Mass and heat transfer control in the GMAW process utilizing feedback linearization and sliding mode observer

International Communications in Heat and Mass Transfer 111 (2020) 104410

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Mass and heat transfer control in the GMAW process utilizing feedback linearization and sliding mode observer

T

Saeed Soltania, Mohammad Eghtesadb, Yousef Bazargan-Laric,

⁎

a

Department of Mechatronic Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran School of Mechanical Engineering, Shiraz University, Shiraz, Iran c Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran b

ARTICLE INFO

ABSTRACT

Keywords: Sliding mode observer Nonlinear control GMAW Heat transfer Mass transfer

Controlling the gas metal arc welding (GMAW) process is investigated utilizing feedback linearization with heat input, detaching droplet diameter and melting rate as controlled variables while accounting for actuator saturation. The control algorithm is employed using a state space model with eight dynamic states. Stability and internal dynamics of the process are studied thoroughly. Sliding mode state observers are implemented to determine the arc current and contact tip to work piece distance as well as the stick out length (which constitute the internal states of the system). Several simulation reports are provided to demonstrate successful regulation and tracking of detaching droplet diameter, heat input rate and melting rate. Finally, simulations indicate that the method is capable of overcoming parametric uncertainty.

1. Introduction Gas Metal Arc Welding (GMAW) has gained substantial significance in the weld industry since it became commercially available in 1948 [1]. During the welding process the arc is protected from atmospheric contaminants by a shielding gas [2]. It offers numerous advantages when compared to the other types of welding including but not limited to: relatively high speed operation, absence of heavy slag on the work piece, better overall weld quality, possibility of welding different thicknesses, ability to join a wide variety of materials, continuous welding and no limit on electrode length [1,3–5]. Some studies have also focused on dissimilar materials joining using GMAW [6]. Recent interest in GMAW for dissimilar material joining as evidenced by the computational models developed for the process [7] and extensive studies of the joint properties [8] have set the field for even broader developments in the years ahead. However, exposure to gases that can cause serious health issues, and difficulties associated with maintaining the system in close proximity to its desired operating point makes the process difficult for the operator. Furthermore, weld geometry and thermal properties of the molten pool need to be controlled in order to minimize weld defects and achieve a successful joint. Therefore, suitable controllers should be utilized to automate the process and obtain desired weld quality. In the GMAW process, an electric arc provides the heat source. The welding is achieved by utilizing a consumable continuous filler ⁎

Corresponding author. E-mail address: [email protected] (Y. Bazargan-Lari).

https://doi.org/10.1016/j.icheatmasstransfer.2019.104410

0735-1933/ © 2019 Published by Elsevier Ltd.

electrode and does not involve pressure. The resultant arc produces drastic heat, melting the electrode and the base metal, thus resulting the weld formation [2]. The welding characteristics are related to the mass and heat transferee during the GMAW process to the weld pool [5,9]. There are four major control parameters for the GMAW process, namely, electrode extension (stick-out length), electrode diameter, welding current and arc voltage [3]. In practical GMAW process, however, the main controlled variables are arc current Ia and arc voltage Ua which affect many of the weld features. A number of other GMAW variables also have impact on mass and heat transfer, namely, melting rate (MR), heat input (P), and gas composition. Controlling the mass and heat transfer is of special interest since they have significant impact on weld geometry and thermal properties of the welding pool. However, this would be a difficult task since coupling exists between these two variables [5,10]. This has resulted in efforts to separate the effects of these variables to ease the control task 3. This problem has been undertook using the pulsed current control. In pulsed GMAW the goal is to attain features of metal spray transfer mode and in the meantime keep the heat level as low as possible [11,12]. Although, this has been proven to be an effective method, the relatively high costs associated with producing high frequency pulsed currents with large amperage make it unsuitable for many applications where cost is a top priority. In [13,14] authors have utilized a PI controller to regulate the arc current and thereby control heat input over the unit length of the base

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m. s−1 (Torch travel speed respectively) − (Defined as the transversal cross-sectional area of deposited metal) S (−) m. s−1 (Weld wire speed) Aw (−) m2 (Wire cross section) ηg (−) − (Mass transfer efficiency) MR (−) mm3. s−1 (Melting rate) ls (−) M (Stick out) xd (−) m. s−1 (Droplet displacement) vd (−) m. s−1 (Droplet velocity) md (−) Kg (Droplet mass) Voc (−) V (Open circuit voltage) Vc (−) m. s−1 (Torch speed) V1 (−) V (Armature voltage of welding wire DC motor) V2 (−) V (Armature voltage of torch DC motor) FT (−) N (Resultant force acting on droplet) RL (−) Ω (Electrode electrical resistance) M (Droplet radius) rd (−) P (−) Watt (Product of welding current and voltage difference between contact tip and work piece) uct (−) V (Voltage difference between contact tip and work piece) Fg (−) N (Gravitational force) Fem (−) N (Electromagnetic force) N (Plasma drag force) Fd (−) Fmf (−) N (Momentum flux force) Iac (202) A (Critical transition current) λ1 (0.0504) − (Observer Factor) λ2 (1) − (Observer Factor) λ3 (0.31198) − (Observer Factor)

Nomenclature

R (−) G (−)

Symbol (Value) Unit (Description) γ (1.2) N. m−1 (Surface tension of liquid steel) θ (90) ° (Angle of conducting zone with drop) C1 (3.3 × 10−10) m3. s−1. A−1 (Melting rate constant 1) C2 (0.78 × 10−10) m3.s−1.1.A−2 (Melting rate constant 2) τ1 (50 × 10−3) s (Weld wire DC motor time constant) k1 (1) m. v−1. s−1 (Weld wire DC motor steady state gain) τ2 (80 × 10−3) s (Torch DC motor time constant) k2 (1) m. v−1. s−1 (Torch DC motor steady state gain) Ra (0.0237) Ω (Arc resistance) Ea (400) V. m−1 (Arc length factor) ρ (0.43) Ω. m−1 (Resistivity of the electrode) g (9.8) m. s−2 (Gravity) ρw (7860) Kg. m−3 (Density of the liquid electrode material (Steel)) ρp (1.6) Kg. m−3 (Density of the plasma (Argon gas)) V0 (12) V (Constant charge zone) Rs (6.8 × 10−3) Ω (Total wire resistance) Ls (306 × 10−6) H (Total inductance) μ0 (4π × 10−7) H. m−1 (Permeability of free space) kd (3.5) Kg. s−2 (Spring constant of drop) bd (0.8 × 10−7) Kg. s−1 (Damper constant of drop) rw (0.005) M (Electrode radius) vp (10) m. s−1 (Shielding gas velocity) H (−) Watt (Heat transfer) η (0.66 to 0.85) − (Heat transfer efficiency) us (−) V (Voltage difference across stick out) ua (−) V (Arc voltage) I (−) A (Arc current) metal (H) and the cross section of deposited metal at steady state (G). In this method, however, any changes in the GMAW inputs will affect the control task adversely. In [15], a model predictive control (MPC) based on ARMarkov PFC [16] has been proposed to control welding current and arc voltage in a linearized GMAW process. In [17], authors have implemented decentralized PI controllers and then used them with MPC in a two-layer control architecture with cascade configuration to achieve closed loop stability. However, they used a linearized model and did not account for the nonlinearities in the system. In a recent work, the drop detachment frequency was controlled implementing the welding current as the control signal [18]. They have used a three-layer cascade control method to regulate the deposition area, heat input and voltage to current ratio as the performance indicators. They also claimed to have reduced the interaction effects on the current channel using a decoupled controller in the internal loop. While a significant portion of the works in controlling the GMAW process have used linearized models, a number of other researchers have utilized nonlinear models of the system with different orders [19–22]. Feedback linearization has been exploited to improve the control performance, compensate for nonlinear dynamics of the system, and also decouple the control variables [21,23–27]. However, most of the works in this regard, have proposed simplified models of the actual system which might have shortcomings in presence of heavy uncertainties and parametric variations. Several approaches have focused on sliding mode control (SMC) to utilize the robustness of SMC algorithms. Most recently, super twisting control algorithm, which is part of higher order sliding mode (HOSM) control algorithms, has been implemented in [28] where authors have assumed the distance between contact tip and the work piece and also the weld speed to be held constant, which might not be the case in practical GMAW process. The motivation behind the present work, is to take advantage of both feedback linearization and sliding mode observer at the same time, while approaching the control task from a different perspective. While

most of the works to date have tried to control mass and heat transfer to work piece through manipulation of voltage and arc current in either continuous or pulsed form, in this paper we consider melting rate MR, heat input P, and detaching droplet diameter Dbdb, as controlled variables. These parameters are directly related to the weld quality, G (cross section of deposited metal) and H (heat input per unit length). Moreover, regulating the detaching droplet diameter helps in maintaining metal transfer mode, and regulating droplets transfer [29,30]. This approach lets us perform the control task without disregarding any of the nonlinear terms in governing equations and also take all of the important variables into account. Although the resultant model is more complex than some of the earlier studies, it is still computationally tractable and the added complexity provides us with higher fidelity. In Section 2, the proposed nonlinear model of the system is presented along with governing equations for mass and heat transfer to the work piece. The operating conditions are also discussed in this section. Section 2 describes the controller design method. This section starts with an overview of feedback linearization followed by internal dynamics description and an investigation of zero dynamics stability. The section is concluded by the sliding mode observer design and its implementation. Section 4 is dedicated to simulation results. The results are presented in three subsections, namely, regulation, tracking, and sliding mode observer. Next section covers uncertainty analysis which is followed by concluding statements. 2. Modeling the GMAW process In the context of mathematical modeling, there is always a compromise between model complexity and its accuracy. As such, proper models that provide the required accuracy with minimal complexity are sought [31]. As discussed earlier a number of different models have been proposed for the GMAW process and several extensive studies have focused on arc properties in various arc welding processes. 2

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Parameters related to the droplet and its detachment for the pulsed mode are described in [32,33]. The detaching droplet can be modeled as a spring dashpot system [34]. Melting rate, arc length, melting droplet, forces acting on the droplet, droplet detachment, electric circuit used for welding and the shielding gas are among contributing factors to obtaining the equations governing the GMAW process. The model which will be presented shortly is slightly more complex than the models used in most of earlier studies. However, it can be readily implemented with use of state observers which will be discussed in later sections. Here, the governing equations for mass and heat transfer to the work pieces are presented followed by an eighth-order nonlinear state space model of the process. Finally, the operating point of the process along with controlling variables and controlled outputs are described.

1 Voc Ls

I=

S=

1

vc =

md =

G=

R

=

g MR

R

+ xd + Ra I

Ea (lc

ls )

V0

(k1 V1

S)

(k2 V2

vc )

1 2

(7) (8) (9)

+ C2 I 2ls )

w (C1 I

(10)

FT

kd xd md

bd vd

(11)

RL =

(1)

w

ls +

1 (rd + xd ) 2

(12)

where rd is the droplet radius. Since stick out is much larger than the amount of melted welding wire, i.e.:

1 3md 2 4 w

1/3

+ xd

ls

(13)

we can assume the electrode electrical resistance to be given by:

Rw =

(2)

ls +

1 3m d 2 4 w

1/3

+ xd

ls

Therefore, the arc current dynamics can be described by:

where S, A, ηg, and MR are wire cross section, weld wire speed, mass transfer efficiency, and melting rate, respectively. Melting rate is defined by the relation:

MR = C1 I + C2 ls I 2

(3)

In the above equation, C1 and C2 are constants related to melting rate, ρ is the specific electrical resistance of the welding wire and ls is the stick out. Taking all the above equations into account, it is evident that mass and heat transfer to work piece are directly related to melting rate and heat input. Meanwhile, weld surface roughness and arc stability are significantly dependent on transfer mode of molten metal and regulation of metal droplets. In order to maintain the transfer mode and to regulate the droplets transference, controlling heat and mass transmission is essential. As such, melting rate, heat input, and detaching droplet diameter are chosen as controlled variables in this paper. 3. Nonlinear state-space representation of GMAW process Here we use the same sixth-order state space representation which was proposed in [9,20] with addition of two states to obtain a rather more accurate model of the process. The added dimensions to state space represent driving motors of welding torch and welding wire spool [17]. The GMAW process is considered to work in projected metal spray transfer mode, since it offers several advantages. In this transfer mode, arc stability is achieved while heat transfer to work piece is kept at low level and regularity of metal droplets transfer is maintained [36]. The resulting state space model of the process is stated as follows:

ls = S + vc

c c1 I + 2 2 I 2ls rw2 rw

(5)

Detailed description of constituents of FT will be given in what follows as was presented in [37]. Schematic of the model is illustrated in Fig. 1. Electrode electrical resistance, RL, is given by:

Where η, us, ua I, and R are heat transfer efficiency (the efficiency factor has a value between 0.66 and 0.85 for the GMAW process [35]), voltage difference across stick out, arc voltage, arc current, and torch travel speed, respectively. P is the product of voltage and welding current difference between the contact tip and the work piece which is almost equal to the power provided by the system supply. The weld reinforcement G, defined as the transversal cross-sectional area of deposited metal, is given by [18]: g AS

1/3

x d = vd

To date, a significant number of studies have focused on mass and heat transfer to the work piece [31–34]. Heat transfer per unit length of the base metal (H) is given by [14]:

I (us + ua ) R

1 3md 2 4 w

(6)

1

2.1. Mass and heat transfer to the work piece

P = R

ls +

lc = vc

vd =

H=

Rs +

(4)

Fig. 1. Schematic of the model used for GMAW process. 3

(14)

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Voc

I=

RL I

Va

Rs I

Fm =

(15)

Ls

µ0 I 2 4

[(rw / d )2

(1

(rw / d )2 ) 2],

1

As mentioned earlier, controlling inputs and controlled variables are as follow:

Fs = 2 rw

U = [V1 Voc V2 ] Y = [MR Ddbd P ]

4. Controlling the GMAW process

(16)

ls I 2 + Ra I 2 + Ea lc I

Ea ls I + V0 I

(17)

Where uct is voltage difference between contact tip and the work piece. Due to inherent complexity and difficulty associated with measuring the output variables, three estimators are utilized to estimate those variables. To estimate heat input and melting rate, Eqs. (3) and (18) have been employed respectively. The droplet detaches when surface tension force can no longer hold the droplet in contact to the electrode. As such, equating input force on droplet FT and surface tension force Fs [38], leaves us with the following relation:

Ddbd =

2rw (Fs e sin

Fg Fd Fm (kµ0 I 2 /4 ))/(µ0 I 2 /4 )

1 4

1 cos

1

+

2 2 ln cos ) 2 1 + cos

(1

(19)

In designing controller for the proposed model, we have utilized MIMO (Multi Input-Multi Output) feedback linearization. To do so, the output is differentiated with respect to time until all inputs appear in the outputs or their derivatives:

y2 = ls I 2 + 2 ls II + 2Ra II + Elc I + Elc I

4 3

y3 =

Fem =

µ0 I 2 4

×

1

1 + cos( ) (1

Cd (rd2

rw2 ) 2

2 p vp

Els I + V0 I b2

4ac

4

0 . 274Cd md p vp2 md

1/3

md g

T1 + T2

+ T3 T4

w

(29) Therefore, transformation matrix D, will be:

(22)

1 r sin( ) + ln d 4 rw

1 µ0 I 2

e

c1 + 2c2 Ils c2 I 2 Ls 2ls I + V0 Els + Elc + 2IRa I 2 Ea I I2 Ls 0 Q 0 c2 I 2

2 2 ln cos( ))2 1 + cos( )

D=

(23)

Fd =

2rw sin( )

(21) 3 w rd

Els I

(28)

(20)

Fg = md g

(27)

y1 = C1 I + C2 ls I 2 + 2C2 ls II

where each component can be calculated as follows:

md =

(26)

4.1. Controller design

is used to estimate the detaching droplet diameter, Ddbd. Several forces act on the droplet during welding which have been presented in numerous studies. As presented in [37], total external force acting on droplets is the sum of gravitational, electromagnetic, plasma drag and momentum flux forces. In other words, we have:

FT = Fg + Fem + Fd + Fmf

(25)

(18)

This relation in which k is given by:

k=

rw rd

Controlling the GMAW process can be divided into controlling the weld pool and the droplet, and adjusting the arc and mass and heat transfer to the work piece. Controlling the weld pool can be regarded as the outer control loop while controlling the arc and the droplet is treated as the inner control loop Fig. 2. GMAW is carried out by either a human or a robot. In manual GMAW the outer control loop is completed by the person who does the welding and is dependent on the person’s skill. However, in both manual and automated welding, a control system is required for the inner loop to obtain enhanced weld properties. Therefore, in this paper the focus is on designing the inner loop control system. As described earlier, we use feedback linearization for controller design.

Heat input is calculated using the following:

P = Iuct = I (us + ua )

rw / d =

(24)

(30)

Where T1, T2, T3, T4, and Q are given in appendix A.

Fig. 2. GMAW control can be divided into an inner control and an outer control in a cascade coupled system. The outer control addresses the weld pool, while the inner control addresses the electrode and the arc. Adopted from [20]. 4

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A=

3Els2 I

2Elc ls Ic 2 + 2Els2 Ic2 Ls

4ls I 2R

2ls I 2Rs

+

3Elc IRa

ls I 3c22 2

Elc IRs + 3Els IRa

vd =

Ls

W 2ls I 2Rs c2

+

3IRa V0

2ls IV0 c2

IR s V0

IRa c1

ls Ic1 Ls

Elc c1 + Els c1

2ls2 I 2 2 + 2E2lc ls 2I 2Ra R s Ls

(35)

x d = vd

I 3c1 c 2

re2

IRs c1

2Elc V0 + 2Els V0

kd xd md

bd vd

(36)

By applying zero inputs, we obtain:

V0 c1 E2lc2

FT

(37)

x d = vd

+

0 3ls IV0

Els IR s

kd xd

Ls 3Elc ls I

V02

E2ls2

2I 2Ra2

I 3c

Ls

2 2 1 + EI c1 + Els I c 2 2 rw

0

ls I 3c2

vd = (31)

A)

(32)

The above inputs transform the output equation to a simple form presented by Eq. (33) letting the external dynamics to be controlled easily using linear controllers.

yiri

e . dt

2 p vp

+ gmd

md

(38)

For nonlinear systems, sliding mode observers, Thaus method and Raghavans method are used for state estimation and observation [40]. In this paper, we use a sliding mode observer to estimate I, lc, and ls since their direct measurement is costly and difficult (except for I). md can be simply found by integrating the corresponding state equation. Moreover, by using Eq. (15) for electrode electrical resistance and also Eq. (16) for I, the remaining state variables, namely, xd and vd, do not affect mass and heat transfer to work piece. Therefore, their estimation is not required.

State variables, md, la, ls, and lcv will be controlled using a PID controller.

u = kp e + k v e + kI

md

2 rw

4.4. State observer

(33)

= vi

0.333

Since vd is a constant, we only have to show that the last two equations in the above set represent a stable system. The above set of equations represent a quadratic equation whose homogeneous part is stable indeed. The stability of the nonhomogeneous part is shown by the fact that md which is among the zero-inputs is constant. Therefore, the entire internal dynamics is shown to be stable. The stability of internal dynamics is also shown in Fig. 3 and Fig. 4 where boundedness of the internal dynamics is observed. Note that the increasing trend observed in the droplet position is due to the droplet dynamics that result in detachment. Also, note that the amplitude of the oscillations in Fig. 3 decreases continuously.

Where W is given in appendix A. D is a nonsingular matrix. As it is apparent in the above equations, total relative degree of the system is 4 where Ia, ls, lc and md have all appeared in derivatives and therefore are among external dynamics of the system. The remaining state variables, namely, xd and vd are internal dynamics. Therefore, inputs can be calculated according to Eq. (32):

u1 u2 = D 1 (v u3

bd vd + Cd

3md 4

(34)

4.2. Internal dynamics Dynamics of a system in which input-output linearization has been carried out, would decompose into an external (input-output) part and an internal or unobservable part. Since designing the input v and the output y behave in the desired manner, the external part consists of a linear relation between y and v (or equivalently the controllability canonical form between y and v) [39]. However, the question that arises is whether the internal part of the system dynamics would also behave as desired or more specifically, whether they remain bounded. Every time input-output linearization is performed, internal dynamics of the system should be examined carefully to make sure that the whole system will behave in the desired manner.

4.4.1. Sliding mode observer design Measuring the outputs and inputs is the sole purpose of an observer approximating the unmeasurable conditions of the systems. This process is followed by the recently published method applied [41]. Thus, repetitive contents are omitted. Considering the GMAW system, the observer takes the form:

4.3. Zero dynamics stability investigation To overcome difficulties put forth by nonlinear systems compared to linear ones, a so-called zero-dynamics is defined for a nonlinear system. The zero-dynamics is defined to be the internal dynamics of the system when the system output is kept at zero by the system input. Therefore, stability of zero-dynamics yields internal dynamics stability. To find zero-dynamics of the system, the zero-inputs (inputs for which the outputs will remain zero at all times) are calculated as follows:

Ia = 0 (External) ls = 0 (External) lc = 0 (External) md > 0.003855 (External) Internal dynamics variables are xd and vd with the following dynamics:

Fig. 3. Droplet position (xd). 5

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4.5. Saturation All physical control systems are subject to some sort of actuator limitations. For instance, the motor can only provide torque or power in a limited range. This kind of actuator limitation, ultimately limits the achievable closed-loop performance of the system. In many cases actuator constrains limit the system performance even more than modeling uncertainties [42]. Previously, saturation was ignored most of the times and it was hoped that the designed controller performs well in presence of saturation. However, this might lead to poor performance or even catastrophic failure in some cases. In the present work, we determine the following conditions on the input variables to ensure that actuator saturation does not happen:

0.1 < S < 0.1 (m . s 1) 10 < VOC < 40 (V ) 0.1 < VC < 0.1 (m . s 1) 4.6. Parametric uncertainty analysis

Fig. 4. Droplet velocity (vd).

x1 = S + vc

x2 =

1 (Voc Ls

c1 (x2 + x 22 x1) + rw2

(Rs + x1 + Ra ) x2

1 sgn(x1

E (x3

x1)

x1)

V0) +

The nonlinear control law developed in previous sections, is based on the model describing the system dynamics. However, all variables and parameters in the model are subject to some degree of uncertainty mainly due to measurements. The controller must be capable of stabilizing the system in the presence of such uncertainties to make sure that the system operates safely. In the present work, we have accounted for some of the most important uncertainties in the model. In particular, the following uncertainties are being considered:

(39) 2sgn 2 (x2

x1) (40)

x3 = vc +

3sgn 2 (x3

x3)

(41)

Where:

x2 = x2 + 1 sgn(x1 x3 = x3 + 2 sgn(x2 1 = 0.0504 2 = 1 3 = 0.31198

x1 ) x2 )

Rs :+30% 15%

Ls :+30% 30%

Vo:+30% 30% rw:+20% 30%

:+30% 30% vp:+30% 30%

Ra :+30% 30% uo :+30% 30% +30% : p 30%

c1:+30% 15%

c2:+30% 15%

:+30% 15%

E :+30% 30%

kd:+30% 30%

+20% w : 30% +30% cd: 30%

bd :+30% 30%

:+30% 15%

5. Simulation results and discussion The control architecture used for the GMAW process is presented in Fig. 5. To illustrate the performance of the designed controller, simulations are carried out with full state feedback and sliding mode observer.

This observer was implemented with the model and the simulation results will be presented in the following section.

Fig. 5. Control architecture used for the GMAW process. 6

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5.1. Full state feedback

presented in Section 3. Fig. 12 shows regulation of detaching droplet diameter. We observe that the transients of the system with nominal parameter values lie between those obtained with parameters corresponding to upper and lower bounds. Results for tracking of melting and heat input rates are shown in Fig. 13 and Fig. 14, respectively. The same trend with respect to parameter variation is observed, namely, curves corresponding to nominal values lie between those corresponding to upper and lower bounds on the parameters. Integral of absolute error (IAE) for each of the cases presented herein is shown in Table 1. We note that the results indicate IAE is lower than that of the PI controllers and the two-layer control architecture proposed in [17].

Regulation of the droplet diameter about the desired operating point is shown in Fig. 6. As seen in the figure, the droplet diameter is perfectly regulated using the proposed control method. The inset shows the initial transient period with more details. The simulation represented here has been carried out by implementing variable step method for nonlinear set of Eqs. (4) to (11). Desired heat input and melting rate are changed in a staircase like manner with amplitudes of 3.5 mm3sec−1 and 300W respectively. Fig. 7 shows tracking of melting rate while Fig. 8 illustrates that of the heat input rate. The insets are provided to show the fast transients more clearly. As shown in the figures, the performance of the control system far exceeds that of previously proposed architectures. Melting rate response exhibits some spikes which are due to the step changes in the heat input.

5.4. Observer based controller Fig. 15 illustrates successful regulation of detaching droplet diameter with observer based controller. The inset clearly shows fast convergence of the observer state to the true value. Fig. 16 shows the observer error associated with droplet diameter regulation. Tracking of command signals to melting rate and heat input to the system with the observer based controller is shown in Figs. 17 and 18 respectively. The insets are provided to show the transients more clearly. Figs. 19 and 20 depict the observer error associated with melting rate and heat input, respectively. As shown in the figures, the observer error converges to zero within 0.3 seconds.

5.2. Full state feedback with actuator saturation As described in the Section 3, we impose constraints on input variables to ensure actuator saturation does not happen. Fig. 9 illustrates regulation of detaching droplet diameter when the limits are taken into account. The results for tracking melting and heat input rates are shown in Fig. 10 and Fig. 11 respectively. Again we observe that both regulation and tracking tasks are achieved with fast transients and zero steady-state error. As we noticed in the previous results, a step change in the heat input results in fast transients of melting rate which now appear to be smaller in magnitude; this is due to the introduction of actuator saturation.

6. Conclusions In this paper, a new control architecture for GMAW process was developed based on feedback linearization method. A modified state space model was used to implement the controller. Actuator saturation was taken into account through inequality constraints on torch speed, weld wire speed and open circuit voltage. Sliding mode state observers were designed to estimate the internal states of the system that are not amenable to direct measurements. Simulations results show asymptotic convergence of the observer dynamics to the actual state dynamics with

5.3. Full state feedback with actuator saturation and parametric uncertainty As discussed in Section 3, there is some uncertainty associated with the physical parameters of the system. The results here illustrate the ability of the proposed control scheme to achieve regulation and tracking tasks as long the parameters are bounded within the range

Fig. 6. Detaching droplet diameter regulation with full state feedback. 7

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Fig. 7. Staircase command tracking of melting rate.

Fig. 8. Staircase command tracking of heat input.

8

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Fig. 9. Detaching droplet diameter regulation with actuator limits.

Fig. 10. Staircase command tracking of melting rate with actuator limits.

9

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Fig. 11. Staircase command tracking of heat input with actuator limits.

Fig. 12. Detaching droplet diameter regulation with actuator limits and parametric uncertainty Results for nominal parameter values and upper and lower bound parameter values are shown for comparison.

10

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Fig. 13. Staircase command tracking of melting rate with actuator limits and parametric uncertainty - Results for nominal parameter values and upper and lower bound parameter values are shown for comparison.

Fig. 14. Staircase command tracking of heat input with actuator limits and parametric uncertainty - Results for nominal parameter values and upper and lower bound parameter values are shown for comparison. Table 1 Integral of absolute error for various scenarios considered here. Scenarios Full Full Full Full

state state state state

Pc feedback feedback feedback feedback

without actuator saturation with actuator saturation with actuator saturation and upper bound parametric uncertainty with actuator saturation and lower bound parametric uncertainty

190.1 192.8 272 209

11

MR

Ddbd −10

1.6 × 10 1.21 × 10−9 7.57 × 10−9 1.0 × 10−10

2.17 2.17 1.15 6.87

× × × ×

10−4 10−4 10−3 10−6

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Fig. 15. Detaching droplet diameter regulation with observer based compensator.

Fig. 16. Observer error associated with detaching droplet diameter regulation.

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Fig. 17. Staircase command tracking of melting rate with observer based compensator.

Fig. 18. Staircase command tracking of melting rate with observer based compensator.

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Fig. 19. Observer error associated with melting rate.

Fig. 20. Observer error associated with heat input.

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arbitrary initial conditions. Overall, the simulations provided herein indicate that the algorithm is successful in regulation and tracking tasks with both full state feedback and sliding mode observers. Finally, an uncertainty analysis is carried out to understand the capabilities of the proposed method in dealing with parametric uncertainties. The control

scheme was shown to be successful in presence of uncertainties that are common in real world applications. Designing observers for the case with parametric uncertainty and actuator saturation is the focus of ongoing work in our group and the results will be discussed in a separate article.

Appendix A. The variables used in representation of Eqs. (29) to (31) are provided here. In Eq. (29) T1 through T5 are given by

(1

2) (1.21rw

µ 0 II

2/3

md

1 3

T11

)

2

w

T1 =

(A.1)

2 µ0 I 2

2m rw d

0.808

+

5/3

md

2m ) 1.21 T11 rw d

3 M

w

w

T2 =

2(1

md

5/3

kµ 0 II 2

w

w

4

8 I2 Fs µI 3

T3 =

T4 = e

0.5Cd

2/3

rw2

w kµ 0 I 2 4

Fs Fg Fd Fmf

2/3

md

2 p vp

md g

T5

(A.3)

µ0 I 2 4

/

(1

2) (1.21rw

µ0 I 2

md

0.825

(A.2)

1 3

(A.4)

T11

)

2

kµ0 Iw2

w

T5 =

(A.5)

4

Where:

10.902rw2

T11 = 9

( ) md

2/3

(A.6)

w

In Eq. (30) Q is given by:

e

µ0 I 2

e

1 2 I

( )

e md

1 2Q1 1 + 3I e 4 1 cos( )

(e ) e Q=

md 2/3 0.5 2Cd p vp 3.3 w

8

2r w µ0 I 2

2 2Cd p vp rw2 µ0 I 2

e

2 2.2rw md 2/3 I w

( )

2/3

cos( ))8 (1 + cos( ))2

3.3(1

w 2 2C d p vp

0.5rw

+ 4rw md g

Q2 sin(

0.33rw3 md

+ Q2 sin(

)I 2

( ) md

w

8rw2

2

0.99rw3 md

) µ 0 I 3Ls

5/3

Q1 Q2 sin( ) I 2

+ w Ls

rw

2rw3

2 2C d p vp / I Q2 sin( ) I 2Ls

+

( ) md

w

5/3

w Ls

3rw Q1 Q2 sin( ) I 2Ls (A.7)

Where:

Q1 =

10.08rw

9

( ) md

2/3

w

md g 4

Q2 = e µ0 I 2

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And in Eq. (31) W is determined by:

e

2r w µ0 I 2

(e ) e 1 2 I

( )

e

1 2Q1 1 + 3I e 4 1 cos( )

2/3

md

cos( ))8 (1 + cos( )) 2

3.3(1

w (Ils c2

+ c1) + 0.998rw3 md (V0 )I 2

Q1 Q2 sin( +

( )

w

0.33rw3 I 2ls

8rw2

2.2rw2 md 2/3 I w

md 2/3 2 0.5 2Cd p vp 3.3 2 2Cd p vp rw w 2 µ0 I 2 µ I 0 e e

8

2

I (Ra + Rs

w Ls

(V0 + I (Ra + Rs + ls ) + E (lc Q2 sin( ) µ 0 I 3Ls

5/3

w

ls ))

4rw md g (V0 + I (Ra + Rs + ls ) + E (lc

W=

ls ) + E (lc + ls ))

( ) md

+

ls ))

2rw

g (c 1

w

c2 ls I )

Q2 sin( ) µ 0 I 0.5rw

2 2C d p vp (IV0

+ Ra ) + Rs )

Q2 sin( ) µ 0 I 3Ls +

0.22rw3 (Ils c2 + c1 ) Q2 sin( )

3rw (V0 + I (Ra + Rs + ls ) + E (lc Q1 Q2 sin( ) ILs

( ) md

5/3

w

rw

0.167rw Cd p vp2 (c1 + c2 ls ) Q2 sin( ) µ 0 I ( md /

(

V0 I

+ Ra + Rs + ls +

1/3 w)

Q2 sin( ) I 2Ls 2 2C d p vp )(Ils

+ E (l c

E (l c

ls ) I

)

Q2 sin( ) Ils

0.33rw3 md (V0 + I (Ra + Rs + ls ) + E (lc

(

ls ))

w

( ) md

ls ))

5/3

w

ls ))(2rw3 + 0.5rw ) + 2rw3 Q2 sin(

) I 3L

2 2C d p vp (IRs

+ V0 )

s µ0

(A.8)

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