High Gain Observer for Series-Parallel Resonant Converter
OUADIA EL MAGUIRI1*, ABDELMOUNIME EL MAGRI2, FARCHI ABDELMAJID1
1IMMII-lab, University Hassan 1st, Faculty of Science and Technology, Settat 2SSDIA-lab, University of Hassan II, ENSET-M, Casablanca,
MOROCCO
*Corresponding author: Ouadia_elmaguiri@yahoo.fr
Abstract: - We are considering the problem of state observation in series-parallel resonant converters (LCC).
This is a crucial issue in (LCC) output voltage control as the control model is nonlinear and involves nonphysical state variables, namely real and imaginary parts of complex electrical variables. An interconnected high gain observer is designed to get online estimates of these states. The observer is shown to be globally asymptotically stable. The global stability of the observer is analytically treated using the lyapunov theory;
finally we present numerical simulation to illustrate the performance of the suggested approach.
Key-Words: - Resonant converter, Averaging approximation, state variables observation, lyapunov theory, interconnected high gain observer.
1 Introduction
In recent years, the conventional pulse-width modulated (PWM) converters are well studied and are still widely used such as power systems for computers, telephone equipment, and battery chargers. However, as mentioned in [1-3], the inefficient operation of PWM converters at very high frequencies imposes a limit on the size of reactive components of the converter and consequently on power density. In fact, the turn-on and turn-off losses caused by PWM rectangular voltage and current waveforms limit the operating frequency and produces hard switching. This yields in power losses in electrical switches and increases the potential for electromagnetic interference (EMI).
To eliminate or at least mitigate the adverse effects of hard switching, most existing works have studied these converters along with circuitry arrangements to modify the current and voltage at the switches during the commutations [4-6].
Since the emergence of the resonant technology, major research efforts have been conducted to apply the enhanced features of resonant converters to practical applications. Robotics, electrostatic precipitators, X-ray power supply are a few examples in witch resonant converters are used today. These applications are treated respectively in [7], [8],[9] and [10].
The main advantages of resonant converters are well known as lower switching losses improving thus the conversion efficiency, lower electromagnetic interference; the size and weight are greatly reduced due to high operating frequency. In fact, resonant
converters can run in either the zero-current- switching (ZCS) or zero-voltage-switching (ZVS) mode. That means that turn-on or turn-off transitions of semiconductor devices can occur at zero crossings of tank voltage or current waveforms, thus reducing or eliminating some of the switching loss mechanisms. Since the losses are proportional to switching frequency, resonant converters can operate at higher switching frequencies than comparable PWM converters.
As mentioned in [11], Series and series- parallel resonant converters are variable structure systems.
In fact, they are linear piecewise systems whose global behavior is strongly nonlinear. This nonlinear aspect is one of the reasons of the difficulties encountered when computing control laws, in [12], this problem is avoided by linearizing the system around a steady point and then applying the linear control techniques. An efficient method for the analysis of the series-parallel resonant converter has been proposed by [13]. There the output rectifier, filter capacitor and load are substituted by an RC load model. Unfortunately, these approaches gives only mediocre performances, to obtain a robust and high quality performances, we have to consider the nonlinearity of the circuits structure when modeling the converter, such step implies the obtention of an efficient mathematical model witch describes the dynamical behavior of the converter accurately and later, the definition of a suitable control law.
The point is that these models generally involve state variables that all are not accessible to measurement. Then, they cannot be used in control unless they allow the construction of observers.
State observation has yet to be solved for resonant converters (or was seldom been dealt with). In the present work, it is shown that such an issue can be solved for the (LCC) resonant DC/DC converter.
The focus is made on the circuit of Fig 1.
Based on an extension of the first harmonic analysis proposed in [14],[15] and [16], a fifth order large signal nonlinear model that can describe the transient behavior of the converter and is useful in the development of nonlinear controllers is designed for the considered circuit. There, the output power is controlled by duty-cycle variation. However, most of involved state-variables turn out to be non- accessible to measurement. Therefore, an interconnected high gain observer is developed and shown, under mild assumptions, to be globally exponentially convergent. The global exponential convergence feature makes the proposed observer readily utilizable in the converter control. The first step in the observer design is the construction of a state diffeomorphism map leading to a transformed model that fits the required form. Using this special form, an interconnected high gain observer can be designed in a rather straight way under some global Lipschitz assumptions on the controlled part [17- 19]. The gain of the proposed observer is issued from a differential Lyapunov equation.
The paper is organized as follows: mathematical modeling of the series resonant converter is addressed in Section II; theoretical design of the state observer is coped with in Section III; a global stabilities analysis of established observer is treated in section IV. The performances are illustrated by simulation in Section V; a conclusion and reference list end the paper.
2 Modeling series-parallel resonant converters
Resonant converters contain resonant L-C networks whose voltage and current waveforms vary sinusoidally during one or more subintervals of each switching period. The resonant network has the effect of filtering higher harmonic voltages such that a nearly sinusoidal current appears at the input of the resonant network [9]. Depending on how the resonant networks are combined with other circuit configurations, one can obtain several types of resonant converters. The studied series - parallel resonant DC-to-DC converter is illustrated by Fig 1.
Fig 1. Series parallel resonant converter under study
A state-space representation of the system is the following:
( ) ( ) ( )
s
AB cs cp
Ldi v t v t v t
dt
(1)
cs ( )
s s
C dv i t dt
(2) ( ) 2
o o
o T
dv v
C abs i
dt R (3)
where vcs and is denote the resonant tank voltage and current respectively; vo is the output voltage supplying the load (here a resistor R), L and Cs designate respectively the inductance and capacitance of the series resonant tank. The parallel resonant capacitor is designed Cp. In order to simplify the analysis it will be assumed that: all the components are ideal and have no losses and that the voltage E is constant and has no ripple The (SPRC) converter modeling is based upon the following assumptions:
Assumption 1: The voltage v and current cs is are approximated with good accuracy by their (time varying) first harmonics
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.
If one observes the waveforms of Fig. 2. one can see that the resonant current isis almost sinusoidal.
However, waveformsvAB,iT and vcp do not have sinusoidal shape. That means their spectrum have high-order harmonics. Since the active power transferred to the load is dependent on the voltage vAB and the resonant currentis. As the resonant current is (almost) sinusoidal, i.e., only has the first harmonic component in its spectrum, the high-order
A is
vcs
Cs
Cp vcp
io
vo
Co
Co
iT
E
harmonics of vAB will be multiplied by zero when one calculates the instantaneous power. Thus, the high-order harmonics of vAB do not contribute to the power transfer to the load. For this reason, the design procedure based on the first harmonic analysis produces good results.
-
VAB
is
iT
vcp
t
t
t
t
E
E max is
max is
2 Vo
2 Vo
d
Fig 2 Time behavior of the characteristicvoltages and currents
A control/observation oriented model can be obtained applying to (1-3) the first harmonic approximation procedure. This is developed in the next section.
3. 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 is approximated by the Fourier series expansion of the function , defined in the interval . Mathematically, one has the following standard expressions:
(4a)
(4b)
with . The coefficients undergo the following equation:
(5)
In the case is generated by a controlled nonlinear system where denotes the control signal, it follows from (5) that:
(6) Applying (6) with to equations (1) to (3), we obtain the following „first harmonic‟ equations :
(7)
(8)
(9) Given the waveforms and shown in fig 2, the fundamental term is calculated as:
(10) The next step is to find the Fourier representation of the term , which represents the average current of the output rectifier. As shown in Fig.2, the current is equal to the resonant current when the voltage across the parallel capacitor is clamped to and is equal to zero otherwise. one gets
(11) where is the peak value of the resonant current
and is the rectifier conduction angle.
The equation of the rectifier conduction angle can be written as a function of state variables, inputs and circuit parameters as follows:
(12)
From the fact that and ,
equation (11) and (12) can be rewritten as
(13) and
) (t x
) (
) ,
ˆ(t s xt T s x
def s[ T0, ]
( )
( , ) ( ) k ( ) jk s t T s
k
x t s x t T s x t e
( )
0
( ) 1 ( ) s
T jk t T s
xk t x t T s e ds
T
T / 2
xk
( ) ( )
k
s k
k
d x d
x t jk x t
dt dt
) (t x
) , ( ux f
x u
( , ) ( ) ( )
k
s k
k
d x f x u t jk x t
dt
1 k
1 1
1 1 1
1
s s s cs cp AB
s
d i j i v v v
dt L
1 1 1
1
cs s cs s
s
d v j v i
dt C
0 0
0
1 2
( ) o
o T
o
d v
v abs i
dt C R
vAB is
AB 1
v
1 sin cos( ) 1
AB
v E d j d
( )T 0
abs i
iT is
o/ 2 v
0
max max
0
( ) 1 sin( ) sin( )
T 2 s s
abs i i d i d
max 1 cos
is
max
is
is
1 max
cos s p o 1
s
C V i
max 2 1
s s
i i vo 0 Vo
1 0
2
( )T is 1 cos
abs i
(14) The new state variables are: , and . The state variables and are complex Fourier coefficients that can be rewritten with real variables by separating the real and imaginary parts of the equations (7) and (8). Thus, the state variables can be written as:
, , (15)
The voltage across the parallel capacitor can be also written as a function of real variables:
. and are expressed as a function of the existing state variables and by:
(16) (17) where
(18) (19) Substituting (15) in (7)-(9) yields the following state-space representation:
(20) (21) (22) (23)
(24) where
, (25)
; , (26)
and
(27)
where , In the above model, the only quantities that are accessible to measurements are:
, , (28)
That is, the variables must be estimated using some measurable quantities. To this end, an observer is built up in the next section
4 high gain observer synthesis
There is no systematic method to design an observer for a given nonlinear control system but several designs are available for nonlinear systems with specific structures. This is particularly the case for nonlinear systems that can be seen as the interconnection of several subsystems, where each subsystem satisfies specific conditions. The idea is to first design an observer for each subsystem supposing known the state of the others. Then, a global observer is developed for the whole nonlinear system combining the observers obtained separately.
4.1. Model transformation
We propose the change of coordinates defined by:
(29)
with
(30)
Using (30), it follows from (20)-(24) that the new state vector undergoes the following equations:
(31) (32)
(33)
(34)
1 0
1
cos 1
2
s p o
s
C v i
s 1
i vcs 1 v0 0
s 1
i vcs 1
1 2
s 1
i x j x vcs 1x3j x4 Vo x7
5 6
cp 1
v x j x x5 x6
x1 x2
5 1 2
1
s p
x x x
C
6 2 1
1
s p
x x x
C
1
sin 2
2
sin2
3 5
1
2 sin
s
x x
dx E
x u
dt L L L
2 4 6
1 cos( ) 1
s
dx x x E
x u
dt L L L
3 1
4 s
s
dx x
dt x C
4 2
3 s
s
dx x
dt x C
2 2
1 2
7 2 2 7
1 cos
o o
x x
dx x
dt C RC
5 1 2
1
s p
x x x
C
6
2 1
1
s p
x x x
C
1sin 2
2 sin2
1 7
2 2
1 2
cos 1
2
sC xp
x x
def
u d
1 7
y x y2 x12x22 y3 x32x42
4 3 2 1,x ,x ,x x
6
:IR5 IR
1 1
2 2
3 3
4 4
5 7
6
( ) x z
x z
z
x x z x
x z x z
z
1 7
2 1
3 2
2 2
4 1 2
5 3
6 4
z x z x z x
z x x
z x z x
z
1 1 4
2 2
1 cos
o o
z z z
RC C
5 2 3
2 3
( )
s sin
s p
z z z E
z z u
L L C L
3 2
6
3 s 2 cos 1
s p
z z
z E
z z u
L L C L
2 5 3 6
4 4
4
( )
s p
z z z z
z z
L z L C
2 3
4
sin( ) cos( ) 1
E u z u z
Lz
(35) (36) The above model can be rewritten in the form of two interconnected subsystem:
(∑1) (37)
(∑2) (38)
The above subsystems are given the following compact forms:
(39) and
(40) where:
, (41)
(42)
(43)
(44)
(45)
(46)
(47)
4.2. Observer design
In this Section, an interconnected observer will be designed for the interconnected system (39)-(40).
Such design is performed under the following assumption:
Assumption 1. The signals and are bounded and regularly persistent to guarantee the observability of the subsystems (39) and (40), see e.g. (Besançon and Hammouri, 1998).
Under the above assumption, we propose the following observer candidate for the interconnected systems (39)-(40), see e.g.(Besançon and Hammouri, 1998):
(48)
(49) where and are a symmetric positive definite matrix that are solution of the Lyapunov equations:
(50) (51)
2
5 s 6
s
z z z
C
3
6 s 5
s
z z z
C
1 1 4
5 2 3
2 3
3 2
6
3 2
2 2
1 cos
( )
sin
cos 1
o O
s
s p
s
s p
z z z
RC C
z z z E
z z u
L L C L
z z
z E
z z u
L L C L
2 5 3 6
4 4
4
2 3
4 2
5 6
3
6 5
( )
sin( ) cos( ) 1
s p
s s
s s
z z z z
z z
L z L C
E u z u z
Lz z z z
C z z z
C
1 1 1 1 2
1 1 1
( ) ( , , )
Z A y Z g u y Z y C Z
2 2 1 2 2 1
2 2 2
( , ) ( , , )
Z A y Z Z g u y Z y C Z
3
3 2 1
1 IR
z z z
Z
3
6 5 4
2 IR
z z z
Z
y y
T def
z z
Ty 1 2 1 4
1
2 0 0
( ) 0
0
o
s
s p s p
s
s p s p
RC
A y L C L C
L C L C
3 2
4 4
2( , 1) 0 0
0 0
s p
s s
z z
L C L z L z
A y Z
1 0 0
2 1C C
4
5
1 2
6
2 1 cos
( , , ) sin
cos 1
O
C z
z E
g u y Z u
L L
z E
L L u
2 3
4
2
2 1
3
sin( ) cos( ) 1
( , , )
s
s
E u z u z
Lz g u y Z z
C z C
, 1
,y Z
u Z2
1
1 1 1 1 2 1 1 1 1
1 1
ˆ ( ) ˆ ( , ,ˆ ) ( ˆ )
ˆ ˆ
Z A y Z g u y Z S CT y y y C Z
1
2 2 1 2 2 1 2 2 2
2 2
ˆ ( ,ˆ ˆ) ( , ,ˆ ) ( ˆ )
ˆ ˆ
Z A y Z Z g u y Z S CT y y y C Z
S1 S2
1 1 1 1T( ) 1 1 1( ) 1T 1
S S A y S S A y C C
2 2 2 2T( ,ˆ1) 2 2 2( ,ˆ1) 2T 2
S S A y Z S S A y Z C C
where and are arbitrary positive real design parameters. The other notations are defined as follows:
, (52)
(53)
(54)
(55)
The convergences of the interconnected observer defined by (48)-(49) are investigated in the next section.
5 Global stability analysis
Let us introduce the estimation errors:
(56) (57) It readily follows from (39), (40), (48) and (49) that the above errors undergo the following differential equations:
(58)
(59) Rearranging terms on the right sides of (58) and (59) one easily gets:
(60)
(61) Now let consider the Lyapunov function candidate:
(62) with
, (63)
From (62) and (63), it is clear that the time-derivative of is:
(64) Using (60)-(61), one obtains from (64) that:
(65)
The right side of (65) can be upper bounded as follows:
(66)
Now, from the fact that and are globally lipschitz with respect to uniformly with
respect to . And is globally
lipschitz with respect to uniformly with respect to we gets
(67) (68) (69) where and are the Lipschitz constants of the functions respectively;
On other hand we have the following Inequalities (70) (71)
(72)
(73) where and denote the largest eigenvalue of the positive definite matrices and , respectively;
and are the upper bound of the state vectors and , respectively (i.e. Assumption 1).
Using (67)-(73), we get from (66) that:
(74)
with
(75) As and are bounded and positive definite, one has
(76) (77)
1 2
z z z
TZˆ1 ˆ1 ˆ2 ˆ3 Zˆ2
zˆ4 zˆ5 zˆ6
T3 2
4 4
2 1
ˆ ˆ
( ,ˆ ) 0 0
0 0
s p
s s
z z
L C L z L z
A y Z
4
5
1 2
6
2 1 cos ˆ ˆ
( , , ) sin
ˆ cos 1
O
C z
z E
g u y Z u
L L
z E
L L u
2 3
4
2
2 1
3
ˆ ˆ
sin( ) cos( ) 1 ˆ ˆ
( , , )
ˆ
s
s
E u z u z
Lz g u y Z z
C z C
1 1 ˆ1
e Z Z
2 2
2 Z Zˆ
e
1 1 1( ) 1 1( , , 2) 1 1T 1 1
e A y Z g u y Z S C C e
1( )ˆ1 1( , , ˆ2) A y Z g u y Z
1 2 2( , 1) 2 2( , , 1) 2 2T 2 2
e A y Z Z g u y Z S C C e
2( , ˆ ˆ1) 2 2( , ,ˆ1) A y Z Z g u y Z
1
1 [ 1( ) 1 1T 1]1 1( , , 2) 1( , ,ˆ2) e A y S C C e g u y Z g u y Z
1
2 [ 2( , 1) 2 2T 2] 2 2( , , 1) e A y Z S C C e g u y Z
2( , ,ˆ1) [ 2( , 1) 2( ,ˆ1)]ˆ2
g u y Z A y Z A y Z Z
2
1 V
V V
1 1 1 1
1(e ) e S e
V T V2(e2)e2TS2e2
V
1 1 1 1 1 1 2 2 2 2 2 2
2 T T 2 T T
V e S e e S e e S e e S e
1T( 1 1 1T 1)1 2T( 2 2 2T 2)2
Ve S C C e e S C C e
2 2 2 1 2 ˆ1 2
2e ST A y Z( , ) A y Z( , )Z
1 1 1 2 1 ˆ2
2e ST g u y Z( , , ) g u y Z( , , )
2 2 2 1 2 ˆ1
2e ST g u y Z( , , ) g u y Z( , , )
1 1 1 1 2 2 2 2
T T
V e S e e S e
2 2 2 1 2 ˆ1 2
2e S A y Z( , ) A y Z( , ) Z
1 1 1 2 1 ˆ2
2e S g u y Z( , , ) g u y Z( , , )
2 2 2 1 2 ˆ1
2 e S g u y Z( , , ) g u y Z( , , )
2( , 1)
A y z g u y Z2( , , 1) Z1
( , )u y g u y Z1( , , 2) Z2
( , )u y
2( , 1) 2( ,ˆ1) 1 1
A y Z A y Z k e
2( , , 1) 2( , ,ˆ1) 2 1
g u y Z g u y Z k e
1( , , 2) 1( , ,ˆ2) 3 2
g u y Z g u y Z k e
3, 5
k k k7
1, 2, 2, g A g
1 4
S k
2 5
S k
1 6
Z k
2 7
Z k k4 k5
S1 S2 k6 k7
Z1 Z2
1 1 1 1 2 2 2 2
T T
V e S e e S e
1 1 2 2 1 2 3 1 2
2 e e 2 e e 2 e e
1 k k k5 1 7, 2 k k1 3, 3 k k2 4
S1 S2
2 1 1 max 1 1 1 2 1 1
min(S)e eTSe (S)e
2 2 2 max 2 2 2 2 2 2
min(S )e eTSe (S )e
where and are respectively the minimum and the maximum eigenvalue of ,
. In view of (76)-(77), inequality (74) implies
(78)
with
(79) (80)
, (81)
Applying the well known inequality
with and , one gets
. This, together with (78) yields:
(82) Up to now, the design parameters and are arbitrary. Let them be chosen such that:
(83) (84) Then, (82) gives:
(85) with:
(86) It is readily seen from (85) that is negative definite, implying the global asymptotic stability of the error system (58)-(59). The result thus established is summarized in the following theorem.
Theorem 1 (main result). Consider the error system described by (58)-(59) obtained applying the interconnected observer (48) to (51) to the system (39)-(40) subject to Assumptions 1. If the observer parameters and are chosen as in (83)-(84), then the error system is globally exponentially stable. Consequently, the estimates will converge exponentially fast to their true values , whatever the initial conditions . Remarks 1.
1) From (30) it is readily seen, that the estimates of the are given by:
Then Theorem 1 implies that these estimates converge exponentially to their true values.
2) The observer (48)-(51) is a high gain type, inequality (85), together with (86), shows that the estimates convergence speed depends on the design parameters and : the larger these parameters, the more speedy the convergence. On the other hand, excessive values of and make the
observer too sensitive to output noise inherent to practical situations. Therefore, the choice of the observer parameters must be a compromise between the rapidity of estimates convergence and sensitivity of estimates to output noise.
6 Simulation results
In order to illustrate the performance of the proposed observer, digital simulations using MATLAB/SIMULINK are performed. the LCC resonant converter is given the following characteristics:
Table 1: numerical values of the LCC characteristics
parameter Symbol value unit
Inductor L 16x10-6 H
Capacitor Capacitor
Cs Cp
48x10-6 15x10-9
F F Capacitor
Resistance
Co
R
1x10-6 100
F Ω
The DC voltage source is fixed to E=50V. The initial value of the state vectors and parameter estimates are chosen as follows:
x(0)=[0.1 0.3 -0.5 -0.2 0]T (87)
, (88)
The LCC converter is controlled in open-loop applying a variable control signal (duty-cycle). Fig (3) shows the magnitude of the complex coefficients obtained via the averaging method when the system starts up in open loop with predetermined duty cycle and
switching frequency .
The results for the resonant converter, the output voltage y1 across capacitor Co, the resonant current y2
and resonant voltage y3, are shown in Fig. 3. Fig 4 show the resulting state variables . The zoom of fig 5 illustrate the behavior of the state variables over the first 5x10-6 s period. Figure 6 show the state estimation errors ;
; and obtained
with , . It is seen that the estimates state variables converge well to their true values within about 200μs, confirming Theorem 1. This is further illustrated by the zoom of Fig. 7. Note that the convergence rate depends on the value of the observer gain.
)
min(Si
max(Si)
Si
2 ,
1 i
1 1 2 2 2( 1 2 3) 1 2
V V V V V
) ( ) (
~
2 min 1 min
1
1 S S
) ( ) (
~
2 min 1 min
2
2 S S
) ( ) (
~
2 min 1 min
3
3 S S
2
2ab a2b V1
a b V2
2 1 2
2 V1 V V V
1 ( 1 2 3)
1
2 ( 1 2 3)
2V V V
1 2
1 ( 1 2 3) 0
2 ( 1 2 3) 0
V V
1 1 2 3 2 1 2 3
min ,
V
1 2
2 1,Zˆ Zˆ
2 1,Z
Z Z1(0),Z2(0)
xi
1 2 2 3 3 5 4 6
ˆ ˆ , ˆ ˆ , ˆ ˆ , ˆ ˆ x z x z x z x z
1 2
1 2
Zˆ (0) [1.5 3 1]T
1 Zˆ (0) [1.5 3 1]T
2
2 256
s
fs kHz
4 3 2 1,x ,x ,x x
1 1
1 z ˆz
er
2 2
2 z ˆz
er er3 z3ˆz3 er4 z4ˆz4
1 200
2 120
