A predictive controller for induction motors using an ... - Springer Link

J Control Theory Appl 2013 11 (3) 367–375 DOI 10.1007/s11768-013-2152-5

A predictive controller for induction motors using an unknown input observer for online rotor resistance estimation and an update technique for load torque Hanen CHEHIMI† , Salim HADJ SA¨ID, Faouzi MSAHLI Department of Electrical Engineering, National Engineering School of Monastir, 5019 Monastir, Tunisia

Abstract: This paper deals with the design of an output feedback predictive controller for induction motors. The fundamental interest of the proposed controller is the capability of decoupling the mechanical speed and the rotor fluxes, without degradation against the variation of rotor resistance and load torque. Hence, the contribution is to apply two estimation procedures in order to achieve this goal. Namely, an unknown input observer (UIO) is used for the constant time estimation whereas a heuristic solution is exploited for the load torque update. Moreover, rotor flux components are recovered as an unavailable state of the system. Effectiveness of the proposed observers and the performance of the controller are confirmed by simulation results. Keywords: Predictive controlle; UIO; Induction motors; Parameters estimation

1 Introduction Rapid improvements in the industrial field combined with severe and thorny requirements have been largely responsible for the development of the output feedback controller, namely direct field oriented control (DFOC), output feedback linearization, sliding mode (SMO), high gain and backstepping methods, etc. [1–2]. In particular, the different advantages proposed by the nonlinear model predictive control (NMPC) have witnessed a steady growth for their wide use. Nonetheless, several issues in that framework are still open like the online optimization requirement. Indeed, it is generally nonconvex, and its computational burden grows exponentially with the decision variable. The nonlinear generalized predictive control (NGPC), proposed in [3–4], is an interesting topic to tackle and resolve the computational challenges in NMPC strategies. Indeed, an explicit solution in continuous time is given. Furthermore, an asymptotic convergence is ensured for systems having a finite relative degree. Induction motors (IMs) drive was generally proposed as an interesting benchmark problem for a nonlinear control [2,5–6]. Indeed, it is well-known that the IMs are the most employed workhorse in industry. They enjoy several inherent benefits, such as being rugged, reliable, compact, efficient and less expensive compared to other electrical machines used in similar applications. However, many reasons justify the difficulty of the control algorithms design, namely the nonlinearity and the MIMO nature of the motor model. In addition, in several applications and more particularly in the electric traction, the high-speed capability of the machine is required. As a result, the induction motor undergoes significant degradations in terms of performance,

in particular at the level of the available torque. Therefore, the means of field weakening was proposed to overcome this problem [7]. All previous difficulties are strongly coupled with the unknown load disturbances and parameters uncertainties to create a tempting challenge. Rotor resistance Rr is one of the most interesting parameters which deserve the attention; and we see this in a large number of works that have been developed by using different approaches to estimate this critical parameter [2, 8–10]. Due to its strong dependence on operating conditions, the resistance value can vary up to 100%, predominantly caused by rotor heating, and it can be hardly recovered by using thermal model temperature sensors. Specifically, the perfect knowledge of the rotor constant time is a key for achieving a successful control algorithm [6, 11–12]. The sensitive calculation of the controller is not only caused by its error but also by the unavailability of both the rotor flux ψr and load torque TL . The estimation of the load torque during the induction motor control is important also to ensure a good operation quality. The dynamics of the previous parameter may occur due to changes in solids conveying, melting, mixing, melt conveying, etc. Hence, it is useful for the detection of possible incipient faults and updating the working conditions of mechanical loads. When parameters and state mismatch are not taken into account, the efficiency of the motor drive decreases and cannot answer the high precision performance which is usually needed in machine tools or robotics applications. Hence, any variation of state or parameters should be tracked when it occurs. As is frequently the case, the observer incorporation is the favorable solution for recovering electrical and mechanical vari-

Received 1 July 2012; revised 21 November 2012. † Corresponding author. E-mail: [email protected]. Tel.: +216-97-034-024; fax: +216-73-500-514. This work was supported by the National Engineering School of Monastir, Tunisia. c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2013 

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ables uncertainties of the induction motor [13–16]. Tremendous research activities have focused on the observer design for nonlinear dynamic systems, and the result is various approaches and extensions [17–20]. Over the last decade, the unknown input observer (UIO) has been one of the most interesting topics [21–23]. One important reason for the UIO success is the simultaneous estimation for the system state and parameters from only input and output measurements. Based on the LMI approach, Corless and Tu [24] proposed the solution to jointly estimate the missing states and the unknown inputs. Besides, the reduced order observers have been proposed in a relatively recent work. They have as subject to simultaneously estimate the state and the unknown inputs when the latter vary slowly. In the nonlinear observer context, many contributions for some particular classes of nonlinear systems have been proposed in the literature [25–26]. See for instance, Farza et al. [27] extended the triangular class of MIMO systems involving unknown inputs. Moreover, a full-order high-gain observer is suggested for the simultaneous estimation of the nonmeasured states and the unknown inputs. The contribution presented in this paper, is to explore the output feedback predictive controller for the driving induction motor and to study its robustness against the variation of state and parameters. These variations are corrected by the combination between two estimation mechanisms and the state feedback NGPC law. Indeed, profiting of the stator current and speed measurements it is easy to achieve the simultaneous estimation of the internal state variables and the time varying parameters namely, the stator flux components, load torque and the rotor resistance. In addition, the high-speed problem is solved by using the conventional field-weakening method. The paper is organized as follows: the next section presents the nonlinear model of the induction motor. Then, in Section 3, we present the UIO design and the load torque estimation. Using these observers results, in Section 4 one details the synthesis of the output-feedback predictive controller. Simulation results are presented in Section 5 to verify the effectiveness of the proposed controller/observer scheme. Finally, some conclusions are gathered in Section 6.

2 Model of the induction motor The well-known fourth-order model of the IMs, referred to a fixed reference frames, can be presenting the electrical behavior as follows: ⎧ Rr ⎪ ⎪ ⎨ I˙s = b( I2 − pωJ2 )ψr − γ(Rr )Is + aUs , Lr (1) ⎪ R M Rr ⎪ ⎩ ψ˙ r = −( r I2 − pωJ2 )ψr + Is , Lr Lr where Is = [ Isα Isβ ]T , ψr = [ ψrα ψrβ ]T , Us = [ Usα Usβ ]T , respectively, denote the stator currents, the rotor fluxes and the stator input voltages in a fixed stator reference. The paM2 rameters σ, γ, a, b and c are defined by σ = 1 − , Lr Ls

γ(Rr ) =

Rs Rr M 2 1 M + , a = , b = , and 2 σLs σLs Lr σLs σLr Ls

pM , in which Rr and Rs are the resistances, Lr and JLr Ls are the self-inductances, M is the mutual inductance between the stator and rotor, p is the number of pole pairs and Jm is the inertia of the system (motor and load). The subscripts s and r refer to the stator and  rotor.I2 is the 20 −1 dimensional identity matrix and J2 = is a skew1 0 symetric matrix. Taking into account the rotor velocity ω, the mechanical subsystem is written in the following equation: 1 TL . (2) ω˙ = cIsT J2 ψr − Jm It is worth noting that the load torque is considered as a constant unknown disturbance. Moreover, the rotor resistance is assumed to be uncertain with known bounds. Some assumptions will also be considered until further notice. Let us quote hereafter, the subject of this paper:  Design the estimation mechanisms, recovering jointly the unavailable system state (rotor fluxes) and the parameter variable (rotor resistance and load torque), to include them into the control loop in the second step.  Design a control algorithm guaranteeing the tracking of a smooth predefined profile for the motor speed ω, as well as the regulation of the square of the rotor flux modulus ψr 2 at a desired constant value. c=

3

UIO-based rotor resistance estimation

3.1 UIO synthesis The UIO algorithm is related to systems of a specific structure [27], roughly corresponding to the property of the MIMO system presented as follows:  x˙ = f (u, v, x), (3) ¯ = x1 , y = Cx ⎡ 1 ⎤ ⎡ 1 ⎤ x f (u, v, x1 , x2 ) ⎢ x2 ⎥ ⎢ f 2 (u, v, x1 , x2 , x3 ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥, .. x = ⎢ .. ⎥ , f (u, v, x) = ⎢ ⎥ ⎢ q−1 ⎥ ⎢ ⎥ ⎣x ⎦ ⎣ f q−1 (u, v, x) ⎦ xq f q (u, v, x) C¯ = [ In 0n ×n · · · 0n ×n ], 1

1

2

1

q

where  the state x ∈ Rn denotes the state vector with xk ∈ Rnk , q  nk = n; k = 1, . . . , q and p = n1  n2  . . .  nq , k=1

 the input vector w = (u, v) ∈ W the set of bounded absolutely continuous functions with bounded derivatives from R+ into W a compact subset of Rs . One shall suppose that the subvector u ∈ U ⊂ Rs−m of the input W is known while the remaining subvector v ∈ V ⊂ Rm is unknown;  the measured output y ∈ Rp ; and  f (u, v, x) ∈ Rn with f k (u, v, x) ∈ Rnk .

H. Chehimi et al. / J Control Theory Appl 2013 11 (3) 367–375

To start with the unknown input observer design, some hypotheses should be adopted in due courses. In this respect, it is convenient to announce the following assumptions: A1) The state x(t), the control u(t) and the unknown inputs v are bounded, i.e., x(t) ∈ X, u(t) ∈ U and v ∈ V , where X ⊂ Rn , U ⊂ Rs−m and V ∈ Rm are compact sets. A2) There exist αf , βf > 0 such that for all k ∈ {1, . . . , q − 1}, ∀x ∈ X, ∀(u, v) ∈ U × V , αf2 In1  (

∂f k ∂f k T (u, v, x)) ( (u, v, x))  βf2 In1 . ∂xk+1 ∂xk+1

Another important point should be to also be assumed for 1  k  q − 1, for all (u, v) ∈ U × V , the map xk+1 → f k (u, v, x1 , . . . , xk , xk+1 ) from Rnk+1 into Rnk is one to one. 1 1  A3)  The output x can be portioned as follows: x = x11 with x11 ∈ Rm1 , x12 ∈ Rp−m1 and m  x12 m1 < p. Such a partition induces  the following one 1 1 2 f (u, v, x , x ) f 1 (u, v, x1 , x2 ) = 11 that has to satisfy the f2 (u, v, x1 , x2 ) following two conditions:  there exist αv , βv > 0 such that ∀x ∈ X, ∀(u, v) ∈ U ×  ∂f 1 T ∂f 1  1 1 (u, v, x1 , x2 ) (u, v, x1 , x2 )  V , αv2 Im  ∂v ∂v βv2 Im ; and ⎤ ⎡ 1 1 ∂f1 1 2 ∂f1 1 2 (u, v, x (u, v, x , x ) , x ) ⎥ ⎢ 2 ∂v  rank ⎣ ∂x1 ⎦ = n2 +m, 1 ∂f2 ∂f 2 1 2 1 2 (u, v, x , x ) (u, v, x , x ) ∂x2 ∂v 1 2 n1 +n2 , ∀(u, v) ∈ U × V. for all (x , x ) ∈ R A4) The time derivative of the unknown input v(t) is a completely unknown function, ε(t) which is uniformly bounded that sup ε(t)  βε where βε > 0 is a real numt0

ber. If all previous hypotheses are checked, the system observability is clearly shown. Hence, as a preliminary step for the observer design, one introduces a simple notation change 

     1 x ˜1 x x12 1 1 2 x ˜= = = , x ˜ , , x ˜ x ˜2 v x2 C = diag{C1 , C2 }, C1 = [ Im1 0 ], C2 = [ Ip−m1 0 ],   ˜) f˜1 (u, x , f (u, x ˜) = ˜2 ˜) f (u, x   1 1 2 f (u, v, x , x ) 1 1 ˜) = , f˜ (u, x 0 ⎡ ⎤ f21 (u, v, x1 , x2 ) ⎢ 2 ⎥ ⎢ f (u, v, x1 , x2 , x3 ) ⎥ ⎥, ˜) = ⎢ f˜2 (u, x .. ⎢ ⎥ . ⎣ ⎦ f q (u, v, x)

369

    0 ε , ε¯ = . ε= ε 0 Obviously, system (3) is transformed into the following form: ⎧ 1 ˜) + ε, x ˜˙ = f˜1 (u, x ⎪ ⎪ ⎪ ⎨x ˙˜2 = f˜2 (u, x ˜),  (4) ⎪ x11 ⎪ ⎪ y = Cx . ˜= ⎩ x12 The above system can be condensed as follows:  x ˜˙ = f˜(˜ x, u) + ε, y = Cx ˜.

(5)

Therefore, the equation of the observer can also be written in the following form: ˆ˜ − y), (6) ˆ˜˙ = f˜(x ˆ˜, u) − (Λ(x, u, v)) + Δ−1 (θ)K(C x x where Λ+ (x, u, v) isthe left inverse of block diagonal maΛ1 , and trix Λ with Λ( · ) = Λ2 ∂f11 (u, v, x1 , x2 )), ∂v ∂ f¯1 Λ2 = blockdiag(Ip−m1 , 2 (u, x), ∂x ∂ f¯1 ∂ f¯2 (u, x) 3 (u, v, x), . . . , ∂x2 ∂x q−1  ∂ f¯k ∂ f¯1 (u, x) (u, v, x)), 2 k+1 ∂x k=1 ∂x     2Im1 K1 , K1 = , K= K2 Im1 ⎡ ⎤ Cq1 I(p−m1 ) ⎢ 2 ⎥ ⎢ Cq I(p−m1 ) ⎥ q! ⎢ ⎥ with Cqi = K2 = ⎢ .. ⎥ i!(q − i)! . ⎣ ⎦ Cqq I(p−m1 ) for 1  i  q,   Δ1 (θ) Δ(θ) = , Δ2 (θ) 1 1 Δ1 (θ) = blockdiag( q−1 Im1 , 2(q−1) Im1 ), θ θ 1 1 1 Δ2 (θ) = blockdiag( Ip−m1 , 2 Ip−m1 , . . . , q Ip−m1 ), θ θ θ with θ > 0 is a real number representing the only design parameter of the observer. Otherwise, the developed observer is given by the following expression: ⎧ ⎨x ˆ1 , x ˆ2 ) − 2θq−1 (ˆ x11 − x11 ), ˆ˙ 11 = f11 (u, vˆ, x 1 (7) ⎩ vˆ˙ = −θq−1 ( ∂f1 (u, vˆ, x ˆ1 , x ˆ2 ))+ (ˆ x11 − x11 ), ∂v ⎡ ⎤ ⎡ ⎤ ˆ1 , x ˆ2 ) x ˆ˙ 12 f21 (u, vˆ, x ⎢ ˙2 ⎥ ⎢ ⎥ ˆ1 , x ˆ2 , x ˆ3 ) ⎥ ˆ ⎥ ⎢ f 2 (u, vˆ, x ⎢x ⎢ . ⎥=⎢ ⎥ .. ⎢ . ⎥ ⎢ ⎥ . ⎣ . ⎦ ⎣ ⎦ Λ1 = blockdiag(Im1 ,

x ˆ˙ q

ˆ1 , x ˆ2 , . . . , x ˆq ) f q (u, vˆ, x ˆ1 , x ˆ2 , . . . , x ˆq ))+ −(Λ2 (u, vˆ, x

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×Δ−1 ˆ12 − y2 ). 2 (θ)K2 (C2 x In order to appreciate the performance of the unknown input observer, we propose a validation application relative to the studied process. Hereafter, the reconstruction problem of the system states and unknown input, without assuming any model for the unknown inputs, is solved and the joint estimation is successfully achieved for the IM. 3.2 Rotor resistance and flux estimations The purpose of this section is to estimate conjointly the states and the critical parameter such as the ψr and Rr . We should note that, during the design of the recent unknown input observer, TL is considered as a known parameter. This latter assumption will be removed by using the second observer approach. As has been mentioned, let us now apply the UIO design for the induction motor. The full IM model (1)–(2) may be seen as a continous multi-output system, belonging to the class of nonlinear systems (3). Focusing on this fact, one considers the following suitable distribution: ⎡ ⎤ Is ⎢ ⎥  x = ⎣ ψr ⎦ ∈ R5 (n = 5) is the systems state. Thus, ω   Is 1 let us define the coordinate x such as x = ∈ R3 ω (n1 = 3), x2 = ψr ∈ R2 (n2 = 2), for k = 1, 2 and 2  p = n1  n2 , nk = n.  k=1 Is ∈ R3 (p = 3) denotes a vector of directly y = ω measured outputs.  Furthermore, one shall propose that the input vector w = (u, v) ∈ W a compact set of R3 (s = 3). The subvector u = Us ∈ R2 is a known input of w while the remaining subvector v = Rr ∈ Rm (m = 1) is unknown. According to the appropriate choice already proposed for the induction motor, it is easy to see the different elements respectively contracted as follows:     f 1 (u, v, x1 , x2 ) x1 , f (u, v, x) = , x= x2 f 2 (u, v, x) C¯ = [ I3 03×2 ], where I3 is the identity matrix and 03×2 the null matrix. To achieve this end, the next step of basic routines of design algorithm consists in checking the previous hypotheses. In fact, according to the appropriate choice of the two parts of state, the control and the unknown input, it is easy to see that the assumptions A1), A2), A3) and A4) are successively verified. Therefore, one introduces a simple notations change:       x11 x12 x ˜1 1 2 , x ˜ = with , x ˜ = x ˜= 2 x ˜ v x2 x11 = Is ∈ R2 , x12 = ω, v = Rr , x2 = ψr ,

C = diag{C1 , C2 }, C1 = [ I2 0 ], C2 = [ 1 0 ],   f˜1 (˜ x, u) f (˜ x, u) = ˜2 , x, u) f (˜   1 1 2 f (x , x , u, v) 1 1 x, u) = , f˜ (˜ 0   1 1 2 f (x , x , u, v) x, u) = 2 2 . f˜2 (˜ f (x, u, v) Then, the model (3) can be rewritten under the following form: ⎤ ⎧   ⎡ R r ⎪ 1 ⎪ I −pωJ )ψ −γ(R )I +aU b( x ˙ 2 2 r r s s⎦ ⎪ ⎪ , x ˜˙ 1 = 1 = ⎣ Lr ⎪ ⎪ v ˙ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎤ ⎡ ⎪ 1 ⎪   T ⎨ J ψ − T cI 1 L 2 r s x˙ ⎥ (8) ⎢ Jm x ˜˙ 2 = 22 = ⎣ R ⎦, ⎪ M R r r ⎪ x˙ ⎪ −( I − pωJ )ψ + I 2 2 r s ⎪ ⎪ Lr ⎪ ⎪   Lr ⎪ ⎪ 1 ⎪ x1 ⎪ ⎪ ⎪y = C x ˜= . ⎩ x12 Obviously, according to the previous obtained system (8), the observer equation (7) is identified as follows: ∂f11 1 2 (x , x , u, v)), ∂v 1 ¯ ∂f Λ2 = blockdiag(I2 , 2 (x, u)),   ∂x  2 2I2 , K2 = ,  K1 = 1 I2 1 1  Δ1 (θ) = blockdiag( , 2 ), θ θ 1 1 Δ2 (θ) = blockdiag( I2 , 2 I2 ). θ θ  Λ1 = blockdiag(1,

As a result, the appropriate observer for the induction motor can be written in the following form: ⎧⎧ ⎨x ⎪ x1 , x ˆ2 , u, vˆ) − 2θ(ˆ x11 − x11 ), ˆ˙ 11 = f11 (ˆ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ vˆ˙ = −θ2 ( ∂f1 (ˆ ⎪ x1 , x ˆ2 , u, vˆ))+ (ˆ x11 − x11 ), ⎪ ⎪ ⎪  ⎨    ∂v x ˆ˙ 12 f21 (ˆ x1 , x ˆ2 , u, vˆ) (9) = ⎪ 2 2 1 2 3 ⎪ ˙ ⎪ (ˆ x , x ˆ , x ˆ , u, v ˆ ) f x ˆ ⎪ ⎪ ⎪ ⎪ ⎪ x1 , x ˆ2 , u, vˆ))+ −(Λ2 (ˆ ⎪ ⎪ ⎩ −1 ×Δ2 (θ)K2 (C2 x ˆ12 − y2 ). Before giving a final expression of the UIO, let us recall the suitable partition of state and unknown input of the IM model as follows: n = 5, n1 = 3, n2 = 2, q = 2, p = 3, s = 3, m = 2, m1 = 1, x11 = y1 = Is , x12 = y2 = ω, T 2 x1T x1 = [x1T 1 2 ] , x = ψr , v = Rr . Thus, according to (9), the appropriate observer can be de-

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signed in the following form: ⎧⎧ ˆ ⎪ ⎪⎪ ⎪ Iˆ˙ = b( Rr I − pˆ ⎪ ⎪ ω J2 )ψˆr − (Rs a + Rˆr b)Iˆs 2 s ⎪ ⎪ ⎪ ⎨ Lr ⎪ ⎪ ⎪ ⎪ +aUs − 2θ(yˆ1 − y1 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b ⎪ ⎪ ⎪ ⎩ Rˆr = −θ2 ( I2 ψˆr − bIˆs ) + (yˆ1 − y1 ), ⎪ ⎪ L ⎪ r ⎤ ⎡ ⎪ ⎪ 1   ⎪ ⎪ TL cIˆsT J2 ψˆr − ⎨ ω ˙ˆ Jm ⎥ ⎢ ˆr ˆr ⎦ ˙ =⎣ R R ˆ ⎪ ψ ˆ ˆ r ⎪ −( I2 − pˆ ω J2 )ψr + M Is ⎪ ⎪ ⎪ Lr Lr ⎪ ⎪ ⎡ ⎤−1 ⎪ ⎪     1 ⎪ + ⎪ ⎪ 0 ⎥ ⎪ 1 0 0 2 ⎢ ⎪ ⎪ − ⎣θ 1 ⎦ ⎪ T ˆ ⎪ 0 c I 1 J ⎪ 2 s 0 2 ⎪ ⎪ ⎪ θ ⎪ ⎩ ×(ˆ y − y ). 2

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It should be emphasized that here, no actual coordinate change is visualized at the level of the estimate load torque equation. The result of the proposed heuristic solution is quite simple to be interconnected with the UIO in a preliminary step, before implementing them in the control loop.

5 (10)

2

Equation (10) presents the result of the UIO design method applied to IM. In particular, the whole state and the unknown inputs are simultaneously estimated by using the real number θ, representing the only design parameter of the observer. This parameter should be chosen as a compromise between the speed of convergence and noise rejection.

4 Load torque estimation The main contribution of this article is the design of an UIO for the IMs. However, the performance of this observer is limited if the estimation of flux and rotor resistance is obtained. The load torque knowledge is also required for a high performance control. The dynamics of the previous parameter may occur due to changes in solids conveying, melting, mixing, melt conveying, etc. Hence, it is useful for the detection of possible incipient faults and updating the working conditions of mechanical loads. Thus, to overcome this problem, another observer is proposed which will be combined with the previous observer. The solution consists to change reference system by referring to the field oriented control approach [28–29]. Indeed, it is easy to rewrite the IMs model in a rotating reference frame of axes (d, q), where the axis is oriented like the rotor flux vector. As far as the stator current components are concerned, it is well known for its dependence of the flux magnitude and the motor torque, they are directly proportional. They can be obtained by using the latter equation:      cos θs sin θs Isα Isd = , (11) Isq − sin θs cos θs Isβ where the θs is the angle between the rotating d axis and the fixed α axis defined as follows: ψrβ θs = arctan( ). (12) ψrα Now, the Isq current expression provided by (11) is used to determinate the load torque estimation, but from an error point of view. In fact, we have dTˆL = λTL ((Iˆsα −Isα ) sin θˆs −(Iˆsβ −Isβ ) cos θˆs ), (13) dt with λTL > 0 and ψˆrβ ). (14) θˆs = arctan( ψˆrα

Synthesis of the output feedback predictive controller

The IMs is considered as a case study of the application of the predictive controller. In fact, the principal idea in this section is to track the prescribed trajectories and respect the certainty equivalence principle. Updating, the difficulty of the unavailable states and parameter is already solved in the last section. The design procedure starts by introducing a mild change of variables. Through the function Φ, assumed as a globally lipschitzian diffeomorphism in z, the IMs equations (1) are given into a special form that will 5 5 be ⎡ easier ⎤ Indeed, let Φ : R → R , x = ⎤ to work ⎡with. 1   z Iˆs z11 ⎢ˆ ⎥ ⎢ 2⎥ ∈ R2 and ⎣ ψr ⎦ → z = ⎣ z ⎦, where z 1 = z21 ˆ ω ˆ θs   2 z z 2 = 12 ∈ R2 , such as z2 ⎧ 1 2 2 ˆ , z21 = ψˆr 2 = ψˆrα + ψˆrβ , z1 = ω ⎪ ⎪ ⎪ ⎪ 2 pM ˆT ˆ ⎨ z1 = I J2 ψr , z22 = IˆsT ψˆr , (15) Lr s ⎪ ˆrβ ⎪ ψ ⎪ ⎪ ). ⎩ θˆs = arctan( ψˆrα Using this transformation, one can write the initial model (15) in the new coordinates under the following representation: ⎧ ˆr ˆr 1 2 1 ˆ MR R ⎪ 1 ⎪ ⎪ z˙1 = z1 − z22 − 2 z21 , TL , z˙21 = 2 ⎪ ⎪ Jm Jm Lr Lr ⎪ ⎪ ⎪ ˆr ⎪ bp2 M 1 1 R p2 M 1 2 ⎪ 2 2 ⎪ z z − (γ + )z − z z z˙1 = − ⎪ ⎪ Lr 1 2 Lr 1 Lr 1 2 ⎪ ⎪ ⎪ ⎪ pM  1 ⎪ ⎪ ⎪ − z1 (U1 sin θˆs − U2 cos θˆs ), ⎪ ⎪ σL L r s ⎪ ⎪ ⎨ ˆr ˆr 2bR Lr 1 2 R (16) z1 z1 − (γ + z˙22 = − + )z 2 ⎪ Lr M Lr 2 ⎪ ⎪ ⎪ ⎪ ⎪ ˆr 1 ⎪ MR Lr 2 2 ⎪ 2 2 ⎪ + ⎪ 1 ((z2 ) + ( pM z1 ) ) ⎪ L z ⎪ r 2 ⎪ ⎪ 1  1 ⎪ ⎪ ˆ ˆ z + ⎪ 1 (U1 cos θs + U2 sin θs ), ⎪ ⎪ σLs ⎪ ⎪ ⎪ ˆ z2 R ⎪ ⎪ ⎩ θˆ˙s = pz11 + r 11 , y = z 1 . p z2 The above system of form (16) can be condensed as follows: ⎧ 1 1 ˆ ˆ 2 ⎪ ⎪ z˙ = A(Rr )z + ϕ(z , TL ), ⎪ ⎪ ˆ r ), ⎪ ⎨ z˙ 2 = B(z, θˆs )U + g(z, R 2 ˆ (17) Rr z 1 1 ˆ˙ ⎪ ⎪ ⎪ θs = pz1 + p z21 , ⎪ ⎪ ⎩ y = z1, where z = [z 1T z 2T ]T ∈ R4 is the state vector, y = z 1 ∈

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R2 is the measured output, and u ∈ U ⊂ R2 . ⎡ ⎤ ⎡ 1 ⎤ −TˆL 0 ⎢ ⎥ J ⎥ ˆ r ) = ⎢ Jm ⎥, ˆr) = ⎢ , ϕ(z 1 , TˆL , R A(R ⎣ m ⎦ ˆ ⎣ ˆ Rr Rr ⎦ 0 2M −2z21 Lr ⎡ ⎤Lr pM pM  sin θˆs cos θˆs ⎦ − 1 , z21 ⎣ Lr B(z, θˆs ) = Lr σLs cos θˆs sin θˆs ˆ r ) = [ g1 (z) g2 (z) ]T , g(z, R 2 2 ˆ r ) = − Kp M z 1 z 1 − (γ + Rr )z 2 − p M z 1 z 2 , g1 (z, R 1 2 Lr Lr 1 Lr 1 2 ˆ r ) = − 2KRr z 1 + Lr z 1 z 2 − (γ + Rr )z 2 g2 (z, R Lr 2 M 1 2 Lr 2 M Rr 1 2 2 M Rr 1 Lr 2 2 z ) . + (z ) + ( Lr z21 2 Lr z21 pM 1 1T 1T T zd2 ] be the referLet zd = [zd1T zd2T ]T with zd1 = [zd1 ence trajectories corresponding to the mechanical speed and the modulus rotor flux. In fact, ˆ r )), zd2 = A−1 (z˙d1 − ϕ(zd1 , TˆL , R 2 ˆ ˆ B(z, θs )Ud = z˙d − g(zd , Rr ). In the spirit of the control law proposed in [10], the analytic solution of predictive control, without taking into account any constraints, can be given by the following expression:

¯ (z, R ˆr) = U

⎧ Umin , if u < Umin , ⎪ ⎪ ⎪ ⎨ B(z, θˆ )−1 (z˙ 2 − g(z , R ˆ r ) + δ(ez )), s d ⎪ ⎪ ⎪ ⎩

d

if Umin < u < Umax , Umax , if u > Umax ,

(18) where ˆ r )K1 e1 + (K2 + dϕ(R ˆ r ))e2 δ(ez ) = −(A−1 (R z z −1 ˆ +A1 Δϕ(K2 + dϕ(Rr ))), ˆ ˆ ˆ r ) = (0, − 2Rr e1 )T , dϕ(R ˆ r ) = diag{0, − 2Rr }, Δϕ(R z2 Lr Lr 10 10 5 5 }, K2 = diag{ , }. K1 = diag{ 2 , 3Tp 3Tp2 2Tp 2Tp Umin and Umax are respectively the lower and the upper control bounds. For the output feedback controller, we note that the resulting state estimate of the UIO (10) and load torque observer (13) will replace the unavailable state and parameters given in (18).

where ψrn and ωrn are respectively the nominal value of the rotor flux and speed. The proposed methods have been tested with MATLAB/Simulink environment. The IM parameters used during simulation are summarized in Table 1. Table 1 IM parameters used in simulations. Active power Number of pole pairs Rotor per-phase inductance Stator per-phase inductance Stator per-phase resistance Mutual inductance Moment of inertia

1.5 kW 2 0.464 H 0.464 H 5.72 Ω 8.99 H 0.0049 kg · m2

The tuning parameters of the output feedback controller are as follows:  Trapezoidal profile for the speed reference trajectory.  The norm of flux vector is constant: ψ2reference = 0.8 Weber2 .  Prediction horizon of the predictive controller: Tp = 75 ms.  UIO design parameters: θ1 = 300.  Parameter of load torque estimation: λTL = 150. Figs. 1 and 2 show the simulation results obtained in an open loop. Both graphs of the actual and estimated values are compared respectively for the dynamic behavior of the rotor speed, the rotor resistance and the load torque. Despite some overshoots during the initial transient, one can remark the good tracking of the induction motor variables. Hence, we can confirm the combining effectiveness between the UIO and the load torque estimator. The rotor flux estimation is depicted in Fig. 2 (c)–(d) where estimation error converges quickly to zero.

6 Simulation results In this section, we illustrate the effectiveness of our proposed control, combined with the estimation techniques. In addition, the means of field weakening, proposed to overcome the overspeed problem is tested. Indeed, the principle of this conventional method is to set the rotor-flux reference inversely proportional to the rotor speed (‘1/ωr ’ method). The principle of this technique is presented according to the expression below:  if ωr  ωrn , ψrn , ωrn ψr = , if ωr > ωrn , ψrn ωr

Fig. 1 Estimated variables in the open loop: (a) angular speed response ω, (b) rotor resistance tracking Rr , and (c) load torque estimation TL .

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The block scheme chart used set-up to test the output feedback predictive controller with observers is presented in Fig. 3. Indeed, the synthesized output feedback predictive controller is tested of point of view the driving induction motor capability and tracking the prescribed trajectories (ωd , ψrd 2 ). The robustness against the variation of state and parameters is considered in our study. Hence, profiting only of the stator current and speed measurements, the combination between the both estimation mechanisms guaranteed the simultaneous estimation of the internal state variables and the time varying parameters. As results, the stator flux components, load torque and the rotor resistance are considered as inputs for the controller algorithm, which generates the required stator voltage for satisfying the tracking objective.

Fig. 2 Stator current and rotor flux estimations.

Fig. 3 Block diagram of the output feedback predictive controller with UIO structure.

The performances of whole system are illustrated in Fig. 4. Graph (a) shows the response of the mechanical speed relative to the trapezoidal reference trajectory, a negligible error is observed. The reference trajectory of angular speed allows to test the effectiveness and robustness of the ‘Controller + Observer’ in the different possible regimes, that can take place during the induction motor drive. Indeed, the mechanical speed is carried to a great value (200 rad/s) from 2 until 3 s, to test the overspeed regime. Moreover, low operating speed which represent a challenge for induction machines is taking account in the interval between 4.8 and 6.8 s. Hence, the comparison between the reference speed and the value provided by the predictive controller, confirms its good quality in the different steady state. We see also in (b) that the flux modulus tracks correctly its reference even during the conventional field-weakening phenomenon. Graph (c) demonstrates the estimation of the time varying of the rotor resistance with a variation of +100% of its rated value. Despite, the little fluctuations presented in the esti-

mation, according to the variation speed, a rallying behavior between the guess and the true values is achieved. We see also in (d), an acceptable performance for the load torque estimation.

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Fig. 6 (α, β) components of voltage vectors.

7

Fig. 4 Estimated variables in the open loop: (a) angular speed response, (b) rotor resistance tracking, (c) load torque estimation, and (d) performance for the load torque estimation.

The resulting components of the stator currents, rotor flux and voltage vectors are reported in Figs. 5 and 6, and all of them stay in the predefined bounds.

Conclusions

In this paper, the problem of the predictive controller for the induction motor with the lack of knowledge about state and the actual values of some critical parameters, is studied. The major contribution is the robustness of the proposed design towards uncertainties in the rotor resistance and load torque, which are usually the causes of some deterioration in high performances IMs drives. This is achieved by using simultaneously two estimation mechanisms. The simulation results prove the performances of the proposed control scheme, as well as the good capabilities of the observers to online estimation of unmeasured variables and parameters. Therefore, the theoretical test creates a tempting project which deserves to be implemented in real-time. References [1] M. Ghanes, G. Zheng. On sensorless induction motor drives: sliding mode observer and output feedback controller. IEEE Transactions on Industrial Electronics, 2009, 56(9): 3404 – 3413. [2] M. Barut, M. G. S. Bogosyan. Speed sensorless direct torque control of IMs with rotor resistance estimation. Energy Conversion and Management, 2005, 46(3): 335 – 349. [3] W. Chen. Predictive control of general nonlinear systems using approximation. IEE Proceedings-Control Theory and Applications, 2004, 151(2): 137 – 144. [4] W. Chen, D. Ballance, P. Gawthrop. Optimal control of nonlinear systems: a predictive control approach. Automatica, 2003, 39(4): 633 – 641. [5] H. G. Lez, M. A. Duarte-Mermoud, I. Pelissier, et al. A novel induction motor control scheme using IDA-PBC. Journal of Control Theory and Applications, 2008, 6(1): 59 – 68. [6] M. Montanari, S. Peresada, A. Tilli. A speed-sensorless indirect field oriented control for induction motors based on high gain speed estimation. Automatica, 2006, 42(10): 1637 – 1650. [7] S. H. Kim, S. K. Sul. Voltage control strategy for maximum torque operation of an induction machine in the field-weakening region. IEEE Transactions on Industrial Electronics, 1997, 44(4): 512 – 518. [8] G. Kenn, T. Ahmed-Ali, F. Lamnabhi-Lagarriguec, et al. An improved rotor resistance estimator for induction motors adaptive control. Electric Power Systems Research, 2011, 81(4): 930 – 941. [9] G. Kenn, T. Ahmed-Ali, F. Lamnabhi-Lagarriguec, et al. Nonlinear systems time-varying parameter estimation: Application to induction motors. Electric Power Systems Research, 2008, 78(11): 1881 – 1888.

Fig. 5 Stator current and rotor flux estimations.

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[27] M. Triki, M. Farza, T. Maatoug, et al. Unknown inputs observers for a class of nonlinear systems. International Journal of Sciences and Techniques of Automatic control and computer engineering, 2010, 4(1): 1218 – 1229. [28] F. Blaschke. The principle of field orientation as applied to the new transvector closed loop control system for rotating field machines. Siemens Review, 1972, 39: 84 – 90. [29] C. Aurora, A. Ferrara. Design and experimental test of a speed/flux sliding mode observer for sensorless induction motors. Proceedings of the American Control Conference. Piscataway: IEEE, 2007: 5881 – 5886. Hanen CHEHIMI received her B.E. and M.S. degrees in Automatic from the National Engineering School of Monastir, Tunisia, in 2007 and 2009, respectively. She is currently a Ph.D. candidate at Electrical Engineering in National Engineering School of Monastir. His research interests are focused on nonlinear control, theoretical aspects of nonlinear observer design and control of induction motors. E-mail: [email protected]. ¨ was born in Monastir, Tunisia Salim HADJ SAID in 1978. He received his M.S. degree in Automatic and Signal Processing and his Ph.D. in Electric Engineering from National Engineering School of Tunis, in 2004 and 2009, respectively. He is currently an assistant professor of Automatic at Preparatory Institute for Engineering Studies of Monastir, Tunisia. His research interests include state observation, predictive and adaptive control of linear and nonlinear systems. E-mail: [email protected]. Faouzi M’SAHLI received his B.S. and M.S. degrees from ENSET, Tunis, Tunisia in 1987 and 1989, respectively. In 1995, he obtained his Ph.D. degree in Electrical Engineering from ENIT, Tunisia. He is currently a professor of Electrical Engineering at National School of Engineers, Monastir, Tunisia. His research interests include modeling, identification, predictive and adaptive control of linear and nonlinear systems. He has published over 80 technical papers and co-author of a book ‘Identification et commande numrique des procds industriels’, Technip editions, Paris, France. E-mail: [email protected].