energy-efficient controller design for a redundantly …molong/pdf/pub/j/tmech15.pdfhybrid feed...

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2015.2500165, IEEE/ASMETransactions on Mechatronics

TMECH-02-2015-4329 1

Abstract— This paper presents a method for designing a

controller that achieves the best positioning performance while maximizing the energy efficiency of a redundantly-actuated hybrid feed drive. A two degree-of-freedom controller, consisting of a feed forward (FF) controller for tracking and a feedback (FB) controller for regulation, is assumed. It is shown that the ideal FF controller, which achieves perfect tracking with maximum efficiency, is not always stable. Therefore, a method for designing a stable FF controller that achieves perfect tracking with near optimal efficiency is proposed. Furthermore, the optimal relationship between FB control inputs, which guarantees maximum efficiency for any specified regulation performance, is derived. Two approaches for using the derived optimal relationship to synthesize a FB controller that achieves the best positioning performance while maximizing efficiency are proposed. Simulations and machining experiments are conducted to demonstrate the effectiveness of the proposed energy-efficient FF and FB controller design methods. Significant improvements in energy efficiency (without sacrificing positioning performance) are reported.

Index Terms— Energy Efficiency; Control Structure Design; H2/H∞ Optimization; Hybrid Feed Drive; Redundant Actuation

I. INTRODUCTION

eed drives are used for coordinated motion delivery in manufacturing machines. Consequently, their positioning

accuracy and speed are critical to the quality and productivity of a variety of manufacturing processes [1]. Moreover, they account for a significant portion of the energy consumption of manufacturing machines [2], [3]. Therefore, a lot of work has been done in the literature related to improving their energy efficiency, e.g., [4]–[10]. It is however recognized that, to achieve truly sustainable manufacturing, improvements in energy efficiency must be achieved without unduly sacrificing quality and productivity [11].

Most machine tools utilize screw drives (SDs) for actuating their feed axes [1]. The reason is that SDs are cost effective and have high mechanical advantage which allows them to support high cutting (i.e., machining) forces with very low energy consumption [12], [13]. The speed and accuracy of SDs are

Manuscript received on February 6, 2015. This work is funded by the

National Science Foundation’s CAREER Award #1350202: Dynamically Adaptive Feed Drives for Smart and Sustainable Manufacturing.

M. Duan and C.E. Okwudire are with University of Michigan, Ann Arbor, MI 48109 USA. (e-mail: [email protected]; [email protected]).

however limited because of mechanical issues like vibration, wear, backlash and geometric errors of the screw and associated mechanical components [1], [13]. To mitigate these shortcomings, linear motor drives (LMDs) are increasingly being resorted to [13]–[18]. LMDs can achieve higher speeds and accelerations than SDs and are not subject to the inaccuracies caused by geometric errors, wear and structural deformations arising from the screw and other mechanical components like bearings, couplers and nuts that are connected to it [1], [13]. LMDs are therefore generally more precise than SDs. However, because they provide no mechanical advantage, they consume a lot more electrical energy to support cutting forces than SDs, thus significantly increasing the energy consumption of the machine [15].

When two or more actuators provide complementary benefits (as in SDs and LMDs), it is not uncommon to resort to hybrid feed drives (HFDs). In HFDs, dissimilar but complementary actuators are synergistically combined to meet conflicting requirements with little or no tradeoff. HFDs are most commonly designed with actuators that act in series to achieve long-range coarse motion and short-range fine motion, thus improving positioning speed and precision [19]–[24]. A relatively uncommon configuration of HFDs is the parallel HFD, where two dissimilar actuators are coupled together in parallel. For example, Shinno et al. [25] developed a tilting platform consisting of a pneumatic actuator and a couple of voice coil motors, in order to simultaneously generate high-precision and high-torque motions. Frey et al. [26] proposed a parallel HFD consisting of an SD and an LMD, which was controlled to actively damp the axial vibration of the SD. Okwudire and Rodgers [27] proposed a parallel HFD which synergistically combined an LMD and an SD to achieve speeds and accuracies similar to LMDs while consuming up to 80% less energy [27], [28]. However, the controller design technique they employed was ad hoc and could not guarantee that the best positioning performance and/or energy efficiency was achieved.

Well-established controller design techniques, like LQR [5] and mixed sensitivity H2/H∞ synthesis [29], can be employed to systematically design energy optimal controllers. Such techniques depend on user-defined weighting functions or matrices which are used to trade off conflicting requirements (e.g., positioning performance and energy efficiency) [5], [29], [30]. However, the methods do not provide a clear means of

Energy-efficient Controller Design for a Redundantly-actuated Hybrid Feed Drive with

Application to Machining

Molong Duan, Student Member, IEEE, Chinedum E. Okwudire, Member, IEEE

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METHODOLOGY

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2-DOF contFF controller, C

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ingly, xr1 is giv

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ositioning perfofectly track xd, ix1, needs not feference commng constraint bng that the HFDmass model, Gect tracking:

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ven by

s proposed in th

Control Structure f

ntroller

em, where d isine tool is profollow (whereormance is fori.e., xd −x2 = 0.follow xd. Accmand to the HFboils down to tD’s dynamics

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22 2 0.ffG u

ariation of a d is a specified

ntroller must ge

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(5)

function or d function of

enerate uff to

(6)

uators of the ed when its

(7)

fect tracking rol law as

(8)



TMECH-02-2015-4329 4

1 11 1 12 2

2 1 111 22 21 12 .

r ff ff

ff d

x G u G u

G G G G G x

(9)

One major challenge that arises with the energy optimal FF controller (given by (8)) is that, even though each element of G is minimum phase, the transfer function Gff may have non-minimum phase (NMP) zeros, as such, it may not have a stable inverse. There are several methods in the literature for calculating a stable approximation for the inverse for a transfer function with NMP zeros (e.g., [39]–[41]). Amongst them, the most notable is Tomizuka’s zero phase error tracking control (ZPETC) [39]. ZPETC determines a stable inverse such that there are no errors in phase across all frequencies; however, it gives rise to errors in gain compared to the exact inverse. If the numerator and denominator polynomials of Gff are represented by Bff and Aff, respectively, its stable inverse with zero phase error is given by

12

( ) ( ),

( ) (0)ff ff u

ff zpetcffs ffu

A s B sG

B s B

(10)

where Bffs and Bffu represent the stable and unstable portions of Bff. Note that (10) represents the continuous-time equivalent of Tomizuka’s discrete-time ZPETC [39].

The problem with the approximate inverse of (10) is that it sacrifices energy optimality as well as perfect tracking because it satisfies neither (5) nor (7); this is also the case for the other similar approximate inversion methods available in the literature (e.g., [40], [41]). We therefore seek a minimum phase approximation for Gff that can guarantee perfect tracking while maintaining near energy optimality. To determine such a near energy optimal (NEO) FF controller, let the minimum energy condition of (7) be re-written as

2 12

1 2

,ff ffff

ff ff

u uK

u u

(11)

where Kff is a gain. The implication of (11) is that energy optimality is sacrificed by making the variation of the energy functional in (7) to be non-zero by introducing Kff. Combining the perfect tracking constraint of (5) with (11), we get

2 1 11 22 21 2 2

2 1 221 22 21 22

, ;

.

ff ff ff ff ff neo d

ff neo ff

u K G G u u G x

G G K G G G

(12)

It can be shown that there always exists a Kff œ [0, ∞) that makes Gff−neo to be minimum phase. To do this, let us re-write Gff−neo as

1 2 2 222 22 21 21 22 .ff neo ffG G G G K G G

(13)

Therefore, the zeros of Gff−neo can be obtained from the the roots of the polynomial

2 2 222 21 21 22num num num ,ffG G K G G (14)

where num is a function that returns the numerator of the transfer function in its argument. Notice that (14) is in the standard form for root locus analysis. When Kff = 0, the zeros of Gff−neo are the same as those of Gff. However, as Kff →∞, the zeros of Gff−neo approach the zeros of G22G21 which, according to (3), are always minimum phase. Since the objective functional of (6) is quadratic, near optimality is achieved by selecting the smallest value of Kff that makes the poles of

G−1ff−neo stable, and provides sufficient damping to prevent

ringing due to poorly damped poles [39]. Note that if Gff is minimum phase, Kff = 0 becomes the default solution for Kff, in which case Gff−neo equals Gff, which is the energy optimal solution. Given Kff, xr1 can be calculated as

1 11 1 12 2

2 1 111 22 21 12 .

r ff ff

ff ff neo d

x G u G u

G K G G G G x

(15)

B. Design of Energy-efficient FB Controller

For the regulation problem, xr and uff are ignored such that u = ufb can be written as

,fb u Kz (16)

where z is the extended state vector of the HFD defined as

T

2 1 2 1 2 2 2; .i ix x x x x x x dt z (17)

Observe that z contains all four states of the two-mass model of the HFD, plus an additional state, x2i, representing the integral of the table’s position signal, added to ensure zero steady-state regulation of the table’s position. Let us assume that K is a full state FB matrix which can be expressed as

11 12 13 14 15

21 22 23 24 25

,K K K K K

K K K K K

K (18)

where K11…K25 are its gains. This means that the general form of Cfb, indicated in Fig. 3, can be written as

12 14 11 13 15

22 24 21 23 25

( ) .fb

K K s K s K K ss

K K s K s K K s

C (19)

In the rest of this section, we will show that there always exists some redundancy in the general form of Cfb, in terms of achieving the best FB positioning performance. We therefore propose a systematic method for using the available redundancy to maximize actuator efficiency without sacrificing positioning performance.

1) Redundancy in Achieving Best Positioning Performance

Recall that it is the precise positioning of the table (i.e., x2) that is of primary concern for the HFD. Therefore, the goal of the FB controller is to minimize the effect of disturbance d on e2. This objective can be written as

min s.t. stability constraints,e dJ W S (20)

where ║·║ represents a suitable norm (e.g., H2 or H∞), We is a scalar weighting function of s that indicates the frequencies in e2 that are of utmost importance, and Sd is defined as a scaled version of the closed loop (CL) disturbance transfer function from d to e2, given by the expression

10 1 ,d fb d

S I GC GW (21)

where I is the identity matrix and Wd = diag({d1,max, d2,max}) is a matrix that scales d1 and d2 by their respective maximum values, d1,max and d2,max, so that their magnitudes are comparable when calculating the norm of (20) [29].

Let J* represent the minimum value of J obtained using the full state (i.e., unstructured) FB matrix K of (18); it represents the best positioning performance that can be achieved using Cfb defined in (19). The corresponding optimal CL disturbance transfer function, Sd

*, can be expressed as



TMECH-02-2015-4329 5

T* *24 22 1,max

**

2 * *1 1 14 12 2,max

* * 5 * 4 * 3 * 2 * *,5 ,4 ,3 ,2 ,1 ,0

1;

.

dcl

cl cl cl cl cl cl cl

s c K s k K d

D s m s b c K s k K d

D a s a s a s a s a s a

S

(22)

The coefficients of D*cl are functions of K*

11…K*25 and the

plant parameters; their expressions are given in the Appendix. Note that the asterisk attached to the gains indicates that they are the optimal gains. A sufficient (but not necessary) condition to achieve J*, irrespective of the type of norm or weighting function (We) used in (20), is to match the numerator and denominator coefficients of Sd

*. Matching these coefficients mathematically means that the gain matrix K, with 10 degrees of freedom (i.e., 10 gains to be determined), should satisfy 9 linear/bilinear equations; 4 for the numerators of Sd

* and 5 for their common denominator. Therefore, at least 1 gain in K is redundant.

In practice, it is not uncommon for the condition d1,max << d2,max to hold. This is because d1 is primarily the result of Coulomb friction in the bearings of the rotary motor, which is often of much smaller magnitude than the largest cutting force magnitude associated with d2. Even when the Coulomb friction of the bearings is non-negligible, it can be measured and cancelled out reliably through feed forward friction compensation so that the FB controller does not have to deal with it. One can therefore conveniently consider d1,max ≈ 0, such that Sd

* is reduced to a scalar transfer function which can be matched using 7 gains. This leaves at least 3 redundant gains to achieve J*.

2) Use of Redundant Gains for Optimizing Efficiency

The set of all FB control inputs, ufb, that yield the same e2 under the influence of a given disturbance input, d, must satisfy the relationship

2 21 1 1 22 2 2

21 1 22 2 0.

fb fb

fb fb

x G u d G u d

G u G u

(23)

where δd1 = δd2 = δx2 = 0 because they are each specified, even if unknown, functions of time. Also, the most efficient control inputs must minimize the energy functional,

2 2

1 2 1 21 22 2

1 2 1 2

0.fb fb fb fbfb fb

m m m m

u u u udt u u

K K K K

(24)

Notice that (23) and (24) are the same constraints expressed for the FF controller in (5) and (7), respectively. Combining (23) and (24) we get

1 2 21

2 22

; .fb

fb

u G

u G (25)

The implication of (25) is that to satisfy a given e2 requirement at maximum efficiency, the control inputs are bound by an optimal transfer function relationship given by β. This result is very powerful and can be used to directly or indirectly determine a FB controller that achieves the best performance, J*, more efficiently than the optimal full state FB controller.

a. Energy-efficient FB Controller Design − Direct Approach

The direct approach seeks to structure K such that (25) is

satisfied (as much as possible) without compromising J*. To do this, note that ufb can be written as

T 1

1 2

,

, ,

fb u

u fb fb dd d

u S d

d S C I GC GW (26)

where 1 2, [ 1, 1]d d represent the scaled values of d1 and d2,

respectively. Combining (26) with (25), we get the relationship that must be satisfied by the elements of the re-structured K in order to maximize efficiency; it is given by

1 2

T T T1 1 2 2 1 2 2 1max

T T T2 1 1 1 2 1 2 2 max

0

0

0

fb fbu u

d

d

K QK p K p K

p K K QK p K

(27)

where

21 1

2 3 22 2

3 2 21 21 1

3 2 4 3 22 2

4 3 2 3 21 1

2

2

2 3

2 3

,

0 1 0 0

1 0 0

,0 0 0

0 0

0 0 0

i i

m s c b s k cs k

cs ks m s c b s ks

m s c b s ks cs ks

cs ks m s c b s ks

m s c b s ks cs ks

s

s s

s s

s s s

s s

K

p p

Q

K T

1 2 3 4 5 1,2 .i i i iK K K K i

(28)

Substituting s = jω into Γ1(s) and Γ2(s) in (27) (where ω represents the frequency content of d), we get a pair of complex-valued equations which are bilinear with respect to the elements of K. The real and imaginary parts of both equations must equal zero at every ω contained in d in order to attain optimal efficiency. Obviously, K may not have enough redundant gains to satisfy the equations for every ω contained in d. Therefore, one can maximize efficiency using the available redundancy by determining the K that minimizes the least-squares objective

H_

stability constraintsmin ( ) ( ) s.t. ,

*ee de d

J j jW J

Γ ΓS

(29)

where Γ = [Γ1(jω), Γ2(jω)]T and the superscript H represents the Hermitian transpose. If needed, the least squares objective in (29) can be weighted to emphasize some frequencies over others.

b. Energy-efficient FB Controller Design − Indirect Approach

The traditional approach for enforcing an energy efficient structure in K is to optimize for energy efficiency with positioning performance constraints; i.e.,

stability constraintsmin s.t. ,

*ee in u ue d

JW J

W S

S (30)

where Su is the transfer function matrix from input d to output ufb as defined in (26), Wu is a diagonal weighting matrix whose elements reflect knowledge about the relative efficiencies of



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MECH-02-2015

e actuators. Tdicates the knoe same performanner. The weigdirectly determtimization; a pucture for K, wpractice, Wu is

owever, the opeory, by selectib-optimality thotice that the sfference is thatforce (as much1 and ufb2. Onlationship indirces using a monstration ofSection IV. It must be n

sumption of a en made purelyethod for thentrollers). The plied to the deth observers).

B controller desFD described inother HFDs. Tal input, contrrformance defper).

Stability Const

Thus far, only sumed to be ac has been nstraints canno

L system determros, hence the gthout causing nstraints, the -modeled highsume that Ga iequency respoequencies, ω. Le system’s looccording to thestem is stable iakes Np counthout passing tMost often, it at there are suf

confining thnsitivity functi

max at frequenc

stability is likel

5-4329

The addition owledge that themance can beghting matrix, mines the st

poor choice of which cannot as typically chosptimum structuing Wu = diag(hat could be cretup of (30) ist the objective

h as possible) thn the other hairectly by minsuitable norm

f the direct and

noted before static FB contry based on its

e HFD (whicmethods prop

esign of dynamMoreover, eve

sign methods arn Section II, thThe methods arollable and mfined on one

traints and Hig

the low-order ccurately repreconsidered. Not be enforced mined using Ggains of the FBany stability iactual plant

her-order dynamis available in onse function Let Np denote top transfer fue Generalized if and only if t

nter-clockwise the origin. is desired to enfficient marginhe maximum on, S = (I + Lcies above the

ly to occur [29

of the constraere is redundane achieved in Wu, is very imtructure of KWu could lead

attain the best esen by intuitionure for K can([1, β]), therebreated by a pos very similar te in (29) is deshe optimal relaand, (30) enfonimizing the

m of Su, weigindirect appro

leaving this roller in (16) irelevance to thh seeks to a

posed in this semic controllers en though the re presented in

hey can be applare valid as loninimum phaseof its outputs

gher-order Dyn

dynamics of thesented by its Note, howevein (20), (29) o. This is becau controller can ssues [29]. Todynamics Ga,

mics, must be cthe form of a (FRF) measu

the number of unction given Nyquist Theorthe Nyquist cu

encirclement

nforce robust sns of stability. singular value

L)−1, below a spe CL bandwi

9]; i.e.

int related to ncy in K such ta more effici

mportant becausK used in d to a sub-optimenergy efficienn or trial and errn be obtained,by eliminating oor choice of Wto (29). The onsigned to direcationship betweorces the optimfeedback cont

ghted by Wu. oaches is provid

section that is arbitrary. It hhe desired contavoid high-orection can also

(e.g., controllproposed FF athe context of

lied more broang as the plantwith position(e.g., x2 in t

namics

he HFD, whichtwo-mass mod

er, that stabilor (30) based ose G has no NMbe infinitely h

o enforce stabil, which includconsidered. Letplant model oured at discrunstable polesby L = GaC

rem [29], the urve of det(I +s of the ori

stability, mean This is achieves of the pecified threshdth, ωBW, wh

J* that ient se it the

mal ncy. ror. in the

Wu. nly ctly een mal trol

A ded

the has trol der be lers and the dly t is ing this

h is del, lity

on a MP igh lity des t us or a rete s in Cfb. CL L)

gin

ing ved CL old

here

Simusection FB conprototypconstanand Km

convert2π/l, wi328.8, iefficienmass sidentifieFRFs experimFRFs arcritical

F

I

m1 [kg] m2 [kg]

Fig. 5. C

A. Evalu

Simuperform(NEO) Fis not assumedand nodescribeperfect dependiEvaluatreveals located

[ I

IV. SIMU

ulation and exto demonstrate

ntroller design pe of the HFD

nts for its rotary

m2 = 21 N/√Wted from its noith lead l = 5 indicating two

ncies of the twosystem whose ed by curve fitof the two-

mentally measure in good agreaxial/torsional

Fig. 4. HFD Protot

IDENTIFIED PARAM

616.2 b46.3 b

Comparison of Two

uation of FF C

ulations are mance and enerFF control metconsidered ind to be perfec

o disturbances ed, the propos

tracking withing on whetherting Gff using

that it has a at s = 196 ±

1m( )]

BW

j

L

ULATIONS AND E

xperiment resue the effectivenmethods. Fig.

D discussed iny and linear moW , respectiveominal value in

× 10−3 m. Tho orders of mao actuators. Th

parameters, tting measured -mass model ured FRFs (i.e.eement up to 10l mode of the H

type used for Simu

TABLE I METERS OF THE HF

b1 [kg/s] 44b2 [kg/s] 83

o-mass Model with

Controllers usin

conducted torgy efficiency thod proposed in the simulatitly described b

are introduced NEO FF coh near optimr Gff defined in

g the reportedcomplex conj

± 314j. Theref

max .

EXPERIMENTS

ults are preseness of the prop

4 shows an inn Section II.Aotors are Km1 = ely, where Km

n Nm/√W usinherefore, κ2 = (agnitude diffee HFD is modereported in TFRFs. Fig. 5 c(i.e., G(jω)

, Ga(jω)). As s00 Hz, hence GHFD, occurring

ulations and Exper

FD’S TWO-MASS M

4.8 c [kg/3.3 k [N/μ

th Experimentally

ng Simulations

o evaluate thof the near enin Section III.Aions because by its two-mased. Under theontroller is abl

mal or optimaln (8) has NMP d parameters ojugate pair of fore, the gain

6

(31)

nted in this posed FF and n-house built

A. The motor 380.8 N/√W

m1 has been ng a factor of (Km1/Km2)

2 = erence in the eled as a two Table I, are compares the ) with the seen, the two

G captures the g at 41 Hz.

riments

MODEL

/s] 5777.2 μm] 3.1469

Measured FRF

he tracking ergy optimal

A. FB control the plant is ss model, G, e conditions le to achieve l efficiency, zeros or not. of the HFD

f NMP zeros Kff must be



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MECH-02-2015

lected to ensurar energy optimKff is varied in le pair in G−1

ff

ff−neo) becomes int marked Kff

gure; it represmparative simu

Fig. 6.

For tracking temm, with ki

5×106 mm/s3 and snap, respectg. 7(a); its pos

mitted due to spdy FF (RB FF)e proposed Necifying xr1 = x

ffu G

he implication oe responsibilityat the two-masegant and it guase error FF anchmark becaquirement stipuFig. 7 (b)-(d) ae table, and theF and the propol three contro

otice, however,MD. As a resultf = 1000. In thost efficient. Hsults in undesirves rise to poorergy formulatiorly damped rmulation of (herefore, the onlue of Kff that is−1

ff−neo. Fig. 8 acking the refernging from 30ry high but dro

5-4329

re a stable invmality. Fig. 6 sthe interval [0

ff−neo (correspostable for valu= 1000 has bee

sents a value ulations later in

Root Locus of G−

ests, a desired pinematic limitsnd 5×108 mm/stively, is used.sition, accelerapace limitation) [42] is used aEO FF controxr2 = xd such th

T1d d

mx x

m

of (32) is that iy to move the mss system movuarantees perfeapproximation ause it does nulated for FF coand Table II coe power/heat waosed NEO FF, ollers achieve , that RB FF det, it is 71.5% le

heory, NEO FFHowever, notice

rable ringing orly damped polion of (12) dopoles/zeros (m

(8) misses thenus falls on thes high enough tplots the tota

rence trajectory.3 to 1000. No

ops rapidly as th

verse of Gff−neo

shows the root, ∞). It is foundnding to the Nues of Kff ≥ 30en highlighted

of Kff whichn this section.

−1ff−neo as a Functio

position trajects of 100 mm/s4 for velocity, Its velocity pr

ation, jerk and ns. A FF technas the benchmaoller. RB FF at

21 1

22 2

.d

m s b sx

m s b s

in RB FF eachmass to which ites as one rigidect tracking. N

of (10) is nonot meet the ontrol. ompare the tracasted by each awith Kff = 30.perfect tracki

epends heavily ess efficient thaF with Kff = 30e from the figuof the control sles in G−1

ff−neo. oes not recognmuch like thee effect of NMe controller deto avoid poorlyal heat energyy of Fig. 7(a) uotice that the ehe value of Kff

o while providt locus of G−1

ff−

d that the unstaNMP zero pair0.3. Note that on purpose in

h is utilized

n of Kff

tory, xd, of stro/s, 1×104 mmacceleration, jrofile is shownsnap profiles

nique called riark for evaluatis generated

(3

h actuator is givt is attached, sud body. It is v

Note that the zot selected as

perfect track

cking error (e2)actuator using R3 and Kff = 10ing, as expecton the ineffici

an NEO FF, us0.3 should be ure that Kff = 3signals becauseThe near optim

nize the effecte optimal enerMP poles/zero

esigner to selecy damped polesy of NEO FF using values of energy is initiaf increases and

ing f−neo able r in the the for

oke m/s2,

erk n in are gid ing by

2)

ven uch ery ero the ing

) at RB 00. ted. ient ing the 0.3 e it mal

of rgy os). ct a s in

in f Kff ally the

poles becontinubecomecontinubetween0.03 J.energy(For insslightly that as lin G−1

ff−

Fig. 7. R

COM

MaxH

HeTo

Fig. 8. To

B. Evalu

SimuindirecttransfercontrollunstructThe pra(which assumedoptimizdetermiH∞ normand Pos

Equatiovery lowfrequenenforce

ecome more daues to reduce ases very low aftues until Kff =n the energy atIt must be nothighly depend

stance see the y different deslong as Kff is tu−neo, the energy

Reference VelocitActuator Po

MPARISON OF TRAC

CONTROLLE

x Tracking Error [nHeat Energy SD [J]eat Energy LMD [otal Heat Energy [J

otal Heat Energy oReference T

uation of FB C

ulations are cot, the direct anr function relatiler design. Thetured) state FBactical scenariis the cutting d. MATLAB’s

zation based oine the Sd

* that m. A weightinstlethwaite [29

eW

on (33) represenw frequencies bncies (i.e., freque integral action

amped. For thiss Kff is increaser about Kff =

= 1000 and bt Kff = 1000 andted that the reds on the systAppendix for ired trajectory

uned to providey performance

ty Profile and Comower of FF Contro

TABLE II

CKING ERRORS AND

ERS BASED ON SIM

RB FF

nm] 0.000 ] 3.967 J] 7.398 J] 11.365

of Proposed NEO Trajectory of Fig. 7

Controllers usin

onducted to evnd indirect metionship, β, derie benchmark iB controller dio where d1,max

force limit of s hinfstruct comon a structuredminimizes the

ng function, W] is adopted; it

1

1,

nn

B

nn

B

s M

s A

nts a filter of oby a factor of Muencies much hn. We select n =

s particular cassed but the rate50. The decreaeyond, but thd that at Kff = 1lationship betwem and desirethe energy vs.

y). Nonethelesse sufficiently dremains near o

mparison of Trackiollers (Simulation)

D ENERGY EFFICIE

MULATION RESULTS

NEO FF (Kff = 30.3) (

0.000 27.918 24.358 52.276

FF as a Function o7(a)) – Simulation

ng Simulations

valuate EE dirthods for usingived in Sectionis selected as tdiscussed in Sx = 0 and d2,m

the HFD of Fimmand (which d gain matrix

e objective of (2We, suggested b

t is given by

,

order n, designeM/A comparedhigher than ωB

= 2, M = 2, A =

7

se, the energy e of decrease ase in energy he difference 10,000 is just ween Kff and ed trajectory. Kff plot of a s, it appears

damped poles optimal.

ing Errors and

ENCY OF FF

S

NEO FF (Kff = 1000)

0.000 3.242 0.001 3.243

of Kff (based on

n

rect and EE g the optimal n III.B for FB the full (i.e., ection III.B.

max = 400 N ig. 4 [27]) is performs H∞ ) is used to 20) using the

by Skogestad

(33)

ed to weight d to very high B), so as to = 5 × 10−6 and



TMECH-02-2015-4329 8

ωB = 200π rad/s (indicating that we want integral action to taper off beyond 100 Hz).

The HFD is assumed to be perfectly modeled by G, and the table-side disturbance force, d2, is assumed to be a harmonic cutting force at 60 Hz, with an additional DC component; it is given by

2( ) 200 200sin(120 ) [N].d t t (34) However, as discussed in Section III.C, applying the optimization of (20) to G could yield infinitely large gains. Therefore, a 7th order Butterworth low-pass filter, with cutoff frequency of 1000 Hz, is applied to each element of G before carrying out the H∞ optimization in order to have finite stability limits. The default stability settings of MATLAB’s hinfstruct command are maintained for the optimization. The resulting Sd

*(1,2) (scaled by 1/d2,max) is plotted in Fig. 9(a) and the elements of the optimal full state FB gain matrix (i.e., K*) are given in Table III. Note that only Sd

*(1,2) is shown in the figure because Sd(1,1) = 0 based on the assumption that d1,max = 0.

Fig. 9(b) shows the optimal transfer function relationship, β, for the HFD. It requires the magnitude of the ratio ufb1/ufb2 to be 50.3 dB at ω = 0, and to be 23.5 dB at ω = 120π rad/s. In employing β for energy efficiency optimization of K, we have chosen to use Sd

* instead of its infinity norm, J*, as the performance constraint in (29) and (30). This is done in order to ensure that the exact same regulation performance as the optimal full state FB controller is achieved at all frequencies. Therefore, using (22), we derive 7 constraint equations to ensure that the poles and zeros of Sd

*(1,2) are matched with 7 out of the 10 gains of K, leaving us with 3 redundant gains. To obtain the energy efficient controller using the direct method (i.e., EE direct), one of the remaining 3 gains is used to perfectly enforce (27) at ω = 0. In theory, the last two gains could be used to satisfy the real and imaginary parts of (27) at ω = 120π rad/s, but the equations are highly nonlinear and difficult to solve analytically. Therefore, the remaining two gains are used to minimize Jee-d (i.e., (29)) at ω = 120π rad/s. The elements of the resulting gain matrix are given in Table III.

To determine the energy efficient controller using the indirect method (i.e., EE indirect), MATLAB’s hinfstruct command is used to minimize the H∞ norm of (30). A weighting function given by Wu = Wωdiag([1, β]) is selected, where Wω is a weighting filter designed to emphasize frequencies ω = 0 and ω = 120π rad/s contained in d2. It is defined as

2

2 2,

2n

en n

W Ws s

(35)

Note that We is the exact same filter defined in (33); it is used in Wω to place a higher weight on frequencies close to ω = 0. The second-order filter multiplying We is designed with ωn = 120π rad/s and ζ = 8.2×10−6, such that its magnitude at resonance is equal to We(0). The constraint equations that were used to match the poles and zeros of Sd

*(1,2) for the EE direct method are used to structure K within MATLAB’s hinfstruct function so that Sd

*(1,2) is also matched perfectly in EE indirect. The elements of the resulting FB gain matrix are reported in Table III.

Fig. 9(a) compares the magnitudes of Sd(1,2)/d2,max for the

EE direct and EE indirect controllers to the original Sd

*(1,2)/d2,max determined based on the optimal state FB controller; the plots are coincident, indicating that all three methods achieve the best positioning performance at all frequencies. Fig. 9 (c) and (d) respectively compare Su(1,2)/d2,max and Su(2,2)/d2,max for the three methods. Notice that Su(1,2)/Su(2,2) = 50.3 dB for the EE direct controller, meaning that it perfectly matches β at ω = 0. However, at ω = 120π rad/s, it achieves a Su(1,2)/Su(2,2) ratio of 32 dB instead of the 23.5 dB stipulated by β, due to errors in the least squares solution applied at that frequency. The EE indirect controller does not perform as well as the EE direct in terms of matching β. This is because, in determining the gain matrix for the EE indirect controller, the hinfstruct command was initialized with the gain matrix of the optimal full state FB controller (i.e., K*). Comparing the elements of the full state FB controller’s gain matrix to those of the EE indirect controller in Table III, one can conclude that the gradient-based algorithm used in hinfstruct may have gotten stuck in a local minimum close to its initial value, thus preventing it from reaching a better solution. This is a well-known limitation of gradient based methods when performing non-convex optimizations. It could be mitigated by testing various initial values in hopes of finding other local minima. Note that the direct method was also initialized with the elements of the full state FB matrix, when performing the nonlinear least-squares optimization of (29). Therefore, its solution could also get stuck in local minima around its starting point, but it appears from Table III that its gains are quite different from those of the full state FB matrix.

The expression 2 2

2

21 2.max 2.max0 120

1 (2, ) 1 (2, )200 200 ,

2

T Tu u

i mi

i i

K d d

S S

(36)

representing the sum of the average power at ω = 0 and ω = 120π rad/s of each actuator’s steady-state response to d2(t) given in (34), is used to calculate the average power dissipated in heat by each FB controller. The results are shown in Table IV. EE direct and EE indirect are respectively 60% and 33% more efficient than the full state FB method. Fig. 10 compares the ratio of the average power of the EE direct and EE indirect based controllers, relative to that of the full state FB controller, using a disturbance input of d2(t) = sin(2πft). The EE direct controller is much more efficient than the full controller from f = 0 through f = 44 Hz, after which it becomes less efficient. Note that its relative efficiency is 8 dB worse at f = 60 Hz, due to the aforementioned errors in the least square solution. The EE indirect controller, on the other hand, is more efficient than the full controller at all frequencies, but only by a very small (0.6 dB) margin (except at f = 0 and f = 60 Hz where it shows slightly better relative efficiencies of 4.2 dB and 0.8 dB, respectively).



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OMPARISON OF ELE

FULL STATE FB, E

Full State FB −8

EE direct +9EE indirect −4

Full State FB −2.

EE direct −5EE indirect −2.

g. 9. Comparison ote FB and the Ener

and

COMPARISON OF E

Average PowerDissipated in He

SD [W] LMD [W] Total [W]

Fig. 10. ComparisoDirect and EE Ind

Machining Exp

Machining exrformance andsign methodsototype of Fig. 20 3-axis milli

ot in an AISI 1y cutting param00 rpm (i.e., aintain a reasoajectory, xd, is pm/min (i.e., 5 m62.5 mm/s2, 3such low spe

ntributions of d are indisting

onsequently, thachining tests.

5-4329

TABL

EMENTS OF FB GA

EE DIRECT AND EE

K11 K12 .2×105 −3.1×106

.2×105 −3.1×106

.5×105 −3.1×106

K21 K22 4×1010 +4.2×107

.2×107 −7.6×104

3×1010 +7.0×107

of Positioning Perfrgy-efficient FB Cd EE Indirect App

TABL

ENERGY EFFICIEN

EQUATION (36)

r eat

Full State FB

1.3381.8383.1

on of Efficiencies direct Methods (Re

Controller) –

periments

xperiments ard energy efficie. To perform4 is mounted o

ing machine to018 steel workmeters for the60 Hz) is selonable feed pprescribed to tramm/s), with ac125 mm/s3 and

eeds and accelRB FF and NEguishable fromhe effect of

LE III AIN MATRICES CAL

E INDIRECT METHO

K13 6 −3.1×108 −56 +3.8×109 −56 −1.8×108 −5

K23 7 +5.2×107 +94 +1.1×108 −87 +6.8×107 +1

formance and ConControllers designeproaches (Simulatio

LE IV

NCY OF FB CONTRO

) (SIMULATION)

B EE direct

36 125.8080 26.31

6 152.11

of Controllers deselative to Efficienc– Simulation

e conducted ency of the pro

m the experimon the x axis ofo cut a 15 mm kpiece. Table Ve operation. A ected for the

per tooth, the avel at a maximcceleration, jerd 6.25×105 mmerations, the thEO FF are neg

m noise in theFF is not co

LCULATED USING T

ODS (SIMULATION

K14 K15 5.8×103 −9.3×105.8×103 +8.5×105.8×103 −6.2×10

K24 K25 9.5×104 +1.6×108.9×101 +1.6×101.3×105 +1.6×10

ntrol Effort of the Fed using the EE Dion)

OLLERS BASED ON

EE indirect

0 0.511 254.441 254.95

signed using the EEcy of Full State FB

to evaluate oposed controlments, the Hf a FADAL VMlong, 1 mm de

V summarizes spindle speedcutting tests. desired posit

mum speed of 3rk and snap limm/s4, respectiveheoretical energligible (< 6 me control signaonsidered in

THE

N)

05 06 05

05 05 05

Full irect

N

E

B

the ller FD

MC eep the

d of To ion 300 mits ely. rgy

mJ), als. the

SpindlTool NumbFeed pFeed rLubric

The

optimizplant dfriction easily cassumedplaced aFig. 5)appearinRobust maximufrequen

2.65. MAT

simulatioptimizis givenfunctionOptimizout the HreportedFB gain

Feed harmonTherefosame prconstraigains ar120π raindirectthe optimits initia(22), foreportedindirect

COMPAR

FULL S

Full StaEE D

EE Ind

Full StaEE D

EE Ind

Fig.

direct ausing th

le speed

er of flutes per tooth rate cation

optimal full zing the H∞ nordynamics, Ga(j

in the bearingcancelled by feed, and d2,max at the 847 Hz , and at the ng in Ga(2,2)

stability conum singular vncies higher th

TLAB’s hinfstrions, could n

zation in (20) un as a FRF atn modeled in zation (PSO) TH∞ optimizatiod in [44]. The en matrix are proforces in milli

nics at the spinore, for the EErocedure as useined to match re used for enf

ad/s, representint method, MATmization of (30al values; cons

ollowing the exd in Section t controllers are

RISON OF ELEMENT

STATE FB, EE DIRE

K11

ate FB 4.65×107

Direct 5.48×107

direct 4.80×107

K21 ate FB 1.52×107

Direct 2.54×106

direct 6.16×106

11(a) compareand EE indireche full state FB

TABLE V

CUTTING PARAME

3600 r3/8” di4 0.02 m300 mNone

state FB corm of (20) with(jω), substitutegs of the rotaryed forward actis selected as resonance pea400 Hz and in order to i

nstraints are evalues of the

han ωBW = 628

truct commandnot be employusing Ga insteat discrete freqs-domain. Th

Toolbox in MAon, based on a telements of theovided in Tabling typically handle and toothE direct controed in Section IVSd

*(1,2) usingforcing the β cng the lowest hTLAB’s PSO t0) based on Ga

straints are addxact same procIV.B. The gae reported in T

TABLE VI TS OF FB GAIN MA

ECT AND EE INDIR

K12 7 4.92×105 1.07 4.92×105 8.97 4.92×105 8.5

K22 7 6.15×105 1.06 1.59×105 1.6 1.57×104 1.0

es Sd(1,2)/d2,m

ct controllers toB controller. A

ETERS

rpm ia. HSS end mill

mm/tooth mm/min

ontroller is dh the actual (i.eed for G. Th

y motor is smaltion. Therefore

400 N. Notcak appearing in487 Hz reson

increase stabilenforced by e sensitivity f8 rad/s (100 H

d, which was yed for carryad of G. This isquencies, not aherefore, a ParATLAB [43] is utechnique simile resulting optimle VI. ave a DC portioh passing frequoller, we folloV.B. The eleme

g (22), and the constraint at ωharmonic in d2

toolbox is used, with the elem

ded to match Sedure as in the

ains of the EETable VI.

ATRICES CALCULA

RECT METHODS (E

K13 K14

06×108 1.29×1090×107 1.29×1057×107 1.29×10K23 K24

02×107 4.85×1012×107 1.02×1006×107 -4.25×10

ax generated uo Sd

*(1,2)/d2,m

An almost perf

9

designed by e., measured) he Coulomb ll and can be

e, d1,max = 0 is ch filters are n Ga(1,1) (in nance peaks lity margins. limiting the function for Hz) to max =

used in the ying out the s because Ga as a transfer rticle Swarm used to carry lar to the one mal full state

on, as well as uencies [45]. ow the exact ents of K are 3 redundant = 0 and ω = . For the EE

d to carry out ments of K* as Sd

*(1,2) using e simulations E direct and

ATED USING THE

EXPERIMENTS)

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TMECH-02-2015-4329 11

have been used to demonstrate the benefits of the proposed approaches in significantly improving energy efficiency without sacrificing positioning performance for a redundantly-actuated HFD. The proposed FF and FB controller design methods are general enough to apply to any plant (HFD) that is dual input, controllable and minimum phase with positioning performance defined on one of its outputs. Future work will seek to generalize the proposed methods to plants with more than two inputs. Performance and stability issues that could arise when the HFD studied in this paper switches rapidly between its two modes of operation will also be investigated.

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PENDIX

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