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Boundary Control of Reacting Species in Semi-infinite and Finite Diffusion Processes
17th IFAC Workshop on Control Applications of Optimization 17th IFAC Workshop onOctober Control Applications Optimization Yekaterinburg, Russia, 2018 of 17th IFAC Workshop on Control 15-19, Applications ofonline Optimization Available at www.sciencedirect.com 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, 2018 of Optimization Yekaterinburg, Russia, October 15-19, 2018 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, 2018 of Optimization Yekaterinburg, Russia, October 15-19, 2018
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IFAC PapersOnLine 51-32 (2018) 73–78
Boundary Control of Reacting Species in Semi-infinite and Finite Diffusion Processes Boundary Control of Reacting Species in Semi-infinite and Finite Diffusion Processes Boundary Semi-infinite and Finite Diffusion Processes Boundary Control Control of of Reacting Reacting Species Species in in Semi-infinite and Finite Diffusion Processes R. Semi-infinite Tenno Boundary Control of Reacting Species in and Finite Diffusion Processes R. Tenno
R. R. Tenno Tenno Aalto University, School of Electrical Engineering R. Tenno Aalto University, Electrical Engineering PO BoxSchool 15500,of Aalto, Finland Aalto University, School of Electrical Engineering Aalto University, School ofAalto, Electrical Engineering PO Box 15500, Finland PO Box 15500, Aalto, Finland Aalto University, School of Electrical Engineering PO Box 15500, Aalto, Finland Abstract: Boundary control of oxidizing species at the cathode surface is considered in an PO Box 15500, Aalto, Finland Abstract: Boundary control of at the cathodeand surface in an electrochemical system, which isoxidizing the plantspecies of two electrodes plant is of considered three electrodes Abstract: control of species at the surface is in Abstract: Boundary Boundary control of isoxidizing oxidizing species at electrodes the cathode cathodeand surface isof considered considered in an an electrochemical system, which the plant of two plant three electrodes submerged in thesystem, stirred and unstirred electrolytes. The feedforward controls are found using electrochemical which the plant of electrodes and plant three electrodes Abstract: Boundary control of is at exact the cathode surface isof in an electrochemical system, which isoxidizing theelectrolytes. plantspecies of two two electrodes and plant of considered three electrodes submerged in the stirred and unstirred The exact feedforward controls are found using ideas of motion planning, Laplace techniques that the end the inverse submerged in the stirred and unstirred The exact feedforward controls are found using electrochemical system, which istransform theelectrolytes. plantand of other two electrodes and inplant of solve three electrodes submerged in the stirred and unstirred electrolytes. The exact feedforward controls are found using ideas of motion planning, Laplace transform and other techniques that in the end solve the inverse problem forinthe time-variable reference of species at thefeedforward cathode surface. The uncertainty of ideas of planning, Laplace transform and other that the end solve the submerged thegiven stirred and unstirred electrolytes. The techniques exact controls are found using ideas of motion motion planning, Laplace transform and other techniques that in in the endThe solve the inverse inverse problem for the given time-variable reference of species at the cathode surface. uncertainty of surface reaction existing inLaplace the system is suppressed with two modifications ofend controls. In the first problem for the given time-variable reference of species at the cathode surface. The uncertainty of ideas of motion planning, transform and other techniques that in the solve the inverse problemreaction for the given time-variable reference of species attwo themodifications cathode surface. The uncertainty of surface existing in the system is suppressed of controls. In the first case, thereaction feedforward control is extended with aspecies PIwith feedback that usessurface. measured in the process surface existing in the system is suppressed with two modifications of controls. In the first problem for the given time-variable reference of at the cathode The uncertainty of surface reaction existing in the system is suppressed with two modifications of controls. In the first case, the feedforward control is extended with aa PI feedback that uses measured in the process electric current passing through the system. In the second case, an adaptive feedforward control is case, the feedforward control is extended with PI feedback that uses measured in the process surface reaction existing in the system is suppressed with two modifications of controls. In the first case, thecurrent feedforward control isthe extended with a PI feedback that uses measured in thecontrol process electric passing through system. In the second case, an adaptive feedforward is established for the case when the exchange current density is unknown drifting parameter and is electric passing through system. In the second case, an feedforward case, thecurrent feedforward control isthe extended a PI feedback that uses measured in thecontrol process electric current passing through the system.with In the second case, an adaptive adaptive feedforward control is established for the case when the exchange current density is unknown drifting parameter and is estimated as it changes using a conditionally Gaussian filter. In both cases, the process control is established for the case when the exchange current density is unknown drifting parameter and electric current passing theexchange system. In the second case, an adaptive feedforward established for the case through when athe current density isInunknown drifting parameter and is is estimated as it changes conditionally Gaussian filter. both cases, the process control simulated and revealed tousing be effective. estimated as it changes using a conditionally Gaussian filter. In both cases, control established for the case when the exchange current density is unknown drifting parameter and is estimated and as itrevealed changestousing a conditionally Gaussian filter. In both cases, the the process process control is simulated be effective. simulated and revealed be effective. estimated as it(International changesto a conditionally Gaussian filter. In both cases, process control is simulated and revealed tousing beFederation effective. © 2018, IFAC of Automatic by control Elsevierthe Ltd. All rights reserved. Keywords: Distributed system, stochastic Control) process, Hosting boundary simulated and revealed parameter to be effective. Keywords: Keywords: Distributed Distributed parameter parameter system, system, stochastic stochastic process, process, boundary boundary control control Keywords: Distributed parameter system, stochastic process, boundary control Keywords: Distributed parameter system, stochastic process, boundary control 1. Introduction i (t ) D x c(t , x) 0 i (t ) . 1. Introduction (3) 1. Introduction nii ((ettF)) .. D x cc((tt ,, xx)) 0 (3) 1. Introduction D (3) The 1. problem solved in this paper stems from an industrial D xx c(t , x) 00 n Introduction (3) i (eetF ) . n F The problem solved in this paper stems from an industrial n F D c ( t , x ) . (3) e x In the case of 0unstirred electrolytes (semi-infinite diffusion), plating process, where the production maximization is crucial. The problem solved in this paper stems from an industrial ne F electrolytes (semi-infinite diffusion), The problem solved inthethis paper stems from an is industrial In the case of unstirred plating process, where production maximization crucial. condition at the electrolytes outer boundary is givendiffusion), as the In case (semi-infinite This is process, closelysolved related tothis thepaper plating rate from maximization and the plating where production maximization crucial. The problem stems an is industrial In the thecondition case of of unstirred unstirred electrolytes (semi-infinite diffusion), plating process, whereinthe the production maximization is crucial. the at outer boundary is given as the This is closely related to the plating rate maximization and concentration inunstirred bulkthe solution the condition at the outer boundary is given this in turn means maximization of the current density. This is closely related to the plating rate maximization and In the case of electrolytes (semi-infinite diffusion), plating process, where the production maximization is crucial. the conditionin at the outer boundary is given as as the the This is closely relatedmaximization to the platingofrate maximization and concentration bulk solution this in turn means the current density. concentration in bulk solution the condition at the outer boundary is given as the However, a substantial rise of the current density ends up with this in turn means maximization of the current density. This is closely related to the plating rate maximization and concentration in bulk solution c ( t , ) c . (4a) this in turn means maximization of the current density. bulk However, aaofsubstantial rise of the current density ends up with concentration in bulk solution depletion the oxidizing species at the the cathode surface c ( t , ) c However, substantial rise of the current density ends up with . (4a) this in turn means maximization of current density. bulk . c((tt ,, (4a) However, aofsubstantial rise ofspecies the current density ends up with depletion the at the cathode surface ))agitated ccbulk bulk . electrolyte a finite diffusion takes (4a) which leads to aoxidizing reduced plating rate and production. depletion the oxidizing at the cathode surface place However, aof substantial rise ofspecies the current density ends up with In ccwell depletion of the oxidizing species at the cathode surface ( t , ) c . (4a) bulk which leads to aaoxidizing reduced plating rate and production. In well agitated electrolyte aathe finite diffusion takes place Therefore, reactant concentration control is a close objective which leads to reduced plating rate and production. In well agitated electrolyte finite diffusion takes place of stagnation layer. In such within a certain thickness depletion of the species at the cathode surface which leads to a concentration reduced plating rateis aand production. In wella certain agitatedthickness electrolyte athe finite diffusion takes placeaa Therefore, reactant control close objective of stagnation layer. In such within casewell thea inner boundary condition remains the layer. same, while thea that helps toreactant avoid the depletionplating while maximizing the current In Therefore, concentration control is a close objective of the stagnation In such within certain thickness which leads to a reduced rate and production. agitated electrolyte a finite diffusion takes place Therefore, reactant concentration control is a closethe objective of theremains stagnation layer. In such a within a inner certain thickness case the boundary condition the same, while the that helps to avoid the depletion while maximizing current outer boundary immobile (fixed spatially condition) case the inner boundary condition remains the same, while the density through stabilisation of the boundary concentration at that helps to avoid the depletion while maximizing the current Therefore, reactant concentration control is a close objective of the stagnation layer. In such a within a certain thickness case the inner boundary condition remains the same, while the that helps to avoid the depletion while maximizing the current outer boundary immobile (fixed spatially condition) through stabilisation of the boundary concentration at outer boundary immobile (fixed spatially condition) adensity wisely chosen reference profile. The concentration control is case density through stabilisation of the boundary concentration at the inner boundary condition remains the same, while the that helps to avoid the depletion while maximizing the current outer through stabilisation of theThe boundary concentration at c(t boundary , ) cbulk .immobile (fixed spatially condition) (4b) aadensity wisely chosen reference profile. control is outer immobile (fixed spatially condition) transient control problem. Itofaims to concentration bring the concentration wisely chosen reference profile. concentration control is through stabilisation theThe boundary concentration at cc((tt boundary ,, ) c . (4b) bulk aadensity wisely chosen reference profile. The concentration control is ccbulk . (4b) transient control problem. It aims to bring the concentration 3 c(t , )) c(t,x) the boundary from a bulk solution concentration to a transient control problem. It aims to bring the concentration bulk . is the concentration (mol/m ) of species(4b) aat wisely chosen reference profile. The concentration control is In (1)-(4), in the aattransient control from problem. It aims to bring the concentration c(t , ) c(t,x) cbulk . is the concentration (mol/m33) of species(4b) the boundary a bulk solution concentration to a In (1)-(4), in the level in from a completely controlled manner. at the boundary a bulk solution concentration to a is the bulk solution depth x of the stagnation layer (m), c aprescribed transient control problem. It aims to bring the concentration In (1)-(4), c(t,x) is the concentration (mol/m ) of species in the 3 bulk at the boundary from a bulk controlled solution concentration to a In (1)-(4), c(t,x) is the 3concentration (mol/m ) of species in the 2 prescribed level in a completely manner. is the bulk solution depth x of the stagnation layer (m), c bulk is3 the ), D is the diffusivity (m /s), F is concentration (mol/m prescribed level in a completely controlled manner. bulk solution depth x of the stagnation layer (m), c at the boundary from a bulk solution concentration to a In (1)-(4), is the 3concentration (mol/m ) of species in the 2 prescribed level in a completely controlled manner. bulk solution depth x ofc(t,x) the (mol/m stagnation (m), cbulk bulk is the 3), Dlayer 2/s), F is the diffusivity (m is the concentration number of electrons Faraday constant (96485 C/mol), ndiffusivity is (m F is concentration 3), D 2/s), 2. The problem statements prescribed level in a completely controlled manner. e iscthe depth x of the (mol/m stagnation layer (m), bulk is the ), D is the the diffusivity (mbulk /s), Fsolution is the the concentration (mol/m is the number of electrons Faraday constant (96485 C/mol), n 3 2 2. The problem statements e involved in an (mol/m electrode reaction and isnumber the(m current density the of electrons Faraday constant (96485 C/mol), n 2. The problem statements ), D is the diffusivity /s), F is the concentration e isi(t) is the number of electrons Faraday constant (96485 C/mol), n 2. The problem statements 2 e i(t) is the current density involved in an electrode reaction and ) of an electrode reaction that satisfies a bidirectional (A/m involved in an electrode and i(t) is the current density General electrodeposition obeys convection, migration and is the number of electrons Faraday constant (96485 C/mol), n 2. The problem statements 2 e i(t) is the current density involved an electrode electrode reaction reaction and 2) ofinan that satisfies aa bidirectional (A/m General electrodeposition obeys convection, and Butler-Volmer (B-V) equation [1,and 2] i(t) reaction that satisfies (A/m 2) ofinan diffusion laws. However, when observing the migration mass-transfer involved an electrode electrode is the current density General electrodeposition obeys convection, migration and ) of an electrode reaction that satisfies a bidirectional bidirectional (A/m General electrodeposition obeys convection, migration and Butler-Volmer (B-V) equation [1, 2] 2 diffusion laws. However, when observing the mass-transfer Butler-Volmer (B-V) equation [1, 2] effects close toHowever, an electrode surface, thethediffusion does ) of an electrode reaction that satisfies a bidirectional (A/m diffusion laws. when observing mass-transfer General electrodeposition obeys convection, migration and Butler-Volmer equation 2] diffusionclose laws.toHowever, when surface, observingthethediffusion mass-transfer k 1(B-V) effects an electrode does c(t , 0) [1, E ( t ) equation k E (2] t) prevail. For that reason, concentration of the reacting species is Butler-Volmer effects to an electrode does diffusionclose laws. However, when surface, observing thediffusion mass-transfer c(t , 0) e[1, i(t ) i0 ek 1(B-V) (5) . E ( t ) effects close to an electrode surface, the diffusion does k E ( t) prevail. For that reason, concentration of reacting species is c ( t , 0) k 1 E ( t ) k E ( t) i ( t ) i e e . (5) c modelled by athat diffusion equation: prevail. For reason, concentration of reacting species is 0 bulk effects close to an electrode surface, the diffusion does c ( t , 0) k 1 E ( t ) ii((tt )) ii0 ee ee k E (t ) .. (5) prevail. For that reason,equation: concentration of reacting species is c modelled by a diffusion (5) bulk c ( t , 0) 0 k 1 E ( t ) c k E ( t ) modelled by diffusion prevail. reason,equation: concentration of reacting species is e cbulk i ( t ) i e . (5) the 2 modelledFor by aathat diffusion equation: 0 bulk 2 In (5), i is the exchange current density (A/m ), 0 cbulk current density (A/m22), is modelled t c(t , xby ) aDdiffusion 2x c(t , x) .equation: (1) is the exchange In (5), i is the 0 apparent coefficientcurrent and k isdensity the temperature voltage 22x cc((tt ,, xx)) .. (1) In (5), i0transfer is (A/m is t c(t , x) D (1) In is the the exchange exchange current density (A/m22), ), voltage is the the -1(5), i0transfer apparent coefficient and k is the temperature tt cc((tt ,, xx)) D D 2xx c(t , x) . (1) ). The mass-transfer effect is considered through (V apparent coefficient and kk is the temperature i0transfer is the exchange current density (A/m ), including voltage is the -1(5), apparent transfer coefficient and is the temperature voltage (t , x) D x c(t , x) . of species is assumed to (1) be In Thet cinitial distribution -1). The mass-transfer effect is considered through including (V the of oxidizing near the including electrode ).concentration The mass-transfer effect is through (V apparent coefficient kspecies is the temperature voltage The initial distribution of species is assumed be Thetransfer mass-transfer effectand is considered considered through including (V-1 homogeneous in the stagnation layer and equal to theto bulk The initial distribution of species is assumed to be the concentration oxidizing species near the electrode -1). surface c(t,0) in the of model (5). is E(t) is the voltage applied to an concentration of oxidizing species near the electrode The initial distribution of species is equal assumed to bulk be the ). The mass-transfer effect considered through including (V homogeneous in the stagnation layer and to the the concentration of oxidizing species near the electrode solution concentration: homogeneous in the stagnation layer and to the surface c(t,0) in the model (5). E(t) is the voltage applied to an The initial distribution of species is equal assumed to bulk be electrochemical system (V), considered as the control. In twosurface c(t,0) in the model (5). E(t) is the voltage applied to an homogeneous in the stagnation layer and equal to the bulk the concentration of oxidizing near the electrode solution concentration: surface c(t,0) in the model (5). E(t)species is theasvoltage applied to an solution concentration: electrochemical system (V), considered the control. In twohomogeneous in the stagnation layer and equal to the bulk electrode plant, the applied voltage is the difference between electrochemical system (V), considered as the control. In twosolution concentration: c(0, x) cbulk . surface c(t,0) in system model (5). E(t) is theasvoltage applied to an (2) electrochemical (V), considered the control. In twoelectrode plant, the applied voltage is the difference between solution concentration: cc(0, xx)) ccbulk .. (2) anodic andplant, cathodic overpotentials each ofthe them is once electrode the applied is the difference between electrochemical system (V), voltage considered as control. In more two(0, (2) bulk electrode plant, the applied voltage is the difference between x) cbulkon. the cathode is given by the first Fick law (2)and anodic and cathodic overpotentials each of them once more Thecc(0, condition between the electrode potential andis equilibrium anodic and cathodic overpotentials of them is once more electrode the applied voltageeach is the between (0, x) cbulkon. the cathode is given by the first Fick law (2)and difference anodic andplant, cathodic overpotentials each of difference them isequilibrium once more The condition difference between the electrode potential and states that the molar flux of reacting species is proportional to The condition on the cathode is given by the first Fick law and difference between the electrode potential and equilibrium U ( t ) can be measured in potential. The equilibrium potential anodic and cathodic overpotentials each of them is once more The condition on the flux cathode is givenspecies by the first Fick law and difference between the electrode potential and equilibrium states that the molar of reacting is proportional to U ((tt )) can be measured in potential. The equilibrium potential the cathode current density: states that the molar flux of reacting species is proportional to The condition on the cathode is given by the first Fick law and U can be measured in potential. The equilibrium potential difference between the electrode potential and equilibrium practice with to a potential referenceUelectrode SHE) states that the molardensity: flux of reacting species is proportional to potential. (t ) can be(versus measured in The respect equilibrium the cathode current the cathode current density: practice with respect to a reference electrode (versus SHE) states that the molar flux of reacting species is proportional to U ( t ) can be measured in potential. The equilibrium potential practice with respect to a reference electrode (versus SHE) the cathode current density: practicebywith respect to rights a reference 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting Elsevier Ltd. All reserved.electrode (versus SHE) the cathode current density: Copyright © 2018 IFAC 73 practice with respect to a reference electrode (versus SHE) Peer review under responsibility of International Federation of Automatic Control. Copyright © 2018 IFAC 73 Copyright © 2018 IFAC 73 10.1016/j.ifacol.2018.11.356 Copyright © 2018 IFAC 73 Copyright © 2018 IFAC 73
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that is submerged in electrolyte at the cathode surface in the case of three-electrode system. The device shown in Fig 1 is used for the boundary concentration c (t , 0) evaluation from the indirect measurement U (t ) . The evaluation procedure is explained later in subsection 4.1, where the Nernst relationship (25) and its inverted stochastic modification (26) are introduced.
solvable ordinary differential equation. Incorporating the corresponding boundary conditions yields the solution in a complex argument variable S
1 1 x cref ( S , x) cbulk cmin e S S
cbulk . S
(8)
Here cref (S,x) is the Laplace transform of cref (t,x) and the subscript ref indicates correspondence to the target profile. Equation (8) is the solution of motion planning problem in the Laplace domain for the model (1)-(3) and the reference concentration (6) at the boundary (7). The controller can be derived without the analytical inverse transformation of (8) in the whole subdomain but on the controlled boundary. Indeed, according to the Neumann boundary condition (3) the controller is governed by the flux of species at the boundary. Differentiating cref (S,x) with respect to x and taking controlled boundary x = 0, yields
E (t )
U (t )
S D
Fig. 1. Scheme and apparatus of three-electrode measurements.
x cref ( S , x)
We are seeking for a control that brings the concentration at the boundary from the bulk solution concentration cbulk to a lower level cmin near the cathode surface in a fully controlled manner. The prescribed reference is set by a time-varying function that declines exponentially cref (t ) cmin cbulk cmin e
t
.
0
x cref (t , x)
(6)
D
0
(9)
cbulk cmin je t erf j t .
(10)
In (10), symbol j stands for the imaginary unit. It is known that jerf ( jz ) erfi( z ) for any complex argument z [4], where the function erfi is the imaginary error function defined as z 2 t2 erfi z (11) e dt .
0
Consequently, taking real value z t j 0 the desired flux at the boundary can be written in the terms of the imaginary error function as
Motion planning
The problem of designing a voltage E(t), which is to be applied to (1)-(5) forces the concentration at the boundary to follow a given reference trajectory, is a boundary control problem known also as a motion planning problem [3]. To solve the motion planning problem one should find a reference profile cref (t,x) in the whole subdomain of the stagnation layer. This profile can be found by replacing in (1)(4) the boundary condition (3) with
c(t , 0) cref (t ) .
.
The solution in time domain can be found as the inverse Laplace transforms of (9)
Here cref is the reference trajectory of the concentration at the boundary, cmin and are the positive-valued constants, that specify the desired concentration at the boundary at steady state and the rate of tracking to steady state respectively. We are not concerned with the question of choosing cmin and . In this paper, we will focus on the development of a control law to follow the reference trajectory (6).
3.
cbulk cmin 1 S D S S
x cref (t , x )
D
0
cbulk cmin e t erfi t .
(12)
In contrast to the error function, the imaginary error function rapidly diverges to infinity as z grows. However, the function
F z e z erfi z is bounded for all real z-values. To find the boundary control the boundary condition (3) can be utilized. In the tracking mode the flux at the boundary also obeys the first Fick law 2
(7)
The motion planning problem is solved further in the case of semi-infinite diffusion and finite-length diffusion processes for the reference trajectory given by (6).
D x cref (t , x) 0
iref (t ) ne F
,
(13)
where iref(t) is the current density corresponding to the desired reference concentration. Assuming the bidirectional electrode model (5) k 1 Eref (t ) cref (t ) k Eret (t ) iref (t ) i0 e e (14) , cbulk
3.1 Semi-infinite diffusion Perhaps, the simplest way to discovery the reference trajectory in the whole subdomain in the case of semi-infinite diffusion is to apply the Laplace transform to the diffusion equation (1) with the Dirichlet boundary conditions (7) and (4a). This transform reduces the diffusion equation (1) into a
the tracking control can be found as follows. Define a function 74
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y eax cebx f x; a, b, c .
flux at the boundary satisfies the first Fick law (13). Assuming the bidirectional model (14) leads to the following boundary control
(15)
In this definition, a and b are the arbitrary nonnegative constants and c is the strictly positive. The latter function is invertible, partly 2 f 1 y;1,1,1 ar sinh( y), generally, inverse
ne FD cbulk cmin i0 2 e t e n Dt Eref (t ) f 1 1 e t 2 n2 D n 1 cref (t ) k (1 ), k , cbulk
is a continuous bounded function, denoted by x f 1 y; a, b, c .
(16)
Given y, a, b, and c, the inverse function above can be fast computed using the bisection method, Newton’s method or other simple method. The control, that forces the concentration at the boundary to follow the prescribed reference trajectory is the continuous bounded function (17). ne F D cbulk cmin e t erfi i 0 Eref (t ) f 1 c (t ) k (1 ), k , ref cbulk
t ;
Fig. 2. The feedforward control system. This represents the motion planed control for maintenance of the boundary concentration at given target.
w(t, x) cref (t, x) cref (t ) 1 x cbulk cref (t ) ,
The designed feedforward boundary controllers (17) and (23) do not tolerate extensive uncertainties. To deal with uncertainties (unmodeled convection, migration and other effects) they should be accompanied with a feedback.
(18)
4.
one can lift the boundary data to homogeneous boundary that contributes a result: a source-like term
cbulk cmin 1
1
x
(19)
t w(t , x) D 2x w(t , x) R(t , x) ,
(20)
With the zero initial condition w(0, x) = 0 and the homogeneous boundary conditions w(t, 0) = 0 and w(t, ) = 0. This problem can be solved analytically using the separation of the variables method. This eventually yields the following solution (21)
cbulk cmin
e t en Dt 2
n 1
n
D 2 n
sin n x , n
n
4.1 PI modified feedforward control The feedforward controls (17) and (23) can be improved with a simple feedback, if instead of cref (t) its PI modified values c f (t ) are used t 1 c f (t ) K p cref (t ) (t ) cref (s) ( s) ds (24) T i 0 ( t ) is the measured concentration at the boundary In (24), (mol/m3), Kp is the regulator gain, and Ti is the integral time (s). The concentration c (t , 0) at the boundary can be measured indirectly as the potential U (t ) or voltage between electrodes in a three electrodes system [1]. This voltage and concentration are related to each other as shown below in the Nernst relationship
.
Using (21), the reference profile given as the solution of the original problem can be expressed as
cref (t, x) w(t, x) cref (t ) 1 x cbulk cref (t ) . Consequently, the flux at the boundary is given by cbulk cmin e t en Dt t 1 e 2 2 n D n 1 2
x cref (t , x) 0
Uncertain system control
Effective use of the feedforward controls (17) and (23) requires a good knowledge of the model, which is not the case. Usually, the model is known poorly and is more complex. For that reason, the deterministic controls are modified below with a feedback from measured in the process data and with adjustment of the model to unknown stochastic drift.
in the diffusion equation
w(t , x) 2
(23)
(17)
Like that of semi-infinite diffusion we calculate the reference profile cref (t,x) inside the whole subdomain from the diffusion equation (1),(2) with the Dirichlet boundary conditions (7) and (4b). Using a new variable
R(t, x) e
; .
The control system. The structure of the control system developed for maintenance of the concentration of species at the tracking reference at the boundary is depicted in Fig. 2.
3.2 Finite-length diffusion
t
75
. (22)
U (t ) U eq
Once again to develope the boundary control, the inner boundary condition (3) is utilized. In the tracking mode the 75
1 c(t , 0) ln . k cbulk
(25)
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where is the measure current density (A/m2), B is a given parameter that specifies accuracy of the measurements, V is a generalized white noise that can be expressed as another Wiener process V in Ito integral and At is the time-varying function specified by (5) as
Here U (t ) is the electrode potential and Ueq is the equilibrium potential measured versus a reference electrode [1]. If the equation (25) is inverted (solved with respect to the concentration) and additive white noise error is added, which is more relevant to the real case, the measurement signal can be expressed as in (26).
At e
(t ) cbulk e
k U (t ) Ueq
r (t )
k 1 E ( t )
(26)
c(t , 0) k E ( t ) e . cbulk
(29)
In the case of three-electrode system the boundary concentration can be found from the Nernst equation and if replaced in (29) gives
Here is the measured indirectly boundary concentration, is the independently distributed standard Gaussian variable of zero mean and unit variance ~ N(0, 1), and r is the accuracy of measurements.
At ek 1 E (t ) e
k U (t ) Ueq
ek E (t ) .
(30)
The unknown drift of the exchange current density can be estimated from the measured data using the best mean-square filter [7] specified for the system (27) and (28):
The feedback control system. The structure of the control system developed for uncertain model for maintenance of the concentration of species at the tracking reference at the boundary is depicted in Fig. 3. The feedforward controller is extended with the PI-feedback. This system obliges threeelectrode measurements to be implemented.
dmt a0 a1mt dt At
t B2
dy At mt dt ,
m(0) m0 , (31)
where mt is the conditional mean mt M{i0 (t ) Ft } of the unknown drift and t is the mean variance (accuracy)
2 d 2a1 t b2 At2 t2 , dt B
4.2 Adaptive control: unknown stochastic drift
i01
To deal with the uncertainties of electrode kinetics an adaptive control can be applied. One can use the certainty equivalent approach that replaces the unknown parameters with their estimates in the control laws (17) and (23) or cautious controls applied that uses variance of the estimates as well (applied in this paper). The exchange current density i0 and apparent transfer coefficient are usually poorly known as well as the diffusivity coefficient D. The model is nonlinear by the latter two parameters. They can be estimated from a relatively complex Zakai equation [5, 6]. The model is linear by the exchange current density i0; we assume that it changes in time as a linear stochastic process (t):
i0 , i
(33)
2 0
where is a small parameter The mathematical expectation of the concave function (33) is upper-bounded with the function of mean and variance M
i0 m . i m2
(34)
2 0
Alternatively, a more complex exact expectation can be expressed through the imaginary part of the complex 2
argument probability function ( z) e z erfc( jz) as follows
(27)
M
with a partly unknown initial condition 0 ~ N(m0, 0) specified by a Gaussian distribution with a given mean m0 and covariance 0. They serve for initial guess. In (27), a0, a1 are the parameters specifying dynamics of the process, b is the parameter specifying other effects of unmodeled kinetics along with a Wiener process W. The measured current density i (t ) of (5) can be rewritten in terms of the drifting (t) exchange current density i0 and measurements errors, as
(t ) At (t ) BV (t ) ,
(32)
To avoid the potential problems like division by zero in implementation of (17) and (23) while the estimate mt is applied instead of i0, the inverse of the exchange current density is approximated by
Fig. 3. The feedback modified feedforward control system.
d a0 a1 (t ) dt bdW , (0) 0
(0) 0 .
i0 1 Im ( z ) , i02 2
z 2
m
j
.
(35)
According to the cautious control approach the hyperbolic function of unknown exchange current density in (17) and (23) is replaced by its upper-bounded function (33) estimate (34) (alternatively estimate (35)), which leads to the controls ne F D mt cbulk cmin e t erfi 2 m t t Eref (t ) f 1 c (t ) k (1 ), k , ref cbulk
(28) 76
t ;
(36)
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R. Tenno / IFAC PapersOnLine 51-32 (2018) 73–78
for the semi-infinite diffusion process, and ne FDmt cbulk cmin 2 mt t 2 e t e n Dt Eref (t ) f 1 1 e t 2 n2 D n 1 cref (t ) k (1 ), k , cbulk
c(t , 0) cbulk
;
2 ne F
77
h N kh D k 1
n k 1 n k . (40)
The finite-diffusion. The concentration estimate at the boundary for the finite-length diffusion can be found similarly from the analytical solution by replacement of i ( ) with measurements ( ) in another convolution-like solution of [7]
(37)
n 1 2 1n e t D d , (41) n 1 0 ne F t D
c(t , 0) cbulk
for the finite-length diffusion process.
2 2
( )
t
where is the measured current density (A/m2).
The adaptive control system. The structure of the adaptive control system developed for tracking of the drift and for maintenance of the concentration of species at the tracking reference at the boundary is depicted in Fig. 4. This feedforward controller is adaptive to unknown stochastic 3drift. This system obliges three-electrode system measurements to be implemented.
5.
Control simulation
The control effect was tested in simulation. The reaction occurring at the electrode was assumed to be a single step reaction with two exchanged electrons (n = 2). The values of kinetic and mass-transfer parameters utilized in the simulation are listed in Table 1. TABLE I THE PARAMETER VALUES IN SIMULATION Symbol
D
The semi-infinite diffusion. The semi-infinite diffusion process at the boundary was simulated as (38) along with the concentration measurements (16) and then controlled by the PI modified feedforward controller (24) that forces the concentration at the boundary to follow the reference trajectory (6) with cmin = 100 mol/m3, cbulk = 750 mol/m3 and = 1. The measurement accuracy (standard deviation) of concentration at the boundary was assumed to be r = 5 mol/m3. The PI controller parameters Kp = 4 and Ti = 100 were selected from the best performance of the simulation. The tracking of the boundary concentrations with such a controller is illustrated in Fig. 5 as well as the tracking errors and control signals in Figs 5 and 6 respectively.
Fig. 4. The adaptive, feedback, feedforward control system.
In a two-electrode system the Nernst inverted part is missing in Fig. 4. In this case, the boundary concentration required in (29) should be found from solved diffusion model as explained below. 4.3 The boundary concentration solved using model The semi-infinite diffusion. The boundary concentration can be evaluated by past current density measurements, if the current density is replaced with measurements in the analytical solution (38) found for the semi-infinite diffusion problem with Laplace method ne F D
t
0
d .
800
(38)
700
Concentration, molCu(II)/m3
c(t , 0) cbulk
i ( )
t
Following Nicholson and Shain [8], the convolution integral in (38) can be approximated as t
0
i ( ) t
N
d 2 h i kh k 1
n k 1 n k ,
42 A/m2 0.25 4∙10-10 m2/s
Exchange current density (mean) Apparent transfer coefficient Diffusivity
i0
1
Value
QUANTITY
(39)
where N is the number of samples and h is the discretization step (s). Replacing in the equation (39) the modelled current density with the measured current density gives the desired estimate of the concentration at the boundary:
600 500 c(t,0) cref(t)
400 300 200 100 0 0
2
4
Time, sec
6
Fig. 5. The concentration tracking at the boundary. 77
8
10
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R. Tenno / IFAC PapersOnLine 51-32 (2018) 73–78
modified, can deal with uncertainties and unmodeled dynamics. The effectiveness of the proposed controls is verified by means of simulations.
50
c(t,0) - c ref(t)
15
30
10
20
Tracking error, molCu(II)/m3
Tracking error, molCu(II)/m3
40
10 0 -10 -20 0
2
4
Time, sec
6
8
10
c(t,0) - c ref(t) cest(t,0) - c ref(t) 5
0
-5
Fig. 6. The tracking error in the case of feedback feedforward control. -10 0
0
100
-0.15
90 Exchange current density, A/m2
Voltage, V
-0.1
-0.2 -0.25 -0.3
2
4
Time, sec
6
8
10
Time, sec
6
8
10
i0 drift
80
i0 estimate
70 60 50 40
Fig. 7. The applied voltage: PI feedback feedforward control.
30 -2 10
The finite-diffusion. The finite-diffusion process was simulated along with the measured (28) and controlled by adaptive controller (37) that forces the concentration at the boundary to follow the reference trajectory (6). The exchange current density i0 was assumed to be an unmeasured stochastic process (27) with the known mean: -a0/a1 = 42, dynamics a1 = -1 and standard deviation b = 3 of noise. This process started in simulation from the initial value i0(0) = 50. The measurement accuracy (standard deviation) of the current density measurements was assumed to be B = 10 A/m2. Calculation by the filter (31), (32) started from the guess values m0 100 and 0 400 . The concentration tracking error of simulation is illustrated in Fig. 8 and the exchange current density estimated in Fig. 9. The tracking error is shown in two cases when the drifting parameter is known and when it is estimated from the electric current measurements.
10
-1
Time, sec
10
0
10
1
Fig. 9. Drift of the exchange current density and its estimate.
Literature [1] [2] [3] [4] [5]
[6]
[7]
6.
4
Fig. 8. The concentration tracking errors of the adaptive control when the drift i0 is known c(t,0) and when estimated cest(t,0).
-0.05
-0.35 0
2
Conclusion [8]
This study has shown that the feedforward controllers for the semi-infinite diffusion and finite-length diffusion processes nonlinear at the boundary can be derived in the form of semi-explicit control function. These controllers, if
[9]
78
Bard, A.J., and Faulkner, L.R. (2001) Electrochemical Methods: Fundamentals and Applications. John Wiley&Sons, New-York, 856 p. Newman, J., and Thomas-Alyea, K. (2004). Electrochemical Systems. John Wiley & Sons, New Jersey, 672 p. Krstic, M., and Smyshlyaev, A. (2008) Boundary Control of PDEs. SIAM, Philadelphia, 192 p. Lebedev, N. (1965) Special Functions and Their Applications. PrenticeHall, 308 p. Mendelson, A., and Tenno, R. (2009) Electrode kinetics parameters estimation using Zakai equation. Journal of Process Control, 19(10), pp. 1698-1706. Tenno, R., and Mendelson, A. (2010) Adaptive boundary concentration control using Zakai equation. International Journal of Control, 83(6), 1287-1295. Tenno, R. (2016) Neumann boundary controls for finite diffusion process. International Journal of Control, 90(12), 2786-2798. Liptser, R.S., and Shiryayev, A.N. (2000) Statistics of Random Processes. II, Springer, New York, 417 p. Nicholson, R.S., and Shain, I. (1964) Theory of stationary electrode polarography: single scan and cyclic methods applied to reversible, irreversible, and kinetic systems. Analytical Chemistry, 36(4), pp. 706723.
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IFAC PapersOnLine 51-32 (2018) 73–78
Boundary Control of Reacting Species in Semi-infinite and Finite Diffusion Processes Boundary Control of Reacting Species in Semi-infinite and Finite Diffusion Processes Boundary Semi-infinite and Finite Diffusion Processes Boundary Control Control of of Reacting Reacting Species Species in in Semi-infinite and Finite Diffusion Processes R. Semi-infinite Tenno Boundary Control of Reacting Species in and Finite Diffusion Processes R. Tenno
R. R. Tenno Tenno Aalto University, School of Electrical Engineering R. Tenno Aalto University, Electrical Engineering PO BoxSchool 15500,of Aalto, Finland Aalto University, School of Electrical Engineering Aalto University, School ofAalto, Electrical Engineering PO Box 15500, Finland PO Box 15500, Aalto, Finland Aalto University, School of Electrical Engineering PO Box 15500, Aalto, Finland Abstract: Boundary control of oxidizing species at the cathode surface is considered in an PO Box 15500, Aalto, Finland Abstract: Boundary control of at the cathodeand surface in an electrochemical system, which isoxidizing the plantspecies of two electrodes plant is of considered three electrodes Abstract: control of species at the surface is in Abstract: Boundary Boundary control of isoxidizing oxidizing species at electrodes the cathode cathodeand surface isof considered considered in an an electrochemical system, which the plant of two plant three electrodes submerged in thesystem, stirred and unstirred electrolytes. The feedforward controls are found using electrochemical which the plant of electrodes and plant three electrodes Abstract: Boundary control of is at exact the cathode surface isof in an electrochemical system, which isoxidizing theelectrolytes. plantspecies of two two electrodes and plant of considered three electrodes submerged in the stirred and unstirred The exact feedforward controls are found using ideas of motion planning, Laplace techniques that the end the inverse submerged in the stirred and unstirred The exact feedforward controls are found using electrochemical system, which istransform theelectrolytes. plantand of other two electrodes and inplant of solve three electrodes submerged in the stirred and unstirred electrolytes. The exact feedforward controls are found using ideas of motion planning, Laplace transform and other techniques that in the end solve the inverse problem forinthe time-variable reference of species at thefeedforward cathode surface. The uncertainty of ideas of planning, Laplace transform and other that the end solve the submerged thegiven stirred and unstirred electrolytes. The techniques exact controls are found using ideas of motion motion planning, Laplace transform and other techniques that in in the endThe solve the inverse inverse problem for the given time-variable reference of species at the cathode surface. uncertainty of surface reaction existing inLaplace the system is suppressed with two modifications ofend controls. In the first problem for the given time-variable reference of species at the cathode surface. The uncertainty of ideas of motion planning, transform and other techniques that in the solve the inverse problemreaction for the given time-variable reference of species attwo themodifications cathode surface. The uncertainty of surface existing in the system is suppressed of controls. In the first case, thereaction feedforward control is extended with aspecies PIwith feedback that usessurface. measured in the process surface existing in the system is suppressed with two modifications of controls. In the first problem for the given time-variable reference of at the cathode The uncertainty of surface reaction existing in the system is suppressed with two modifications of controls. In the first case, the feedforward control is extended with aa PI feedback that uses measured in the process electric current passing through the system. In the second case, an adaptive feedforward control is case, the feedforward control is extended with PI feedback that uses measured in the process surface reaction existing in the system is suppressed with two modifications of controls. In the first case, thecurrent feedforward control isthe extended with a PI feedback that uses measured in thecontrol process electric passing through system. In the second case, an adaptive feedforward is established for the case when the exchange current density is unknown drifting parameter and is electric passing through system. In the second case, an feedforward case, thecurrent feedforward control isthe extended a PI feedback that uses measured in thecontrol process electric current passing through the system.with In the second case, an adaptive adaptive feedforward control is established for the case when the exchange current density is unknown drifting parameter and is estimated as it changes using a conditionally Gaussian filter. In both cases, the process control is established for the case when the exchange current density is unknown drifting parameter and electric current passing theexchange system. In the second case, an adaptive feedforward established for the case through when athe current density isInunknown drifting parameter and is is estimated as it changes conditionally Gaussian filter. both cases, the process control simulated and revealed tousing be effective. estimated as it changes using a conditionally Gaussian filter. In both cases, control established for the case when the exchange current density is unknown drifting parameter and is estimated and as itrevealed changestousing a conditionally Gaussian filter. In both cases, the the process process control is simulated be effective. simulated and revealed be effective. estimated as it(International changesto a conditionally Gaussian filter. In both cases, process control is simulated and revealed tousing beFederation effective. © 2018, IFAC of Automatic by control Elsevierthe Ltd. All rights reserved. Keywords: Distributed system, stochastic Control) process, Hosting boundary simulated and revealed parameter to be effective. Keywords: Keywords: Distributed Distributed parameter parameter system, system, stochastic stochastic process, process, boundary boundary control control Keywords: Distributed parameter system, stochastic process, boundary control Keywords: Distributed parameter system, stochastic process, boundary control 1. Introduction i (t ) D x c(t , x) 0 i (t ) . 1. Introduction (3) 1. Introduction nii ((ettF)) .. D x cc((tt ,, xx)) 0 (3) 1. Introduction D (3) The 1. problem solved in this paper stems from an industrial D xx c(t , x) 00 n Introduction (3) i (eetF ) . n F The problem solved in this paper stems from an industrial n F D c ( t , x ) . (3) e x In the case of 0unstirred electrolytes (semi-infinite diffusion), plating process, where the production maximization is crucial. The problem solved in this paper stems from an industrial ne F electrolytes (semi-infinite diffusion), The problem solved inthethis paper stems from an is industrial In the case of unstirred plating process, where production maximization crucial. condition at the electrolytes outer boundary is givendiffusion), as the In case (semi-infinite This is process, closelysolved related tothis thepaper plating rate from maximization and the plating where production maximization crucial. The problem stems an is industrial In the thecondition case of of unstirred unstirred electrolytes (semi-infinite diffusion), plating process, whereinthe the production maximization is crucial. the at outer boundary is given as the This is closely related to the plating rate maximization and concentration inunstirred bulkthe solution the condition at the outer boundary is given this in turn means maximization of the current density. This is closely related to the plating rate maximization and In the case of electrolytes (semi-infinite diffusion), plating process, where the production maximization is crucial. the conditionin at the outer boundary is given as as the the This is closely relatedmaximization to the platingofrate maximization and concentration bulk solution this in turn means the current density. concentration in bulk solution the condition at the outer boundary is given as the However, a substantial rise of the current density ends up with this in turn means maximization of the current density. This is closely related to the plating rate maximization and concentration in bulk solution c ( t , ) c . (4a) this in turn means maximization of the current density. bulk However, aaofsubstantial rise of the current density ends up with concentration in bulk solution depletion the oxidizing species at the the cathode surface c ( t , ) c However, substantial rise of the current density ends up with . (4a) this in turn means maximization of current density. bulk . c((tt ,, (4a) However, aofsubstantial rise ofspecies the current density ends up with depletion the at the cathode surface ))agitated ccbulk bulk . electrolyte a finite diffusion takes (4a) which leads to aoxidizing reduced plating rate and production. depletion the oxidizing at the cathode surface place However, aof substantial rise ofspecies the current density ends up with In ccwell depletion of the oxidizing species at the cathode surface ( t , ) c . (4a) bulk which leads to aaoxidizing reduced plating rate and production. In well agitated electrolyte aathe finite diffusion takes place Therefore, reactant concentration control is a close objective which leads to reduced plating rate and production. In well agitated electrolyte finite diffusion takes place of stagnation layer. In such within a certain thickness depletion of the species at the cathode surface which leads to a concentration reduced plating rateis aand production. In wella certain agitatedthickness electrolyte athe finite diffusion takes placeaa Therefore, reactant control close objective of stagnation layer. In such within casewell thea inner boundary condition remains the layer. same, while thea that helps toreactant avoid the depletionplating while maximizing the current In Therefore, concentration control is a close objective of the stagnation In such within certain thickness which leads to a reduced rate and production. agitated electrolyte a finite diffusion takes place Therefore, reactant concentration control is a closethe objective of theremains stagnation layer. In such a within a inner certain thickness case the boundary condition the same, while the that helps to avoid the depletion while maximizing current outer boundary immobile (fixed spatially condition) case the inner boundary condition remains the same, while the density through stabilisation of the boundary concentration at that helps to avoid the depletion while maximizing the current Therefore, reactant concentration control is a close objective of the stagnation layer. In such a within a certain thickness case the inner boundary condition remains the same, while the that helps to avoid the depletion while maximizing the current outer boundary immobile (fixed spatially condition) through stabilisation of the boundary concentration at outer boundary immobile (fixed spatially condition) adensity wisely chosen reference profile. The concentration control is case density through stabilisation of the boundary concentration at the inner boundary condition remains the same, while the that helps to avoid the depletion while maximizing the current outer through stabilisation of theThe boundary concentration at c(t boundary , ) cbulk .immobile (fixed spatially condition) (4b) aadensity wisely chosen reference profile. control is outer immobile (fixed spatially condition) transient control problem. Itofaims to concentration bring the concentration wisely chosen reference profile. concentration control is through stabilisation theThe boundary concentration at cc((tt boundary ,, ) c . (4b) bulk aadensity wisely chosen reference profile. The concentration control is ccbulk . (4b) transient control problem. It aims to bring the concentration 3 c(t , )) c(t,x) the boundary from a bulk solution concentration to a transient control problem. It aims to bring the concentration bulk . is the concentration (mol/m ) of species(4b) aat wisely chosen reference profile. The concentration control is In (1)-(4), in the aattransient control from problem. It aims to bring the concentration c(t , ) c(t,x) cbulk . is the concentration (mol/m33) of species(4b) the boundary a bulk solution concentration to a In (1)-(4), in the level in from a completely controlled manner. at the boundary a bulk solution concentration to a is the bulk solution depth x of the stagnation layer (m), c aprescribed transient control problem. It aims to bring the concentration In (1)-(4), c(t,x) is the concentration (mol/m ) of species in the 3 bulk at the boundary from a bulk controlled solution concentration to a In (1)-(4), c(t,x) is the 3concentration (mol/m ) of species in the 2 prescribed level in a completely manner. is the bulk solution depth x of the stagnation layer (m), c bulk is3 the ), D is the diffusivity (m /s), F is concentration (mol/m prescribed level in a completely controlled manner. bulk solution depth x of the stagnation layer (m), c at the boundary from a bulk solution concentration to a In (1)-(4), is the 3concentration (mol/m ) of species in the 2 prescribed level in a completely controlled manner. bulk solution depth x ofc(t,x) the (mol/m stagnation (m), cbulk bulk is the 3), Dlayer 2/s), F is the diffusivity (m is the concentration number of electrons Faraday constant (96485 C/mol), ndiffusivity is (m F is concentration 3), D 2/s), 2. The problem statements prescribed level in a completely controlled manner. e iscthe depth x of the (mol/m stagnation layer (m), bulk is the ), D is the the diffusivity (mbulk /s), Fsolution is the the concentration (mol/m is the number of electrons Faraday constant (96485 C/mol), n 3 2 2. The problem statements e involved in an (mol/m electrode reaction and isnumber the(m current density the of electrons Faraday constant (96485 C/mol), n 2. The problem statements ), D is the diffusivity /s), F is the concentration e isi(t) is the number of electrons Faraday constant (96485 C/mol), n 2. The problem statements 2 e i(t) is the current density involved in an electrode reaction and ) of an electrode reaction that satisfies a bidirectional (A/m involved in an electrode and i(t) is the current density General electrodeposition obeys convection, migration and is the number of electrons Faraday constant (96485 C/mol), n 2. The problem statements 2 e i(t) is the current density involved an electrode electrode reaction reaction and 2) ofinan that satisfies aa bidirectional (A/m General electrodeposition obeys convection, and Butler-Volmer (B-V) equation [1,and 2] i(t) reaction that satisfies (A/m 2) ofinan diffusion laws. However, when observing the migration mass-transfer involved an electrode electrode is the current density General electrodeposition obeys convection, migration and ) of an electrode reaction that satisfies a bidirectional bidirectional (A/m General electrodeposition obeys convection, migration and Butler-Volmer (B-V) equation [1, 2] 2 diffusion laws. However, when observing the mass-transfer Butler-Volmer (B-V) equation [1, 2] effects close toHowever, an electrode surface, thethediffusion does ) of an electrode reaction that satisfies a bidirectional (A/m diffusion laws. when observing mass-transfer General electrodeposition obeys convection, migration and Butler-Volmer equation 2] diffusionclose laws.toHowever, when surface, observingthethediffusion mass-transfer k 1(B-V) effects an electrode does c(t , 0) [1, E ( t ) equation k E (2] t) prevail. For that reason, concentration of the reacting species is Butler-Volmer effects to an electrode does diffusionclose laws. However, when surface, observing thediffusion mass-transfer c(t , 0) e[1, i(t ) i0 ek 1(B-V) (5) . E ( t ) effects close to an electrode surface, the diffusion does k E ( t) prevail. For that reason, concentration of reacting species is c ( t , 0) k 1 E ( t ) k E ( t) i ( t ) i e e . (5) c modelled by athat diffusion equation: prevail. For reason, concentration of reacting species is 0 bulk effects close to an electrode surface, the diffusion does c ( t , 0) k 1 E ( t ) ii((tt )) ii0 ee ee k E (t ) .. (5) prevail. For that reason,equation: concentration of reacting species is c modelled by a diffusion (5) bulk c ( t , 0) 0 k 1 E ( t ) c k E ( t ) modelled by diffusion prevail. reason,equation: concentration of reacting species is e cbulk i ( t ) i e . (5) the 2 modelledFor by aathat diffusion equation: 0 bulk 2 In (5), i is the exchange current density (A/m ), 0 cbulk current density (A/m22), is modelled t c(t , xby ) aDdiffusion 2x c(t , x) .equation: (1) is the exchange In (5), i is the 0 apparent coefficientcurrent and k isdensity the temperature voltage 22x cc((tt ,, xx)) .. (1) In (5), i0transfer is (A/m is t c(t , x) D (1) In is the the exchange exchange current density (A/m22), ), voltage is the the -1(5), i0transfer apparent coefficient and k is the temperature tt cc((tt ,, xx)) D D 2xx c(t , x) . (1) ). The mass-transfer effect is considered through (V apparent coefficient and kk is the temperature i0transfer is the exchange current density (A/m ), including voltage is the -1(5), apparent transfer coefficient and is the temperature voltage (t , x) D x c(t , x) . of species is assumed to (1) be In Thet cinitial distribution -1). The mass-transfer effect is considered through including (V the of oxidizing near the including electrode ).concentration The mass-transfer effect is through (V apparent coefficient kspecies is the temperature voltage The initial distribution of species is assumed be Thetransfer mass-transfer effectand is considered considered through including (V-1 homogeneous in the stagnation layer and equal to theto bulk The initial distribution of species is assumed to be the concentration oxidizing species near the electrode -1). surface c(t,0) in the of model (5). is E(t) is the voltage applied to an concentration of oxidizing species near the electrode The initial distribution of species is equal assumed to bulk be the ). The mass-transfer effect considered through including (V homogeneous in the stagnation layer and to the the concentration of oxidizing species near the electrode solution concentration: homogeneous in the stagnation layer and to the surface c(t,0) in the model (5). E(t) is the voltage applied to an The initial distribution of species is equal assumed to bulk be electrochemical system (V), considered as the control. In twosurface c(t,0) in the model (5). E(t) is the voltage applied to an homogeneous in the stagnation layer and equal to the bulk the concentration of oxidizing near the electrode solution concentration: surface c(t,0) in the model (5). E(t)species is theasvoltage applied to an solution concentration: electrochemical system (V), considered the control. In twohomogeneous in the stagnation layer and equal to the bulk electrode plant, the applied voltage is the difference between electrochemical system (V), considered as the control. In twosolution concentration: c(0, x) cbulk . surface c(t,0) in system model (5). E(t) is theasvoltage applied to an (2) electrochemical (V), considered the control. In twoelectrode plant, the applied voltage is the difference between solution concentration: cc(0, xx)) ccbulk .. (2) anodic andplant, cathodic overpotentials each ofthe them is once electrode the applied is the difference between electrochemical system (V), voltage considered as control. In more two(0, (2) bulk electrode plant, the applied voltage is the difference between x) cbulkon. the cathode is given by the first Fick law (2)and anodic and cathodic overpotentials each of them once more Thecc(0, condition between the electrode potential andis equilibrium anodic and cathodic overpotentials of them is once more electrode the applied voltageeach is the between (0, x) cbulkon. the cathode is given by the first Fick law (2)and difference anodic andplant, cathodic overpotentials each of difference them isequilibrium once more The condition difference between the electrode potential and states that the molar flux of reacting species is proportional to The condition on the cathode is given by the first Fick law and difference between the electrode potential and equilibrium U ( t ) can be measured in potential. The equilibrium potential anodic and cathodic overpotentials each of them is once more The condition on the flux cathode is givenspecies by the first Fick law and difference between the electrode potential and equilibrium states that the molar of reacting is proportional to U ((tt )) can be measured in potential. The equilibrium potential the cathode current density: states that the molar flux of reacting species is proportional to The condition on the cathode is given by the first Fick law and U can be measured in potential. The equilibrium potential difference between the electrode potential and equilibrium practice with to a potential referenceUelectrode SHE) states that the molardensity: flux of reacting species is proportional to potential. (t ) can be(versus measured in The respect equilibrium the cathode current the cathode current density: practice with respect to a reference electrode (versus SHE) states that the molar flux of reacting species is proportional to U ( t ) can be measured in potential. The equilibrium potential practice with respect to a reference electrode (versus SHE) the cathode current density: practicebywith respect to rights a reference 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting Elsevier Ltd. All reserved.electrode (versus SHE) the cathode current density: Copyright © 2018 IFAC 73 practice with respect to a reference electrode (versus SHE) Peer review under responsibility of International Federation of Automatic Control. Copyright © 2018 IFAC 73 Copyright © 2018 IFAC 73 10.1016/j.ifacol.2018.11.356 Copyright © 2018 IFAC 73 Copyright © 2018 IFAC 73
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that is submerged in electrolyte at the cathode surface in the case of three-electrode system. The device shown in Fig 1 is used for the boundary concentration c (t , 0) evaluation from the indirect measurement U (t ) . The evaluation procedure is explained later in subsection 4.1, where the Nernst relationship (25) and its inverted stochastic modification (26) are introduced.
solvable ordinary differential equation. Incorporating the corresponding boundary conditions yields the solution in a complex argument variable S
1 1 x cref ( S , x) cbulk cmin e S S
cbulk . S
(8)
Here cref (S,x) is the Laplace transform of cref (t,x) and the subscript ref indicates correspondence to the target profile. Equation (8) is the solution of motion planning problem in the Laplace domain for the model (1)-(3) and the reference concentration (6) at the boundary (7). The controller can be derived without the analytical inverse transformation of (8) in the whole subdomain but on the controlled boundary. Indeed, according to the Neumann boundary condition (3) the controller is governed by the flux of species at the boundary. Differentiating cref (S,x) with respect to x and taking controlled boundary x = 0, yields
E (t )
U (t )
S D
Fig. 1. Scheme and apparatus of three-electrode measurements.
x cref ( S , x)
We are seeking for a control that brings the concentration at the boundary from the bulk solution concentration cbulk to a lower level cmin near the cathode surface in a fully controlled manner. The prescribed reference is set by a time-varying function that declines exponentially cref (t ) cmin cbulk cmin e
t
.
0
x cref (t , x)
(6)
D
0
(9)
cbulk cmin je t erf j t .
(10)
In (10), symbol j stands for the imaginary unit. It is known that jerf ( jz ) erfi( z ) for any complex argument z [4], where the function erfi is the imaginary error function defined as z 2 t2 erfi z (11) e dt .
0
Consequently, taking real value z t j 0 the desired flux at the boundary can be written in the terms of the imaginary error function as
Motion planning
The problem of designing a voltage E(t), which is to be applied to (1)-(5) forces the concentration at the boundary to follow a given reference trajectory, is a boundary control problem known also as a motion planning problem [3]. To solve the motion planning problem one should find a reference profile cref (t,x) in the whole subdomain of the stagnation layer. This profile can be found by replacing in (1)(4) the boundary condition (3) with
c(t , 0) cref (t ) .
.
The solution in time domain can be found as the inverse Laplace transforms of (9)
Here cref is the reference trajectory of the concentration at the boundary, cmin and are the positive-valued constants, that specify the desired concentration at the boundary at steady state and the rate of tracking to steady state respectively. We are not concerned with the question of choosing cmin and . In this paper, we will focus on the development of a control law to follow the reference trajectory (6).
3.
cbulk cmin 1 S D S S
x cref (t , x )
D
0
cbulk cmin e t erfi t .
(12)
In contrast to the error function, the imaginary error function rapidly diverges to infinity as z grows. However, the function
F z e z erfi z is bounded for all real z-values. To find the boundary control the boundary condition (3) can be utilized. In the tracking mode the flux at the boundary also obeys the first Fick law 2
(7)
The motion planning problem is solved further in the case of semi-infinite diffusion and finite-length diffusion processes for the reference trajectory given by (6).
D x cref (t , x) 0
iref (t ) ne F
,
(13)
where iref(t) is the current density corresponding to the desired reference concentration. Assuming the bidirectional electrode model (5) k 1 Eref (t ) cref (t ) k Eret (t ) iref (t ) i0 e e (14) , cbulk
3.1 Semi-infinite diffusion Perhaps, the simplest way to discovery the reference trajectory in the whole subdomain in the case of semi-infinite diffusion is to apply the Laplace transform to the diffusion equation (1) with the Dirichlet boundary conditions (7) and (4a). This transform reduces the diffusion equation (1) into a
the tracking control can be found as follows. Define a function 74
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y eax cebx f x; a, b, c .
flux at the boundary satisfies the first Fick law (13). Assuming the bidirectional model (14) leads to the following boundary control
(15)
In this definition, a and b are the arbitrary nonnegative constants and c is the strictly positive. The latter function is invertible, partly 2 f 1 y;1,1,1 ar sinh( y), generally, inverse
ne FD cbulk cmin i0 2 e t e n Dt Eref (t ) f 1 1 e t 2 n2 D n 1 cref (t ) k (1 ), k , cbulk
is a continuous bounded function, denoted by x f 1 y; a, b, c .
(16)
Given y, a, b, and c, the inverse function above can be fast computed using the bisection method, Newton’s method or other simple method. The control, that forces the concentration at the boundary to follow the prescribed reference trajectory is the continuous bounded function (17). ne F D cbulk cmin e t erfi i 0 Eref (t ) f 1 c (t ) k (1 ), k , ref cbulk
t ;
Fig. 2. The feedforward control system. This represents the motion planed control for maintenance of the boundary concentration at given target.
w(t, x) cref (t, x) cref (t ) 1 x cbulk cref (t ) ,
The designed feedforward boundary controllers (17) and (23) do not tolerate extensive uncertainties. To deal with uncertainties (unmodeled convection, migration and other effects) they should be accompanied with a feedback.
(18)
4.
one can lift the boundary data to homogeneous boundary that contributes a result: a source-like term
cbulk cmin 1
1
x
(19)
t w(t , x) D 2x w(t , x) R(t , x) ,
(20)
With the zero initial condition w(0, x) = 0 and the homogeneous boundary conditions w(t, 0) = 0 and w(t, ) = 0. This problem can be solved analytically using the separation of the variables method. This eventually yields the following solution (21)
cbulk cmin
e t en Dt 2
n 1
n
D 2 n
sin n x , n
n
4.1 PI modified feedforward control The feedforward controls (17) and (23) can be improved with a simple feedback, if instead of cref (t) its PI modified values c f (t ) are used t 1 c f (t ) K p cref (t ) (t ) cref (s) ( s) ds (24) T i 0 ( t ) is the measured concentration at the boundary In (24), (mol/m3), Kp is the regulator gain, and Ti is the integral time (s). The concentration c (t , 0) at the boundary can be measured indirectly as the potential U (t ) or voltage between electrodes in a three electrodes system [1]. This voltage and concentration are related to each other as shown below in the Nernst relationship
.
Using (21), the reference profile given as the solution of the original problem can be expressed as
cref (t, x) w(t, x) cref (t ) 1 x cbulk cref (t ) . Consequently, the flux at the boundary is given by cbulk cmin e t en Dt t 1 e 2 2 n D n 1 2
x cref (t , x) 0
Uncertain system control
Effective use of the feedforward controls (17) and (23) requires a good knowledge of the model, which is not the case. Usually, the model is known poorly and is more complex. For that reason, the deterministic controls are modified below with a feedback from measured in the process data and with adjustment of the model to unknown stochastic drift.
in the diffusion equation
w(t , x) 2
(23)
(17)
Like that of semi-infinite diffusion we calculate the reference profile cref (t,x) inside the whole subdomain from the diffusion equation (1),(2) with the Dirichlet boundary conditions (7) and (4b). Using a new variable
R(t, x) e
; .
The control system. The structure of the control system developed for maintenance of the concentration of species at the tracking reference at the boundary is depicted in Fig. 2.
3.2 Finite-length diffusion
t
75
. (22)
U (t ) U eq
Once again to develope the boundary control, the inner boundary condition (3) is utilized. In the tracking mode the 75
1 c(t , 0) ln . k cbulk
(25)
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where is the measure current density (A/m2), B is a given parameter that specifies accuracy of the measurements, V is a generalized white noise that can be expressed as another Wiener process V in Ito integral and At is the time-varying function specified by (5) as
Here U (t ) is the electrode potential and Ueq is the equilibrium potential measured versus a reference electrode [1]. If the equation (25) is inverted (solved with respect to the concentration) and additive white noise error is added, which is more relevant to the real case, the measurement signal can be expressed as in (26).
At e
(t ) cbulk e
k U (t ) Ueq
r (t )
k 1 E ( t )
(26)
c(t , 0) k E ( t ) e . cbulk
(29)
In the case of three-electrode system the boundary concentration can be found from the Nernst equation and if replaced in (29) gives
Here is the measured indirectly boundary concentration, is the independently distributed standard Gaussian variable of zero mean and unit variance ~ N(0, 1), and r is the accuracy of measurements.
At ek 1 E (t ) e
k U (t ) Ueq
ek E (t ) .
(30)
The unknown drift of the exchange current density can be estimated from the measured data using the best mean-square filter [7] specified for the system (27) and (28):
The feedback control system. The structure of the control system developed for uncertain model for maintenance of the concentration of species at the tracking reference at the boundary is depicted in Fig. 3. The feedforward controller is extended with the PI-feedback. This system obliges threeelectrode measurements to be implemented.
dmt a0 a1mt dt At
t B2
dy At mt dt ,
m(0) m0 , (31)
where mt is the conditional mean mt M{i0 (t ) Ft } of the unknown drift and t is the mean variance (accuracy)
2 d 2a1 t b2 At2 t2 , dt B
4.2 Adaptive control: unknown stochastic drift
i01
To deal with the uncertainties of electrode kinetics an adaptive control can be applied. One can use the certainty equivalent approach that replaces the unknown parameters with their estimates in the control laws (17) and (23) or cautious controls applied that uses variance of the estimates as well (applied in this paper). The exchange current density i0 and apparent transfer coefficient are usually poorly known as well as the diffusivity coefficient D. The model is nonlinear by the latter two parameters. They can be estimated from a relatively complex Zakai equation [5, 6]. The model is linear by the exchange current density i0; we assume that it changes in time as a linear stochastic process (t):
i0 , i
(33)
2 0
where is a small parameter The mathematical expectation of the concave function (33) is upper-bounded with the function of mean and variance M
i0 m . i m2
(34)
2 0
Alternatively, a more complex exact expectation can be expressed through the imaginary part of the complex 2
argument probability function ( z) e z erfc( jz) as follows
(27)
M
with a partly unknown initial condition 0 ~ N(m0, 0) specified by a Gaussian distribution with a given mean m0 and covariance 0. They serve for initial guess. In (27), a0, a1 are the parameters specifying dynamics of the process, b is the parameter specifying other effects of unmodeled kinetics along with a Wiener process W. The measured current density i (t ) of (5) can be rewritten in terms of the drifting (t) exchange current density i0 and measurements errors, as
(t ) At (t ) BV (t ) ,
(32)
To avoid the potential problems like division by zero in implementation of (17) and (23) while the estimate mt is applied instead of i0, the inverse of the exchange current density is approximated by
Fig. 3. The feedback modified feedforward control system.
d a0 a1 (t ) dt bdW , (0) 0
(0) 0 .
i0 1 Im ( z ) , i02 2
z 2
m
j
.
(35)
According to the cautious control approach the hyperbolic function of unknown exchange current density in (17) and (23) is replaced by its upper-bounded function (33) estimate (34) (alternatively estimate (35)), which leads to the controls ne F D mt cbulk cmin e t erfi 2 m t t Eref (t ) f 1 c (t ) k (1 ), k , ref cbulk
(28) 76
t ;
(36)
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for the semi-infinite diffusion process, and ne FDmt cbulk cmin 2 mt t 2 e t e n Dt Eref (t ) f 1 1 e t 2 n2 D n 1 cref (t ) k (1 ), k , cbulk
c(t , 0) cbulk
;
2 ne F
77
h N kh D k 1
n k 1 n k . (40)
The finite-diffusion. The concentration estimate at the boundary for the finite-length diffusion can be found similarly from the analytical solution by replacement of i ( ) with measurements ( ) in another convolution-like solution of [7]
(37)
n 1 2 1n e t D d , (41) n 1 0 ne F t D
c(t , 0) cbulk
for the finite-length diffusion process.
2 2
( )
t
where is the measured current density (A/m2).
The adaptive control system. The structure of the adaptive control system developed for tracking of the drift and for maintenance of the concentration of species at the tracking reference at the boundary is depicted in Fig. 4. This feedforward controller is adaptive to unknown stochastic 3drift. This system obliges three-electrode system measurements to be implemented.
5.
Control simulation
The control effect was tested in simulation. The reaction occurring at the electrode was assumed to be a single step reaction with two exchanged electrons (n = 2). The values of kinetic and mass-transfer parameters utilized in the simulation are listed in Table 1. TABLE I THE PARAMETER VALUES IN SIMULATION Symbol
D
The semi-infinite diffusion. The semi-infinite diffusion process at the boundary was simulated as (38) along with the concentration measurements (16) and then controlled by the PI modified feedforward controller (24) that forces the concentration at the boundary to follow the reference trajectory (6) with cmin = 100 mol/m3, cbulk = 750 mol/m3 and = 1. The measurement accuracy (standard deviation) of concentration at the boundary was assumed to be r = 5 mol/m3. The PI controller parameters Kp = 4 and Ti = 100 were selected from the best performance of the simulation. The tracking of the boundary concentrations with such a controller is illustrated in Fig. 5 as well as the tracking errors and control signals in Figs 5 and 6 respectively.
Fig. 4. The adaptive, feedback, feedforward control system.
In a two-electrode system the Nernst inverted part is missing in Fig. 4. In this case, the boundary concentration required in (29) should be found from solved diffusion model as explained below. 4.3 The boundary concentration solved using model The semi-infinite diffusion. The boundary concentration can be evaluated by past current density measurements, if the current density is replaced with measurements in the analytical solution (38) found for the semi-infinite diffusion problem with Laplace method ne F D
t
0
d .
800
(38)
700
Concentration, molCu(II)/m3
c(t , 0) cbulk
i ( )
t
Following Nicholson and Shain [8], the convolution integral in (38) can be approximated as t
0
i ( ) t
N
d 2 h i kh k 1
n k 1 n k ,
42 A/m2 0.25 4∙10-10 m2/s
Exchange current density (mean) Apparent transfer coefficient Diffusivity
i0
1
Value
QUANTITY
(39)
where N is the number of samples and h is the discretization step (s). Replacing in the equation (39) the modelled current density with the measured current density gives the desired estimate of the concentration at the boundary:
600 500 c(t,0) cref(t)
400 300 200 100 0 0
2
4
Time, sec
6
Fig. 5. The concentration tracking at the boundary. 77
8
10
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modified, can deal with uncertainties and unmodeled dynamics. The effectiveness of the proposed controls is verified by means of simulations.
50
c(t,0) - c ref(t)
15
30
10
20
Tracking error, molCu(II)/m3
Tracking error, molCu(II)/m3
40
10 0 -10 -20 0
2
4
Time, sec
6
8
10
c(t,0) - c ref(t) cest(t,0) - c ref(t) 5
0
-5
Fig. 6. The tracking error in the case of feedback feedforward control. -10 0
0
100
-0.15
90 Exchange current density, A/m2
Voltage, V
-0.1
-0.2 -0.25 -0.3
2
4
Time, sec
6
8
10
Time, sec
6
8
10
i0 drift
80
i0 estimate
70 60 50 40
Fig. 7. The applied voltage: PI feedback feedforward control.
30 -2 10
The finite-diffusion. The finite-diffusion process was simulated along with the measured (28) and controlled by adaptive controller (37) that forces the concentration at the boundary to follow the reference trajectory (6). The exchange current density i0 was assumed to be an unmeasured stochastic process (27) with the known mean: -a0/a1 = 42, dynamics a1 = -1 and standard deviation b = 3 of noise. This process started in simulation from the initial value i0(0) = 50. The measurement accuracy (standard deviation) of the current density measurements was assumed to be B = 10 A/m2. Calculation by the filter (31), (32) started from the guess values m0 100 and 0 400 . The concentration tracking error of simulation is illustrated in Fig. 8 and the exchange current density estimated in Fig. 9. The tracking error is shown in two cases when the drifting parameter is known and when it is estimated from the electric current measurements.
10
-1
Time, sec
10
0
10
1
Fig. 9. Drift of the exchange current density and its estimate.
Literature [1] [2] [3] [4] [5]
[6]
[7]
6.
4
Fig. 8. The concentration tracking errors of the adaptive control when the drift i0 is known c(t,0) and when estimated cest(t,0).
-0.05
-0.35 0
2
Conclusion [8]
This study has shown that the feedforward controllers for the semi-infinite diffusion and finite-length diffusion processes nonlinear at the boundary can be derived in the form of semi-explicit control function. These controllers, if
[9]
78
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