Fault-Tolerant Torque Control of BLDC Motors - IEEE Xplore

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 2, FEBRUARY 2011

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Fault-Tolerant Torque Control of BLDC Motors Farhad Aghili, Senior Member, IEEE

Abstract—Fault tolerance is critical for servomotors used in high-risk applications, such as aerospace, robots, and military. These motors should be capable of continued functional operation, even if insulation failure or open-circuit of a winding occur. This paper presents a fault-tolerant (FT) torque controller for brushless dc (BLdc) motors that can maintain accurate torque production with minimum power dissipation, even if one of its phases fails. The distinct feature of the FT controller is that it is applicable to BLdc motors with any back-electromotive-force waveform. First, an observer estimates the phase voltages from a model based on Fourier coefficients of the motor waveform. The faulty phases are detected from the covariance of the estimation error. Subsequently, the phase currents of the remaining phases are optimally reshaped so that the motor accurately generates torque as requested while minimizing power loss subject to maximum current limitation of the current amplifiers. Experimental results illustrate the capability of the FT controller to achieve ripple-free torque performance during a phase failure at the expenses of increasing the mean and maximum power loss by 28% and 68% and decreasing the maximum motor torque by 49%. Index Terms—BLDC motors, brushless motors, fault detection, fault tolerant control, optimal control, precision motion control, quadratic programming, torque control.

NOMENCLATURE a b c1 , . . . , cN ik im ax j L p, q R vk vF k vˆF k , v˜F k wk z θ λ k σk

Electrical time-constant of the motor. Filter time-constant. Fourier coefficients of the torque shape function. Current of the kth phase. Maximum allowable current. √ −1. Self-inductance of the phases. Number of phases and pole pairs. Resistance of the phases. Voltage of the kth phase. Filtered version of the kth phase voltage. Estimation of vF k and the estimation error. Perturbation signal in kth phase. Window width. Motor angle. Lagrangian multiplier. Fault signature signal associated with the kth phase. Covariance of the voltage estimation error.

Manuscript received March 29, 2010; revised May 25, 2010; accepted July 14, 2010. Date of current version February 4, 2011. Recommended for publication by Associate Editor S. Williamson. The author is with the Space Exploration of the Canadian Space Agency (CSA), Saint-Hubert, QC J3Y 8Y9, Canada (e-mail: farhad.aghili@ asc-csa.gc.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2010.2060361

ς τ, τd τp eak (θ) τm ax φk ω · · p eak · rm s

Threshold. Actual and command torques. Maximum motor torque at specific rotor angle θ. Maximum motor torque available over all rotor angles. Torque shape function of the kth phase. Angular velocity. Euclidean norm of a vector. Maximum magnitude of a signal. RMS norm of a signal. I. INTRODUCTION

RUSHLESS dc (BLdc) motors are commonly used in applications, such as robotics and automation, aerospace, vehicles, computed numerically controlled machines, manufacturing, and military. The importance of employing reliable motor drive systems in many of these application is paramount. In particular, for space application, it is highly desirable to use fault-tolerant (FT) actuators to drive the joints of a manipulator or wheels of a rover. Permanent-magnet synchronous motors, also known as BLdc motors, are commonly used as the drives of servo systems. BLdc motors are composed of a rotor containing a series of permanent magnets and the armature, which remains static while the electric power is distributed by an electronically controlled commutation system, instead of a mechanical commutator using brushes. Control approaches for generating accurate torque with electric motors and their underlying models have been studied by several researchers [1]–[8]. BLdc motors offer several advantages over brushed dc motors making them suitable for use as servomotors. Those include higher reliability, efficiency, and longer lifetime because of the absence of electrical and friction losses as well as erosion due to brushes. Moreover, BLdc motors can be completely sealed off and protected from dirt, oil, grease, and other types of foreign matter. Nevertheless, even a BLdc motor is subject to a variety of failures, which may occur during its operation. The incorporation of fault tolerance into a BLdc motor drive system involves the following tasks: 1) detection and identification of fault and 2) remedial strategies to recover from the fault. There has been a lot of research done in the field of fault detection and diagnosis of BLdc motors [9]–[17]. Fault detection of square waveform kind of BLdc motors based on parameters estimation of both electrical and mechanical systems is proposed in [9]. Diagnosis of electrical motors operating under constant speed has been studied in [10]. Analytic wavelet transform is used for detecting dynamics eccentricity in BLdc motors operating under varying speed [13]. Other researchers used neural network, particle and Kalman filtering, and fuzzy

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and instantaneously so that the phase currents can be treated as the control inputs. Then, the voltage equation of the phases is given by vk = L

Fig. 1.

Architecture of the FT torque controller.

systems for fault diagnosis of BLdc motors [11], [12], [14], [15]. Incorporating design features in the BLdc motors to make it inherently fault tolerant was discussed in [18]. This approach is applicable to BLdc motors with trapezoidal back electromotive forces (EMFs). Experimental verification of remedial strategies against failures occurring in the inverter power devices of a permanent-magnet synchronous motor drive is presented in [19] and [20]. Here, the basic idea is to incorporate a fourth inverter pole, with the same topology and capabilities of the other conventional three poles. This paper presents a novel FT ripple-free torque controller for BLdc motors operating under a single-phase failure. Multiple failures can be also recovered for motors with four or more phases. Unlike other approaches, the controller is applicable for BLdc motors with any phase waveform. Furthermore, the phase currents are optimally reshaped during a fault to minimize power dissipation while the maximum phase current limitation is taken into account. The FT torque controller is schematically illustrated in Fig. 1. First, an observer based on Fourier coefficients of the motor waveforms is employed to detect the faulty phases. Then, the optimal current values of the remaining phases are determined by solving a quadratic programming problem so that the motor can maintain producing ripple-free torque while minimizing power loss subject to maximum current limitation of the current amplifiers. A closed-form solution of optimal phase currents has been found rendering the control approach suitable for real-time implementation. The drawbacks of the FT torque controller are also discussed. These are: the FT controller tends to increase the power loss and decrease the maximum attainable motor torque under a fault condition. This paper is organized as follows: Section I reviews the modeling of BLdc motors based on Fourier coefficients of phase waveforms. Fault detection of motor phase is presented in Section II, while optimal reshaping of motor phase currents during phase failure is presented in Section III. Finally, experimental results demonstrating the performance of the FT torque controller are given in Section IV.

II. BLDC MOTOR MODEL We assume that there is negligible cross-coupling between the phase torques and there is no reluctance torque. In addition, we assume that the phase currents can be controlled accurately

d ik + Rik + φk (θ)ω + wk , dt

k = 1, . . . , p

(1)

where vk is the kth phase voltage, ik is the kth phase currents, R and L are the resistance and inductance of the phases, θ is rotor angle, ω is rotor speed, and wk represents an additive perturbation signal due to occurrence of a fault in kth phase. The last term in the right-hand side (RHS) of (1) are the back EMF of the phases, which are equal to the rotor speed times the phase shape functions φk (θ) ∀k = 1, . . . , p. Note that φk (θ) is a periodic function of rotor angle θ in rotary electric motors. Since successive phase windings are shifted by 2π/p, the kth torque shape function can be constructed as 2π(k − 1) φk (θ) = φ qθ + (2) p where q is the number of the pole pairs. Since φ is a periodic function with spatial frequency 2π/q, it can be effectively approximated through the truncated complex Fourier series as φ(θ) =

N

cn ej n q θ

(3)

n =−N

√ where j = −1, {c1 , . . . , cN } are the Fourier coefficients, and N can be chosen arbitrary large. The torque developed by a single phase is a function of the phase current and the rotor angle as τk (ik , θ) = ik φk (θ),

k = 1, . . . , p.

(4)

The motor torque τ is the superposition of all phase-torque contributions, τ = φT (θ)i

(5)

where φ = [φ1 , . . . , φp ]T and i = [i1 , . . . , ip ]T . The torque-control problem is to solve the aforementioned equation in terms of current ik (θ, τd ) as a function of motor position, given a desired motor torque τd . Given a scalar torque set point, (5) permits infinitely many (position dependent) phase current wave forms. Since the continuous mechanical power output of electrical motors is limited primarily by heat generated from internal copper losses, it makes sense to use the freedom in the phase-current solutions to minimize power losses Ploss = Ri2 .

(6)

Current saturation is the other limitation that should be considered. Let im ax > 0 be the maximum equivalent phase current corresponding to a linear-phase current-torque relationship, i.e., (4) is valid, or to current limit of the servo amplifier. Then, the phase currents must satisfy |ik | ≤ im ax

∀k = 1, . . . , p.

(7)

AGHILI: FAULT-TOLERANT TORQUE CONTROL OF BLDC MOTORS

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III. VOLTAGE OBSERVER The main objective of an observer-based fault-detection system is to generate some residual signals, which are sensitive to a group of faults. To this end, we will develop a simple voltage observer based on (1) and the input signals. However, such an observer will require the measurement of rotor speed ω, which may not be available. To get away from the speed measurement, we define functions Φk (θ) such that dΦk (θ) φk (θ) = . dθ

(8)

Thus, Φk (θ) =

N −jcn j n (q θ +ϕ k ) e nq

(8)

n =−N

where ϕk = 2π(k − 1)/p. Subsequently, by using the chain rule, we can rewrite the back-EMF term in RHS of (1) as dΦk dΦk dθ = , k = 1, . . . , p. (9) dθ dt dt Therefore, we can write the estimations of the phase voltages as φk (θ)ω =

d (Lik + Φk ) + Rik , k = 1, . . . , p. dt Now, consider the following stable and proper filters vˆk =

1 1 + bs ˘ 3 (s) = s G 1 + bs

(10)

˘ 2 (s) = R 1 + as G 1 + bs

˘ 1 (s) = G

where b > 0 is the filter time-constant, and L R is the time-constant of the motor. It is worth noting that if the time-constants of the filter and the motor are equal, i.e., b = a, ˘ 2 (s) = R. Now, define then the second filter simply becomes G the filtered version of the measured phase voltages and their estimations as a=

vF k (t) = G1 (t) ∗ vk (t)

(11a)

vˆF k (t) = G1 (t) ∗ vˆk (t),

k = 1, . . . , p

(11b)

˘ 1 (s) and ∗ stands where G1 is the impulse response of filter G for the convolution integral. Substituting vˆF from (10) in (11b) gives the values of the voltage estimation as vˆF k (t) = G2 (t) ∗ ik (t) + G3 (t) ∗ Φk (t)

(12)

where G2 and G3 are the impulse responses of the corresponding ˘ 2 (s) and G ˘ 3 (s). In the following, we develop proper filters G a model-based fault-detection scheme based on the voltageestimation error ˆ F − vF ˜F = v v

(13)

ˆ F are given from measurement (11a) and in which v F and v observer (12). In view of (1), (11a), and (12), we get ˜ F k = G1 ∗ wk , v

k = 1, . . . , p

(14)

which means that in absence of any fault, i.e., wk ≡ 0, the residuals are supposed to remain at zero. However, the perturbation is not identically zero during a fault condition and that gives rise to the residual error. Therefore, we use the covariance of the residual error to measure the severity of the fault. An Ergodic vF2 k ] in the sliding approximation of the covariance σk = E[˜ sampling window with finite length z can be obtained from σk (tn ) =

n 1 2 v˜ (tm ), w m =n −z F k

k = 1, . . . , p

(15)

where z is chosen empirically for smoothing data. The covariances (15) can be recursively computed by ⎧n − 1 1 2 ⎪ if n < z ⎨ n σk (tn −1 ) + n v˜F k (tn ), σk (tn ) = ⎪ 1 2 ⎩ v (tn ) − v˜F2 k (tn −z )), otherwise. σk (tn −1 ) + (˜ z Fk (16) Now, we can define the signature vector = [1 , . . . , p ]T , in which 1, if σk ≤ ς ∀k = 1, . . . , p. (17) k = 0, otherwise Here, ς is the chosen threshold level that should be empirically selected to determine the severity of the fault. The values of the threshold and the width of the sampling window are the tuning parameters of the FTC system, and they should be chosen empirically to give some statistical smoothing while retaining the system sensitivity to faults. Note that the FT system can quickly recover the torque-production performance when a fault occurs, if the values of the threshold ς is chosen as small as possible. This is because the controller can switch to the remedied phasecurrent waveform in a short period of time. However, it is well known that the false alarm rate is a function of the threshold value, i.e., the lower the threshold value, the higher is the rate of the false alarm. In other words, a small value of the threshold may trigger a false fault detection. Therefore, an adequate value of threshold should compromise quick fault recovery and reliable fault detection. IV. TORQUE CONTROL OF BLDC MOTOR WITH FAULTY PHASES A. Optimal Reshaping of Phase Currents The torque controller should not energize phases, which are detected to be faulty. Therefore, the commutation function considered for the control design purpose is defined as ⎡ ⎤ φ1 1 ⎢ . ⎥ ψ = ⎣ .. ⎦ . (18) φp p It will be shown later that the torque controller automatically sends zero-current commands to phases of which the waveform is zero. The following derivations present the optimal phase currents i∗k (θ, τd ), which generate the desired torque (5) and minimize the power losses (6) subject to the constraints (7). By setting τ = τd in (5), the problem of finding optimal phase

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currents that minimize power losses subject to the constraints is formulated by the quadratic programming problem min subject to:

T

i i

(19a)

h = ψ T i − τd = 0

(19b)

g1 = |i1 | − im ax ≤ 0 .. . gp = |ip | − im ax ≤ 0.

(19c)

Since all the functions are convex, any local minimum is a global minimum as well. Now, we seek the minimum point i∗ = [i∗1 , i∗2 , . . . , i∗p ]T satisfying the equality and inequality constraints. Before we pay attention to the general solution, it is beneficial to exclude the trivial solution, i∗k = 0. If the kth torque shape function is zero, that phase contributes no torque regardless of its current. Hence, ψk = 0 =⇒ i∗k = 0

∀k = 1, . . . , p

(20)

immediately specifies the optimal phase currents at the crossing point. By excluding the trivial solution, we deal with a smaller set of variables and number of equations in our optimization programming. Therefore, we have to find the optimal solution corresponding to the nonzero part. Hereafter, without loss of generality, we assume that all torque shape functions are nonzero. Now, by defining the function L = f + λh + μT g

(21)

where f = iT i, g = [g1 , g2 , . . . , gp ]T ∈ Rp , λ ∈ R, and μ = [μ1 , μ2 , . . . , μp ]T ∈ Rp . Let i∗ provide a local minimum of f (i) satisfying the equality and inequality constraints (19b) and (19c). Assume that column vectors ∇i g|i=i ∗ are linearly independent. Then, according to the Kuhn–Tucker theorem [21], there exist μk ≥ 0 ∀k = 1, . . . , p such that ∇i L = 0 μk gk (i∗k )

=0

(22a) ∀k = 1, . . . , p.

(22b)

Denoting sgn(·) as the sign function, we can show that ∇i g = diag sgn(i1 ), sgn(i2 ), . . . , sgn(ip ) in which the columns are linearly independent. The only pitfall is ik = 0, where the sign function is indefinite. We assume that the optimal solutions, i∗k are nonzero because ψk = 0. This assumption will be relaxed later. Substituting f , h, and g into (22) yields 2i∗k + λψk + μk sgn(i∗k ) = 0 μk (|i∗k | − im ax ) = 0,

(23) k = 1, . . . , p.

(24)

Equations (23) and (24) together with (19b) constitute a set of 2p + 1 nonlinear equations with 2p + 1 unknowns i∗ , λ, and μ to be solved in the following. Since μk gk (i∗k ) = 0 while μk ≥ 0 and gk (i∗k ) ≤ 0, we can say that μk = 0 for |ik | < im ax , and that μk ≥ 0 for |ik | = im ax . Therefore, (23) can be written in

the following compact form T (i∗k ) = −0.5λψk

∀k = 1, . . . , p.

(25)

The mapping T : D → R, and D(x) = {x ∈ R : |x| ≤ im ax }, is defined by x, |x| < im ax (26) T (x) = x + 0.5sgn(x)μ, |x| = im ax where μ is any positive number. It is apparent that the mapping is invertible on D, that is, there exists a function T −1 (x) such that T −1 (T (x)) = x ∀x ∈ D. In other words, the variable i∗k in (25) can be determined uniquely if the RHS of the equation is given. The inverse of the mapping is the saturation function, i.e., T −1 (·) ≡ sat(·), defined by x, |x| ≤ im ax (27) sat(x) = sgn(x)im ax , otherwise. Now, (25) can be rewritten as i∗k = sat(−0.5λψk )

∀k = 1, . . . , p.

(28)

The aforementioned equation implies that i∗k = 0, as ψk = 0, which relaxes the assumption we made earlier. The second result is that the larger the magnitude of the torque shape function |ψk |, the larger the magnitude of the optimal current i∗k . If the phases are labeled in descending order |i∗1 | ≥ |i∗2 | ≥ . . . ≥ |i∗p | (29) the optimal phase currents from i∗1 to i∗p must be saturated consecutively. We use this fact to calculate the optimal phase currents consecutively in the same order, starting with i∗1 . In case saturation of a phase occurs, (28) implies that only knowing the sign of λ is enough to calculate the associate phase current. One can infer from (25) and (19b) that |ψ1 | ≥ |ψ2 | ≥ . . . ≥ |ψp |

=⇒

sgn(τd ) = sgn(−λ).

(30)

Therefore, if i∗1 saturates, then i∗1 = sgn(−ψ1 λ)im ax = sgn(i1 τd )im ax .

(31)

If i∗1 does not saturate, i.e., |i∗1 | < im ax , then neither does {i2 , . . . , ip }[see (29)]. Let λ(1) represents the Lagrangian multiplier when i∗1 does not saturate, then the Lagrangian multiplier can be calculated by substituting phase currents from i∗k = −0.5λψk into (19b) −2τd λ(1) = p 2 k =1 ψk

(32)

which, in turn, can be substituted in (28) to obtain the optimal phase current ψ 1 τd ∗ i1 = sat p . (33) 2 k =1 ψk Since the denominator in (33) is always positive, by virtue of (31), one can infer that (31) provides the optimal solution for the saturation case as well. Analogously, i∗2 can be calculated, if


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ψ1 i∗1 − τd is treated as the known parameter in (19b). In general, the ith phase current can be calculated by induction as follows: since up to (i − 1)th phase currents have been already solved, we have ψi i∗i + · · · + ψp i∗p = τd − (ψ1 i∗1 + · · · + ψi−1 i∗i−1 )

(34)

where the value of the RHS of the aforementioned equation is known. The Lagrangian multiplier associated with the case of unsaturated i∗i can be found from (28) and (34) as ψ i∗ ) −2(τd − i−1 p k =12 k k . λ(i) = (35) k =i ψk Finally, substituting (35) in (28) gives the optimal phase currents, which produce the desired torque precisely, while minimizing power losses, subject to the constraints of current saturation ⎧ ψ 1 τd ⎪ ⎪ sat , if k = 1 ⎪ ⎪ ψ2 ⎨ i∗k = k −1 ∗ ⎪ τ − ψ ψ i ψ ⎪ k d k m m ⎪ p m =1 ⎪ , if k = 2, . . . , p. ⎩ sat 2 m =k ψm (36)

Fig. 2.

Motor prototype mounted on the dynamometer.

until, in the limit, all phases are saturated. Then, the peak torque τp eak corresponding to maximum allowable phase current im ax is p τp eak (θ) = |ψk (θ)|im ax . (37) k =1

Under normal conditions, the peak torque is given by τp eak (θ) =

B. Implementation of Faut-Tolerant Torque Controller The amplifier drive system of the phases must be independent so that fault condition in one drive of motor phase do not affect normal operation of the drivers of the other phases. The proposed commutation strategy does not rely on any condition for the phase-torque-angle waveforms, e.g., having balanced phases that impose the Kirchhoff’s Current Law (KCL) constraint at the floating neutral node. In particular, the condition of having balanced phases doe not apply when at least one phase fails. Therefore, ik = 0, which means the floating neutral node must be grounded (see Fig. 1). Assume that vector [ψk (θ1 ), ψk (θ2 ), . . . , ψk (θn )]T ∈ Rn represents the discrete torque shape functions corresponding to n measurements of the phase torque (with unit current excitation) and positions. The torque-control algorithm is implemented as follows. 1) For a given rotor position θ, calculate the shape functions ψk from the Fourier coefficients and the phase shifts according to (2) and (3). 2) From the voltage observer (12), determine the signature vector σ, according to (16) and (17). Subsequently, construct the modified shape functions ψk , according to (18). 3) Sort ψk s such that |ψ1 | ≥ |ψ2 | ≥ · · · ≥ |ψp | and calculate the optimal currents from (36); Go to step i.

p

|φk (θ)|im ax .

(38)

k =1

Note that the peak torque varies from one rotor angle to another. Therefore, only the low torque value is available over all rotor angles without having saturation-induced torque ripple. Thus, τm ax = min τp eak (θ). θ

V. EXPERIMENT In order to evaluate the performance of the fault-tolerant torque controller, experiments were conducted on a three-phase synchronous motor with nine pole pairs. The experimental setup is illustrated in Fig. 2. The motor used for the testing is the McGill/MIT synchronous motor [22]. Three independent current servo amplifiers (Advanced Motion Control 30A20AC) control the motor’s excitation currents as specified by the electronic commutator. The amplifier’s rated current and voltage are 15 A and 190 V, respectively, with a switching rate of 22 kHz. The phase voltages are attenuated by resistor branches and then sensed by isolation amplifiers (model AD 210 from Analog Device). The isolation amplifier system eliminates the possibility of leakage paths and ground loops between the power servo amplifiers and the data-acquisition system by providing complete transformer isolation. All analog sensor signals are passed through antialiasing filters and connected to a multichannel dataacquisition system.

C. Maximum Motor Torque Under Normal and Fault Conditions

A. Identification of Motor Parameters

The control algorithm presented in earlier section permits torque among phases when some phases saturate. In this case, the maximum torque depends on the saturation of the largest phase-torque function. The controller increases the torque contribution of the unsaturated phases when one phase saturates,

The actual values of the winding resistance and selfinductance are measured by a Wheatstone bridge instrument (see Table I). The Fourier coefficients of the back-EMF shape functions are measured by a specially designed dynamometer setup (see Fig. 2). The electric motor and a hydraulic rack and

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TABLE I ELECTRIC PARAMETERS OF THE MOTOR

Fig. 3.

TABLE II SPATIAL HARMONICS OF THE PERIODIC FUNCTION φ(θ)

Back-EMF shape functions of the phases.

pinion rotary motor are mounted on the rigid structure of the dynamometer. The hydraulic motor’s shaft is connected to that of the direct drive motor via a torque transducer (Himmelstein MCRT 2804TC) by means of two couplings, which relieve bending moments or shear forces due to small axes misalignments. The speed of the hydraulic motor is controlled by a pressurecompensated flow control valve. The hydraulic pressure is set sufficiently high so that the hydraulic actuator regulates the angular speed regardless of the applied motor torque. An adjustable cam and two limit switches detect the two rotational extremes and activate a solenoid valve through a programming logic controller unit to reverse the direction. Since the motor has nine pole pairs, the torque trajectory is periodic in position with a fundamental spatial frequency of nine cycles/r, and thus, the torque pattern repeats every 40◦ . The phase-shape functions φk s are shown in Fig. 3. The significant spatial frequency components appear at the 1st, 11th, and 13th harmonics, as shown in Table II. B. Performance Test The objective of this section is to demonstrate that the FT torque controller can achieve continued ripple-free torque performance under a single-phase failure. Again, the motor shaft is rotated by the hydraulic actuator while the motor torque is monitored by the torque transducer. In this experiment, the desired torque is set to τd = 10 Nm

Fig. 4.

Phase currents in absence of fault corresponding to τ d = 10 Nm.

while the parameters of the fault-detection system are set to ς = 25 V2

z = 4 s.

Figs. 4 and 5 show the optimal phase currents under normal and fault conditions (one broken phase), respectively. Note that the remaining two phases during the fault produces the same torque 10 Nm as the three phases. To emulate the fault condition, the current circuit of the motor’s third phase is virtually broken by sending zero signal to the enable port of the corresponding power amplifier. Time histories of the voltage residual error and its covariance are illustrated in Figs. 6 and 7, respectively. It is apparent from the graphs that the fault occurs at t = 9.9 s, and subsequently, it is detected at t = 11.5 s. It is worth pointing out that the small but nonzero errors during the normal condition are due to the lumped effects of the unmodeled dynamics, parametric uncertainties, and the sensor noise. However, the


Fig. 5.

Fig. 6.

Fig. 7.

Phase currents with occurrence of fault corresponding to τ d = 10 Nm.

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Fig. 8.

Motor torque before and after occurrence of the fault.

Fig. 9.

Power dissipation under normal and fault conditions.

Time history of the voltage residual error.

Time history of the covariance of the residual error.

perturbation cased by the fault gives rise to the magnitude of the residual errors. Trajectory of the motor torque before and after the fault occurrence is shown in Fig. 8. It is evident from the graph that fault-tolerant torque controller achieve ripple-free torque as soon as the fault is detected at t = 11.5 s. This can be useful in practice when there is a need to continue operating the motor even in the case of a phase failure. However, the price is a higher power loss and lower maximum torque as comparatively demonstrated in Figs. 9 and 10. Fig. 10 shows the graphs of the maximum achievable torque with respect to maximum phase current im ax = 15 A. The solid line and the dashed line depict the maximum attainable torque with respect to normal and fault conditions, respectively. As described in Section III-C, the torque saturation points differs from one position to another. The summary of the results are reported in Table III. In this particular case, the FT torque controller is able to generate the requested torque 10 Nm, even with the failure of one of its phases at the expense of increasing the mean and maximum of power loss by 28% and 68%, respectively, while reducing

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minimizing power loss subject to maximum current limitation of the current amplifiers has been formulated as a quadratic programming problem. Subsequently, a closed-form solution has been found. The shortcomings of that FT torque controller regarding increasing the power loss and decreasing maximum attainable motor torque while the motor operates under fault condition has been discussed. Experimental results demonstrated that the FT controller can achieve accurate torque control performance during a phase failure at the expenses of increasing the mean and maximum power loss by 28% and 68%, while decreasing the maximum motor torque by 49%. REFERENCES

Fig. 10. Maximum attainable motor torque corresponding to im a x = 15 A under normal and fault conditions.

TABLE III CHARACTERISTICS OF THE FTC UNDER NORMAL AND FAULT CONDITIONS

the maximum torque capability of the motor by 49%. The table also shows that the rms norm1 and peak norm of the current of the healthy phases are increased by 59% and 88%, respectively, during the a phase failure. This means that a recovery at the rated torque is not possible unless both amplifiers and the motor are designed in such way that they can cope with the enhanced phase currents and overheating. Furthermore, during a fault, the phase voltages are increased as result of the increased phase current. Therefore, the phase voltages reach to their saturation limits at a lower speed during a fault compared to the normal operation. Consequently, the rated speed also decreases when operating with a phase failure. VI. CONCLUSION A novel FT ripple-free torque controller has been proposed for BLdc motors operating under a phase failure. An observer based on Fourier coefficients of the motor waveforms has been developed to estimate the phase voltages, which is used to calculate the residual voltage errors from measured phase voltages. The covariances of the voltage errors are then used to detect the faulty phases. Finding the optimal current values of the remaining phases so that the motor produces ripple-free torque while

1 The

rms norm of signal i is defined as irm s =

T 1 T

t= 0

i2 dt

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Farhad Aghili (SM’07) received the B.Sc. degree in mechanical engineering and the M.Sc. degree in biomedical engineering from the Sharif University of Technology, Tehran, Iran, in 1988 and 1991, respectively, and the Ph.D. degree in mechanical engineering from McGill University, Montreal, QC, Canada, in February 1998. During 1994–1997, he was a Research Engineer at MPB Technologies, Montreal. In January 1998, he joined the Canadian Space Agency, Saint-Hubert, QC, where he is currently a Federal Research Scientist. He has authored or coauthored more than 100 papers in journals and conference proceedings, and holds six patents and patent applications in U.S. and Canada. His research interests include multibody dynamics, robotics, space robotics, vision system, control systems, and mechatronics systems. Dr. Aghili is a member of AIAA. He won the best paper award in 2007 ASME/IEEE International Conference on Mechatronic and Embedded Systems and received the CSA Inventor & Innovator Certificate six times.