Observer design for induction motor: an approach based on the mean value theorem

Abstract

In this paper, observer design for an induction motor has been investigated. The peculiarity of this paper is the synthesis of a mono-Luenberger observer for highly coupled system. To transform the nonlinear error dynamics for the induction motor into the linear parametric varying (LPV) system, the differential mean value theorem combined with the sector nonlinearity transformation has been used. Stability conditions based on the Lyapunov function lead to solvability of a set of linear matrix inequalities. The proposed observer guarantees the global exponential convergence to zero of the estimation error. Finally, the simulation results are given to show the performance of the observer design.

Keywords

observer design / differential mean value theorem (DMVT) / sector nonlinearity transformation / linear matrix inequalities (LMI) / induction motor

Introduction

The induction motor (IM) is the most widely used motor in industrial driving system applications due to its performance characteristics. The IM is rugged, reliable, and low cost and has simple hardware structure. However, it is very difficult to achieve high performance with the IM, due to the intrinsic nonlinear coupling between the dynamics of the electrical and mechanical parts.

In the available literature, most of the works are based on linear models of the studied systems [ 1, 2]. However, the use of nonlinear models seems very interesting and appropriate because it allows an accurate representation of the system on a wide operating range. But the disadvantage of the nonlinear approach is the lack of a unified and general solution for observer design. For this reason, the authors are oriented to specific classes of nonlinear systems, such as Lipschitz systems. Many approaches have been then elaborated, for example, using the immersion, Lie algebraic transformations, etc [ 3, 4], and high gain observer, etc [ 5- 7].

To achieve a more accurate and robust speed estimation performance of the induction motor, many strategies were proposed in literature such as adaptive methods, sliding-mode, extended Kalman filter, and extended Luenberger observer. A robust adaptive observer for sensorless IM was designed based on the linearized dynamic equation and linear matrix inequality method [ 8]. Sliding mode observers to estimate rotor flux were proposed [ 9, 10]. An extended Kalman filter method was adapted to estimate the rotor flux of induction motor [ 11]. Nonlinear Luenberger observers were proposed for sensorless vector control of IM [ 12]. A nonlinear observer was used to estimate the IM flux which proved to be satisfactory [ 13]. An approach based on the nonlinear transformation of the original system into a linear was proposed [ 14, 15], but it was very difficult to achieve this due to the strong conditions under which these transformations existed.

The Takagi-Sugeno (TS) fuzzy approach was extensively used to nonlinear systems [ 16- 18]. The basic idea was to decompose the model of a nonlinear system into a set of linear subsystems with associated nonlinear weighting functions [ 19].

The class of Lipschitz nonlinear systems was also studied [ 20, 21]. An adaptive resilient observer for a Lipschitz nonlinear system was designed [ 20], and an observer based on differential mean value theorem (DMVT) for a nonlinear system was proposed [ 21]. Another linear matrix inequalities (LMI)-based observer design for a class of Lipschitz nonlinear dynamical systems was also suggested [ 22, 23].

The main idea of this paper is still to compute the observer gain in order that the linear part counteracts the effect of the nonlinear part. Contrarily to the other methods, the use of the mean value theorem makes it possible to obtain a solution, even for large Lipschitz constant.

Recently, a new observer based DMVT has been presented [ 24] for a TS fuzzy system with unmeasurable premise variables.

In this paper, the problem of observer design for induction motor is considered. An approach combining the DMVT and the sector nonlinearity transformation is proposed.

First, the IM model is written as a TS system. Then, based on the mean value theorem, the state estimation error dynamics are transformed to a quasi-LPV system. Next, the sector nonlinearity transformation is used to transform the quasi-LPV system in a TS system. Finally, the stability of the latter is studied with the Lyapunov theory.

Observer design

Problem statement

This section presents an efficient methodology to design observers for the class of nonlinear systems; consider the following nonlinear system described in Ref. [ 25]:

(1) \{\begin{array}{l}\dot{\mathit{x}}(t)=\mathit{f}(\mathit{x}(t))+\mathit{g}(\mathit{x}(t))\mathit{u},\hfill \\ \mathit{y}=\mathit{C}\mathit{x}(t),\hfill \end{array}

where \mathit{x}\in R^n is the state vector, \mathit{u}\in R^p is the input vector, and \mathit{y} is the output measurement vector.

Equation (1) could be represented as a TS model as

(2) \{\begin{array}{l}\dot{\mathit{x}}\left(t\right)={\displaystyle \sum _{i=1}^r{\mathit{h}}_i}(\mathit{x}\left(t\right)\left({\mathit{A}}_i\mathit{x}\left(t\right)+{\mathit{B}}_i\mathit{u}\left(t\right)\right),\hfill \\ \mathit{y}=\mathit{C}\mathit{x}\left(t\right).\hfill \end{array}

The proposed observer is described in Ref. [ 25] as

(3) \dot{\widehat{\mathit{x}}}(t)={\displaystyle \sum _{i=\mathrm{1}}^r{\mathit{h}}_i}(\widehat{\mathit{x}}(t)\left[{\mathit{A}}_i\widehat{\mathit{x}}(t)+{\mathit{B}}_i\mathit{u}(t)+{\mathit{L}}_{\mathrm{0}}\left(\mathit{y}(t)-\widehat{\mathit{y}}(t)\right)\right].

Let us introduce the following matrices

(4) {\mathit{A}}_{\mathrm{0}}={\displaystyle \frac{\mathrm{1}}{r}}{\displaystyle \sum _{i=\mathrm{1}}^r{\mathit{A}}_i},{\mathit{B}}_{\mathrm{0}}={\displaystyle \frac{\mathrm{1}}{r}}{\displaystyle \sum _{i=\mathrm{1}}^r{\mathit{B}}_i},{\overline{\mathit{A}}}_i={\mathit{A}}_i-{\mathit{A}}_{\mathrm{0}},{\overline{\mathit{B}}}_i={\mathit{B}}_i-{\mathit{B}}_{\mathrm{0}}.

Then, it is easy to rewrite Eq. (2) into the Lipchitzien form as

(5) \{\begin{array}{l}\dot{\mathit{x}}(t)={\mathit{A}}_{\mathrm{0}}\mathit{x}(t)+{\mathit{B}}_{\mathrm{0}}\mathit{u}(t)+{\displaystyle \sum _{i=\mathrm{1}}^r{\mathit{h}}_i}(\mathit{x}(t)\left({\overline{\mathit{A}}}_i\mathit{x}(t)+{\overline{\mathit{B}}}_i\mathit{u}(t)\right),\hfill \\ \mathit{y}=\mathit{C}\mathit{x}(t).\hfill \end{array}

The matrices {\mathit{A}}_{\mathrm{0}} and {\mathit{B}}_{\mathrm{0}} play the role of nominal values of the system.

The state Eq. (3) of the observer can also be presented as

(6) \dot{\widehat{\mathit{x}}}(t)={\mathit{A}}_{\mathrm{0}}\widehat{\mathit{x}}(t)+{\mathit{B}}_{\mathrm{0}}\mathit{u}(t)+{\displaystyle \sum _{i=\mathrm{1}}^r{\mathit{h}}_i}(\widehat{\mathit{x}}(t)\left({\overline{\mathit{A}}}_i\widehat{\mathit{x}}(t)+{\overline{\mathit{B}}}_i\mathit{u}(t)\right)+{\mathit{L}}_{\mathrm{0}}\left(\mathit{y}(t)-\widehat{\mathit{y}}(t)\right).

Let us define a new function Φ(x,u)

(7) \mathbf{\Phi }(\mathit{x},\mathit{u})={\displaystyle \sum _{i=\mathrm{1}}^r{\mathit{h}}_i}(\mathit{x}(t)\left({\overline{\mathit{A}}}_i\mathit{x}(t)+{\overline{\mathit{B}}}_i\mathit{u}(t)\right).

The dynamic of the state estimation error e(t) is given by

(8) \{\begin{array}{l}\mathit{e}(t)=\mathit{x}(t)-\widehat{\mathit{x}}(t),\hfill \\ \dot{\mathit{e}}(t)=({\mathit{A}}_{\mathrm{0}}-{\mathit{L}}_{\mathrm{0}})\mathit{e}(t)+\left(\mathbf{\Phi }(\mathit{x},\mathit{u})-\mathbf{\Phi }\left(\widehat{\mathit{x}},\mathit{u}\right)\right).\hfill \end{array}

Note that the stability analysis of Eq. (8) cannot be directly achieved. The objective is to find the gain L₀ of Eq. (6) that stabilizes the state estimation error in Eq. (8).

DMVT

In this section, the mean value theorem for vector functions is presented in order to develop the observer gain in the next section [ 24].

Lemma 1 Consider a vector function \mathit{}\mathit{f}:{\mathbf{R}}^n\to {\mathbf{R}}^q as

(9) \mathit{f}(\mathit{x})={\displaystyle \sum _{i=\mathrm{1}}^q{\mathit{e}}_q(i)}{\mathit{f}}_i(\mathit{x}),

where f_i is the ith component of f.

And e_q is defined by

(10) {\mathit{e}}_q\left(i\right)={\left(\begin{array}{ccccccc}0& \cdots & 0& 1& 0& \cdots & 0\\ 1& i-1& i& i+1& q\end{array}\right)}^T.

Theorem 1

Consider f_i:{\mathbf{R}}^n\to \mathbf{R} . Let \mathit{b}\in {\mathbf{R}}^n . f_i is assumed to be differentiable on [a, b]. Then there exists a constant vector {\mathit{\xi }}^i\in [a,b], such that

(11) f_i(\mathit{a})-f_i(\mathit{b})={\displaystyle \frac{\partial f_i({\mathit{\xi }}^i)}{\partial \mathit{x}}}(\mathit{a}-\mathit{b}).

Applying the theorem to Eq. (9), it is obtained for \mathit{a}\mathit{,}\mathit{}\mathit{b}\in {\mathbf{R}}^n

(12) f_i(\mathit{a})-f_i(\mathit{b})={\displaystyle \sum _{i=\mathrm{1}}^n{\displaystyle \sum _{j=\mathrm{1}}^n{\mathit{e}}_n(i){\mathit{e}}_{n+m}^T(i)}}{\displaystyle \frac{\partial f_i({\mathit{\xi }}^i)}{\partial x_j}}(\mathit{a}-\mathit{b}).

The observation problem consists of finding a gain L₀ such that the observer error converges exponentially and asymptotically toward zero.

Comparing Eq. (5) with Eq. (7), it can be found that

(13) \mathbf{\Phi }(\mathit{x},\mathit{u})=(\mathit{f}(\mathit{x})-{\mathit{A}}_{\mathrm{0}}\mathit{x})+(\mathit{g}(\mathit{x})\mathit{u}-{\mathit{B}}_{\mathrm{0}}\mathit{u}).

From Eq. (13), and using the DMVT, there exists

\mathit{\xi }(t)\in \left[\mathit{x},\widehat{\mathit{x}}\right],

such that:

(14) \mathbf{\Phi }(\mathit{x})-\mathbf{\Phi }(\widehat{\mathit{x}})={\displaystyle \frac{\partial \mathbf{\Phi }(\mathit{\xi })}{\partial \mathit{x}}}\left(\mathit{x}-\widehat{\mathit{x}}\right).

Then, Eq. (14) could be written as

(15) \mathbf{\Phi }(\mathit{x})-\mathbf{\Phi }(\widehat{\mathit{x}})={\displaystyle \sum _{i=\mathrm{1}}^n{\displaystyle \sum _{j=\mathrm{1}}^n{\mathit{e}}_n(i){\mathit{e}}_n^T(j)}}{\displaystyle \frac{\partial {\mathbf{\Phi }}_i({\mathit{\xi }}^i)}{\partial {\mathit{x}}_j}}\left(\mathit{x}-\widehat{\mathit{x}}\right).

Injecting Eq. (15) into Eq. (8), the dynamics of the observer error becomes

(16) \dot{\mathit{e}}(t)=\left({\mathit{A}}_{\mathrm{0}}-{\mathit{L}}_{\mathrm{0}}\mathit{C}+{\displaystyle \sum _{i=\mathrm{1}}^n{\displaystyle \sum _{j=\mathrm{1}}^n{\mathit{e}}_n(i){\mathit{e}}_n^T(j)}}{\displaystyle \frac{\partial {\mathrm{\Phi }}_i({\mathit{\xi }}^i)}{\partial x_j}}\right)\mathit{e}.

Assumption

It is assumed that the functions \mathbf{\Phi }(\mathit{x}) is a differentiable function satisfying

(17) {\alpha }_{ij}\le {\displaystyle \frac{\partial {\mathrm{\Phi }}_i({\mathit{\xi }}^i)}{\partial x_j}}\le {\beta }_{ij}.

where \partial {\mathrm{\Phi }}_i({\mathit{\xi }}^i)/\partial x_j can be written as

(18) {\displaystyle \frac{\partial {\mathrm{\Phi }}_i({\mathit{\xi }}^i)}{\partial x_j}}={\delta }_{ij}^{\mathrm{1}}({\mathit{\xi }}^i){\alpha }_{ij}+{\delta }_{ij}^{\mathrm{2}}({\mathit{\xi }}^i){\beta }_{ij},

where

(19) \{\begin{array}{l}\mathrm{0}\le {\delta }_{ij}^{\mathrm{1}}={\displaystyle \frac{\partial {\mathrm{\Phi }}_i({\mathit{\xi }}^i)/\partial x_j-{\alpha }_{ij}}{{\beta }_{ij}-{\alpha }_{ij}}}\le \mathrm{1},\hfill \\ \mathrm{0}\le {\delta }_{ij}^{\mathrm{2}}={\displaystyle \frac{{\beta }_{ij}-\partial {\mathrm{\Phi }}_i({\mathit{\xi }}^i)/\partial x_j}{{\beta }_{ij}-{\alpha }_{ij}}}\le \mathrm{1},\hfill \\ {\delta }_{ij}^{\mathrm{1}}+{\delta }_{ij}^{\mathrm{2}}=\mathrm{1.}\hfill \end{array}

Consequently

(20) \dot{\mathit{e}}(t)=\left({\mathit{A}}_{\mathrm{0}}-{\mathit{L}}_{\mathrm{0}}\mathit{C}+{\displaystyle \sum _{i=\mathrm{1}}^n{\displaystyle \sum _{j=\mathrm{1}}^n{\displaystyle \sum _{k=\mathrm{1}}^{\mathrm{2}}{\delta }_{ij}^k}{\mathit{e}}_n(i){\mathit{e}}_n^T(j)}}{\alpha }_{ij}^k\right)\mathit{e}(t),

where {\alpha }_{ij}^{\mathrm{1}}={\alpha }_{ij} and {\alpha }_{ij}^{\mathrm{2}}={\beta }_{ij}.

Finally, the dynamic state estimation error can be represented as

(21) \dot{\mathit{e}}(t)={\displaystyle \sum _{i=\mathrm{1}}^r{\mu }_i(\mathit{\xi }(t))}{\mathit{H}}_i\mathit{e}(t),

(22) {\mathit{H}}_i={\mathit{A}}_i^{*}+{\mathit{A}}_{\mathrm{0}}-{\mathit{L}}_{\mathrm{0}}\mathit{C},

where r\le {\mathrm{2}}^{n^{\mathrm{2}}} , and {\mathit{A}}_i^{*} depends on {\alpha }_{ij}^k .

The stability of the state estimation error in Eq. (21) is studied by the quadratic Lyapunov function defined by

(23) V(\mathit{e}(t))={\mathit{e}}^T(t)\mathit{P}\mathit{e}(t),\mathit{P}\mathit{=}{\mathit{P}}^T>0.

The state estimation error asymptotically converges toward zero if there exists a matrix \mathit{P}\mathit{=}{\mathit{P}}^T>\mathbf{0} such that the following LMIs are feasible (for more details see [ 24])

(24)

where the observer gain is given by

(25) {\mathit{L}}_{\mathrm{0}}={\mathit{P}}^{-\mathrm{1}}\mathit{N}.

Dynamical model of induction motor

Let

(i_{sd},i_{sq}),({\mathrm{\Psi }}_{rd},{\mathrm{\Psi }}_{rq}),{\omega }_s,(u_{sd},u_{sq})

denote the components of the stator currents, rotor fluxes, electrical speed of stator, and the stator voltages, respectively.

The electromagnetic dynamic model of the IM in the synchronous d-q reference frame can be described by Ref. [ 2].

(26) \dot{\mathit{x}}=\mathit{f}(\mathit{x})+\mathit{g}(\mathit{x})\mathit{u}+\mathit{v},

where

\mathit{f}\left(\mathit{x}\right)=\left(\begin{array}{l}-\gamma i_{sd}+{\omega }_s{i}_{sq}+{\displaystyle \frac{k_s}{{\tau }_r}}{\mathrm{\Psi }}_{rd}+k_s{n}_p{\omega }_m{\mathrm{\Psi }}_{rq}\hfill \\ -{\omega }_s{i}_{sd}-\gamma i_{sq}-k_s{n}_p{\omega }_m{\mathrm{\Psi }}_{rd}+{\displaystyle \frac{k_s}{{\tau }_r}}{\mathrm{\Psi }}_{rq}\hfill \\ {\displaystyle \frac{M}{{\tau }_r}}{i}_{sd}-{\displaystyle \frac{1}{{\tau }_r}}{\mathrm{\Psi }}_{rd}+\left({\omega }_s-n_p{\omega }_m\right){\mathrm{\Psi }}_{rq}\hfill \\ {\displaystyle \frac{M}{{\tau }_r}}{i}_{sq}-\left({\omega }_s-n_p{\omega }_m\right){\mathrm{\Psi }}_{rd}+{\displaystyle \frac{1}{{\tau }_r}}{\mathrm{\Psi }}_{rq}\hfill \\ {\displaystyle \frac{n_{p}M}{J{L}_r}}\left({\mathrm{\Psi }}_{rd}{i}_{sq}-{\mathrm{\Psi }}_{rq}{i}_{sd}\right)-{\displaystyle \frac{f}{J}}{\omega }_m\hfill \end{array}\right),\mathit{g}\left(\mathit{x}\right)={\left(\begin{array}{ccccc}{\displaystyle \frac{1}{\sigma L_s}}& 0& 0& 0& 0\\ 0& {\displaystyle \frac{1}{\sigma L_s}}& 0& 0& 0\end{array}\right)}^T,

\mathit{x}={\left(\begin{array}{lllll}{i}_{sd}\hfill & i_{sq}\hfill & {\mathrm{\Psi }}_{rd}\hfill & {\mathrm{\Psi }}_{rq}\hfill & {\omega }_m\hfill \end{array}\right)}^{\mathrm{T}},\mathit{u}={\left(\begin{array}{ll}{u}_{sd}& u_{sq}\end{array}\right)}^T,\mathit{v}={\left(\begin{array}{lllll}\mathrm{0}\hfill & \mathrm{0}\hfill & \mathrm{0}\hfill & \mathrm{0}\hfill & -{\displaystyle \frac{C_r}{J}}\hfill \end{array}\right)}^{\mathrm{T}}

\gamma =\left({\displaystyle \frac{\mathrm{1}}{\sigma {\tau }_s}}+{\displaystyle \frac{\mathrm{1}-\sigma }{\sigma {\tau }_r}}\right),k_s={\displaystyle \frac{M}{\sigma L_s{L}_r}},{\tau }_r={\displaystyle \frac{L_r}{R_r}},{\tau }_s={\displaystyle \frac{L_s}{R_s}},\sigma =\mathrm{1}-{\displaystyle \frac{M^{\mathrm{2}}}{L_s{L}_r}}.

The motor parameters are stator resistance and inductance (R_s, L_s), rotor resistance and inductance (R_r, L_r), moment of inertia, mutual inductance M, friction coefficient f and number of poles pair n_p.

The electrical speed reference and the expression of the open-loop control are written as

(27) \{\begin{array}{l}{u}_{sdc}=\gamma L_s\left({\displaystyle \frac{d{i}_{sdc}}{dt}}+\gamma i_{sdc}-{\omega }_{sc}{i}_{sqc}-{\displaystyle \frac{k_s{\mathrm{\Psi }}_{rdc}}{{\tau }_r}}\right),\hfill \\ u_{sqc}=\gamma L_s\left({\displaystyle \frac{d{i}_{sqc}}{dt}}+\gamma i_{sqc}+{\omega }_{sc}{i}_{sdc}+{\displaystyle \frac{k_s{n}_p{\omega }_m{\mathrm{\Psi }}_{rdc}}{{\tau }_r}}\right),\hfill \end{array}

(28) {\omega }_{sc}=n_p{\omega }_{mc}+{\displaystyle \frac{M}{{\tau }_r{\mathrm{\Psi }}_{rdc}}}{i}_{sqc}.

Open loop field oriented control (OL-FOC)

To control the machine flux and torque independently, a flux oriented control has been used, which is obtained by letting the rotor flux vector to be aligned in the d-axis.

The electrical speed of the stator with electromagnetic field-oriented is obtained as [ 19]

(29) {\omega }_s=n_p{\omega }_m+{\displaystyle \frac{M}{{\tau }_r{\mathrm{\Psi }}_{rdc}}}{i}_{sq}.

Replace Eq. (29) in Eq. (26), it comes

(30) \{\begin{array}{l}\dot{\mathit{x}}(t)=\mathit{A}(\mathit{x})\mathit{x}+\mathit{B}\mathit{u}+\mathit{v},\hfill \\ \mathit{y}=\mathit{C}\mathit{x},\hfill \end{array}

where

\mathit{A}=\left(\begin{array}{lllll}-\gamma & n_p{\omega }_m+{\displaystyle \frac{M}{{\tau }_r{\mathrm{\Psi }}_{rdc}}}{i}_{sq}& {\displaystyle \frac{k_s}{{\tau }_r}}& k_s{n}_p{\omega }_m& \mathrm{0}\\ -\left(n_p{\omega }_m+{\displaystyle \frac{M}{{\tau }_r{\mathrm{\Psi }}_{rdc}}}{i}_{sq}\right)& -\gamma & -k_s{n}_p{\omega }_m& {\displaystyle \frac{k_s}{{\tau }_r}}& \mathrm{0}\\ {\displaystyle \frac{M}{{\tau }_r}}& \mathrm{0}& -{\displaystyle \frac{\mathrm{1}}{{\tau }_r}}& {\displaystyle \frac{M}{{\tau }_r{\mathrm{\Psi }}_{rdc}}}{i}_{sq}& \mathrm{0}\\ \mathrm{0}& {\displaystyle \frac{M}{{\tau }_r}}& -{\displaystyle \frac{M}{{\tau }_r{\mathrm{\Psi }}_{rdc}}}{i}_{sq}& -{\displaystyle \frac{\mathrm{1}}{{\tau }_r}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& {\displaystyle \frac{n_{p}M}{J{L}_r}}{i}_{sq}& -{\displaystyle \frac{n_{p}M}{J{L}_r}}{i}_{sd}& -{\displaystyle \frac{f}{J}}\end{array}\right),

\mathit{B}={\left(\begin{array}{lllll}{\displaystyle \frac{\mathrm{1}}{\sigma L_s}}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& {\displaystyle \frac{\mathrm{1}}{\sigma L_s}}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right)}^T,\mathit{C}=\left(\begin{array}{lllll}\mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right).

T-S fuzzy model representation of the induction motor

The method based on nonlinear sector transformation makes it possible to exactly transform the system (26) into the T-S model with 8 sub-models [ 18].

The chosen premise variables are given by

(31) \{\begin{array}{l}{z}_{\mathrm{1}}(t)=i_{sd}(t),\hfill \\ z_{\mathrm{2}}(t)=i_{sq}(t),\hfill \\ z_{\mathrm{3}}(t)={\omega }_m(t).\hfill \end{array}

Then the T-S fuzzy model can be written as [ 19]

(32) \dot{\mathit{x}}(t)={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{8}}{h}_i}(z(t)({\mathit{A}}_i\mathit{x}(t)+{\mathit{B}}_i\mathit{u}(t)+\mathit{v}(t)),

where h_i(z(t)) are the weighting functions depending on the variables z_i(t) and satisfy the convexity property

(33) \{\begin{array}{l}{\displaystyle \sum _{i=\mathrm{1}}^r{h}_i}(z(t))=\mathrm{1}\forall i\in \{\mathrm{1},\mathrm{2},\cdots ,n\},\hfill \\ \mathrm{0}\le (z(t))\le \mathrm{1.}\hfill \end{array}

Observer design for induction motor

After stabilizing the system, an observer is proposed to estimate the unknown states for IM, assuming that only the stator currents are assumed to be measured.

The basic design steps for the mean value theorem observer are summarized below.

1)Calculate the matrix A₀ from the T-S representation.

2)Form the matrix \partial \mathit{f}(\mathit{\xi })/\partial \mathit{x} .

3)Calculate the matrices {\mathrm{\Psi }}_i .

4)And solve the linear matrix inequality.

The state estimation error is given as Eq. (21), where {\mathit{A}}_i^{*} are obtained from

(34) {\displaystyle \frac{\partial \mathit{f}}{\partial \mathit{x}}}(\mathit{\xi })={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{5}}{\displaystyle \sum _{j=\mathrm{1}}^{\mathrm{5}}{\mathit{e}}_n^T(i){\mathit{e}}_n(j)}}{\displaystyle \frac{\partial f_i}{\partial x_j}}(z^i)

{\displaystyle \frac{\partial \mathit{f}}{\partial \mathit{x}}}(\mathit{\xi })=\left(\begin{array}{lllll}-\gamma & {\displaystyle \frac{\partial f_{\mathrm{1}}}{\partial x_{\mathrm{2}}}}(\mathit{\xi })& {\displaystyle \frac{k_s}{{\tau }_r}}& {\displaystyle \frac{\partial f_{\mathrm{1}}}{\partial x_{\mathrm{4}}}}(\mathit{\xi })& {\displaystyle \frac{\partial f_{\mathrm{1}}}{\partial x_{\mathrm{5}}}}(\mathit{\xi })\\ {\displaystyle \frac{\partial f_{\mathrm{2}}}{\partial x_{\mathrm{1}}}}(\mathit{\xi })& {\displaystyle \frac{\partial f_{\mathrm{2}}}{\partial x_{\mathrm{2}}}}(\mathit{\xi })& {\displaystyle \frac{\partial f_{\mathrm{2}}}{\partial x_{\mathrm{3}}}}(\mathit{\xi })& {\displaystyle \frac{k_s}{{\tau }_r}}& {\displaystyle \frac{\partial f_{\mathrm{2}}}{\partial x_{\mathrm{5}}}}(\mathit{\xi })\\ {\displaystyle \frac{M}{{\tau }_r}}& {\displaystyle \frac{\partial f_{\mathrm{3}}}{\partial x_{\mathrm{2}}}}(\mathit{\xi })& -{\displaystyle \frac{\mathrm{1}}{{\tau }_r}}& {\displaystyle \frac{\partial f_{\mathrm{3}}}{\partial x_{\mathrm{4}}}}(\mathit{\xi })& {\displaystyle \frac{\partial f_{\mathrm{3}}}{\partial x_{\mathrm{5}}}}(\mathit{\xi })\\ \mathrm{0}& {\displaystyle \frac{\partial f_{\mathrm{4}}}{\partial x_{\mathrm{2}}}}(\mathit{\xi })& {\displaystyle \frac{\partial f_{\mathrm{4}}}{\partial x_{\mathrm{3}}}}(\mathit{\xi })& -{\displaystyle \frac{\mathrm{1}}{{\tau }_r}}& \mathrm{0}\\ {\displaystyle \frac{\partial f_{\mathrm{5}}}{\partial x_{\mathrm{1}}}}(\mathit{\xi })& {\displaystyle \frac{\partial f_{\mathrm{5}}}{\partial x_{\mathrm{2}}}}(\mathit{\xi })& {\displaystyle \frac{\partial f_{\mathrm{5}}}{\partial x_{\mathrm{3}}}}(\mathit{\xi })& {\displaystyle \frac{\partial f_{\mathrm{5}}}{\partial x_{\mathrm{4}}}}(\mathit{\xi })& -{\displaystyle \frac{f}{J}}\end{array}\right),

{\displaystyle \frac{\partial f_{\mathrm{1}}}{\partial x_{\mathrm{2}}}}(\mathit{\xi })=n_p{\xi }_{\mathrm{5}}+\mathrm{2}{k}_{\mathrm{1}}{\xi }_{\mathrm{2}},

{\displaystyle \frac{\partial f_{\mathrm{1}}}{\partial x_{\mathrm{4}}}}(\mathit{\xi })={\displaystyle \frac{\partial f_{\mathrm{2}}}{\partial x_{\mathrm{3}}}}(\mathit{\xi })=k_s{n}_p{\xi }_{\mathrm{5}},

{\displaystyle \frac{\partial f_{\mathrm{1}}}{\partial x_{\mathrm{5}}}}(\mathit{\xi })=k_s{n}_p{\xi }_{\mathrm{4}}+n_p{\xi }_{\mathrm{2}},

{\displaystyle \frac{\partial f_{\mathrm{2}}}{\partial x_{\mathrm{1}}}}(\mathit{\xi })=-n_p{\xi }_{\mathrm{5}}-k_{\mathrm{1}}{\xi }_{\mathrm{2}},

{\displaystyle \frac{\partial f_{\mathrm{2}}}{\partial x_{\mathrm{2}}}}(\mathit{\xi })=-\gamma +k_{\mathrm{2}}{\displaystyle \frac{\partial f_{\mathrm{5}}}{\partial x_{\mathrm{4}}}}(\mathit{\xi })=-\gamma -k_{\mathrm{1}}{\xi }_{\mathrm{1}},

{\displaystyle \frac{\partial f_{\mathrm{2}}}{\partial x_{\mathrm{5}}}}(\mathit{\xi })=-k_s{n}_p{\xi }_{\mathrm{3}}-n_p{\xi }_{\mathrm{1}},

{\displaystyle \frac{\partial f_{\mathrm{3}}}{\partial x_{\mathrm{2}}}}(\mathit{\xi })=-k_{\mathrm{2}}{\displaystyle \frac{\partial f_{\mathrm{5}}}{\partial x_{\mathrm{1}}}}(\mathit{\xi })=k_{\mathrm{1}}{\xi }_{\mathrm{4}},

{\displaystyle \frac{\partial f_{\mathrm{3}}}{\partial x_{\mathrm{4}}}}(\mathit{\xi })=-{\displaystyle \frac{\partial f_{\mathrm{4}}}{\partial x_{\mathrm{3}}}}(\mathit{\xi })=k_{\mathrm{2}}{\displaystyle \frac{\partial f_{\mathrm{5}}}{\partial x_{\mathrm{3}}}}(\mathit{\xi })=-k_{\mathrm{1}}{\xi }_{\mathrm{2}},

{\displaystyle \frac{\partial f_{\mathrm{4}}}{\partial x_{\mathrm{2}}}}(\mathit{\xi })={\displaystyle \frac{M}{{\tau }_r}}-k_{\mathrm{2}}{\displaystyle \frac{\partial f_{\mathrm{5}}}{\partial x_{\mathrm{2}}}}(\mathit{\xi })=-{\displaystyle \frac{M({\mathrm{\Psi }}_{rdc}+\mathrm{1})}{{\tau }_r{\mathrm{\Psi }}_{rdc}}}{\xi }_{\mathrm{3}}.

Note that it is not necessary to compute the functions μ_i because they are not required for the observer design. Only the matrices {\mathbf{\Psi }}_i are needed.

If a solution exists to the LMI constraints (24), then, the stability of the system is guaranteed.

The gain L₀ of the observer which guarantees the exponential convergence is calculated by solving the LMI problem given by Equation (25).

{\mathit{L}}_{\mathrm{0}}=\left(\begin{array}{ll}-\mathrm{1095.7}\hfill & -\mathrm{895.3}\hfill \\ -\mathrm{403.04}\hfill & \mathrm{102.19}\hfill \\ -\mathrm{9.4541}\hfill & \mathrm{7.721}\hfill \\ -\mathrm{32.049}\hfill & -\mathrm{45.56}\hfill \\ \mathrm{16150}\hfill & \mathrm{17254}\hfill \end{array}\right).

Simulation results

To evaluate the proposed observer for rotor flux and speed estimation for IM under open loop FOC control, computer simulations have been conducted under the Matlab/Simulink environment. The parameters of the used IM are as follows:

Mutual inductance M = 0.4475 H;

Moment of inertia J = 0.0293 kg·m²;

Stator resistance R_s = 9.65 Ω;

Rotor resistance R_r = 4.3047 Ω;

Stator inductance L_s = 0.4718 H;

Rotor inductance L_r = 0.4718 H;

Pole pair n_p = 2.

The premise variables are bounded as

\{\begin{array}{l}-\mathrm{200}\le \omega (t)\le \mathrm{200},\hfill \\ -\mathrm{6}\le i_{sd}(t)\le \mathrm{6},\hfill \\ -\mathrm{6}\le i_{sq}(t)\le \mathrm{6.}\hfill \end{array}

The overall scheme of the proposed observer for speed estimation is shown in Fig. 1.

The states and their estimates are depicted in Figs. 2-4 (a) and the error dynamics are illustrated in Figs. 2-4 (b).

The actual and estimated speed response is shown in Fig. 2(a) (blue line is the real state) and the speed estimation error is illustrated in Fig. 2 (b). The {\mathrm{\Psi }}_{rd} and {\mathrm{\Psi }}_{rq} are shown in Figs. 3 and 4, respectively.

As shown in Figs. 2-4, the observed signal is close to its real value when the initial conditions are (–5 –5 0 0 –157)^T.

The rotor speed error is approximately 0.1% of the real state at most after a short time, which is very acceptable in practical applications.

The error between the real flux and the estimated flux tends to zero through time (approximately 0.2 s).

From Figs. 2 to 4, it can be found that the estimation errors have good convergence rate. These results demonstrate and confirm the highlight effectiveness of the proposed observer. However, low cost and fast digital signal processors capable of implementing relatively complex algorithms are available in the market that makes this method suitable for high performance applications.

Conclusions

In this paper, a nonlinear observer based on DMVT combined with the sector nonlinearity transformation for induction motor has been developed. First, the nonlinear model of induction motor has been transformed into a T-S fuzzy representation, which derives from the sector nonlinearity approach. Next, the DMVT has been used which allows writing the dynamics of the observer error as a LPV system. The stability conditions are expressed in terms of LMI. The simulation results are provided to verify the validity of the observer design.

References

[1] Gomez-Gutierrez D, Ramirez-Prado G, Ramirez Trevino A, Ruiz-Leon J. Joint state-mode observer design for switched linear systems. In: Proceedings of IEEE International Conference on Emerging Technologies and Factory Automation. Hamburg, Germany, 2008, 1408-1415

[2] Sadaka H, Shafai B, Sipahi R. Robust PI observer design for linear time-delay systems. In: Proceedings of IEEE International Conference on Control Applications, (CCA) & Intelligent Control. Saint Petersburg, Russia, 2009, 1209-1213

[3] Guerra R M, Luviano-Juarez A. Rincon-Pasaye J J. On nonlinear observers: A differential algebraic approach. In: Proceedings of American Control Conference. New York, USA, 2007, 1682-1686

[4] .Babaali M, Egerstedt M, Kamen W E. An observer for linear systems with randomly-switching measurement equations. In: Proceedings of American Control Conference. Arlington, USA, 2003, 1879-1884

[5] Rafaralahy H, Zasadzinski M, Boutayeb M, Darouach M. State observer design for descriptor bilinear systems. In: Proceedings of UKACC International Conference on Control. Institution of Engineering and Technology, UK, 1996, 843-848

[6] Ball A A. Khalil H K. High-gain observers in the presence of measurement noise: A nonlinear gain approach. In: Proceedings of IEEE Decision and Control Conference, Cancun, Mexico, 2008, 2288-2293

[7] Sayem H, Braiek N B, Hammouri H. Trajectory tracking of bilinear systems using high gain observer. In: Proceedings of International Conference on Electrical Engineering and Software Applications, Hammamet, Tunisia, 2013, 1-6

[8] Hasegawa M. Robust-adaptive-observer design based on γ-positive real problem for sensorless induction-motor drives. IEEE Transactions on Industrial Electronics, 2006, 53(4): 76-85

[9] Proca A B, Keyhani A. Sliding mode flux observer with online rotor parameter estimation for induction motors. IEEE Transactions on Industrial Electronics, 2007, 54(2): 716-723

[10] Shi H, Feng Y. A hybrid sliding mode flux observer for induction motor drive. In: Proceedings of the 30th Chinese Control Conference. Yantai, China, 2011, 762-767

[11] Joen S H, Oh K K, Choi J Y,. Flux observer with online tuning of stator and rotor resistances for induction motor. IEEE Transactions on Industrial Electronics, 2002, 49(3): 653-664

[12] Savoia A, Mengoni M, Zarri L, Casadei D. A nonlinear luenberger observer for sensorless vector control of induction motors. In: Proceedings of 2011 International Aegean Conference on Electrical Machines and Power Electronics, İstanbul, Turkey2011, 544-549

[13] Park C W, Lee S. Nonlinear observer based control of induction motors. Electrical Engineering, 2007, 92(2): 107-113

[14] Besançon G. Nonlinear Observers and Applications. Springer, 2007

[15] Besançon G, Ticlea A. An immersion-based observer design for rank-observable nonlinear systems. IEEE Transactions on Automatic Control, 2007, 52(1): 83-88

[16] Wang Y, Chai T, Zhang Y. State observer-based adaptive fuzzy output-feedback control for a class of uncertain nonlinear systems. Information Sciences, 2010, 180(24): 5029-5040

[17] Takagi T, Sugeno M. Fuzzy identification of system and its applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics, 1985, 15(1): 116-132

[18] Tanaka K, Wang H O. Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. New York: Wiley-Interscience, 2001

[19] Allouche M, Chaabane M, Souissi M, Mehdi D, Tadeo F. State feedback tracking control for indirect field-oriented induction motor using fuzzy approach. International Journal of Automation and Computing, 2013, 10(2): 99-110

[20] Pourgholi M, Majd V J. A nonlinear adaptive resilient observer design for a class of Lipschitz systems using LMI. Circuits, Systems, and Signal Processing, 2011, 30(6): 1401-1415

[21] Zemouche A, Boutayeb M, Bara G I. Observer design for nonlinear systems: an approach based on the differential mean value theorem. In: Proceedings of 44th IEEE Conference on Decision and Control, and the European Control Conference. Seville, Spain, 2005, 12-15

[22] Zemouche A, Boutayeb M, Bara G I. Observer for a class of Lipschitz systems with extension to H∞ performance analysis. Systems & Control Letters, 2008, 57(1): 18-27

[23] Shen Z, Zhao J, Xu J, Gu X S. Nonlinear unknown input observer design by LMI for lipschitz nonlinear systems. In: Proceedings of the 8th World Congress on Intelligent Control and Automation, Jinan, China, 2010, 3450-3454

[24] Ichalal D, Arioui H, Mammar S. Observer design for two-wheeled vehicle: A Takagi-Sugeno approach with unmeasurable premise variables. In: Proceedings of 19th Mediterranean Conference on Control and automation. Corfu, Greece, 2011, 934-939

[25] Ichalal D, Marx B, Maquin D, Ragot J. On observer design for nonlinear Takagi-Sugeno systems with unmeasurable premise variable. In: Proceedings of International Symposium on Advanced Control of Industrial Processes. Hangzhou, China, 2011, 353-358

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg