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Eigenvector placement in state space control of induction heating device
Eigenvector placement in state space control of induction heating device J. Egalon*, S. Caux*, P. Maussion*, O. Pateau** *LAPLACE/INPT-UPS-CNRS – 2 rue Camichel BP 7122 – 31071 Toulouse - France Université de Toulouse (e-mail: [email protected]). **EDF – Eco-efficiency & Industrial Processes Dept, Av des Renardières, 77250, Moret sur Loing - France (e-mail: [email protected])
Abstract: Metal industries still use furnace that have low efficiency and are based on thermal heating produced by combustion process. To have better cost production and low gas emission, induction heating is a way to be explored. This paper presents an industrial multiphase induction heating process focusing on the inductor’s current control. A model of the electrical power part is described linking the power sources, the inverter currents and the power density produced to heat the metal piece. Due to the structure of the global power system, couplings between currents should be well managed to control the temperature. A state space representation is used not only to control in closed loop the dynamics of both currents but also a specified eigenvector placement is forced to get uncoupling and robust behavior of the global system. The results obtained in simulation show good performances and fulfill the industrial constraints. Keywords: current control, inverter, decoupling structure 1. INTRODUCTION Due to ecologic and economic restriction, there is also a great interest in metal industries to increase efficiency of heating process with low CO2 emission. Some of these industries still use thermal furnace to produce metal devices, hot sealing, thermal forming and so one. The target application is an important temperature elevation of long metal piece, and first of all, only a static heating will be presented here. To replace classical furnace using fossil fuels, there is a great interest in utilizing induction heating process. There is some application using open loop control, using magnetic flux concentrator and/or moving screens to obtain the desired power density injected in the material thus the desired temperature (Souley 2011). The coils used here, can create a transversal or longitudinal flux resulting from the design and placement of the inductor. It is well known that depending on the structure, the number of coils, their shapes and their arrangements, the current power density injected is directly linked to the magnitude and the phase of the currents (Acero 2008). In this paper, a multi coil induction heating process is presented. The process presents three concentric inductors producing transversal magnetic flux, heating the metal piece placed inside. Couplings shown by the structure create a great challenge in current management and control theory (Fujita 2007). A disturbance rejection controller or a non linear decoupling is not so easy in such device, because the inverter output waveform and the coupling effects are at the same resonant frequency. The paper presents the inverter structure proposed to create the three desired instantaneous currents on the three inductors and proposes an advanced control structure in state space to obtain decoupling and correct imposition of the inductor’ currents.
The first part explains the power structure of the process and the relationships between inverter' currents, inductor' currents and power density in the metal piece. The state space representation is formulated in part two, and the imposed eigenstructure placement is exposed in part three. In the last part, results in simulation are presented both to validate the performances and also to reach a certain robustness of the proposed control. 2. INDUCTION HEATING SYSTEM DESCRIPTION 2.1 Experimental setup The heating process can be described in Fig1, as a three level process. The first level consists in controlling the inverter’ currents delivered to the load, producing the currents in the three inductors (Ii). The second one is the load part itself (metal piece in the middle of the inductors) receiving the transversal flux from the inverters at the resonant frequency (fr=1500Hz). The magnetic field generated creates a power density (Dpi) that heats the piece depending on its thermal characteristics to obtain the final temperature (θi) but not treated here (Tudorache 2008).
Fig. 1. Complete induction heating process to control.
An innovative solution consists of multi-coil system which has been demonstrated and experienced in (Spagnolo 2009), where the multi-phase inductor is supplied by currents adjusted in magnitude and phase. The proposed study is explained on a three transversal concentric inductors, placed face to face (Fig2-a), but the optimized and final solution is a larger device (a six inductors structure as shown in Fig2–b)
Fig. 2.: Induction heating device: a– 3inductors (EDF), b– 6inductors (CELES). 2.2 System modelling The three-inductors heating process can be described in Fig. 3, presenting a voltage Vi, a current Ii and the mutual terms linking the field produced through reaction Mij.
A multi-phase induction process is thus more complex and a global current management should be made to manage the power density, due to the interactions between the converters and the inductors, between the inductors themselves and between the inductors and the work piece to be heated. From a controlled dc current source, the goal is to control the three inductor’ currents because the inverter’ currents (Iond1, Iond2, Iond3) supply the global RLC parts but only the inductor’ currents (I1, I2, I3) produce the load piece heating. First of all, an identification of RLC parameters should be made, not only to obtain a correct description of the dynamic behaviour of the process, but also to define the resonant accordance to the specified frequency fr=1500Hz (Forest 2007). This system can be represented by a 4 phases electrical circuit including the material to be heated (indexed 4), which is subject to a current flow but in short circuit. A matrix description of the system is given in (1) where sinusoidal currents I1, I2 and I3 feed the three coils while I4 is the current through the material to be heated. 0
(1)
- , , : coupling terms between the inductors i and the material to be heated (load) - , : resistance and inductance for inductor i - , : disc plate resistance and inductance - : mutual inductance between inductor i and j
From the 4th line comes the 3x3 reduced matrix (2), between inductor voltages and current . Fig. 3. Three-inductors coupling through the load. Several types of power supply have been proposed for mono or multi-inductor systems. They may consist of a combination of conventional structures such as H-bridge on each inductor or nested structures supplying several inductors (Yong-Chae 1999, Fujita 2007). The three inductor’ currents here are generated by three current resonant inverters, which are presented in Fig. 4, where K1a is the IGBT 1 (respectively 2,3,4) for inverter a (respectively b,c) to be controlled, and the associated diode (D1a), Ci is the resonant condensator for Inductor i (i=1, 2, 3). K4a
D4a
K3a
D3a
Inverter 1 K2a
Current source
K1a
D2a
C1
D1a
V1
K1c
K4c
D1c
D4c
L Inverter 2
Disc plate
V2
C2
V3 C3
Current control circuit
D2c
D1c
K2c
K1c
K2c
K3c
D2c
D3c
Inverter 3
D3c
K3c
D4c
K4c
Fig. 4. Current Power Electronic’s structure of the heating device.
! . . . # $! #
)))) & )))) & )))) & )))) )))) )))) ⟹ &̅ ()))) & & & * &+, , , )))) )))) & )))) & &
(2)
As shown by (3) and (4), the terms , and -, are nonlinear, regarding the pulsation . It is also the case with the temperature . that influences the parameter representing the material resistance. )))) & +,+ /
², ², . . 1 !3# 1 /
4²56,7 .57,8 .97 4²56,7 .57,8 .<7 ))))) & +,, 3 : : : = 3, : : : = (4) 97 ;<7 4
97 ;<7 4
Equations (3) and (4) reflect that the presence of the work piece represented by , L4 and Mi4 increases the resistance of the diagonal terms and creates a mutual resistance in the impedance coupling terms. Consequently, measurements and calculations are required to obtain the real and imaginary parts of the global impedances including the coupling terms. The values of the impedance matrix terms are given by (5) which were calculated according to a “pseudo – energy” method from V and I measurements (Souley 2009).
33.1 j244.7 25.9 j43.8 21.5 j24.3 &̅ > 25.8 j43.6 67.3 j247.3 65.9 j113.7 H IΩ 20.9 j21.4 65.2 j111.4 107.1 j568.2
with:
(5) Fig. 5, shows for phases 1 and 2, the inverter current waveforms and the currents in the corresponding inductor. Due to the LC resonant circuits, the inductor currents are quite sinusoidal despite square wave supply inverter currents. The specific control angle calculation (Fig. 5) derives from an optimization procedure which is presented in (Souley 2011). Indeed this has been made possible by the knowledge of the induced current distributions. I_ind1
I_inv1
400
0 -200
ϕ21
-400
α1=0° I_ind2
I_inv2
300 200
δ2
100 0 -100 -200
α2
-300 0.0692
0.0694
0.0696
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 ∙ L 0 0 1 0 L 0 0 0 0 L 0 a 0 0 0 0 0 0 0 0 0 0
0 0 0 ∙ 0 0 L 0 0 1 0 0 1 0 L 0 0 0 0 0 L a
0 1 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 0
0 0
0 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0
1 0 0 0
Matrices A and B exist whatever the parameters are, and the system is controllable and observable. In this study, the current and voltage measurements are considered effective. After presenting the specified control structure, a robustness study is added to confirm the obtained performance.
δ1
200
A 0 0 0 B 0 0 0
0.0698
Time (s)
Fig. 5. Phase 1 and 2 inverter and inductor current waveforms
The control structure presented in Fig. 6, is quite classical but presents full matrix K and H and should be adapted to the defined induction process (Skogestad 2005). The feedback gain tuning K and pre-control matrix H could be pointed out. The contribution of both matrices should not only place the closed loop poles of Af (Af =A-BK), but also insure the decoupling between input, output, modes and mutual terms.
Writing all transfer functions as (6) describing the behaviour from Iondi and Ii and also Ij disturbing Iondi, a representation by bloc diagrams on 3 phases, allows to focus on the important coupling effects. The coupling terms add disturbances for the other phases with magnitude and frequencies that cannot be classically rejected. The process also requires an accurate control in magnitude and phase of each instantaneous current. K
1 !6# . L . M . L . M 1
NOP K,
L . M. !, . M ,#
. L . M² . L . M 1
QRSTS U V W
The proposed control structure, should decoupled the inductor’ currents, to obtain an independent control (with its own dynamic behaviour), thus in order to obtain a final accurate power density in the work piece. 3. CONTROL STRUCTURE In the state space representation (7) the vector X is 6dimensionnal presenting three control inputs U, to control the three outputs Y. The Multi-Input Multi-Output coupled system can now be treated to be as efficient as possible and the main goal is to decoupled inductor’ currents in the output Y. X - Y. - Z. [ \ L. 1 0 0 0 0 1 0 0 ]^_ - , [ >]^_ H, \ > H, L 0 0 1 0 0 0 0 1 ]^_ 0 0 0 0 0 0 0 0
dim(X)=n=6, dim(U)=m=3
0 0 0 0 1 0
0 0
0 0 0 1
(7)
Fig. 6. Complete eigenstructure control in state space. The MIMO system should be controlled globally and a short theory is resumed hereafter in state space representation introducing the theoretical tools used in the control synthesis. 3.1 Controller synthesis In the state space representation a pre-multiplication by a matrix H is possible only if this matrix presents a constant determinant and is not depending on time. The classical use leads to a transformation on the state X, in order to work in diagonalized space instead of a full matrix. The diagonal system permits to extract eigenvalues λ (called also modes or dynamics of the system). This multiplication can also be seen as a base transformation giving new properties or easy computation in this new reference frame. Manipulating new states in the virtual new space, should be easier and using the inverse transformation leads to the real values. All changes on variables is also possible on output only if the transformation is not singular (i.e.: conserving dimension). In matrix theory, the eigenvalues and eigenvectors of a matrix A, can be computed in different ways: eigenvalues
cancelling the determinant, right and left eigenvectors as expressed in (8). det!g.
Y# 0, Yh g h , i Y g i
(8)
-1
The modal matrix is V=[v1…vn] its inverse is V =[u1…un].
A transformation - . j based on the eigenvectors allows to write (7) as (9) in diagonal space, with : P a disturbance vector (noise) added to be general, with G a matrix applying noise on the states. The control law in closed loop (Fig7), introduces: [ Kklmn – p- so, noticing Λ the diagonal closed loop dynamic matrix equivalent to (A-BK) derives (9) jX Λj r
a ZKklmn u Lj
a s. t
(9)
In practice, V (dim 6 by 6), describes the repartition of the dynamics (dim n=6 modes λi) into the state X, V-1 represents how the modes are activated by the reference E (dim m=3) and/or disturbance P and CV represents how the modes act on the outputs. Considering the components on line k in Vi, due to -
v | … |^ y. j, shows that the relation fkVi=0 cut the link between mode jU and state xi. The function f is the direction on inputs and the control consists in defining K to have the closed loop desired dynamics and eigenvector placement in order to insure the desired decoupling. In the three-inductors case presented, f is 3-dimensionnal corresponding to the 3 current’ directions and is expressed as (10): f 1
0 0 , f 0 1
0 , f 0 0
1
(10)
Where: wi =-K.vi is the input direction acting on the states (including K unknown) dim m by 1. Writing (11) with matrix form, (12) is obtained and allows to indicate that the solution is in the kernel of the defined matrix with large dimension and can be sub-divised in two sub-space M, N. i.e.: in the case studied here, n=6, m=3, i=[1..6] so: N is dim 6 by 6 and M dimension is 3 by 6, for the total dimension 6 by 6+3 Ker([Y
-
vi= _λ .zi and wi= _λ .zi.
That means in a multi-input system, even if K can have multiple values, vi should be computed accurately in the frame made by the kernel of matrix [Y g I, Z] respecting the constrains ( fkVi=0 for the forsaken decoupling of the induction process) and written in (13). Y ~
the excitation input i and the dynamic j are separated if : i ZK{| 0 the dynamics i is eliminated from the state j if : { h 0 the dynamics i is eliminated from the output j : { Lh 0
The matrix K is m by n presenting m.n degrees of freedom even if only n dynamics should be placed (classically used in pole placement). Here all equations are used for decoupling purposes, and H contributes too. That opens the field to impose not only behavior on each state, but also insure a certain independent behavior between each state. This decoupling procedure is called eigenstructure placement. (Skogestad2005). 3.2 Control analysis Matrix K The K and H computation should insure pole placement and respect the constraints introduced for decoupling . The closed loop right eigen formulation can be established as (11) for each mode λi from 1 to n=6: Zp#h
g h 0 !A
g I#h BQ
g {W
Z U 3 = 0 0 U
(13)
To solve the system (13), desired eigenvector vid (dim 6x1) and directions wid (dim 3x1) are computed solving kernel of (13) for each i mode. Implying dim W=3x6 and dim V=6x6 and expression (14): K=-W.V-1
It is important to notice that the terms ‘eliminate/separate’ dynamics, actually means: cancelling this dynamic and not only considering fast and slow mode approximation.
!Y
(12)
With : N a subspace containing possible eigenvectors zi. That shows whatever the correction K is, when modes g are chosen fixing the closed loop dynamics, upper part of (12) is satisfied. The following relationships are obtained:
Considering this, three constraints can be listed: -
_λ g I, Z] )=~ _λ
(11)
(14)
where K (dim m by n=3 by 6) is determined using n degrees of freedom to impose n eigenvalues and let n.(m-1) to place eigenvectors including fk (f2 and f3 are used here I1 is chosen as reference here). 3.3 Control analysis Matrix H Matrix H only acts on the constraints insuring that the input (Eref) will not add its linked modes to the state and thus, on the output ever controlled by K. In the same way, if input i should not act on mode λj, the relation (15) is obtained: i ZK{| 0
(15)
Of course, the problem arises when there are more constraints than equations (13) and (15) should impose. An optimization algorithm based on least square algorithm can easily be used or inversion of the non square matrix N should be studied. In both cases, differences arise from the desired eigenvectors vid and the one computable and usable vi that satisfies all constraints vi=N(λi).zi.
Considering from (12), λ6 = inv([Y g I]).B. _λ , is a n by (n+m) non square matrix. Consequently, an approached solution should be computed : -
By minimization of a criterion J expressed in (16), representing the best possible solution zi approaching the eivenvectors vid in the
minimization of the square distance, so the least square method could be used. -
J ‖v
N!λ #. z
‖
(16)
Table 1. Poles placed and obtained Poles desired obtained
1 0.7401 0.7401
2 0.7393 0.7393
3 0.7386 0.7386
4 0.7397 0.7397
5 0.739 0.739
6 0.7382 0.7382
By using pseudo inverse formulation (17):
v N!λ #N!λ # N!λ # N!λ # . v a
Table 2. Performances obtained
(17)
where : T is for matrix transpose.
Phase 1
To be noticed the solution of least square form (16) corresponds to (16) with vi= N!λ #.zi.
From (11) and (12) the second equation ( C(A-λI).N+B.M=0 ), allows also to compute H (dim 3 by 3) with (17) and C=Id . In classical case, the pre-control square matrix H is solved using the solution vi in (18) and left-pseudo inverse formulation for M : ! C!A
BK#a . B# M
. K ∶ K ! . # ′
a
(18)
NB: The pole placement is now possible placing also constraints to quasi-cancel the link between input, output and modes. The eigenstructure is now completely defined from K and H to be tested on the simulated induction heating process.
Phase 2
Phase 3
Gain
with coupling 4.67
With decoupling 0.91
With coupling 2.89
With decoupling 0.91
with coupling 3.39
With decoupling 0.91
Phase
-48.6o
-32,4o
-16.20
-32.40
21.60
-32.40
Fig. 7, shows phase currents, and a zoom is made to focus attention on the good behavior of the current. The transient corresponds to the temperature profile imposed to rise up from ambient temperature to the desired one (300°C corresponding to 150A in phase3 represented here). Thus at time 0.03s, a little reference variation is introduced to test the performance of the control law. The performance should be on the magnitude and phase control of the current. Increasing the dynamics of poles provides more accurate currents (also in steady state), but a compromise has to be pointed out, in order to obtain a good control with the controller implemented in discrete time.
4. RESULTS AND ROBUSTNESS In the practical case studied in this paper, n=6 dynamics of the closed loop are imposed around ωbf=3*(2.π.1500)=30000rd/s. Exactly λ1=-30100rd/s, λ2=30200rd/s, λ3=-30300rd/s, λ4=-30400rd/s, λ5=-30500rd/s, λ6=-30600rd/s are set to fulfill certain conditions : to be as quick as possible, have quite the same dynamics on each current, avoiding solving matrix problems, imposing distinct poles and let open the possibility to transpose the control in discrete time (see TABLE1). A hierarchy is arbitrary imposed because the λ are quite the same, to let 2 modes λ on each inductor currents Iond linked to each phase I. In our case λ1 and λ4 are kept linked to the dynamic from Iond1 to I1. In the same way λ2 and λ5 to I2/Iond2, λ3 and λ6 to I3/Iond3 Finally K and H are a result to be used in the control scheme: . >. .
. . . . . . . . . 9.45 1.76 0.81 K >0.92 5.51 2.46H 0.22 1.3 7.02
. . .
. . H .
4.1 Results and global performances
Using sampled formulation of the state space, using Te=10µs (Fe = 100 kHz) the 6-discrete poles (exp(λiTe)) are given in Table 1. The poles are imposed and eigenvectors are near the requested ones. Performance obtained on current magnitudes and phases at f=1500 Hz are given in Table 2.
Fig. 7: Desired and obtained current on phase2 (complete signal and partial zoom) A gain (0.91) at f=1500Hz is slightly different from unity gain because no integrator is present. To overcome this problem, it could be easy to modify the gains in H to obtain accuracy in magnitude on the currents. A certain phase is introduced but may be checked at the system level, depending on its impact on the temperature profile and he accuracy requested for the metal piece heated.
4.2 Parameters variation and robustness A quick study is conduced to test the robustness of the controller in presence of load parameter variation. The load piece and the design of inductors as can provide variations: on R, L or M, also on C which should be accorded to the resonant frequency but the parameter dispersion on capacitance values with a certain tolerance can be of paramount importance. A maximum +20% and -20% variations of each parameters have been conduced and Fig. 8, Fig. 9 and Fig. 10, show the variations of performance in percent of the nominal level providing less than 10% effect, fulfilling the industrial requirement.
Fig. 8: R variations and performance on currents
Fig. 9: L variations and performance on currents
Futur works consist in applying the proposed control structure on the experimental device, optimized the pole placement to increase robustness, and test on the six inductors device. REFERENCES Acero J. et al. (2008), « The domestic induction heating appliance: An overview of recent research », in Applied Power Electronics Conf. and Exposition, 2008. APEC 2008. IEEE, 2008, p. 651–657 Forest F., Faucher S., Gaspard J.-Y., Montloup D., Huselstein J.-J., and Joubert C., (2007): « Frequency-Synchronized Resonant Converters for the Supply of Multiwinding Coils in Induction Cooking Appliances », IEEE Transactions on Industrial Electronics, vol. 54, no. 1, p. 441-452, févr. 2007. Fujita H., Uchida N., et Ozaki K. (2007), « Zone Controlled Induction Heating (ZCIH) A New Concept in Induction Heating », in Power Conversion Conference - Nagoya, 2007. PCC ’07, 2007, p. 1498-1504 Skogestad S. and Postlehwaite I. (2005), Multivariable feedback control: an analysis and design, John Wiley and Sons, 2005 Souley M., Egalon J., Caux S., Pateau O., Maussion P. (2011), "Modeling of a multi-phase induction system for metal disc heating", in Proc. of Electrimacs 2011, 5-9 June 2011, Paris Souley M., Spagnolo A., Pateau. O, Paya B, Maussion P. (2009): “Characterization methodology for the impedance matrix of multi-coil induction heating device “, 6th International Conference on Electromagnetic Processing of Materials EPM 2009; October 19-23, 2009, Dresden, Germany. Spagnolo A., Forzan M., Lupi S., Pateau O., et Paya B. (2009), « Space control optimisation of multi-coil transverse flux induction heating of metal strips », in Proceedings of the International symposium HES-10 Heating by Electromagnetic Sources, Padua, 2009 Tudorache T. et Fireteanu V. (2008), « Magneto-thermalmotion coupling in transverse flux heating », COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 27, no. 2, p. 399-407, 2008 Yong-Chae Jung (1999), « Dual half bridge series resonant inverter for induction heating appliance with two loads », Electronics Letters, vol. 35, no. 16, p. 1345-1346, 1999
Fig. 10: C variations and performance on currents 5. CONCLUSIONS A state space representation of an induction process is presented in this paper. The power process should be controlled in current whatever the couplings are between inductors. A eigenstructure state feedback is presented and adapted in the induction case studied. The control obtained has fixed gains and can be easily implemented on a ship controller. Performances obtained respect the industrial process requirement, allows the three current management and provides a certain robustness to parameter variations.
AKNOWLEDGMENT This work is a part of the Innovative Solutions for Induction Systems project, funded by the French Research National Agency with EDF EPI, CELES, LAPLACE, Arts Mines, SIMAP, ATYS CONSULTING GROUP and CNRT as partners.
Abstract: Metal industries still use furnace that have low efficiency and are based on thermal heating produced by combustion process. To have better cost production and low gas emission, induction heating is a way to be explored. This paper presents an industrial multiphase induction heating process focusing on the inductor’s current control. A model of the electrical power part is described linking the power sources, the inverter currents and the power density produced to heat the metal piece. Due to the structure of the global power system, couplings between currents should be well managed to control the temperature. A state space representation is used not only to control in closed loop the dynamics of both currents but also a specified eigenvector placement is forced to get uncoupling and robust behavior of the global system. The results obtained in simulation show good performances and fulfill the industrial constraints. Keywords: current control, inverter, decoupling structure 1. INTRODUCTION Due to ecologic and economic restriction, there is also a great interest in metal industries to increase efficiency of heating process with low CO2 emission. Some of these industries still use thermal furnace to produce metal devices, hot sealing, thermal forming and so one. The target application is an important temperature elevation of long metal piece, and first of all, only a static heating will be presented here. To replace classical furnace using fossil fuels, there is a great interest in utilizing induction heating process. There is some application using open loop control, using magnetic flux concentrator and/or moving screens to obtain the desired power density injected in the material thus the desired temperature (Souley 2011). The coils used here, can create a transversal or longitudinal flux resulting from the design and placement of the inductor. It is well known that depending on the structure, the number of coils, their shapes and their arrangements, the current power density injected is directly linked to the magnitude and the phase of the currents (Acero 2008). In this paper, a multi coil induction heating process is presented. The process presents three concentric inductors producing transversal magnetic flux, heating the metal piece placed inside. Couplings shown by the structure create a great challenge in current management and control theory (Fujita 2007). A disturbance rejection controller or a non linear decoupling is not so easy in such device, because the inverter output waveform and the coupling effects are at the same resonant frequency. The paper presents the inverter structure proposed to create the three desired instantaneous currents on the three inductors and proposes an advanced control structure in state space to obtain decoupling and correct imposition of the inductor’ currents.
The first part explains the power structure of the process and the relationships between inverter' currents, inductor' currents and power density in the metal piece. The state space representation is formulated in part two, and the imposed eigenstructure placement is exposed in part three. In the last part, results in simulation are presented both to validate the performances and also to reach a certain robustness of the proposed control. 2. INDUCTION HEATING SYSTEM DESCRIPTION 2.1 Experimental setup The heating process can be described in Fig1, as a three level process. The first level consists in controlling the inverter’ currents delivered to the load, producing the currents in the three inductors (Ii). The second one is the load part itself (metal piece in the middle of the inductors) receiving the transversal flux from the inverters at the resonant frequency (fr=1500Hz). The magnetic field generated creates a power density (Dpi) that heats the piece depending on its thermal characteristics to obtain the final temperature (θi) but not treated here (Tudorache 2008).
Fig. 1. Complete induction heating process to control.
An innovative solution consists of multi-coil system which has been demonstrated and experienced in (Spagnolo 2009), where the multi-phase inductor is supplied by currents adjusted in magnitude and phase. The proposed study is explained on a three transversal concentric inductors, placed face to face (Fig2-a), but the optimized and final solution is a larger device (a six inductors structure as shown in Fig2–b)
Fig. 2.: Induction heating device: a– 3inductors (EDF), b– 6inductors (CELES). 2.2 System modelling The three-inductors heating process can be described in Fig. 3, presenting a voltage Vi, a current Ii and the mutual terms linking the field produced through reaction Mij.
A multi-phase induction process is thus more complex and a global current management should be made to manage the power density, due to the interactions between the converters and the inductors, between the inductors themselves and between the inductors and the work piece to be heated. From a controlled dc current source, the goal is to control the three inductor’ currents because the inverter’ currents (Iond1, Iond2, Iond3) supply the global RLC parts but only the inductor’ currents (I1, I2, I3) produce the load piece heating. First of all, an identification of RLC parameters should be made, not only to obtain a correct description of the dynamic behaviour of the process, but also to define the resonant accordance to the specified frequency fr=1500Hz (Forest 2007). This system can be represented by a 4 phases electrical circuit including the material to be heated (indexed 4), which is subject to a current flow but in short circuit. A matrix description of the system is given in (1) where sinusoidal currents I1, I2 and I3 feed the three coils while I4 is the current through the material to be heated. 0
(1)
- , , : coupling terms between the inductors i and the material to be heated (load) - , : resistance and inductance for inductor i - , : disc plate resistance and inductance - : mutual inductance between inductor i and j
From the 4th line comes the 3x3 reduced matrix (2), between inductor voltages and current . Fig. 3. Three-inductors coupling through the load. Several types of power supply have been proposed for mono or multi-inductor systems. They may consist of a combination of conventional structures such as H-bridge on each inductor or nested structures supplying several inductors (Yong-Chae 1999, Fujita 2007). The three inductor’ currents here are generated by three current resonant inverters, which are presented in Fig. 4, where K1a is the IGBT 1 (respectively 2,3,4) for inverter a (respectively b,c) to be controlled, and the associated diode (D1a), Ci is the resonant condensator for Inductor i (i=1, 2, 3). K4a
D4a
K3a
D3a
Inverter 1 K2a
Current source
K1a
D2a
C1
D1a
V1
K1c
K4c
D1c
D4c
L Inverter 2
Disc plate
V2
C2
V3 C3
Current control circuit
D2c
D1c
K2c
K1c
K2c
K3c
D2c
D3c
Inverter 3
D3c
K3c
D4c
K4c
Fig. 4. Current Power Electronic’s structure of the heating device.
! . . . # $! #
)))) & )))) & )))) & )))) )))) )))) ⟹ &̅ ()))) & & & * &+, , , )))) )))) & )))) & &
(2)
As shown by (3) and (4), the terms , and -, are nonlinear, regarding the pulsation . It is also the case with the temperature . that influences the parameter representing the material resistance. )))) & +,+ /
², ², . . 1 !3# 1 /
4²56,7 .57,8 .97 4²56,7 .57,8 .<7 ))))) & +,, 3 : : : = 3, : : : = (4) 97 ;<7 4
97 ;<7 4
Equations (3) and (4) reflect that the presence of the work piece represented by , L4 and Mi4 increases the resistance of the diagonal terms and creates a mutual resistance in the impedance coupling terms. Consequently, measurements and calculations are required to obtain the real and imaginary parts of the global impedances including the coupling terms. The values of the impedance matrix terms are given by (5) which were calculated according to a “pseudo – energy” method from V and I measurements (Souley 2009).
33.1 j244.7 25.9 j43.8 21.5 j24.3 &̅ > 25.8 j43.6 67.3 j247.3 65.9 j113.7 H IΩ 20.9 j21.4 65.2 j111.4 107.1 j568.2
with:
(5) Fig. 5, shows for phases 1 and 2, the inverter current waveforms and the currents in the corresponding inductor. Due to the LC resonant circuits, the inductor currents are quite sinusoidal despite square wave supply inverter currents. The specific control angle calculation (Fig. 5) derives from an optimization procedure which is presented in (Souley 2011). Indeed this has been made possible by the knowledge of the induced current distributions. I_ind1
I_inv1
400
0 -200
ϕ21
-400
α1=0° I_ind2
I_inv2
300 200
δ2
100 0 -100 -200
α2
-300 0.0692
0.0694
0.0696
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 ∙ L 0 0 1 0 L 0 0 0 0 L 0 a 0 0 0 0 0 0 0 0 0 0
0 0 0 ∙ 0 0 L 0 0 1 0 0 1 0 L 0 0 0 0 0 L a
0 1 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 0
0 0
0 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0
1 0 0 0
Matrices A and B exist whatever the parameters are, and the system is controllable and observable. In this study, the current and voltage measurements are considered effective. After presenting the specified control structure, a robustness study is added to confirm the obtained performance.
δ1
200
A 0 0 0 B 0 0 0
0.0698
Time (s)
Fig. 5. Phase 1 and 2 inverter and inductor current waveforms
The control structure presented in Fig. 6, is quite classical but presents full matrix K and H and should be adapted to the defined induction process (Skogestad 2005). The feedback gain tuning K and pre-control matrix H could be pointed out. The contribution of both matrices should not only place the closed loop poles of Af (Af =A-BK), but also insure the decoupling between input, output, modes and mutual terms.
Writing all transfer functions as (6) describing the behaviour from Iondi and Ii and also Ij disturbing Iondi, a representation by bloc diagrams on 3 phases, allows to focus on the important coupling effects. The coupling terms add disturbances for the other phases with magnitude and frequencies that cannot be classically rejected. The process also requires an accurate control in magnitude and phase of each instantaneous current. K
1 !6# . L . M . L . M 1
NOP K,
L . M. !, . M ,#
. L . M² . L . M 1
QRSTS U V W
The proposed control structure, should decoupled the inductor’ currents, to obtain an independent control (with its own dynamic behaviour), thus in order to obtain a final accurate power density in the work piece. 3. CONTROL STRUCTURE In the state space representation (7) the vector X is 6dimensionnal presenting three control inputs U, to control the three outputs Y. The Multi-Input Multi-Output coupled system can now be treated to be as efficient as possible and the main goal is to decoupled inductor’ currents in the output Y. X - Y. - Z. [ \ L. 1 0 0 0 0 1 0 0 ]^_ - , [ >]^_ H, \ > H, L 0 0 1 0 0 0 0 1 ]^_ 0 0 0 0 0 0 0 0
dim(X)=n=6, dim(U)=m=3
0 0 0 0 1 0
0 0
0 0 0 1
(7)
Fig. 6. Complete eigenstructure control in state space. The MIMO system should be controlled globally and a short theory is resumed hereafter in state space representation introducing the theoretical tools used in the control synthesis. 3.1 Controller synthesis In the state space representation a pre-multiplication by a matrix H is possible only if this matrix presents a constant determinant and is not depending on time. The classical use leads to a transformation on the state X, in order to work in diagonalized space instead of a full matrix. The diagonal system permits to extract eigenvalues λ (called also modes or dynamics of the system). This multiplication can also be seen as a base transformation giving new properties or easy computation in this new reference frame. Manipulating new states in the virtual new space, should be easier and using the inverse transformation leads to the real values. All changes on variables is also possible on output only if the transformation is not singular (i.e.: conserving dimension). In matrix theory, the eigenvalues and eigenvectors of a matrix A, can be computed in different ways: eigenvalues
cancelling the determinant, right and left eigenvectors as expressed in (8). det!g.
Y# 0, Yh g h , i Y g i
(8)
-1
The modal matrix is V=[v1…vn] its inverse is V =[u1…un].
A transformation - . j based on the eigenvectors allows to write (7) as (9) in diagonal space, with : P a disturbance vector (noise) added to be general, with G a matrix applying noise on the states. The control law in closed loop (Fig7), introduces: [ Kklmn – p- so, noticing Λ the diagonal closed loop dynamic matrix equivalent to (A-BK) derives (9) jX Λj r
a ZKklmn u Lj
a s. t
(9)
In practice, V (dim 6 by 6), describes the repartition of the dynamics (dim n=6 modes λi) into the state X, V-1 represents how the modes are activated by the reference E (dim m=3) and/or disturbance P and CV represents how the modes act on the outputs. Considering the components on line k in Vi, due to -
v | … |^ y. j, shows that the relation fkVi=0 cut the link between mode jU and state xi. The function f is the direction on inputs and the control consists in defining K to have the closed loop desired dynamics and eigenvector placement in order to insure the desired decoupling. In the three-inductors case presented, f is 3-dimensionnal corresponding to the 3 current’ directions and is expressed as (10): f 1
0 0 , f 0 1
0 , f 0 0
1
(10)
Where: wi =-K.vi is the input direction acting on the states (including K unknown) dim m by 1. Writing (11) with matrix form, (12) is obtained and allows to indicate that the solution is in the kernel of the defined matrix with large dimension and can be sub-divised in two sub-space M, N. i.e.: in the case studied here, n=6, m=3, i=[1..6] so: N is dim 6 by 6 and M dimension is 3 by 6, for the total dimension 6 by 6+3 Ker([Y
-
vi= _λ .zi and wi= _λ .zi.
That means in a multi-input system, even if K can have multiple values, vi should be computed accurately in the frame made by the kernel of matrix [Y g I, Z] respecting the constrains ( fkVi=0 for the forsaken decoupling of the induction process) and written in (13). Y ~
the excitation input i and the dynamic j are separated if : i ZK{| 0 the dynamics i is eliminated from the state j if : { h 0 the dynamics i is eliminated from the output j : { Lh 0
The matrix K is m by n presenting m.n degrees of freedom even if only n dynamics should be placed (classically used in pole placement). Here all equations are used for decoupling purposes, and H contributes too. That opens the field to impose not only behavior on each state, but also insure a certain independent behavior between each state. This decoupling procedure is called eigenstructure placement. (Skogestad2005). 3.2 Control analysis Matrix K The K and H computation should insure pole placement and respect the constraints introduced for decoupling . The closed loop right eigen formulation can be established as (11) for each mode λi from 1 to n=6: Zp#h
g h 0 !A
g I#h BQ
g {W
Z U 3 = 0 0 U
(13)
To solve the system (13), desired eigenvector vid (dim 6x1) and directions wid (dim 3x1) are computed solving kernel of (13) for each i mode. Implying dim W=3x6 and dim V=6x6 and expression (14): K=-W.V-1
It is important to notice that the terms ‘eliminate/separate’ dynamics, actually means: cancelling this dynamic and not only considering fast and slow mode approximation.
!Y
(12)
With : N a subspace containing possible eigenvectors zi. That shows whatever the correction K is, when modes g are chosen fixing the closed loop dynamics, upper part of (12) is satisfied. The following relationships are obtained:
Considering this, three constraints can be listed: -
_λ g I, Z] )=~ _λ
(11)
(14)
where K (dim m by n=3 by 6) is determined using n degrees of freedom to impose n eigenvalues and let n.(m-1) to place eigenvectors including fk (f2 and f3 are used here I1 is chosen as reference here). 3.3 Control analysis Matrix H Matrix H only acts on the constraints insuring that the input (Eref) will not add its linked modes to the state and thus, on the output ever controlled by K. In the same way, if input i should not act on mode λj, the relation (15) is obtained: i ZK{| 0
(15)
Of course, the problem arises when there are more constraints than equations (13) and (15) should impose. An optimization algorithm based on least square algorithm can easily be used or inversion of the non square matrix N should be studied. In both cases, differences arise from the desired eigenvectors vid and the one computable and usable vi that satisfies all constraints vi=N(λi).zi.
Considering from (12), λ6 = inv([Y g I]).B. _λ , is a n by (n+m) non square matrix. Consequently, an approached solution should be computed : -
By minimization of a criterion J expressed in (16), representing the best possible solution zi approaching the eivenvectors vid in the
minimization of the square distance, so the least square method could be used. -
J ‖v
N!λ #. z
‖
(16)
Table 1. Poles placed and obtained Poles desired obtained
1 0.7401 0.7401
2 0.7393 0.7393
3 0.7386 0.7386
4 0.7397 0.7397
5 0.739 0.739
6 0.7382 0.7382
By using pseudo inverse formulation (17):
v N!λ #N!λ # N!λ # N!λ # . v a
Table 2. Performances obtained
(17)
where : T is for matrix transpose.
Phase 1
To be noticed the solution of least square form (16) corresponds to (16) with vi= N!λ #.zi.
From (11) and (12) the second equation ( C(A-λI).N+B.M=0 ), allows also to compute H (dim 3 by 3) with (17) and C=Id . In classical case, the pre-control square matrix H is solved using the solution vi in (18) and left-pseudo inverse formulation for M : ! C!A
BK#a . B# M
. K ∶ K ! . # ′
a
(18)
NB: The pole placement is now possible placing also constraints to quasi-cancel the link between input, output and modes. The eigenstructure is now completely defined from K and H to be tested on the simulated induction heating process.
Phase 2
Phase 3
Gain
with coupling 4.67
With decoupling 0.91
With coupling 2.89
With decoupling 0.91
with coupling 3.39
With decoupling 0.91
Phase
-48.6o
-32,4o
-16.20
-32.40
21.60
-32.40
Fig. 7, shows phase currents, and a zoom is made to focus attention on the good behavior of the current. The transient corresponds to the temperature profile imposed to rise up from ambient temperature to the desired one (300°C corresponding to 150A in phase3 represented here). Thus at time 0.03s, a little reference variation is introduced to test the performance of the control law. The performance should be on the magnitude and phase control of the current. Increasing the dynamics of poles provides more accurate currents (also in steady state), but a compromise has to be pointed out, in order to obtain a good control with the controller implemented in discrete time.
4. RESULTS AND ROBUSTNESS In the practical case studied in this paper, n=6 dynamics of the closed loop are imposed around ωbf=3*(2.π.1500)=30000rd/s. Exactly λ1=-30100rd/s, λ2=30200rd/s, λ3=-30300rd/s, λ4=-30400rd/s, λ5=-30500rd/s, λ6=-30600rd/s are set to fulfill certain conditions : to be as quick as possible, have quite the same dynamics on each current, avoiding solving matrix problems, imposing distinct poles and let open the possibility to transpose the control in discrete time (see TABLE1). A hierarchy is arbitrary imposed because the λ are quite the same, to let 2 modes λ on each inductor currents Iond linked to each phase I. In our case λ1 and λ4 are kept linked to the dynamic from Iond1 to I1. In the same way λ2 and λ5 to I2/Iond2, λ3 and λ6 to I3/Iond3 Finally K and H are a result to be used in the control scheme: . >. .
. . . . . . . . . 9.45 1.76 0.81 K >0.92 5.51 2.46H 0.22 1.3 7.02
. . .
. . H .
4.1 Results and global performances
Using sampled formulation of the state space, using Te=10µs (Fe = 100 kHz) the 6-discrete poles (exp(λiTe)) are given in Table 1. The poles are imposed and eigenvectors are near the requested ones. Performance obtained on current magnitudes and phases at f=1500 Hz are given in Table 2.
Fig. 7: Desired and obtained current on phase2 (complete signal and partial zoom) A gain (0.91) at f=1500Hz is slightly different from unity gain because no integrator is present. To overcome this problem, it could be easy to modify the gains in H to obtain accuracy in magnitude on the currents. A certain phase is introduced but may be checked at the system level, depending on its impact on the temperature profile and he accuracy requested for the metal piece heated.
4.2 Parameters variation and robustness A quick study is conduced to test the robustness of the controller in presence of load parameter variation. The load piece and the design of inductors as can provide variations: on R, L or M, also on C which should be accorded to the resonant frequency but the parameter dispersion on capacitance values with a certain tolerance can be of paramount importance. A maximum +20% and -20% variations of each parameters have been conduced and Fig. 8, Fig. 9 and Fig. 10, show the variations of performance in percent of the nominal level providing less than 10% effect, fulfilling the industrial requirement.
Fig. 8: R variations and performance on currents
Fig. 9: L variations and performance on currents
Futur works consist in applying the proposed control structure on the experimental device, optimized the pole placement to increase robustness, and test on the six inductors device. REFERENCES Acero J. et al. (2008), « The domestic induction heating appliance: An overview of recent research », in Applied Power Electronics Conf. and Exposition, 2008. APEC 2008. IEEE, 2008, p. 651–657 Forest F., Faucher S., Gaspard J.-Y., Montloup D., Huselstein J.-J., and Joubert C., (2007): « Frequency-Synchronized Resonant Converters for the Supply of Multiwinding Coils in Induction Cooking Appliances », IEEE Transactions on Industrial Electronics, vol. 54, no. 1, p. 441-452, févr. 2007. Fujita H., Uchida N., et Ozaki K. (2007), « Zone Controlled Induction Heating (ZCIH) A New Concept in Induction Heating », in Power Conversion Conference - Nagoya, 2007. PCC ’07, 2007, p. 1498-1504 Skogestad S. and Postlehwaite I. (2005), Multivariable feedback control: an analysis and design, John Wiley and Sons, 2005 Souley M., Egalon J., Caux S., Pateau O., Maussion P. (2011), "Modeling of a multi-phase induction system for metal disc heating", in Proc. of Electrimacs 2011, 5-9 June 2011, Paris Souley M., Spagnolo A., Pateau. O, Paya B, Maussion P. (2009): “Characterization methodology for the impedance matrix of multi-coil induction heating device “, 6th International Conference on Electromagnetic Processing of Materials EPM 2009; October 19-23, 2009, Dresden, Germany. Spagnolo A., Forzan M., Lupi S., Pateau O., et Paya B. (2009), « Space control optimisation of multi-coil transverse flux induction heating of metal strips », in Proceedings of the International symposium HES-10 Heating by Electromagnetic Sources, Padua, 2009 Tudorache T. et Fireteanu V. (2008), « Magneto-thermalmotion coupling in transverse flux heating », COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 27, no. 2, p. 399-407, 2008 Yong-Chae Jung (1999), « Dual half bridge series resonant inverter for induction heating appliance with two loads », Electronics Letters, vol. 35, no. 16, p. 1345-1346, 1999
Fig. 10: C variations and performance on currents 5. CONCLUSIONS A state space representation of an induction process is presented in this paper. The power process should be controlled in current whatever the couplings are between inductors. A eigenstructure state feedback is presented and adapted in the induction case studied. The control obtained has fixed gains and can be easily implemented on a ship controller. Performances obtained respect the industrial process requirement, allows the three current management and provides a certain robustness to parameter variations.
AKNOWLEDGMENT This work is a part of the Innovative Solutions for Induction Systems project, funded by the French Research National Agency with EDF EPI, CELES, LAPLACE, Arts Mines, SIMAP, ATYS CONSULTING GROUP and CNRT as partners.