aisc 187 - h∞ control of an induction heating invertervoltage controlled oscillator) ... the...

I. Dumitrache (Ed.): Adv. in Intelligent Control Systems & Computer Science, AISC 187, pp. 235–250. springerlink.com © Springer-Verlag Berlin Heidelberg 2013

H∞ Control of an Induction Heating Inverter

Abstract. The industry strongly relies on PI controllers for induction heating in-verters due to their simple design. Because correct load parameter estimation is almost impossible and their effect over electrical variables is not negligible, as it is presented in the paper, it is necessary a variation tolerant control algorithm. The present work proposes such advanced algorithms, based on different augmented plant models and on H∞ design method. The paper also focuses on the practical approach of the controller implementation on an experimental pilot. The advan-tages of the robust controller are justified by simulation results and experimental data, as compared to the classical PI controller solution.

Keywords: induction heating, robust H∞ control, pilot plant, embedded control.

1 Introduction

Induction heaters are modern, economic and environment friendly equipments. The heated work-piece is placed in an inductor, which generates high intensity electro-magnetic field inducing eddy currents. Through Joule-Lenz effect of these currents heat is generated directly in the working material. For a given inductor and work-piece geometry, the heating profile strongly depends on the skin and proximity ef-fects, which are influenced by the physical properties of the heated material [1].

For low power (<25kW) and high frequency (>10kHz) applications, inverters with series resonant load are preferred due to their simple design.

The output inverter variables (load current, developed power, capacitor voltage or phase shifts) can be controlled through inverter supply voltage, or through driv-ing strategies of the power transistors of inverters.

For the present study the second approach was adopted, output variables being controlled by the inverter bridge, in order to avoid complex and expensive power supplies.

The industrial applications strongly relies on PI controllers, for frequency con-trolled inverters [2, 3, 4], and variable frequency PWM (Pulse Width Modulation) inverters [5].

Although PI controller studies are very common, are also known methods based on Fuzzy controllers [6] or adaptive controllers [7]. They have small impact due to the high amount of calculus implied by these advanced control algorithms.

236 T. Szelitzky and E.H. Dulf

Fig. 1. Inverter block diagram

The general scheme of the induction heater is exemplified in figure 1. The in-verter uses a DC power supply. The inverter generates a variable frequency square wave, which supplies the load. The values of the resulted load current are acquired and delivered to the controller which drives the power transistors through a VCO (Voltage Controlled Oscillator) and proper transistor drivers.

The present work conceives, design and implement a new version of a H∞ ro-bust controller dedicated to the above described process. After a short introduc-tion, the paper justifies the difficulty of correct load parameter estimation and emphasizes the effect upon electrical variable, highlighting the necessity of a vari-ation tolerant control algorithm. Based on conventional models, using the first harmonic of Fourier expansion, the next section develops an “upgraded” model with uncertain parameters, enabling the robust controller synthesis. Based on the augmented plant model, on the weighting functions and on the mixed sensitivity algorithm, the H∞ controller is designed and implemented. This new controller will improve general performance of the heating equipment, compared to the usual, well-known controllers. Are emphasized the advantages of the H∞ control-ler by comparing the simulation results with the experimental data of the laborato-ry induction heating equipment. The last section of the paper presents concluding remarks.

2 Load Parameter Estimation, and Their Influence on System Performance

2.1 Parameter Effects on Electric Variables

Inductor analytical design methods were developed in the first half of the XX cen-tury and became most popular in the 1960-1970s. The procedure is based on the similarities between the inductor and high leakage transformers. The design me-thod is based on the use of correction factors, which asks for a strong knowledge and experience of the design engineer.

H∞ Control of an Induction Heating Inverter 237

The development of the actual computing power leads to newer and easier cir-cuit parameter estimation methods. The current possibilities of existing numerical methods and of the computers hardware enable the development of some advanced algorithms, the most used for inductor modeling being [8]:

- finite element method; - boundary element method; - mutual impedance method, etc.

These methods can be used together or separately in order to model electromag-netic fields and heat flow. Some dedicated programs for this kind of problems are: Ansoft’s MAXWELL, Consol, Flux, Elta and FEMM. For this study the open source FEMM-method (Finite Element Method in Magnetics) was used, which is dedicated to solve the steady-state electromagnetic and heat flow problems [9].

The major disadvantage of this approach relies in the appropriate mesh sizing. For this particular application, a dependency between mesh size and circuit impedance is shown in figure 2.

Fig. 2. FEMM meshing and equivalent circuit impedance

It can be observed a high variation of circuit parameters as function of mesh size, making difficult an accurate parameter estimation, which will make the con-troller design more unreliability. To minimize the possibility of system instability while ensuring fast dynamics, robust control of induction heating systems is considered a viable solution.

2.2 Parameter Effects on Load Current

It is a general observation that, during the heating evolution, the load’s circuit pa-rameters vary. These variations are generated by the changes with temperature of the work-piece physical parameters, with effect on the skin and proximity pheno-mena. These parameter changes can affect current, power and phase shifts in the circuit. A secondary interdependency is established between the switching


frequency of the power supply and the load parameters. With the increase of the switching frequency the load’s equivalent resistance grows while its inductivity decreases in accord to the inverse relation between frequency and skin depth. This influence is relative weak, but must be considered in the case of frequency controlled inverters.

a b

Fig. 3. Square wave duty ratio and switching frequency effects over load current for various loads

Almost all dependencies listed above can be incorporated in one single varia-ble: circuit quality factor, which make possible the study of current, power, vol-tage and phase displacement evolutions.

Figure 3 presents the effects of the duty ratio and of the switching frequency on load current for different values of the quality factor, the data being normalized. If quality factor fluctuations arise (due to new batch or unequal temperature distribu-tion), inductor current variations can occur. These can be compensated by chang-ing the main input variables: square wave duty cycle and switching frequency, since these variables are controllable through the inverter. The increase of duty ra-tio increases the load current, while the frequency increase has a diminution effect. The circuit has a band pass filter characteristic, which implies electric variable at-tenuation in each direction of frequency deviation from resonant frequency. For a given resonant frequency f0, by frequencies greater than 1.7*f0, the current variations with frequency are insignificant. For desired lower current values, a secondary control strategy is required.

3 Inverter Modeling

The series RLC circuit with a quality factor over two and powered by an AC source presents band-pass filter properties. Therefore, if the circuit is excited with


a rectangular signal (signal with a infinite harmonic spectrum), the effects located in the pass band will be more pronounced than the effects of harmonics outside this band. Hence for model construction purposes the rectangular voltage power supply with variable frequency and duty cycle can be replaced by a sinusoidal voltage source with variable amplitude and frequency.

A demonstrative example is given in Figure 4. The series RLC circuit was ex-cited using a rectangular wave and the first (sinusoidal) harmonic. Measuring the load current, acceptable differences can be observed for the two excitation cases. If load quality factor is further increased, the differences between the effects of the two excitations will decrease. This property maintains as far as the rectangular signal’s frequency and band pass of the filter (load) coincide or are close to each other [10]. Based on this observation, several models have been deduced. The most flexible [11] and most convenient will be used to augment with uncertainties enabling the design of robust controllers.

Fig. 4. Differences between square wave and sinusoidal wave excitation

For easier manipulation, the variable duty cycle rectangular signal is centered be-fore Fourier decomposition to accomplish the anti-symmetry property, which enables the approximation of the original signal only through sine functions of form:

)tωsin()β2

πsin(U

π4

u dcAB ⋅⋅= (1)

Applying Kirchoff’s law, it can be obtained the following equations:

dt

duCi

Riudt

diLu

CAB

ABCAB

AB

=

++= (2)


Considering the load current and the capacitor voltage:

)tωsin()t(u)tωcos()t(u)t(u

)tωsin()t(i)tωcos()t(i)t(i

sin,Ccos,CC

sin,ABcos,ABAB

+=+=

(3)

the matrix form of the model is:

u

0

0

)β2

πsin(

L

1

π4

0

x

L

R

L

1ω0

C

100ω

ω0L

R

L

1

0ωC

10

x ⋅

+⋅

−−−

−

−−= (4)

Choosing the system variables as:

Tcos,ABcos,Csin,ABsin,C ]iuiu[x = (5)

and the system output variable the load current, defined as:

2i

2iy,iiy

2cosAB

2sinAB

RMS2

cosAB2

sinABMAX +=+= (6)

and taking into account the possible disturbances for all variables, results the fol-lowing linearized system:

xII

I0

II

I0y

ω

I

U

I

U

β

0

0

)β2

πcos(

L

U20

U

0

0

)β2

πsin(

L

1

π4

0

x

L

R

L

1ω0

C

100ω

ω0L

R

L

1

0ωC

10

x

C

cos,AB2

sin,AB2

cos,AB

cos,AB2

sin,AB2

sin,AB

B

sin,AB

sin,C

cos,AB

cos,C

BB

A

321

++=

−−

+

+⋅

+⋅

−−−

−

−−=

(7)

Since the paper is focused on frequency controlled inverter, the matrixes B1 and B2, which are useful in PAM (Pulse Amplitude Modulation) and PWM (Pulse Width Modulation) control, respectively, are neglected.


To compute the system parameters in the static operational point, the Cramer rule can be applied as:

3SOP BA

1x ⋅= (8)

The linearization point is commonly chosen based on general usage of the equip-ment. Mathematically, for higher accuracy of the model over a wider range of switching frequencies and duty factors, it is recommended to set the linearization point for a duty factor of 25% and the switching frequency in the range (1.02-1.06*f0).

4 Robust Control

The rapid development of computers in the last decades led the introduction of new concepts in system engineering. In the modern automation, are considered models which approximate the plant, but are taken into account also the differenc-es between the model and the real plant. These models have to be simple to facilitate the analysis and controller design, but complex enough to include all plant behavior. [12] The differences between the mathematical model and the process are called "modeling errors" or "uncertainties" [13]. The presence of un-certainty is imminent, resulting from voluntary or accidental neglect of high fre-quency dynamics and un-modeled nonlinearities, from parameter variations and component tolerances, etc [14].

The aim of the robust control synthesis is to design a fixed structure controller which ensures stability and acceptable performances in the presence of disturbances and noise for the nominal system but also for the system family in the considered range of uncertainties. The easiest method to ensure this is to take into account the "size" of system family signals, which is represented by the norm of signals.

The norms are defined in Hilbert space, and if the system is linear and the out-put signals are bounded, the norms represent gains, amplifications [15]:

( )∞

∞∞∞ =

u

ysupG

u,

(9)

Figure 5 represents the infinity norm of a frequency controlled inverter with RLC load, considering as output variable the load current. The maxima of this norm, indifferent of batch temperature, are obtained at the resonant frequency, where the circuit amplification is maxim. The maximum of the infinity norm is (1.032e-3) at the melting point of the work piece for a switching frequency of ~7.4kHz, while the minimum is obtained at room temperature at 15 kHz. The separate group of values from the right side of figure represents the system norms without work-piece where the maximum (4.129-3) was obtained at 6.9 kHz, and the minimum at 15 kHz.


0

5000

10000

15000

0

500

1000

15000

1

2

3

4

5

x 10−3

Switching Frequency (Hz)Batch Temperature (°C)

Infin

ity N

orm

Fig. 5. Infinity norm of the uncontrolled system

4.1 Robust Controller Synthesis

Robust control was firstly developed for the defense industry, mainly for airplane and rocket guidance. These systems vary their flight characteristics and dynamics with the change of weight (cargo), altitude, speed etc. The controllers minimize the norm of the closed loop system, minimization which takes into consideration the system uncertainties. To design the controller, the uncertainties have to be in-troduced in the model in order to form the augmented plant model.

In the case frequency controlled inverters with load current as output variable the nominal model is based on equation 7 and is presented as block diagram in figure 6. The character X denotes all auxiliary parameters which depend by the working (linearization) point. For the augmented model the number of uncertain-ties can rise up to 9, depending on the model complexity: three variable parame-ters (L, R and ω) and six parameters depending on operation point, located in matrixes B and C of equation 7. [16]

To keep system order as low as possible, three different cases will be studied: - system with two uncertainties: L and R - system with three uncertainties: L, R and ω - system with all nine uncertainties: L, R, ω and six linearization uncertainties.

The easiest way to introduce the uncertainties is by replacing the concerned para-meters with upper LFT in uncertainty, figure 7.


Fig. 6. Inverter block diagram

Fig. 7. Introduction of parameters with uncertainties

Two different uncertainty representations were used (additive and multiplicative) in each case in order to compare the performances of the controlled plants. The structures of these additive and multiplicative uncertainties are presented in figure 8.

Fig. 8. Additive and multiplicative uncertainties

Depending on the used modeling approach, the uncertainty matrixes from figure 7 are the following: - For additive uncertainties:


=

=

=

=

X1

10M;

ω1

10M;

R1

10M;

L/11

10M XωRL (10)

- For multiplicative uncertainties:

=

=

=

−−

=Xp

X0M;

ωp

ω0M;

Rp

R0M;

L/1p

L/1pM

XX

ωω

RR

L

LL . (11)

The augmented model of the process in all cases takes the standard form used in robust control:

wDuDuC z

wDuDuCy

wBuBAxx

22212

12111

21

++=++=

++= (12)

Two weighting functions are chosen (Wp and Wu) to represent the frequency cha-racteristics of some external (output) disturbance d and performance requirement (including consideration of control-effort constraint) level, which modify the mi-nimization problems to [17]:

1u

1p

)GKI(Kw

)GKI(wmin −

−

++

(13)

This approach is known in the literature as mixed sensitivity problem or S-KS problem. The performance weighing function is defined as a low pass filter ampli-fying the low frequency errors, while, Wu, the control signal weighting function, is defined as a high pass filter, ensuring the control signal limits.

1sT

sKw;

1sT

Kw

u

uu

p

pp +

=+

= (14)

The controller design follows the algorithm described in [14]:

1. Define D1*,D*1, Rn and nR~

as:

[ ]

−=

−=

==

00

0IγDDR

~

00

0IγDDR

D

DDDDD

1p2

T1*1*n

1m2

*1T*1n

21

111*1211*1

(15)

2. Solve the two Riccati equations

1)GKI(S −+= (16)


3. Construct the feedback and the observer gain matrixes

[ ] 1n

TT1*121211

T1

T*1

1n

2

12

11

R~

)YCDB(LLL

)XBCD(R

F

F

F

−

−

+−=

+−=

(17)

4. Compute 2121 C,C,B,B,A,Z

)F(CDC );F(CDFC

D)LZ(BB ;D)LZ(BZLB

)F(CBBFAA YX);γ(IZ

1222121221121

1212221112221

12212

+−=+−=

+=++−=

+−+=−= −

(18)

5. Build the central controller

=

111

1

DC

BAK (19)

For each of the six cases (two uncertainty representation for each considered case), the weighting functions parameter are selected in accord to Table 1.

Table 1. Weighting function parameters

Nr of uncer-tainties

Type of uncer-tainties

Kp Tp Ku Tu

2 Additive 8536.59 7.44 0.002 0.002 2 Multiplicative 4146.34 24.39 0.003 0.010 3 Additive 51219.5 5.61 0.000 0.009 3 Multiplicative 975.61 28.54 0.001 0.000 9 Additive 4512.19 91.46 0.007 0.439 9 Multiplicative 1524.390 280.488 0.001 1.000

The resulting structure of the H∞ controllers is

00

11

22

33

44

55

65

00

11

22

33

44

55

regzazazazazazaza

zbzbzbzbzbzbH

+++++++++++

=∞ , (20)

with the parameters presented in Table 2.


Table 2. Controller parameters

Nr of un-cert.

Type of un-cert.

Numerator

b5 b4 b3 b2 b1 b0 2 Ad. -30.05 127.3 -229.3 219.32 -111.4 24.06 2 Mult. -215.70 477.2 -396.5 169.14 -38.329 3.700 3 Ad. -19.594 92.4 -176.59 170.61 -83.412 16.505 3 Mult. -7.495 -4.235 2.953 3.197 0.988 0.106 9 Ad. -0.828 3.915 -7.492 7.253 -3.553 0.705 9 Mult. -0.543 -0.209 0.349 0.304 0.089 0.009 Denominator Ad. a6 a5 a4 a3 a2 a1 a0

2 Ad. 1 -4.32 7.890 -7.63 3.906 -0.85 0.000

2 Mult. 1 -2.23 1.870 -0.8 0.180 -0.02 0.000 3 Ad. 1 -5.376 12.15 -14.8 10.22 -3.80 0.596 3 Mult. 1 0.288 -0.68 -0.49 -0.11 -0.001 0.002 9 Ad. 1 -5.71 13.69 -17.6 12.89 -5.06 0.836 9 Mult. 1 -0.61 -1.03 0.083 0.396 0.147 0.017

5 Simulations and Experimental Results

In order to test each designed controllers, an experimental low power prototype induction heating inverter was built, figure 9. The block diagram of the control loop is presented in figure 10.

Fig. 9.The experimental plant


For this particular application the two BSM75GB120DN2 modules mounted in an H bridge connection is fed from a 30V DC power supply. Each power transistors is driven by a UCC37321, which receives the signals through two optocouplers.

Fig. 10. The block diagram of the control loop

The two optocoupler approach was selected to ensure safe operations by requir-ing two signals (enable and gate signals) to open a transistor, minimizing the acci-dental short circuits. Prior to galvanic separations, a delay and signal shaper circuit was designed, which ensure the safe transition between open and blocked state of the power transistors. The pulse generator receives the driving signals from a 4046N’s VCO (Voltage Controlled Oscillator) part, which obtains the control sig-nals from a PIC32MX440F512H through an IO (Input Output) board. This inter-face converts the digital signals provided by the microcontroller through an R-2R ladder, and passes through a series of op-amps for convenient VCO control. The load circuit consists of three elements: matching transformer, capacitor and the in-ductor. The capacitor is mounted in series with the matching transformer, which has a 16:1 transformation ratio. The inductor is a three turn solenoid with an inter-nal diameter of 60mm. The work piece is a 40mm iron rod inserted in the middle of the coil. In series with the primary winding, a current transformer is mounted, with a transformation ratio of 5:1. The feedback is fed to an AD736 RMS-DC converter, which output is introduced - after a few mathematical operations - in the analog to digital converters of the microcontroller. The mathematical operations are required due to AD736’s negative output and the microcontroller’s analog to digital converter low input range (0-3.3V).

The designed robust controllers are tested using Matlab® simulations. In figure 11.a are presented the step response of the closed loop system using the aug-mented models with nominal values and of the closed loop with PI controller and classical model, while in figure 11.b are presented the same step responses in worst case. It can be observed that in nominal case the PI controller is even better than the robust controller, but in the worst case simulation the robust controller remains stable, the performances depending on the used uncertainty model, while the system with PI controller became unstable.


0 0.5 1 1.5 2 2.5 3

x 10−3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec)

Cur

rent

(A

)

Ad. 2 uncertMult. 2 uncertAd. 3 uncert.Mult. 3 uncertAd. 9 uncert.Mult. 9 uncert.PI Control

a a

a b

Fig. 11. Step response of the closed loop using augmented models with nominal values (a) and in worst case (b)

The same good performances of PI and robust controllers can be observed from figure 12 in the nominal case with the experimental plant.

a b

Fig. 12. Experimental step response of the closed loop using robust (a) and PI (b) controller with nominal parameters

Figure 13 presents the experimental worst case scenario. The experimental re-sult obtained with the best robust controller is plotted in figure 13.a, highlighting the same stability as in simulation results.

a b

Fig. 13. Experimental step response of the closed loop using robust (a) and PI (b) controller in worst case


The output using PI controller has an oscillatory characteristic, due to electric protection equipment, figure 13.b. Without these safety features the output would be unstable, as in the simulation result.

6 Conclusions

In the literature are detailed different PI controller designs for the frequency con-trolled induction heating inverters with good results. Advanced control strategies are studied only in a few article and presented as control structures with difficult implementation possibilities. The present work proposed to overcome this gap and to design a series of robust H∞ controller, based on augmented models with dif-ferent complexity and two type of uncertainty (additive and multiplicative). The validation of the controllers is done using Matlab® simulations. The realized pilot scale experiments prove the viability of the proposed advanced control and ask for future research in the domain.

Acknowledgement. The authors are grateful to eng. Peter Bela for his support and contri-bution in the experimental part of the present work.

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aisc 187 - h∞ control of an induction heating invertervoltage controlled oscillator) ... the...

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