parameters so that stability and other control demands are ensured.

r Level of the Hierarchy t

The impulse response matrix of the proposed model (1)-(2) can be obtained using the inverse Laplace transformation H(t)=L-1 {H(s)} . To this end, the following notation is introduced:

yp (t)=[y p (t+ lIt), ... ,y p (t+ Ly It)]T , Ym (t)=[y m (t+ lIt), ... ,y m (t+L y It)]T r(t)=[r(t+lIt), ...,r(t+L y 1t)]T , q(t)=[q(t+lIt), ... ,q(t+L y It)]T u(t) = [u(tlt),._,u(t +Lu _lIt)]T, w(t)=[w(t+ lit), ... , w(t+ Ly It)] T \)(t)=[u(t+lIt), ...,u(t+L y It)f, ii(t)=[u(t-I), ... ,u(t-L N +I)]T Then, from equations (3)-(7) one obtains:

Ym (t)=H·u(t)+ B·ii(t) Yp (t)= Ym (t)+q(t) r(t)=w(t)+V(t) ,

4. DESIGN OF THE 2-LEVEL CONTROL SYSTEM

where the matrices it: and H are defined in (Clarke, 1990) and so the cost function (8) can be rewritten

as:

Several fundamental requirements are imposed on the control design. Among them are: - asymptotic stability of the closed-loop system, - zero-steady state error, limited settling time, and minimal overshoot in response to command inputs, - robustness to unmodeled dynamics and parameter variations of the process. The proposed structure of the closed-loop predictive control scheme which guarantees stability is illustrated in Fig. 3.

J(t)=nll yp

(t)-r(t)II~ +llu(t)ll~ }

where

Q--diag[O,O,...,O,Q(L o ),Q(L o +1),...,Q(L y )] R;::diag[R(O),R(I),....R(Lu -I)]

224

(9)

5. SIMULATION RESULTS AND CLOSED-LOOP PERFORMANCE

Optimizing the cost function (9) one gets from oJ(t)/oo(t)=O, the optimal solution:

u(t)=[R+B:T .Q.it:]-I[w(t)+U(t)-H.u(t)-q(t)]

Extensive simulation studies have been carried out to

(10)

demonstrate the effectiveness of the 2-level GPC approach. Zero initial conditions (start-up of the operation) were assumed and use was made of both the 2-level GPC approach and the I-level approach, in order to confirm the influence and the increased effectiveness of the closed-loop system under the action of the coordinator. A representative set of results are provided in Figs. 4-6.

Note that the assumptions made imply that J(t) is convex with respect to u(t) with

and so the problem is always solvable with the criterion being minimized. Only, the first block u(tlt) of the extended vector u(t) is used as the control input for the process as it is implied by the moving horizon technique.

I

U l'

I

4.2. The

r

SO

100

T

Level of the Hierarchy

I ••

1 SO

( •• C

I. 4)

lO 0

2S 0

Fig.4(a). Nugget cross section (NS): I-level approach

The efficiency of the GPC controller of the first level depends strongly on the following set of parameters: LN (modeling horizon), Ly (prediction horizon), L" (defines together with Ly the coincidence horizon), L., (control horizon), Q, R (the weights matrices) and the parameter "a" for fast and slow control. According to the Clarke and Mohtadi theorem (Clarke, 1990), the coordinator is built here with the aid of the following search rules: Step 1. Start with LN sufficiently large. Give initial values to the parameter Ita" and the weight matrices Q, R Lo=n (the order of the model). Ly= 2n-I (the minimum value required for stability). L.,=Ly-L,,+I, (the minimum value required for stability). Step 2. If the error history meets the specifications go to step 8. Stml. Increase Ly. ~ Increase L.,. ~. If the error history meets the specifications go to step 8. ~ Increase the weights of Q and reduce the weights of R S!m...1. If the error history meets the specifications go to step 8. Step 8. Reduce the parameter "a" so that the settling time is minimized and the overshoot does not exceed the previous values. ~ End The coordinator is called every time there is a change in the outputs away from their nominal conditions, in order to produce the GPC controller with the set of parameters 9=[L." Ly, Lu. a, Q, R] that best fit the desired closed-loop specifications. The error history e=[Y(l)-w(l), ... , y(tc)-W(tc)]T can be easily produced by an in-process numerical simulation (l( is the simulation time limit) of the GPC controller, while the best choice of the parameters is specified, without having effect on the overall controller performance since this simulation process is very fast.

-.h

II

I

"

I ••

I

so

200

25.

Fig.4(b). Nugget cross section (NS): 2-level approach

.... -.l

lO

I ••

ISO

100

25.

T ••• (", •• 4)

Fig.S(a). Heat affected zone (HZ): I-level approach

I'-

5.

ltt

.,.

T'•• (I.e • • • )

Fig.S(b). Heat affected zone (HZ): 2-level approach

225

1"

.50

.

.,

overall welding process, using robots, the following problems have to be faced: (i) seam following control, (ii) welding characteristics control. The present paper has addressed the second problem for the GMA process through a MIMO hierarchical generalized predictive control technique that assures closed-loop stability. Several simulation tests were carried out which show that the effectiveness and robustness of the technique is much superior than standard optimal control, particularly when large modeling errors exist. Also our MBPC scheme was shown to be much faster than adaptive control (e.g. (Doumanidis, 1991».

.,.

...... · 10

,,,.. ...

I.

L

· 100

· lIt · 120

.so

100

T , ••

I SO

HI

(I, c . . . . )

,>0

Fig.6(a). Cooling rate (CR): I-level approach

.

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, .,. 1\ .

•7 •

· to

u

· 100

r

REFERENCES

·110 · 120 "

00

T Ji • •

J"

( t . c • • ,. )

'OO

"

Bitmead, R, M. Gevers, and V. Wertz, (1990). Adaptive Optimal Control, Prentice Hall International Series in Systems and Control Engineering. Clarke, D.W., (1990). Generalized Predictive Control and its Application. In: Computer

Fig.6(b). Cooling rate (CR): 2-level approach Both model mismatch and uncertainty are assumed to exist in the internal model used by the GPC controller. The initial learning period takes a little time, but satisfactory tuning is achieved after some time period as shown in Figs. 4 through 6 . In all cases, the disturbances influence the thermal process at two different times and the outputs return to their nominal conditions by the action of the GPC controller. In Figs 4(a), 5(a) and 6(a) this is achieved without the second level of coordination with: Ly=5, Lo=2, L,..=3, a=O.5 and Q=I, R=10021, (I is the identity matrix of proper dimensions). In Figs 4(b), 5(b) and 6(b) the same regulation is achieved with the simultaneous action of the coordinator that produces two different sets of parameters for the different disturbances: Ly=7, 0 Lo=2, L,..=4, a=O.4 and Q=I, R=10 31, (for the first distwbance) and Ly=6, Lo=2, L,..=3, a=O.3 and Q=I, 02 R= 10 1 (for the second). One can notice that when the coordinator is used the overall operation is upgraded. The overshooting phenomena are minimized and the system settling time is decreased in all three outputs, whereas zero steady-state error is achieved. Moreover, when modeling errors are involved, the GPC controller was shown to give much more satisfactory results in comparison with pure optimal control techniques. All of our simulation results have verified the fleXIbility and effectiveness of the 2-level GPC approach.

Integrated Design of Controlled Industrial Systems (Richalet, 1., Tzafestas, S. eds), Proc. ClM-Europe Workshop, Brussels, April 26-27. Doumanidis, C. and D.E Hardt, (1989). A model for In-Process Control of Thermal Properties During Welding. In: ASME J. Dyn. Syst. Meas. and Control, VoL 11t, pp. 40-50. Doumanidis, C. and D.E. Hardt, (1991). Multivariable Adaptive Control of Thermal Properties During Welding. In: ASME J. Dyn. Syst. Meas. and Control, VoL 13, pp. 82-92. and Doumanidis, C., (1994). Multiplexed Distributed Control of Automated Welding. In: IEEE Control Systems Magaz, pp. 13-24. Henderson, D.E., P.V. Kokotovich, 1.L Schiano and D.S. Rhode, (1991). Adaptive Control of an Arc Welding Process. In: Proe. American Control Conference, Volt, pp. 723-728. Keyser RM.C. De, (1990). Model Based Predictive Control TooIbox, In: Computer Integrated

DeSign of Controlled Industrial Systems (Richalet, J., Tzajestas, S. eds), Proe. CIMEurope Workshop, Brussels, April 26-27. Nishar D .V., lL Schiano, W.R Perkins and RA. Weber, (1994). Adaptive Control of Temperature in Arc Welding. In: IEEE Control Systems Magaz., pp. 4-12, Aug. Richalet, 1., (1990). Model Based Predictive Control in the Context of Integrated Design, In:

Computer Integrated Design of Controlled Industrial Systems (Richalet, J., Tzafestas. S. eds), Proe. CIM-Europe Workshop, Brussels,

6. CONCLUSIONS

April 26-27..

Welding is a complex process which to perform manually requires a considerable amount of experience and skill. To simplify and automate the

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