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Process control for plasma electrolytic removal of TiN coatings
Surface & Coatings Technology 199 (2005) 198 – 204 www.elsevier.com/locate/surfcoat
Process control for plasma electrolytic removal of TiN coatings Part 2: Voltage control E.V. Parfenova,T, R.R. Nevyantsevaa, S.A. Gorbatkovb b
a Ufa State Aviation Technical University, 12 K. Marx Street, 450000 Ufa, Russia Russian Institute of Finances and Economics, Ufa branch, 69/1 Socialisticheskaya Street, 450006 Ufa, Russia
Available online 16 February 2005
Abstract The paper discusses two main problems of creating a control system for plasma electrolytic removal of titanium nitride coating from stainless steel hardware. The first feedback problem of unobservability of the surface state has been solved using indirect identification with an informative model of the process. The informative model has been developed on the base of the current signal processing and correlation analysis. DC component of the current and power spectral density at 1 kHz are in correlation with the surface state parameters. The second problem—inverse ill-posed control problem has been solved using inverse neural network, an object with artificial intelligence. For the solution of the inverse problem, the surface state and feedback parameters were used as the neural net inputs and voltage as the output. Because of approximation power of the neural network, the inverse problem has been resolved, and the treatment program for coating removal has been generated. Computational and practical tests show that the control system allows to intentionally change the state of the surface with 10% accuracy. D 2005 Elsevier B.V. All rights reserved. Keywords: Automation; Coating removal; Plasma electrolysis; Polishing; Titanium nitride; Neural networks; Ill-posed inverse problems
1. Introduction Plasma electrolysis can be used not only for hardening and polishing of metal surfaces, coatings deposition, but also in the field of renovation technologies for removal of coatings with deteriorated properties [1,2]. Successful renovation demands coating removal with high accuracy, especially in repair technology for jet aircraft engine blades. One of the best ways to improve processing accuracy is to create a control system for the process. Since there is no feedback control systems known for the plasma electrolysis, this research is the first attempt to create an approach to the process control and automation. The phenomenon of plasma electrolysis has been investigated for a significant number of applications, but feedback control is still an issue. The main problem is that the surface state, which is usually
T Corresponding author. Tel.: +7 3472 558551; fax: +7 3472 229909. E-mail address: [email protected] (E.V. Parfenov). 0257-8972/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.surfcoat.2004.10.144
characterized by roughness, microhardness, etc., is unobservable during the processing. This problem can be solved with indirect identification, which require parameters, both observable and controllable, to be found. The second problem that arises is an inverse control problem, which usually appears to be ill-posed. The paper proposes an approach for dealing with these problems on the example of creating a control system for plasma electrolytic removal of TiN coating from stainless steel samples. This paper is a logical continuation of Part 1, which is dedicated to the phenomenology of the process and to the issue of the duration control of the process. 1.1. Indirect identification: informative model The indirect identification of complex technological processes is usually based on a correlation between observable and unobservable parameters which both required to be controllable. For the search and classification of these parameters, a phenomenological model of the
E.V. Parfenov et al. / Surface & Coatings Technology 199 (2005) 198–204
199
Table 1 Parameters classification Parameter
Acronym
Position in experiment design
Input/Output position
Controllability/ Observability
Voltage (V) Initial temperature of the electrolyte (8C)
U T0
Factor Factor
Yes/Yes No/Yes
Duration of the processing (s) Percentage of area completely free of coating (relative units) Normalized average roughness (relative units) Average microhardness (Gpa) Temperature of the electrolyte (8C) Current (A)
t St
Factor Response
Input Constant parameter Time Output
– Yes/No
Ra HV T i
Response Response Response Response
Output Output Output Output
Yes/No Yes/No Yes/Yes Yes/Yes
coating removal process could be employed. The model is presented in Part 1, where a description of the system under consideration is also provided. One of the main results obtained from the phenomenological model is voltage controlled mechanism which yields bmatteQ surface after the coating removal and gives a direction for further polishing of the surface. This mechanism involves TiN oxidation to TiO and TiO2 in the bubble boiling mode, removal of oxides and polishing in the same mode. The classification of the parameters is shown in Table 1. As seen from the table, current and electrolyte temperature appear to be the only observable parameters. From this limited set, the current turns to be the most perspective for the search of informative parameters for indirect identification. As shown in Ref. [3], the current has both the DC and AC components with significant magnitude. The DC component (I) has the maximal value at the beginning of the treatment and exponentially falls with time; parameters of the exponent depend on the treatment conditions and the surface state. Therefore, the DC component could be one of the feedback parameters. The AC component has been analyzed in Part 1, where the power spectral density in the bandwidth 500H1500 Hz ( p 1) has been chosen as a feedback parameter. Correlation analysis analogous to that in Part 1 shows that the chosen feedback parameters are in correlation with the unobservable surface state parameters. Table 2 contains significant coefficients of paired correlation that has been discovered. As seen from lines 1 and 2, DC component (I) and power spectral density ( p 1) are in correlation with time derivatives of S t and Ra respectively. As seen from line 3, percentage of the area completely free of coating (S t ) is in correlation with microhardness (HV); therefore only one of
these parameters is independent. The same situation is in line 4: the DC component (I) is in correlation with the electrolyte temperature (T). Overall, two pairs of the parameters appear to be independent: S t and Ra for the surface state, I and p 1 for the feedback. Finally, the results obtained have been arranged into an informative model of the process (Fig. 1). This model solves the problem of indirect identification of the surface state during the processing. 1.2. Inverse control problem solution: intelligent control system Now having feedback parameters, the only question remains: how should the voltage be driven so that by the end of the treatment the coating would be removed and the surface would have roughness equal to or better than the initial? This control problem is inverse to the problem that is being solved by experiment design and further interpolation. Also this problem can be treated as a problem of constraint optimization, where parameter S t goes to a maximum and the value of Ra is a constraint for the search. This dual representation helps to decompose the problem. First, an inverse model should be created; second, a realizable treatment program should be designed. The inverse model should be capable of mapping the set of output parameters into the input set. This problem is illposed in the sense of Adamar [4], since for a correct problem all of the three following conditions must be satisfied. First, the solution of the problem must exist. Second, the solution must be unique. Third, the solution must be continuously dependent upon variations of the problem inputs. If any of these conditions fails to be satisfied, the problem becomes ill-posed. Two former conditions typically cause the ill-posedness of the process control inverse problems. For example, as
Table 2 Correlation between parameters No.
Parameter 1
1. 2. 3. 4.
I p1 St I
Coefficient of paired correlation 0.80 0.76 0.61 0.95
Parameter 2 dS/dt dRa/dt HV T
Controls:
St
k1 d dt
I
Ra
k2 d dt
p1
U T0 t
Fig. 1. Informative model of the system.
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Table 3 Fragment of the set M
1
2
Discrete time (n)
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
Factors
Surface state
Lag
Feedback
Duration t (s)
Voltage U (V)
Temperature T 0 (8C)
S t [n]
Ra[n]
S t [n 1]
Ra[n 1]
I (A)
p 1 (W)
0 15 30 45 60 75 90 105 0 15 30 45 60 75 90 105
179 179 179 179 179 179 179 179 320 320 320 320 320 320 320 320
56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56
0.00 0.10 0.11 0.14 0.16 0.19 0.22 0.24 0.00 0.03 0.03 0.04 0.04 0.05 0.06 0.06
1.00 0.86 0.86 0.86 0.87 0.88 0.89 0.91 1.00 1.05 1.05 1.05 1.05 1.06 1.06 1.06
0.00 0.00 0.10 0.11 0.14 0.16 0.19 0.22 0.00 0.00 0.03 0.03 0.04 0.04 0.05 0.06
1.00 1.00 0.86 0.86 0.86 0.87 0.88 0.89 1.00 1.00 1.05 1.05 1.05 1.05 1.06 1.06
5.0 6.0 8.5 3.5 2.7 2.3 2.1 1.6 4.2 4.0 2.2 1.8 1.5 1.4 1.3 1.2
0.058 0.025 0.030 0.033 0.026 0.026 0.018 0.018 0.019 0.017 0.016 0.017 0.016 0.016 0.012 0.012
seen from Fig. 7 in Part 1, for one set of the factors, say, U=300 V, T 0=50 8C, t=600 s, a definite output exist: S=0.5, Ra=0.7. It is possible that the output would be the same for a set of other factor levels, such as U=350 V, T 0=70 8C, t=600 s. Now, if the input and the output are inverted, then for one input of the inverse model, S=0.5, Ra=0.7, there are two valid outputs, i.e., two solutions. Moreover, the solution may not even exist if the input is set at S=1.0, Ra=0.1. Thus, the regularization of the inverse model should be performed. The regularization of the ill-posed control problem has been performed in the sense of Tikhonov [4]. The problem is correct in the sense of Tikhonov if all of the three following conditions are satisfied. First, the solution of the problem must exist on a set M. Second, the solution must be unique within the set M. Third, the solution must be continuously dependent upon variations of the problem inputs within the set M. In the case of experimental investigation of the process control, the set of experimental data or its subset could be treated as the set M. Let us discuss this regularization in the case of the plasma electrolytic coating removal. First, for the existing of the solution, the treatment program must be realizable on the set M. This requires taking into consideration the phenomenological model. Second, for the solution being unique on the set M, the set must be constructed in such way that all its entries are distinct. Third, for the solution being continuously dependent upon variations of the inputs within the set M, the model must employ continuous and preferably smooth functionals. The second and the third conditions demand highly adequate and smooth approximation of the dependencies within the set M. In Ref. [5] it has been shown that this kind of approximation cannot be performed with conventional regression analysis, which deals with polynomial equations and requires significantly distinct input values. These input values are usually obtained by means of experiment design,
where the factors’ values are set to distinct levels. In the case of the inverse problem, no experiment design could be performed unless its solution is known. Nevertheless, the highly adequate and smooth approximation could be achieved by using an artificial neural network. Being an object with artificial intelligence, which makes the model resistant to noise and even contradictions [6], in this case a neural network can be treated as an advanced regression. From the wide variety of different kinds of neural networks, a general regression neural network, which is a net with a radial basis function, has been chosen for its ability to smooth interpolation [7]. The neural net has one hidden layer with a Gaussian activation function, which width controls the model adequacy. Before being used, any neural network needs a set of data to be trained on. In this case, the neural network must be trained on the set M, having all output parameters in the input and voltage U in the output. This network will be called binverse.Q Let us proceed with construction of the set M. This set must contain all known information and could be any Inverse neural network
St [n] Ra [n] ••• 1250 neurons
No. of realization
St [n-1] Ra [n-1] I [n]
U [n]
p1 [n] T0 Input layer
Hidden layer
Output layer
Fig. 2. Inverse neural network model structure.
St
B
C Ra and St in relative units
Ra and St in relative units
A
U, V
E.V. Parfenov et al. / Surface & Coatings Technology 199 (2005) 198–204
Time,s
201
St
Time,s
Time,s
Fig. 3. Inverse model test: (A) modeled trajectory of parameters St and Ra for U=180 V, T 0=70 8C; (B) the inverse problem solution U[n]; (C) modeled trajectory for U[n] and T 0=70 8C.
combination of data that was obtained either experimentally, or computationally. The set M is a table that contains data for all realizations tabulated in discrete time n every Dt=15 s (Table 3). This sampling interval has been calculated after applying Nyquist sampling theorem [8] to a typical curve of DC component. Columns U and T 0 contain factors’ values. Columns S t [n] and Ra[n] contain their measured values at the beginning and the end of the realization, and calculated values in between. The calculated data were obtained from modeling of the surface state with a direct neural network model as described in Part 1 and Ref. [9]. Columns I, p 1 contain measured over Dt DC component and averaged over Dt power spectral density in the bandwidth 500H1500 Hz. Columns S t [n 1] and Ra[n 1] contain lag variables that represent the state of the surface on the previous time step in the discrete time n. Introducing these variables and excluding duration t from the inverse neural net makes this model dynamic, since this way it becomes a set of discrete time nonlinear differential equations. The inverse neural network, which structure is shown in Fig. 2, has been successfully trained on the set M. The
a
adequacy of the inverse problem solution is illustrated in Fig. 3. Chart (A) corresponds to modeled trajectory of parameters S t and Ra for U=180 V, T 0=70 8C. Chart (B) clearly illustrates that for the whole operational range the inverse problem solution U[n] appear to be almost constant at 180 VF10%. Chart (C) is the modeled trajectory for U[n] and T 0=70 8C. The inverse neural network obtained is trained over the whole set of data. Any new data would increase the neural network capabilities; therefore, additional training should be performed in case of considering other work piece size, material, etc. This means that the neural network should be upgraded, not completely retrained. This will not change the phenomenological model, since it describes basic physical and chemical processes of the plasma electrolysis. With inverse and direct models of the process developed, it is now possible to create the treatment program and complete the control system. After analysis of the phenomenological model, the following program has been proposed (Fig. 4). For this program initial temperature of the electrolyte T 0=70 8C has been chosen. In full accordance
c
U, V
S, relative units
Ra, relative units
b
Time,s
Time,s
Time,s
Fig. 4. Generating treatment program for the coating removal: solid line—desired trajectory; dashed line—realizable trajectory.
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b
U, V
Ra, relative units
a
Time,s
Time,s
Fig. 5. Generating treatment program for the polishing: solid line—desired trajectory; dashed line—realizable trajectory.
with the phenomenological model, the roughness after the coating removal is higher than the initial (RaN1.0, see Fig. 4c), which requires separate inverse and direct models and feedback parameters for the process of polishing. Analogous investigation has been performed for the process of plasma electrolytic polishing of the steel surface at the electrolyte temperature 90 8C. It has been established that the average power spectral density p 1 is in correlation with the roughness Ra of the surface being polished. The coefficient of paired correlation is 0.95. For this process, analogous inverse neural network has been created with inputs Ra[n], Ra[n 1], p 1[n] and output U[n]. Generation of the treatment program for polishing is shown in Fig. 5.
Finally, the control system for the plasma electrolytic coating removal has the following structure (Fig. 6). The treatment program is fed to the intelligent controller, where S t [n 1] and Ra[n 1] are acquired with the unittime delay block (z 1). Calculated U[n] is transformed into U(t) with a digital to analog converter (DAC) every Dt of 15 s. Current of the plasma electrolysis is measured with an analog to digital converter (ADC) and transferred both to a low pass filter (LPF) and to a band pass filter (BPF) to obtain I[n] and p 1[n] respectively. These feedback signals are also fed to the controller. The treatment program block, the intelligent controller and the feedback block are embedded in an operator’s personal computer.
Intelligent Controller T0
T0
Treatment Program
T0
S t [n]
St[n]
Ra[n]
Ra[n]
U [n]
S t[n-1]
−1
z
Ra[n-1]
U(t )
DAC
Inverse neural network model
n
Clock p1 [n]
I [n]
LPF
Feedback
0... 1 Hz
i (t )
BPF
ADC
500..1500 Hz
Fig. 6. Structure of the control system.
Plasma Electrolytic Coating Removal
St (t )
Ra(t)
E.V. Parfenov et al. / Surface & Coatings Technology 199 (2005) 198–204
Percentage of the area completely free of coating
Voltage control curve
1
350 Polishing
S, relative units
Coating removal
300
U, V
203
250
200
100 0
200
400 Time, s
600
0,8
0,4 0,2 0
800
Polishing
Coating removal
0,6
0
200
400 600 Time, s
800
Average roughness
Average power spectral density at 1 kHz 1,2
0,20
Ra, relative units
0,16
p1, W
0,12 0,08 0,04
1,0
0,8
0,6
0 0
200
400
Time,s
600
800
0
200
400
Time,s
600
800
Fig. 7. Control system tests results: solid line—calculated realizable trajectory; dashed line—experimentally recorded trajectory; — experimental points.
Operational ranges of the control system are the following: U=150H400 V, T 0=20H90 8C, i=0H10 A. Computational and practical tests of the control system show that it is capable of driving the surface of a treated sample over the desired realizable trajectory (Fig. 7). This fact is supported, first, by the stable following the desired trajectory. Second, measurements of the surface parameters after the treatment show that the calculated trajectory lays within the 10% neighborhood of the experimental points. The surface micrographs are presented in Fig. 8. Third, the recorded trajectory of the feedback signal-average power spectral density p 1 has either 10% mean square deviation from the calculated one.
It is supposed that with sufficient amount of information used for the neural network training, the intelligent control system should be robust. Nevertheless, the serious issue of the control system robustness is currently under investigation. The results of pilot experiments with different shape of the samples, other electrolyte, material and coating gives a hope that the control system would be robust.
2. Conclusion On the base of the phenomenological model, an informative model of the process has been created. It has
Fig. 8. Surface micrographs after the voltage controlled treatment: (a) 480 s; (b) 840 s.
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been shown that the process of the coating removal could be sufficiently described with the percentage of the area that is completely free of the coating and average roughness of the surface. A robust correlation between these parameters and the DC component and the average power spectral density in the bandwidth 500. . .1500 Hz has been established. Thus, the main problem of creating feedback control has been resolved. An inverse mathematical output–input model of the process has been created with the means of artificial intelligence. The model resolves the ill-posed inverse control problem. It has been employed for direct control of the process with regard of the actual surface state that is observed through the established feedback parameters. An automated control system for the process has been constructed. Computational and practical tests show that the control system allows to intentionally change the state of the surface with 10% accuracy, and, therefore, to successfully remove TiN coating for further replating. The method for the process control of plasma electrolytic coating removal is protected with Russian Patent [10].
References [1] A.L. Yerokhin, X. Nie, A. Leyland, A. Matthews, S.J. Dowey, Surf. Coat. Technol. 122 (1999) 73. [2] N.A. Amirkhanova, R.R. Nevyantseva, V.A. Belonogov, T.M. Timergazina, Russian Patent 2094546, Bull. Inv. 30 (1997). [3] R.R. Nevyantseva, S.A. Gorbatkov, E.V. Parfenov, A.A. Bybin, Surf. Coat. Technol. 148 (2001) 30. [4] A.N. Tikhonov, Nonlinear Ill-posed Problems, Chapman and Hall, New York, 1998. [5] R.A. Badamshin, N.D. Bublik, S.A. Gorbatkov, G.S. Nevostruev, A.V. Nikitin, E.V. Parfenov, IEEE Proceedings of the 3rd International Symposium SIBCONVERS’99, Tomsk, May 18–20, Tomsk State University of Control Systems and Radioelectronics, 1999, p. 494. [6] S. Haykin, Neural Networks: A Comprehensive Foundation, MacMillan, New York, 1994. [7] P.D. Wasserman, Advanced Methods in Neural Computing, Van Nostrand Reinhold, New York, 1993. [8] A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing, Prentice Hall, New Jersey, 1999. [9] E.V. Parfenov, S.A. Gorbatkov, R.R. Nevyantseva, D.A. Sosnovski, Proceedings of the Russian Conference CIT’03, Saint Petersburg, April 3–4, Saint Petersburg State Electrotechnical University, 2003, p. 258 (in Russian). [10] A.I. Mikhailovsky, R.R. Nevyantseva, E.V. Parfenov, A.A. Bybin, Russian Patent 2202451, Bull. Inv. 11, 2003.
Process control for plasma electrolytic removal of TiN coatings Part 2: Voltage control E.V. Parfenova,T, R.R. Nevyantsevaa, S.A. Gorbatkovb b
a Ufa State Aviation Technical University, 12 K. Marx Street, 450000 Ufa, Russia Russian Institute of Finances and Economics, Ufa branch, 69/1 Socialisticheskaya Street, 450006 Ufa, Russia
Available online 16 February 2005
Abstract The paper discusses two main problems of creating a control system for plasma electrolytic removal of titanium nitride coating from stainless steel hardware. The first feedback problem of unobservability of the surface state has been solved using indirect identification with an informative model of the process. The informative model has been developed on the base of the current signal processing and correlation analysis. DC component of the current and power spectral density at 1 kHz are in correlation with the surface state parameters. The second problem—inverse ill-posed control problem has been solved using inverse neural network, an object with artificial intelligence. For the solution of the inverse problem, the surface state and feedback parameters were used as the neural net inputs and voltage as the output. Because of approximation power of the neural network, the inverse problem has been resolved, and the treatment program for coating removal has been generated. Computational and practical tests show that the control system allows to intentionally change the state of the surface with 10% accuracy. D 2005 Elsevier B.V. All rights reserved. Keywords: Automation; Coating removal; Plasma electrolysis; Polishing; Titanium nitride; Neural networks; Ill-posed inverse problems
1. Introduction Plasma electrolysis can be used not only for hardening and polishing of metal surfaces, coatings deposition, but also in the field of renovation technologies for removal of coatings with deteriorated properties [1,2]. Successful renovation demands coating removal with high accuracy, especially in repair technology for jet aircraft engine blades. One of the best ways to improve processing accuracy is to create a control system for the process. Since there is no feedback control systems known for the plasma electrolysis, this research is the first attempt to create an approach to the process control and automation. The phenomenon of plasma electrolysis has been investigated for a significant number of applications, but feedback control is still an issue. The main problem is that the surface state, which is usually
T Corresponding author. Tel.: +7 3472 558551; fax: +7 3472 229909. E-mail address: [email protected] (E.V. Parfenov). 0257-8972/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.surfcoat.2004.10.144
characterized by roughness, microhardness, etc., is unobservable during the processing. This problem can be solved with indirect identification, which require parameters, both observable and controllable, to be found. The second problem that arises is an inverse control problem, which usually appears to be ill-posed. The paper proposes an approach for dealing with these problems on the example of creating a control system for plasma electrolytic removal of TiN coating from stainless steel samples. This paper is a logical continuation of Part 1, which is dedicated to the phenomenology of the process and to the issue of the duration control of the process. 1.1. Indirect identification: informative model The indirect identification of complex technological processes is usually based on a correlation between observable and unobservable parameters which both required to be controllable. For the search and classification of these parameters, a phenomenological model of the
E.V. Parfenov et al. / Surface & Coatings Technology 199 (2005) 198–204
199
Table 1 Parameters classification Parameter
Acronym
Position in experiment design
Input/Output position
Controllability/ Observability
Voltage (V) Initial temperature of the electrolyte (8C)
U T0
Factor Factor
Yes/Yes No/Yes
Duration of the processing (s) Percentage of area completely free of coating (relative units) Normalized average roughness (relative units) Average microhardness (Gpa) Temperature of the electrolyte (8C) Current (A)
t St
Factor Response
Input Constant parameter Time Output
– Yes/No
Ra HV T i
Response Response Response Response
Output Output Output Output
Yes/No Yes/No Yes/Yes Yes/Yes
coating removal process could be employed. The model is presented in Part 1, where a description of the system under consideration is also provided. One of the main results obtained from the phenomenological model is voltage controlled mechanism which yields bmatteQ surface after the coating removal and gives a direction for further polishing of the surface. This mechanism involves TiN oxidation to TiO and TiO2 in the bubble boiling mode, removal of oxides and polishing in the same mode. The classification of the parameters is shown in Table 1. As seen from the table, current and electrolyte temperature appear to be the only observable parameters. From this limited set, the current turns to be the most perspective for the search of informative parameters for indirect identification. As shown in Ref. [3], the current has both the DC and AC components with significant magnitude. The DC component (I) has the maximal value at the beginning of the treatment and exponentially falls with time; parameters of the exponent depend on the treatment conditions and the surface state. Therefore, the DC component could be one of the feedback parameters. The AC component has been analyzed in Part 1, where the power spectral density in the bandwidth 500H1500 Hz ( p 1) has been chosen as a feedback parameter. Correlation analysis analogous to that in Part 1 shows that the chosen feedback parameters are in correlation with the unobservable surface state parameters. Table 2 contains significant coefficients of paired correlation that has been discovered. As seen from lines 1 and 2, DC component (I) and power spectral density ( p 1) are in correlation with time derivatives of S t and Ra respectively. As seen from line 3, percentage of the area completely free of coating (S t ) is in correlation with microhardness (HV); therefore only one of
these parameters is independent. The same situation is in line 4: the DC component (I) is in correlation with the electrolyte temperature (T). Overall, two pairs of the parameters appear to be independent: S t and Ra for the surface state, I and p 1 for the feedback. Finally, the results obtained have been arranged into an informative model of the process (Fig. 1). This model solves the problem of indirect identification of the surface state during the processing. 1.2. Inverse control problem solution: intelligent control system Now having feedback parameters, the only question remains: how should the voltage be driven so that by the end of the treatment the coating would be removed and the surface would have roughness equal to or better than the initial? This control problem is inverse to the problem that is being solved by experiment design and further interpolation. Also this problem can be treated as a problem of constraint optimization, where parameter S t goes to a maximum and the value of Ra is a constraint for the search. This dual representation helps to decompose the problem. First, an inverse model should be created; second, a realizable treatment program should be designed. The inverse model should be capable of mapping the set of output parameters into the input set. This problem is illposed in the sense of Adamar [4], since for a correct problem all of the three following conditions must be satisfied. First, the solution of the problem must exist. Second, the solution must be unique. Third, the solution must be continuously dependent upon variations of the problem inputs. If any of these conditions fails to be satisfied, the problem becomes ill-posed. Two former conditions typically cause the ill-posedness of the process control inverse problems. For example, as
Table 2 Correlation between parameters No.
Parameter 1
1. 2. 3. 4.
I p1 St I
Coefficient of paired correlation 0.80 0.76 0.61 0.95
Parameter 2 dS/dt dRa/dt HV T
Controls:
St
k1 d dt
I
Ra
k2 d dt
p1
U T0 t
Fig. 1. Informative model of the system.
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Table 3 Fragment of the set M
1
2
Discrete time (n)
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
Factors
Surface state
Lag
Feedback
Duration t (s)
Voltage U (V)
Temperature T 0 (8C)
S t [n]
Ra[n]
S t [n 1]
Ra[n 1]
I (A)
p 1 (W)
0 15 30 45 60 75 90 105 0 15 30 45 60 75 90 105
179 179 179 179 179 179 179 179 320 320 320 320 320 320 320 320
56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56
0.00 0.10 0.11 0.14 0.16 0.19 0.22 0.24 0.00 0.03 0.03 0.04 0.04 0.05 0.06 0.06
1.00 0.86 0.86 0.86 0.87 0.88 0.89 0.91 1.00 1.05 1.05 1.05 1.05 1.06 1.06 1.06
0.00 0.00 0.10 0.11 0.14 0.16 0.19 0.22 0.00 0.00 0.03 0.03 0.04 0.04 0.05 0.06
1.00 1.00 0.86 0.86 0.86 0.87 0.88 0.89 1.00 1.00 1.05 1.05 1.05 1.05 1.06 1.06
5.0 6.0 8.5 3.5 2.7 2.3 2.1 1.6 4.2 4.0 2.2 1.8 1.5 1.4 1.3 1.2
0.058 0.025 0.030 0.033 0.026 0.026 0.018 0.018 0.019 0.017 0.016 0.017 0.016 0.016 0.012 0.012
seen from Fig. 7 in Part 1, for one set of the factors, say, U=300 V, T 0=50 8C, t=600 s, a definite output exist: S=0.5, Ra=0.7. It is possible that the output would be the same for a set of other factor levels, such as U=350 V, T 0=70 8C, t=600 s. Now, if the input and the output are inverted, then for one input of the inverse model, S=0.5, Ra=0.7, there are two valid outputs, i.e., two solutions. Moreover, the solution may not even exist if the input is set at S=1.0, Ra=0.1. Thus, the regularization of the inverse model should be performed. The regularization of the ill-posed control problem has been performed in the sense of Tikhonov [4]. The problem is correct in the sense of Tikhonov if all of the three following conditions are satisfied. First, the solution of the problem must exist on a set M. Second, the solution must be unique within the set M. Third, the solution must be continuously dependent upon variations of the problem inputs within the set M. In the case of experimental investigation of the process control, the set of experimental data or its subset could be treated as the set M. Let us discuss this regularization in the case of the plasma electrolytic coating removal. First, for the existing of the solution, the treatment program must be realizable on the set M. This requires taking into consideration the phenomenological model. Second, for the solution being unique on the set M, the set must be constructed in such way that all its entries are distinct. Third, for the solution being continuously dependent upon variations of the inputs within the set M, the model must employ continuous and preferably smooth functionals. The second and the third conditions demand highly adequate and smooth approximation of the dependencies within the set M. In Ref. [5] it has been shown that this kind of approximation cannot be performed with conventional regression analysis, which deals with polynomial equations and requires significantly distinct input values. These input values are usually obtained by means of experiment design,
where the factors’ values are set to distinct levels. In the case of the inverse problem, no experiment design could be performed unless its solution is known. Nevertheless, the highly adequate and smooth approximation could be achieved by using an artificial neural network. Being an object with artificial intelligence, which makes the model resistant to noise and even contradictions [6], in this case a neural network can be treated as an advanced regression. From the wide variety of different kinds of neural networks, a general regression neural network, which is a net with a radial basis function, has been chosen for its ability to smooth interpolation [7]. The neural net has one hidden layer with a Gaussian activation function, which width controls the model adequacy. Before being used, any neural network needs a set of data to be trained on. In this case, the neural network must be trained on the set M, having all output parameters in the input and voltage U in the output. This network will be called binverse.Q Let us proceed with construction of the set M. This set must contain all known information and could be any Inverse neural network
St [n] Ra [n] ••• 1250 neurons
No. of realization
St [n-1] Ra [n-1] I [n]
U [n]
p1 [n] T0 Input layer
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Output layer
Fig. 2. Inverse neural network model structure.
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C Ra and St in relative units
Ra and St in relative units
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Fig. 3. Inverse model test: (A) modeled trajectory of parameters St and Ra for U=180 V, T 0=70 8C; (B) the inverse problem solution U[n]; (C) modeled trajectory for U[n] and T 0=70 8C.
combination of data that was obtained either experimentally, or computationally. The set M is a table that contains data for all realizations tabulated in discrete time n every Dt=15 s (Table 3). This sampling interval has been calculated after applying Nyquist sampling theorem [8] to a typical curve of DC component. Columns U and T 0 contain factors’ values. Columns S t [n] and Ra[n] contain their measured values at the beginning and the end of the realization, and calculated values in between. The calculated data were obtained from modeling of the surface state with a direct neural network model as described in Part 1 and Ref. [9]. Columns I, p 1 contain measured over Dt DC component and averaged over Dt power spectral density in the bandwidth 500H1500 Hz. Columns S t [n 1] and Ra[n 1] contain lag variables that represent the state of the surface on the previous time step in the discrete time n. Introducing these variables and excluding duration t from the inverse neural net makes this model dynamic, since this way it becomes a set of discrete time nonlinear differential equations. The inverse neural network, which structure is shown in Fig. 2, has been successfully trained on the set M. The
a
adequacy of the inverse problem solution is illustrated in Fig. 3. Chart (A) corresponds to modeled trajectory of parameters S t and Ra for U=180 V, T 0=70 8C. Chart (B) clearly illustrates that for the whole operational range the inverse problem solution U[n] appear to be almost constant at 180 VF10%. Chart (C) is the modeled trajectory for U[n] and T 0=70 8C. The inverse neural network obtained is trained over the whole set of data. Any new data would increase the neural network capabilities; therefore, additional training should be performed in case of considering other work piece size, material, etc. This means that the neural network should be upgraded, not completely retrained. This will not change the phenomenological model, since it describes basic physical and chemical processes of the plasma electrolysis. With inverse and direct models of the process developed, it is now possible to create the treatment program and complete the control system. After analysis of the phenomenological model, the following program has been proposed (Fig. 4). For this program initial temperature of the electrolyte T 0=70 8C has been chosen. In full accordance
c
U, V
S, relative units
Ra, relative units
b
Time,s
Time,s
Time,s
Fig. 4. Generating treatment program for the coating removal: solid line—desired trajectory; dashed line—realizable trajectory.
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b
U, V
Ra, relative units
a
Time,s
Time,s
Fig. 5. Generating treatment program for the polishing: solid line—desired trajectory; dashed line—realizable trajectory.
with the phenomenological model, the roughness after the coating removal is higher than the initial (RaN1.0, see Fig. 4c), which requires separate inverse and direct models and feedback parameters for the process of polishing. Analogous investigation has been performed for the process of plasma electrolytic polishing of the steel surface at the electrolyte temperature 90 8C. It has been established that the average power spectral density p 1 is in correlation with the roughness Ra of the surface being polished. The coefficient of paired correlation is 0.95. For this process, analogous inverse neural network has been created with inputs Ra[n], Ra[n 1], p 1[n] and output U[n]. Generation of the treatment program for polishing is shown in Fig. 5.
Finally, the control system for the plasma electrolytic coating removal has the following structure (Fig. 6). The treatment program is fed to the intelligent controller, where S t [n 1] and Ra[n 1] are acquired with the unittime delay block (z 1). Calculated U[n] is transformed into U(t) with a digital to analog converter (DAC) every Dt of 15 s. Current of the plasma electrolysis is measured with an analog to digital converter (ADC) and transferred both to a low pass filter (LPF) and to a band pass filter (BPF) to obtain I[n] and p 1[n] respectively. These feedback signals are also fed to the controller. The treatment program block, the intelligent controller and the feedback block are embedded in an operator’s personal computer.
Intelligent Controller T0
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Treatment Program
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S t [n]
St[n]
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DAC
Inverse neural network model
n
Clock p1 [n]
I [n]
LPF
Feedback
0... 1 Hz
i (t )
BPF
ADC
500..1500 Hz
Fig. 6. Structure of the control system.
Plasma Electrolytic Coating Removal
St (t )
Ra(t)
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Percentage of the area completely free of coating
Voltage control curve
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350 Polishing
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Coating removal
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Fig. 7. Control system tests results: solid line—calculated realizable trajectory; dashed line—experimentally recorded trajectory; — experimental points.
Operational ranges of the control system are the following: U=150H400 V, T 0=20H90 8C, i=0H10 A. Computational and practical tests of the control system show that it is capable of driving the surface of a treated sample over the desired realizable trajectory (Fig. 7). This fact is supported, first, by the stable following the desired trajectory. Second, measurements of the surface parameters after the treatment show that the calculated trajectory lays within the 10% neighborhood of the experimental points. The surface micrographs are presented in Fig. 8. Third, the recorded trajectory of the feedback signal-average power spectral density p 1 has either 10% mean square deviation from the calculated one.
It is supposed that with sufficient amount of information used for the neural network training, the intelligent control system should be robust. Nevertheless, the serious issue of the control system robustness is currently under investigation. The results of pilot experiments with different shape of the samples, other electrolyte, material and coating gives a hope that the control system would be robust.
2. Conclusion On the base of the phenomenological model, an informative model of the process has been created. It has
Fig. 8. Surface micrographs after the voltage controlled treatment: (a) 480 s; (b) 840 s.
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been shown that the process of the coating removal could be sufficiently described with the percentage of the area that is completely free of the coating and average roughness of the surface. A robust correlation between these parameters and the DC component and the average power spectral density in the bandwidth 500. . .1500 Hz has been established. Thus, the main problem of creating feedback control has been resolved. An inverse mathematical output–input model of the process has been created with the means of artificial intelligence. The model resolves the ill-posed inverse control problem. It has been employed for direct control of the process with regard of the actual surface state that is observed through the established feedback parameters. An automated control system for the process has been constructed. Computational and practical tests show that the control system allows to intentionally change the state of the surface with 10% accuracy, and, therefore, to successfully remove TiN coating for further replating. The method for the process control of plasma electrolytic coating removal is protected with Russian Patent [10].
References [1] A.L. Yerokhin, X. Nie, A. Leyland, A. Matthews, S.J. Dowey, Surf. Coat. Technol. 122 (1999) 73. [2] N.A. Amirkhanova, R.R. Nevyantseva, V.A. Belonogov, T.M. Timergazina, Russian Patent 2094546, Bull. Inv. 30 (1997). [3] R.R. Nevyantseva, S.A. Gorbatkov, E.V. Parfenov, A.A. Bybin, Surf. Coat. Technol. 148 (2001) 30. [4] A.N. Tikhonov, Nonlinear Ill-posed Problems, Chapman and Hall, New York, 1998. [5] R.A. Badamshin, N.D. Bublik, S.A. Gorbatkov, G.S. Nevostruev, A.V. Nikitin, E.V. Parfenov, IEEE Proceedings of the 3rd International Symposium SIBCONVERS’99, Tomsk, May 18–20, Tomsk State University of Control Systems and Radioelectronics, 1999, p. 494. [6] S. Haykin, Neural Networks: A Comprehensive Foundation, MacMillan, New York, 1994. [7] P.D. Wasserman, Advanced Methods in Neural Computing, Van Nostrand Reinhold, New York, 1993. [8] A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing, Prentice Hall, New Jersey, 1999. [9] E.V. Parfenov, S.A. Gorbatkov, R.R. Nevyantseva, D.A. Sosnovski, Proceedings of the Russian Conference CIT’03, Saint Petersburg, April 3–4, Saint Petersburg State Electrotechnical University, 2003, p. 258 (in Russian). [10] A.I. Mikhailovsky, R.R. Nevyantseva, E.V. Parfenov, A.A. Bybin, Russian Patent 2202451, Bull. Inv. 11, 2003.