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This technical report presents an optimal torque control of brushless DC motor with any back-emk waveform that minimizes power dissipation subject to voltage and current

limits of the motor’s drivers/amplifiers. When one or more phases reach the voltage or/and current limits, the controller optimally reshapes the stator currents of the remaining phases for continuing accurate torque production. This allows the motor to operate above the rated speed and torque that would be achieved without current reshaping. In the event that an open-circuit or short-circuit of a winding occurs, the torque controller can also isolate the faulty phase in order to generate torque as requested given the voltage and current constraints of the healthy phases. A closed-form solution for the optimal phase currents at give angular position, velocity, and desired torque is obtained rendering the control algorithm suitable for real-time implementation. Experimental results illustrate the capability of the controller to achieve precise torque production during voltage/current saturation of the motor’s drivers or a phase failure.

limits of the motor’s drivers/amplifiers. When one or more phases reach the voltage or/and current limits, the controller optimally reshapes the stator currents of the remaining phases for continuing accurate torque production. This allows the motor to operate above the rated speed and torque that would be achieved without current reshaping. In the event that an open-circuit or short-circuit of a winding occurs, the torque controller can also isolate the faulty phase in order to generate torque as requested given the voltage and current constraints of the healthy phases. A closed-form solution for the optimal phase currents at give angular position, velocity, and desired torque is obtained rendering the control algorithm suitable for real-time implementation. Experimental results illustrate the capability of the controller to achieve precise torque production during voltage/current saturation of the motor’s drivers or a phase failure.

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CSA-SE-TN-001 This-Page-Intentionally-Left-Blank ii

CSA-SE-TN-001 Preface This report, CSA-SE-TN-001, must be approved by the undersigned. Any proposed changes to the initial release version of this document shall be forwarded to the CSA Space Exploration (SE) Division and implemented in co-operation with the CSA SE Division’s document change process. Prepared By: August 24, 2012 Farhad Aghili, Research Scientist Date iii

CSA-SE-TN-001 Revision History Rev. IR Description Date Initial Release of the document 24-08-2012 iv Initials FA

CSA-SE-TN-001 Abstract This technical report presents an optimal torque control of brushless DC motor with any back-emk waveform that minimizes power dissipation subject to voltage and current limits of the motor’s drivers/ampliﬁers. When one or more phases reach the voltage or/and current limits, the controller optimally reshapes the stator currents of the remaining phases for continuing accurate torque production. This allows the motor to operate above the rated speed and torque that would be achieved without current reshaping. In the event that an open-circuit or short-circuit of a winding occurs, the torque controller can also isolate the faulty phase in order to generate torque as requested given the voltage and current constraints of the healthy phases. A closed-form solution for the optimal phase currents at give angular position, velocity, and desired torque is obtained rendering the control algorithm suitable for real-time implementation. Experimental results illustrate the capability of the controller to achieve precise torque production during voltage/current saturation of the motor’s drivers or a phase failure. v

CSA-SE-TN-001 Contents Preface iii Abstract v 1 Motor Model with Current and Voltage Limits 2 2 Optimal Reshaping of Phase Currents 2.1 Quadratic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fault-Tolerant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 9 3 Experiment 3.1 Identiﬁcation of Motor Parameters . . 3.2 Performance Test . . . . . . . . . . . 3.3 Motor Torque-Velocity Characteristic 3.4 Single-phase Failure . . . . . . . . . vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10 10 13 20

CSA-SE-TN-001 List of Figures 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 The piecewise linear function of the Lagrangian multiplier. . . . . . . . . The motor, ampliﬁer, sensors, and controller diagram. . . . . . . . . . . . The back-emf waveform. . . . . . . . . . . . . . . . . . . . . . . . . . . The waveform of the cogging torque. . . . . . . . . . . . . . . . . . . . . The ̺ and s variables corresponding to the phase current and voltage limits The virtual torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case I: phase currents. . . . . . . . . . . . . . . . . . . . . . . . . . . . Case I: phase voltages. . . . . . . . . . . . . . . . . . . . . . . . . . . . Case I: motor torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case II: phase currents. . . . . . . . . . . . . . . . . . . . . . . . . . . . Case II: phase voltages. . . . . . . . . . . . . . . . . . . . . . . . . . . . Case II: motor torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum attainable motor torque . . . . . . . . . . . . . . . . . . . . . Case III: phase currents when phase 1 is open circuit. . . . . . . . . . . . Case III: phase voltages when phase 1 is open circuit. . . . . . . . . . . . Case III: motor torque when phase 1 is open circuit. . . . . . . . . . . . . vii 6 8 11 11 14 15 15 16 16 17 17 18 18 19 19 20

List of Figures 1 Introduction Brushless DC motors are commonly used as the drives of servo systems in a wide range of industrial applications from robotics and automation to aerospace and military. Accurate and ripple-free torque control of brushless motors is essential for precision control of such servo systems. In brushless motors, the electric power is distributed by an electronically controlled commutation system, instead of a mechanical commutator found in brushed DC motors. The conventional electronic commutator incorporates a feedback from the rotor angular position into a control system, which excites the stator coils of the motor in a speciﬁc order in order to rotate the magnetic ﬁeld generated by the coils to be followed along by the rotor. Conventional drivers of brushless motors produce sinusoidal current waveforms for smooth motor operation. However, non-ideal motors are not with a perfectly sinusoidally distributed magneto-motive force, and hence a sinusoidal commutation can result in torque ripple. It has been proved that suppressing the torque ripple of the motor drive of a servo system can signiﬁcantly improve system performance by reducing speed ﬂuctuations (Park et al., 2000; Aghili et al., 2003). Commercial high-performance electric motors reduce the pulsating torque by increasing the number of motor poles. However, such motors tend to be expensive and bulky due to construction and assembly of multiple coil windings. Control approaches for accurate torque production in electric motors and their underlying models have been studied by several researchers (Murai et al., 1989; Matsui et al., 1993; Taylor, 1994; French and Acarnley, 1996; Kang and Sul, 1999; Park et al., 2000; Aghili et al., 2003; Wang et al., 2003; Xu and Rahman, 2004; Ozturk and Toliyat, 2011; Grabner et al., 2010; Tsai et al., 2011; Aghili, 2011a,b). It was assumed in these works that the phase currents can be controlled accurately and instantaneously and therefore the currents can be treated as the control inputs. Then, the waveforms of the motor phase currents are adequately preshaped so that the generated torque is equal to the requested torque. However, when the motor’s drivers have ﬁxed rated current and voltage limits, some of them may not be able to deliver the current inputs dictated by the electronic commutator that may occur when the motor operates at high torque or speed. Consequently, the performance of the torque production may signiﬁcantly deteriorate as a result of the phase current distortions caused by the voltage or current saturation of the ampliﬁers. Flux weakening allows a machine to operate above the base speed in constant-power, high speed region when there is a ﬁxed inverter voltage and current (EL-Refaie et al., 2004). Below the rated speed, all of the stator currents can be used to produce torque. Above the rated speed, a part of the stator current must be used to oppose the permanent magnet ﬂux while the remaining portion is used to produce torque. Several authors have addressed ﬂux weakening in PM machines (Sebastiangordon and Slemon, 1987; Jahns, 1987; Sneyers et al., 1985; Schiferl and Lipo, 1990). However, this techniques can deal with electric motors with prefect sinusoidal back-emf waveform and, in addition, phase current limits are not taken into account. This work presents a closed-form solution for optimal excitation currents for accurate torque control of brushless motors with any waveform that minimizes power dissipation http://www.asc-csa.gc.ca CSA-SE-TN-001

1. Motor Model with Current and Voltage Limits 2 subject to currents and voltage limits of the motor’s drivers. When the motor terminal voltages or/and phase currents reach their saturation levels, the controller automatically reshapes the excitation currents in such a way that the motor generates torque as requested. This optimal management of motor’s excitation currents can signiﬁcantly increase the rated speed and torque of the motor in the face of the voltage and current limits of the drivers. In addition, the torque controller can be use as a remedial strategy to compensate for a phase failure by optimally reshaping the currents of the remaining healthy phases for accurate torque production. 1 Motor Model with Current and Voltage Limits The electromagnetic torque developed by a single phase is a function of the phase current and the rotor angle, while cogging torque is only a function of the rotor angle. If a motor operates in a linear magnetic regime, then the torque τ of the motor with p phases is given by τ (θ, i) = φT (θ)i + τcog (θ), (1) where τcog is the cogging torque, θ is the angular position of the rotor, and vectors φ ∈ Rp and i ∈ Rp are deﬁned as: φ(θ) = [φ1 (θ), · · · , φp (θ)]T i = [i1 , · · · , ip ]T Here, ik and φk (θ) are the instantaneous current and back-emf shape function of the kth phase. In rotary electric motors, the shape functions are periodic functions of rotor angle. 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 motor poles. Furthermore, since φ is a periodic function with spacial frequency 2π/q, it can be effectively approximated through the truncated complex Fourier series as N cn ejnqθ φ(θ) = (3) n=−N √ where j = −1, {c1 , · · · , cN } are the Fourier coefﬁcients, and N can be chosen arbitrary large. Similarly, the cogging torque can be approximated by another ﬁnite Fourier series as N bn ejnqθ , τcog (θ) = (4) n=−N where {b1 , · · · , bN } are the corresponding Fourier coefﬁcients. The torque control problem is: Given a desired motor torque τd , solve the torque equation (1) in terms of the phase currents, i. Given a scalar torque set point, mapping (1) http://www.asc-csa.gc.ca CSA-SE-TN-001

1. Motor Model with Current and Voltage Limits 3 permits inﬁnitely many phase current waveforms. 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 = R i 2 , (5) where R is the resistance of the phases. However, due to voltage and current limits of the drivers, minimization of (5) must be subject to the following set of 2p inequality constraints: |ik | ≤ imax |vk | ≤ vmax (6a) ∀k = 1, · · · , p. (6b) where vk is the kth phase voltage, and imax and vmax are, respectively, the current and voltage limits of the motor’s drivers. In the following development, we will show that the above set of 2p inequalities can be equivalently reduced to a set of p inequalities if the motor inductance is assumed negligible. The voltage of the winding terminal of an electric motor is the superposition of the back-emf and the ohmic voltage drop if the inductance of the stator coils is negligible (Hendershot and Miller, 1994; Moseler and Isermann, 2000; Lee et al., 2003). That is vk = Rik + ek k = 1, · · · , p (7) where ek is the back-emf of the phases, which is equal to the rotor speed, ω, times the back-emf shape functions, φk (θ), i.e., ek = φk (θ)ω. Using the relation between the terminal voltages and the phase currents in (7), we can rewrite inequalities (6b) as ik + ω vmax φk (θ) − ≤0 R R ∀k = 1, · · · , p (8) which are equivalent to: −vmax − ωφk (θ) vmax − ωφk (θ) ≤ ik ≤ R R k = 1, · · · , p (9) Using (6a) in (9) yields: −vmax − ωφk (θ) > −imax R vmax − ωφk (θ) < imax R ∀θ ∈ R, k = 1, · · · , p Therefore, the condition for the existence of a solution for inequalities (6a) and (9) can be expressed by vmax + Rimax (10) max |φk (θ)| < θ,k ω http://www.asc-csa.gc.ca CSA-SE-TN-001

2. Optimal Reshaping of Phase Currents 4 In other words, if condition (10) is satisﬁed, then inequalities (6a) and (9) can be combined in the following form: ik ≤ ik ≤ ¯k i (11) where the lower- and upper-bound current limits are −vmax − ωφk (θ) ik (θ, ω) = max − imax , R (12) vmax − ωφk (θ) ¯k (θ, ω) = min imax , i R Finally, the constraint inequality on the phase currents pertaining to both current and voltage limitations can be expressed by |ik − ̺k (θ, ω)| − σk (θ, ω) ≤ 0 ∀k = 1, · · · , p (13) where 1 i ̺k (θ, ω) = (¯k + ik ) 2 (14) 1 i σk (θ, ω) = (¯k − ik ) 2 are the current offset and gain associated with the kth phase. The inequality constraints (13) imposed on the phase currents are equivalent to both voltage and current limits (6). Notice that parameters ̺k and σk are not constants rather they are at given mechanical states θ and ω. However, in the following analysis, we drop the position and velocity arguments for the simplicity of notation. 2 Optimal Reshaping of Phase Currents 2.1 Quadratic Programming The derivations in this section present the optimal phase currents i∗ = [i∗ , · · · , i∗ ]T , 1 p which generate the desired torque τd and minimize the power losses (5) subject to the constraints (6). By setting τ = τd in (1) and using the equivalent inequality constraints (13), the problem of ﬁnding optimal instantaneous currents can be equivalently formulated as the following quadratic programming problem: min iT i (15a) T (15b) subject to: h = φ i + τcog − τd = 0 |i1 − ̺1 | g1 = −1≤0 σ1 . . . gp = http://www.asc-csa.gc.ca (15c) |ip − ̺p | −1 ≤0 σp CSA-SE-TN-001

2.1 Quadratic Programming 5 Note that all the instantaneous variables in the above equality and inequality constants, i.e., φk , ̺k , and σk , are at given rotor angular position, θ, and velocity ω. Since all the functions are convex, any local minimum is a global minimum as well. Now, we seek the minimum point i∗ satisfying the equality and inequality constraints. Before we pay attention to the general solution, it is beneﬁcial to exclude the trivial solution, i∗ = 0. If the kth torque k shape function is zero, that phase contributes no torque regardless of its current. Hence, φk = 0 =⇒ i∗ = 0 k ∀k = 1, · · · , p (16) 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 non-zero. Now, Deﬁning the function L = f + λh + µT g, (17) where f = iT i, g = [g1 , g2, · · · , gp ]T , λ ∈ R, and µ = [µ1 , µ2, · · · , µp ]T . Let i∗ provide a local minimum of f (i) satisfying the equality and inequality constraints (15b) and (15c). Assume that column vectors ∇i g|i=i∗ are linearly independent. Then according to the Kuhn-Tucker theorem (Kuhn and Tucker, 1951), there exist µk ≥ 0 ∀k = 1, · · · , p such that ∇i L = 0 µk gk (i∗ ) k =0 (18a) ∀k = 1, · · · , p. (18b) Denoting sgn(·) as the sign function, we can show that ∇i g = diag sgn(i1 − ̺1 ) sgn(ip − ̺p ) ,··· , , σ1 σp 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∗ are non-zero because k φk = 0. This assumption will be relaxed later. Substituting f , h and g from (15) into (18) yields 2σk i∗ + λσk φk + µk sgn(i∗ − ̺k ) = 0 k k µk (|i∗ k − ̺k | − σk ) = 0 (19a) (19b) Equations (19a) and (19b) together with (15b) 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∗ ) = 0 k while µk ≥ 0 and gk (i∗ ) ≤ 0, we can say that µk = 0 if |ik − ̺k | < σk , while µk > 0 if k |ik − ̺k | = σk . Therefore, equations (19a) can be written in the following compact form 1 Tk (i∗ − ρk ) = − λφk − ̺k k 2 http://www.asc-csa.gc.ca ∀k = 1, · · · , p. (20) CSA-SE-TN-001

2.1 Quadratic Programming 6 30 theta=100 20 psi 10 0 −10 −20 −30 −40 −30 −20 −10 0 lam 10 20 30 40 Figure 1: The piecewise linear function of the Lagrangian multiplier. The mapping Tk : Dk → R, in which Dk (x) = {x ∈ R : |x − ̺k | ≤ σk }, is deﬁned by Tk (x) = x x+ µk sgn(x 2σk − ̺k ) |x − ̺k | < σk |x − ̺k | = σk (21) It is apparent that the mapping is invertible on D, that is there exists a function T −1 (x) −1 such that Tk (Tk (x)) = x ∀x ∈ D. In other words, the variable i∗ in (20) can be k determined uniquely if the right-hand-side of the equation is given. The inverse of the −1 mapping is the saturation function, i.e. Tk (·) ≡ σk sat(·/σk ) where x>1 1 sat(x) = (22) x −1 ≤ x ≤ 1 −1 x < −1 In view of the function deﬁnition, we can rewrite (20) as i∗ = ̺k + σk sat k −0.5λφk − ̺k σk ∀k = 1, · · · , p. (23) Upon substitution of the optimal phase currents from (23) into the torque equation (15b), we arrive at p −0.5λφk − ̺k = τ ∗, (24) φk σk sat σk k=1 http://www.asc-csa.gc.ca CSA-SE-TN-001

2.1 Quadratic Programming 7 where τ ∗ = τd − τcog − τ̺ and p φj ̺j τ̺ = (25) j=1 is the torque offset calculated at any position θ. Note that τ̺ does not have any physical meaning but can be interpreted as the torque associated with non-zero mean values of the phase current limits. Since λ is the only unknown variable in (24), ﬁnding the optimal values of the phase currents boils down to solve the algebraic equation (24) for the Lagrangian multiplier. Apparently, the left-hand side of (24) is a piecewise linear function of λ. Thus, (24) can be concisely written as ψ(λ) = τd − τcog − τ̺ . (26) As an illustration, the piecewise linear function for a motor system which will be described in Section 3 is depicted in Fig.1 for a particular motor angle. The slop of the piecewise linear function abruptly decreases each time one of the phases saturates and when all phases saturate the slop become zero, in that case there is no solution for λ. Therefore, the piecewise linear function is invertible only if the following conditions are satisﬁed ψmin + τcog − τ̺ ≤ τd ≤ ψmax + τcog − τ̺ (27) Then, the corresponding Lagrangian multiplier can be obtained by λ = ψ −1 (τd − τcog − τ̺ ) (28) Finally, substitution of (28) into (23) yields the optimal phase currents as: i∗ = ̺k + σk sat k −0.5ψ −1 (τd − τcog − τ̺ )φk − ̺k . σk (29) The motor, ampliﬁer derives and the torque controller are depicted in Fig. 2. In summary, the optimal torque control algorithm subject to current and voltage limits of the drivers may proceed by the following steps: (1) Given angular position θ, calculate the shape functions, φk , and cogging torque, τcog , form their Fourier coefﬁcients according to (2), (3), and (4). (2) Given angular velocity ω, compute σk , ̺k , and τ̺ from the current and voltage limits, imax and vmax , according to (12), (14), and (25). (3) Calculate the optimal currents from (29); Go to step i. http://www.asc-csa.gc.ca CSA-SE-TN-001

http://www.asc-csa.gc.ca 4π pq 2π pq 0 Fourier series [c1 , · · · , cn ] 0 Fourier series [c1 , · · · , cn ] 0 Fourier series [c1 , · · · , cn ] 0 normal fault normal fault normal fault Fourier series [b1 , · · · , bn ] φ3 φ2 φ1 _ _ σ1 (θ, ω) σ3 (θ, ω) ̺3 (θ, ω) σ2 (θ, ω) ̺2 (θ, ω) scale 1 − 2 ψ(·)−1 + + + piecewise linear function + ̺1 (θ, ω) oﬀset τd τ∗ + + + + + + 1 σ3 1 σ2 1 σ1 +1 +1 +1 σ3 σ2 σ1 + + _ + _ + i∗ 1 i∗ 2 i∗ 3 i ≤ imax v ≤ vmax Current ampliﬁers τ M velocity sensor −1 −1 −1 Limiter θ Load position sensor ω 2.1 Quadratic Programming 8 Figure 2: The motor, ampliﬁer, sensors, and controller diagram. CSA-SE-TN-001

2.2 Fault-Tolerant 9 2.2 Fault-Tolerant The optimal solution given in the previous section can be readily used as a remedial strategy to recover from a phase failure. The idea is that when an open-circuit or short-circuit fault occurs, the faulty phase is isolated and the currents of the remaining healthy phases are optimally reshaped for accurate torque production. It is clear from (16) that when the instantaneous back-emf of a phase takes a zero value then the corresponding phase current becomes trivially zero. Therefore, the faulty phase can be easily isolated by setting the value of its back-emk shape function to zero in the optimization formulation. Now, suppose that the motor’s shape functions for the optimization programming (15) is modiﬁed according to φk for normal phase (30) φk = 0 for faulty phase Then, the optimal solution (29) ensures that the instantaneous currents of faulty phase is zero, while the remaining phases generate the desired torque satisfying the optimality conditions. 3 Experiment In order to evaluate the performance of the optimal torque controller, experiments were conducted on a three-phase synchronous motor with 9 pole pairs. Three independent current servo ampliﬁers (Advanced Motion Control 30A20AC) control the motor’s excitation currents as speciﬁed by the torque controller. The electric motor and a hydraulic rack and pinion rotary motor are mounted on the rigid structure of a dynamometer. The hydraulic motor’s shaft is connected to that of the electric 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 pressure compensated ﬂow control valve. The hydraulic pressure is set sufﬁciently high so that the hydraulic actuator can always regulate the angular speed regardless of the electric motor torque. In other words, the operating speed of the electric motor is independently set by the hydraulic actuator. 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. A multi-channel data acquisition system acquires the analog data at the sampling rate of 1 kHz. Errors between the measured rotor position, used by the torque controller for current commutation, and the true rotor position will result in torque ripple. One source of this error is measured quantization. In order to minimize the torque-ripple induced by this quantization as much as possible, the motor uses a high-resolution encoder with 0.001◦ resolution for measurement of angular position. http://www.asc-csa.gc.ca CSA-SE-TN-001

3.1 Identiﬁcation of Motor Parameters 10 3.1 Identiﬁcation of Motor Parameters Torque shape functions are measured by using the hydraulic dynamometer (Aghili et al., 2007). To this end, the torque trajectory data versus position was registered during the rotation, while one phase was energized with a constant current. The value of the winding resistance measured by a wheat-stone bridge instrument is R = 2.54Ω. (31) The torque-angle data are registered within almost one rotation while the phase current is kept constant. But the current is incremented at the end of each rotation stroke by 1A until an ensemble of torque proﬁles belonging to the span of [−15, 15] A is obtained. By taking the average of two sequences of the motor torques corresponding to when the motor shaft is rotated clockwise and counter-clockwise, the effect of the bearing friction torque in the motor torque measurement is compensated. Now at every motor position θ, the current load cases ij s and the corresponding measured torques τ j (θ) are related by 1 1 1 τ (θ) i1 1 τ 2 (θ) i2 1 1 φ1 (θ) (32) . . . . . τcog (θ) = . . . . τ u (θ) iu 1 1 Then, φ1 (θ) and τcog (θ) can be obtained at every given position by using the pseudoinverse of the above matrix equation. Since the motor has nine pole pairs, the torque trajectory is periodic in position with a fundamental spatial-frequency of 9 cpr (cycles/revolution) and thus the torque pattern repeats every 40 degrees. The Fourier coefﬁcients can by calculated by the inverse fourier series θo + 2π q cn = φ1 (θ)e−jnqθ θ=θo θo + 2π q bm = τcog (θ)e−jmqθ θ=θo The phase shape functions, φk , and the cogging torque are shown in Fig.3 and the corresponding Fourier coefﬁcients are given in Table 1. 3.2 Performance Test The proposed optimal torque controller subject to phase current and voltage limits has been implemented on the three-phase motor. The objective of this section is to demonstrate that the optimal torque controller can deliver accurate torque production in the face of http://www.asc-csa.gc.ca CSA-SE-TN-001

3.2 Performance Test 11 2 phase 1 phase 2 phase 3 1.5 φ (Nm/Amp) 1 0.5 0 −0.5 −1 −1.5 −2 100 110 120 130 140 Motor angle (deg) 150 160 150 160 Figure 3: The back-emf waveform. 0.5 0.4 0.3 τcog (Nm) 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 100 110 120 130 140 Motor angle (deg) Figure 4: The waveform of the cogging torque. http://www.asc-csa.gc.ca CSA-SE-TN-001

3.2 Performance Test 12 Table 1: Harmonics of the back-emf, φ(θ), and cogging torque, τcog (θ). cn bn (Nm/A) (Nm) 1 0.2730 + 0.7270j 2 −0.0002 − 0.0002j 0.0236 − 0.0476j −0.0010 + 0.0009j −0.0000 + 0.0002j 0.0001 − 0.0007j Harmonics no. 3 4 5 6 7 8 9 10 11 12 13 14 15 −0.0004 − 0.0004j 0.0002 + 0.0001j 0.0000 − 0.0000j 0.0003 + 0.0001j 0.0000 − 0.0000j −0.0002 + 0.0000j 0.0001 + 0.0000j 0.0001 − 0.0007j 0.0006 + 0.0003j 0.0000 + 0.0003j −0.0008 − 0.0004j 0.0000 − 0.0006j 0.0002 − 0.0001j −0.0006 + 0.0001j −0.0199 − 0.0021j −0.0010 + 0.0009j −0.0022 + 0.0004j −0.0003 − 0.0002j 0.0000 + 0.0001j −0.0000 − 0.0003j 0.0009 − 0.0008j 0.0001 − 0.0001j http://www.asc-csa.gc.ca 0.0863 + 0.0122j −0.0006 + 0.0001j CSA-SE-TN-001

3.3 Motor Torque-Velocity Characteristic 13 current and voltage saturation. It has been shown (Aghili et al., 2003) that non-constrained optimization of motor torque leads to the following solution of phase currents ik (θ) = φk (θ) (τd − τcog ). φ(θ) 2 (33) For a comparative result, the performances of both constrained optimization torque controller (29) and non-constrained optimization torque controller (33) are presented. It is worth noting that, in the case of non-constrained optimization torque controller, the maximum torque is reached as soon as one phase saturates. On the other hand, the constrained optimization algorithm increases the torque contribution of the unsaturated phases when one phase saturates, until, in the limit, all phases are saturated. The motor shaft is rotated by the hydraulic actuator at a constant speed while the motor torque is monitored by the torque transducer. The maximum phase voltage and current are speciﬁed as: imax = 10 Amp vmax = 40 Volt (34) In the ﬁrst part of the experiment, the desired torque and the rotation speed of the hydraulic motor are set to: C ASE I: τd = 10 Nm ω = 21 rad/s (35) The average of the current limits and the unbiased current limits for each phase and the corresponding bias torque at every motor angle are shown in Figs. 5 and 6, respectively. Figs. 7 and 8 illustrate the phase current waveform and the time-history of the subsequent terminal voltages, respectively. It is clear from the latter ﬁgure that the phase voltages reach their limits in this experiment, but the phase currents are far from the current limit. The motor torque is shown in Fig. 9. At low velocity and high desired torque, it is most likely that the phases current saturation occur rather than the phase voltage saturation. As an illustration, in the second part of the experiment the desired torque and the shaft speed are set to C ASE II: τd = 25 Nm ω = 2 rad/s (36) It is apparent from the values of phase currents and voltages in Figs. 10 and 11 that the phase currents reach their limit while the terminal voltages are not saturated. 3.3 Motor Torque-Velocity Characteristic The control algorithm presented in previous section permits torque sharing among phases when some phases saturate. This results in a considerable increase in the attainable maximum motor torque because the torque controller automatically increases the torque contribution of the unsaturated phases when one phase saturates. This is clearly demonstrated http://www.asc-csa.gc.ca CSA-SE-TN-001

3.3 Motor Torque-Velocity Characteristic 14 4 phase 1 phase 2 phase 3 3 2 ̺ (Amp) 1 0 −1 −2 −3 −4 100 110 120 130 140 150 160 Motor angle (deg) 11 phase 1 phase 2 phase 3 10 s (Amp) 9 8 7 6 5 100 110 120 130 140 150 160 Motor angle (deg) Figure 5: The ̺ and s variables corresponding to the phase current and voltage limits imax = 10 Amp, vmax = 40 Volt, and ω = 21 rad/s. http://www.asc-csa.gc.ca CSA-SE-TN-001

3.3 Motor Torque-Velocity Characteristic 15 −6 −6.5 τ̺ (Nm) −7 −7.5 −8 −8.5 −9 100 110 120 130 140 Motor angle (deg) 150 160 Figure 6: The virtual torque. 5 phase 1 phase 2 phase 3 4 Phase current (Amp) 3 2 1 0 −1 −2 −3 −4 −5 100 110 120 130 140 150 160 Motor angle (deg) Figure 7: Case I: phase currents. http://www.asc-csa.gc.ca CSA-SE-TN-001

3.3 Motor Torque-Velocity Characteristic 16 50 phase 1 phase 2 phase 3 40 Terminal voltage (Volt) 30 20 10 0 −10 −20 −30 −40 −50 100 110 120 130 140 Motor angle (deg) 150 160 Figure 8: Case I: phase voltages. 12 11 10 Torque (Nm) 9 8 7 6 τd = 10 Nm 5 ω = 21 rad/s 4 3 0 sat. is taken into account sat. is not taken into account 0.5 1 1.5 2 2.5 Time (s) 3 3.5 4 4.5 5 Figure 9: Case I: motor torque with and without taking the phase current and voltage saturation into account. http://www.asc-csa.gc.ca CSA-SE-TN-001

3.3 Motor Torque-Velocity Characteristic 17 phase 1 phase 2 phase 3 Phase current (Amp) 10 5 0 −5 −10 100 110 120 130 140 Motor angle (deg) 150 160 Figure 10: Case II: phase currents. 50 phase 1 phase 2 phase 3 40 Terminal voltage (Volt) 30 20 10 0 −10 −20 −30 −40 −50 100 110 120 130 140 Motor angle (deg) 150 160 Figure 11: Case II: phase voltages. http://www.asc-csa.gc.ca CSA-SE-TN-001

3.3 Motor Torque-Velocity Characteristic 18 27 26 Torque (Nm) 25 24 23 τd = 25 Nm ω = 2 rad/s 22 21 sat. is taken into account sat. is not taken into account 20 0 5 10 15 20 Time (s) 25 30 35 40 Figure 12: Case II: motor torque with and without taking the phase current and voltage saturation into account. 30 Maximum torque (Nm) 20 10 constrainted optimization non−constrainted optimization Case I operating point Case II operating point 0 −10 −20 −30 −30 −20 −10 0 10 20 30 ω (rad/s) Figure 13: Maximum attainable motor torque with constrained and non-constrained optimization of motor torques. http://www.asc-csa.gc.ca CSA-SE-TN-001

3.3 Motor Torque-Velocity Characteristic 19 phase 2 phase 3 Phase current (Amp) 10 5 0 −5 −10 100 110 120 130 140 150 160 Motor angle (deg) Figure 14: Case III: phase currents when phase 1 is open circuit. 50 phase 2 phase 3 40 Terminal voltage (Volt) 30 20 10 0 −10 −20 −30 −40 −50 100 110 120 130 140 150 160 Motor angle (deg) Figure 15: Case III: phase voltages when phase 1 is open circuit. http://www.asc-csa.gc.ca CSA-SE-TN-001

3.4 Single-phase Failure 20 12 11 10 Torque (Nm) 9 8 7 6 5 4 3 0 with compensation without compensation 0.5 1 1.5 2 2.5 Time (s) 3 3.5 4 4.5 5 Figure 16: Case III: motor torque when phase 1 is open circuit. in Fig. 13 which depicts the maximum attainable motor torques corresponding to the solutions of the constrained optimization (29) and non-constrained optimization (33). The graphs indicate that the maximum torque capability is boosted by 20% when the phase saturation is considered in the phase current shape function. 3.4 Single-phase Failure The optimal torque controller can produce accurate torque even under operation of a single phase failure. In this experiment, the current circuit of the motor’s ﬁrst phase is virtually broken by sending zero signal to the enable port of the corresponding power ampliﬁer. Thus φ1 (θ) ≡ 0 ∀θ ∈ R (37) The objective is to produce the same torque 10 Nm as the three phases by using only the remaining two phases, i.e., τd = 10 Nm C ASE III: ω = 21 rad/s (38) φ1 ≡ 0 (open-cuircuit) The currents and voltages of the two normal phases are shown in Figs. 15 and 14. The dashed line in Fig. 16 depicts motor torques when one phase is open circuit. The dashed lines in the ﬁgures shows motor torque produced by a controller assuming all three phases are normal. This results in drastic torque ﬂuctuation because the healthy phases do not compensate for lacking torque of the faulty phase. The solid line in the ﬁgure shows that http://www.asc-csa.gc.ca CSA-SE-TN-001

References 21 the motor still produces the constant desired torque when the torque controller is designed based on the two healthy phases. References Aghili, F.: 2011a, Fault-tolerant torque control of BLDC motors, Power Electronics, IEEE Transactions on 26(2), 355 –363. Aghili, F.: 2011b, Optimal joint torque control subject to voltage and current limits of motor drivers, Proc. of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), San Francisco, CA, pp. 3784–3791. Aghili, F., Buehler, M. and Hollerbach, J. M.: 2003, Experimental characterization and quadratic programming-based control of brushless-motors, IEEE Trans. on Control Systems Technology 11(1), 139–146. Aghili, F., Hollerbach, J. M. and Buehler, M.: 2007, A modular and high-precision motion control system with an integrated motor, IEEE/ASME Trans. on Mechatronics 12(3), 317–329. EL-Refaie, A., Novotny, D. and Jahns, T.: 2004, A simple model for ﬂux weakening in surface pm synchronous machines using back-to-back thyristors, Power Electronics Letters, IEEE 2(2), 54–57. French, C. and Acarnley, P.: 1996, Direct torque control of permanent magnet drives, IEEE Trans. on Industry Applications 32(5), 1080–1088. Grabner, H., Amrhein, W., Silber, S. and Gruber, W.: 2010, Nonlinear feedback control of a bearingless brushless dc motor, Mechatronics, IEEE/ASME Transactions on 15(1), 40–47. Hendershot, J. R. and Miller, T. J. E.: 1994, Design of Brushless Permanent-Magnet Motors, Magna Physics and Clarendon, Oxford, U.K. Jahns, T. M.: 1987, Flux-weakening regime operation of an interior permanent-magnet synchronous motor drive, Industry Applications, IEEE Transactions on IA-23(4), 681– 689. Kang, J.-K. and Sul, S.-K.: 1999, New direct torque control of induction motor for minimum torque ripple and constant switching frequency, IEEE Trans. on Industry Applications 35(5), 1076–1082. Kuhn, H. W. and Tucker, A. W.: 1951, Nonlinear programming, Proc. Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley: University of California Press, pp. 481–492. http://www.asc-csa.gc.ca CSA-SE-TN-001

References 22 Lee, J.-H., Kim, D.-H. and Park, I.-H.: 2003, Minimization of higher back-emf harmonics in permanent magnet motor using shape design sensitivity with b-spline parameterization, Magnetics, IEEE Transactions on 39(3), 1269–1272. Matsui, N., Makino, T. and Satoh, H.: 1993, Autocompensation of torque ripple of direct drive motor by torque observer, IEEE Trans. on Industry Application 29(1), 187–194. Moseler, O. and Isermann, R.: 2000, Application of model-based fault detection to a brushless dc motor, Industrial Electronics, IEEE Transactions on 47(5), 1015–1020. Murai, Y., Kawase, Y., Ohashi, K., Nagatake, K. and Okuyama, K.: 1989, Torque ripple improvement for brushless dc miniature motors, Industry Applications, IEEE Transactions on 25(3), 441–450. Ozturk, S. and Toliyat, H.: 2011, Direct torque and indirect ﬂux control of brushless dc motor, Mechatronics, IEEE/ASME Transactions on 16(2), 351–360. Park, S. J., Park, H. W., Lee, M. H. and Harashima, F.: 2000, A new approach for minimum-torque-ripple maximum-efﬁciency control of BLDC motor, IEEE Trans. on Industrial Electronics 47(1), 109–114. Schiferl, R. and Lipo, T. A.: 1990, Power capability of salient pole permanent magnet synchronous motors in variable speed drive applications, Industry Applications, IEEE Transactions on 26(1), 115–123. Sebastiangordon, T. and Slemon, G. R.: 1987, Operating limits of inverter-driven permanent magnet motor drives, Industry Applications, IEEE Transactions on IA-23(2), 327– 333. Sneyers, B., Novotny, D. W. and Lipo, T. A.: 1985, Field weakening in buried permanent magnet ac motor drives, Industry Applications, IEEE Transactions on IA-21(2), 398– 407. Taylor, D. G.: 1994, Nonlinear control of electric machines: An overview, IEEE Control Systems Magazine 14(6), 41–51. Tsai, C.-W., Lin, C.-L. and Huang, C.-H.: 2011, Microbrushless dc motor control design based on real-coded structural genetic algorithm, Mechatronics, IEEE/ASME Transactions on 16(1), 151–159. Wang, Y., Cheng, D., Li, C. and Ge, Y.: 2003, Dissipative Hamiltonian realization and energy-based L2-disturbance attenuation control of multimachine power systems, IEEE Trans. on Automatic Control 48(8), 1428–1433. Xu, Z. and Rahman, M. F.: 2004, A variable structure torque and ﬂux controller for a DTC IPM synchronous motor drive, IEEE 35th Annual Power Electronics Specialists Conference, PESC 04., pp. 445–450, Vol. 1. http://www.asc-csa.gc.ca CSA-SE-TN-001

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