Reduction of torque ripple in brushless motor drives by tailored current control

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motor drives by tailored current control

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Nowadays, permanent magnet synchronous motor (PMSM), well-known as AC brushless motor, are widely used in several industrial elds for their high power density, high torque, low maintenance, high reliability and high eciency . These kind of machines are suitable in every eld where it is nec-essary to obtain a fast dynamic response and accurate positioning capability as well as a smooth speed control. From the point of view of the control to obtain a constant torque, considering sinusoidal linked uxes the control sys-tem imposes constant current references in the synchronous reference Park frame. Anyway, when the linked uxes do not exhibit a sinusoidal trend vs position, the simple assumption that a constant reference value for the trans-formed vector current permits to achieve a constant electromagnetic torque is no more valid. In fact, when the layout of magnets and rotor magnetic core is not optimized with respect to the winding structure torque ripple emerges as more or less signicant problem, since it may aect the accuracy of speed and position control while also determining an additional fatigue mechanical stress and possibly generating acoustic noise. To suppress such undesired torque oscillations, the current reference should be then properly modied and this aects the implementation of the control system, as a more sophisticated current control loop is required. Therefore, conventional PI controllers usually employed for such loops turn out no more suitable, as the compensation of torque ripple harmonics requires high current con-trol bandwidth. A possible solution is represented by the repetitive concon-trol

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or rejection of periodic signals. The typical example of repetitive control application is the rejection of periodic disturbances in optical disk drives. In this thesis repetitive control is used to reduce the torque ripple, where the d-q current references have been modied to obtain a constant torque despite of the intrinsic harmonics of the uxes. The repetitive controller has been coupled with a conventional PI, to ensure the perfect tracking for both the constant reference and the compensation reference. The document is organized in ve dierent chapters. The rst chapter is an analytical model-ing of the torque ripple phenomenon startmodel-ing from the analysis of the main functions of a long drum device. The second chapter contains the theory of the repetitive control, from the basic concept to the stability criterion. The third chapter contains the modeling of the brushless device that is able to take into account the ripple torque phenomenon as well as the control scheme implemented for the simulation in MATLAB/SIMULINK environment. The fourth chapter is a summary of the simulations results. The last chapter contains the conclusion.

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Chapter 1 Theoretical analysis

1.1 Formulation

This chapter aim to reach the expression that is able to take into account the torque ripple phenomenon in a isotropic permanent magnet synchronous motor. The approach is based on the theoretical derivation by analytical modeling and this analysis highlights the main causes of the torque ripple in a permanent magnet synchronous motor (PMSM).

1.2 Proposed modeling

1.2.1 Torque

The goal of this subsection is to reach the expression of the torque that contains the information about the torque ripple. In connection with the

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the brushless motor is an electromecanical device with just nM = 1

mechani-cal degrees of freedom) and i vector of the currents owing in the phase) the electromagnetic torque produced may be calculate as electromagnetic wrench

c_W

E(χ, i)over pole pair p: c_T

E(χ, i) = p ·cWE(χ, i) (1.1)

where the wrench c_W

E(χ, i) is: c_W E(χ, i) = ∂c_C E(χ, i) ∂χ (1.2)

with the coenergy state function c_C E(χ, i)

c

CE(χ, i) = i T

·cΨ(χ, i) −cEE(χ, i) (1.3)

wherec_{Ψ(χ, i)} _{is the linkage uxes with the winding and cE}

E(χ, i)is the

electromagnetic energy stored in the device. The equation 1.2 assumes this expression: c_W E(χ, i) = ∂ i T ·c_{Ψ(χ, i) −}c_E E(χ, i) ∂χ = ∂ i T ·c_{Ψ(χ, i)} ∂χ − ∂ c_E E(χ, i) ∂χ (1.4) in conclusion the complete expression of the torque might be obtained by the following equation:

c_T E(χ, i) = p ·   ∂i T ·c_{Ψ(χ, i)} ∂χ − ∂ cEE(χ, i) ∂χ   (1.5)

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and as stated in the [2] this is valid even in case of transformed variable by an uniform parametric pseudo-linear orthonormal transformations, so:

c TE(χ, i0) = p ·   ∂i0 T _·c_Ψ0_{(χ, i}0₎ ∂χ − ∂ cE_E0 (χ,0_i) ∂χ   (1.6)

It is important to highlight that electromagnetic energy c_E

E(χ, i) = c_E0

E(χ, i0) What may be inferred from the 1.5 and 1.6 is that to evaluate

correctly the torque and hence even the torque ripple it is necessary to focus the on the current vector i(i0), on the c_Ψ(c_Ψ0) and on thec_E

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1.2.2 Treatment

The proposed treatment, according to the [1], provides the expression of the terms of wrench seen above, dened as in [3], of an electric machine. This section is developed, considering a device with the phases only on the stator and the magnet only on the rotor. From the generalized analysis the attention will focus on the isotropic motors.

The basic quantities used in this analysis are the vector i, owing in to the phases, and the and the vector Ψ, built with the linkage uxes with the winding that form the phases. Considering a simplied cross section of the device such as in Figure 1.1

×1 • 10 × 2 • 20 N S + × 3 •3 0

Figure 1.1: Cross section of a simplied device

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quan-tities may be reasonable referred to the position along to the tangential direc-tion in the above mendirec-tioned cross secdirec-tion of the electric device by 2 angular reference frames respectively positioned to stator and rotor. The associated coordinates are in order γ and β. Assuming to select the position of the ori-gin of the rotor frame and identifying the relative angular position between the to reference frame with α, as in Figure 1.2, the two angular quantities result correlated. α β γ Stator frame Rotor frame

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Looking at Figure 1.3 εM εG ΩM ΩG Rotor Stator Magnet Air Gap

Figure 1.3: Zoom of the main area

To point out the characteristic function in the considered section the mapping variable λ may be introduced:

λ = β 2π =

γ − α

2π (1.7)

it is now necessary to highlight the following quantities to provide a global sight of the problem in the cross section:

→ µG(λ, α) the permeability of the airgap

→ µM(λ, α) the permeability of the permanent magnet

→ εG(λ, α) radial thickness of the airgap

→ εE(λ, α) radial thickness of the magnets

→ AG(λ) magnetically equipotential surface of the the airgap with radial

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→ AM(λ) magnetically equipotential surface of the the airgap with radial

distance from the revolutionary axis ρM(λ, α)

→ ΩG(λ) the perimeter of the surface AG(λ) in plane of the cross section

→ ΩM(λ) the perimeter of the surface AM(λ) in plane of the cross section

To describe completely the geometrical aspects it is necessary to introduce the further following quantities as:

τG(λ, α) = dAG(λ, α) l · dλ (1.8) and τM(λ, α) = dAM(λ, α) l · dλ (1.9)

The two τ represents the radius of the equipotential surfaces mentioned, respectively τG is the radius of the airgap surface and τM is the radius of the

magnet surface.

The constitutive magnetic equations of the airgap layer and the magnet one are: _               BG(λ, α, i) = µG(λ, α) · HG(λ, α, i) BM(λ, α, i) = µM(λ) · HM(λ, α, i) + HC(λ) (1.10)

Applying the Gauss law and the Amperé law in the main region and tak-ing into account that this analytical process aims to point out the expression

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of the ux state function

c_{Ψ(α, i) =}c_Ψ

P(α, i) +cΨL(i) (1.11)

that provides the linked uxes in function of the position and the current(the left "c" indicates that the state variable is the current), it may be possible to calculate the main function that features the device. In this analysis the leakage uxes are independent from the position and it is in general true. With the same logic presented in [1] the main function might be here resumed: 1. No load uxes ΨP 0(α) = l Z 1 0 µE(λ, α)NE(λ, α)FE0(λ, α) dλ (1.12)

2. Principal inductance matrix LP(α) = l Z 1 0 µE(λ, α)NE(λ, α)NE T (λ, α) dλ (1.13)

The characteristic function in 1.12 e and 1.13 the internal function may be classied as follows:

1. µE(λ, α)is the equivalent permeability

µE(λ, α) = µG(λ, α)

τG(λ, α)

εE(λ, α) (1.14)

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→ εE(λ, α) = εG(λ, α) + εG(λ, α) τG(λ, α) τM(λ, α) µG(λ, α) µM(λ, α)

2. NE(λ, α) is the vector of the equivalent winding function where NEK

is the k-th element of the vector for the k-phase.

3. FE(λ, α) is the equivalent m.m.f due to the permanent magnet and it

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1.2.3 Fourier decomposition

The equivalent function obtained, that in general exhibit periodical trend, may be decomposed with the Fourier series. The equivalent coercive eld and the equivalent winding descriptive function have a period of 2π

p instead

the the equivalent permeability has double periodicity correlated with the others, however the decomposition is over the 2π

p period. Using the Fourier

series it is possible to achieve this new formulation of those function: FE(λ, α) = +∞ X h=1 h FC · cos [2πph (λ − λ0(α))] (1.15) NEK(λ, α) = +∞ X h=1 h_N K · cos [2πph (λ − λK(α))] (1.16) µE(λ, α) = +∞ X h=0 h_µ E· cos [2πph (λ − λE(α))] (1.17) with: λK(α) = γK− α 2π (1.18) and γK(α) = γ1 + K − 1 3 · 2π (1.19)

with γ1 angular position of the symmetry axis of the winding function of the

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1.2.4 Main functions

Substituting the decomposed equivalent function in the integral inductance function and the linkage uxes function and considering that:

Z 2π 0

cos k(a + b) da = 0 (1.20)

and the Werner relation:

cos(a) · cos(b) = 1

2[cos(a + b) + cos(a − b)] (1.21) considering an isotropic brushless the equivalent permeability is a con-stant and it is µE = 0µE, and with the previous consideration the uxes

are: c ΨP 0 =       ... c_Ψ P 0K ...       = 1 2·l· 0 µE             ... P+∞ h=1 h_N K ·hFc· cos {2πph [λK(α) − λ0(α)]} ...             (1.22) The element of the inductance matrix LP KH( that correlates the linkage

ux with the phase K due to the current in the phase H) is LP KH = +∞ X h=1 h NK·hNH · cos {2πph [λK(α) − λH(α)]} (1.23)

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of the equivalent coercive eld then:

h_λ

0 = 0 ∀h (1.24)

and even if the origin of stator frame is coincident with the symmetry axis of the winding function of the phase 1( magnetic axis of the phase 1), then:

1

λ1 = −

α

2π (1.25)

With the canonical choices: 1. λ0 = β0 2π = 0 (1.26) 2. λ1 = β1 − α 2π β1=0 = −α 2π (1.27)

The vector of the uxes 1.22 may be written as:

c_Ψ P 0=       ... c_Ψ P 0K ...       =1 2· l · 0_µ E              ... P+∞ h=1 h_N K·hFc· cos 2πph α 2π − K − 1 3 ...              k = 1, 2, 3 (1.28)

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Highlighting the fundamental component: 1,c_Ψ P 0(α) = 1 2· l · 0_µ E               1_N K·1Fc· cos h 2πpα 2π i 1_N K·1Fc· cos 2πp α 2π− 2 − 1 3 1_N K·1Fc· cos p α 2π− 3 − 1 3               =1 2· l · 0_µ E               1_N K·1Fc· cos (pα) 1_N K·1Fc· cos 2πpα −2 3π 1_N K·1Fc· cos pα −2 3π               (1.29) and indicating c_Ψ P 0s = 1 2 · l · 0_µ E ·1NK·1Fc and introducing, σ3(x) =       cos(x) cos(x − 2 3) cos(x − 4 3)       (1.30) then 1,c_Ψ P 0(α) =cΨP 0s               cos (pα) cos 2πpα − 2 3π cos pα −2 3π               =cΨP 0s σ(pα) (1.31)

Looking at the structural properties of the equivalent winding functions and the equivalent f.m.m due to the magnet it is possible to state that they have the 2 n-symmetry property ad then

f (x) + f (x +T

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the integral properties: C2υ = 1 T Z T 0 f (x)e−2υωx_{dx = 0, υ ∈ Z} (1.33)

So the even harmonics are null. Further the harmonics may be classied as follows:

Groups υ Harmonics h = 3υ υ = 1, 3, . . . h = 3, 9, . . . h = 6υ − 1 υ = 1, 2, . . . h = 5, 11, 17, . . . h = 6υ + 1 υ = 1, 2, . . . h = 7, 13, . . .

In conclusion the ux functions has no even harmonics and the odd may be split i 3 dierent groups. So the uxes: dening,

σ+₃(x) =       cos(x) cos(x + 2 3) cos(x + 4 3)       (1.34)

and showing that:

σT₃(y) · σ+₃(x) = 3

2cos(y + x) (1.35) and

σT₃(y) · σT₃(x) = 3

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c_Ψ P 0 =1,cΨP 0s· σ3(pα) + X h=3υ 3υ,c_Ψ P 0s· cos(3υpα)1+ + X h=6υ−1 6υ−1,c_Ψ P 0s· σ+3((6υ − 1)pα) + X h=6υ+1 6υ+1,c_Ψ P 0s· σ3((6υ + 1)pα)

It is very simple to that the matrix inductance is constant in reference to the position with the assumption of µE =0µE and the expression is:

LSSP =0LSP σ3(0) σ3( 2 3π) σ3( 4 3π) (1.37) where the 0_L

SP = l ·1NS2·0µE to simplied the scenario or in more accurate

way may be evaluated 0_L SP =

Pn=n∗

h=1 l ·hNS2·0µE where n∗ is an arbitrary

great number. In this document it will be considered the rst approximation. From the 1.37, for a linear magnetic device, is possible say:

∂LSSP

∂χ = 0 (1.38)

where χ = α.

So, the wrench can be reduced to:

c_W

E(χ, i) =

∂ iT ·c_Ψ P 0(χ)

∂χ (1.39)

The vector i contains the current in the 3 phases of the stator in base variable.

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1.2.5 Transformed variables

To simplied the scenario it is possible to recur to the Park-Clarke Transfor-mation. Considering the Park transformation matrix:

TP(θ) = r 2 3 σ(θ) σ(θ + π 2) 1 √ 2 (1.40) T_PT(θ) = r 2 3       σT_(θ) σT_{(θ +} π 2) 1T √ 2       (1.41)

To calculate the transformed uxes: Ψ0 = T_PT(θ) ·0ΨP 0 = = r 3 2 1,c_Ψ P 0s    cos(θ − pα) cos(θ + π₂ − pα) 0   + + r 3 2 X h=3υ 3υ,c ΨP 0s    0 0 3 · cos(3υα)   + + r 3 2 X h=3υ 3υ−1,c_Ψ P 0s    cos(θ + (3υ − 1)pα) cos(θ + π₂ + (3υ − 1)pα) 0   + + r 3 2 X h=3υ 3υ+1,c ΨP 0s    cos(θ − (3υ + 1)pα) cos(θ + π₂ − (3υ + 1)pα) 0    (1.42)

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and with the canonical θ = pα the 1.42 become: Ψ0 = = r 3 2 1,c ΨP 0s    cos(0) cos(π₂) 0   + + r 3 2 X h=3υ 3υ,c_Ψ P 0s    0) 0 3 · cos(3υα)   + + r 3 2 X h=6υ 6υ−1,c_Ψ P 0s    cos(6υpα) −cos(6υpα + π 2) 0   + + r 3 2 X h=6υ 6υ+1,c_Ψ P 0s    cos(6υpα) cos(6υpα +π₂) 0    (1.43)

The 1.43 shows that the only components that contributes to the 3-th component of the transformed uxes are the harmonics of the group 3υ. The components of the groups 3υ −1 and of the groups 3υ +1 contribute to the d and q axes. In general there are all the three components d,q and z. However the third element which is due to 3υ harmonics that in general is negligible and it is also null when the 3 n-symmetry is veried P2

k=0f (x + k

3T )where

T is the period. In this situation the C3υ = 0 ∀υ ∈ Z and then the third

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The wrench may be expressed as c_W E(χ, i) = ∂ iT ·c_Ψ P 0(χ) ∂χ = i T ·∂ c_Ψ P 0(χ) ∂χ = i 0T · ∂ c_Ψ0 (χ) ∂χ = c_W0 E(χ, i 0 ) (1.44) where cΨ 0 (χ) ∂χ = c_M0 0 c_M0 0 = + r 3 2 1,c_Ψ P 0s       sen(θ − pα) sen(θ + π₂ − pα) 0       + + r 3 2 X h=3υ 3υ ·3υ,cΨP 0s       0 0 −3 · sen(3υpα)       + + r 3 2 X h=6υ (6υ − 1) ·6υ−1,cΨP 0s       −sen(θ + (6υ − 1)pα) −sen(θ + π 2 + (6υ − 1)pα) 0       + + r 3 2 X h=6υ (6υ + 1) ·6υ+1,cΨP 0s       sen(θ − (6υ + 1)pα) sen(θ + π₂ − (6υ + 1)pα) 0       (1.45)

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and with the same canonical choice previously presented χ = pα c M₀0 = + r 3 2 1,c ΨP 0s       sen(0) sen(π₂) 0       + + r 3 2 X h=3υ 3υ ·3υ,cΨP 0s       0 0 −3 · sen(3υpα)       + + r 3 2 X h=6υ (6υ − 1) ·6υ−1,cΨP 0s       −sen(6υpα) −sen(6υpα + π 2) 0       + + r 3 2 X h=6υ (6υ + 1) ·6υ+1,cΨP 0s       −sen(6υpα) −sen(6υpα −π 2) 0       (1.46)

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Indicating: 1. 1,c_M 0 = r 3 2 1,c_Ψ P 0s 2. 3υ,c_M 0 = r 3 2(3υ) 3υ,c_Ψ P 0s 3. 6υ−1,c_M 0 = r 3 2(6υ − 1) 6υ−1,c_Ψ P 0s 4. 6υ+1,c_M 0 = r 3 2(6υ + 1) 6υ+1,c_Ψ P 0s

Referring the harmonics of M0 to the fundamental 1,cM0 it is possible to

highlight the ratio: 1. 3υ_{Γ =} 3υ,cM0 1,c_M 0 (1.47) 2. 6υ−1_{Γ =} 6υ−1,cM0 1,c_M 0 (1.48) 3. 6υ+1 Γ = 6υ+1,c_M 0 1,c_M 0 (1.49) In many application the local approach to approximate the the signal of reference is used to control the inverter that feeds the motor. For instance in PWM control (when no overmodulation occurs) the current generated may be approximated with the fundamental component. With the typical connection of the brushless motor feed with a symmetrical voltages the current does not have the third component indeed i1−3υ(t) + i2−3υ(t) + i3−3υ(t) = 0 and then

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The transformed current: i0 = T_PT(θ) · i =       Isd Isq 0       (1.50)

Returning to the Wrench expression:

c_W0 E(χ, i 0 ) = i0T · ∂ c_Ψ0 (χ) ∂χ = Isd Isq 0 ·c_M0 0 = Isq · 1,c_M 0+ + Isq ·1,cM0 " X h=6υ 6υ−1_{Γ −}6υ+1_{Γ cos(6υpα)} # + − Isd ·1,c_M 0 " X h=6υ 6υ−1_{Γ +}6υ+1_{Γ sen(6υpα)} #

With the normal control approach Isd = 0 so the ripple Wrench is:

Rc_W0 E = Isq · 1,c_M 0 " X h=6υ 6υ−1_{Γ −}6υ+1_{Γ cos(6υpα)} # (1.51) and the torque ripple is:

RTE = p · Isq · 1,c M0 " X h=6υ 6υ−1 Γ −6υ+1Γ cos(6υpα) # (1.52) So to eliminate the ripple is necessary to provide a 6υ harmonics of current able to produce a torque ripple TEC(compensation torque) opposite to the

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so the compensation current for the 6υ alternating torque ripple is:

i6υ(t) = − 6υ−1Γ −6υ+1Γ Isqcos [6υα(t)] υ = 1, 2, . . . (1.54)

1.3 Discussion

The analysis developed yields that there are torque ripple components only at the frequencies of integer multiples of six times lagrangian variable ωRIP P LE =

6pαwhere α is the relative position between the rotor and stator frame. Fur-ther, in general the only components of the linkage uxes that may contribute to the torque ripple are the component belong to the groups 3υ, 6υ − 1 and 6υ + 1 and with the canonical scenario with PMSM star-connected the com-ponents reduce that are part of groups 6υ − 1 and 6υ + 1.

It has been highlighted that to eliminate the torque ripple, with the hy-pothesis of Isd = 0, due to the harmonics of the linkage phase uxes the reference might be obtained as the sum of constant term and an alternating term.

isq= Isq+

X

6υ

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Chapter 2 Repetitive Control Theory

2.1 Introduction

As seen in the previous chapter the controller should follow a reference sig-nal that is the sum of a constant reference with a sinusoidal reference. It is necessary to achieve a zero steady-state error both for the constant reference and for the sinusoidal reference and hence the repetitive controller has been analysed to achieve this purpose.

The repetitive controller theory here introduced is coherent with the treat-ment presented in [6].

The repetitive control is based on the internal model principle and theo-retically the output of a stable feedback system can track the periodic ref-erence signal or/and eliminate the periodic disturbance. In this chapter the repetitive control is investigated to achieve a zero( in theory) error on the sinusoidal reference

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2.2 Classication

The repetitive control consists in a controller that reproduce, internally to the close-loop, the model necessary to reject the disturbance or to track the reference. There are mainly three dierent schemes that may implement the logic: direct type, parallel type, and plug-in type (Figure 2.1) . The quantities r, e, y, GP, GC and GRC are respectively the period reference, the

error, the system output, the transfer function of the plant, the conventional controller and the repetitive controller .

(a) Direct type

(b) Parallel type

(c) Plug-in type

Figure 2.1: Possible structure of repetitive control system

In the direct connection the controller is connected directly with the sys-tem plant, this kind of solution is suitable for plant (GP) internally stable

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over-come this problem additional compensator able to stabilize the plant might be inserted and this purpose is achieved with parallel conguration and plug-in type.

The parallel conguration is realized connecting the repetitive controller in parallel with the traditional controller( for instance a PI controller), the out-puts of the GRC and of the GC are summed to form the actuating signal

for the plant GP. This scheme holds the advantages both the conventional

control in dynamic response and the repetitive control in steady state per-formance. Further the plug-in type has fast learning speed and quick conver-gence, due to its modication of the error signal rather than the actuating signal. Based on this obvious pro this kind of solution it will be used later. The plug-in type has become the most commonly used structure in the recent years. The repetitive control is placed between the error signal e and the GC

controller . The design may be reach without disturbing the original control system, ana this is particularly attractive when the conventional controller should be design alone.

2.3 Foundamentals

The internal model principle forms the theoretical foundation of the repetitive control theory. According to the IMP the perfect tracking of the period reference might be achieved in steady state if the model is included in the system.

An example of the IMP is the case when the reference of type 1 closed-loop control system is the unit step signal, so if the integrator 1

s is implemented

in the system the steady state error is zero and so on such as if the signal is a ramp to obtain a zero steady state error it is necessary that the model includes 1

s2.

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refer-2.3.1 Continuous time domain

In continuous domain the necessity to have an internal model with a period-icity of TP is achieved by a time delay in the Laplace domain e−sTP and the

scheme to realize the purpose is the one shown in Figure 2.2, the time delay term is put into the forward path of a positive unity feedback loop.

Figure 2.2: Period model in continuous domain

The transfer function is: B(s) A(s) =

e−sTP

1 − e−sTP (2.1)

By solving the 2.1 the position the position of the close-loop function may be obtained. The position of these poles on the Laplace plane is shown in Figure 2.3. The poles are located on the imaginary axis and their position of the k-th is individuated by

pk = jkω ∀k ∈ Z (2.2)

where ω = 2π TP

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2.3.2 Discrete time domain

In the discrete time domain to implement the periodic model is relatively simple. A delay Z−N _{is employed in the forward path of a positive unity}

feedback loop (Figure 2.4). N is the number of the sample points during one single signal period.

N has to be an integer, and so the sampling frequency has to be selected as an exacted multiple of the periodic signal frequency.

Figure 2.4: Periodic model in discrete domain

N = ωS ω = fS fP = TP TS (2.3) The transfer function of the implemented scheme is:

B(s) A(s) =

Z−N

1 − Z−N (2.4)

By solving the characteristic equation of the 2.5 the position of the z-domain poles may be founded. The N poles are located on the z-z-domain unity circle and the k-th pole is individuated by:

zk = ejk 2π N ∀k ∈ −N − 1 2 , . . . , + N 2 (2.5) By this realization the perfect tracking of a periodic signal would be achieved.

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Figure 2.5: Position of the repetitive controller

2.3.3 Considerations

The physical realization of the continuous repetitive controller is unrealistic. In most application repetitive control is realized by a digital approach. In this scenario the limited number of the poles(N) limits naturally the high frequency compensated. The action of the discrete solution is limited to the components below the Nyquist frequency. However the digital implementa-tion is the simplest that could be requested indeed it may be realized just by a delay of N sampling period.

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2.4 Plug-in Repetitive controller

In a plug-in scheme the conventional controller is designed rst to stabilize the plant and provide the basic logic of control without using the periodic reference. The, the repetitive controller is applied for removing the periodic signal in the error signal. The Figure 2.6 shows the complete plug-in discrete repetitive control system,

Figure 2.6: Block diagram of the complete pug-in repetitive control system

where:

→ R(z) is the reference input; → Y (z) is the system output;

→ GC(z)is the conventional controller;

→ GP(z) is the transfer function of the plant;

→ E(z) is the error signal transfer function; → U (z) is the actuating signal;

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→ D(z) is generalized external noise;

The function of the repetive controller assumes the following form: GRC(z) =

F (z) E(z) =

kRCGf(z)z−N

1 − Q(z)z−N (2.6)

Taking into account the GRC(z) in the whole system and it possible to

calculate the close loop transfer function: Y (z) R(z) = 1 − Q(z)z−N + kRCGf(z)z−N GC(z)GP(z) 1 − Q(z)z−N _{+ (1 − Q(z)z}−N _{+ k} RCGf(z)z−N) GC(z)GP(z) (2.7) With the same logic it is possible to highlight the disturbance transfer function: Y (z) D(z) = 1 − Q(z)z−N 1 − Q(z)z−N _{+ (1 − Q(z)z}−N _{+ k} RCGf(z)z−N) GC(z)GP(z) (2.8)

Combining the two expressions 2.7 and 2.8 the error signal in z-domain can be expressed: E(z) = (R(z) − D(z)) 1 − Q(z)z −N (1 + GC(z)GP(z)) 1 − z−N Q(z) − kRCGf(z)GC(z)GP(z) 1 + GC(z)GP(z) (2.9)

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Based on the 2.9, an equivalent scheme for the error may be built as in Figure 2.7.

Figure 2.7: Equivalent system block for the 2.9

2.4.1 Stability analysis

The stability analysis of repetitive control needs to be carefully indeed the pe-riodic model introduces poles located all on the unity circle in the z-domain( on the imaginary axis in the continuous domain) that is the stability bound-ary. The function of the repetitive controller is very complex and an analysis that utilizes the traditional method such as Routh Hurwitz and Gain & Phase margin criterion is very dicult. To overcome this problem, the gain theorem is used to derive the stability conditions.

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Small gain theorem

Considering a typical feedback control system shown in Figure 2.8 the small gain theorem states that if the close loop system is bounded input, the R(z), D(z) are bounded and the loop gain |GC(z)GP(z)|is smaller than unity the

output is bounded stable.

Figure 2.8: Typical control system

Stability condition

With reference to the Figure 2.7 is conducted. The scheme contains three dierent sections connected in cascade and each has to be stable to assure the whole stability.

Sections:

• the rst part is realized by a Q(z) and with a delay z−N_.

The Q(z) is either a moving average lter which passes the low and the medium frequencies attenuating the high frequencies or a close to unity constant (in general 0.98).

• the second part is the denominator 1 + GC(z)GP(Z), which is the

characteristic equation of the system without the repetitive controller. GC(z) is used to stabilize the system hence the stability of this part

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the small gain theorem, assuming that the other two parts are stable,the third part will be stable if the the entire loop(Figure 2.7) is smaller than unity. The second part of the loop is the delay so to achieve a value smaller than 1 it is necessary that the rst part of the feedback is < 1.

|S(ejωT s_{)| =} Q(ejωT s) −kRCGf(e jωT s_)G C(ejωT s)GP(ejωT s) 1 + GC(ejωT s)GP(ejωT s) < 1 (2.10) the relation above has to be veried ∀ ω ∈ [0, ωnyq].

The interpretation of the relation 2.10 is represented below:

Figure 2.9: Vector representation of the relation 2.10

As it is possible to see from the Figure 2.9 to achieve the request of the small gain theorem is necessary to select in the right way the the Q(z), kRC

and Gf(z).

In conclusion for the stability in the plug-in type repetitive control the following conditions should be satised:

1. All roots of the 1 + GC(z)GP(z) = 0are located inside the unity circle;

2. Equation 2.10 is guaranteed for all the frequencies below the ωnyq = _Tπ_s

Zero error tracking

Assuming the two stability conditions are both satised, the overall system is thereby stable. For sake of simplicity choosen Q(z) = 1 the error may be

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written as follows: E(z) = (R(z) − D(z)) (1 − z −N₎ (1 + GC(z)GP(z)) 1 − z−N 1 − kRCGf(z)GC(z)GP(z) 1 + GC(z)GP(z) (2.11) and substituting z = ejωTs, E(ejωTs_{) =} R(e jωTs_{) − D(e}jωTs) (1 − (ejN ωTs₎₎ (1 + GC(ejωTs)GP(ejωTs)) 1 − (ejN ωTs₎ 1 −kRCGf(e jωTs_)G C(ejωTs)GP(ejωTs) 1 + GC(ejωTs)GP(ejωTs) (2.12) the 2.12 is null when the numerator is null and hence the condition

1 − (ejωTs_{) = 0} (2.13)

an it is veried when the input signal has a period of angular frequency ω = ωk =

2kπ TP

< ωnyq ∀k (2.14)

It may be concluded that lim

ω→ωk

E(ejωTS) = 0 (2.15)

The relation 2.15 implies that zero error may be achieved, theoretically if the input signal has a angular frequency that is an integer multiple of ω = 2πfS

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Figure 2.10: Pole-zero map of a periodic signal generator withQ(z) = 0.6

2.4.2 Filter Q(z) Design

Repetitive control system is extremely sensitive to model uncertainties typi-cally in the high frequencies, since the poles are equally distributed on unity circle. Two dierent types of lter are used to enhance the robustness of the system.

Close to unity constant

The close to unity constant is the most common form of the robustness lter Q(z). The concept is to put inside the unity circle the all poles on the stability boundary. It is easily implemented in digital control. For instance one may consider a discrete periodic signal generator with N = 20 and sampling frequency of 20[kHz]. By setting the Q(z) = 0.6 and to show the inuence of this choice the Figure 2.10 is presented.

This choice pull-back the poles, before on the unity boundary, on a cir-cumference with the radius r = eln(Q)

N . This adjustment has the advantage

to put the poles all inside the unity-circle, but the drawback is that it is im-possible to achieve a zero error. As a result the stability has been increased, while the ability to reject the harmonics has been sacriced to the entire frequency band. Generally the constant is chosen between 0.9 and 0.99.

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Figure 2.11: Bode responses of the moving average lter

Moving average lter

Another common solution for the Q(z) is the moving average lter, which has low-pass characteristics and zero phase shift. Its expression is given in 2.16 where q and m are the parameters determined by the designer. Generally, a larger q0 brings a higher pass band.

Q(z) = Pm i=1qi zi₊ 1 zi + q0 2Pm i=1qi+ q0 (2.16) Three design examples of a moving average lter are provided in 2.17, 2.18, 2.19 where the sampling frequency is 20[kHz]. Figure 2.11 shows the bode responses of the three moving average lters.

Q1(z) = 0.125z2_{+ 0.75z + 0.125} z (2.17) Q2(z) = 0.25z2_{+ 0.5z + 0.25} z (2.18) Q3(z) = 0.0625z4+ 0.25z3+ 0.375z2 + 0.25z + 0.0625 z2 (2.19)

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Figure 2.12: Pole-zero map of the periodic signal generator with Q3(z)

with Q1(z) and Q2(z), Q3(z)has the lowest cut-o frequency. This fact

im-plies that the repetitive control system with Q3(z)adopted as the robustness

lter will have a larger stability margin and a worse high frequency regulation ability.

The discrete periodic signal generator with N of 20 and sampling fre-quency of 20kHz is tested with Q3(z) implemented in the feedback path to

examine its inuence on the pole position. Figure 2.12 presents the pole-zero map of the generator.

It can be seen that the use of Q3(z)has pulled all the poles inward except

for the one locating at the point (1, 0). By employing it as the robustness lter, the stability margin of the repetitive control system can be signicantly increased, while the ability of the high frequency regulation is degraded. The selection of moving average lter is based on the system model accuracy. A poor modelling at high frequency requires a strict lter design (i.e. low cut-o frequency). Generally, Q1(z) and Q2(z) are sucient for most applications.

Discussion

The use of the close-to-unity constant as the robustness lter pulls all the poles from the unity circle edge towards the centre, while the use of the moving average lter mainly aects the high frequency poles. However, it has been experienced that there is no signicant dierence between these

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two choices in practical applications [19]. From the implementation point of view, the closeto-unity constant can be employed and tuned easier in digital control platforms.

Two aspects should be taken into account: one is that the robustness lter is not necessary when the system is perfectly modelled, for example in sim-ulation environment. The second is that the introduction of the robustness lter will aect the design of the repetitive control gain kRC, which will be

discussed later.

2.4.3 Stability Filter G

f

(z)

Design

The stability lter Gf(z)and the repetitive control gain kRC are coupled and

related directly to the second stability condition of a plug-in type repetitive control system. In order to assure the system closed-loop stability, they need to be designed in conjunction so that the locus of vector S(ejT s₎stay within

the unity circle. Two approaches are commonly used to design Gf(z).

Zero Phase Error Tracking Compensator

Considering a general plug-in repetitive control system a conventional con-troller GC(z) is employed to stabilize the plant GP(z) and provide basic

regulation. Hence, the rst stability condition can be guaranteed. According to the additive property of LTI system, the closed-loop response from E(z) to F (z) can be derived by setting both R(z) and D(z) to be 0, as expressed in 2.20. E(z) F (z) = −GC(z)GP(z) 1 + GC(z)GP(z) (2.20) Therefore, from the viewpoint of GRC(z), the plug-in repetitive control

system can be re-drawn to an equivalent negative feedback system as pre-sented in Figure 5.13. Note that −E(z) is the output signal, and 0 is the system reference for zero tracking error.

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Figure 2.13: Equivalent block diagram of the plug-in repetitive control system from the viewpoint of GRC(z).

The plant seen by GRC(z)can be represented by the following form,

GPRC = GC(z)GP(z) 1 + GC(z)GP(z) = k(z − z1) . . . (z − zn) (z − p1) . . . (z − pn) (m ≥ n) (2.21) where k is the lumped gain; p1, p2,...pm and z1, z2,...zn are the m poles

and n zeros of the original closed-loop system without GRC(z).

Since GC(z) stabilizes GP(z) properly, all the poles are located inside the

unity circle. Without loss of generality, GPRC(z) is assumed to contain u

non-erasable zeros and n-u erasable ones. Equation can be further expressed as,

GPRC(z) =

kN um+(z)N um−(z)

Den(z) (2.22)

where Den(z) represents the denominator of the transfer function; Num+_(z)

and Num−_(z)_{are the erasable and non-erasable parts of the numerator,respectively.}

It is also essential to assume that Num+_{(z)N um}−_(z)_{and 1−Q(z)z −N (i.e.}

GRC(z) denominator as expressed) are relatively prime in order to ensure

the inclusion of the internal model for periodic signals. Under these three assumptions, the zero phase error tracking compensator may be designed in the following structure,

Gf =

Den(z)N um−(1_z)

kN um+_{(z) [N um}−_(1)]2 (2.23)

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[N um−(1)]2 is calculated by substituting z in Num−(z) with 1. Den(z), N um+_(z)_{and k are used to cancel the poles, erasable zeros and gain of (5.23),}

while Num−_(1/z) _{and [Num}−_(1)]2 _{compensate the phase and magnitude}

ef-fects of u non-erasable zeros. With this compensator employed as the stability lter, the phase shift introduced by (5.23) is reduced to 0 for all frequencies, and the second stability condition is always satised for 0 < kRC < 2.

Based on this design, the repetitive controller can be derived by substituting Gf(z), GRC(z) = F (z) E(z) = kRCz−N 1 − Q(z)z−N · Den(z)N um−(1_z) kN um+_{(z) [N um}−_(1)]2 (2.24)

It has to be pointed out that when GPRC(z) contains more poles than

zeros (m > n), the resultant Gf(z) has a number of zeros greater than the

number of poles. However, since it is in cascade with an N periods delay block (i.e. z−N_{), there is no causality problem for its implementation.}

If the system is a minimum-phase one, the numerator of GPRC(z)consists of

only erasable zeros, i.e. no zeros outside the unity circle. In this case, both N um−(1/z) and [Num−(1)]2 are equal to 1, so that the designed zero phase error tracking compensator is just the inverse form of GPRC(z).

Gf(z) =

Den(z)

k · N um+_{(z) · 1}2 =

1 + GC(z)GP(z)

GC(z)GP(z) (2.25)

Time Advance Compensator

Although the use of the zero phase error compensator can ensure the stability of the repetitive control system, the design procedure is complicated and te-dious, especially for systems with a high order transfer function. In addition, when the original control system is based on an analogic implementation, the accurate plant model is dicult to obtain.

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purpose of this lter is to balance the phase delay introduced by GPRC(z)

and thereby improve the system stability margin. Its expression is given in 2.26

Gf(z) = zTA (2.26)

where TA stands for the time advance step chosen by the designer. The

transfer function of the repetitive controller with the time advance compen-sator can be obtained as in

GRC(z) =

F (z) E(z) =

kRCz−N +TA

1 − Q(z)z−N (2.27)

Under this condition, GRC(z)needs to be designed by tuning TA and kRC

together, such that the magnitude of S(ejωTs₎ stays within the unity circle

centred at (0, 0) for the entire frequency band (till the Nyquist frequency). Similarly, there is no causality problem for the implementation of this lter, because it is in cascade with an N periods delay element and N is far greater than TA.

Discussion

If the zero phase error tracking compensator is selected, the resultant repet-itive control system is suciently stable for 0 < kRC < 2. The shortage of

this compensator is that the design procedure and digital implementation are quite complicated when the system has a high order transfer function. Con-versely, the use of the time advance compensator is more preferable in the case an accurate system model is not available. Due to its simple structure, the design is straightforward. Also, it can be implemented easily in digital control platforms. However, as kRC is coupled with TA, a stability range for

the values of kRC needs to be evaluated through the expression interpreted

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Chapter 3 Modeling

This chapter aims to build a model of the brushless motor in d-q reference system which is able to reproduce the torque ripple disturbance. After the modeling of the electrical device the schemes of the control system are pro-vided. This analysis is conducted according with [3], [2] and [1].

3.1 Equivalent brushless model

With the Park transformation, it is possible to highlight the expression of the transformed voltage:

v0 = v0R+ v0E z }| { v0EL+ v 0 EM (3.1)

this expression still includes the resistive and the electromagnetic terms in the transformed reference system. The quantities in the above expression are: → v0 E = v 0 L+ v 0

AM transformed electromagnetic voltage

→ v0

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→ v0

AM pseudo-motional additional voltage

3.1.1 Apparent inductive voltage

This term is a part of the electromagnetic term. It results equal to the product of the transformed dierential inductance matrix by time derivative of transformed currents. v0L = L 0 D(i 0 (t), χ(t), θ(t)) · d i 0 dt (3.2) with: L0_D(i0(t), χ(t), θ(t)) = T_PT(θ) · LD(i(t), χ(t)) · TP(θ) (3.3)

in case of device magnetically linear and isotropic: LD(i(t), χ(t)) = L(χ(t)) = L = =0LSP · h σ(0) σ(2 3π) σ( 4 3π) i (3.4)

and substituting the 3.4 in the 3.3, L0_D(i0 (t), χ(t), θ(t)) = L0(χ(t), θ(t)) = = L0 =    Lsd 0 0 0 Lsq 0 0 0 Lsz    (3.5)

The resultant inductance matrix is diagonal and with previously mentioned hypothesis of isotropic device Lsd = Lsq .

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3.1.2 Pseudo-motional additional voltage

The v0

AM terms is named pseudo-motional voltage and it is a product of the

matrix c_A0

M(i

0

, χ(t), θ(t)) by the time derivative of the parameter θ.

v0 AM =cA 0 M(i 0 , χ(t), θ(t)) · dθ dt (3.6) with: c_A0 M(i 0 , χ(t), θ(t)) = L0_D(i0 (t), χ(t), θ(t)) · T_PT(θ) · dTP(θ) dt · D[i 0 ] (3.7) and with the hypothesises above mentioned:

c A0_M(i0, χ(t), θ(t)) =    −Lsdisq +Lsqisd 0    (3.8)

Therefore using the isotropic hypothesis Lsd = Lsqthe 3.9 may be

rewrit-ten in this interesting way:

c A0_M(i0, χ(t), θ(t)) =    −Lsqisq +Lsdisd 0    (3.9)

3.1.3 Motional transformed voltage

The v0

EM is the transformed term of the motional voltage in basic variable

v0EM =cM 0 (i0, χ, θ) · dχ dt (3.10) where the c_M0_(i0 , χ, θ) = ∂ c_Ψ0 ∂χ (3.11)

and with the hypothesises mentioned of the linearity and isotropic device, it is possible to highlight

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Thec_M0 0(i

0

, χ, θ)has been founded in the rst chapter and is below rewritten:

c_M0 0(i 0 , χ, θ) =         −1,c_M 0[P_h=6υ(6υ−1Γ +6υ+1Γ) sen(6υpα)] 1,c_M 0+1,cM0[ P h=6υ(6υ−1Γ −6υ+1Γ) cos(6υpα)] −3 ·1,c_M 0[ P h=3υ 3υ_{Γsen(3υpα)]}         =    Md Mq Mz    (3.13) One may highlight that1,c_M

0 = Ψsd = M q and this is the general

expres-sion for the second element of the vector c_M0

0(i

0

, χ, θ).

In the common case, no harmonic terms are considered, the vector is:    0 Ψ0sd 0    (3.14)

The 3.13 is rather important and without stronger hypothesises it may not be simplied more.

3.1.4 Apparent motional voltage

The pseudo-motional voltage term and the motional transformed voltage may be grouped together for their relation with a parameter that is an angular variable. v0_M = v0EM + v 0 AM =cM 0 (i0, χ, θ) · dχ dt + c_A0 M(i 0 , χ, θ) · dθ dt (3.15) with the canonical choice θ = χ and adding the two terms 3.9 and 3.13

v0_M =cM0(i0 , χ, θ) +cA0_M(i0 , χ, θ)·dχ dt =    Md− Lsq· Isq Mq+ Lsd· Isd Mz   · dχ dt =    Msd Msq Msz   · dχ dt (3.16)

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By considering that χ = pα where α is the relative position between the reference system of stator and the reference system of the rotor, the 3.16 may be expressed in the following way:

v0_M =    Msd Msq Msz   p · dα dt =    vM sd vM sq vM sz    (3.17)

The below circuits provide the general equivalent block of the electric part of the model for the axis d and q.

The apparent motional term components are formally represented as an ex-ternal disturb. The Figure 3.1 and the Figure 3.2 provides general model for the electric part of the device.

vsd + 1 Lsd vM sd − Rd R dt isd −

Figure 3.1: Equivalent scheme for the d circuit

vsq + 1 Lsq vM sq − Rq R dt isq −

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The complete scheme of the brushless motor, comprehensive of the me-chanical part, is presented in Figure 3.3. The block used in this pattern are coherent with previous treatment. This scheme contains, within the elec-tric parts (above mentioned and represented), the mechanical part of the apparatus and it may be noted that the following quantities have been used: → WE = Msd· isd+ Msq· isq electromagnetic wrench(presented in rst

Chap-ter); → p pole pair;

→ TE electromagnetic torque;

→ TL external torque load;

→ J inertia;

→ b friction coecient;

→ ωM mechanical velocity inrad_s

The device is interconnected with the external environment by two dif-ferent port the electrical port and the mechanical port. The inputs of the electrical port are the transformed voltage vsd and vsq and the outputs are

the transformed currents isd and isq. For the mechanical interface the input

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vsd + 1 _Lsd Rd R dt isd vR d − M sd × × p isd vsq + 1 _Lsq Rq R dt isq M sq × vR q − × p isq + + p W E TL TE + − 1 J R dt ωM b − − vM sq − vM sd Figure 3.3: Complete brushless d-q ci rc uit

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3.2 Control strategy

According to the Chapter 1 the current reference is a combination of a con-stant term with a periodic terms. The rst term is necessary to balance the external torque while the second provides the compensation to the intrinsic torque ripple of the device. The strategy to achieve this purpose is the most common where:

→ isd = Isd = 0;

→ isq is the current to achieve the torque request.

Considering the requests the isq assumes the following expression:

isq= Isq+ +∞ X υ=1 6υ_i sq· cos(6υpα) (3.18)

Isq and P+∞_υ=16υisq· cos(6υpα)are respectively the constant term and the

periodic term of the reference signal.

The scheme control may be split into two dierent schemes one for the d axis and the other for the q axis. Taking into account the previous consider-ation the equivalent control systems for the d axis is presented in Figure 3.4.

IsdR = 0 + CIsd(z) vM sd + + Conv. vM sd − + Kd 1 + sTsd Isd −

Figure 3.4: Scheme of control of the d-axis

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Cω (z ) K T TE R I ∼ sqR + Isq R + + R E P (z ) + CI sq (z ) vM sq + + C onv . vM sq − + K q 1 + sT sq Isq − × M 0 sq p TE L − + 1 J 1 s _b ωM − Figure 3.5: Sc heme of con trol of the q-axis

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3.2.1 Conventional controllers

The conventional current controller for the d and q axis are designed as usual just to response to the request of the dynamic, and in general are PI controller. With the same concept the speed controller has been designed. PI current controller

The design of the current PI controller is not aected by the repetitive con-troller. Hence the implementation of the current controller depends just from the transfer function of the brushless motor. The transfer function of the brushless for the q axis assumes the following expression:

GQ =

KQ

z − pQ (3.19)

where the gain KQ and the pole pQ depend on the parameters of the device

Rss and Lsq.

The PI is designed to obtain a faster response of the q-axis current loop.

GP I = Ki · TC z z − 1 + Kp = (TC · Ki+ Kp) · z − Kp TCKi+Kp z − 1 (3.20) The choice is:

Kp

TCKi+ Kp

= pQ

and considering the close loop transfer function that follows the previous consideration:

GQCL=

(TCKi+ Kp) · Kp

z − [1 − (TCKi+ Kp) · Kq] (3.21)

the pole pQCL= [1 − (TCKi+ Kp) · Kq] is chosen to achieve a faster

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PI speed controller

This controller is designed focusing the attention on the stability within the dynamic necessities of the whole speed loop. One may note that the parameters of the speed PI might not be the same of the case when no repetitive controller occurs. This implementation should be done after the dimensioning of the repetitive controller.

3.2.2 Repetitive controller

The repetitive controller has been designed according to the chapter two and the scheme is reported in Figure 3.7.

kRC + z−N Q(z) Gf(z) +

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N delay

The modelling of the Repetitive controller starts from the block z−N_.

z−N

Figure 3.7: z−N _block

Here it may be noted that N has to be evaluate to compensate the right frequency of the torque ripple. The torque ripple frequency is

f6υp=

ω

2π · p · 6υ (3.22) and it means that the poles have to be placed on the unity circle bound-ary with an angular position according with 3.23, indicating with ω6υp, the

expression is:

ω6υp· TC =

2πk

N k = [0, . . . , N − 1] (3.23) With this condition the poles are located exactly in the position to have a zero steady state error for the periodic reference signal of frequency f6υp So

N:

N = 2πk ω6υp· TC

(3.24) TC is the sampling time of the discrete system.

The constraint over N is that being the the block a delay in z-domain N ∈ Z+. To compensate an arbitrary frequency that does not produce a in-teger one may assume to use the nearest inin-teger to the real number achieved by the 3.24.

It is interesting to highlight that two degree of freedom are available and they are the sampling time TC and the N delay. This result is valid with a

xed sampling time another possibility might be to select proper sampling time to reach a better compensation. However changing the sampling time is not so easily to implement and in this document it has not been considered.

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Stability lter

The stability lter is the block on the positive feedback and the equivalent block is shown in Figure 3.8.

Q(z)

Figure 3.8: Q(z) block

The stability lter has been chosen such as a moving average lter. This choice is due to maintain high compensation capability for the lower fre-quency, this selection can theoretically provide zero steady error for the lower harmonics. The expression of the moving average lter is here represented.

Q(z) = Pm i=1qi zi₊ 1 zi + q0 2Pm i=1qi+ q0 (3.25) Note the order of the lter is m − 1.

Compensation lter

This part of the repetitive controller is necessary to stabilize the entire current loop.

Gf(z)

Figure 3.9: Gf(z) block

According to the Chapter 2, considering that the system(current con-troller and and electric transfer function) is a minimum-phase one the ex-pression of the lter is:

Gf(z) =

1 + GC(z)GP(z)

GC(z)GP(z)

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and, GC(z) = KI· TC z z − 1+ KP (3.28) Stability constant

The kRC is the constant that is shown in Figure 3.10

kRC

Figure 3.10: kRC block

The stability constant kRC has been selected in the allowed range

indi-cated in Chapter 2 kRC = 1. This choice permits is due to that the cascade

of the blocks above analysed have already stabilized the system loop. Discussion

The mentioned choices permit to stabilize the entire current loop. Some constraints are highlight on the speed controller and indeed it is necessary to reduce the parameter of the speed PI achieved without the repetitive controller, so the speed loop reduce its performances but does not aect a lot the dynamic.

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Chapter 4 Simulations

In this chapter some results of the proposed control are presented and some consideration will be done about the control. The dynamic control scheme are implemented in MATLAB/SIMULINK enviroment.

4.1 Brushless motor

Device analysed:

→ Brushless isotropic motor; → Vn = 150[V ]nominal voltage;

→ fn= 150[Hz] nominal frequency;

→ Rss = 0.5[Ω]stator resistance;

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→ J = 1.20608e−1 _{mechanical inertia;}

→ Ψ0sd= 0.147 [W b] direct no load-ux;

→ b = 9.6252e−3_[N m

rad] friction coecient;

→ 5_{Γ −}7_{Γ = 0.033}_;

→ 5_{Γ +}7_{Γ = 0.021}_;

→ Tid = 4.6 [ms] q axis time constant;

→ Tiq = 4.6 [ms] q axis time constant;

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4.2 Control design

The design of the controllers may be split in three dierent phases:

1. design of the controller of the current loop without repetitive controller; 2. implementation of the Repetitive controller;

3. design of the velocity loop controller with the constraints due to the repetitive control.

4.2.1 PI design

According to the control strategy presented in the previous Chapter the pa-rameters of the conventional controllers have been selected. The choices are summarized below:

→ integral constant of the current PI of the q-axis Kid = 1250

→ proportional constant of the current PI of the q-axis Kpd = 5.75

→ integral constant of the current PI of the q-axis Kiq = 1250

→ proportional constant of the current PI of the q-axis Kpq = 5.75

→ integral constant of the current PI of the q-axis Kp−ω = 1250

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4.2.2 Repetitive controller

The repetitive controller has been designed taking into account the frequency of the torque ripple it is necessary to compensate. Almost the all opera-tive frequencies has been tested during the simulations. The strategy fol-lowed aims to reach the frequency of the current compensation by the delay N(analysed in the previous chapter).

→ sampling time TC = 100[µs]

→ mechanical speed ω = 30[rad s ]

→ angular frequency of the torque ripple ω = 1800[rad s ]

→ delay N = 35

→ order of the stability lter m − 1 = 4 → repetitive constant KRC = 1

According with the choices listed above the all poles are compensate more for high frequencies as shown in Figure 4.1.

Pole−Zero Map Real Axis Imaginary Axis −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1 −0.5 0 0.5 1

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The frequency response is provided in Figure 4.2. −20 0 20 40 Magnitude (dB) 102 103 104 105 Phase (deg) Bode Diagram Frequency (rad/s) REP PI

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4.3 Results and comparison

In this section the results are listed. A comparison has been done between the control system with the repetitive control and the one without.

4.3.1 Current tracking

The current reference is due to the aggregate of a constant term and an alternating term. In Figure 4.3 the tracking of the reference with the PI controller is provided. Due to the limited bandwidth of PI it does not track properly the reference and hence a steady-state current error occurs.

2.48 2.485 2.49 2.495 2.5 6.5 7 7.5 Time[s] Reference current Real current Real current Reference current

Figure 4.3: Current tracking without the Repetitive controller

The result of the action of the aggregate control system (Pi and Repetitive together) is plotted in Figure 4.4. The behaviour of the control system shows an almost perfect current tracking.

4.3.2 Torque

The speed control loop provides a torque reference of TE = 10.3 [N m] and

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2.48 2.485 2.49 2.495 2.5 6.8 6.9 7 7.1 7.2 7.3 Time[s] Reference current Real current Real Current Reference Current

Figure 4.4: Current tracking with the Repetitive controller

Figure 4.5 :

RIP P LE TE =

Tpp

TE

· 100 = 9.51% (4.1) where Tpp is the Torque peak to peak.

2.44 2.45 2.46 2.47 2.48 2.49 2.5 9.8 10 10.2 10.4 10.6 10.8 Time[s] Torque/Load torque [Nm]

Figure 4.5: Estimate motor torque without the Repetitive controller

The result with the aggregate controllers, the torque ripple is almost negligible(Figure 4.6).

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2.44 2.45 2.46 2.47 2.48 2.49 2.5 9.8 10 10.2 10.4 10.6 10.8 Time[s] Torque/Load torque[Nm]

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4.3.3 Phase currents

To complete the scenario the phase current are reported in both cases in Figure 4.7 and Figure 4.8.

2.45 2.455 2.46 2.465 2.47 2.475 2.48 2.485 2.49 2.495 2.5 −5 0 5 Time[s] Phase currents

Figure 4.7: Estimate phase currents without repetitive controller

2.45 2.46 2.47 2.48 2.49 2.5 −5 0 5 Time[s] Phase currents

Figure 4.8: Estimate phase currents with repetitive controller

Considering that the 6-th harmonic in synchronous reference(Park trans-formation) in basic domain becomes the sum of both the 5-th harmonic and 7-th harmonic while as well-known the DC component in the transformed domain becomes the fundamental component with angular frequency of pω.

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Chapter 5 Conclusion

This thesis addressed the problem of torque ripple generated by drives em-ploying isotropic permanent magnet synchronous motors under conventional q-axis current control, due to harmonics in linked uxes determined by non-optimal matching of winding structure and magnets layout. In particular, the possibility to modify the structure of a conventional drive control sys-tem, based on PI controllers, by adding a repetitive controller adopting the "plug-in" conguration, was investigated. Such investigation was developed by obtaining rst an analytical expression of the torque ripple by properly applying a general theory able to consider arbitrary structure of the wind-ings and layouts of the magnets. Such expression was then analysed to obtain a suitably modied current reference prole able to compensate the torque ripple. Such reference prole was then processed by a modied current con-trol loop, which employs a repetitive concon-troller in a "plug-in" architecture. A smooth torque production was obtained, in steady state conditions, by adding a suited periodic signal to the basic control signal generated by the

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PI controller used for the q-axis current reference. The Repetitive controller designed properly to this application is aimed to eliminate the high-frequency periodic current error.

The result, supported by simulations, is an almost perfect current tracking on the entire range of speed and hence a smooth torque has been achieved.

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Appendix A

Repetive implementetion

% **================================================================** % || **---** || % ||---||* REPETITIVE CONTROL DEFINITION *||---|| % ||---|| Stability Filter ||---|| % ||---|| Repetitive ||---|| % || **---** || % || Description: this function calcuates the numerator coefficients|| % || and denominator coefficients of the repetitive controller that || % || it has all poles internally a the unity circle || % ||---|| % || --> Of: stability filter order || % || --> w: velocità di rotazione del motore || % || --> h: armonica da compensare || % || --> p: numero di paia di poli ||

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% **---**

function [ numRe, denRe] = CoefRep( Of, w, h, p) % Of: stability filter order

% w: velocità di rotazione del motore % h: armonica da compensare

% p: numero di paia di poli

Fsample=10; %[kHz] Sample frequency Ts=1/(Fsample*1000); % Sample time

z=tf('z',Ts); % definizione dominio z Resto=mod((2*pi),(w*Ts*h*p)) % calcolo del resto N1=(2*pi-Resto)/(w*Ts*h*p) % calcolo risultato intero % esponente per ripetitivo

% valutazione del resto per decidere se approssimare per % ECCESSO o DIFETTO

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N=int32(N1+1) % ECCESSO end

% stability filter coefficients (Q) for i=1:Of+1

aFiltro(i)=(nchoosek(Of,i-1))/(2^Of); end;

% Repetitive control coefficients %compensator Filter G(f) Ord_Gf=1; k=0.2473; Par_Gf=[1,-0.753]/k; ANA=round(N+(Of/2)+1+Ord_Gf); for i=1:ANA+1

if i>=ANA && i<=ANA+1

numRe1(i)=Par_Gf(ANA+2-i); else numRe1(i)=0; end end % numerator

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AN=round(N+(Of/2)+1);

%numRe2=eye(1,AN+Ord_Gf); % crea vettore con elemento % a11 = 1 numRe=fliplr(numRe1); % denominator AL=round(2*N+(Of/2)+1); for j=1:AL if j==AL aRepet(j)=1;

elseif j>=(N+1) && j<=(N+Of+1) IND=(j-(N+1)+1); aRepet(j)=-aFiltro(IND); else aRepet(j)=0; end end;

denRe=fliplr(aRepet); % devo invertire end

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Appendix B

Parameters

% Questo file racchiudde i dati della macchina Brushless % dl laboratorio a Nottingham

%======================================================================= % ---->PARAMETRI DELLA

MACCHINA<---Rss = 0.5; % resistenza di statore in Ohm Lsd = 2.3e-3; % Induttanza asse D in H

Lsq = 2.3e-3; % Induttanza asse Q in H J = 1.2608e-1; % Inerzia del rotore p = 10; % Paia di poli

Fi0=0.12;

Fi0sD =sqrt(3/2)*Fi0; % Flusso asse D in Wb Fi0sQ = 0; % Flusso asse Q in Wb

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b=9.6252e-3; % Coefficiente di attrito viscoso Tt=J/b; % Costante di tempo meccanica q=1; % smorzamento anello di corrente R_a=1; % Abilita Ripple 1 Disabilita 0

%****** ---->@@ PARAMETRI RIPPLE DI COPPIA @@<---******** M6sd=0.033; % Coefficiente di mutua di 6a armonica Asse D M6sq=0.021; % Coefficiente di mutua di 6a armonica Asse Q %============================================================== % ---->PARAMETRI DEL CONVERTITORE DI POTENZA<---Vdc= 100; % Tensione del bus DC

fp = 10000; % Frequenza della portante in Hz Tinv = 1/fp; % costante di tempo del convertitore

Vf=Vdc*0.353; % Valore efficace max di alimentazione Wn=Vf/(p*Fi0sD); % Velocità meccanica max

% ---->PARAMETRI SISTEMA DI CONTROLLO<---Tc=Tinv;

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%Wm(t)--> IqR(t) ******************************** %============================================================== % ---->PARTE ELETTRICA<---s=tf('s'); Kqm=1/Rss; Kdm=1/Rss; Tsd=Lsd/Rss; Tsq=Lsq/Rss;

Kt =p*Fi0sD; % Costante di Coppia

Tid = Lsd/Rss; % Costante di tempo integrale % PI asse D

Kd = 1/(4*Tinv*Kdm*q^2); % Costante asse D Tiq = Lsq/Rss; % Costante di tempo

%integrale PI asse Q Kq = 1/(4*Tinv*Kqm*q^2); % Costante asse Q % ================= RIPORTO I PARAMETRI SECONDO % =============== CONVENZIONI MATLAB

Kid = Kd; % Guadagno integrale CORRENTE asse D *** Kpd = Kd*Tid; % Guadagno proporzionale CORRENTE asse D Kiqi = Kq; % Guadagno integrale CORRENTE asse Q Kpqi = Kq*Tiq; % Guadagno proporzionale CORRENTE asse Q

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% =============================================== % ---> Funzione di TRASF motore asse Q<---G_q=Kqm/(1+Tsq*s); % CONTINUO

G_qC=minreal(c2d(G_q,Tc)); % DISCRETO

% ---> Funzione di TRASF motore asse

D<---G_d=Kdm/(1+Tsd*s); % Funzione di TRASF motore asse D % ---> Funzione di Trasferimento dell'INVERTER <---G_inv=1/(1+Tinv*s);

G_invC=c2d(G_inv,Tc);

% ---> Funzione di Trasferimento dell'INVERTER + ASSE Q <--G_SE=minreal(G_inv*G_q);

G_SEC=minreal(c2d(G_SE,Tc));

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<---% ---> Funzione di Trasferimento PARTE MECCANICA <---G_t=(1/b)/(1+Tt*s);

%{

============================================================ nota bene risulta presente nell'anello di velocita la

costante di coppia

che quindi va considerata nel calcolo della stabilita %}

K_coppia=1/(Fi0sD*p);

% --> Funzione di trasferimento del PI di Asse Q<---- ANELLO di

% CORRENTE Q

G_PiQ=Kpqi+(Kiqi/s);

G_PiQC=minreal(c2d(G_PiQ,Tc));

%---> Funzione di trasferimento del PI di Asse D<---- ANELLO di

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G_PiD=Kpd+(Kid/s);

%--> Funz. Trasf. anello APERTO-CORRENTE %---sistema+ controllore ASSE Q

% Funz. Trasf. anello aperto sistema+ controllore Asse Q G_iQ=minreal(G_PiQ*G_inv*G_q);

% ---> Funz. Trasf. anello APERTO-VELOCITA' <---G_m=(1/Kt)*(G_iQ/(1+G_iQ))*G_t;

%============================================================= %============================================================= % Funz. Trasf. anello aperto sistema+ controllore Asse D

G_iD=G_PiD*G_inv*G_d; G_Sq=c2d(G_inv*G_q,Tc); % discretizzazione

%della G_iQ per applicare il controllo ripetitivo G_iQC=c2d(G_iQ,Tc);

%bode(G_iQ)

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% | | | | || % --->| G_PiQC |--->| G_qC |---> || % | | | | || % --- --- || % || || % || || % \/ || % || % --- || % | | || % --->| G_OL |---> || % | | || % --- || %========================================================= G_OL= minreal(G_PiQC * G_qC) G_CL= minreal(G_PiQC * G_qC/(1 +G_PiQC * G_qC)) %necessario per il controllo ripetitivo