observer design for a class of nonlinear systems

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proof see (Gauthier et al., 1992)) that single output control af£ne systems which are observable for every input can be transformed locally almost everywhere.
Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain

OBSERVER DESIGN FOR A CLASS OF NONLINEAR SYSTEMS - APPLICATION TO AN INDUCTION MOTOR L. Rossignol ∗ M. Farza ∗ M. M’Saad ∗ J.F. Massieu ∗ R. Alvarez Salas ∗∗



Laboratoire d’Automatique de Proc´ed´es E.A. 2611, L.A.P., I.S.M.R.A., Universit´e de Caen, 6, Boulevard Mar´echal Juin, F-14050 CAEN cedex, France. [email protected] ∗∗ Laboratoire d’Automatique de Grenoble UMR 5528, L.A.G., E.N.S.I.E.G., Domaine universitaire BP 46, F-38402 Saint-Martin-d’HÁ eres, France.

Abstract: A simple observer is proposed for a large class of multi-output nonlinear systems. The main characteristic of the proposed observer is its implementation and performance tailoring simplicity. Indeed, the gain of this observer does not require the resolution of any dynamical system and is analytically given. Moreover, the performance speci£cation is made through the choice of a unique parameter. The performances of the proposed observer are demonstrated through a real experiment dealing with the estimation of the rotor ¤ux, the motor speed and the load torque in an induction motor. Keywords: Nonlinear systems, Nonlinear observer, Induction motor

1. INTRODUCTION This paper deals with the design of observers for a special class of nonlinear systems satisfying some regularity assumptions. The general framework of this observer design is particularly based on the contributions of (Bornard and Hammouri, 1991; Gauthier et al., 1992; Farza et al., 1998; Busawon et al., 1998). In (Gauthier et al., 1992), the authors have designed an observer for single output control af£ne systems which are observable for every input (uniformly observable). It is shown in (Gauthier and Bornard, 1981) (for a short proof see (Gauthier et al., 1992)) that single output control af£ne systems which are observable for every input can be transformed locally almost everywhere into a canonical observable form. This canonical form is composed of a £xed linear dynamic component and a triangular controlled one. Using this canonical form,

the authors in (Gauthier et al., 1992) have designed a high gain observer for such systems under some global Lipschitz assumptions on the controlled part. The gain of the proposed observer is issued from an algebraic Lyapunov equation. An extension of this observer synthesis to the multi-output case is given in (Farza et al., 1998; Busawon et al., 1998) for a larger class of systems which are observable for every input. More precisely, the authors have considered uniformly observable multi-output systems which are split into a time-varying linear part and a nonlinear controlled part. Under some structure assumptions, the authors have shown that the technique involved in (Gauthier et al., 1992) can be generalized for such systems. A further extension of this last result for systems with locally regular inputs is given in (Bornard and Hammouri, 1991). But here the observer gain involves the integration of some differential Lyapunov equation.

In the present paper, we use the general strategy of observer design adopted in (Farza et al., 1998; Busawon et al., 1998) to construct a high gain observer for a larger class of nonlinear systems under similar regularity assumptions. This paper is organized as follows. The next section is devoted to the observer design for a class of multi-output nonlinear systems. In section 3, the performances of the proposed observer are highlighted through a real experiment dealing with the estimation of the rotor ¤ux, the motor speed and the load torque in an induction motor.

2. OBSERVER DESIGN We consider multi-output nonlinear systems which can be described as follows: ½ x˙ = F (s, x)x + G(u, s, x) + ε(t) (1) y = Cx where

 x(1)  x(2)    x =  .  ∈ IRn with x(k) ∈ IRnk , k = 1, . . . , q  ..  x(q)

and p = n1 ≥ n2 ≥ . . . ≥ nq ,

q X

nk = n; the input

k=1 m

u ∈ U a compact subset of IR , the output y ∈ IRp , s(t) is a known signal. The functions G and F assume triangular forms i.e.   (1) (1) (u, s, x )   G(2) (u, s, x(1) , x(2) )   ..  G(u, s, x) =  .    G(q−1) (u, s, x(1) , . . . , x(q−1) )  G(q) (u, s, x) G

with G(k) (u, s, x) ∈ IRnk , k = 1, . . . , q;  (1) 0 F1 (s, x

 0   .. F (s, x) =  .   0  0

0

...

)



0

... 0 .  . . .  F2 (s, x(1) , x(2) ) . .   ..  . 0   .. . 0 Fq−1 (s, x)  ...

0



is block diagonal and each Fk , k = 1, . . . , q − 1, denotes a nk × nk+1 rectangular matrix;   0  (k)   ..  ε1  .     ε(k)  2   0   ε(t) =  (k0 )  with ε(k) =  .  ∈ IRnk  ε  ..   .  (k)  ..  εn k ε(q)

(k)

k = k0 , . . . , q and each εi , i = 1, . . . , nk is an unknown bounded real-valued function which may depend on x, s, u, uncertain parameters, etc.; C = [Ip , 0, . . . , 0] where Ip is the p × p identity matrix.

The observation problem we are concerned with is well posed if the following set of assumptions holds: (A1) There exist a set of controls U ∈ C ∞ (U, IRm ) and two compact sets K1 ⊂ K2 such that every trajectory x(t) associated to any u ∈ U and issued from K1 lies into K2 . This means, in particular, that we only deal with bounded inputs and bounded trajectories. (A2) There exist two positive constants α, β such that for every k ∈ 1, . . . , q − 1, ∀u ∈ IRm , ∀x ∈ IRn , ∀t ≥ 0, 0 < α2 Ink +1 ≤ Fk (s, x)T Fk (s, x) ≤ β 2 Ink +1 where Ink +1 is the (nk + 1) × (nk + 1) identity matrix. (A3) The function ε is bounded. (A4) The signal s(t) and its time derivative s(t) ˙ are bounded. (A5) The functions G(i) (u, s, x), i = 1, . . . , q are global Lipschitz with respect to x uniformly in u and s. (A6) The functions Fi (s, x),i = 1, . . . , q−1 are global Lipschitz with respect to x uniformly in u. Before giving our main theorem, we shall make some remarks and introduce notations used hereafter. (1) For every ξ ∈ IRn , t ≥ 0, let Λ(s(t), ξ) be the block diagonal matrix de£ned by Λ(s(t), ξ) = diag [In1 , F1 (s(t), ξ), # q−1 Y Fi (s(t), ξ) F1 (s(t), ξ)F2 (s(t), ξ), . . . ,

(2)

S + AT S + SA − C T C = 0

(3)

i=1

where In1 denotes the n1 × n1 identity matrix. By Assumption (A2), Λ(s(t), ξ) is left invertible. Indeed, its left-inverse shall be denoted by Λ+ (s(t), ξ) in the sequel. (2) Let S be the unique solution of the algebraic Lyapunov equation :



0  .. .   where A =  ...  .  .. 0

I n1 0 .. . I n1 .. . ...

... .. . .. . .. . ...

0 .. . 0 I n1 0

        

is n1 q × n1 q square matrix and C = [In1 , 0 . . . , 0] is n1 × n1 q. It can be shown that the explicit solution of (3) is given by: j−1 S(i, j) = (−1)i+j Ci+j−2 In1 , f or 1 ≤ i, j ≤ q

where Cji =

j! i!(j − i)!

(4)

and S is symmetric positive de£nite. Moreover, Cholesky’s decomposition of S is given by S = U T U where the upper triangular matrix U is given by:

i−1 U (i, j) = (−1)i+j Cj−1 In1 , f or 1 ≤ i ≤ j ≤ q (5)

¯ 3) Set S(s(t), ξ) = ΛT (s(t), ξ)SΛ(s(t), ξ) and C¯ = CΛ(s(t), ξ). On one hand, it is easy to see that S¯ is an invertible square matrix and one can show that: S¯−1 = Λ+ U −1 ΛΛ+ (U −1 )T (Λ+ )T On the other hand, we note that C¯ is constant. Indeed, we have: C¯ = [In , 0n ×n , 0n1×n , . . . , 0n ×n ] (6) 1

1

2

3

1

q−1

where 0n1 ×nk denotes the n1 × nk null matrix, k = 2, . . . , q − 1. Now, one can easily check that: i−1 U −1 (i, j) = Cj−1 In1 , f or 1 ≤ i ≤ j ≤ q (7)   H1  ..  Moreover, if we denote by H =  .  = S¯−1 C¯ T ,

Hq

we have:

q−1

H 1 = I n1 +

Hi =

j

X Y j=1

à i−1 Y

Ã

Fk

k=1

k=1

!+

2≤i≤q−1 Hq =

Ãq−1 Y

k=1

Fk

Fk

+

q−1 X



j

Y

k=1

Cji−1

j=i

Ã

Fk

!+

j Y k=i

Fk



¯ x ¯ x θS(s, ˆ) + F T (s, x ˆ)S(s, ˆ) T ¯ ¯ ¯ + S(s, x ˆ)F (s, x ˆ) − C C = 0.

(9)

¯ x, s) = ΛT (ˆ where S(ˆ x, s)SΛ(ˆ x, s) and C¯ is the constant matrix de£ned by (6). Set e(t) = x ˆ(t) − x(t), then : ¡ ¢ ¯ x e˙ = F (s, x ˆ) − θ∆θ S(s, ˆ)−1 C¯ T C¯ e + F (s, x ˆ)x − F (s, x)x + G(u, s, x ˆ) − G(u, s, x) − ε(t)

Set e¯ = ∆−1 θ e. We obtain ¡ ¢ ¯ x e¯˙ = θ F (s, x ˆ) − S(s, ˆ)−1 C¯ T C¯ e¯ + ∆−1 ˆ)x − F (s, x)x) θ (F (s, x

+ ∆−1 ˆ) − G(u, s, x)) − ∆−1 θ (G(u, s, x θ ε(t) The last equality comes from the fact that : ∆−1 ˆ)∆θ = θF (s, x ˆ) θ F (s, x ¯ x Consider the quadratic function V (¯ e) = e¯T S(s, ˆ)¯ e, then

j Y

k=1

Fk

!+

!+

¯ x ˙e V˙ = 2¯ eT S(s, ˆ)e¯˙ + 2¯ eT ΛT S Λ¯ ¡ T ¢ ¯ x = θ 2¯ e S(s, ˆ)F (s, x ˆ)¯ e − 2¯ eT C¯ T C¯ e¯ ¯ x + 2¯ eT S(s, ˆ)∆−1 (F (s, x ˆ)z − F (s, x)x) θ

¯ x + 2¯ eT S(s, ˆ)∆−1 ˆ) − G(u, s, x)) θ (G(u, s, x T T ˙ T ¯ + 2¯ e Λ S Λ¯ e − 2¯ e S(s, x ˆ)∆−1 ε(t)

Bearing in mind the contributions of (Bornard and Hammouri, 1991; Gauthier et al., 1992; Farza et al., 1998; Busawon et al., 1998), a souhaitable candidate observer for system (1) under assumptions (A1) to (A6) is given by : x ˆ˙ = F (s(t), x ˆ)ˆ x + G(u, s(t), x ˆ) (8) ¯ − θ∆θ S(s(t), x ˆ)−1 C¯ T (C¯ x ˆ − y) where S¯ and£C¯ are given above and ¤ ∆θ = diag In1 , θIn2 , θ2 In3 , . . . , θ q−1 Inq for some θ > 0 and Ink denotes the nk × nk identity matrix. Indeed, we have the following main result:

θ

≤ −θV − θkC¯ e¯k2 ¯ x + 2kS(s, ˆ)¯ ekk∆−1 (F (s, x ˆ)x − F (s, x)x) k θ

¯ x + 2kS(s, ˆ)¯ ekk∆−1 ˆ) − G(u, s, x))k θ (G(u, s, x δ ˙ e + 2kS(s, ¯ x + 2¯ eT ΛT S Λ¯ ˆ)¯ ek k0 −1 θ by equation (9). Therefore, µ ¡ 2 ˙ V = −θV + 2˘ κ λmax (S)k¯ ek k∆−1 F (s, x ˆ)x θ ¢

−F (s, x)x k +

k∆−1 θ

(G(u, s, x ˆ) − G(u, s, x)) k

+2η˘ κλmax (S)k¯ ek2 + 2˘ κ2 λmax (S)

δ k¯ ek θk0 −1



Theorem. Consider system (1) under assumptions (A1) to (A6) and system (8). Then, one has:

˙ z , s)k, λmax (S) is the largest eigenwhere η = sup kΛ(ˆ

∃θ0 > 0; ∀θ > θ0 ; ∃λθ > 0; ∃µθ > 0; ∃Mθ > 0;

value of S and κ ˘ = max(1, β, β 2 , . . . , β q−1 ) where β is given in Assumption (A2). Now, assume that θ ≥ 1, then, because of the triangular structure and the Lipschitz assumption on G, one can show that :

∀u ∈ U ; ∀ˆ x(0) ∈ Rn ; one has : kˆ x(t) − x(t)k ≤ λθ e−µθ t kˆ x(0) − x(0)k + θ q−k0 Mθ δ where δ is the upper bound of kεk. Moreover, one has lim µθ = +∞ and lim Mθ = 0.

θ→∞

θ→∞

Proof: Firstly, one can easily check that : Λ(ˆ z , s)F (s, x ˆ) = AΛ(s, x ˆ) Thus, by multiplying the left and the right side of equation (3) by ΛT (ˆ x, s) and Λ(ˆ x, s) respectively, we obtain :

t≥0

(G(u, s, x ˆ) − G(u, s, x)) k ≤ ζ1 k¯ ek k∆−1 θ

for some constant ζ1 which does not depend on θ (see (Gauthier et al., 1992)). Similarly and since the state z is assumed to be bounded (assumption (A1)), we have : k∆−1 (F (s, x ˆ)x − F (s, x)x) k ≤ ζ2 k¯ ek θ

for some constant ζ2 which does not depend on θ. δ Hence, V˙ ≤ −θV + c1 V + θkc02−1 where c1 = 2σ 2 (S)

κ ˘2 (ζ1 + ζ2 + κη), κ2

κ = min(1, α, α2 , . . . , αq−1 ), α given in assumption p κ ˘2 (A2), and c2 = 2 σ(S) λmax (S) with σ(S) = κ s λmax (S) ; λmin (S) is the smallest eigenvalue of S. λmin (S) Thus, d dt

³p

V (¯ e(t))

Consequently, p

´

≤−

h ³

V (¯ e(t)) ≤ exp − +

(θ − c1 ) p c2 δ V (¯ e(t)) + k −1 2 2θ 0

θ − c1 2

´ ip

V (¯ e(0))

t

c2 δ 1 − exp − θ k0 −1 (θ − c1 )

h

h ³

θ − c1 2

´ ii t

Now taking θ0 = max {1, c1 } and θ > θ0 , we obtain : ´ i h ³ k¯ e(t)k ≤ σ(S) +

κ ˘ θ − c1 exp − κ 2 c2 δ

θ k0 −1 (θ

− c1 )κ

t k¯ e(0)k

(10)

p

λmin (S)

On the other hand, for θ ≥ 1, we have : ke(t)k ≤ k¯ e(t)k ≤ θ q−1 ke(t)k

(11)

Combining (10) and (11), and replacing c2 by its above expression, we obtain : ´ i h ³ ke(t)k ≤ θ q−1 σ(S) +2

θ q−1

θ k0 −1

θ − c1 κ ˘ exp − κ 2 κ ˘ σ 2 (S) δ. κ (θ − c1 )

t ke(0)k

Now, it is easy to see that θ0 , λθ , µθ and Mθ needed by the theorem are: θ0 = max {1, c1 }, λθ = θq−1 σ(S) κκ˘ , 2

σ (S) 1 µθ = θ−c and Mθ = 2 κκ˘ (θ−c This ends the proof 2 1) of the theorem.

Remark : Observe that for ε(t) = 0, the convergence of the estimation error is an exponential one. In the case where kε(t)k 6= 0 but bounded by δ, the asymptotic estimation error is bounded and the corresponding upper bound is as small as δ. Moreover, if the uncertain term intervenes in the last block only (i.e. k0 = q), the bound of the estimation error can be made as small as desired by choosing values of θ high enough. It is worth noticing that the scalar θ has to be speci£ed bearing in mind the unavoidable compromise between fast convergence and noise sensitivity.

3. APPLICATION TO THE INDUCTION MOTOR It is well-known that most control systems for induction motors require the knowledge of the rotor ¤ux as well as that of the angular speed (Leonard, 2001). Since these informations, in particular the ¤ux, are

not easily accessible, many research efforts have been focused on their estimation in the past few years. Different alternatives have been studied in order to design observers for induction motor using Luenbergerlike observer (Verghese and Sanders, 1988; Martin and Rouchon, 2000), nonlinear observers (Busawon et al., 1998; Lubineau et al., 2000), sliding modebased observer (Benchaib et al., 1999), etc. However, in most of these works, the speed measurements has been supposed to be available and the objective of these works was to provide on-line estimates of the rotor ¤uxes only. Note that, in the work of (Ortega et al., 1998), the authors proposed passivity-based control of induction motors which does not make use of the rotor ¤ux but the angular speed is assumed to be measured. Many other works deal with sensorless control of AC motors drives (Holtz, 1996). Most of these works are based on open loop estimation models which performance reduces as the mechanical speed reduces. Moreover, this performance depends on how precisely the model parameter can be matched to the corresponding parameter in the controlled machine. In order to improve the robustness against parameter mismatch and signal noise, some authors have proposed adaptive observers where the ¤ux are estimated using a closed loop observers while the angular speed and load torque are treated as time-varying parameters which are adapted through open loop models based on the machine equations. Here, we shall give a closed loop nonlinear observer which allows to estimate the rotor ¤ux as well as the angular speed and the load torque from the measurements of the stator currents. Indeed, consider the α-β Park’s model for an induction motor:

µ ¶  1 1  ˙  Isα = K Φrα + pΩΦrβ − γIsα + usα   Tr σLs  ¶ µ   1 1   ˙   Isβ = K −pΩΦrα + T Φrβ − γIsβ + σL usβ  r s  µ ¶  ˙Φrα = − 1 Φrα + pΩΦrβ + M Isα   ¶ Tr µ Tr   M 1   ˙  Isβ Φrβ = − −pΩΦrα + Φrβ +   Tr Tr      Ω˙ = − 1 TL + pM (Φrα Isβ − Φrβ Isα ) J JLr (12) where Isα , Isβ are stator currents; usα , usβ are stator voltages; Φrα , Φrβ are rotor ¤ux; Ω is the angular speed; J is the moment of inertia; TL is the load torque; M is the mutual inductance and p is the number of pairs of poles. The parameters Tr , σ, K and γ Lr M2 are de£ned as follows: T r = ,σ = 1− , Rr Ls Lr Rs Rr M 2 M where Rs , Rr are ,γ = + K = σLs Lr σLs σLs L2r stator (resp. rotor) per-phase resistance and Ls , Lr are stator (resp. rotor) per-phase inductance. Model (12) can be written as follows :

¶ µ ¶ µ µ ¶ µ ˙ ¶ 1 usα Isα Isα r1   + −γ = K   Isβ r I˙sβ σLs usβ    2    M  µ ¶  I −r +  r˙1   2 Tr sβ   = p   Ω  M  r˙2  r − I  1 sα   Tr  M  I −r +  1  1 Tr sα    +   + εr (t)   Tr −r + M I   2 sβ   Tr    pM 1  ˙  T + (Φ Ω = −  L rα Isβ − Φrβ Isα )  J JLr   T˙L = εTL (t)   (13) 1 µ ¶ Φrα + pΩΦrβ r1   =  Tr where r = , 1 r2 −pΩΦrα + Φrβ Tr µ ¶ µ ¶ εr 1 Φrβ ˙ εr (t) = = Ω(t) and εTL (t) is a εr 2 −Φrα real-valued unknown and bounded function. It is easy to see is under µ that ¶system (13) µ ¶ form (1) I Φ sα rα with x(1) = , x(2) = , x(3) = Isβ Φrβ Ω and x(4) = TL . As a result, one can design an observer of the form (8) in order to estimate r1 , r2 (and consequently Φrα and Φrβ ), Ω and TL . However, before using such observer, let us investigate how realistic are assumptions (A1) to (A6). The equations of physical systems (in particular those of system (13)) make sense only on a physical bounded domain ν (i.e. the trajectories of system (13) lie into ν). On such a compact domain ν, assumptions (A1) and (A3) to (A6) as well as the right inequality of Assumption (A2) are satis£ed. Note according to assumption (A3), the acceleration is assumed to be bounded and this is in accordance with practical situations. Let us now examine the left inequality of Assumption (A2) (i.e. the lower bound of FkT Fk , k = 1, . . . , q − 1). In our application, we clearly have: 1 J   M ¶ µ −r + I Φ˙ rβ  2 Tr sβ  = p F2 = p . ˙ M −Φrα Isα r1 − Tr where I2 is the 2 × 2 identity matrix. Accordingly, Assumption (A2) is trivially satis£ed by F 1 and F3 and it has to be discussed this ´ ³ for F2 , only. Indeed, assumption requires that Φ˙ 2rα (t) + Φ˙ 2rβ (t) remains far away from zero along the experiment. This is clearly satis£ed in potential applications of inductor motors where the objective consists in controlling the speed and the ¤ux norm. Indeed, the nullity of the above quantity occurs when the motor is used as a courant generator which is not generally the aim of such applications. As a result, assumptions (A1) to (A6) are satis£ed and an observer of the form (8) can be used. q = 4, F1 = KI2 , F3 =

3.1 Real experiment results The observer performances have been validated using off-line real data obtained from a 7.5 kW experimental controlled induction motor device at the Laboratoire d’Automatique de Grenoble (Von Raumer, 1994; Lubineau et al., 2000). Our objective consists in comparing the real values of the rotor ¤ux, the motor speed and the load torque (off-line available data) with those provided by the nonlinear observer. The norm of the real stator voltages applied to the motor with the resulting current norm are shown on £gures 1 and 2, respectively and they are used as inputs of the observer. The following parameter values are used for the numerical integration of the observer equations: p = 2; Ls = 97.7 mH; M = 91 mH; Lr = 91 mH

J = 0.22 Kg.m2 ; Rs = 0.63 Ohm; Rr = 0.4 Ohm The estimated motor speed is compared to the real one on £gure 3. Notice that thought the estimated speed has been intentionally initialized at a value which is relatively far from the real initial one, both curves coincides after about 0.1 s. We also remark the good agreement between real and estimated load torques as pointed out by £gure 4. This clearly highlights the observer ability to deal with abrupt load variations. Concerning the rotor ¤ux and since these quantities are not available for measurement, we have reported in £gure 5 the set-point at which the ¤ux norm is regulated. In order to have more con£dence in the obtained estimated ¤ux norm, we have also reported in this £gure an another estimation of the ¤ux norm which was obtained by an observer where the speed was assumed to be measured (see e.g. (Busawon et al., 1998)). Again, we remark the good agreement between the obtained speed sensorless ¤ux estimate and other values.

350

||Us|| (V) 300

250

200

150

100

50

0

0

0.5

1

1.5

2 Time (s)

Fig. 1. Norm of stator voltage

2.5

3

3.5

4

Conclusion : A simple observer has been designed for a large class of nonlinear systems. The appealing features of the proposed observer are its implementation and calibration simplicity. Of practical interest, the performances of the proposed observer have been demonstrated through real data obtained from a real experiment dealing with the estimation of the rotor ¤ux, angular speed and load torque in an induction motor.

35

|| Is || (A) 30

25

20

15

10

5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

Fig. 2. Norm of stator current 200

Ω (rad/s)

150

100

50

0

REAL

ESTIMATED −50

0

0.5

1

1.5

2

2.5

3

3.5

4

2.5

3

3.5

4

Time (s)

Fig. 3. Angular speed TL (N.m) 80

60

REAL

40

20

0

−20

−40

ESTIMATED

−60 0.5

1

1.5

2 Time (s)

Fig. 4. Load torque 1.4

ESTIMATED WITH SPEED MEASUREMENT

1.2

SET POINT

1

0.8

0.6 ESTIMATED

0.4

|| Φr || (Wb) 0.2

0

0

0.5

1

1.5

Fig. 5. Norm of rotor ¤ux

2

2.5

3

3.5

4

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