Adaptive Sliding Mode Observer Design for Switching Power Converters

Adaptive Sliding Mode Observer Design for Switching Power Converters Jianxing.Liu ∗ S.Laghrouche ∗ M.Harmouche ∗ M.Wack ∗ ∗

Universit´e de Technologie de Belfort-Montb´eliard (UTBM), Belfort, France (e-mail: [email protected]).

Abstract: In this paper, a nonlinear adaptive super-twisting(STW) sliding mode observer is proposed for a three-cell converter. The aim is to reduce the number of sensors in such system by estimating all the capacitor voltages only with the measurement of load current. The gains of super-twisting algorithm are allowed to adapt to ensure the establishment of the sliding mode in the presence of uncertainties with unknown bounds. The hybrid behavior of the converter is also taken into account when designing the observer. The recent concept of Z(TN ) − observability is applied to analysis the observability of the switching system in order to obtain the estimation of the voltage across each capacitor. Simulation results show the efficiency and robustness of the proposed observer with respect to load resistance variation. Keywords: Adaptive, Super-Twisting(STW) Sliding Mode Observer, Multi-Cell Observer, Lyapunov Function. 1. INTRODUCTION

(1995). Benmansour et al. (2009) has also proposed supertwisting sliding mode observer without adaptive gains.

Power electronics have been developed through the last decade, which is carried out due to the developments of the semiconductor of power components and new systems of energy conversion Erickson and Maksimovi´c (2001). Many of those systems present a hybrid dynamics, among these systems, multi-cell converters are based on the association in series of the elementary cells of commutation Gateau et al. (2002), such structure makes possible to share the constraints in voltage and reduce the losses produced from the commutations of the power semiconductors. To benefit as much as possible from the large potential of the multicell structure, the voltage of each cell should be distributed as E/p where E is the source voltage and p is the number of cells. These voltages are generated when a suitable control of switches is applied in order to obtain a specific value. The control inserted by the switches allows to cancel the harmonics at the cutting frequency and reduce the ripple of the chopped voltage. However, these properties will be lost if the voltages of the capacitors become far from the desired value, so it is always valuable to get the knowledge of the capacitor voltages. The estimation of the capacitor voltages by means of an observer becomes an attractive and economical solution which can reduce the number of sensors.

In this paper, an instantaneous model which describes fully the hybrid behavior of the multi-cell converter is studied. An adaptive STW sliding mode observer is proposed in order to estimate the voltage across the capacitors using only the load current, which extends the work of Benmansour et al. (2009) to the adaptive case. The main disadvantage of the classical STW algorithm is the requirement of the information of the boundaries of the disturbance gradient. This is difficult because in many practical cases this boundary can not be easily estimated. The proposed observer is able to solve this problem with its gains adapting to the uncertainties with unknown boundary.

Modeling is a very important issue in such systems. Usually, classical control methods lie on the average model, and Pulse-Width-Modulation(PWM) strategy is used to translate continuous controller signal into switch signals. While instantaneous model accurately represents the state of each cell during a switching period, all the harmonic phenomenon will be present in this model. Several observers based approaches have been developed for nonlinear systems such as, high order sliding mode observers Ghanes et al. (2009), adaptive observers Marino and Tomei

Multi-cell converter is based on the association of a certain number cells, with the object of adapting the energy between the source and the load with best efficiency and minimum energy losses. Figure(1) depicts the topology of a converter with p independent commutation cells associated to an inductive load. The current flows from the source E to the output I through different converter switches. The multi-cell converter shows a hybrid behavior due to discrete variables(switching logic) and continuous variables(currents and voltages) Barbot et al. (2007). It

The paper is organized as follows. In Section II, the model of multi-cell converter and its characteristics are presented. In Section III, the observability of multi-cell converter is studied with the new concept of Z(TN ) − observability. Section IV is devoted to the design of the proposed observer for estimating the capacitor voltages. Section V gives illustrative simulations results under load variation condition. 2. MULTI-CELL CONVERTER MODELLING

Sp

{

S j +1

S2

S1

where

L E

c p -1

Vc p-1

cj

Vc j

c1

Vc1

I Vs

R

Fig. 1. Multicell converter on RL load is important to note that at the output, there are (p + 1) (p − 1)E E , E). This requires a unique voltlevels (0, , · · · , p p E age switch constraint of , and it is necessary to ensure p an equilibrated distribution of the capacitor voltages. The reference voltage of the j th capacitor is: E Vcj ref = j (1) p The dynamics of the p-cells converter is given by the following differential equations  p−1 ∑  Vcj   ˙ = − R I + E Sp −  (Sj+1 − Sj ) I   L L L   j=1    ˙ I Vc1 = (S2 − S1 ) c 1   ..    .    I    V˙ cp−1 = (Sp − Sp−1 ) cp

(2)

where I is the load current, cj is the j th capacitor, Vcj is the voltage of the j th capacitor and E is the voltage of the source. Each commutation cell is controlled by the binary input signal Sj ∈ {0, 1}. Sj = 1 means that the upper switch of the jth cell is on and the lower switch is off, whereas Sj = 0 means that the upper switch is off and the lower switch is on. By the following definition of the discrete control input: {

uj = Sj+1 − Sj , up = Sp

j = 1, . . . , p − 1

(3)

Assuming that only the load current I can be measured, the system(2) can be represented as follows:  p−1  ∑ Vcj  E R  ˙  I + u − uj I = − p   L L L   j=1    I   V˙ c1 = u1 c1 ..    .    I   V˙ c up−1  p−1 =   c p−1   y=I Describing the model(4) in a state affine equation as:

(4)

x˙ = f (x, u) = A(u)x + B(u) y = h(x, u) = Cx

[ ]T x = I Vc1 · · · Vcp−1   u1 up−1 R − · · · − −  L L L    u1   0 · · · 0   c1 A(u) =  . .. . . ..    . .  . . .    up−1 0 ··· 0 cp−1 ]T [ E B(u) = up 0 · · · 0 L C = [1 0 · · · 0]

(5) (6)

(7)

(8) (9)

3. HYBRID OBSERVABILITY ANALYSIS Considering the system (5), it shows that there are several operating switching modes which make the system unobservable. For instance, if u1 = u2 = · · · = up−1 = 0 or u1 = u2 = · · · = up−1 = 1, the voltages Vcj (j = 1, . . . , p − 1) become completely unobservable. These operating switching modes are not affected by the capacitor voltages. Fortunately, this condition will not occur for all control sequences, otherwise it will lose the physical interest. Because they represent particular situations in witch the multi-cell converter is not switching. The other operating switching modes of the multi-cell converter make the system observable. Now, for any other sequence of the corresponding input {u1 , u2 , · · · , up } applied to the system (4), the control sequence becomes sufficiently periodic. The observability matrix of system(5) is not of full rank, the recently suggested Z(TN ) − Observability concept will be introduced to solve this problem Kang et al. (2010). 3.1 Z(TN )-Observability The multi-cell converter belongs to a particular class of switching hybrid systems, in such systems the observation concept is linked to the switching sequence. Thus, it is important to give the following definitions Kang et al. (2010). Definition 1. A hybrid time trajectory is a finite or infinite sequence of intervals TN = ΓN i=0 , such that • Γi = [ti,0 , ti,1 ), f or all 0 ≤ i < N ; • For all i < N, ti,1 = ti+1,1 ; • t0,0 = tini and tN,1 = tend . Moreover, ⟨TN ⟩ is defined as the ordered list of u associated to TN , ui i=0,N where ui is the value of u on the interval Γi Kang et al. (2010). Definition 2. The function z = Z(t, x) is said to be Z−Observable with respect to the hybrid time trajectory TN and ⟨TN ⟩, if for all any two trajectories, (t, x1 , u1 ) and (t, x2 , u2 ) defined in [tini , tend ], the identity h(x1 , u1 ) = h(x2 , u2 ), implies z(t, x1 ) = z(t, x2 ). Lemma 1. Consider the system(5) and a fixed hybrid time trajectory TN and ⟨TN ⟩. Suppose that z = Z(t, x) is always continuous under any admissible control input. Suppose there exists a N +1 linear sequence of projections Pi , i = 0, 1, · · · , N , such that

• For all i < N , Pi Z(t, x) is Z−Observable for t ∈ Γi ; • Rank([P0T , · · · , PNT ]) = dim(z); dP¯i Z(t, x) = 0, for t ∈ Γi where P¯ is the complement • dt of P (projecting z to the variables eliminated by P ). Then, z = Z(t, x) is Z−observable with respect to the hybrid time trajectory TN and ⟨TN ⟩ Bejarano et al. (2010). Remark 1. In the Lemma 1, the third condition requires that the components of Z which are not observable in Γi must remain constant within this time interval. The hybrid time trajectory TN and ⟨TN ⟩ Bejarano et al. (2010) affects the observability property in a similar way as an input. Lemma 2. Consider the system(5) and taking z = x. Then, z is Z−observable with respect to the hybrid time trajectory TN and ⟨TN ⟩ = {u0 , · · · , uN }, if the vectors {u0 , · · · , uN } generate the space Rp−1 . Remark 2. It can be noted from Lemma 2 that after p − 1 time intervals, the measurement of the load current I assures to obtain (p − 1) linearly independent equation with respect to the voltages in the (p−1) capacitors, which enables to estimate the capacitor voltages Defoort et al. (2011). 4. OBSERVER DESIGN FOR THREE CELL CONVERTER An adaptive STW observer is designed in order to estimate the capacitor voltages of the multi-cell converter by only using the measurement of the current I and the knowledge of the control input sequence u. It is assumed that: (1) There exist TN such that z = x is Z−observable with respect to the hybrid time trajectory of system(4). (2) There exists a constant τ > 0 such that |ti,1 −ti,0 | > τ for every interval Γi . 4.1 Adaptive Super-Twisting Algorithm Description The super-twisting algorithm Levant (1993) is one of the popular algorithms among the second order sliding mode algorithms. A novel aspect of the formulation is that the gains will be allowed to adapt to ensure the establishment of the sliding mode in the presence of uncertainties with unknown bounds as proposed in Alwi and Edwards (2011). Consider a simple differential equation, x(t) ˙ = f (t, x)

{

with measured output y(t) = x(t). Assume that |f˙(t, x)| ≤ σ for some unknown constant σ > 0. The following observer is proposed: { 1 z˙1 (t) = −λ(t)|e1 (t)| 2 sign(e1 (t)) + z2 (t) (11) z˙2 (t) = −α(t)sign(e1 (t)) where e1 (t) = z1 (t) − x(t), and with adaptive gain λ(t), α(t). Defining e2 (t) = z2 (t)−f (t, x), substituting (10) into (11) yields the error dynamic system: { 1 e˙ 1 (t) = −λ(t)|e1 (t)| 2 sign(e1 (t)) + e2 (t) (12) e˙ 2 (t) = −α(t)sign(e1 (t)) − f˙(t, x) The gains α(t), λ(t) are chosen with the adaptive law as:

(13)

where L(t) = r(t) + l is time varying scalar, l is a fixed positive scalar ,and the varying r(t) is adapted as follows: { 1 β|e1 (t)| 2 if r(t) ≤ rmax r(t) ˙ = (14) 0 else where β > 0 is a positive design constant. The adaptive law in (14) allows r(t) to increase until the value rmax is reached. The convergence of the error system (12) has been proved in Alwi and Edwards (2011). 4.2 Adaptive STW Observer Design for the System Recalling the state-affine systems of the form (5) as discussed in Hammouri and de Leon Morales (1990): { ΣB :

x(t) ˙ = A(u)x(t) + B(u) y(t) = Cx(t)

(15)

where x ∈ Rn is the state vector of the system, u ∈ Rm is the control vector and y ∈ Rp is the output vector. A(u) is a matrix dependent nonlinearly on the input u, B(u) represents a vector which depends nonlinearly on u. It is assumed that the input u is persistently exciting such that there exist β, γ, T > 0 fulfilling the following inequality: ∫ t+T βI ≤ ΦTu (τ, t)C T CΦu (τ, t)dτ ≤ γI (16) t

where Φu denotes the transition matrix for system: { Σ0 :

x(t) ˙ = A(u)x(t) y(t) = Cx(t)

(17)

Remark 3. Since u remains constant on the time interval Γi , therefore, the matrix A(u) is constant. The persistent excitation can be checked by verifying that the current crosses each capacitor in different control sequences. It is interesting to note that the relationship between the current crosses through the capacitor and the measured load current from the topology of multi-cell in figure(1), which can be described by the following equations: {

(10)

√ α(t) = 2L(t) λ(t) = 4L(t)

Ic1 = I(S2 − S1 ) = Iu(1) Ic2 = I(S3 − S2 ) = Iu(2)

(18)

Substitute equation(18) into equation(4), one can get,  2  ˙c1 = − R Ic1 + u(1)u(3) E − u (1) Vc − u(1)u(2) Vc  I  1 2  L L L L   I    V˙ c1 = c1 c1 (19) u(2)u(3) u(1)u(2) u2 (2) R   ˙ I + E − V − V I = −  c2 c c2 c2 1   L L L L   I   V˙ c2 = c2 c2 The adaptive STW sliding mode observer is designed as follows:

 u(1)u(3) R ˙   E Iˆc1 = − Ic1 +   L L     u2 (1) ˆ u(1)u(2) ˆ   − Vc1 − Vc2 + µ(e1 )    L L    Ic1 ˙   − µ(e1 )  Vˆc1 = c1 (20)   Iˆ˙ = − R I + u(2)u(3) E  c2 c2   L L    2  u(1)u(2)  ˆc − u (2) Vˆc + µ(e2 )  − V  1 2  L L     I   Vˆ˙ c2 = c2 − µ(e2 ) c2 where the errors and the super-twisting algorithm with adaptive gains are defined as follows:   e1 = Ic1 − Iˆc1     e = I − Iˆ 2 c2 c2 (21) ˆ  eVc1 = Vc1 − Vc1     eVc2 = Vc2 − Vˆc2 ∫  1   µ(ei ) = αi sign(ei )dt + λi |ei | 2 sign(ei )  (22) αi = αi (t)    λi = λi (t) where i = {1, 2}, and the dynamics of the adaptive gains are the same as(13,14). Equation (19) and (20) yield the dynamics of the observation error as:  e˙ 1 = −µ(e1 ) + φ1     u2 (1) u(1)u(2)    φ˙ 1 = − µ(e1 ) − µ(e2 ) L L (23)   e˙ 2 = −µ(e2 ) + φ2   2    φ˙ 2 = − u(1)u(2) µ(e1 ) − u (2) µ(e2 ) L L where  u2 (1) u(1)u(2)   φ1 = − eVc1 − eVc2 L L (24) 2   φ = − u(1)u(2) e − u (2) e 2 Vc1 Vc2 L L The proposed nonlinear observer will be analyzed using Lyapunov function proposed in D´avila et al. (2011) which ensures the convergence of the error of the system(23) to zero. Consider (23) as a standard super-twisting algorithm with a bounded perturbation: { 1 e˙ i = −λi |ei | 2 sign(ei ) + ϕi (25) ϕ˙ i = −αi sign(ei ) + φ˙ i It is assumed that |µ(e1 )| ≤ σ1 , |µ(e2 )| ≤ σ2 for some unknown constants σ1 > 0 and σ2 > 0, then the following inequality will be attained with the condition of switch control signals |u| ≤ 1: 1 σ1 σ2 1 (26) + =σ |φ˙ i | ≤ |µ(e1 )| + |µ(e2 )| ≤ L L L L Then, with the result of Alwi and Edwards (2011), one can get the following convergence: { e˙ 1 = e1 = 0 (27) e˙ 2 = e2 = 0

With equations (27,24,23), one can get,  2   µ(e1 ) = − u (1) eVc1 − u(1)u(2) eVc2 L L2 (28)   µ(e2 ) = − u(1)u(2) eV − u (2) eV c1 c2 L L Then, substitute (28) into (21), one can get the observation error dynamics of the capacitor voltages,  2   e˙ Vc1 = − u (1) eVc1 − u(1)u(2) eVc2 L L2 (29)   e˙ V = − u(1)u(2) eV − u (2) eV c1 c2 c2 L L It is assumed that the system (29) satisfies the following assumptions, Assumption 1. The exists a constant ϕM > 0 such that for all t ≥ 0, and all u ∈ D, where D ∈ R2 is a closed compact subset, ∥Ψ(t, u)∥ ≤ ϕM , where [ ]T u1 (t) u2 (t) √ Ψ(t, u) = √ (30) L L Assumption 2. There exists a constant T1 , µ > 0 such that ∫ t+T1 Ψ(τ, u)ΨT (τ, u)dτ ≥ µI > 0, ∀t ≥ 0. (31) t

From the results of Lor´ıa and Panteley (2002), Panteley and Lor´ıa (2000), one can conclude that the estimation errors eVc1 , eVc2 of the system (29) converge to zero exponentially. 5. SIMULATION RESULTS Computer simulation is performed to verify the effectiveness and behavior of the proposed estimation strategy to estimate the capacitor voltages. A three-cell converter connected to an RL load which is modeled by the system (4), the main parameters used in the simulation are shown in Table(1), load resistance varied at time 0.02s to demonstrate the robustness of the proposed observer when handle with varying conditions. The inputs of the Table 1. Main Parameters Of Simulation System Parameters DC voltage(E) Capacitors(c1 , c2 )

Values 150V 40*10−6 F

Load resistance(R) Load Inductor(L) The chopping frequency The sampling period

131 −→ 200Ω 10−3 H 5kHz 5us

0.02s

switches uj , j = {1, 2, 3} are generated for the hybrid time trajectory given in Figure(2). In the simulation results presented bellow, only a load current measurement is used for the capacitor voltages estimation, the supply voltage E must also be used to impose the equal distribution of the voltage constraints on each cell(during steady conditions 2E E = 50V, Vc2 = = 100V ). In the following Vc1 = 3 3 figures which highlight the efficiency of the proposed observer, it can be seen that the estimations of the capacitor voltages Vc1 , Vc2 converge to their reference values exactly. Figures(3) show the results obtained with the adaptive STW sliding mode observer . Simulation results show that

100

50 45

90

Vˆc1

80

35

70

30

60 (V)

(V)

40

Vc1

25 20

Vˆc2

50 40

30.8

79

30

15 30.8

10

20

29.8 0.0198

5 0

Vc2

0

0.02

0.02

0.04

t(s)

0.06

0.08

78

10

0.0202

0

0.1

0.0198 0

0.02

(a) Vc1 and its estimation Vˆc1

0.02 0.04

t(s)

0.0202 0.06

0.08

0.1

(b) Vc2 and its estimation Vˆc2

Fig. 3. Capacitors voltages Vc1 , Vc2 and their estimations Vˆc1 , Vˆc2 100

45

Vc1

40

Vˆc1

90

Vc2

80

Vˆc2

35

70

30

60 (V)

(V)

50

25 30.7

20

50 78.8

15

30

10

20

5 0

79.2

40

0

77.8 0.0198

10

29.7 0.0198

0.02

0.02

0.04

t(s)

0.0202

0.06

0.08

0.1

ˆ = 1.5R (a) Vc1 and its estimation Vˆc1 when R

0

0

0.02

0.02 0.04

t(s)

0.06

0.0202 0.08

0.1

ˆ = 1.5R (b) Vc2 and its estimation Vˆc2 when R

ˆ = 1.5R Fig. 4. Capacitors voltages Vc1 , Vc2 and their estimations when R

U1

1 0 −1

0

0.5

1

1.5

2

2.5 −4

x 10

U2

1 0 −1

0

0.5

1

1.5

2

2.5 −4

x 10

U3

1 0.5 0

0

0.5

1

t

1.5

2

of the voltages across the capacitors in the multi-cell converter in order to reduce the number of sensors in such systems. It is suitable and possible to observe the capacitor voltage with the new concept of observability as the system is assumed to be Z(TN ) observable on an interval when all switch control signals keep constant. However, the voltage across the capacitors can not be estimated instantaneously, but can be done after some time. Simulation results demonstrate the performance of the proposed observer, the robustness of the observer was also verified by the resistance variation at time t = 0.02s.

2.5 −4

x 10

Fig. 2. Applied switch control inputs the trajectories of the proposed observer converge to the capacitor voltages in steady condition. Figure(4) shows the robustness of the proposed observer with the variation of the load resistance variation 50% at time t = 0.02s. 6. CONCLUSION In this article, an adaptive STW(Super Twisting) sliding mode observer was proposed for the estimation problem

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Benmansour, K., Djemai, M., Tadjine, M., and Boucherit, M. (2009). On observability and high order sliding mode and adaptive observers design for a multicell chopper. In Variable Structure Systems, 2008. VSS’08. International Workshop on, 285–290. IEEE. D´avila, A., Moreno, J., and Fridman, L. (2011). Optimal lyapunov function selection for reaching time estimation of super twisting algorithm. In Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on, 8405–8410. IEEE. Defoort, M., Djemai, M., Floquet, T., and Perruquetti, W. (2011). Robust finite time observer design for multicellular converters. International Journal of Systems Science, 42(11), 1859–1868. Erickson, R. and Maksimovi´c, D. (2001). Fundamentals of power electronics. Springer Netherlands. Gateau, G., Fadel, M., Maussion, P., Bensaid, R., and Meynard, T. (2002). Multicell converters: active control and observation of flying-capacitor voltages. Industrial Electronics, IEEE Transactions on, 49(5), 998 – 1008. Ghanes, M., Bejarano, F., and Barbot, J. (2009). On sliding mode and adaptive observers design for multicell converter. In American Control Conference, 2009. ACC’09., 2134–2139. IEEE. Hammouri, H. and de Leon Morales, J. (1990). Observer synthesis for state-affine systems. In Decision and Control, 1990., Proceedings of the 29th IEEE Conference on, 784 –785 vol.2. Kang, W., Barbot, J.P., and Xu, L. (2010). On the observability of nonlinear and switched systems. In C.F.Z.Y. Ghosh Bijoy; Martin (ed.), Emergent Problems in Nonlinear Systems and Control, volume 393 of Lecture Notes in Control and Information Sciences,. Springer. Levant, A. (1993). Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58(6), 1247–1263. Lor´ıa, A. and Panteley, E. (2002). Uniform exponential stability of linear time-varying systems: revisited. Systems & Control Letters, 47(1), 13–24. Marino, R. and Tomei, P. (1995). Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems. Automatic Control, IEEE Transactions on, 40(7), 1300–1304. Panteley, E. and Lor´ıa, A. (2000). Uniform exponential stability for families of linear time-varying systems. In Decision and Control, 2000. Proceedings of the 39th IEEE Conference on, volume 2, 1948–1953. IEEE.