Observation of state variables in resonant DC-DC converter using the high gain design approach

Observation of state variables in resonant DC-DC converter using the high gain design approach F. Giri, F. Liu, O. El maguiri*, H. El Fadil, GREYC Lab, University of Caen, France * Corresponding author, [email protected] ______________________________________________________________________________________________________ Abstract: We are considering the problem of state observation for in series resonant converters (SRC). This is a crucial issue in SRC output voltage control because the control model is nonlinear and involves nonphysical state variables, namely real and imaginary parts of complex electrical variables. A high gain observer is designed that ensures global exponential convergence of state estimates to their true values. The observer design involves no resolution of complex dynamical or algebraic equation. The global and exponential features of state estimate convergence make the observer utilizable in control. Keywords: series resonant converters, state variables observation, high gain observer. ______________________________________________________________________________________________________

1.

S

INTRODUCTION

eries and parallel resonant DC-to-DC converters, and their various variants, have been given a great deal of interest in the power electronic literature. Compared to (hard) switched converters, SRC converters present several advantages e.g. they provide much higher power supplies. Besides, power losses are considerably reduced (improving thus the conversion efficiency) in SRC converters these do not involve switched components. However, SRC converters are more complex to control as they involve much more nonlinear dynamics. Furthermore, they are supplied by bipolar square signal generators and, consequently, the switching frequency is generally the only available control variable. These considerations make SRC modeling a particularly hard task. Instead of highly complex generalpurpose models, we generally seek simpler control-oriented models, [1]. A modeling approach, based on generalized averaging, was developed in [2] for series resonant DC-toDC converters. Small signal models for series and parallel resonant converters were developed in [3]. Sampled-data modeling and digital control has been addressed in [4]. The point is that these models generally involve state variables that all are not accessible to measurement. Therefore, they cannot be used in control unless they allow the construction of observers. State observation has yet to be solved for SRCs. In the present work, it is shown that such an issue can be solved for DC-to-DC SRCs. The focus is made on the circuit of Fig 1 which is largely considered in specialized literature. Following the first harmonic approach developed in [1], a fifth order state-space model is designed for the considered circuit. Most of involved state-variables turn out to be nonaccessible to measurement. Then, an observer of the high gain type is developed. It is well known that high gain observers ensure, under mild assumptions, global exponential convergence of state estimates to their true values. Such global and exponential features make the proposed observer readily utilizable in control schemes. The general framework

of the observer design is based on [5]-[6]. The major step in such design is the construction of a signal diffeomorphism map leading to a transformed model that fits a canonical observable form. This canonical form is composed of a fixed linear dynamic component and a triangular controlled one. Using this special form, a high gain observer can be designed in a rather straight way under some global Lipschitz assumptions on the controlled part. The gain of the proposed observer is issued from an algebraic Lyapunov equation. The paper is organized as follows: mathematical modeling of the series resonant converter is addressed in Section 2 and 3; theoretical design of the state observer is coped with in Section 4; the observer performances are illustrated by simulation in Section 5; a conclusion and reference list end the paper. 2.

MODELING SERIES RESONANT CONVERTERS

The studied series resonant DC-to-DC converter is illustrated by Fig 1. A state-space representation of the system is the following:

di

$ #v # vo sgn(i ) " E sgn(sin(! t )) dt dv C $i dt v dv C0 o $ abs(i ) # o dt R L

(1) (2) (3)

where v and i denote the resonant tank voltage and current respectively; vo is the output voltage supplying the load (here a resistor R ); the power source supplying the converter is characterized by a constant amplitude E and a varying

switching frequency ! ( rd / s ) ; L and C designate

d xk

respectively the inductance and capacitance of the resonant

dt

tank; the resonant frequency is denoted !0 $ 1/ LC . As the supply source amplitude E is constant, the pulsation ! turns out to be the only possible control variable.

i

v

t

vo

The SRC converter modeling is based upon the following assumptions: Assumption 1: The voltage v and current i are accurately approximated by their (time varying) first harmonics (denoted V1 and I1 e j* respectively).

Applying (6) with k $ 1 to equations (1) to (3), we obtain the following ‘first harmonic’ nonlinear model of the SRC [1]:

!

Fig 1. Series resonant converter under study The model (1) is not suitable for controller design because the control signal ! comes in nonlinearly. Then, it cannot be based on to get an observer for control purpose. A control/observation oriented model can be obtained applying to (1) the first harmonic approximation procedure introduced in [3]. This is developed in the next section. FIRST HARMONIC APPROXIMATION

This approach relies on the assumption that the solution of a nonlinear oscillator system can be expanded in a Fourier series with time-varying coefficients. Then, a solution x(t ) is approximated by the Fourier series expansion of the function

dI1 1 2 2E $ # j ! I1 " [#V1 # V0 e j+ # j ] dt L ) ) dV1 1 $ # j ! V1 " I1 dt C V dVo 4 abs( I1 ) # o $ ) C0 RCo dt

xˆ (t , s ) $ x(t # T " s ) $ ' x k (t )e

jk! (t #T " s )

(4a)

k $ #&

1 xk (t ) $ T with

! $ 2) / T

d xk dt

$

T

(

0

x(t # T " s)e

. The coefficients

d x dt

# jk! (t #T " s )

xk

ds

(4b)

undergo the following equation:

(t ) # j k ! x k (t )

(5)

k

In the case where x(t ) is generated by a controlled nonlinear system x! $ f ( x, u ) where u denotes the control signal, it follows from (5) that:

(7) (8) (9)

The ‘harmonic’ model (7)-(9) is quite representative of the SRC due to Assumptions 1 and 2. Its advantage, compared to (1)-(3), is that the control signal ! comes in linearly. However, it is still not suitable for control/observer design because it involves complex variables and parameters. To get a convenient state-space model, introduce the following notations: I 1 $ x1 " j x2 , V1 $ x3 " j x4 , Vo $ x5 (10) Substituting (10) in (7)-(9) yields the following state-space representation:

def

xˆ (t , s) $ x(t # T " s) , defined in the interval s % [0, T ] . Mathematically, one has the following standard expressions: "&

(6)

Assumption 2: The time scale of the output filter is much larger than the resonant tank so that the ripple appearing in the output voltage can be neglected and vo can be accurately approximated by its DC-component, i.e. vo $ Vo where Vo is the DC-component of the (time varying) Fourier series of vo .

E

3.

$ f ( x, u ) k (t ) # j k ! xk (t )

x!1 $ x2 u #

x3 2 x5 # L )L

x! 2 $ # x1 u #

x4 2 x5 # L )L

x1 x12

(11)

" x22 x2

x12

"

x22

#

2E )L

x1 C x x!4 $ # x3 u " 2 C x 4 x!5 $ x 2 " x22 # 5 )Co 1 RCo x!3 $ x4 u "

(12) (13) (14) (15)

def

where u $ ! . In the above model, the only quantities that are accessible to measurements are: x5 $ Vo ,

x12 " x22 $ I 1 ,

x32 " x42 $ V1

(16)

That is, the variables x1 , x2 , x3 , x4 must be estimated using measurable quantities. To this end, an observer is built up in the next section. HIGH GAIN OBSERVER DESIGN

4.

z!7 $ z5 u " y $ [ z1

Our objective is to design an observer that provides estimates of the state variable x1 , x2 , x3 , x4 such that the estimation errors converge exponentially to zero. First, the model (11)(15) is transformed so that it fits the form considered in the high gain design method, e.g. [5], [6]. To this end, we propose the change of coordinates + : IR 5 , IR7 defined by:

(17)

z $ 3z1

/ $ " $ " $ x1 x 3 " x 2 x 4 $ x2 $ x4 $ x1 $ x3 . x 22 x 42

(18)

Using (18), it follows from (11)-(15) that the new state vector z undergoes the following equations: z!1 $

# z3 2 x5 2 E z 4 # # L z1 ) L ) L z1

z z! 2 $ 3 z2 C

z!3 $ #

z 22 z12 2 z3 x5 2 E z5 " # # L C ) L z1 )L

2

z 6 4 ; z $ z7 T

2 #1 0 Lz F1 ( z , u ) $ 0 1 0 1 0Cz 1 2 1

z4 4

T

(28)

4

(29)

# 2E / -

) L z1 -

2 # 2E 0 F2 ( z1 , z 2 , u ) $ 0 ) L 0 #1 0 1 L

0

(30)

.

/ 0 # u.

(31)

(32)

2 # 2 x5 / 0 G1 ( z 1 , u ) $ 0 ) L 0 0 1 . 2 z 22 z12 2 z 3 x 5 0# " # 0 C ) L z1 G2 ( z 1 , z 2 , u) $ 0 L # 2 x5 z 4 2E 00 # ) L z1 )L 1 z4 2 / 0 C 1 2 3 G3 ( z , z , z , u ) $ 0 x z 2 0 5 6 z u # 4 0 ) L z1 -. 1

(33) / -.

(34)

(35)

z / 2 G 4 ( z , u ) $ 00 z 5 u " 6 -C . 1 4

' nk $ 7

(36) and z k % IR nk , G k % R nk ,

(19)

It readily follows that

(20)

Fk % IR n k 5 n k "1 ( k $ 1, ..., 4 ) and z $ z 1 z 2 z 3 z 4 % IR7 . With the above notations, the system (19)-(27) can be given the following compact form:

(21)

z!4 $ # z6 u #

z5 2 x5 z 4 2 E # # L ) L z1 ) L

(22)

z!5 $ # z 7 u "

z4 C

(23)

z 2x z z!6 $ z 4 u # 7 # 5 6 L ) L z1

z $ 3z 5 3

(27)

z 2 4 ; z $ 3z 3 T

2 #u / F3 ( z1, z 2 , z 3 , u ) $ 0 1 0# 1 L.

with: x12 x 32

(26)

n1 $ n2 $ n3 $ 2 ; n4 $ 1 1

2z 0 1 0 0 z2 0 z3 0 0 z4 0z 0 5 0 z6 0 1 z7

(25)

Let us introduce the following notations and definitions:

4.1. PRELIMINARY MODEL TRANSFORMATION

2 z1 / 0 2 x1 / 0 z2 0 0z 0 x2 0 30 x $ x3 , z $ + ( x ) $ 0 z 4 0 0 0 x4 0 z5 0 0 z6 1 x5 . 00 -1 z7 .

z6 C z2 ]T

k $1

3

96 z! $ F ( z , u ) z " G (u, z ) 8 67 y $ z1 $ C z

T

(37)

with :

3

G ( z, u ) $ G1 ( z1 , u ) G2 ( z1 , z 2 , u ) G3 ( z 1 , z 2 , z 2 , u ) G 4 ( z , u )

(24)

4

4

T

(38)

?0 F1 ( z1 , u ) < 0 0 = : 1 2 0 0 F2 ( z , z , u ) =0 : F ( z, u ) $ = : (39) 1 2 3 0 0 0 ( , , , ) F z z z u = : 3 = : 0 0 0 0 > ; C $ 3I 25 2

025 2

025 2

02 5 2 4

(40)

where I 25 2 denotes the 2 5 2 identity matrix and 0i5 j the

i 5 j null matrix. In view of (38)-(40), the new model (37) fits the form required in the high gain observer design. 4.2. SECONDARY MODEL TRANSFORMATION

Following [5]-[6], we first introduce a change of coordinates which transforms the system (37) into a special form for which an observer is developed. Then, the equations of the obtained observer are expressed in the original coordinates. Let @ (u , z ) be the block diagonal matrix defined by: @ (u, z ) $ diag 3I 252

F1

F1 F2

F1 F2 F3 4

(41)

3

@ (u , z ) $ diag I 252

( F1 )

"

( F1 F2 )

"

( F1 F2 F3 )

"

4 (42)

2E u / ) L z1 0 .

2 6Eu 2Eu F 1F 2 F 3 $ 0 # 0 ) L2 z ) L C z2 1 1 and C z2 / 2 0 F1" $ 00 # L) z1 # C) z 2 -0 2E . 1 2E

(43a)

/ .

T

(43b)

(44a)

2 0 0 ( F1F2 ) $ 0 L) z 1 0 0 2E u 1 "

# LC) z 2 / 2E C) z 2 E u -.

? #3 ( F1 F2 F3 ) $ = 2 >= 2 E L ) A u z1 where

(45a)

? < ? z1 < ? w1 < z1 = : = 2: = 2: 2 F1 z z w : z $ = 3: , w $ = 3: $ @ z $ = = F F z3 : =z : =w : = 1 2 4: = 4: = 4: >= z ;: >= w ;: >= F1 F2 F3 z ;:

(45b)

where wk % R 2 ( k $ 1, ..., 4 ). It follows from (42) that B is to one. Let B c denotes its inverse i.e. B : w , z $ B c (w) . Then, the dynamics of w is given by:

one c

C! C! C! z! " u! $ 3F ( z, u ) z " G (u, z )4 " C! u! Cu Cz Cu Cz

which, in view of (45b), yields: C! C! 2 C! / w! $ @F ( z , u ) z " 0 G (u , z ) " u! # @ - F ( z, u ) z " Cz Cu 1 Cz .

C! C! C! G (u, z ) " u! # @)@" A@ z " Cz Cz Cu

where (46) follows from the fact that @F $ A@ equivalently, F $ @" A@ with: ?0 =0 A$= =0 = >0

(46) or,

0< 0 :: I2 : : 0 0 0; is a 8 5 8 square matrix. Substituting @#1 w to z in (46), one obtains: I2 0 0

0 I2 0

C! 9 (u , ! c ( w)) u! 6w! $ A w " D( w, u ) " E ( w, u ) " Cu 8 67 y $ C w $ w1

(47)

where C $ 3I 25 2 025 2 025 2 025 2 4 is a 2 5 8 matrix and:

(44b) "

(44d)

B : R7 , R 8

$ A@ z " (

It can be checked that: 4E 2 0 2 F 1 F 2 $ 0 ) L z1 0 2E 0# 1 ) L C z2

9 1 " L4) 2 z12 ) 2 L2C 2 z 22

Consider the following change of coordinates:

w! $

Then the pseudo-inverse of @(u, z ) has the form: "

A$

< : 2 E LC ) A u z 2 ;: 1

? C! D( w, u ) $ = (u , ! c ( w)) @" (u , ! c ( w)) # > Cz

@(u , ! c ( w)) @" (u, ! c ( w))4 A w

(44c)

E ( w, u ) $

C! (u , ! c ( w)) G (u, ! c ( w)) Cz

(48a) (48b)

4.3 . OBSERVER DESIGN

zˆ 3 $ #

Let GF be the block diagonal matrix defined by:

? GF $ diag = I 25 2 >

I 25 2

F

I 25 2

F

2

I 25 2 < F 3 :;

where F H 0 is a real number and let S F $

(49) 1

GF S 1 GF where F S1 is a symmetric positive definite matrix that is solution of the Lyapunov equation: S 1 " AT S 1 " S 1 A # C T C $ 0 It can be shown that: S1 (i, j ) $ (#1)i" j Ci "j #j1#2 I 2 , for i I 4, j I 4

j! $ i!( j # i )! Then, an observer candidate for the system (46) is: C! w!ˆ (t ) $ Awˆ " D(u , wˆ ) " E (u, wˆ ) " (u , ! c ( wˆ )) u! (t ) Cu # F G#F1 S1#1C T (C wˆ # y ) where :

zˆ4 $ # zˆ6u #

(56c)

zˆ5 2 x5 zˆ4 2E 2 L) zˆ1 # # # 6F 2 0 ( zˆ1 # z1 ) L )L zˆ1 )L 1 2E

#

C) zˆ 2 / ( zˆ2 # z 2 ) 2E .

(56d)

(50)

(51)

#

(52)

zˆ5 $ # zˆ7 u "

(53)

(54a)

GF #1 $ diag[ I 252 F I 252 F 2 I 252 F 3 I 252 ]

(54b)

It is proved under mild assumptions that, for sufficiently large F , the estimation error wˆ (t ) # w(t ) converges exponentially to zero, whatever the initial condition wˆ (0) (see e.g. [6]). In the original coordinates, the observer (53) takes the following form: (55)

It is proved that the estimation error zˆ (t ) # z (t ) , just as

wˆ (t ) # w(t ) , converges exponentially to zero, whatever the initial condition zˆ (0) (see e.g. [6]). Using the notations (28)-(29), the observer (55) is now explicitly expressed in terms of the variables z i ( i $ 1, " , 7 ) :

C) zˆ 2 / ( zˆ 2 # z 2 ) Eu .

(56e)

(56f)

2 ( zˆ 2 # z 2 ) zˆ 3( zˆ1 # z1 ) /(56g) z!ˆ 7 $ zˆ 5 u " 6 # F 4 0 # 0 2 E ) L C zˆ 2 Aˆ u 2 E L2) C zˆ Aˆ u C 1 1 .

with

Aˆ $

S1#1 $ col[C41 I 2 C42 I 2 C 43 I 2 C44 I 2 ]

zˆ! $ F ( zˆ, u ) zˆ " G (u , zˆ ) # F @" ( zˆ, u )G#F1 S1#1C T C ( zˆ # z )

# 6F 2 C zˆ 2 ( zˆ 2 # z 2 )

zˆ 4 LC) zˆ2 " 4F 3 ( zˆ2 # z 2 ) C 2E 2 L) zˆ1 2 x zˆ zˆ 2E # 6F 3 00 ( zˆ1 # z1 ) zˆ6 $ zˆ 4u # 7 # 5 6 # L )L zˆ1 )L 1 2E u

with C ij

zˆ 22 zˆ12 2 zˆ 3 x 5 2 Ez 5 " # # L C )L zˆ1 )L

9 4

L)

2 2 zˆ1

"

1

(57)

() L C zˆ 2 ) 2

Remark. It is shown (see e.g. [6]) that the observer (53) (and so (55)) is globally exponentially convergent provided that C! D(u , w) , E ( w, u ) and (u, ! c ( w)) are globally Lipschitz Cu with respect to w uniformly in u and the control signal u and its time derivative u! are bounded. In the present work, the observation problem is dealt with in open loop. Accordingly, all signals are bounded and, consequently, the state vector w remains in a compact subset of IR 8 . Then, it can readily be checked that the above Lipschitz assumption is satisfied.

5.

SIMULATION RESULTS

The performances of the high gain observer designed in section IV, are now illustrated through simulation. The parameters of the serie resonnant converter are given the numerical values of Table 1.characteristics: Table 1 parameter Symbol value Load resistance R 4.7 Inductor L 0.9 5 10 -3 Capacitor C 130 5 10 #6 Capacitor Co 2.4 5 10 #3

zˆ 2 2 x 2 Ezˆ4 z!ˆ1 $ # 3 # 5 # # 4F ( zˆ1 # z1 ) L zˆ1 )L )Lzˆ1

(56a)

The DC voltage source is fixed to E $ 20V . The initial states of x and z are respectively:

zˆ zˆ!2 $ 3 # 4F ( zˆ2 # z 2 ) C zˆ 2

(56b)

x(0) $ 30.35 # 0.75 # 5 # 8 54 T

unit

J H F F

(58)

zˆ (0) $ 30.3 0.3 0.2 0 0 0.25 0.5 04 T

(59)

The control signal u is constant equal to 20 5 10 3 rd / s . Figure (2) shows the trajectory of all state estimates obtained with F $ 500 . The corresponding estimation errors are shown in Fig (3). It is seen that all error converges to zero after 3 ms. The convergence rate depends on F : the larger F the more rapid the estimate convergence. In practical applications, the choice of F is a compromise between convergence rapidity and noise sensitivity.

resolution of any differential or algebraic equation. A crucial step in the observer design was the preliminary model transformation of Subsection IV.A. This led to the new system representation (19)-(26) that assumes the canonical form for the high gain observer design. The observer gain is explicitly specified through the choice of a single constant parameter F . The proposed observer is readily utilizable in control applications, due to its global exponential convergence feature.

REFERENCES 2

3 x1 x1 obs

1

-2

-2 0

0.5

1

1.5 2 time [s]

2.5

3

-3

3.5

0

0.5

1

-3

x 10

10

1.5 2 time [s]

2.5

3

3.5 -3

x 10

10 x3 x3 obs

5

x4 x4 obs

5

0

-5

-5

0

0.5

1

1.5 2 time [s]

2.5

3

-10

3.5

0

0.5

1

-3

x 10

1.5 2 time [s]

2.5

3

3.5 -3

x 10

Fig 2. Comparison between observed and simulated transients in x1 , x2 , x3 and x4 with F $ 500 3

3 x1-x1 obs

2

[V]

[V]

-1

-2

-2 0

1

2

3 time [s]

4

5

-3

6

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0

-1

0

1

2

-3

x 10

10

3 time [s]

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x4-x4 obs 5 [A]

[A]

6 -3

10

0

-5

-10

5 x 10

x3-x3 obs 5

0

-5

0

1

2

3 time [s]

4

5

6

-10

0

1

-3

x 10

2

3 time [s]

4

5

6 -3

x 10

Fig 3: Tracking errors with ( F $ 500 ) 6.

[5] J. P. Gauthier and I. A. K. Kupka. ‘Observability and observers for nonlinear systems’. SIAM J. Control and Optimization, Vol. 32, No. 4, pp. 975-994, 1994 [6] M. Farza, M. M’saad and L. Rossignol . ”Observer

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0

[4] M.E. Elbuluk, G.C. Verghese and J.G. Kassakian. ‘Sampled-data modeling and digital control of resonant converters’. IEEE Trans. Power Electron, vol 3, pp 344354, 1988.

x2-x2 obs

2

1

-3

[2] S.R. Sanders, J.M. Noworolski, X. Z. liu, and G.C. Verghese. ‘Generalized averaging method for power conversion circuits’. IEEE Trans. Power Electronics, vol.6, pp 251-259, apr 1991 [3] V. Vorperian,”approximate small signal analysis of the series and parallel resonant converters” IEEE Trans. Power Electron., vol.4, pp 15-24, 1989.

0

[A]

[A]

0 -1

-3

-10

[1] A.F. Witulski and R. Erickson. ‘Small signal equivalent circuit modeling of a series resonant converter’. Proc. IEEE PESC, pp 693-704, 1987.

1 [V]

[V]

0 -1

-4

x2 x2 obs

2

CONCLUSION

We have designed a high gain observer for SRC DC-to-DC converters. The observer design does not involve the