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Z(Tn)-Observability and control for SEPIC converter
Imen Mrad, Jean-Pierre Barbot, Lassaad Sbita
To cite this version:
Imen Mrad, Jean-Pierre Barbot, Lassaad Sbita. Z(Tn)-Observability and control for SEPIC converter.
International federation of automatic control : IFAC World congres 2017, Jul 2017, Toulouse, France.
�hal-01573253�
Z(Tn)-Observability and control for SEPIC converter
I. Mrad∗ JP.Barbot∗∗ L. Sbita∗
∗National Engineering School of Gabes,St Omar Ibn-Elkhattab, Zrig, Gabes 6029,Tunisia. (e-mail: imen.mrad@ensea.fr)
(e-mail: lassaad.sbita@enig.rnu.tn)
∗∗Quartz EA 7393, ENSEA, 6 Avenue du Ponceau, 95014 Cergy Pontoise, France. (e-mail: barbot@ensea.fr)
Abstract:In this paper, a hybrid observer for a class of DC-DC power converter is proposed.
The considered class of switched system including one or several unobservable subspace. Thus, the state vector of a converter is not observable at any time. The aim is to solve this problem of observability by taking into account the hybrid dynamic behavior appearing in the converter. Such problem can be tacked by a new observability concept (Z(TN)-Observability).
An application to a Single-Ended Primary Inductor Converter (SEPIC) is presented. After recalling to hybrid dynamic behavior of the SEPIC, an observability analysis is investigated.
Following this analysis, a hybrid observer is designed based on homogeneous observer coupled with an estimator. Finally some illustrative results are given in order to show the efficiency of the designed observer.
Keywords:Observability, hybrid dynamical system, hybrid observer, Z(TN)-observability, homogeneous observer.
1. INTRODUCTION
A Hybrid Dynamic SystemHDSis a system that switches between divers operation modes, where each mode is gov- erned by its own dynamic law, it’s modeled or defined as a combination of continuous and discrete event subsystems.
Recently this type of system is encountered in different fields, in communications networks Xiao et al. (2000), in manufacturing Pepyne and Cassandras (2000), autopilot Koo et al. (1998), computer synchronization Parlitz et al.
(1992), automotive engine control Balluchi et al. (2000), chemical processes Seader and Henley (2011) and traffic control Tomlin et al. (1998),...The formalism of HDSsis very general, it includes different classes of models. This paper is focused on Switched Linear Systems (SLS), it’s interested in the observability of switched linear systems DC DC power converters, which have unobservable con- figurations linked to particular switching sequences. Two main approaches exist to deal with this problem, the first one is based on the average model, it is an approach that remains efficient Gensior et al. (2006) Spinu et al. (2012).
However, this method is valid only in a frequency range for which the average model approximates well system behaviour. Currently, the use of hybrid dynamical model is more efficient in a wide frequency range or for variable frequency. Then one can solve control problems, as stabil- ity and observability. Moreover, the synthesis of a hybrid observer becomes possible under few conditions Barbot et al. (2007) Tanwani et al. (2011). In this context many studies have treated the problem of hybrid observability of systems with unobservable operating modes during switch- ing cycles Babaali (2004) Zhao and Sun (2010). Indeed, the hybrid observability shows that a switched system
which has an unobservable subspaces for some operation modes, is observable in a hybrid sense. Thus, it can be proved that the hybrid observability of a switched system does not necessarily imply the existence of a procedure for reconstructing the state, in the case where the process goes through an operation mode for which a subspace is unobservable. However, hybrid observers synthesis which takes into account unobservable states remains a scientific problem, and very few results exist in the literature. The consideration of such problem in observers’ synthesis was proposed in Ghanes et al. (2009) in a frame work of multicellular chopper as the serial multicellular converters, nevertheless this work studies a particular case of switched systems, whose unobservable states are constant. Besides, in Tanwani et al. (2013) the authors treat the synthesis of hybrid observers in general with either observable or un- observable subspace, but the disadvantage of this observer is that its synthesis and its implementation in practice is difficult. In this paper, the analysis of hybrid observability for the Single-Ended Primary Inductor Converter (SEPIC) converter is investigated, using the concept of Z(TN)- observability Kang and Barbot (2007), this new concept takes into consideration of five factors: partial state ob- servability, observability with partial model of a dynam- ical system, estimation of unknown input and unknown parameters, systems with algebraic constraints and par- tial time observability. In this work, Z(TN)-observability takes into account the partial observability of SEPIC, with respect to the time, considering a time sequence (related to the so called hybrid time trajectory), after which we can reconstruct all the state vector for SEPIC. Finally, a hybrid observer is proposed based on the homogenous observer algorithm Perruquetti et al. (2008) coupled with
an estimator.
The paper is organised as follows: The SEPIC converter modelling is presented in section 2; The analysis of a hybrid observability of SEPIC converter using the concept of Z(TN) observability can be found in Section 3. Section 4 is devoted to the observer design. To illustrate the theo- retical results, simulation results are presented in Section 5. Finally, a conclusion is given in Section 6.
2. SEPIC CONVERTER MODELLING
SEPIC is a DC-DC power converter that can provide an output voltage either greater then, either less then, or equal to the input voltage continuous and constant.
Unlike the buck-boost converter, SEPIC can maintains the same polarity and the same ground reference for the input and the output Jaafar et al. (2009), moreover it is unidirectional with respect to the current. This circuit is used in several applications such as digital cameras, mobile phones, CD / DVD drives, PDAs, GPS systems, ....SEPIC has a fourth order model with four storage elements: 2 capacitors and 2 inductors as show in Fig. 1. It is similar to cuk and zeta topology Jaafar et al. (2009) Niculescu et al.
(2007). This converter is controlled through the switching element q (figure 1) and the output voltage is the only measured value.
Fig. 1. SEPIC Circuit
Generally for the circuit modelling, the capacitor voltage and inductor current are considered as the states variables.
x= [iL1,Vc1,iL2,Vc2]T = [x1, x2, x3, x4]T y=x4
x˙ =fq(x) y=x4 We have Forq= 0
Fig. 2. SEPIC open (q= 0)
f0
˙
x1=−rL1 L1
x1− 1 L1
x2− 1 L1
x4+ 1 L1
Ve
˙ x2= 1
C1
x1
˙
x3=−rL2
L2x3+ 1 L2x4
˙ x4= 1
C2
x1− 1 C2
x3− 1 RC2
x4 Forq= 1
Fig. 3. SEPIC closed (q=1)
f1
˙
x1=−rL1
L1
x1+ 1 L1
Ve
˙ x2= 1
C1x3
˙
x3=− 1 L2
x2−rL2
L2
x3
˙
x4=− 1 RC2x4
Looking for the hybrid dynamical model of the SEPIC converter, we introduce the discrete variableq={0,1}
f =f0(1−q) +f1q we obtain:
˙ x1
˙ x2
˙ x3
˙ x4
=
−rL1
L1
q−1
L1 0 q−1 L1 1−q
C1 0 q
C1 0
0 −q
L2
−rL2
L2
1−q L2
1−q C2
0 q−1 C2
−1 C2R
x1 x2
x3 x4
+ 1
L1 0 0 0 T
Ve
y= ( 0 0 0 1 ) [x1 x2 x3 x4]T =Vs 3. OBSERVABILITY ANALYSIS Let us consider the following class of system:
ξ˙=fq(t, ξ, u),q∈Q,, ξ∈ <n, u∈ <m
Y =hq(t, ξ, u) (1)
WhereQis a finite index set, fq < × <n× <m→ <n and hq(t, ξ, u) are sufficiently smooth, all dwell time intervals, [ti,0, ti,1], between two switchings of the structure satisfy ti,1−ti,0> τminfor someτmin>0.τminexclusion of Zeno phenomena Goebel et al. (2004). For the input uin any time interval [ti,0, ti,1[⊆[tini, tend[,the input u(t) is equal toqi.e in 0,1
For switched systems, the concept of observability and
observer design are strongly related to the sequence of switching, thus it is important to recall the definition of hybrid time trajectory
Definition 1. Goebel et al. (2009) Lygeros et al. (2003) A hybrid time trajectory is a finite or infinite sequence of intervalsTN = (Ii)Ni=0, such that:
-Ii= [ti,0, ti,1[, for 0≤i < N;
- For alli < N ti,1=ti+1,0 -t0,0=tini ettN,1=tend
Moreover,hT Niis the ordered list ofqassociated toT N (q0, ..., qN) withqi the value ofqduring the time interval II.
The concept of Z(TN)-Observability was proposed for nonlinear systems and switching systems. This notion has the particularity of taking into account the partial observability of the state and the hybrid time trajectory, the observability for each partial model and deduces the observability or unobservability with respect to switching sequence and the estimation of the parameters by a dynamic extension.
Definition 2. Kang and Barbot (2007) Consider a system (1) and a variable z = Z(t, ξ, u). Let (t, ξ1(t), u1(t)) be a trajectory in U with a hybrid time trajectory T N and hT Ni. Suppose for any trajectory, (t, ξ2(t), u2(t)), in U with the sameT N andhT Ni, the equality:
h(t, ξ1(t), u1(t)) =h(t, ξ2(t), u2(t)) (2) implies
Z(t, ξ1(t), u1(t)) =Z(t, ξ2(t), u2(t)) (3) Then we say thatz=Z(t, ξ, u) is Z(TN)-observable along the trajectory (t, ξ1(t), u1(t)).
For a fixed hybrid time trajectory T N and hT Ni, if z = Z(t, ξ, u) is Z(TN)-observable along all trajectories in U, then, z =Z(t, ξ, u) is said to be Z(TN)-observable in U.
Suppose for any trajectory (t, ξ(t), u(t)) inU, there always exists an open setU1⊂Uso that (t, ξ(t), u(t)) is contained in U1 andz=Z(t, ξ, u) is Z(TN)-observable inU1. Then, z=Z(t, ξ, u) is said to be locally Z(TN)-observable.
Hereafter, the dimension of z variable is denoted by nz. Before to recall a useful proposition it’s important to define linear projectionP:
P :
z1 ... znz
→
δ1 0 0. . . 0 0 δ2 0 . . . 0 . . . . . 0 0 0 . . . δnz
z1 ... znz
where δi,i = 1,2, . . . ,nz, is zero or one. The complement of P is called ¯P (projecting z to the variables eliminated byP).
Proposition 1. Kang et al. (2009) Consider the system (1) and a fixed hybrid time trajectory T N and hT Ni. Let U be an open set in time-state-control space. Suppose Z(t, ξ(t), u(t)) ∈ <nz is always continuous under any admissible control input.
Suppose there exists a sequence of projectionsP i, i = 0,1, . . . ,nzsuch that
(1) given any 0 ≤i ≤N,P iZ(t, xi, u) is Z-observable in U on the subintervalt∈[ti,0, ti,1[;
(2) Rank
P0T. . . PNT
=nz; (3) dP¯iZ(t,ξ(t),u(t))
dt = 0 fort∈[ti,0, ti,1[ and (t, ξ(t), u(t))∈ U.
Then,z=Z(t, ξ, u) is Z(TN)-observable inU with respect to the hybrid time trajectoryT N andhT Ni.
3.1 observability of the SEPIC
Considering the observability matrix based on q of this hybrid system:
y=x4
˙
y= 1−q C2
x1+q−1 C2
x3− 1 C2Rx4
¨ y=
−rL1 L1
(1−q)− 1
C22R(1−q)
| {z }
e
x1
+
− 1 C2L1
(1−q)2− 1 C2L2
(q2−q)
| {z }
f
x2
+
− rL2 C2L2
(q−1)− 1
C22R(q−1)
| {z }
g
x3
+
− 1
C2L1(1−q)2− 1
C2L2(1−q)2− 1 C2
2R2
| {z }
h
x4
...y =
−rL1
L1
e+(1−q) C1
f+(1−q) C2
h
| {z }
a
x1
+
q−1
L1 e− q L1g
| {z }
b
x2+
q
C1f−rL2
L2g+q−1 C2 h
| {z }
c
x3
+
q−1
L1 e+1−q L2 g− 1
RC2h
| {z }
d
x4
dO(4) =
0 0 0 1
1−q C2
0 q−1 C2
− 1 C2R
e f g h
a b c d
forq= 1
e=f =g=a=b=c= 1−q C2 = 0
thenx1,x2x3 are unobservable.
Nevertheless the system forq= 0 is completely observable.
According to definition 2, it can be said that this system is observable in the hybrid sense if at least the hybrid time trajectory contains one mode withq= 0.
4. OBSERVER DESIGN
In this paper, the proposed method is based on the Homogeneous Observer Bernuau et al. (2015) Perruquetti et al. (2008), it converges in finite time and reaches the real dynamics. To present the observer design some definitions are given seeP erruquettiet al.(2008)
Definition 3. A functionG:<n−→ < is homogeneous of degreedwith respect to the weights (r1, ..., rn)∈ <n>0
G(λr1x1, ...., λrnxn) =λdG(x1, ...., xn) for allλ >0
Definition 4. A vector field g is homogeneous of degreed with respect to the weights (r1, ..., rn) ∈ <n>0 if, for all 1≤i≤ntheithcomponentgiis a homogeneous function of degreeri+d, that is,
gi(λr1x1, ...., λrnxn) =λri+dgi(x1, ...., xn) 4.1 Problem formulation
Let us consider system with the canonical observable form z˙ =Az+f(y, u,u, ..., u˙ r)
y=Cz (4)
wherez∈ <n the state,r∈N >0, and
A=
a1 1 0 0 0 a2 0 1 0 0 ... 0 0 1 0 an−1 0 0 0 1 an 0 0 0 0
C= ( 1 0 0 0 )
The homogeneous observer for this system is designed as:
˙ˆ z1
˙ˆ z2
...
˙ˆ zn−1
˙ˆ zn
=A
ˆ z1
ˆ z2
... ˆ zn−1
ˆ zn
+f(y, u,u, . . . , u˙ (n))
−
k1de1cα1 k2de1cα2
... kn−1de1cαn−1
knde1cαn
Where deicαi = |ei|αisign(ei) and ei = zi−zˆi for i ∈ 1, ..., nz tends to zero in finite time Perruquetti et al.
(2008). Let α1=α
Thus the observation errors dynamics is given by:
˙
e1=e2+χ1(e1)
˙
e2=e3+χ2(e1) ...
˙
en−1=en+χn−1(e1)
˙
en=χn(e1)
(5)
Where function χi=kide1cαi.
Lemma 1. If α > 1 − n−11 the system (5) is homo- geneous of degree α − 1 with respect to the weights {(i−1)α−(i−2)}1≤i≤nandαi=iα1−(i−1),1≤i≤n.
Proof. For the proof see Perruquetti et al. (2008).
4.2 hybrid Observer for the SEPIC converter
A hybrid observer is synthesized, which is characterized by two coupled operation modes. The first mode is the observation of the system states based on a homogeneous observer for q = 0, and the second one for q = 1 is the estimate of the 3 first unobservable states by knowing their
dynamique.
The hybrid observer given by:
˙ˆ
x1 = −rL1
L1xˆ1−q−1
L1 xˆ2−q−1 L1 xˆ4+ 1
L1Ve
− k1dx4−xˆ4cα(1−q)
˙ˆ
x2 = 1−q C1
ˆ x1− q
C1
ˆ
x3−k2dx4−xˆ4c2α−1(1−q)
˙ˆ
x3 = − q L2
ˆ x2−rL2
L2
ˆ
x3+1−q L2
ˆ x4
− k3dx4−xˆ4c3α−2(1−q)
˙ˆ
x4 = 1−q
C2 xˆ1−q−1
C2 ˆx3− 1
RC2xˆ4−k4dx4−xˆ4c4α−3 Withα= 0.68,k1= 24,k2= 35k3= 50 andk4= 24
5. SIMULATION RESULT
The hybrid observer was tested on the model of the SEPIC converter in simulation using MATLAB. The observer model is implemented in open loop with computation step equal to 5 10−6sand a SEPIC switching frequency 20kHz.
The parameters of the circuit are given in table 1.
TABLE I : Parametres of the SEPIC P arameter V alue(unit)
L1 2.3 10−3H L2 330 10−6H C1 190 10−6F C2 190 10−6F
R1 2.134 Ω
R2 0.234 Ω
Ve 20
Fig. 4. CurrentiL1
Fig. 5. VoltageVC1
Fig. 5 shows that the tension ˆVC1 estimated by the observer converges in finite time to the real valueVC1 . It is noted in Fig. 4 that the current ˆiL1 estimated by the observer converges in finite time to iL1, and we observe
Fig. 6. CurrentiL2
from the the Fig. 6 that the current ˆiL2 estimated by the observer converges to iL2
6. CONCLUSION
In this paper a hybrid observer is proposed for switched linear system with unobservable subspace in specific mode.
Two types of observer are coupled; the first one is a homogeneous observer, and the second one is an estimator.
Some simulations work are performed to test the proposed observer structure. The simulation results show the fin- ished time convergence. But it must be mentioned that in this observer structure design qis considered known at each instant of time. In a future work it will be assumed that this stateqis unknown in order to take into account fault of switch driver. This case can be considered us a left inversion problem withqas input Vu and Liberzon (2008).
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