Fuzzy sensor fault-tolerant control of an induction motor

Abstract

The sensor faults in the induction motor can cause the degradation of the performances of the system and even lead sometimes to the instability. This paper focuses on the design of a state space fuzzy observer that simultaneously estimate descriptor system states and sensor faults. By using the estimates of descriptor states and faults, and the linear matrix inequality (LMI) technique, a fault-tolerant control scheme is worked out. The developed approach takes into account the stability of the closed-loop system and the design of non-linear fuzzy inference systems based on TakagiSugeno (TS) fuzzy models. The TS fuzzy model is employed to approximate the nonlinear induction motor in the synchronous d-q frame rotating with field-oriented technique. The output fuzzy controller is able to maintain the damaged system at some acceptable level of performance even in the intermittent loss of sensor measurement signal. The gains of the observer and the controller are obtained by solving a set of Linear Matrix Inequalities (LMIs). Finally, simulation and experimental results are given to illustrate the effectiveness of the proposed approach.

Keywords

Induction motor sensor faults TakagiSugeno model descriptor observer linear matrix inequalities LMIs

1 Introduction

The advances in power electronics and digital signal processor (DSP) technology make the field-oriented control technique widely used for high-performance induction motor drive for many industrial applications. The success of this well-known control strategy requires the perfect knowledge of the measurement states: Stator currents and rotor speed. The optical encoders, the tachgenerators and the three hall-effect current sensors do occasionally fail and their resulting accidental downtime can be very expensive for users which can damage production quality and affect the safety of operator. These sensors faults can appear as a bias, an intermittent sensor connection or a complete loss of the sensor. They can lead to closed loop instability if no proper action is performed. As a result, many research have been recently devoted to study fault-tolerant control (FTC) applied to the induction machine [1 –7], able to maintain the stability of the system and some acceptable degree of performance not only when the system is fault-free, but also when there components mal-functions. However, the proposed fault tolerant control law requires a perfect knowledge of the location and the magnitude of incipient faults. Therefore, a fault detection and isolation bloc (FDI) has been integrated in the control scheme which has, by simple processing input/output data, the ability to detect the presence of an incipient fault and to isolate it from other faults or disturbances. The most commonly FDI techniques used traditional diagnosis methods are the motor current signature analysis (MCSA) [7, 8, 7, 8], wavelet analysis [9] and Parks vector approach [10, 11]. the variable-frequency PWM drive applied in the field oriented control technique and used to achieving the perfect decoupling control, would make the accurate detection of spectral components from faults very difficult. Other techniques used the physical model of the motor in which residual output is generated, by comparing the expected behavior of the system with the measured behavior, where the expected behavior is obtained from a model of the system [1, 3].

Recently, substantial research efforts have been made in artificial intelligence (AI) such as neural networks and fuzzy logic to deal with the problems of diagnosis and fault tolerant control applied to the induction motors [3 , 12–15]. For example, based on the bilinear matrix modeling and using the Takagi-Sugeno (TS) fuzzy approach, Lopez-Torribio (2000) propose in [4] a generalized observer scheme (GOS) to detect and to isolate sensor faults. The basic idea is to use a bank of TS fuzzy observers to generate residuals for each monitored system output signal. Benbouzid in [3] presents an active fault-tolerant-control systems for a high-performance control of induction-motor drive used in automotive applications. A fuzzy-logic controller is proposed to maintain the system to operating in satisfactory condition after sustaining the faults. A fuzzy switchover is integrated in the control scheme that ensures the controller-transition smoothness with no abrupt change in the torque in case of sensorfailures.

This paper presents a fault tolerant control for induction drive able to offset the effect of the sensor faults. Our first goal is to develop a TS fuzzy observer to estimate simultaneously the system states and the fault signal by using the known input u (t) and the measurement output y (t). The second goal is to work out an IFO controller able to takes the necessary action to maintain the controller performance. The information obtained from the TS fuzzy observer should be used in the controller redesign. The diagnostic scheme can be depicted in Fig. 1. The proposed fuzzy fault tolerant control design problem is parameterized in terms of a linear matrix inequality problem which can be solved very efficiently using the convex optimization techniques.

This paper is organized as follows: In Section 2, an open-loop control strategy is presented which includes a physical model of the induction motor. The nonlinear induction motor is represented by an equivalent T-S type fuzzy model in Section 3. Section 4 is reserved to the fault tolerant control scheme. In Section 5, simulation and experiments results are given to highlight the effectiveness of the proposed design observer. The last section gives a conclusion on the main works developed in this paper.

2 Open-loop control

2.1 The induction motor model

Under assumptions of linear magnetic circuits, the equivalent two-phase model of the induction motor represented in the synchronously (d-q) rotating frame is given as

$\begin{matrix} \dot{x} (t) & = & f (x (t)) + Bu (t) + Dw (t) \\ y (t) & = & Cx (t) \end{matrix}$ (1) In which is the state vector, u (t) = (u _sd u _sq) ^T is a control vector and y = (i _sd i _sq ω _m) ^T is a vector of measurable variables. The state variables are defined as follows: ω _m is the rotor speed, (Ψ _rd, Ψ _rq) are the components of the rotor fluxes, (i _sd, i _sq) are the components of the stator currents. The vector-valued function f (x (t)) and the constant matrices B and D are, respectively, $\begin{matrix} f (x (t)) & = & (\begin{matrix} - γ i_{sd} + ω_{s} i_{sq} + \frac{K_{s}}{τ_{r}} Ψ_{rd} + K_{s} n_{p} ω_{m} Ψ_{rq} \\ - ω_{s} i_{sd} - γ i_{sq} - K_{s} n_{p} ω_{m} Ψ_{rd} + \frac{K_{s}}{τ_{r}} Ψ_{rq} \\ \frac{M}{τ_{r}} i_{sd} - \frac{1}{τ_{r}} Ψ_{rd} + (ω_{s} - n_{p} ω_{m}) Ψ_{rq} \\ \frac{M}{τ_{r}} i_{sq} - (ω_{s} - n_{p} ω_{m}) Ψ_{rd} - \frac{1}{τ_{r}} Ψ_{rq} \\ \frac{n_{p} M}{J L_{r}} (Ψ_{rd} i_{sq} - Ψ_{rq} i_{sd}) - \frac{f}{J} ω_{m} \end{matrix}) \\ B & = & {(\begin{matrix} 0 & \frac{1}{σ L_{s}} & 0 & 0 & 0 \\ \frac{1}{σ L_{s}} & 0 & 0 & 0 & 0 \end{matrix})}^{T} D = {(\begin{matrix} 0 & 0 & 0 & 0 & - \frac{1}{J} \end{matrix})}^{T} \end{matrix}$ where the positive constants defined in the model are related to the electrical parameters of IM as follows: $\begin{matrix} τ_{r} & = & \frac{L_{r}}{R_{r}} τ_{s} = \frac{L_{s}}{R_{s}} \\ σ & = & 1 - \frac{M^{2}}{L_{s} L_{r}} K_{s} = \frac{M}{σ L_{s} L_{r}} \end{matrix}$ $γ = (\frac{1}{σ τ_{s}} + \frac{1 - σ}{σ τ_{r}})$ With C _r is the input unknown load torque. ω _s is the speed of the (d,q) rotating frame. J is the moment of inertia, (R _s, R _r) are the stator and the rotor resistances, (L _s, L _r) are the inductances, M is the mutual inductance, f is the friction coefficient and n _p the number of poles pairs.

2.2 Indirect field oriented control strategy

Indirect field oriented control (IFOC) is a well established and widely applied control technique when dealing with high performance induction motor drives. In this case two outputs must be controlled: rotor speed ω _m and d-axis rotor flux (Ψ _rd). These variables are supposed to track two reference signals denoted, respectively, ω _mr (t) and Ψ _rdr (t). A further control objective consists of having the q-axis rotor flux Ψ _rq (t) asymptotically converging to zero, a property known as steady-state flux decoupling.

The Indirect field oriented control strategy consists of designing a dynamic feedback controller, such that for all initial states x (0), and for all possible load torque C _r we have [2 , 17]

$lim_{t \to \infty} (ω_{mr} (t) - ω_{m} (t)) = 0$ (2) $lim_{t \to \infty} (Ψ_{rdr} (t) - Ψ_{rd} (t)) = 0$ (3) $lim_{t \to \infty} Ψ_{rq} (t) = 0$ (4) Equations (3) and (4) imply that the (d, q) frame rotating at speed ω _s tends to have the d-axis coincident with the rotating rotor flux vector as t goes to infinity, i.e. field orientation is achieved.

2.3 Reference model

In this section, we adopt the indirect field oriented control strategy in the synthesis of the reference model. In an ideally decoupling of the IM, the state variable x (t) converges to the reference state x _r (t) = (i _sdr, i _sqr, Ψ _rdr, 0, ω _mr) in the steady-state. By replacing the reference signals x _r (t) in Equation (1), yields [17]: ${\begin{matrix} \frac{d}{dt} i_{sdr} = - γ i_{sdr} + ω_{s} i_{sqr} + \frac{K_{s}}{τ_{r}} Ψ_{rdr} + \frac{1}{σ L_{s}} u_{sdr} \\ \frac{d}{dt} i_{sqr} = - ω_{s} i_{sdr} - γ i_{sqr} - K_{s} ω_{mr} Ψ_{rdr} + \frac{1}{σ L_{s}} u_{sqr} \\ 0 = - (ω_{s} - n_{p} ω_{mr}) Ψ_{rdr} + \frac{M}{τ_{r}} i_{sqr} \\ \frac{d}{dt} Ψ_{rdr} = \frac{M}{τ_{r}} i_{sdr} - \frac{1}{τ_{r}} Ψ_{rdr} \\ \frac{d}{dt} ω_{mr} = \frac{n_{p} M}{J L_{r}} (Ψ_{rdr} i_{sqr}) - \frac{f}{J} ω_{mr} - \frac{1}{J} C_{r} \end{matrix}$ (5) The last two equations in (2) lead to the stator current reference: ${\begin{matrix} i_{sdr} = \frac{Ψ_{rdr}}{M} + \frac{τ_{r}}{M} \frac{d}{dt} Ψ_{rdr} \\ i_{sqr} = \frac{J L_{r}}{n_{p} M Ψ_{rdr}} (\frac{C_{r}}{J} + \frac{f}{J} ω_{mr} + \frac{d}{dt} ω_{mr}) \end{matrix}$ (6) According to (2), the electrical speed reference of the stator is given as follows: ${ω_{s}}_{r} = n_{p} ω_{mr} + \frac{M}{τ_{r} Ψ_{rdr}} i_{sqr}$ (7) The first two equations in (2), lead to the open-loop control vector u _r (t) = (u _sdr u _sqr) ^T: ${\begin{matrix} u_{sdr} = σ L_{s} (\frac{d}{dt} i_{sdr} + γ i_{sdr} - ω_{sr} i_{sqr} - \frac{K_{s}}{τ_{r}} Ψ_{rdr}) \\ u_{sqr} = σ L_{s} (\frac{d}{dt} i_{sqr} + γ i_{sqr} + ω_{sr} i_{sdr} + K_{s} ω_{mr} Ψ_{rdr}) \end{matrix}$ (8)

3 TS fuzzy model of induction motor

A Takagi-Sugeno model allows to the representation of the behavior of the induction motor by the interpolation of a set of linear submodels. Each submodel contributes to the global behavior of the nonlinear system through weighting functions [18, 19]. The IFOC technique, when is implanted using the rotating (d, q) reference frame, allows for an independent control of the electromagnetic torque and rotor flux. In this condition, the rotor flux vector is aligned to the d-axis and we have [2, 17] ${\begin{matrix} Ψ_{rd} = Ψ_{rdr} \\ Ψ_{rq} = 0 \end{matrix}$ (9) To ensure condition (9), the synchronous speed of the (d,q) rotating frame must be chosen as $ω_{s} = n_{p} ω_{m} + \frac{M}{τ_{r} Ψ_{rdc}} i_{sq}$ (10) Substituting Equation (10) in the induction motor model (1) leads to the following equivalent state space: ${\begin{matrix} \dot{x} (t) = A (x (t)) x (t) + Bu (t) + Dw (t) \\ y = C x (t) \end{matrix}$ (11) Where: $A (x (t)) = (\begin{matrix} - γ & ω_{s} & \frac{K_{s}}{τ_{r}} & K_{s} n_{p} ω_{m} & 0 \\ - ω_{s} & - γ & - K_{s} n_{p} ω_{m} & \frac{K_{s}}{τ_{r}} & 0 \\ \frac{M}{τ_{r}} & 0 & - \frac{1}{τ_{r}} & \frac{M}{τ_{r} Ψ_{rdc}} i_{sq} & 0 \\ 0 & \frac{M}{τ_{r}} & - \frac{M}{τ_{r} Ψ_{rdc}} i_{sq} & - \frac{1}{τ_{r}} & 0 \\ 0 & 0 & \frac{n_{p} M}{J L_{r}} i_{sq} & - \frac{n_{p} M}{J L_{r}} i_{sd} & - \frac{f}{J} \end{matrix})$ $C = (\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})$

The sector of nonlinearities of the terms z _j = x _j ∈ (z _jmin, z _jmax) of the matrix A(x(t)) with j = 1, 2, 3 are: ${\begin{matrix} z_{1} (t) = i_{sd} (t) \\ z_{2} (t) = i_{sq} (t) \\ z_{3} (t) = ω_{m} (t) \end{matrix}$ (12) A convex polytypic transformation for the premise variables can be performed. Each nonlinear terms z ₁ (t), z ₂ (t) and z ₃ (t) can be represented in the following forms: ${\begin{matrix} z_{1} = F_{1, min} (z_{1}) i_{s d_{max}} + F_{1, max} (z_{1}) i_{s d_{min}} \\ z_{2} = F_{2, min} (z_{2}) i_{s q_{max}} + F_{2, max} (z_{2}) i_{s q_{min}} \\ z_{3} = F_{3, min} (z_{3}) ω_{m_{max}} + F_{3, max} (z_{3}) ω_{m_{min}} \end{matrix}$ (13) Therefore the membership functions can be calculated as $\begin{matrix} F_{1, min} (z_{1}) & = & \frac{z_{1} - i_{s d_{min}}}{i_{s d_{max}} - i_{s d_{min}}} \\ F_{1, max} (z_{1}) & = & 1 - F_{1, min} (z_{1}) \\ F_{2, min} (z_{2}) & = & \frac{z_{2} - i_{s q_{min}}}{i_{s q_{max}} - i_{s q_{min}}}, \\ F_{2, max} (z_{2}) & = & 1 - F_{2, min} (z_{2}) \\ F_{3, min} (z_{3}) & = & \frac{z_{3} - ω_{m_{min}}}{ω_{m_{max}} - ω_{m_{min}}}, \\ F_{3, max} (z_{3}) & = & 1 - F_{3, min} (z_{3}) \end{matrix}$ Using this transformation, r = 2³ submodels characterized by p = 3 premise variables can be generated. Each linear submodel is described by a fuzzy IF-THEN rule defined in the following form:

Rule Ri

IF (z ₁ is F _1,k) and (z ₂ is F _2,l) and (z ₃ is F _3,f)

THEN $\dot{x} (t) = A_{i} x (t) + Bu (t) + Dw (t)$ .

such as k, l, f∈ { min , max }, i = 1, 2, . . . , r

According to the state space representatin (1), the global fuzzy model with sensors faults f _s (t) is described by the following ${\begin{matrix} \dot{x} (t) = \sum_{i = 1}^{8} h_{i} (z (t)) (A_{i} x (t)) + Bu (t) + Dw (t) \\ y (t) = Cx (t) + D_{s} f_{s} (t) \end{matrix}$ (14) where $\begin{matrix} λ_{i} (z_{1}, z_{2}, z_{3}) = \prod_{k, l, f = 1}^{2} F_{1}^{k} (z_{1} (t)) F_{2}^{l} (z_{2} (t)) F_{3}^{f} (z_{3} (t)) \\ h_{i} (z (t)) = \frac{λ_{i} (z_{1}, z_{2}, z_{3})}{\sum_{i = 1}^{r} λ_{i} (z_{1}, z_{2}, z_{3})} \\ \sum_{i = 1}^{r} h_{i} (z (t)) = 1 \end{matrix}$ The global fuzzy model (14) is inferred as ${\begin{matrix} \dot{x} (t) = \sum_{i = 1}^{8} h_{i} (z (t)) (A_{i} x (t)) + B_{d} \bar{u} (t) \\ y (t) = Cx (t) + D_{s} f_{s} (t) \end{matrix}$ (15) in which B _d = [B D] and $\bar{u} (t) = [u (t) w (t)]^{T}$

In the following section, we present the descriptor observer design able to estimate simultaneously the state variables of the motor and the sensor faults.

4 Descriptor observer design

In this section, we present a state-space observer written in the descriptor form able to estimate simultaneously the system states x (t) and the fault signal f _s (t) by using the known input u (t) and the measurement output y (t). First, we considered the TS fuzzy descriptor model of the induction motor with sensor faults as follows [20]: ${\begin{matrix} \bar{E} \dot{\bar{x}} = \sum_{i = 1}^{r} h_{i} (z (t)) ({\bar{A}}_{i} \bar{x} (t)) + \bar{B} \bar{u} (t) + \bar{D} x_{s} (t) \\ y (t) = \bar{C} \bar{x} (t) = C_{0} \bar{x} (t) + x_{s} (t) \end{matrix}$ (16) Where x _s (t) = D _s f _s (t), $\bar{x} (t) = (\begin{matrix} x (t) \\ x_{s} (t) \end{matrix})$ , $\bar{E} = (\begin{matrix} I_{n} & 0 \\ 0 & 0 \end{matrix})$

${\bar{A}}_{i} = (\begin{matrix} A_{i} & 0 \\ 0 & - I_{p} \end{matrix})$ , $\bar{B} = (\begin{matrix} B_{d} \\ 0 \end{matrix})$ , $\bar{D} = (\begin{matrix} 0 \\ I_{p} \end{matrix})$

$\bar{C} = (\begin{matrix} C & I_{p} \end{matrix})$ , $C_{0} = (\begin{matrix} C & 0 \end{matrix})$

The fault x _s (t) is considered as an auxiliary state of the augmented system (16). Next, an interesting descriptor estimator observer is proposed as the following structure: ${\begin{matrix} E \dot{z} (t) = \sum_{i = 1}^{r} h_{i} (z (t)) (N_{i} z (t) + G_{i} \bar{u} (t)) \\ \hat{\bar{x}} (t) = z (t) + Ly (t) \end{matrix}$ (17) Where $\hat{\bar{x}} (t) \in R^{n + p}$ is the estimate of the augmented descriptor state $\bar{x} (t)$ . In order to establish the conditions for the asymptotic convergence of the observer (17), let us define the state estimation errors: $\bar{e} (t) = {[\begin{matrix} e (t) & e_{s} (t) \end{matrix}]}^{T} = \bar{x} (t) - \hat{\bar{x}} (t)$ (18) The dynamic of the state estimation error can be evaluated by combining the Equations (16) and (17) as follows:

$\begin{matrix} (\bar{E} + EL \bar{C}) \dot{\bar{x}} (t) - E \dot{\hat{\bar{x}}} (t) \\ = \sum_{i = 1}^{r} h_{i} (z (t)) [({\bar{A}}_{i} + N_{i} L C_{0}) \bar{x} (t) \\ - N_{i} \hat{\bar{x}} (t) + (\bar{D} + N_{i} L) x_{s} (t) + (\bar{B} - G_{i}) \bar{u} (t)] \end{matrix}$ (19) By considering the following assumptions: $E = \bar{E} + EL \bar{C}$ (20) $N_{i} = {\bar{A}}_{i} + N_{i} L C_{0}$ (21) $\bar{D} = - N_{i} L$ (22) $G_{i} = \bar{B}$ (23) the state estimation error vector $\bar{e} (t)$ of the observer (17) goes to zero asymptotically and the error dynamic Equation (19) becomes: $E \dot{\bar{e}} = \sum_{i = 1}^{r} h_{i} (z (t)) N_{i} \bar{e} (t)$ (24) In order to satisfy the constraints of the Equations (20–23), the observer gains N _i, L and E are chosen as follows [17]: $N_{i} = (\begin{matrix} A_{i} & 0 \\ - C & I_{p} \end{matrix})$ (25) $E = (\begin{matrix} I_{n} & 0_{n \times p} \\ M_{a} C & M_{a} \end{matrix})$ (26) $L = (\begin{matrix} 0 \\ I_{p} \end{matrix})$ (27) Where $M_{a} \in R^{p \times p}$ is a full rank matrix which will be determined to verify the stability condition of the fuzzy observer (17). The error dynamic Equation (19) can be rewriting as follows $\dot{\bar{e}} (t) = \sum_{i = 1}^{r} h_{i} (z (t)) S_{i} \bar{e} (t)$ (28) Where

$\begin{matrix} S_{i} & = & E^{- 1} N_{i} \\ = & (\begin{matrix} A_{i} & 0_{p \times p} \\ - C A_{i} - M_{a}^{- 1} C & - M_{a}^{- 1} \end{matrix}) \end{matrix}$ (29)

5 Sensor fault tolerant control design

In the presence of sensor faults, the faulty measurements influence the closed-loop behaviour and corrupt the state estimation. Then, the tracking error between the reference input and the measurement will no longer be equal to zero. In order to minimize the effect of the sensor faults on the system performance, a fuzzy sensor fault tolerant control scheme is proposed (Fig. 1). In this scheme, the nominal controller u _n (t) is enriched with an additional control inputs u _add (t) able to offset the effect of any sensor fault.

The compensation for a sensor fault effect on the closed-loop system can be achieved by using the follwing fuzzy control law $u (t) = \sum_{i = 1}^{8} h_{i} (z (t)) K_{i} (y_{comp} (t) - y_{r} (t))$ (30) Where, the compensated output is described by the following expression:

$\begin{matrix} y_{comp} (t) & = & y_{mes} (t) - {\hat{x}}_{s} (t) \\ = & (y (t) + x_{s} (t)) - {\hat{x}}_{s} (t) \\ = & y (t) + (0 I_{p}) \bar{e} (t) \end{matrix}$ (31) Using Equation (31), the fuzzy controller (30) can be written as the following expression:

$\begin{matrix} u (t) & = & \sum_{i = 1}^{8} h_{i} (z (t)) K_{i} C (x (t) - x_{r} (t)) \\ + \sum_{i = 1}^{8} h_{i} (z (t)) K_{i} (0 I_{p}) \bar{e} (t) \\ = & u_{n} (t) + u_{add} (t) \end{matrix}$ (32) Before, to look for the controller and observer gains, some assumptions are made:

The reference speed ω _mr, the reference rotor flux Ψ _rdr and the external disturbance are smooth and bounded.

${\dot{x}}_{r} (t) = 0$

Considering the tracking error, given as follows:

e_{r} (t) = x_{r} (t) - x (t)

(33) Substituting (30) into (14) and differencing the tracking error (33), we get:

{\dot{e}}_{r} (t) = \sum_{i = 1}^{8} h_{i} (z (t)) (\begin{matrix} (A_{i} + B K_{i} C) e_{r} (t) \\ + B K_{i} (0 I_{p}) \bar{e} (t) \\ + {\bar{D}}_{i} \bar{w} (t) \end{matrix})

(34) Where

\bar{w} (t) = [x_{r} (t) w (t)]^{T} and \bar{D_{i}} = [A_{i} D]

Let us now define the augmented vector $\tilde{e} (t) = {[e (t) e_{s} (t) e_{r} (t)]}^{T}$ , from which the following augmented system is obtained: $\dot{\tilde{e}} = \sum_{i = 1}^{8} h_{i} (z (t)) [{\bar{M}}_{i} \tilde{e} (t) + {\bar{F}}_{i} \bar{w} (t)]$ (35) Where $\begin{matrix} {\bar{M}}_{i} & = & [\begin{matrix} A_{i} & 0 & 0 \\ - (C A_{i} + M_{a}^{- 1} C) & - M_{a}^{- 1} & 0 \\ 0 & B K_{i} & (A_{i} + B_{i} K_{i} C) \end{matrix}] \\ \bar{F_{i}} & = & [\begin{matrix} 0 & 0 \\ 0 & 0 \\ A_{i} & D \end{matrix}] \end{matrix}$

In order to attenuate the effect of the disturbance w(t) on the tracking error e _r (t) below a desired level ρ ², we consider the H _∞ performance defined as follows [21]: $\int_{0}^{\infty} e_{r}^{T} (t) Q e_{r} (t) dt \leq ρ^{2} \int_{0}^{\infty} w^{T} (t) w (t) dt$ (36) Q is a positive definite weighting matrix, and ρ is a prescribed attenuation level. If we consider the augmented system, the H _∞ performance becomes [21]: $\int_{0}^{t_{f}} {\tilde{e}}^{T} (t) \bar{Q} (t) \tilde{e} (t) dt \leq ρ^{2} \int_{0}^{t_{f}} {\bar{w}}^{T} (t) \bar{w} (t) dt$ (37) Where $\bar{Q} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & Q \end{matrix}]$ The main result on sensor fault-tolerant control (32) for the TS fuzzy system (14) with H _∞ performance (36) is summarized in the following theorem.

Theorem [21] Consider the fuzzy system described by (14). The sensor fault-tolerant control index ρ is achieved, if there exists a symmetric and positive definite matrix $\bar{P} = {\bar{P}}^{T} > 0$ such that the following matrix inequalities hold:

$\begin{matrix} {\bar{M}}_{i}^{T} \bar{P} + \bar{P} {\bar{M}}_{i} + \frac{1}{ρ^{2}} \bar{P} {\bar{F}}_{i} {\bar{F}}_{i}^{T} \bar{P} + \bar{Q} < 0, \\ i \in {1, 2, . . ., 8} \end{matrix}$ (38) Inequality (38) presents bilinear matrix inequality (BMI) conditions. For this reason, we propose a method that consists of solving these BMIs in two stages, such that every stage is considered as a standard LMI [5]. To facilitate the resolution, the matrix variable is chosen diagonal with respect to appropriate matrix blocks: $\bar{P} = [\begin{matrix} P_{1} & 0 & 0 \\ 0 & P_{2} & 0 \\ 0 & 0 & P_{3} \end{matrix}]$ By substituting $\bar{P}$ and $\bar{Q}$ into (38) and using the change of variables $Z = P_{2} M_{a}^{- 1}$ , the inequality (38) becomes: $[\begin{matrix} T J_{11} & T J_{12} & 0 \\ T J_{21} & T J_{22} & T J_{23} \\ 0 & T J_{32} & T J_{33} \end{matrix}] < 0$ (39) Where $\begin{matrix} T J_{11} & = & A_{i}^{T} P_{1} + P_{1} A_{i} \\ T J_{12} & = & {TJ}_{21}^{T} = - C A_{i} P_{2} - C^{T} Z^{T} \\ T J_{22} & = & - Z - Z^{T} \\ T J_{23} & = & {TJ}_{32}^{T} = (P_{3} B K_{i})^{T} \\ T J_{33} & = & (A_{i} + B K_{i} C)^{T} P_{3} + P_{3} (A_{i} + B K_{i} C) \\ + \frac{1}{ρ^{2}} P_{3} (A_{i} A_{i}^{T} + D D^{T}) P_{3} + Q \end{matrix}$ Unfortunately, the conditions (39) are not linear with respect to the variables P ₃ and K _i. In order to solve them with the classical LMI approaches, we can follow the two-step procedures:

Congruence TJ ₃₃ with $P_{3}^{- 1}$ and considering the change of variable X ₃ = P _3^-1 and $S = P_{3}^{- 1} {QP}_{3}^{- 1}$ , the condition TJ ₃₃ is equivalent to the following LMI:

$\begin{matrix} A_{i} X_{3} + X_{3} A_{i}^{T} + B K_{i} C X_{3} + X_{3} C^{T} K_{i}^{T} B^{T} \\ + \frac{1}{ρ^{2}} [A_{i} A_{i}^{T} + D D^{T}] + S < 0 \end{matrix}$ (40)

Then, we consider the change of variable CX ₃ = WC, K _i W = N _i and using the Schurs complement, the condition (40) becomes: $[\begin{matrix} (\begin{matrix} A_{i} X_{3} + X_{3} A_{i}^{T} + B N_{i} C \\ + C^{T} N_{i}^{T} B^{T} + S \end{matrix}) & A_{i} & D \\ A_{i}^{T} & - ρ^{2} I & 0 \\ D^{T} & 0 & - ρ^{2} I \end{matrix}] < 0$ (41)

The resolution of inequality (41) using LMI tools leads to X ₃, N _i, W = CX ₃ (C ^T C) ^-1 C ^T and the output controllers K _i = N _i W ^-1.

By substituting P ₃ and K _i into (39), the last condition becomes a standard LMI and we can easily solve P ₁, P ₂, Z and M _a = Z ^-1 P ₂.

6 Simulation and experiment results

In this section, the performance of the proposed sensor Fault-tolerant control law will be first tested by numerical simulations and secondly experimentally implemented. The IM is 1.1 KW three phases asynchronous motor with 50 Hz and 380 V voltage supply whose electrical and mechanical parameters are shown in Table 1.

6.1 Simulation results

In all the simulation results which will be presented in the following, we have fixed a constant speed referenceω _mr = 100 rd/s and we have taken a constant load torque of value C _r = 5 Nm applied at time t = 1 sec. The reference flux Ψ _rdr starts from zero and increases to the rated constant value 1 Wb. Sensors faults are carried out added to the outputs of the machine. These faults are biases which appear and disappear after a short period. The first fault concerns the phase(a) sensor current and the second faults affects the sensor speed, in which a negative bias of value 50 rd/s is occurs on the sensor measuring appear between the time t = 7 sec and t = 9 sec.

From LMI (41), we get the fault-tolerant controller:

$\begin{matrix} K_{1} & = & [\begin{matrix} - 64.358 & - 3.121 & 1.572 \\ - 3.121 & - 64.072 & - 32.936 \end{matrix}] \\ K_{2} & = & [\begin{matrix} - 64.069 & 3.121 & 1.573 \\ 3.121 & - 64.354 & - 33.080 \end{matrix}] \\ K_{3} & = & [\begin{matrix} - 8703.726 & - 411.644 & - 410.733 \\ - 411.644 & - 8667.225 & - 4251.985 \end{matrix}] \\ K_{4} & = & [\begin{matrix} - 8665.674 & 411.310 & 419.373 \\ 411.644 & 8702.144 & - 4278.814 \end{matrix}] \\ K_{5} & = & [\begin{matrix} - 8703.727 & 411.644 & 410.733 \\ 411.644 & 8667.225 & - 4251.985 \end{matrix}] \\ K_{6} & = & [\begin{matrix} - 8665.674 & 411.310 & 419.373 \\ 411.644 & 8702.144 & - 4278.814 \end{matrix}] \\ K_{7} & = & [\begin{matrix} - 64.358 & - 3.121 & 1.572 \\ - 3.121 & - 64.072 & - 32.936 \end{matrix}] \\ K_{8} & = & [\begin{matrix} - 64.069 & 3.121 & 1.573 \\ 3.121 & - 64.354 & - 33.080 \end{matrix}] \end{matrix}$

In the presence of sensors faults, the controller receives the faulty measurement while the true value is still equal to its reference value. The tracking error between the measurement variables and its reference value is no longer equal to zero. Therefore, the controller tries to bring back the steady-state error to zero. In absence of fault tolerant controller, the rotor speed increase by the same value of the bias on the faulty sensor, as shown in Fig. 2. In the case where the faulty measurement persists, the rotor speed can diverge, and leading to the instability of the machine. In the other hand, the proposed IFO sensor fault-tolerant controller is able to taken into account the sensor fault in the calculation of the new control law in order to prevent the induction machine from all inaccurately function of the sensors. Figure 2 shows the importance of taking into consideration the potential sensors faults while designing the control law in order to preserve the stability of the induction machine and to maintain the damaged system at some acceptable level of performance.

Figures 4 and 5 report, respectively, the components of the rotor flux (Ψ _rd, Ψ _rq) just when the nominal fuzzy controller is applied without compensation and when it is added by a compensating control faults. It is clear that the proposed fuzzy FTC controller force the d-axis rotor flux to track the reference trajectory even in presence of sensors faults and then achieve the decoupling control characteristic.

Figures 6 and 7 schows, respectively, the d-axis stator current tracking error and the q-axis stator current tracking error in case just the nominal fuzzy controller is present in the control scheme (upper plots) and in case the nominal controller is enriched with the fault compensation term (lower plots). It is worth to note that the presence of stator current sensors faults, if not compensated by means of the fault tolerance unit, generates a large stator tracking error of amplitude 0.5 A presented as a oscillations form.

6.2 Experimental results

Experimental tests are carried out to verify the effectiveness of the proposed fuzzy fault-tolerant controllers. The test bench (Fig. 8) is made up of a three-phase induction motor IM, an insulated gate bipolar transistor (IGBT) source voltage inverter, a DSP controller card, an incremental encoder (1024 pulse) and three hall-effect current sensors (LEM, LTA 50P) for the measurement of stator currents. The fuzzy observer as well as the whole control algorithm has been implemented on Dspace 1104 board using the MATLAB/Simulink software package.The DSP performs data acquisition, generates reference state x _r and the pulse width modulation (PWM) inverter commands. The PWM signals are generated with switching frequency of 5 kHz. Using Range-Kutta solver, the sampling time for controller implementation is set to 333 μs. a set of I/O modules are grouped in the card connector panel CP1104 for the voltage/current measurement, the encoder interface, and the protection of the power inverter.

Figure 9 describes the experimental results when the reference speed and the rotor flux is fixed to 100 rad/s and 1 Wb respectively. The same sensors faults are chosen as the similar case of the simulation results.

Using the developed sensor fault compensation method, the real output follows its reference input as illustrated in Fig. 8 and the machine keeps its nominal performances even in failing mode. In addition, the experiment result show the convergence of the estimated states toward the real states whose proves the effectiveness of the proposed TS observer in terms of state estimation and faults detection.

7 Conclusion

In this paper, a TS fuzzy fault tolerant control scheme is proposed to deal with sensor faults problems for induction motor drives. The proposed control law is based on the TS fuzzy models using the sector nonlinearity approach. A descriptor observer is presented to estimate simultaneously system states and sensor faults. The fuzzy controller gains satisfying the H _∞ performance are determinates using an algorithm based on LMI optimization techniques. A two-step LMI method has been given to solve these matrix inequalities. Simulation and experimental results have shown satisfactory estimation and good tracking performance despite of presence of sensor faults.

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