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Adaptive-gain second-order sliding mode observer design for switching power converters
Control Engineering Practice ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Adaptive-gain second-order sliding mode observer design for switching power converters Jianxing Liu, Salah Laghrouche n, Mohamed Harmouche, Maxime Wack IRTES-SeT, Université de Technologie de Belfort-Montbéliard, 13 rue Thierry Mieg, 90000 Belfort, France
art ic l e i nf o
a b s t r a c t
Article history: Received 27 December 2012 Accepted 25 October 2013
In this paper, an adaptive-gain, Second Order Sliding Mode (SOSM) observer for multi-cell converters is designed by considering it as a type of hybrid system. The objective is to reduce the number of voltage sensors by estimating the capacitor voltages from measurement of the load current. The proposed observer is proven to be robust in the presence of perturbations with unknown boundaries. As the states of the system are only partially observable, a recent concept known as Z(TN)-observability is used to address the switching behavior. Multi-rate simulation results demonstrate the effectiveness and the robustness of the proposed observer with respect to output measurement noise and system uncertainty (load variations). & 2013 Elsevier Ltd. All rights reserved.
Keywords: Sliding mode observer Hybrid systems Observability Multi-cell power converter
1. Introduction In recent years, industrial applications requiring high power levels have used medium-voltage semiconductors (Gerry, Wheeler, & Clare, 2003; Meynard & Foch, 1992; Rech & Pinheiro, 2007; Rodriguez, Lai, & Peng, 2002). Because of the efficiency requirements, the power of the converter is generally increased by boosting the voltage. However, medium-voltage switching devices are not available. Even if they did exist, the volume and the cost of such devices would be substantial (Gateau, Fadel, Maussion, Bensaid, & Meynard, 2002). In this sense, the topology of multilevel converters, which have been studied during the last decade, becomes attractive for high voltage applications (Meynard & Foch, 1992). From a practical point of view, the series of a multi-cell chopper designed by the LEEI (Toulouse, France) (Bensaid & Fadel, D., 2001) leads to a safe series association of components working in a switching mode. This structure offers the possibility of reducing the voltage constraints evenly among each cell in a series. These lower-voltage switches result in lower conduction losses and higher switching frequencies. Moreover, it is possible to improve the output waveforms using this structure (Bensaid & Fadel, M., 2001; Bensaid & Fadel, 2002; Gateau et al., 2002). These flying capacitors have to be balanced to guarantee the desired voltage values at the output, which ensures that the maximum benefit from the multi-cell structure is obtained (Meynard, Fadel, & Aouda, 1997). These properties are lost if the capacitor voltage drifts far from the desired value (Bejarano, Ghanes, & Barbot,
n
Corresponding author. E-mail address: [email protected] (S. Laghrouche).
2010). Therefore, a suitable control of the switches is required to generate the desired values of the capacitor voltages. The control of switches allows the current harmonics at the cutting frequency to be canceled and the ripple of the chopped voltage to be reduced (Defoort, Djemaï, Floquet, & Perruquetti, 2011; Djemaï, Busawon, Benmansour, & Marouf, 2011). Several control methods have been proposed for multi-cell converters, such as nonlinear control based on input–output linearization (Gateau et al., 2002), predictive control (Defaÿ, Llor, & Fadel, 2008), hybrid control (Bâja, Patino, Cormerais, Riedinger, & Buisson, 2007), model predictive control (Defaÿ et al., 2008; Lezana, Aguilera, & Quevedo, 2009) and sliding mode control (Amet, Ghanes, & Barbot, 2011; Djemaï et al., 2011; Meradi, Benmansour, Herizi, Tadjine, & Boucherit, 2013). However, most of these techniques require measurements of the voltages of the capacitors to design the controller. That is, extra voltage sensors are necessary, which increases the cost and the complexity of the system. Hence, the estimation of the capacitor voltages using an observer has attracted great interest (Besançon, 2007). It should be noted that the states of the multi-cell system are only partially observable because the observability matrix never has full rank (Besançon, 2007). Hence, the observability matrix rank condition cannot be employed in an observability analysis of a hybrid system such as the one considered here (Babaali & Pappas, 2005; Vidal, Chiuso, Soatto, & Sastry, 2003). A recent concept, Z(TN)observability (Kang, Barbot, & Xu, 2009), can be used to analyze the observability of a switched hybrid system and is applied in this work because the observability of the converter depends upon the switching control signals. Various observers have been designed for the multi-cell converters based on concepts such as homogeneous finite-time observers (Defoort et al., 2011), super-twisting sliding
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Please cite this article as: Liu, J., et al. Adaptive-gain second-order sliding mode observer design for switching power converters. Control Engineering Practice (2013), http://dx.doi.org/10.1016/j.conengprac.2013.10.012i
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2
mode observers (Bejarano et al., 2010; Ghanes, Bejarano, & Barbot, 2009) and adaptive observers (Bejarano et al., 2010). The concept of observability presented in Kang et al. (2009) gives the condition under which there exists a hybrid time trajectory that makes the system observable. Using this concept, estimates of the capacitor voltages can be obtained from the measurements of the load current and the source voltage by taking advantage of the appropriate hybrid time trajectories. In the works of Ghanes et al. (2009) and Bejarano et al. (2010), based on a set of p 1 linearly independent equations with respect to the voltages in the p 1 capacitors, an algorithm is proposed to estimate the capacitor voltages which employs the pseudo-inverse of a matrix whose elements are the switching signals correspondingly. In this paper, an observability analysis based on the results of Kang et al. (2009) and Bejarano et al. (2010) is performed for the multi-cell converter assuming measurements of the load current and the source voltage under certain conditions of the switching input sequences. Then, a novel adaptive-gain SOSM observer for multi-cell converters is introduced that takes into account certain perturbations (load variations) in which the boundaries of their first time derivatives are unknown. The proposed adaptive-gain SOSM algorithm combines the nonlinear term of the supertwisting algorithm (ST) and a linear term, the so-called SOSML algorithm (Moreno & Osorio, 2008). The behavior of the ST algorithm near the origin is significantly improved compared with the linear case. Conversely, the additional linear term improves the behavior of the ST algorithm when the states are far from the origin. Therefore, the SOSML algorithm inherits the best properties of both the linear and the nonlinear terms. An adaptive law of the gains of the SOSML algorithm is derived via the so-called “time scaling” approach (Respondek, Pogromsky, & Nijmeijer, 2004). The output observation error and its first time derivative converge to zero in finite time with the proposed SOSML observer such that the equivalent output-error injection can be obtained directly. Finally, the resulting reduced-order system is proven to be exponentially stable. That is, the estimates of the capacitor voltages, which are considered as the states of the observer system, converge to the real states exponentially. The main advantages of this paper are as follows:
The estimates of the capacitor voltages are obtained directly
through analyzing the information of the equivalent outputerror injection. Only one parameter of the proposed SOSML algorithm has to be tuned. There are no a priori requirements on the perturbation bounds and the finite time convergence of the output error dynamics is proven via Lyapunov analysis.
This paper is organized as follows. In Section 2, a model of the multi-cell converter and its characteristics are presented. In Section 3, the observability of the multi-cell converter is studied with the concept of Z(TN)-observability. Section 4 discusses the design of the proposed adaptive-gain SOSML observer for estimating the capacitor voltages. Section 5 gives multi-rate simulation results including a comparison with a Luenberger switched observer with disturbances.
2. Modeling of the multi-cell converter The structure of a multi-cell converter is based on the combination of a certain number of cells. Each cell consists of an energy storage element and commutators (Gateau et al., 2002). The main advantage of this structure is that the spectral quality of the output signal is improved by a high switching frequency between
Fig. 1. Multi-cell converter on RL load.
the intermediate voltage levels (McGrath & Holmes, 2007). An instantaneous model that was presented in Gateau et al. (2002) and describes fully the hybrid behavior of the multi-cell converter is used here. Fig. 1 depicts the topology of a converter with p independent commutation cells that is connected to an inductive load. The current I flows from the source E to the output through the various converter switches. The converter thus has a hybrid behavior because of the presence of both discrete variables (the switching logic) and continuous variables (the currents and the voltages). Through circuit analysis, the dynamics of the p-cell converter were obtained as in the following differential equations: 8 p 1V R E c > > > I_ ¼ I þ Sp ∑ j ðSj þ 1 Sj Þ; > > L L L > j ¼ 1 > > > > <_ I V c1 ¼ ðS2 S1 Þ; ð1Þ c 1 > > > ⋮ > > > > I > > _ > : V cp 1 ¼ c ðSp Sp 1 Þ; p where I is the load current, cj is the jth capacitor, V cj is the voltage of the jth capacitor and E is the voltage of the source. Each commutation cell is controlled by the binary input signal Sj A f0; 1g, where Sj ¼ 1 indicates that the upper switch of the jth cell is on and the lower switch is off and Sj ¼0 indicates that the upper switch is off and the lower switch is on. The discrete inputs are defined as follows: ( uj ¼ Sj þ 1 Sj ; j ¼ 1; …; p 1 ð2Þ up ¼ S p : With Eq. (2), the system (1) can be represented as follows: 8 p 1V > > > I_ ¼ R I þ E up ∑ cj uj ; > > > L L > j¼1 L > > > > I > > < V_ c1 ¼ u1 ; c1 ð3Þ > ⋮ > > > > I > > ¼ u ; V_ > > > cp 1 c p 1 p 1 > > > : y ¼ I:
Assuming that only the load current I can be measured, it is easy to represent the system (3) as a hybrid (switched affine) system: ( x_ ¼ f ðx; uÞ ¼ AðuÞx þ BðuÞ; ð4Þ y ¼ hðx; uÞ ¼ Cx; where x ¼ ½I V c1 ⋯ V cp 1 T is the continuous state vector, u ¼ ½u1 u2 ⋯ up T is the switching control signal vector which takes
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for all i o N; t i;1 ¼ t i þ 1;0 ; t 0;0 ¼ t ini and t N;1 ¼ t end .
Table 1 Switching modes and capacitor voltages for a three-cell converter. Mode: ½S1 ; S2 ; S3
V c1
V c2
u1
u2
Observable states
0: [0,0,0] 1: [0,0,1] 2: [0,1,0] 3: [0,1,1] 4: [1,0,0] 5: [1,0,1] 6: [1,1,0] 7: [1,1,1]
⇝ ⇝ ↗ ↗ ↘ ↘ ⇝ ⇝
⇝ ↗ ↘ ⇝ ⇝ ↗ ↘ ⇝
0 0 1 1 1 1 0 0
0 1 1 0 0 1 1 0
I I; V c2 I; V c1 ; V c2 I; V c1 I; V c1 I; V c1 ; V c2 I; V c2 I
BðuÞ ¼
E L up
0
⋯
0
iT
Moreover, 〈T N 〉 is defined as the ordered list of inputs u associated with TN, uii ¼ 0;N , where ui is the value of u on the interval Γi. Definition 2 (Kang et al., 2009). The function z ¼ Zðt; xÞ is said to be Z-observable with respect to the hybrid time trajectory TN and 〈T N 〉 if for any two trajectories ðt; x; uÞ and ðt; x′; u′Þ defined in ½t ini ; t end , the equality hðx; uÞ ¼ hðx′; u′Þ implies that Zðt; xÞ ¼ Zðt; x′Þ.
only discrete values and the matrices A(u), B(u), C are defined as 2 R 3 u L uL1 ⋯ pL 1 6 u1 0 ⋯ 0 7 6 c1 7 7; AðuÞ ¼ 6 6 ⋮ 7 ⋮ ⋱ ⋮ 4 5 up 1 0 ⋯ 0 cp 1 h
3
;
Lemma 1 (Kang et al., 2009). Consider the system (4) and a fixed hybrid time trajectory TN and 〈T N 〉. Suppose that z ¼ Zðt; xÞ is always continuous under any admissible control input. If there exists a sequence of projections P i ; i ¼ 0; 1; …; N, such that
for all i o N, P i Zðt; xÞ is Z-observable for t A Γ i ; rank ð½P T0 ; …; P TN Þ ¼ dimðzÞ; dP i Zðt; xÞ=dt ¼ 0, for t A Γ i , where P is the complement of P (projecting z to the variables eliminated by P),
ð5Þ
then, z ¼ Zðt; xÞ is Z-observable with respect to the hybrid time trajectory TN and 〈T N 〉.
The main objective of this paper is to design an observer based on the instantaneous model (3) that is able to estimate the capacitor voltages using only the measurement of the load current and the associated switching control input (which is assumed to be known).
Remark 1. In Lemma 1, the third condition requires that the components of Z that are not observable in Γi must remain constant within this time interval. The hybrid time trajectory TN and 〈T N 〉 influences the observability property in a way similar to an input.
C ¼ ½1 0 ⋯ 0:
3. Hybrid observability analysis From the instantaneous model of the system (4) with p Z 3, it can be noted that there are several switching modes that make the system unobservable. For instance, if u1 ¼ u2 ¼ ⋯ ¼ up 1 ¼ 0, the voltages V cj ðj ¼ 1; …; p 1Þ become completely unobservable. These switching modes are not affected by the capacitor voltages. Fortunately, these cases are the ones in which the p-cells are not switching and will not occur for all control sequences; otherwise, there is no interest in the physical sense. The observability analysis of the system (4) is based on the measurement of the load current I and the knowledge of the control input sequence u. The so-called observability matrix (Besançon, 2007) is defined as 3 2 3 2 1 0 ⋯ 0 C u 6 CA 7 6 RL uL1 ⋯ pL 1 7 7 6 7 6 6 7 6 2 7 2 6 7 6 7 p 1 Opp ¼ 6 CA 7 ¼ 6 R2 ð6Þ ui Rup 1 7: Ru1 ∑ ⋯ 7 6 7 6 L 2 2 L L 5 4 ⋮ 5 4 i ¼ 1Lci ⋮ ⋮ ⋮ ⋮ CAp 1 With simple computations, it can be shown that rankðOÞ ¼ 2 o p:
ð7Þ
It follows that the continuous states are not observable using only the load current because the observability matrix (7) is not full rank. Because of the switching sequences of the system (4), the observability is strongly linked to the hybrid behavior. Therefore, the recently developed concept of Z(TN)-observability (Kang et al., 2009) is applied to analyze the observability of the hybrid system (4). It is important to note the following definitions. Definition 1 (Kang et al., 2009). A hybrid time trajectory is a finite N or infinite sequence of intervals T N ¼ Γ i ¼ 0 such that
Γ i ¼ ½t i;0 ; t i;1 Þ, for all 0 r i oN;
Table 1 presents the eight possible configurations for a threecell converter. The application of Lemma 1 to the three-cell converter is as follows. We take Zðt; xÞ ¼ ½x2 x3 T ¼ ½V c1 V c2 T . For the discrete switching conditions ½0; 0; 0 and ½1; 1; 1, it can be verified that Zðt; xÞ is not Z(TN)-observable. Fortunately, from (1) the dynamics of V c1 and V c2 are zero, which means that these states remain constant during these time intervals. Next, assume that a trajectory of the system has the status ½1; 0; 0 and ½1; 1; 0 during time intervals Γ1 and Γ2, respectively. Let us define P 1 ¼ ½1 0 and P 2 ¼ ½0 1. We have P 1 Z ¼ x3 ¼ V c2 , dP 1 Z=dt ¼ h i P1 P2 ¼
dV c2 =dt ¼ 0, P 2 Z ¼ x2 ¼ V c1 , dP 2 Z=dt ¼ dV c1 =dt ¼ 0 and rank
2. All the assumptions in Lemma 1 are satisfied; therefore, Zðt; xÞ ¼ ½V c1 V c2 T is Z(TN)-observable. The symbols in Table 1 are defined as follows: ⇝ indicates a constant value, ↗ indicates increase and ↘ indicates decrease. In the next section, an adaptive-gain SOSML observer will be presented for the converter system (3).
4. Adaptive-gain SOSML observer design As discussed in Gateau et al. (2002), active control of the multicell converter requires the knowledge of the capacitor voltages. Usually, voltage sensors are used to measure the capacitor voltages. However, the extra sensors increase the cost, the complexity and the size, especially in high-voltage applications. Moreover, any sensors will introduce the measurement noise which will be directly transposed to the estimated value. Therefore, the design of a state observer using only the measurement of load current and the associated switching inputs is desirable. In this section, an adaptive-gain SOSML observer for the threecell converter (p ¼3) is presented that is robust to perturbations (load variations) for which the boundaries of the first time derivative are unknown. A novel adaptive law for the gains of the SOSML algorithm with only one tuning parameter is designed via the so-called “time scaling” approach (Respondek et al., 2004).
Please cite this article as: Liu, J., et al. Adaptive-gain second-order sliding mode observer design for switching power converters. Control Engineering Practice (2013), http://dx.doi.org/10.1016/j.conengprac.2013.10.012i
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4
The proposed approach does not require the a priori knowledge of the perturbation bounds. ^ the system (3) is rewritten to include the Defining e1 ¼ I I, ~ perturbation f ðe1 Þ, i.e., the load resistance uncertainty (Defoort et al., 2011) 8 R E V c1 V c2 > ~ _ > > I ¼ L I þ L u3 L u1 L u2 þ f ðe1 Þ; > > > < u1 V_ c1 ¼ I; ð8Þ c1 > > > > u2 > > : V_ c2 ¼ I: c2
zone technique (Slotine & Li, 1991) as _lðtÞ ¼ k if je1 j Z ε 0 otherwise
The proposed observer is formulated as 8 > R E V^ c V^ c _ > > I^ ¼ I þ u3 1 u1 2 u2 þ μðe1 Þ; > > > L L L L > < u1 _ ^ V c1 ¼ I þ k1 μðe1 Þ; c1 > > > > _ u2 > > ^ > : V c2 ¼ c I þ k2 μðe1 Þ; 2
Assumption 2. There is a TN such that z ¼ x ¼ ½I V c1 V c2 T is Zobservable under the condition of Lemma 1 (Defoort et al., 2011). ð9Þ
þ kα ðtÞ
0
ð10Þ
2
Eqs. (8) and (9) yield the observation error dynamics as e_ 1 ¼ μðe1 Þ
u1 u2 e2 e3 þ f~ ðe1 Þ; L L
ð12Þ
e_ 2 ¼ k1 μðe1 Þ;
ð13Þ
e_ 3 ¼ k2 μðe1 Þ:
ð14Þ
In this paper, the adaptive gains λðtÞ; αðtÞ; kλ ðtÞ and kα ðtÞ are formulated as 8 pffiffiffiffiffiffiffi > > > λðtÞ ¼ λ0 lðtÞ; > > < αðtÞ ¼ α0 lðtÞ; ð15Þ kλ ðtÞ ¼ kλ0 lðtÞ; > > > > > : kα ðtÞ ¼ kα l2 ðtÞ; 0 where λ0 ; α0 ; kλ0 and kα0 are positive constants to be defined and l(t) is a positive, time-varying, scalar function. The adaptive law of the time-varying function l(t) and the design parameters k1 and k2 are given by _lðtÞ ¼ k if je1 j a 0 ð16Þ 0 otherwise k1 ¼
κ u1
if je1 j ¼ 0
0
otherwise
0
otherwise
;
k2 ¼
where k, the initial value lð0Þ and
κ u2
if je1 j ¼ 0
0
otherwise
;
k2 ¼
κ u2
if je1 j r ε
0
otherwise
ð19Þ
ε is a sufficiently small positive value.
Assumption 1. The system (8) and the observer system (9) are bounded input, bounded state (BIBS) because this is a physical system (Perruquetti & Barbot, 2002).
Theorem 1. Consider the error system (12) under Assumptions 1 and 2. Assume that the perturbation f~ ðe1 Þ satisfies the following condition: and
f~ ð0Þ ¼ 0;
ð20Þ
4α0 kα0 4 8kλ0 α0 þ 9λ0 kλ0 : 2 2
ð21Þ
Proof. The system (12) can be rewritten as ( e_ 1 ¼ λðtÞje1 j1=2 signðe1 Þ kλ ðtÞe1 þ φ1 ; φ_ 1 ¼ αðtÞ signðe1 Þ kα ðtÞe1 þ ϱ1 ; _ where ϱ1 ¼ f~ ðe1 Þ ðu1 =LÞe_ 2 ðu2 =LÞe_ 3 .
ð22Þ
2
e 1 dτ ;
and the adaptive gains λðtÞ; αðtÞ; kλ ðtÞ; kα ðtÞ and the design parameters k1 and k2 are to be defined. Define the observation errors as ( e2 ¼ V c1 V^ c1 ; ð11Þ e3 ¼ V c V^ c : 2
if je1 j r ε
where χ 1 is an unknown positive constant. Then, the trajectories of the error system (12) converge to zero in finite time with the adaptive gains in (15) and (16) satisfying the following condition:
0
t
where
κ u1
_ jf~ ðe1 Þj r χ 1
where μðÞ is the SOSML algorithm (Moreno & Osorio, 2008) Z t μðe1 Þ ¼ λðtÞje1 j1=2 signðe1 Þ þ αðtÞ signðe1 Þ dτ þ kλ ðtÞe1 Z
k1 ¼
ð18Þ
ð17Þ
κ are the positive constants.
Remark 2. In view of practical implementation, the condition e1 ¼ 0 in (16) and (17) cannot be satisfied due to measurement noise and numerical approximations. In order to make the adaptive algorithm (16) and the switching gains (17) practically implementable, one has to modify the condition e1 ¼ 0 by a dead-
Based on Assumption 1, because the input u is bounded, the state does not go to infinity in finite time. Moreover, if I^ is bounded, all the states of the observer are also bounded for a finite time. Consequently, the observation error e1 is also bounded (Perruquetti & Barbot, 2002). It follows from (13) and (14) that e_ 2 and e_ 3 are bounded and satisfy je_ 2 j r χ 2 and je_ 3 jr χ 3 , where χ2 and χ3 are some unknown positive values. From Eq. (20), it is easy to deduce that jϱ1 jr χ 1 þ χ 2 =L þ χ 3 =L ¼ F, where F is an unknown positive value. A new state vector is introduced to represent the system in (22) in a more convenient form for Lyapunov analysis 3 2 3 2 1=2 ζ1 l ðtÞje1 j1=2 signðe1 Þ 7 7 6 7: lðtÞe1 ð23Þ ζ¼6 4 ζ2 5 ¼ 6 4 5
ζ3
φ1
Thus, the system in (22) can be rewritten as 2 k 3 2 l_ 3 2 λ0 3 λ 1 ζ 0 2 0 0 20 2lðtÞ 1 2 6 7 6 _ 7 lðtÞ 6 7 6 l ζ 7: ζ_ ¼ λ0 0 5 ζ þ lðtÞ6 4 0 k λ0 1 7 4 0 5 ζ þ 4 2lðtÞ 25 jζ 1 j 0 0 α0 0 kα0 0 ϱ1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A1
A2
ð24Þ Then, the following Lyapunov function candidate is introduced for the system in (24): Vðζ Þ ¼ 2α0 ζ 1 þkα0 ζ 2 þ 12 ζ 3 þ 12 ðλ0 ζ 1 þ kλ0 ζ 2 ζ 3 Þ2 ; 2
2
2
which can be rewritten as a quadratic form 2 2 4α 0 þ λ 0 λ 0 k λ0 λ0 16 T 2 Vðζ Þ ¼ ζ P ζ ; P ¼ 6 λ k k þ 2k kλ0 α0 λ0 24 0 λ0 λ0 k λ0 2
ð25Þ 3 7 7: 5
ð26Þ
Please cite this article as: Liu, J., et al. Adaptive-gain second-order sliding mode observer design for switching power converters. Control Engineering Practice (2013), http://dx.doi.org/10.1016/j.conengprac.2013.10.012i
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As (25) is a continuous Lyapunov function, the matrix P is positive definite. Taking the derivative of (26) along the trajectories of (24) _l lðtÞ T V_ ¼ ζ Ω1 ζ lðtÞζ T Ω2 ζ þ ϱ1 q1 ζ þ q2 P ζ ; lðtÞ jζ 1 j
2 6
α0 þ 2λ20
0
0
kα0 þ kλ0
Ω2 ¼ kλ0 6 4
0
2
k λ0
0
7 k λ0 7 5;
ð28Þ
1
_l λ ðΩ1 Þ 1=2 λ ðΩ2 Þ F‖q1 ‖2 1=2 V lðtÞ min V þ 1=2 V þ ΔΩ; V_ r lðtÞ min 1=2 2lðtÞ λ ðPÞ max λmax ðPÞ λmin ðPÞ ð29Þ where
ΔΩ ¼ ðð4α0 þ λ ζ
λ
2 6 6 Q ¼6 4
ζ 1 ζ 2 þ 2kα0 kλ0 ζ 2
4α0 þ λ0 þ λ0 kλ0 þ λ20 2
0
ζ2ζ3Þ r ζ Q ζ;
kλ 0 0 k λ0 þ 2
0 λ0 þ kλ0
0
T
3
0
2kα0 kλ0 þ λ 2
0
λ ζ ζ
2 2 0 1 3 kλ0
0
7 7 7: 5
For simplicity, we define F‖q1 ‖2
where γ 1 ; γ 2 ; γ 3 and be simplified as
λ
1=2 min ðPÞ
;
γ3 ¼
λmin ðΩ2 Þ ; λmax ðPÞ
γ4 ¼
λmax ðQ Þ ; 2λmin ðPÞ
ð32Þ
γ4 are all positive constants. Thus, Eq. (31) can !
_l V_ r ðlðtÞγ 1 γ 2 ÞV 1=2 lðtÞγ 3 γ 4 V : lðtÞ
ð33Þ
Because _lðtÞ Z0 such that the terms lðtÞγ 1 γ 2 and lðtÞγ 3 ð_l=lðtÞÞγ 4 are positive in finite time, it follows from (33) that V_ r c1 V 1=2 c2 V;
ð34Þ
where c1 and c2 are positive constants. By the comparison principle (Khalil, 2001), it follows that Vðζ Þ and therefore ζ converge to zero in finite time. Thus, Theorem 1 is proven. □ It follows from Theorem 1 that when the sliding motion takes place, e1 ¼ 0 and e_ 1 ¼ 0. Thus, the output-error equivalent injection μðe1 Þ can be obtained directly from Eq. (12): u L
u L
μðe1 Þ ¼ 1 e2 2 e3 :
150 V 40 μF 131 Ω 10 mH 106 Hz 100 kHz 5 kHz
There exists a constant ϕM 4 0 such that for all t Z0 and all u A D, where D A R2 is a closed, compact subset, such that J Ψ ðt; uÞ J r ϕM , where Ψ ðt; uÞ ¼
qffiffi k Lu1 ðtÞ
qffiffi
k Lu2 ðtÞ
T
:
ð37Þ
t
Ψ ðτ; uÞΨ T ðτ; uÞ dτ Z μI 4 0;
8 t Z0:
ð38Þ
2
ð31Þ
γ2 ¼
DC voltage (E) Capacitors (c1 ; c2 ) Load resistance (R) Load inductor (L) Simulation rate (f1) The sampling rate (f2) The chopping frequency (f3)
u 8 u2 1 > e2 þ e3 ; < e_ 2 ¼ k1 L L ð36Þ > e_ ¼ k u1 e þ u2 e : : 3 2 2 3 L L Proposition 1. Consider the reduced-order system (36) with the switching gains k1 and k2 given by (17). Then, the trajectories of the error system (36) converge to zero exponentially if the following two conditions are satisfied (Loría & Panteley, 2002):
t þ T1
ð30Þ
! ! _l λmax ðQ Þ λ ðΩ1 Þ F‖q1 ‖2 1=2 λmin ðΩ2 Þ V: lðtÞ V V_ r lðtÞ min 1=2 λmax ðPÞ 2lðtÞ λmin ðPÞ λ1=2 λmin ðPÞ max ðPÞ
λmin ðΩ1 Þ ; λ1=2 max ðPÞ
Values
There exist constants T 1 40 and μ 4 0 such that Z
With (30), Eq. (29) becomes
γ1 ¼
System parameters
reduced-order system is obtained:
3
it is easy to verify that Ω1 and Ω2 are positive definite matrices under the condition in (21). Because λmin ðPÞ‖ζ ‖2 r V r λmax ðPÞ‖ζ ‖2 , Eq. (27) can be rewritten as
2 2 0 Þ 1 þ2 0 kλ0
Table 2 Main parameters of simulation model.
ð27Þ
where q1 ¼ ½ λ0 kλ0 2, q2 ¼ ½ζ 1 ζ 2 0, and 2 2 3 λ0 þ 2α0 0 λ0 7 λ 6 2 Ω1 ¼ 0 6 0 2kα0 þ 5kλ0 3kλ0 7 5; 24 λ0 3kλ0 1
5
ð35Þ
Substitute (35) into the error system (13) and (14), the following
Proof. Defining the vector eTV ¼ ½e2 e3 and substituting k1 and k2 in (17) into the system in (36), it follows that " 2 # u1 u2 k u1 T e_ V ¼ ð39Þ eV ¼ Ψ ðτ; uÞΨ ðτ; uÞeV : u22 L u1 u2 Because the switch signals u are p generated bypaffiffiffiffiffiffiffiffiffiffi simple pulse-width ffiffiffiffiffiffiffiffi modulation (PWM), J Ψ ðt; uÞ J r k=L J u J r 2k=L and T1 can be chosen as one period of the switching sequence to verify the condition in (38). Given that the conditions (37) and (38) hold, it follows from Loría and Panteley (2002) that the reduced-order system (36) is exponentially stable. Thus, Proposition 1 is proven. □ Remark 3. The proposed observer (9) is applicable to all converters that fall under the class of systems represented by (4). This class applies to a wide range of hybrid switched-affine multi-cell converter systems (see Kouro et al., 2010). 5. Simulation results The multi-rate simulation of the three-cell converter has been carried out, the simulation parameters are shown in Table 2. The multi-rate approach realizes the achievement of realistic simulation results by taking into account some implementation issues: 1. The observer evaluation rate f2 is less than the simulation rate f1 (the integration was carried out according to the Euler method). 2. The power elements switch rate f3 is less than the observer evaluation rate f2 due to switching loss. The performance of the proposed adaptive-gain SOSML observer was evaluated through multi-rate simulation. To demonstrate
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Fig. 2. Estimate of capacitor voltage V c1 ; V c2 and its errors for adaptive-gain SOSML ((a), (c), (e)) and Luenberger switched observer ((b), (d), (f)), respectively, when the system output is not affected by noise and parameter variations. (a,b) Estimate of Vc1, (c,d) Estimate of Vc2 and (e,f) Estimation of errors eVc1, eVc2.
the improvement of the proposed strategy, the results are compared with a Luenberger switched observer given in Riedinger, Sigalotti, and Daafouz (2010). Furthermore, the parameter variations are taken into account in order to demonstrate the robustness of the proposed observer, i.e., the load resistance is varied up to 50%, the value of capacitors c1 ; c2 are varied 710% and the value of inductor L is varied 720% at time t¼0.25 s, respectively. The system in (8) is rewritten in a form convenient for designing the Luenberger switched observer (Riedinger et al., 2010): 8 R E V^ c V^ c > _ > > I^ ¼ I^ þ u3 1 u1 2 u2 þ κ 0 e1 ; > > L L L L > < u1 _ : V^ c1 ¼ I þ ðκ 1 u1 þ κ 3 u2 þ κ 5 u3 Þe1 ; > c1 > > > > u > : V^_ c2 ¼ 2 I þ ðκ 2 u1 þ κ 4 u2 þ κ 6 u3 Þe1 : c2
The error dynamics of eT ¼ ½e1 e2 e3 are given by Eqs. (8) and (40) e_ ¼ ðA~ 0 þ u1 A~ 1 þ u2 A~ 2 þ u3 A~ 3 Þe;
where A~ i ¼ Ai K i C; i ¼ 0; 1; 2; 3, K T0 ¼ ðκ 0 ; 0; 0Þ; K T1 ¼ ð0; κ 1 ; κ 2 Þ; K T2 ¼ ð0; κ 3 ; κ 4 Þ; K T3 ¼ ð0; κ 5 ; κ 6 Þ. The constant gains κ 0 ; κ 1 ; κ 2 ; κ 3 ; κ 4 ; κ 5 and κ6 are chosen such that T there exists a positive matrix P~ that satisfies A~ i P~ þ P~ A~ i r 0, for i ¼ 0; 1; 2; 3. All the details of the parameters can be found in Riedinger et al. (2010). For simulation purposes, the initial values were chosen as V c1 ð0Þ ¼ 5V;
ð40Þ
ð41Þ
V c2 ð0Þ ¼ 10V;
V^ c1 ð0Þ ¼ 0V;
V^ c2 ð0Þ ¼ 0V:
ð42Þ
The parameters of the adaptive SOSML algorithm given by (15) and (16) were chosen as λ0 ¼ 100; α0 ¼ 50; kλ0 ¼ 80; kα0 ¼ 0:2 and k¼ 8 103. The parameter of the switching gains in (17) was chosen to be κ ¼2.
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Fig. 3. Estimate of capacitor voltage V c1 ; V c2 and its errors for adaptive-gain SOSML ((a), (c), (e)) and Luenberger switched observer ((b), (d), (f)), respectively, when the system output is affected by noise and parameter variations. (a,b) Estimate of Vc1, (c,d) Estimate of Vc2 and (e,f) Estimation of errors eVc1, eVc2.
Fig. 4. System output noise.
Fig. 2 shows the estimates of the capacitor voltages V c1 and V c2 and the errors when the system is not affected by output noise and without parameter variations. Both the adaptive-gain SOSML observer and the Luenberger switched observer can achieve desired performance. The estimates of the capacitor voltages V c1 ; V c2 and the errors when the system is affected by the output noise and under parameter variations are shown in Fig. 3. The system output noise was included to test the robustness of the proposed observer, that is, y ¼ I þ ξðtÞ, where ξðtÞ represents the noise (Bejarano et al., 2010). The function ξðtÞ used in the simulations is shown in Fig. 4. It is clear from the figures that the proposed observer is robust against parameter variations and the effect of the noise is essentially imperceptible. On the other hand, the Luenberger switched observer is more sensitive to the noise and the parameter variations. From Levant (1998), we know that the SOSM observer works as a robust exact differentiator, and for this reason we obtain better performance from the proposed observer
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Fig. 5. Adaptive law l(t) of the SOSML algorithm.
compared with the Luenberger switched observer. Fig. 5 shows that the adaptive law of (15) and (16) is effective under parameter variations. Remark 4. From implementation point of view, the calculations required for the adaptive-gain SOSML (9) are slightly more intensive than those of Luenberger observer. However, the correction term μðe1 Þ and two design parameters k1 ; k2 entails low real-time computational burden, as the computational capabilities of digital computers have greatly increased and the additional processing requirements can be easily accomplished (see Evangelista, Puleston, Valenciaga, & Fridman, 2013; Lienhardt, Gateau, & Meynard, 2007; Oettmeier, Neely, Pekarek, DeCarlo, & Uthaichana, 2009). As μðe1 Þ is calculated only once, regardless of the number of cells, the complexity of the calculation increases linearly with the number of cells. This means that, for an n-cell system with 2n permutations ðn 4 3Þ, the additional computational burden comes only from the calculation of new parameters k3 ; …; kn 1 . 6. Conclusions In this paper, a novel adaptive-gain SOSML observer was presented for a multi-cell power converter system, which belongs to a class of hybrid systems. With the use of Z(TN)-observability, the capacitor voltages were estimated under a certain condition of the input sequences, even though the system did not satisfy the observability matrix rank condition. That is, the system becomes observable in the sense of Z(TN)-observability after several switching sequences. The robustness of the proposed observer and the Luenberger switched observer were compared in the presence of output measurement noise and under parameter variations. It was found that the adaptive-gain SOSML observer was more robust than the Luenberger switched observer. Two main advantages of the proposed method are (1) only one parameter k has to be tuned and (2) a priori knowledge of the perturbation bounds is not required. References Amet, L., Ghanes, M., & Barbot, J. P. (2011). Direct control based on sliding mode techniques for multicell serial chopper. In American control conference (ACC) (pp. 751–756). Babaali, M., & Pappas, G. J. (2005). Observability of switched linear systems in continuous time. In Hybrid systems: Computation and control (pp. 103–117). Springer.
Bâja, M., Patino, D., Cormerais, H., Riedinger, P., & Buisson, J. (2007). Hybrid control of a three-level three-cell DC–DC converter. In American control conference (ACC) (pp. 5458–5463). IEEE. Bejarano, F., Ghanes, M., & Barbot, J.-P. (2010). Observability and observer design for hybrid multicell choppers. International Journal of Control, 83, 617–632. Bensaid, R., & Fadel, D. (2001). Sliding-mode observer for multicell converters. In Proceedings of nonlinear control systems (NOLCOS), IFAC (pp. 1396–1401). Bensaid, R., & Fadel, M. (2001). Floating voltages estimation in three-cell converters using a discrete-time Kalman filter. In 32nd annual IEEE power electronics specialists conference, PESC (pp. 327–332). Bensaid, R., & Fadel, M. (2002). Flying capacitor voltages estimation in three-cell converters using a discrete-time Kalman filter at one third switching period. In Proceedings of the American control conference (pp. 963–968). Besançon, G. (2007). Nonlinear observers and applications, Vol. 363. Springer Verlag. Defaÿ, F., Llor, A., & Fadel, M. (2008). A predictive control with flying capacitor balancing of a multicell active power filter. IEEE Transactions on Industrial Electronics, 55, 3212–3220. Defoort, M., Djemaï, M., Floquet, T., & Perruquetti, W. (2011). Robust finite time observer design for multicellular converters. International Journal of Systems Science, 42, 1859–1868. Djemaï, M., Busawon, K., Benmansour, K., & Marouf, A. (2011). High-order sliding mode control of a DC motor drive via a switched controlled multi-cellular converter. International Journal of Systems Science, 42, 1869–1882. Evangelista, C., Puleston, P., Valenciaga, F., & Fridman, L. (2013). Lyapunov-designed super-twisting sliding mode control for wind energy conversion optimization. IEEE Transactions on Industrial Electronics, 60, 538–545. Gateau, G., Fadel, M., Maussion, P., Bensaid, R., & Meynard, T. (2002). Multicell converters: Active control and observation of flying-capacitor voltages. IEEE Transactions on Industrial Electronics, 49, 998–1008. Gerry, D., Wheeler, P., & Clare, J. (2003). High-voltage multicellular converters applied to AC/AC conversion. International Journal of Electronics, 90, 751–762. Ghanes, M., Bejarano, F., & Barbot, J. P. (2009). On sliding mode and adaptive observers design for multicell converter. In American control conference (ACC) (pp. 2134–2139). Kang, W., Barbot, J., & Xu, L. (2009). On the observability of nonlinear and switched systems. In Emergent problems in nonlinear systems and control (pp. 199–216). Khalil, H. K. (2001). Nonlinear systems (3rd ed.). Prentice Hall. Kouro, S., Malinowski, M., Gopakumar, K., Pou, J., Franquelo, L. G., Wu, B., et al. (2010). Recent advances and industrial applications of multilevel converters. IEEE Transactions on Industrial Electronics, 57, 2553–2580. Levant, A. (1998). Robust exact differentiation via sliding mode technique. Automatica, 34, 379–384. Lezana, P., Aguilera, R., & Quevedo, D. (2009). Model predictive control of an asymmetric flying capacitor converter. IEEE Transactions on Industrial Electronics, 56, 1839–1846. Lienhardt, A.-M., Gateau, G., & Meynard, T. A. (2007). Digital sliding-mode observer implementation using FPGA. IEEE Transactions on Industrial Electronics, 54, 1865–1875. Loría, A., & Panteley, E. (2002). Uniform exponential stability of linear time-varying systems: Revisited. Systems & Control Letters, 47, 13–24. McGrath, B., & Holmes, D. (2007). Analytical modelling of voltage balance dynamics for a flying capacitor multilevel converter. In: Power electronics specialists conference, PESC (pp. 1810–1816). IEEE. Meradi, S., Benmansour, K., Herizi, K., Tadjine, M., & Boucherit, M. (2013). Sliding mode and fault tolerant control for multicell converter four quadrants. Electric Power Systems Research, 95, 128–139. Meynard, T., Fadel, M., & Aouda, N. (1997). Modeling of multilevel converters. IEEE Transactions on Industrial Electronics, 44, 356–364. Meynard, T., & Foch, H. (1992). Multi-level conversion: High voltage choppers and voltage-source inverters. In 23rd annual power electronics specialists conference (pp. 397–403). IEEE. Moreno, J., & Osorio, M. (2008). A Lyapunov approach to second-order sliding mode controllers and observers. In: 47th IEEE conference on decision and control (CDC) (pp. 2856–2861). IEEE. Oettmeier, F. M., Neely, J., Pekarek, S., DeCarlo, R., & Uthaichana, K. (2009). MPC of switching in a boost converter using a hybrid state model with a sliding mode observer. IEEE Transactions on Industrial Electronics, 56, 3453–3466. Perruquetti, W., & Barbot, J. P. (2002). Sliding mode control in engineering, Vol. 11. CRC Press. Rech, C., & Pinheiro, J. (2007). Hybrid multilevel converters: Unified analysis and design considerations. IEEE Transactions on Industrial Electronics, 54, 1092–1104. Respondek, W., Pogromsky, A., & Nijmeijer, H. (2004). Time scaling for observer design with linearizable error dynamics. Automatica, 40, 277–285. Riedinger, P., Sigalotti, M., & Daafouz, J. (2010). On the algebraic characterization of invariant sets of switched linear systems. Automatica, 46, 1047–1052. Rodriguez, J., Lai, J., & Peng, F. (2002). Multilevel inverters: A survey of topologies, controls, and applications. IEEE Transactions on Industrial Electronics, 49, 724–738. Slotine, J.-J. E., & Li, W. (1991). Applied nonlinear control, Vol. 1. NJ: Prentice Hall. Vidal, R., Chiuso, A., Soatto, S., & Sastry, S. (2003). Observability of linear hybrid systems. In Hybrid systems: Computation and control (pp. 526–539).
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Adaptive-gain second-order sliding mode observer design for switching power converters Jianxing Liu, Salah Laghrouche n, Mohamed Harmouche, Maxime Wack IRTES-SeT, Université de Technologie de Belfort-Montbéliard, 13 rue Thierry Mieg, 90000 Belfort, France
art ic l e i nf o
a b s t r a c t
Article history: Received 27 December 2012 Accepted 25 October 2013
In this paper, an adaptive-gain, Second Order Sliding Mode (SOSM) observer for multi-cell converters is designed by considering it as a type of hybrid system. The objective is to reduce the number of voltage sensors by estimating the capacitor voltages from measurement of the load current. The proposed observer is proven to be robust in the presence of perturbations with unknown boundaries. As the states of the system are only partially observable, a recent concept known as Z(TN)-observability is used to address the switching behavior. Multi-rate simulation results demonstrate the effectiveness and the robustness of the proposed observer with respect to output measurement noise and system uncertainty (load variations). & 2013 Elsevier Ltd. All rights reserved.
Keywords: Sliding mode observer Hybrid systems Observability Multi-cell power converter
1. Introduction In recent years, industrial applications requiring high power levels have used medium-voltage semiconductors (Gerry, Wheeler, & Clare, 2003; Meynard & Foch, 1992; Rech & Pinheiro, 2007; Rodriguez, Lai, & Peng, 2002). Because of the efficiency requirements, the power of the converter is generally increased by boosting the voltage. However, medium-voltage switching devices are not available. Even if they did exist, the volume and the cost of such devices would be substantial (Gateau, Fadel, Maussion, Bensaid, & Meynard, 2002). In this sense, the topology of multilevel converters, which have been studied during the last decade, becomes attractive for high voltage applications (Meynard & Foch, 1992). From a practical point of view, the series of a multi-cell chopper designed by the LEEI (Toulouse, France) (Bensaid & Fadel, D., 2001) leads to a safe series association of components working in a switching mode. This structure offers the possibility of reducing the voltage constraints evenly among each cell in a series. These lower-voltage switches result in lower conduction losses and higher switching frequencies. Moreover, it is possible to improve the output waveforms using this structure (Bensaid & Fadel, M., 2001; Bensaid & Fadel, 2002; Gateau et al., 2002). These flying capacitors have to be balanced to guarantee the desired voltage values at the output, which ensures that the maximum benefit from the multi-cell structure is obtained (Meynard, Fadel, & Aouda, 1997). These properties are lost if the capacitor voltage drifts far from the desired value (Bejarano, Ghanes, & Barbot,
n
Corresponding author. E-mail address: [email protected] (S. Laghrouche).
2010). Therefore, a suitable control of the switches is required to generate the desired values of the capacitor voltages. The control of switches allows the current harmonics at the cutting frequency to be canceled and the ripple of the chopped voltage to be reduced (Defoort, Djemaï, Floquet, & Perruquetti, 2011; Djemaï, Busawon, Benmansour, & Marouf, 2011). Several control methods have been proposed for multi-cell converters, such as nonlinear control based on input–output linearization (Gateau et al., 2002), predictive control (Defaÿ, Llor, & Fadel, 2008), hybrid control (Bâja, Patino, Cormerais, Riedinger, & Buisson, 2007), model predictive control (Defaÿ et al., 2008; Lezana, Aguilera, & Quevedo, 2009) and sliding mode control (Amet, Ghanes, & Barbot, 2011; Djemaï et al., 2011; Meradi, Benmansour, Herizi, Tadjine, & Boucherit, 2013). However, most of these techniques require measurements of the voltages of the capacitors to design the controller. That is, extra voltage sensors are necessary, which increases the cost and the complexity of the system. Hence, the estimation of the capacitor voltages using an observer has attracted great interest (Besançon, 2007). It should be noted that the states of the multi-cell system are only partially observable because the observability matrix never has full rank (Besançon, 2007). Hence, the observability matrix rank condition cannot be employed in an observability analysis of a hybrid system such as the one considered here (Babaali & Pappas, 2005; Vidal, Chiuso, Soatto, & Sastry, 2003). A recent concept, Z(TN)observability (Kang, Barbot, & Xu, 2009), can be used to analyze the observability of a switched hybrid system and is applied in this work because the observability of the converter depends upon the switching control signals. Various observers have been designed for the multi-cell converters based on concepts such as homogeneous finite-time observers (Defoort et al., 2011), super-twisting sliding
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mode observers (Bejarano et al., 2010; Ghanes, Bejarano, & Barbot, 2009) and adaptive observers (Bejarano et al., 2010). The concept of observability presented in Kang et al. (2009) gives the condition under which there exists a hybrid time trajectory that makes the system observable. Using this concept, estimates of the capacitor voltages can be obtained from the measurements of the load current and the source voltage by taking advantage of the appropriate hybrid time trajectories. In the works of Ghanes et al. (2009) and Bejarano et al. (2010), based on a set of p 1 linearly independent equations with respect to the voltages in the p 1 capacitors, an algorithm is proposed to estimate the capacitor voltages which employs the pseudo-inverse of a matrix whose elements are the switching signals correspondingly. In this paper, an observability analysis based on the results of Kang et al. (2009) and Bejarano et al. (2010) is performed for the multi-cell converter assuming measurements of the load current and the source voltage under certain conditions of the switching input sequences. Then, a novel adaptive-gain SOSM observer for multi-cell converters is introduced that takes into account certain perturbations (load variations) in which the boundaries of their first time derivatives are unknown. The proposed adaptive-gain SOSM algorithm combines the nonlinear term of the supertwisting algorithm (ST) and a linear term, the so-called SOSML algorithm (Moreno & Osorio, 2008). The behavior of the ST algorithm near the origin is significantly improved compared with the linear case. Conversely, the additional linear term improves the behavior of the ST algorithm when the states are far from the origin. Therefore, the SOSML algorithm inherits the best properties of both the linear and the nonlinear terms. An adaptive law of the gains of the SOSML algorithm is derived via the so-called “time scaling” approach (Respondek, Pogromsky, & Nijmeijer, 2004). The output observation error and its first time derivative converge to zero in finite time with the proposed SOSML observer such that the equivalent output-error injection can be obtained directly. Finally, the resulting reduced-order system is proven to be exponentially stable. That is, the estimates of the capacitor voltages, which are considered as the states of the observer system, converge to the real states exponentially. The main advantages of this paper are as follows:
The estimates of the capacitor voltages are obtained directly
through analyzing the information of the equivalent outputerror injection. Only one parameter of the proposed SOSML algorithm has to be tuned. There are no a priori requirements on the perturbation bounds and the finite time convergence of the output error dynamics is proven via Lyapunov analysis.
This paper is organized as follows. In Section 2, a model of the multi-cell converter and its characteristics are presented. In Section 3, the observability of the multi-cell converter is studied with the concept of Z(TN)-observability. Section 4 discusses the design of the proposed adaptive-gain SOSML observer for estimating the capacitor voltages. Section 5 gives multi-rate simulation results including a comparison with a Luenberger switched observer with disturbances.
2. Modeling of the multi-cell converter The structure of a multi-cell converter is based on the combination of a certain number of cells. Each cell consists of an energy storage element and commutators (Gateau et al., 2002). The main advantage of this structure is that the spectral quality of the output signal is improved by a high switching frequency between
Fig. 1. Multi-cell converter on RL load.
the intermediate voltage levels (McGrath & Holmes, 2007). An instantaneous model that was presented in Gateau et al. (2002) and describes fully the hybrid behavior of the multi-cell converter is used here. Fig. 1 depicts the topology of a converter with p independent commutation cells that is connected to an inductive load. The current I flows from the source E to the output through the various converter switches. The converter thus has a hybrid behavior because of the presence of both discrete variables (the switching logic) and continuous variables (the currents and the voltages). Through circuit analysis, the dynamics of the p-cell converter were obtained as in the following differential equations: 8 p 1V R E c > > > I_ ¼ I þ Sp ∑ j ðSj þ 1 Sj Þ; > > L L L > j ¼ 1 > > > > <_ I V c1 ¼ ðS2 S1 Þ; ð1Þ c 1 > > > ⋮ > > > > I > > _ > : V cp 1 ¼ c ðSp Sp 1 Þ; p where I is the load current, cj is the jth capacitor, V cj is the voltage of the jth capacitor and E is the voltage of the source. Each commutation cell is controlled by the binary input signal Sj A f0; 1g, where Sj ¼ 1 indicates that the upper switch of the jth cell is on and the lower switch is off and Sj ¼0 indicates that the upper switch is off and the lower switch is on. The discrete inputs are defined as follows: ( uj ¼ Sj þ 1 Sj ; j ¼ 1; …; p 1 ð2Þ up ¼ S p : With Eq. (2), the system (1) can be represented as follows: 8 p 1V > > > I_ ¼ R I þ E up ∑ cj uj ; > > > L L > j¼1 L > > > > I > > < V_ c1 ¼ u1 ; c1 ð3Þ > ⋮ > > > > I > > ¼ u ; V_ > > > cp 1 c p 1 p 1 > > > : y ¼ I:
Assuming that only the load current I can be measured, it is easy to represent the system (3) as a hybrid (switched affine) system: ( x_ ¼ f ðx; uÞ ¼ AðuÞx þ BðuÞ; ð4Þ y ¼ hðx; uÞ ¼ Cx; where x ¼ ½I V c1 ⋯ V cp 1 T is the continuous state vector, u ¼ ½u1 u2 ⋯ up T is the switching control signal vector which takes
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for all i o N; t i;1 ¼ t i þ 1;0 ; t 0;0 ¼ t ini and t N;1 ¼ t end .
Table 1 Switching modes and capacitor voltages for a three-cell converter. Mode: ½S1 ; S2 ; S3
V c1
V c2
u1
u2
Observable states
0: [0,0,0] 1: [0,0,1] 2: [0,1,0] 3: [0,1,1] 4: [1,0,0] 5: [1,0,1] 6: [1,1,0] 7: [1,1,1]
⇝ ⇝ ↗ ↗ ↘ ↘ ⇝ ⇝
⇝ ↗ ↘ ⇝ ⇝ ↗ ↘ ⇝
0 0 1 1 1 1 0 0
0 1 1 0 0 1 1 0
I I; V c2 I; V c1 ; V c2 I; V c1 I; V c1 I; V c1 ; V c2 I; V c2 I
BðuÞ ¼
E L up
0
⋯
0
iT
Moreover, 〈T N 〉 is defined as the ordered list of inputs u associated with TN, uii ¼ 0;N , where ui is the value of u on the interval Γi. Definition 2 (Kang et al., 2009). The function z ¼ Zðt; xÞ is said to be Z-observable with respect to the hybrid time trajectory TN and 〈T N 〉 if for any two trajectories ðt; x; uÞ and ðt; x′; u′Þ defined in ½t ini ; t end , the equality hðx; uÞ ¼ hðx′; u′Þ implies that Zðt; xÞ ¼ Zðt; x′Þ.
only discrete values and the matrices A(u), B(u), C are defined as 2 R 3 u L uL1 ⋯ pL 1 6 u1 0 ⋯ 0 7 6 c1 7 7; AðuÞ ¼ 6 6 ⋮ 7 ⋮ ⋱ ⋮ 4 5 up 1 0 ⋯ 0 cp 1 h
3
;
Lemma 1 (Kang et al., 2009). Consider the system (4) and a fixed hybrid time trajectory TN and 〈T N 〉. Suppose that z ¼ Zðt; xÞ is always continuous under any admissible control input. If there exists a sequence of projections P i ; i ¼ 0; 1; …; N, such that
for all i o N, P i Zðt; xÞ is Z-observable for t A Γ i ; rank ð½P T0 ; …; P TN Þ ¼ dimðzÞ; dP i Zðt; xÞ=dt ¼ 0, for t A Γ i , where P is the complement of P (projecting z to the variables eliminated by P),
ð5Þ
then, z ¼ Zðt; xÞ is Z-observable with respect to the hybrid time trajectory TN and 〈T N 〉.
The main objective of this paper is to design an observer based on the instantaneous model (3) that is able to estimate the capacitor voltages using only the measurement of the load current and the associated switching control input (which is assumed to be known).
Remark 1. In Lemma 1, the third condition requires that the components of Z that are not observable in Γi must remain constant within this time interval. The hybrid time trajectory TN and 〈T N 〉 influences the observability property in a way similar to an input.
C ¼ ½1 0 ⋯ 0:
3. Hybrid observability analysis From the instantaneous model of the system (4) with p Z 3, it can be noted that there are several switching modes that make the system unobservable. For instance, if u1 ¼ u2 ¼ ⋯ ¼ up 1 ¼ 0, the voltages V cj ðj ¼ 1; …; p 1Þ become completely unobservable. These switching modes are not affected by the capacitor voltages. Fortunately, these cases are the ones in which the p-cells are not switching and will not occur for all control sequences; otherwise, there is no interest in the physical sense. The observability analysis of the system (4) is based on the measurement of the load current I and the knowledge of the control input sequence u. The so-called observability matrix (Besançon, 2007) is defined as 3 2 3 2 1 0 ⋯ 0 C u 6 CA 7 6 RL uL1 ⋯ pL 1 7 7 6 7 6 6 7 6 2 7 2 6 7 6 7 p 1 Opp ¼ 6 CA 7 ¼ 6 R2 ð6Þ ui Rup 1 7: Ru1 ∑ ⋯ 7 6 7 6 L 2 2 L L 5 4 ⋮ 5 4 i ¼ 1Lci ⋮ ⋮ ⋮ ⋮ CAp 1 With simple computations, it can be shown that rankðOÞ ¼ 2 o p:
ð7Þ
It follows that the continuous states are not observable using only the load current because the observability matrix (7) is not full rank. Because of the switching sequences of the system (4), the observability is strongly linked to the hybrid behavior. Therefore, the recently developed concept of Z(TN)-observability (Kang et al., 2009) is applied to analyze the observability of the hybrid system (4). It is important to note the following definitions. Definition 1 (Kang et al., 2009). A hybrid time trajectory is a finite N or infinite sequence of intervals T N ¼ Γ i ¼ 0 such that
Γ i ¼ ½t i;0 ; t i;1 Þ, for all 0 r i oN;
Table 1 presents the eight possible configurations for a threecell converter. The application of Lemma 1 to the three-cell converter is as follows. We take Zðt; xÞ ¼ ½x2 x3 T ¼ ½V c1 V c2 T . For the discrete switching conditions ½0; 0; 0 and ½1; 1; 1, it can be verified that Zðt; xÞ is not Z(TN)-observable. Fortunately, from (1) the dynamics of V c1 and V c2 are zero, which means that these states remain constant during these time intervals. Next, assume that a trajectory of the system has the status ½1; 0; 0 and ½1; 1; 0 during time intervals Γ1 and Γ2, respectively. Let us define P 1 ¼ ½1 0 and P 2 ¼ ½0 1. We have P 1 Z ¼ x3 ¼ V c2 , dP 1 Z=dt ¼ h i P1 P2 ¼
dV c2 =dt ¼ 0, P 2 Z ¼ x2 ¼ V c1 , dP 2 Z=dt ¼ dV c1 =dt ¼ 0 and rank
2. All the assumptions in Lemma 1 are satisfied; therefore, Zðt; xÞ ¼ ½V c1 V c2 T is Z(TN)-observable. The symbols in Table 1 are defined as follows: ⇝ indicates a constant value, ↗ indicates increase and ↘ indicates decrease. In the next section, an adaptive-gain SOSML observer will be presented for the converter system (3).
4. Adaptive-gain SOSML observer design As discussed in Gateau et al. (2002), active control of the multicell converter requires the knowledge of the capacitor voltages. Usually, voltage sensors are used to measure the capacitor voltages. However, the extra sensors increase the cost, the complexity and the size, especially in high-voltage applications. Moreover, any sensors will introduce the measurement noise which will be directly transposed to the estimated value. Therefore, the design of a state observer using only the measurement of load current and the associated switching inputs is desirable. In this section, an adaptive-gain SOSML observer for the threecell converter (p ¼3) is presented that is robust to perturbations (load variations) for which the boundaries of the first time derivative are unknown. A novel adaptive law for the gains of the SOSML algorithm with only one tuning parameter is designed via the so-called “time scaling” approach (Respondek et al., 2004).
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The proposed approach does not require the a priori knowledge of the perturbation bounds. ^ the system (3) is rewritten to include the Defining e1 ¼ I I, ~ perturbation f ðe1 Þ, i.e., the load resistance uncertainty (Defoort et al., 2011) 8 R E V c1 V c2 > ~ _ > > I ¼ L I þ L u3 L u1 L u2 þ f ðe1 Þ; > > > < u1 V_ c1 ¼ I; ð8Þ c1 > > > > u2 > > : V_ c2 ¼ I: c2
zone technique (Slotine & Li, 1991) as _lðtÞ ¼ k if je1 j Z ε 0 otherwise
The proposed observer is formulated as 8 > R E V^ c V^ c _ > > I^ ¼ I þ u3 1 u1 2 u2 þ μðe1 Þ; > > > L L L L > < u1 _ ^ V c1 ¼ I þ k1 μðe1 Þ; c1 > > > > _ u2 > > ^ > : V c2 ¼ c I þ k2 μðe1 Þ; 2
Assumption 2. There is a TN such that z ¼ x ¼ ½I V c1 V c2 T is Zobservable under the condition of Lemma 1 (Defoort et al., 2011). ð9Þ
þ kα ðtÞ
0
ð10Þ
2
Eqs. (8) and (9) yield the observation error dynamics as e_ 1 ¼ μðe1 Þ
u1 u2 e2 e3 þ f~ ðe1 Þ; L L
ð12Þ
e_ 2 ¼ k1 μðe1 Þ;
ð13Þ
e_ 3 ¼ k2 μðe1 Þ:
ð14Þ
In this paper, the adaptive gains λðtÞ; αðtÞ; kλ ðtÞ and kα ðtÞ are formulated as 8 pffiffiffiffiffiffiffi > > > λðtÞ ¼ λ0 lðtÞ; > > < αðtÞ ¼ α0 lðtÞ; ð15Þ kλ ðtÞ ¼ kλ0 lðtÞ; > > > > > : kα ðtÞ ¼ kα l2 ðtÞ; 0 where λ0 ; α0 ; kλ0 and kα0 are positive constants to be defined and l(t) is a positive, time-varying, scalar function. The adaptive law of the time-varying function l(t) and the design parameters k1 and k2 are given by _lðtÞ ¼ k if je1 j a 0 ð16Þ 0 otherwise k1 ¼
κ u1
if je1 j ¼ 0
0
otherwise
0
otherwise
;
k2 ¼
where k, the initial value lð0Þ and
κ u2
if je1 j ¼ 0
0
otherwise
;
k2 ¼
κ u2
if je1 j r ε
0
otherwise
ð19Þ
ε is a sufficiently small positive value.
Assumption 1. The system (8) and the observer system (9) are bounded input, bounded state (BIBS) because this is a physical system (Perruquetti & Barbot, 2002).
Theorem 1. Consider the error system (12) under Assumptions 1 and 2. Assume that the perturbation f~ ðe1 Þ satisfies the following condition: and
f~ ð0Þ ¼ 0;
ð20Þ
4α0 kα0 4 8kλ0 α0 þ 9λ0 kλ0 : 2 2
ð21Þ
Proof. The system (12) can be rewritten as ( e_ 1 ¼ λðtÞje1 j1=2 signðe1 Þ kλ ðtÞe1 þ φ1 ; φ_ 1 ¼ αðtÞ signðe1 Þ kα ðtÞe1 þ ϱ1 ; _ where ϱ1 ¼ f~ ðe1 Þ ðu1 =LÞe_ 2 ðu2 =LÞe_ 3 .
ð22Þ
2
e 1 dτ ;
and the adaptive gains λðtÞ; αðtÞ; kλ ðtÞ; kα ðtÞ and the design parameters k1 and k2 are to be defined. Define the observation errors as ( e2 ¼ V c1 V^ c1 ; ð11Þ e3 ¼ V c V^ c : 2
if je1 j r ε
where χ 1 is an unknown positive constant. Then, the trajectories of the error system (12) converge to zero in finite time with the adaptive gains in (15) and (16) satisfying the following condition:
0
t
where
κ u1
_ jf~ ðe1 Þj r χ 1
where μðÞ is the SOSML algorithm (Moreno & Osorio, 2008) Z t μðe1 Þ ¼ λðtÞje1 j1=2 signðe1 Þ þ αðtÞ signðe1 Þ dτ þ kλ ðtÞe1 Z
k1 ¼
ð18Þ
ð17Þ
κ are the positive constants.
Remark 2. In view of practical implementation, the condition e1 ¼ 0 in (16) and (17) cannot be satisfied due to measurement noise and numerical approximations. In order to make the adaptive algorithm (16) and the switching gains (17) practically implementable, one has to modify the condition e1 ¼ 0 by a dead-
Based on Assumption 1, because the input u is bounded, the state does not go to infinity in finite time. Moreover, if I^ is bounded, all the states of the observer are also bounded for a finite time. Consequently, the observation error e1 is also bounded (Perruquetti & Barbot, 2002). It follows from (13) and (14) that e_ 2 and e_ 3 are bounded and satisfy je_ 2 j r χ 2 and je_ 3 jr χ 3 , where χ2 and χ3 are some unknown positive values. From Eq. (20), it is easy to deduce that jϱ1 jr χ 1 þ χ 2 =L þ χ 3 =L ¼ F, where F is an unknown positive value. A new state vector is introduced to represent the system in (22) in a more convenient form for Lyapunov analysis 3 2 3 2 1=2 ζ1 l ðtÞje1 j1=2 signðe1 Þ 7 7 6 7: lðtÞe1 ð23Þ ζ¼6 4 ζ2 5 ¼ 6 4 5
ζ3
φ1
Thus, the system in (22) can be rewritten as 2 k 3 2 l_ 3 2 λ0 3 λ 1 ζ 0 2 0 0 20 2lðtÞ 1 2 6 7 6 _ 7 lðtÞ 6 7 6 l ζ 7: ζ_ ¼ λ0 0 5 ζ þ lðtÞ6 4 0 k λ0 1 7 4 0 5 ζ þ 4 2lðtÞ 25 jζ 1 j 0 0 α0 0 kα0 0 ϱ1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A1
A2
ð24Þ Then, the following Lyapunov function candidate is introduced for the system in (24): Vðζ Þ ¼ 2α0 ζ 1 þkα0 ζ 2 þ 12 ζ 3 þ 12 ðλ0 ζ 1 þ kλ0 ζ 2 ζ 3 Þ2 ; 2
2
2
which can be rewritten as a quadratic form 2 2 4α 0 þ λ 0 λ 0 k λ0 λ0 16 T 2 Vðζ Þ ¼ ζ P ζ ; P ¼ 6 λ k k þ 2k kλ0 α0 λ0 24 0 λ0 λ0 k λ0 2
ð25Þ 3 7 7: 5
ð26Þ
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As (25) is a continuous Lyapunov function, the matrix P is positive definite. Taking the derivative of (26) along the trajectories of (24) _l lðtÞ T V_ ¼ ζ Ω1 ζ lðtÞζ T Ω2 ζ þ ϱ1 q1 ζ þ q2 P ζ ; lðtÞ jζ 1 j
2 6
α0 þ 2λ20
0
0
kα0 þ kλ0
Ω2 ¼ kλ0 6 4
0
2
k λ0
0
7 k λ0 7 5;
ð28Þ
1
_l λ ðΩ1 Þ 1=2 λ ðΩ2 Þ F‖q1 ‖2 1=2 V lðtÞ min V þ 1=2 V þ ΔΩ; V_ r lðtÞ min 1=2 2lðtÞ λ ðPÞ max λmax ðPÞ λmin ðPÞ ð29Þ where
ΔΩ ¼ ðð4α0 þ λ ζ
λ
2 6 6 Q ¼6 4
ζ 1 ζ 2 þ 2kα0 kλ0 ζ 2
4α0 þ λ0 þ λ0 kλ0 þ λ20 2
0
ζ2ζ3Þ r ζ Q ζ;
kλ 0 0 k λ0 þ 2
0 λ0 þ kλ0
0
T
3
0
2kα0 kλ0 þ λ 2
0
λ ζ ζ
2 2 0 1 3 kλ0
0
7 7 7: 5
For simplicity, we define F‖q1 ‖2
where γ 1 ; γ 2 ; γ 3 and be simplified as
λ
1=2 min ðPÞ
;
γ3 ¼
λmin ðΩ2 Þ ; λmax ðPÞ
γ4 ¼
λmax ðQ Þ ; 2λmin ðPÞ
ð32Þ
γ4 are all positive constants. Thus, Eq. (31) can !
_l V_ r ðlðtÞγ 1 γ 2 ÞV 1=2 lðtÞγ 3 γ 4 V : lðtÞ
ð33Þ
Because _lðtÞ Z0 such that the terms lðtÞγ 1 γ 2 and lðtÞγ 3 ð_l=lðtÞÞγ 4 are positive in finite time, it follows from (33) that V_ r c1 V 1=2 c2 V;
ð34Þ
where c1 and c2 are positive constants. By the comparison principle (Khalil, 2001), it follows that Vðζ Þ and therefore ζ converge to zero in finite time. Thus, Theorem 1 is proven. □ It follows from Theorem 1 that when the sliding motion takes place, e1 ¼ 0 and e_ 1 ¼ 0. Thus, the output-error equivalent injection μðe1 Þ can be obtained directly from Eq. (12): u L
u L
μðe1 Þ ¼ 1 e2 2 e3 :
150 V 40 μF 131 Ω 10 mH 106 Hz 100 kHz 5 kHz
There exists a constant ϕM 4 0 such that for all t Z0 and all u A D, where D A R2 is a closed, compact subset, such that J Ψ ðt; uÞ J r ϕM , where Ψ ðt; uÞ ¼
qffiffi k Lu1 ðtÞ
qffiffi
k Lu2 ðtÞ
T
:
ð37Þ
t
Ψ ðτ; uÞΨ T ðτ; uÞ dτ Z μI 4 0;
8 t Z0:
ð38Þ
2
ð31Þ
γ2 ¼
DC voltage (E) Capacitors (c1 ; c2 ) Load resistance (R) Load inductor (L) Simulation rate (f1) The sampling rate (f2) The chopping frequency (f3)
u 8 u2 1 > e2 þ e3 ; < e_ 2 ¼ k1 L L ð36Þ > e_ ¼ k u1 e þ u2 e : : 3 2 2 3 L L Proposition 1. Consider the reduced-order system (36) with the switching gains k1 and k2 given by (17). Then, the trajectories of the error system (36) converge to zero exponentially if the following two conditions are satisfied (Loría & Panteley, 2002):
t þ T1
ð30Þ
! ! _l λmax ðQ Þ λ ðΩ1 Þ F‖q1 ‖2 1=2 λmin ðΩ2 Þ V: lðtÞ V V_ r lðtÞ min 1=2 λmax ðPÞ 2lðtÞ λmin ðPÞ λ1=2 λmin ðPÞ max ðPÞ
λmin ðΩ1 Þ ; λ1=2 max ðPÞ
Values
There exist constants T 1 40 and μ 4 0 such that Z
With (30), Eq. (29) becomes
γ1 ¼
System parameters
reduced-order system is obtained:
3
it is easy to verify that Ω1 and Ω2 are positive definite matrices under the condition in (21). Because λmin ðPÞ‖ζ ‖2 r V r λmax ðPÞ‖ζ ‖2 , Eq. (27) can be rewritten as
2 2 0 Þ 1 þ2 0 kλ0
Table 2 Main parameters of simulation model.
ð27Þ
where q1 ¼ ½ λ0 kλ0 2, q2 ¼ ½ζ 1 ζ 2 0, and 2 2 3 λ0 þ 2α0 0 λ0 7 λ 6 2 Ω1 ¼ 0 6 0 2kα0 þ 5kλ0 3kλ0 7 5; 24 λ0 3kλ0 1
5
ð35Þ
Substitute (35) into the error system (13) and (14), the following
Proof. Defining the vector eTV ¼ ½e2 e3 and substituting k1 and k2 in (17) into the system in (36), it follows that " 2 # u1 u2 k u1 T e_ V ¼ ð39Þ eV ¼ Ψ ðτ; uÞΨ ðτ; uÞeV : u22 L u1 u2 Because the switch signals u are p generated bypaffiffiffiffiffiffiffiffiffiffi simple pulse-width ffiffiffiffiffiffiffiffi modulation (PWM), J Ψ ðt; uÞ J r k=L J u J r 2k=L and T1 can be chosen as one period of the switching sequence to verify the condition in (38). Given that the conditions (37) and (38) hold, it follows from Loría and Panteley (2002) that the reduced-order system (36) is exponentially stable. Thus, Proposition 1 is proven. □ Remark 3. The proposed observer (9) is applicable to all converters that fall under the class of systems represented by (4). This class applies to a wide range of hybrid switched-affine multi-cell converter systems (see Kouro et al., 2010). 5. Simulation results The multi-rate simulation of the three-cell converter has been carried out, the simulation parameters are shown in Table 2. The multi-rate approach realizes the achievement of realistic simulation results by taking into account some implementation issues: 1. The observer evaluation rate f2 is less than the simulation rate f1 (the integration was carried out according to the Euler method). 2. The power elements switch rate f3 is less than the observer evaluation rate f2 due to switching loss. The performance of the proposed adaptive-gain SOSML observer was evaluated through multi-rate simulation. To demonstrate
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Fig. 2. Estimate of capacitor voltage V c1 ; V c2 and its errors for adaptive-gain SOSML ((a), (c), (e)) and Luenberger switched observer ((b), (d), (f)), respectively, when the system output is not affected by noise and parameter variations. (a,b) Estimate of Vc1, (c,d) Estimate of Vc2 and (e,f) Estimation of errors eVc1, eVc2.
the improvement of the proposed strategy, the results are compared with a Luenberger switched observer given in Riedinger, Sigalotti, and Daafouz (2010). Furthermore, the parameter variations are taken into account in order to demonstrate the robustness of the proposed observer, i.e., the load resistance is varied up to 50%, the value of capacitors c1 ; c2 are varied 710% and the value of inductor L is varied 720% at time t¼0.25 s, respectively. The system in (8) is rewritten in a form convenient for designing the Luenberger switched observer (Riedinger et al., 2010): 8 R E V^ c V^ c > _ > > I^ ¼ I^ þ u3 1 u1 2 u2 þ κ 0 e1 ; > > L L L L > < u1 _ : V^ c1 ¼ I þ ðκ 1 u1 þ κ 3 u2 þ κ 5 u3 Þe1 ; > c1 > > > > u > : V^_ c2 ¼ 2 I þ ðκ 2 u1 þ κ 4 u2 þ κ 6 u3 Þe1 : c2
The error dynamics of eT ¼ ½e1 e2 e3 are given by Eqs. (8) and (40) e_ ¼ ðA~ 0 þ u1 A~ 1 þ u2 A~ 2 þ u3 A~ 3 Þe;
where A~ i ¼ Ai K i C; i ¼ 0; 1; 2; 3, K T0 ¼ ðκ 0 ; 0; 0Þ; K T1 ¼ ð0; κ 1 ; κ 2 Þ; K T2 ¼ ð0; κ 3 ; κ 4 Þ; K T3 ¼ ð0; κ 5 ; κ 6 Þ. The constant gains κ 0 ; κ 1 ; κ 2 ; κ 3 ; κ 4 ; κ 5 and κ6 are chosen such that T there exists a positive matrix P~ that satisfies A~ i P~ þ P~ A~ i r 0, for i ¼ 0; 1; 2; 3. All the details of the parameters can be found in Riedinger et al. (2010). For simulation purposes, the initial values were chosen as V c1 ð0Þ ¼ 5V;
ð40Þ
ð41Þ
V c2 ð0Þ ¼ 10V;
V^ c1 ð0Þ ¼ 0V;
V^ c2 ð0Þ ¼ 0V:
ð42Þ
The parameters of the adaptive SOSML algorithm given by (15) and (16) were chosen as λ0 ¼ 100; α0 ¼ 50; kλ0 ¼ 80; kα0 ¼ 0:2 and k¼ 8 103. The parameter of the switching gains in (17) was chosen to be κ ¼2.
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Fig. 3. Estimate of capacitor voltage V c1 ; V c2 and its errors for adaptive-gain SOSML ((a), (c), (e)) and Luenberger switched observer ((b), (d), (f)), respectively, when the system output is affected by noise and parameter variations. (a,b) Estimate of Vc1, (c,d) Estimate of Vc2 and (e,f) Estimation of errors eVc1, eVc2.
Fig. 4. System output noise.
Fig. 2 shows the estimates of the capacitor voltages V c1 and V c2 and the errors when the system is not affected by output noise and without parameter variations. Both the adaptive-gain SOSML observer and the Luenberger switched observer can achieve desired performance. The estimates of the capacitor voltages V c1 ; V c2 and the errors when the system is affected by the output noise and under parameter variations are shown in Fig. 3. The system output noise was included to test the robustness of the proposed observer, that is, y ¼ I þ ξðtÞ, where ξðtÞ represents the noise (Bejarano et al., 2010). The function ξðtÞ used in the simulations is shown in Fig. 4. It is clear from the figures that the proposed observer is robust against parameter variations and the effect of the noise is essentially imperceptible. On the other hand, the Luenberger switched observer is more sensitive to the noise and the parameter variations. From Levant (1998), we know that the SOSM observer works as a robust exact differentiator, and for this reason we obtain better performance from the proposed observer
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Fig. 5. Adaptive law l(t) of the SOSML algorithm.
compared with the Luenberger switched observer. Fig. 5 shows that the adaptive law of (15) and (16) is effective under parameter variations. Remark 4. From implementation point of view, the calculations required for the adaptive-gain SOSML (9) are slightly more intensive than those of Luenberger observer. However, the correction term μðe1 Þ and two design parameters k1 ; k2 entails low real-time computational burden, as the computational capabilities of digital computers have greatly increased and the additional processing requirements can be easily accomplished (see Evangelista, Puleston, Valenciaga, & Fridman, 2013; Lienhardt, Gateau, & Meynard, 2007; Oettmeier, Neely, Pekarek, DeCarlo, & Uthaichana, 2009). As μðe1 Þ is calculated only once, regardless of the number of cells, the complexity of the calculation increases linearly with the number of cells. This means that, for an n-cell system with 2n permutations ðn 4 3Þ, the additional computational burden comes only from the calculation of new parameters k3 ; …; kn 1 . 6. Conclusions In this paper, a novel adaptive-gain SOSML observer was presented for a multi-cell power converter system, which belongs to a class of hybrid systems. With the use of Z(TN)-observability, the capacitor voltages were estimated under a certain condition of the input sequences, even though the system did not satisfy the observability matrix rank condition. That is, the system becomes observable in the sense of Z(TN)-observability after several switching sequences. The robustness of the proposed observer and the Luenberger switched observer were compared in the presence of output measurement noise and under parameter variations. It was found that the adaptive-gain SOSML observer was more robust than the Luenberger switched observer. Two main advantages of the proposed method are (1) only one parameter k has to be tuned and (2) a priori knowledge of the perturbation bounds is not required. References Amet, L., Ghanes, M., & Barbot, J. P. (2011). Direct control based on sliding mode techniques for multicell serial chopper. In American control conference (ACC) (pp. 751–756). Babaali, M., & Pappas, G. J. (2005). Observability of switched linear systems in continuous time. In Hybrid systems: Computation and control (pp. 103–117). Springer.
Bâja, M., Patino, D., Cormerais, H., Riedinger, P., & Buisson, J. (2007). Hybrid control of a three-level three-cell DC–DC converter. In American control conference (ACC) (pp. 5458–5463). IEEE. Bejarano, F., Ghanes, M., & Barbot, J.-P. (2010). Observability and observer design for hybrid multicell choppers. International Journal of Control, 83, 617–632. Bensaid, R., & Fadel, D. (2001). Sliding-mode observer for multicell converters. In Proceedings of nonlinear control systems (NOLCOS), IFAC (pp. 1396–1401). Bensaid, R., & Fadel, M. (2001). Floating voltages estimation in three-cell converters using a discrete-time Kalman filter. In 32nd annual IEEE power electronics specialists conference, PESC (pp. 327–332). Bensaid, R., & Fadel, M. (2002). Flying capacitor voltages estimation in three-cell converters using a discrete-time Kalman filter at one third switching period. In Proceedings of the American control conference (pp. 963–968). Besançon, G. (2007). Nonlinear observers and applications, Vol. 363. Springer Verlag. Defaÿ, F., Llor, A., & Fadel, M. (2008). A predictive control with flying capacitor balancing of a multicell active power filter. IEEE Transactions on Industrial Electronics, 55, 3212–3220. Defoort, M., Djemaï, M., Floquet, T., & Perruquetti, W. (2011). Robust finite time observer design for multicellular converters. International Journal of Systems Science, 42, 1859–1868. Djemaï, M., Busawon, K., Benmansour, K., & Marouf, A. (2011). High-order sliding mode control of a DC motor drive via a switched controlled multi-cellular converter. International Journal of Systems Science, 42, 1869–1882. Evangelista, C., Puleston, P., Valenciaga, F., & Fridman, L. (2013). Lyapunov-designed super-twisting sliding mode control for wind energy conversion optimization. IEEE Transactions on Industrial Electronics, 60, 538–545. Gateau, G., Fadel, M., Maussion, P., Bensaid, R., & Meynard, T. (2002). Multicell converters: Active control and observation of flying-capacitor voltages. IEEE Transactions on Industrial Electronics, 49, 998–1008. Gerry, D., Wheeler, P., & Clare, J. (2003). High-voltage multicellular converters applied to AC/AC conversion. International Journal of Electronics, 90, 751–762. Ghanes, M., Bejarano, F., & Barbot, J. P. (2009). On sliding mode and adaptive observers design for multicell converter. In American control conference (ACC) (pp. 2134–2139). Kang, W., Barbot, J., & Xu, L. (2009). On the observability of nonlinear and switched systems. In Emergent problems in nonlinear systems and control (pp. 199–216). Khalil, H. K. (2001). Nonlinear systems (3rd ed.). Prentice Hall. Kouro, S., Malinowski, M., Gopakumar, K., Pou, J., Franquelo, L. G., Wu, B., et al. (2010). Recent advances and industrial applications of multilevel converters. IEEE Transactions on Industrial Electronics, 57, 2553–2580. Levant, A. (1998). Robust exact differentiation via sliding mode technique. Automatica, 34, 379–384. Lezana, P., Aguilera, R., & Quevedo, D. (2009). Model predictive control of an asymmetric flying capacitor converter. IEEE Transactions on Industrial Electronics, 56, 1839–1846. Lienhardt, A.-M., Gateau, G., & Meynard, T. A. (2007). Digital sliding-mode observer implementation using FPGA. IEEE Transactions on Industrial Electronics, 54, 1865–1875. Loría, A., & Panteley, E. (2002). Uniform exponential stability of linear time-varying systems: Revisited. Systems & Control Letters, 47, 13–24. McGrath, B., & Holmes, D. (2007). Analytical modelling of voltage balance dynamics for a flying capacitor multilevel converter. In: Power electronics specialists conference, PESC (pp. 1810–1816). IEEE. Meradi, S., Benmansour, K., Herizi, K., Tadjine, M., & Boucherit, M. (2013). Sliding mode and fault tolerant control for multicell converter four quadrants. Electric Power Systems Research, 95, 128–139. Meynard, T., Fadel, M., & Aouda, N. (1997). Modeling of multilevel converters. IEEE Transactions on Industrial Electronics, 44, 356–364. Meynard, T., & Foch, H. (1992). Multi-level conversion: High voltage choppers and voltage-source inverters. In 23rd annual power electronics specialists conference (pp. 397–403). IEEE. Moreno, J., & Osorio, M. (2008). A Lyapunov approach to second-order sliding mode controllers and observers. In: 47th IEEE conference on decision and control (CDC) (pp. 2856–2861). IEEE. Oettmeier, F. M., Neely, J., Pekarek, S., DeCarlo, R., & Uthaichana, K. (2009). MPC of switching in a boost converter using a hybrid state model with a sliding mode observer. IEEE Transactions on Industrial Electronics, 56, 3453–3466. Perruquetti, W., & Barbot, J. P. (2002). Sliding mode control in engineering, Vol. 11. CRC Press. Rech, C., & Pinheiro, J. (2007). Hybrid multilevel converters: Unified analysis and design considerations. IEEE Transactions on Industrial Electronics, 54, 1092–1104. Respondek, W., Pogromsky, A., & Nijmeijer, H. (2004). Time scaling for observer design with linearizable error dynamics. Automatica, 40, 277–285. Riedinger, P., Sigalotti, M., & Daafouz, J. (2010). On the algebraic characterization of invariant sets of switched linear systems. Automatica, 46, 1047–1052. Rodriguez, J., Lai, J., & Peng, F. (2002). Multilevel inverters: A survey of topologies, controls, and applications. IEEE Transactions on Industrial Electronics, 49, 724–738. Slotine, J.-J. E., & Li, W. (1991). Applied nonlinear control, Vol. 1. NJ: Prentice Hall. Vidal, R., Chiuso, A., Soatto, S., & Sastry, S. (2003). Observability of linear hybrid systems. In Hybrid systems: Computation and control (pp. 526–539).
Please cite this article as: Liu, J., et al. Adaptive-gain second-order sliding mode observer design for switching power converters. Control Engineering Practice (2013), http://dx.doi.org/10.1016/j.conengprac.2013.10.012i