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Nonlinear Adaptive Output Feedback Control of Series Resonant DC-DC Converters
Ouadia Elmaguiri, Fouad Giri, Luc Dugard, H. El Fadil, Fatima Zara Chaoui
To cite this version:
Ouadia Elmaguiri, Fouad Giri, Luc Dugard, H. El Fadil, Fatima Zara Chaoui. Nonlinear Adaptive
Output Feedback Control of Series Resonant DC-DC Converters. ACC 2010 - American Control
Conference, Jun 2010, Baltimore, Maryland, United States. pp.n.c. �hal-00550226�
Nonlinear adaptive output feedback control of series resonant dc-dc converters
O. Elmaguiri, F.Giri*, L. Dugard **, H. El Fadil, F.Z. Chaoui GREYC lab, University of Caen, France
* Corresponding author: [email protected] ** GIPSA, ENSIEG, INPG, Grenoble, France
Abstract. The problem of regulating the output voltage of DC-to-DC series resonant converters (SRC) is addressed. The difficulty is threefold: (i) the converter model involves discontinuous and highly nonlinear terms and is, further, controlled through a modulating frequency signal; (ii) all state variables are not accessible to measurements; (iii) the load is uncertain and may even be varying. An output feedback controller, not necessitating the measurement of the converter state variables, is proposed and shown to ensure semi-global stabilization of the closed-loop system and perfect output reference asymptotic tracking. The controller is developed using the backstepping control approach and the high-gain observer design technique.
I. INTRODUCTION
Series 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.
As they do not involve switched components, power losses are considerably reduced improving thus the conversion efficiency. 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 in general the only available control variable. These considerations make SRC modeling a particularly hard task.
A modeling approach, based on generalized averaging, was developed in [1]. Small signal models for series and parallel resonant converters were developed in [2].
In the present work, following the first harmonic approach [1], a fifth order state-space model is developed for the converter of fig (1). From the control design viewpoint, the difficulty lies in: (i) the system nonlinear and discontinuous nature; (ii) the fact that the control signal (switching frequency) comes in all state variable equations. (iii) the vector state is not completely measurable and it should be estimated. Different control strategies were proposed for the considered class of converters. These include hybrid flatness based control [3], resonant tanks variables based optimal control [4], sliding mode control [5] and passivity based control [6]. In the present work, a new control strategy is developed to cope with the problem of output voltage regulation in SRC converters without assuming the state variables to be measurable and the load to be known.
Following [7], a high gain observer is first designed to get estimates of the state variables that are not accessible to measurements. Then, an adaptive output control law is
designed, using the tuning functions backstepping technique [8], based on the above state observer. It is worth recalling that, unlike linear systems, the separation principle does not systematically apply to nonlinear systems [9]. Furthermore, a parameter projection will be introduced in the parameter adaptive law (estimating online the load) to prevent possible parameter estimate drift that, otherwise, could result due to the presence of state estimation errors. The output adaptive controller thus obtained is formally shown to achieve quite interesting performances. Specifically, the closed-loop system is asymptotically stable and the attraction region size can be made arbitrarily large by conveniently choosing the control design parameters. The output reference tracking error vanishes asymptotically. The unknown load is perfectly identified.
The paper is organized as follows: the studied series resonant converter is described and modeled in Section II;
the state observer is presented in Section III; the adaptive output feedback controller is designed in Section IV and the resulting closed-loop system is analyzed in section V; the controller performances are illustrated by simulation in Section VI; technical proofs are placed in the appendix.
II. SERIES RESONANT CONVERTER MODELLING
The studied series resonant DC-to-DC converter is illustrated by Fig 1. A state-space representation of the system is the following:
)) sgn(sin(
)
sgn(i E t
vo v dt di
L (1)
i dt
C dv (2)
R i v abs dt
C0 dvo () o (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 (in rd/s ); L and C designate respectively the inductance and capacitance of the resonant tank;
As the supply source amplitude E is constant, the pulsation
turns out to be the only possible control variable.
Fig 1. Series resonant converter under study
A control oriented model can be obtained applying to (1)-(3) the first harmonic approximation procedure introduced in [1]. Based upon the following assumption,
A1. The voltage v and current i are approximated with good accuracy by their (time varying) first harmonics (denoted V1 and I1ej respectively).
Doing so, one gets the following more convenient model:
(see [1], [7] for details):
2 ] [ 2
1
0 1 1
1
V V e j E
L I j dt
dI j
(4)
1 1
1 1
I C V j dt
dV (5)
o o o
RC I V abs C dt
dV 4 ( )
1
0 (6)
In the „harmonic‟ model (4)-(6) the control signal comes in linearly. However, it still not suitable for control/observer designs because it involves complex variables and parameters. To get a convenient state-space model, introduce the following notations:
I1x1 jx2, V1x3 jx4, Vo x5 (7) Substituting (7) in (4)-(6) yields the following state-space representation:
2 2 2 1 1 5 3 2 1
x x
x L x 2 L u x x x
(8)
L E 2 x x
x L x 2 L u x x x
2 2 2 1 5 2 4 1
2
(9)
C u x x
x3 4 1 (10)
C u x x
x4 3 2 (11)
o Co
x x x C
x5 4 12 22 5
(12)
where udef and 1/R . The only quantities that are accessible to measurements are:
o
5 V
x , x12 x22 I1, x32x42 V1 (13) III HIGH GAIN OBSERVER
In [7] a high-gain observer has been designed to get accurate estimates of unmeasured variables and shown to be exponentially convergent if all system signals are bounded.
This is defined using the following variable change:
z x IR
IR
: 5 8; (14a)
2 2 2 1
1 x x
z ; z2 x32x42 ; z3 x1x3x2x4 (14b)
2
4 x
z ; z5 x4; z6 x1; z7 x3; z8 x5 (14c) The equation describing the evolution of the new state variables, zi (i1,,7) is omitted for space limitation; it can be found in [7] where the following high gain observer was proposed:
ˆ ) ( ˆ 4 2 ˆ 2 ˆ
ˆ ˆ 1 1
1 4 5
1 2 3
1 z z
z L
z E L x z L
z z
(15a)
ˆ ) ( ˆ 4
ˆ ˆ 2 2
2 3
2 z z
z C
z z (15b)
ˆ ) ˆ ( 2 6
ˆ 2ˆ ˆ
ˆ ˆ 5 2 2 2 2
1 5 3 2 1 2 2
3 Cz z z
L Ez z L
x z C z L
z z
(15c)
(ˆ )
2 6 ˆ 2 ˆ ˆ ˆ 2 ˆ
ˆ 2 1 1 1
1 4 5 5 6
4 z z
E z L L
E z L
z x L u z z
z
(ˆ )
2 ˆ
2 2
2 z z
E z C
(15d) ˆ )
( 2 4 ˆ ˆ ˆ
ˆ5 7 4 3 2 z2 z2
E z LC C
u z z
z
(15e)
(ˆ )
2 6 ˆ 2 ˆ ˆ ˆ 2 ˆ
ˆ 3 1 1 1
1 6 5 7 4
6 z z
u E
z L L
E z L
z x L u z z
z
ˆ (ˆ )
2 2
2 z z
Eu z C
(15f)
u z C L E
z z u
z C L E
z z C
u z z
z
ˆ 2 ˆ
ˆ ) ( 3 ˆ 2 ˆ
ˆ ) ˆ (
ˆ ˆ
1 2
1 1 2
2 4 2
6 5 7
(15g)
2 2 2
1 2
4 ( ˆ )
1 ˆ
ˆ 9
z C L z
L
(15h)
where denotes a design parameter.
The convergence of the above observer has been analyzed in [7] using the following Lyapunov function:
z S z z
Vob(~)~T T 1~ (16)
with z z
z ˆ
~ (17a)
] [ 2 2 2 2 2 2 2 3 22
1
diag I I I I (17b)
where I22 denotes the 22 identity matrix and S1 is a symmetric positive definite matrix that is the unique solution of the Lyapunov equation:
0 C C A S S A
S1 T 1 1 T (18a)
with A and C defined as follows:
i v
vo
t E
0 0 0 0
I 0 0 0
0 I 0 0
0 0 I 0 A
2 2 2
, C
I22022022022
(18b)Theorem 1 ([7]). Consider the system (8)-(12), subject to Assumptions A1-A2, the state variable change (14a-c) and the state observer (15a-h). Suppose all the system and observer state variables to be bounded so that all involved nonlinearities can be supposed to be Lipschitz. Then, the time-derivative of Vob(~z) along the trajectory of z~ satisfies the inequality
ob
ob l V
V ( ) (19)
for some real constant l0 , depending on the Lipschitz coefficients of the different nonlinearities. Consequently, the state estimation error z~zzˆ converges exponentially to zero, whatever the initial condition zˆ(0) , provided the observer gain is sufficiently large
IV. ADAPTIVE CONTROL DESIGN
The load resistance R in model (1-3) is allowed to undergo infrequent jumps. To cope with such a parameter uncertainty the adaptive controller to be designed should involve an on line estimation of the unknown parameter 1/R . The unknown parameter estimate and the corresponding estimation error are denoted ˆ and ~ ˆ , respectively.
Following closely (Kristic et al., 1995) the adaptive controller is designed in three major steps.
Design Step 1. Introduce the tracking error:
x ref
x
e1 5 5 (20)
where x5ref denotes the desired constant output reference.
Achieving the tracking objective amounts to enforcing the error e1 to vanish. To this end, the e1-dynamics need to be clearly defined. Deriving (20) one obtains:
o Co z x C
e1 4 1 5
(21)
The quantity 4 z1 Co
stands as a virtual control input in (21). Consider the following Lyapunov equation:
2 2
1 1
1
~ 2
1 2 ) 1 ,~
(
e e
Vc (22)
where 0 is a design parameter, called adaptation gain.
Time-derivation of Vc1, along the ~) ,
(e1 -trajectory, is:
ˆ ) (
~ ˆ)
( 4 1 1 1 1
1
1 z w we
C e V
o
c
(23)
where w1 denotes the first regressor function defined by:
Co
w1 x5 (24)
one can eliminate ~from V1 using the law ˆ1 with:
1 1
1w e
(25)
Furthermore, e1can be regulated to zero if 4 1 1
Co z where the stabilizing function 1 is defined by:
1 c1e1w1ˆ (26)
where c10 is a design parameter. Since 4z1/Cois not the actual control input, we can only seek the convergence of the error (4z1/Co)1 to zero. Also, we do not take
1
ˆ
as parameter update law. Nevertheless, we retain
1as the first tuning function and tolerate the presence of
~inV1. Introduce the second error variable:
1 1 2
4
z
C e
o
(27) Then, equation (21) becomes using (26) and (27):
~
1 2 1 1
1 ce e w
e (28)
Also, (26) can be rewritten as follows:
) ˆ
~(
1 2 1 2 1 1
1
ce ee
Vc (29)
Design Step 2. The objective now is to make the error variables (e1,e2) vanish asymptotically. To this end, the dynamics of e2 are first determined. Deriving (27) one obtains, using (14a-c), (24) (26) and (28):
4 ˆ 8 ˆ
4
8 5
2 1 5
1 3
1 2
4 2
o o o o
o o C
z x C LC C
x z LC
z z
LC z
e E
( ) ˆ ~ ˆ
2 5 1 1 1
2 1 1 1
Co
w x c w
e e c
c
(30) As the states zi (i3,4) are not available they are replaced in (30) by their estimates provided by (15c-d). Doing so, one gets:
4 ˆ 8 ˆ
4ˆ
8 ˆ 5
2 1 5
1 3
1 2
4 2
o o o o
o o C
z x C LC C
x z LC
z z
LC z e E
( ) ˆ ~ ˆ
2 5 1 1 1
2 1 1 1
Co
w x c w
e e c
c
1 3
1 4 2
4 ~ 8 ~
z z LC z z LC
E
o o
(31)
where ~z3 and ~z4 are the estimation errors of z3 andz4 . Introduce the new error:
2 1 2
4 3
8 ˆ
z LC
z e E
o
(32) Then (31) is rewritten as follows:
~ )
~ , , ˆ (
~
4 3 1 1 1 2 2 2 3
2 e w w z z z
e
(33)
where the second regression function is defined by:
ˆ
2 5 1 1 2
Co
w x c
w (34)
and
) 8 (
4 ˆ
2 1 1 2 1
5 3 1
2 c ce e
LC z x
z
LCo o
4 ˆ)
( ˆ
5
1
o o
o C
x z C C
(35)
1 3 1
4 4 2
3 1 1
4 ~ 8 ~
~ )
~ , , (
z z LC z z LC z E
z z
o o
(36)
Notice that the disturbing term 1(z1,~z3,~z4) vanishes exponentially fast whenever ~z3,~z4 do so. Consider the augmented Lyapunov function:
2 2 1
1 2
1
2 2
) 1 ,~ (
~) , ,
(e e V e e
Vc c (37)
Its derivative along the solution of (20) and (27) is:
ˆ)
( 1 3 2 2 1
2 2 1 1
2
ce e e e w
Vc
1 2 2
2
1 )
ˆ
~(
e w e
(38)
~ can be cancelled in V2 using the update law ˆ2 :
2 1 2 1 2 2 1
2 e
e w w
e
(39)
If 8E zˆ4/(2LCoz1)were the actual control in (31) and the term 1were null, one would get Vc2 c1e12c2e22 by using the above parameter update law and letting:
2 1 2 2 2 1
2
e c e w (40)
As 8E zˆ4/(2LCoz1) is just a virtual control, the above parameter update law is not sufficient. Nevertheless, we retain 2 as a second tuning function. Then, (38) gives:
3 2 2
2 1 2 2 2 2 2 1 1
2 )
ˆ
~( ˆ)
( e e
w e e c e c
Vc
e21(z1,~z3,~z4) (41) Design Step 3. Deriving (32) gives:
2 .
1 4
3 2 ˆ )
8 (
z z LC e E
o
(42) On the other hand, one obtains from (14b) and (15d):
3 1 2 4 3 4
1 4 3 2 1 1 1 6 .
1
4 ˆ ( , ,ˆ ) ~ ˆ ~ 2 ˆ
ˆ
z z z E Lz z z z z z u z z z
z (43)
with
4
1 5
1 3
1 4
1 4 5 5
1 ˆ 2 ˆ ˆ 2 2 ˆ
2 ˆ
z Lz
E L x Lz
z z z L E Lz
z x L z
(ˆ )
2 ) ˆ ˆ ( 2
6 2 ˆ1 1 1 2 z2 z2
E z z C z E
z
L
(44)
Furthermore, it is readily seen from (40) that:
)]
(
[ 1 2 1 1 1 1 1 2 2 2 2
2 w w we we w e w e
e12c2e2 (45) Using (24), (34), (35) and (42), the derivatives on the right side of (45) can be given the following more suitable:
~
1 10
1 e w
e (46)
~ (.) ˆ
1 2 1 20
2e w w
e
(47)
~
1 50 1
o
o C
w C
w x (48)
Co
a x a z z
a
ˆ
~ )
( 10 10 1 5 2
0
2
1 3 2 1 1 1
4ˆ ) (
z LC e z e c c
o
(49)
To alleviate the text, the exact expressions of the newly introduced quantities (i.e. e10 , e20, x50 , z10, ~z10, a0,a1 and a2) are placed in the Appendix A. Substituting (43) and (45) in (42), one obtains:
~) , , ˆ (
) ~ ,ˆ ( ˆ
8
3 ..
1 : 2 3 3 2
1 2
6
3 u z z w g e z z
z LC
z E
e i
o
(50)
where w3 denotes the last regressor function defined by:
2 2 1 2 1 2 1 2 1
3 (1 c w ) (c c w w )w
w
2 1 1 1 2 1 ) 1) 1
ˆ (
( C a w
C c e w e
w o
o
(51)
and
10 2 1 2 1 2 1
2 8 (1 )
e w c LC
E
o
3 1 2 1 2 20
1 10
0 4 ˆ ( )
e w w c c z z LC z
a
o
) )
ˆ (
( 2 1 1 1 2 1 1
50 o
o o
C a C c e w e w x
C
(52)
) (
)
( 2
2 1 2 2 1 2 1 1 3
o
o C
e w C w a w c c w
g (53)
1 2 1 2 3 1
1 2 4 3 4
1 4 2 3
2 8 (~ ˆ ~ 2 ˆ ) ( )
c c ww
Lz z z E Lz z z LC
E
o
a0~z10 (54) Note that the actual control input u appeared for the first time in (50). Notice also that the term in 2 vanishes exponentially fast whenever the ~3,~4
z
z do so. Now, the goal is to find a control law u and adaptive law for ˆ so that the
~) , , ,
(e1 e2 e3 system is asymptotically stable. To this end, consider the augmented Lyapunov function candidate:
2
~) , , (
~) , , , (
2 3 2 1 2 3
2 1 3
e e e V e
e e
Vc c
2
~ 2
3 2
1 2 1
i
e (55)
Using (41) and (50), the derivative of Vc3 turns out to be:
