Adaptive Observer for DC Voltages in a cascaded H-bridge multilevel converter

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Adaptive Observer for DC Voltages in a cascaded H-bridge multilevel converter J. de Le´ on Morales M. T. Mata-Jim´ enez M. F. Escalante Facultad de Ingenier´ıa Mec´ anica y El´ectrica, Universidad Aut´ onoma de Nuevo Le´ on, Av. Universidad s/n; Ciudad Universitaria, C.P. 66451, San Nicol´ as de los Garza, Nuevo Le´ on, M´exico Abstract: Using an instantaneous model of a cascaded H-bridge multilevel converter a nonlinear adaptive observer is proposed. The model of the system is arranged as a set of subsystems, in order to take advantage of the converter structure, for designing the observer. This observer scheme can be used to replace the voltage sensors needed to control and monitoring the system capacitor voltages even in the presence of system parameter uncertainties. Simulation results are used to validate the performance of the proposed observer when the multilevel converter is used as an electrical network compensator. Keywords: Nonlinear observer, Multilevel converter, Estimation, Multilevel STATCOM 1. INTRODUCTION Power electronics based converters are nowadays an essential technology to control and conditioning electrical energy. Advancements in areas as power semiconductors, control electronics or control theory, have enabled new applications or new power converter topologies. To handle high power levels the classical 2-level inverter topology can be replaced by a new breed of power converters called multilevel converters [Lai and Peng 1996, Rodr´ıguez et al. 2010]. Among the various multilevel topologies, cascaded H-bridge multilevel converters are very attractive for electrical network compensation [Peng and Lai 1995, Peng et al. 1998]. It offers a modular and simple topology, composed of several cascaded 2-level H-bridges per phase, in which the DC-side is assured by a capacitor. Proper and safe operation of this type of multilevel power converters requires the voltage across the capacitors to be kept around a nominal voltage level. Capacitor voltage control can be obtained by a feedback control loop using voltage sensors. In one hand, using voltage sensors to measure the capacitor voltages is a relative straightforward approach. In the other hand, as the number of cascaded H-bridges increases it will be necessary to increase the number of voltage sensors and the associated conditioning electronics, resulting in an increased hardware complexity, which will impact negatively the system reliability. As an alternative to physical voltage sensors are the software based sensors, known in the literature as observers. Observers are computational algorithms that take advantage of the mathematical model of the system to reconstruct the system states from the inputs and some measured outputs [Nijmeijer and Fossen 1999, Besancon et al. 2003, Djemai et al. 2005]. Using observers to measure the capacitor voltages will reduce the number of physical sensors reducing system costs and increasing the system reliability as the number of system components is reduced. The estimation 1 This work was supported by CONACyT M´ exico. Project CONACyT Ciencia B´ asica 105799.

978-3-902661-93-7/11/$20.00 © 2011 IFAC

of system states or system parameters using observers has recently been proposed in several applications related to electromechanical and electrical power conversion systems [Lin and Cheng 2010, Kenn et al. 2010, Leon and Solsona 2010]. Moreover, observer are becoming standard in applications such as high performance motor drives, as its use have proved to enhance drive performance and reliability [Lascu et al. 2009, Khalil et al. 2009]. Even though the aforementioned applications of observers are all related or linked to the use of power converters, their main objective is not the observation of the converter states by itself but the observation of variables related to global system performance. As the complexity of power converters has evolved, new requirements in terms of the converter itself have emerged. In the case of multilevel converters, it is required to control and monitor some of the converter internal states, and specifically the voltages across the capacitors being used in the multilevel power structure. Recently, some proposals regarding those problems have been presented [Lienhardt et al. 2005, Almaleki and Jon 2010]. This paper proposes a nonlinear adaptive observer in order to estimate the capacitor voltages in a cascaded H-bridge multilevel inverter in spite of system parameter uncertainties. This approach uses an instantaneous model to reconstruct the missing state information and system parameters which are not available from sensors or whose values are known with uncertainties. The system model is represented as a set of n interconnected subsystems for which it will be possible to design an observer taking advantage of the power converter structure. The paper is organized as follows. In Section 2, the model of the cascaded H-bridge multilevel inverter is presented. Then, the observer design is given in Section 3. In Section 4, simulation results are given in order to illustrate the performance of the proposed observer to estimate the Hbridge capacitors voltages. Finally, some conclusions are given.

2797

10.3182/20110828-6-IT-1002.03669

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

If a capacitor is used at the DC side of each H-bridge, then the capacitor voltage for the j-th bridge vcj can be d described by dt vcj = C1j i(t)Sj where Cj is the capacitance of the j-th H-bridge. Thus, the inverter phase voltage vo (t)

H-Bridge1

vo1

Vdc1

H-Bridgej H-Bridgej

voj vo

Vdcj

Vdc

j

T1j

T3j

T2j

T4j

lc

rc

i(t) voj

vnet

vo (t) H-Bridgen

Vdcn

von

Fig. 1. One leg of a three phase cascaded H-bridge multilevel structure. (a) Serial connection of H-bridges and (b) Single H-bridge 2. MODELLING OF CASCADED H-BRIDGE MULTILEVEL INVERTER

voj (t)

(1)

j=1

If the DC side of each H-bridge has a value of Vdc volts, then the output voltage of each individual H-bridge can take 3 different values : −Vdc , 0 or +Vdc , as a function of the switching states, and thus v0j ∈ {−Vdc, 0, +Vdc }. Taking into account that each H-bridge inverter-leg is composed of a series connected pair of switches, (T1j , T2j ) and (T3j , T4j ), those switches must be commanded in a complementary way. Following this rule, it is possible to define a switching function, Sj , to describe the output voltage of each j th H-bridge. This switching function can take 3 different values: −1, 0 or 1, and is directly related to the switching states. In Table 1, the switching function, T1j OFF OFF ON ON

T3j OFF ON OFF ON

Sj 0 -1 1 0

Table 1. Switching function Sj , as a function of the switching states is shown. It is worth noting that the given switching function can be used to describe the H-bridge output voltage for both bipolar and unipolar modulation, because it fully describes the switching states generated by both types of modulation. Introducing the switching function in (1), then it follows that: n X (2) Vdcj Sj vo (t) = j=1

can be expressed as: vo (t) =

Cascaded H-bridge multilevel inverters are based on serial connection of single phase inverters bridges. A single phase-leg is shown in Fig. 1. As shown, each inverter bridge is supplied by a separate and isolated DC power supply. When it is used to compensate reactive power and harmonics in electrical networks, a simple charged capacitor can provide the needed voltage in the DC side of the j-th inverters. The output voltage of a cascaded H-bridge inverter-leg is obtained by adding the single Hbridge output voltages as follows: n X vo (t) =

Fig. 2. One phase multilevel inverter and electrical network equivalent circuit n X

vcj (t)Sj

(3)

j=1

The equivalent circuit for one phase when the multilevel inverter is connected to the electrical network is shown in Fig. 2. Furthermore, the phase current i(t), as represented in Fig. 2, is given by: d rc 1 i(t) = − [vo (t) − vnet (t)] − i(t) dt lc lc

(4)

where vnet (t) is the network phase voltage, lc and rc are the equivalent inductance and resistance of the linker transformer or reactor, respectively. Now, using (2), (3) and (4), the dynamical model describing the behavior of the j H-bridges connected per phase is given by:  n  1 rc 1 X d  vcj (t)Sj + vnet − i(t) i(t) = −  ΣN :

dt

lc

j=1

   d vcj (t) = 1 i(t)Sj dt

Cj

lc

lc

(5)

for i = 1, . . . , n

The resulting system is an (n + 1)-dimensional nonlinear system depending on unknown parameters rc and Cj , where the output is given by y = i(t) and Sj for j = 1, . . . , n; are the n inputs of the system. An observer is proposed in order to estimate the capacitor capacitance values and voltages of the cascaded H-bridge multilevel inverter without using physical sensors, in addition an adaptive scheme is used for providing an estimate of the equivalent impedance of the linker reactor. Furthermore, assuming that the current i(t) is the only measurable variable of the system (6), from the observability rank condition Isidori [1995], it follows that dim(dO) = 2, where dO is the space of differential elements of the observation space O. It is clear that dO is not of full rank, i.e. the system is not observable. In order to overcome this difficulty a convenient representation of the model is proposed and a methodology taking advantage of this representation is developed in the next section. 3. ADAPTIVE OBSERVER DESIGN In this section an alternative representation of model (6) will be considered, i.e. the original model will be split into a suitable set of n subsystems for which it is possible to design an observer for estimating the capacitor voltages vcj and the capacitance values Cj for j = 1, . . . , n.

2798

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

In order to estimate the capacitance values the state vector can be extended by the vector of constant parameters d dt Cj = 0, introducing the capacitance dynamics, and the state affine form is preserved:  n  1 d 1 X rc   i(t) = − vcj (t)Sj + vnet − i(t)   dt lc lc lc   j=1   d −1 v (t) = C

i(t)Sj

c j dt j  d −1   C = 0   dt j    

y

for j = 1, . . . , n

Real

ΣN LSystem

O1

= i(t)

O2

      

On

Σj :

(7)

y = Cχj

j = 1, . . . , n

where τ stands for rc /lc which is uncertain; χj = (i, vcj , Cj−1 )T represents the extended state of the j-th subsystem, Sj , j = 1, . . . , n; are the instantaneous inputs applied to the system and y = i(t) is the measurable output; with     0 −Sj /lc

0

B=

0 1 0 0

0

−i

  Φ(i) =  0   

 Sj i  

0

!

0

,

C=

0

1 0 0




Zn

and ϕ(S¯j , χ ¯j , vnet ) = −

1 1 vnet − lc lc

n X

vcm Sm

m=1 m6=j

where S¯j = (S1 , . . . , Sj−1 , Sj+1 , . . . , Sn ) and χ ¯j (vc1 , . . . , vcj−1 , vcj+1 , . . . , vcn ).

=

The function ϕ(S¯j , χ ¯j , vnet ) is the interconnection term depending on inputs and states of each subsystem. Notice that the output is the current i(t) and is the same for each subsystem. The new set of represented subsystems will be used to design an observer for the n capacitor voltages and capacitance values. Then, the following system  ˙ ¯j ) Zj = A(Sj , i)Zj + Φ(i)τj + Γj (S¯j , Z   −1 T T −1 T  +{Λ R Λ C + P C }(y − CZj ) j j  j j      T T ˙    Pj = −θj Pj − A (Sj , i)Pj − Pj A(Sj , i) + C C T τˆ˙j = R−1  j Λj C (y − CZj )      ˙ j = {A(Sj , i) − P −1 C T C}Λj + Φ(i)  Λ  j     T T

R˙ j = −ρj Rj + Λj C CΛj

Fig. 3. Observer block diagram is an adaptive observer for the system (7) for j = 1, . . . , n where Λj Rj−1 ΛTj C T + Pj−1 C T is the gain of the observer which depends on the solution of the Riccati equation and the adaptation dynamics (8) for each subsystem, Zj = (ˆı, vˆcj , Cˆj ), and, Z¯j = (ˆvc1 , . . . , vˆcj−1 , vˆcj+1 , . . . , vˆcn , Cˆ1−1 , . . . , −1 ˆ −1 , C ˆ −1 , . . . , C ˆn C ). Furthermore, the estimation error j−1 j+1 εj = Zj − χj converges to zero as t tends to infinite. The parameter θj > 0 is called the forgetting factor determinating the convergence rate of the observer and ρj is the adaptation gain. It is clear that the observability of the system depends on the applied input. Then, the convergence of this observer can be proved assuming that the inputs Sj are regularly persistent, i.e. it is a class of admissible inputs that allows to observe the system (for more details see [Besancon et al. 2003]). This guarantees the observer works and the observer gain is well-defined., i.e. the matrix Pj is nonsingular.

;

Γj (S¯j , χ ¯j ) = Bϕ(S¯j , χ ¯j , vnet )

Oj :

Observer Subsystemn

y = i(t)

Each subsystem can be written in a state affine form as: ( χ˙ j = A(Sj , i)χj + Φ(i)τ + Γj (S¯j , χ ¯j )

 A(Sj , i) =  0 

Z2

Observer Subsystem2

j

1 d vc = Sj i(t) dt j Cj d −1 C = 0 dt j

Z1

(6)

for j = 1, . . . , n

Then, the system (6) can be split in n subsystems of the form:  d 1 1 rc  i(t) = − i(t) − Sj vcj − (S2 vc2 + · · · + Sn vcn − vnet )   dt lc lc lc   {z } |   Γ

Σj :

y

Observer Subsystem1

Interconnected observer

ΣE :

S

Now, a further result based on regular persistence is introduced. Lemma 1. Assume that the input Sj is regularly persistent for system (7) and consider the Lyapunov differential equation P˙j = −θj Pj − AT (Sj , i)Pj − Pj A(Sj , i) + C T C with Pj (0) > 0. Then, ∃θj0 > 0 such that for any symmetric positive definite matrix Pj (0), ∃θj ≥ θj0 , ∃αj , βj > 0, t0 > 0 : ∀t > t0 , αj I < Pj (t) < βj I where I is the identity matrix. The proof of this lemma follows the same steps of Theorem 3.2 in Khalil [1996].

(8)

Now, consider that the system ΣN Le can be represented as a set of the interconnected subsystems as follows:   χ˙ 1 = A(S1 , i)χ1 + Φ(i)τ + Γ1 (S¯1 , χ¯1 ) Σ:



χ˙ n

.. . = A(Sn , i)χn + Φ(i)τ + Γn (S¯n , χ ¯n )

(9)

In general, if each Oj is an exponential observer for Σj for j = 1, . . . , n, then the following interconnected system O 2799

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

 ˙ Z1             Z˙ n         P˙1           P˙n τˆ˙1

O:

       τˆ˙n   ˙1  Λ          ˙n Λ    ˙1  R       

R˙ n

= A(S1 , i)Z1 + +Φ(i)τ1 + Γ1 (S¯1 , Z¯1 ) −1 T 1 T +{Λ1 R−1 1 Λj C + P1 C }(y − CZ1 ) . .. ¯n ) = A(Sn , i)Zn + Φ(i)τn + Γn (S¯n , Z T T −1 T +{Λn R−1 n Λn C + Pn C }(y − CZn ) = −θ1 P1 − AT (S1 , i)P1 − P1 A(S1 , i) + C T C . .. = −θn Pn − AT (Sn , i)Pn − Pn A(Sn , i) + C T C T = R−1 1 Λ1 C (y − CZ1 ) .. . T = R−1 n Λn C (y − CZn ) = {A(S1 , i) − P1−1 C T C}Λ1 + Φ(i) .. . = {A(Sn , i) − Pn−1 C T C}Λn + Φ(i) T = −ρ1 R1 + ΛT 1 C CΛ1 .. . T = −ρn Rn + ΛT n C CΛn

ε˙j = {A(Sj , i) − Pj−1 C T C}εj + ∆Γj (S¯j , χ ¯j , Z¯j ) T T C C(ε + Λ σ ) σ˙ j = −R−1 Λ j j j j j

Next, from Lemma 1 and Assumption 1, let be V =

(10)

εj Pj 0 = ψjT Qj ψj σj | {z } | 0 {zRj } | {z } T ψj

Qj

ψj

is a Lyapunov function for each subsystem Σj . Taking the time derivative of Vj (ψj ), it follows that V˙ j (ψj ) = −θj εj Pj εj − ρj σjT Rj σj + 2εT j Pj ∆Γj −(εj + Λj σj )T C T C(εj + Λj σj )

Since −(εj + Λj σj )T C T C(εj + Λj σj ) ≤ 0

it follows that T T V˙ j (ψj ) ≤ −θj εT j Pj εj − ρj σj Rj σj + 2εj Pj ∆Γj

The following result can be established. Lemma 2. Considering the interconnected system Σ given by (9) and if assumptions 1 and 2 are satisfied. Then, the system O (10) is an observer for system Σ (9). Proof. Defining the dynamics of the estimation error, ǫj = Zj −χj and the dynamics of the parameter adaptation σj = τˆj − τj , we obtain −1 −1 T T ǫ˙j = {A(Sj ) − ΛT j Rj Λj C C − Pj C C}ǫj + Φ(i)σj ¯ ¯ +∆Γj (Sj , χ ¯ j , Zj )

Then, the above expression can be written as ¯ ¯ V˙ j (ψj ) ≤ −ϑj Vj (ψj ) + 2εT j Pj ∆Γ h j (Sj , ¯χ¯j , Zj )¯ i ∆Γ ( S , χ ¯ j j j , Zj ) = −ϑj Vj (ψj ) + 2ψjT Qj 0

|

{z

}

¯j ,χ ¯j ) ∆Γj (S ¯ j ,Z

where ϑj = min (θj , ρj ), ∆Γj (S¯j , χ¯j , Z¯j ) = B ϕ¯j (ε), ϕ¯j (ε) = n P − l1 Sl εvcl , and εvcl = vcl − vˆcl . c l=1 l6=j

Now, adding and subtracting the term we have

∆Γj (S¯j , χ ¯j , Z¯j )T Pj ∆Γj (S¯j , χ ¯j , Z¯j ),

¯j , Z¯j ) V˙ j (ψj ) ≤ −ϑj Vj (ψj ) + 2ψjT Qj ∆Γj (S¯j , χ T ¯ ¯ ¯ ¯j , Zj ) Qj ∆Γj (Sj , χ ¯j , Z¯j ) ±∆Γj (Sj , χ

Next, regrouping the appropriate terms V˙ j (ψj ) ≤ −(ϑj − 1)kψj k2Qj − kψj k2Qj ¯j )k2 +2ψT Qj ∆Γj (S¯j , χ ¯j , Z¯j ) − k∆Γj (S¯j , χ ¯j , Z j

Qj

¯j )k2 +k∆Γj (S¯j , χ ¯j , Z Qj

It follows that ¯j )k2 ¯j , Z V˙ j (ψj ) ≤ −(ϑj − 1)kψj k2Qj + k∆Γj (S¯j , χ Qj

Now, from Assumption 2, the following inequality holds n P λl kψl k2Q , we get ¯j , Z¯j )k2Q ¡ k∆Γj (S¯j , χ j j l=1 l6=j

V˙ j (ψj ) ≤ −(ϑj − 1)kψj k2Qj +

n X

λl kψl k2Qj

(11)

l=1 l6=j

Then, the time derivative of V is given by n X V˙ (ψ) =

V˙ j (ψj )

j=1,

T T −R−1 j Λj C Cǫj

where ∆Γj (S¯j , χ¯j , Z¯j ) = Γj (S¯j , Z¯j ) − Γj (S¯j , χ¯j ), for j = 1, . . . , n. Following the same idea as in Zhang [2002], introducing the following change of variable εj = ǫj − Λj σj

a Lyapunov function for the interconnected system Σ, where ih i h T Vj (ψj ) = [εT j σj ]

Now, we will give the sufficient conditions which ensure the convergence of the interconnected observer O. For that, we introduce the following assumptions: Assumption 1. Assume that the input Sj is a regularly persistent input for subsystem Σj and admits an exponential observer Oj for j = 1, . . . , n. In this case, an observer of the form (10) can be designed and the estimation error will be bounded. Assumption 2. The term Γj (S¯j , χ ¯j ) does not destroy the observability property of subsystem Σj under the action of the regularly persistent input Sj . Moreover, Γj (S¯j , χ ¯j ) is Lipschitz with respect to χ ¯j and uniformly with respect to Sj for j = 1, . . . , n. Remark 1. This assumption does not state how to generate the input Sj which ensures the realization of this assumption. In fact, up to our knowledge, excepting some particular cases (linear systems), the problem characterizing the set of inputs ensuring the persistent excitation condition is still open.

yields

Vj

j=1

is an adaptive observer for the interconnected system Σ (See Fig. 4). The main idea of this approach is to construct an observer for the whole system Σ from the separate observer design of each subsystem Σj .

σ˙ j =

n P

Replacing the expression (11) in the above equation, it follows    n  n   X X V˙ (ψ) ≤

 j=1, 

−(ϑj − 1)kψj k2Qj +

λl kψl k2Qj

l=1 l6=j

.

 

Taking into account that the inputs are regularly persistent, then, from Lemma 1, the matrices Qj are bounded. 2800

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

Using the lemma on equivalence of norms, i.e. it exists a positive constant µl such that kεl k2Ql ≤ µl kεl k2Qj

Then, it follows that  n  X V˙ (ψ) ≤

n

−(ϑj −

j=1,

or

 

1)kψj k2Qj

+

X

λl µl kψl k2Ql

l=1 l6=j

n

V˙ (ψ) ≤ −

X

Invertercontrol signals

∀l = 1, . . . , n.

{(ϑj − 1) − (n − 1)λl µl )} kψj k2Qj

  

. Multilevel PWM

Finally, we have

V (ψ) ≤ V (ψ(t0 ))e−γ(t−t0 )

1

Nonlinear Load 3

iload

vc4

Vcm3

vd* vq*

PIcurrent controllers

vphase

Compensation current calculation 4

5 d-q-0

−γ(t−t0 )

Linear Load 3

2

icomp

a-b-c

for γ = min (γ1 , ..., γn ) where γj = (ϑj − 1) − (n − 1)λj µj . Taking ψ = col(ψ1 , ..., ψn ), it is easy to see that kψ(t)k ≤ Kkψ(t0 )ke

vnet

7

j=1

Vcm2

icomp

Interconnected Observer vc1

6 Vcm1

lc

Multilevel Inverter 4H-bridges perphase

 

rc

Cascaded H-Bridge

id* i*q

This ends the proof. In the next section we validate the performance of the adaptive observer by simulation. In order to use the proposed observer schema a system verifying the assumptions 1 and 2 is considered. 4. SIMULATION RESULTS The proposed observer is validated when the multilevel power converter is used to compensate for reactive power and current harmonics in an electrical network as described in Escalante and Arellano [2006]. The complete system is shown in Fig. 4. Each multilevel inverter leg has 4 H-bridges per phase (Block 1). A capacitor Cn = 50mF (nominal value) per bridge was used. The capacitors are pre-charged at 1.0 kV before normal operation starts. The compensator is coupled to a 2.4 kV (phase voltage) utility network through a link reactor (Block 2), with Rcn = 0.1Ω serial resistance and Lcn = 0.5 mH of inductance. Simulated load is a combination of linear and nonlinear elements (Block 3). For the simulations, parameter values are deviated from its nominal values as follows: C1 = 1.2Cn , C2 = 1.1Cn , C3 = 0.8Cn , C4 = 0.95Cn , Rc = 1.3Rcn and Lc = 1.5Lcn. Introduced deviations are intended to show the observer performance in presence of parameter uncertainties. The switching frequency was fixed at 5 kHz. Simulations were carried out using PSIM from Powersim and MATLAB/SIMULINK from Mathworks, working in co-simulation mode. The observer gains are given by θj = 20 and ρj = 0.1 for j = 1, . . . , n. Figure 5 shows the estimation errors, as shown, observer values follows the real capacitor voltages accurately. For any initial condition, the observer is able to follow the real capacitor voltages accurately. Once the observer transient is over, the estimated voltages are always close to the real values, as shown by the respective calculated errors. This test considered non-nominal system parameter as it is indicated in the figure title. Then, Fig. 6 shows the compensator line-to-line voltage and phase current, here the multilevel compensator nature can be appreciated.

Electrical Network

Fig. 4. Block diagram of a cascaded H-bridge multilevel inverter based compensator and interconnected observer. As shown, the observer keeps tracking of the real capacitor voltages, but its performance is similar to that with nonnominal parameter. This shows that the proposed observer is capable of working properly even with large parameters uncertainties or unknown parameter values. 5. CONCLUSIONS As the power converter evolves toward a more complex and sophisticated structures, new needs in terms of control and monitoring are emerging. The control and monitoring of the voltage of the capacitors in a Cascaded H-bridge multilevel converter is essential for proper operation of the converter. Measuring those voltages becomes expensive and impractical because the high voltages and power levels handled in such applications. Thus the advantages of using an observation technique becomes evident. In this paper, a nonlinear adaptive observer was proposed to estimate the capacitor voltages in a cascaded H-bridge multilevel converter. Proposed observer uses an instantaneous model of the system to estimate the system states even in the presence of parameter uncertainties. This capability has been demonstrated by simulation results, in which the estimation error remains small, at values that are sufficiently accurate for being used for purposes of monitoring and control of the power multilevel structure. Thus, using this type of software sensors, the capacitor voltage sensor can be replaced, which will simplify the whole power system, increasing reliability and reducing system costs. REFERENCES Almaleki, M. and Jon, P. (2010). Sliding mode observation of capacitor voltage in multilevel power converters. In 5th IET Int. Conf. on Power Electronics, Machines and Drives (PEMD 2010), 1–6. Brighton, UK. Besancon, G., de Leon-Morales, J., and Huerta-Guevara, O. (2003). On adaptive observers for state affine systems and application to

2801

Line−to−line voltage (V), phase current, (A)

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

100 90

Estimation error for vc1, (V)

80 70 60 50 40 30 20

Line−to−line voltage Phase current

6000 4000 2000 0 −2000 −4000 −6000

10 0

0.845 0

1

2

3

4

5

0.85

0.855

0.86 0.865 Time(s)

0.87

0.875

0.88

Time(s)

Fig. 6. Cascaded H-bridge multilevel inverter line-to-line voltage and phase current.

100 90

Estimation error for vc2, (V)

80 70 60 50 40 30 20 10 0

0

1

2

3

4

5

3

4

5

3

4

5

Time(s)

100 90

Estimation error for vc3, (V)

80 70 60 50 40 30 20 10 0

0

1

2 Time(s)

100 90

Estimation error for vc4, (V)

80 70 60 50 40 30 20 10 0

0

1

2 Time(s)

(b) Fig. 5. (a) Real and estimated capacitor voltages (b) Estimation errors (Considering non-nominal parameters values: C1 = 1.2Cnom , C2 = 1.1Cnom , C3 = 0.8Cnom , C4 = 0.95Cnom , Cnom = 50mF , Rc = 1.3Rcn , Lc = 1.5Lcn ).

synchronous machines. In 42nd Conf. on Decision and Control, 1009–1015. IEEE, USA. Djemai, M., Manamanni, N., and Barbot, J.P. (2005). Sliding mode observer for triangular input hybrid system. In IFAC World Congress. Escalante, M. and Arellano, J. (2006). Harmonics and reactive power compensation using a cascaded h-bridge multilevel inverter. In IEEE Int. Symp. on Ind. Electronics, 1966 – 1971. Montreal, CA. Isidori, A. (1995). Nonlinear control systems. Springer Verlag. Kenn, G., Simo, R., Lamnabhi-Lagarrigue, F., Arzand, A., and Vannier, J.C. (2010). An online simplified rotor resistance estimator for induction motors. IEEE Trans Control Sys Technol, 18(5), 1188–1194. Khalil, H. (1996). Nonlinear systems. Prentice Hall, 2 edition. Khalil, H., Strangas, E., and Jurkovic, S. (2009). Speed observer and reduced nonlinear model for sensorless control of induction motors. IEEE Tran Control Sys Technol., 17(2), 327 – 339. Lai, J.S. and Peng, F.Z. (1996). Multilevel converters - a new breed of power converters-. IEEE Trans. on Industry Applications, 32(3), 509–517. Lascu, C., Boldea, I., and Blaabjerg, F. (2009). A class of speedsensorless sliding-mode observers for high-performance induction motor drives. IEEE Trans. on Ind. Electr., 56(9), 3394 – 3403. Leon, A. and Solsona, J. (2010). Design of reduced-order nonlinear observers for energy conversion applications. Control Theory and Applications, IET, 4(5), 724 – 734. Lienhardt, A., Gateau, G., and Meynard, T. (2005). Stacked multicell converter (smc): estimation of flying capacitor voltages. In 2005 European Conference on Power Electronics and Applications, 1–10. Dresden, Germany. Lin, Y.W. and Cheng, J.W.J. (2010). A high-gain observer for a class of cascade-feedback-connected nonlinear systems with application to injection molding. IEEE Trans. on mech., 15(5), 714–727. Nijmeijer, H. and Fossen, T.I. (1999). New directions in nonlinear observer design. Number 244 in Lecture Notes in Control and Information Sciences. Springer-Verlag. Peng, F.Z. and Lai, J.S. (1995). A multilevel voltage-source inverter with separate DC sources for static var generation. In Conf. Rec. IEEE-IAS Annu. Meeting, 2541–2548. Lake Buena Vista, FL. Peng, F.Z., McKeever, J.W., and Adams, D.J. (1998). A power line conditioner using cascade multilevel inverters for distribution systems. IEEE Trans. on Ind. Applications, 34(6), 1293–1298. Rodr´ıguez, J., Bernet, S., Steimer, P., and Lizama, I. (2010). A survey on neutral-point-clamped inverters. IEEE Trans. on Industrial Electronics, 57(7), 2219–2230. Zhang, Q. (2002). Adaptive observer for multiple-input-multiple output (mimo) linear time-varying systems. IEEE Trans. on Automatic Control, 47(3), 525 –529.

2802