#### Abstract

This paper deals with the problem of induction motor tracking control against actuator faults and external disturbances using the linear matrix inequalities (LMIs) method and the adaptive method. A direct adaptive fault-tolerant tracking controller design method is developed based on Lyapunov stability theory and a constructive algorithm based on linear matrix inequalities for online tuning of adaptive and state feedback gains to stabilize the closed-loop system in order to reduce the fault effect with disturbance attenuation. Simulation results reveal the merits of proposed robust adaptive fault-tolerant tracking control scheme on an induction motor subjected to actuator faults.

#### 1. Introduction

With technology advances and modern control systems complexity increasing, rotating electrical machines play important roles in many fields especially in industrial processes because of their rigid, rugged, low price, reliable relative simplicity, and easy to maintain behaviors [1, 2]. However, the reliable electric drives are essential in all safety critical applications such as aerospace, transportation, medical, and military applications. In these applications, the reliability of electric drive systems must be ensured, and any failure in motor drives may result in loss of property and human life. Therefore, it is absolutely necessary for the motor drives (utilized in safety critical applications) in order to have a fault-tolerant capability and an ability to produce a satisfactory output torque even in the presence of faults [3–5]. That is why designing reliable drives has received great attention in the recent years.

When a fault occurs in system components including sensors, actuators, and plant, it can cause performance reduction and the closed-loop system instability. Therefore, there is a crucial need to design a class of controllers to compensate the faults effects and guarantee system stability with acceptable performance. FTC design approaches develop controllers in order to guarantee system stability in the presence of faults and disturbances. They are classified as two main classes: passive FTC and active FTC [6–11]. In the passive FTC approach, robust control techniques are utilized to design a fixed controller for maintaining the acceptable system stability and performances throughout normal or faulty cases [11]. The passive FTC approach considers fault as a special kind of uncertainties, and consequently controllers are fixed and designed to be robust against a class of presumed faults. Then designing proper controllers becomes more conservative, and attainable control performance may not be satisfactory. On the other hand, the FTC based on active technique can compensate faults either by selecting a precomputed control law or by synthesizing a new online control strategy [3, 10–12]. Since the active FTC approach provides flexibility to choose various controllers, different suitable controllers can be selected to reach a better performance. The active FTC design approach is based on fault detection and isolation (FDI). The controller reconfiguration is a special case of active FTC systems based on the fault diagnostic information, which is provided by an FDI mechanism [9, 13–15]. Figure 1 shows a closed-loop system with an active FTC strategy including the FDI block.

Another typical approach for fault compensation is based on adaptive tuning. The developments of adaptive fault-tolerant compensation controller have been reported based on model reference adaptive control, where the outputs of the closed-loop system could track the commanded reference outputs [16–20]. The idea adaptive FTC is that of designing a control unit able to automatically offset the effect of the faults, without the need of an explicit FDI process and consequent explicit reconfiguration. Figure 2 shows a closed-loop system with an adaptation mechanism for online tuning of controller parameters.

In the most of mentioned works such as [21, 22], unmatched external disturbance term has not been considered in the control process or it is hard to guarantee the asymptotical stability in the presence of disturbance term. However, the unmatched external disturbance plays an important role and it is able to decline closed-loop system performance. Therefore, studying the FTC design in the presence of unmatched external disturbance seems necessary and challenging.

In this paper, motivated by the development of fault-tolerant tracking control for TPIM, the problem of fault-tolerant tracking control of TPIM with actuator faults in the presence of bounded disturbances is presented in comparison with [11, 16]; a direct adaptive method is proposed for a fixed state feedback fault-tolerant tracking control that guarantees the closed-loop system stability in the presence of actuator faults and unmatched external disturbances. Based on the Lyapunov stability theorem, a novel constructive algorithm based on linear matrix inequalities (LMIs) is developed. For this purpose, we first proposed an adaptation mechanism for online tuning of the controller parameters. Then, a robust fixed state feedback controller is constructed for the closed-loop system.

This paper is organized as follows: in Section 2, the induction motor mathematical model is presented. The fault-tolerant tracking control problem formulation is described in Section 3. Section 4 presents the design method of the direct robust adaptive fault-tolerant tracking controller. In Section 5, the merits of the proposed approach are verified by the simulations on TPIM subjected to the actuator faults and disturbances. Finally, conclusions are given in Section 6.

#### 2. Dynamic Model of an Induction Motor

In this section, the dynamic model of TPIM is presented in a synchronously reference frame by the following equations [23]: where , are the components of the stator current, , are the components of the rotor current, is the rotor speed vector, , are stator voltage, is synchronous speed, is an unknown load torque, is the number of pair poles, is the moment of inertia coefficient, and , , , and are constants which are defined as where and are stator and rotor resistances, and are stator and rotor inductances, and is the mutual inductance.

Equation (1) can be given by the following state-space form: where is the state vector of the induction motor, is a control vector, represents the unknown disturbance function, and , and are known real constant matrixes with appropriate dimensions.

#### 3. Preliminaries and Problem Formulation

In this section, some useful notations and lemma are expressed. The notations in this paper are fairly standard.

In this context stands for the set of real numbers. For a given matrix , denotes its transpose. denotes a unity matrix with appropriate dimension. For given matrices , , the notation denotes the block-diagonal matrix with along the diagonal denoted for brevity. Moreover, the following lemma is used in this paper.

Lemma 1 (see [24] (Rayleigh inequality)). *Consider a nonsingular symmetric matrix and the minimum and maximum eigenvalues of as and , respectively. Using these notations, for any , one can define .**Now, consider the following continuous-time linear system:
**
where is the state vector, is the faulty control input vector, and is a continuous vector function which represents the bounded external disturbances. , , and are known real constant matrixes with appropriate dimensions.**Assume is the output signal of th actuator that is faulty and is the input signal of th actuator. Then, we denote a general actuator fault model by
**
where is the unknown time-varying actuator efficiency factor and and are the known lower and upper bounds of , respectively. Table 1 illustrates the actuator fault modes.**From (5), one can obtain
**
where , Then, the set of operators with the above structure can be denoted by
**
Hence, the dynamic of system (4) with actuator faults (6) can be described as follows:
*

To ensure the achievement of fault-tolerant objectives, the following assumptions in the FTC design are also assumed to be valid.

*Assumption 1. *All pairs are uniformly completely controllable for any actuator fault mode under consideration.

*Assumption 2. *The unmatched external disturbance is bounded.

*Assumption 3. *The reference signal is smooth and bounded.

If we defined the tracking error as according to the previous assumptions and according to (8) and (9), the error dynamics will be as follows: The system (10) can be rewritten as where

*Assumption 4. *The known positive constant is assumed to satisfy the following inequality:

#### 4. Direct Robust Adaptive Fault-Tolerant Tracking Controller Design

In this section, designing a direct adaptive fault-tolerant controller to guarantee closed-loop system stability via adaptive state feedback is developed. In this context, we designed an adaptive law to update the controller parameter when the mode of occurred fault is unknown.

Consider the following FTC law for the system (11): where and are, respectively, the fixed and time-varying matrix gains that will be designed later.

From (11) and (14), the closed-loop system can be presented by where . The following adaptive law is suggested to design the adaptive FTC: where and are suitable positive constants which satisfy the following equation: and is updated by where is a fixed parameter that will be selected by users for achieving better time response and, to compensate fault effects sooner, is any positive constant. In order to ensure the achievements of FTC objectives such as closed-loop stability and disturbance attenuation, the following theorem is considered.

Theorem 2. *Under Assumptions 1–4, the control law (14) for any positive matrix guarantees the closed-loop system (15) asymptotic stability if there exist a positive symmetric matrix and constant matrix for any such that
**
Furthermore with a feasible solution for LMI *

*Remark 3. *Parametric LMI in Theorem 2 is dependent on but, for feasibly solving it, there is no need to know about for any given time of simulation, because we only solve the parametric LMI for lower and upper boundaries of the matrix components. For example, if the system has actuators LMI must be solved for repetition and at last the answer of LMI for simulation will be the convex combination of the answers [9, 12].

*Proof. *For the closed-loop system described by (15), consider the following Lyapunov candidate:
where is the parametric error and its dynamics can be written as
where is constant.

According to (15) the time derivative of the Lyapunov function becomes

By multiplying by the numerator and denominator of (16) and substituting it in (23), (23) can be rewritten as follows:
Considering (17) it will be easy to show that
Since is scalar and by applying Assumption 4 and Lemma 1, one can get
Then, inequality (27) can be derived from (25) and (26) as follows:
where the constant is assumed to satisfy the following inequality:
Considering (27) and (28) and using of error dynamic of controller parameter, the adaptation law (18) can be derived as
Hence the derivative of Lyapunov function will be as
By defining the positive matrix in a way that
we can get
Taking into account and leads to
Then by using Schur’s Lemma, the LMI form of (33) will be in the following form:
Finally from (30), we can get

Then, the global adaptive fault-tolerant compensation control problem with disturbance rejection is solvable. Moreover, the closed-loop system with FTC is asymptotically stable and the states tracking error will converge to zero for every initial state of the TPIM.

The following algorithm summarizes the suggested FTC implementation method.

*Step 1. *Check Assumptions 1–4 to be satisfied in all conditions.

*Step 2. *Solve LMI and compute the matrix and using (34) assuming .

*Step 3. *Compute the gains using (16) and (18).

*Step 4. *Compute robust adaptive control law with (14).

#### 5. Simulation Results

In this section, some numerical simulations have been performed to validate the proposed FTC scheme. The induction motor parameters are given in the appendix.

The system (3) is linearized about an operating point using MATLAB, where

The following test consists of 5 Nm load torque which is applied to the motor at 0.2 sec; also the actuator fault occurred in the following modes.

*Mode 1.* Actuators 1 and 2 are in normal mode; that is,

*Mode 2.* Actuators 1 and 2 are all in loss of effectiveness mode; for example,

*Mode 3.* Actuators 1 and 2 are all in loss of effectiveness mode; that is, in this mode the effectiveness of the actuators was decreased more than Mode 2:
The following constants and initial conditions are taken for simulation:
By solving LMI inequality (34) using LMI optimization algorithm with MATLAB, and can be obtained as follows:

The closed-loop simulation results are reported in Figures 3(a)–3(e) in the presence of faults and disturbances. Figures 3(a) and 3(b) present the steady state stator currents, Figures 3(c) and 3(d) show the rotor currents, and Figure 3(e) illustrates the speed tracking errors. The simulation results illustrate the performances of the developed approach in terms of states tracking error and converged to zero in a short time for any initial condition of the motor. These figures responses are remarkable when a constant load torque of 5 N·m value is applied at s and the actuator faults occurred at s. Indeed, Figures 3(a)–3(d) show that the stator and rotor currents tracking errors are asymptotically decaying to zero despite the actuator occurring fault. In addition, a less tracking error for speed is reported in Figure 3(e) under both healthy and faulty conditions, and this demonstrates that the disturbance and fault rejection are guaranteed truly. In other words, simulation results demonstrate that the proposed systems adaptively reorganize themselves in the event of actuator faults and in the presence of bounded disturbances to sustain the best control performance. However, the fixed state feedback has a simpler structure in comparison with the adaptive state feedback, and then it can be used in practical implementations extensively.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

#### 6. Conclusion

In this paper, the fault-tolerant tracking problem for three-phase induction motor against actuator faults and bounded disturbances is studied. A direct adaptive fault-tolerant tracking controller design method is developed based on the online tuning of adaptive FTC law to reduce the tracking error and to guarantee maximum performance in the event of faults and disturbances. It has been also shown that one of the adaptive state feedback gains can be fixed which is more suitable from a practical point of view. The effectiveness of the proposed method is revealed using numerical simulation on the induction motor in the presence of actuator faults and disturbances.

#### Appendix

The induction motor parameters are selected as 3 HP/2.4 KW, , 60 Hz, A, and RPM. The parameters are Kg·m^{2}, Ω, Ω, Ω, Ω, Ω, and .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.