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Optimum, machine tool axis traverse regulation
ARTICLE IN PRESS
International Journal of Machine Tools & Manufacture 46 (2006) 1835–1853 www.elsevier.com/locate/ijmactool
Optimum, machine tool axis traverse regulation R. Whalley, M. Ebrahimi, A. Abdul-Ameer, S. El-Shalabi School of Engineering, Design and Technology, University of Bradford, UK Received 31 March 2005; received in revised form 11 November 2005; accepted 15 November 2005 Available online 30 January 2006
Abstract A hybrid, distributed parameter model for a milling machine axis drive system is derived from measured results. Spatial variation effects arising from the machine system geometrical changes are incorporated enabling multivariable regulation via feedback control. Design procedures enabling the realization of an optimum, minimum energy, multivariable regulator for this system are presented. Table velocity and lead-screw twist interaction are controlled via armature and field voltage variations, restricting the effect of cutting loads and lead-screw length changes, respectively. The feedback compensators for this system are evaluated and the open and closed-loop model response transients, following reference input variations, are presented. Disturbance suppression and the surface finish capabilities of the closed-loop, minimum energy, system, were investigated. The performance advantages promoted by multivariable minimum effort control are emphasized with the confinement of lead-screw wind-up and cutting load disturbance effects whilst limiting output cross coupling with improving transient behaviour and steady state accuracy. r 2005 Elsevier Ltd. All rights reserved. Keywords: Optimum; Energy; Milling; Machine; Regulative
1. Introduction In order to improve the disturbance rejection capabilities, speed of response and steady state accuracy of machine tools, automatic, feedback control is usually incorporated. This provision also enhances the repeatability of these systems countering wear, environmental and load variations whilst improving surface finish, tolerance limits, integrity and manufacturing efficiency, as in Landerborough and Ulsay [1]. The effectiveness of the control strategy imposed depends upon the fidelity of the machine tool model employed. This realization must replicate machining performance, following input changes, cutting force disturbances, shock and internally generated vibrations. With the advent of broadband, multivariable, distributed parameter modelling methods, as in Whalley et al. [2], the opportunity to include well known dynamic effects, which have been difficult to incorporate hitherto, arises. These perturbations are promoted by spatial dispersion, lead-screw wind-up, interactive torsional and longitudinal loading, deformation and deflection, as shown in Whalley et al. [3]. These detailed realizations enable the formulation of elevated controller design techniques, improving efficiency, accuracy and effectiveness, thereby. Axis regulation strategies, focused on the suppression of disturbances, arising from geometrical, machine tool variations, as well as simultaneously controlling transverse velocity, are now invited. Thereafter, many of the mandatory, precision manufacturing restrictions on temperature, material variations, vibration isolation etc. may be relaxed without compromising machining tolerances, performance or surface finish quality. Equally, regulation robustness, stability, consistency of operation, irrespective of machine configuration, together with the restriction of the effects of arbitrary cutting forces, significantly advantage manufacturing competitiveness. Corresponding author.
E-mail address: [email protected] (M. Ebrahimi). 0890-6955/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.11.011
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Nomenclature A(s) numerator of G(s), matrix ai;j ðsÞ elements of A(s) 1pi, jpm, function ai;j ; bi;j . . . gi;j , coefficients of ai;j ðsÞ, scalar b(s) polynomial function b0 ; b1 bm1 coefficients of bðsÞ, scalar d(s) denominator of G(s), function F outer loop feedback array, matrix f ; f 1 ; f 2 f m outer loop feedback gains, scalar G(s) transfer function array, matrix ¯ GðsÞ system matrix, matrix P(s) pre-compensator array, matrix h feedback path gain, vector h(s) feedback path function, vector Im identity array (m m), matrix J performance index, functional kS/h outer product of k and h, matrix /k,hS inner product of k and h, scalar k forward path gain, vector k(s) forward path function, vector L(s) left (row) factors, matrix n; n1 ; n2 . . . nm1 gain ratios, scalar Q coefficient array, matrix r¯ðsÞ transformed reference input, scalar r(s) transformed reference input, vector R(s) right (column) factors, matrix Ss steady state array, matrix S(s) sensitivity array, matrix u(s) transformed input, vector y(s) transformed output, vector CL(s), CR(s) finite time array, matrix dðsÞ transformed disturbance input, vector T 0 ðsÞ input torque on the lead screw, scalar o1 ðsÞ output lead screw angular velocity, scalar o0 ðsÞ motor speed and input drive end angular velocity of the lead screw, scalar z characteristic impedance of the lead-screw (torsion) ¼ (1.9864 m/s), scalar z¯ characteristic impedance of the lead-screw (tension) ¼ (3.949 107 m/s), scalar GðsÞ propagation function of lead-screw (torsion) ¼ (3.1224 104 s), function GðsÞ propagation function of lead-screw (tension) ¼ (1.9748 104 s), function R termination frictional coefficient (torsion) ¼ (14.65 103 Nm/(rad/s)), scalar ¯ R termination frictional coefficient (tension) ¼ (5.4 106 N/(m/s)), scalar R0 ðsÞ lead-screw input end transfer function (torsion), function F 0 ðsÞ lead-screw input end transfer function (tension), function R1 ðsÞ lead-screw output end transfer function (torsion), function F 1 ðsÞ lead-screw output end transfer function (tension), function s0 ðsÞ input end stress in lead screw, function Fx(s) cutting force in x direction, function f ðsÞ force change on ball nut, function B saddle viscous friction coefficient (1 N/m/s), scalar M saddle mass (307 kg), scalar V a ðsÞ armature voltage, function V f ðsÞ Field voltage, function Tb motor load torque ¼ mf ðsÞ , function E Young’s modulus (200 109 N/m2), scalar fðsÞ angular change in lead-screw at ball-nut end, scalar
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Kp ball nut lateral stiffness (985 106 N/m), scalar k2 back emf constant (0.7639 volt/(rad /s) scalar m ball nut rolling resistance ¼ fiction coefficient ball nut radius ¼ (26 104 m), scalar kL lead-screw and ball nut series stiffness (1.3586 108 N/m), scalar kd angular to linear movement constant (25 104 m/rad), scalar dðsÞ lead-screw length change, function eðsÞ lead-screw strain due to twisting, function vðsÞ saddle velocity, function Lf D.C. motor field inductance, scalar Rf D.C. motor field resistance, scalar t current limiter time constant, scalar k3 armature current-torque constant, scalar Ra D.C. motor armature resistance, scalar fj ðsÞ; wj ðsÞ compensator models, 1pjpm, function
To achieve these objectives, comprehensive modelling studies and multivariable distributed parameter, dynamic analysis, form a necessary prelude to controller design. Additional measurements and hence transducers are also required and the important choice of an appropriate, multiple input–output regulation strategy, must be exercised. In this regard, this submission presents a controller design study for the x-axis, Arrow 500 milling machine system, the configuration of which is shown in Fig. 1. The multivariable, hybrid model for this machine has been validated via measured results and manufacturer’s data enabling the construction of an accurate input–output realization, for analysis purposes.
Spindle motor
Spindle
Fx Cutting force Cutter Lead-screw
v (s)
Work piece
Forward bearing
Saddle velocity
M, B
Slide D.C Motor
Ball nut
d T
Ti Output end torque
To Input end torque
lt Rear bearing
Coupling Va (t) Vf (t) Motor field Voltage
Motor armature Voltage
a0 Angular velocity
0 v0
Output end stress
l
ai i
Input end stress
vi
Longitudinal velocity
Longitudinal velocity Fig. 1. Machine tool x-axis traverse and drive.
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Initially, a simple representation of this system model was employed in the design of a suitable minimum effort, multivariable controller. This regulator is relatively insensitive to parameter variations whilst generating a cooperative feedback strategy, requiring least energy, commensurate with the disturbance suppression, recovery and dynamic response characteristics specified. Evaluation of the controller for a range of feedback gains and pre-compensators was undertaken. Thereafter, the effect of input and load disturbances will be investigated and the performance of this system under closed-loop conditions will be presented prior to incorporation and integration with the manufacturing system. 2. Axis dynamic model A detailed derivation of the distributed-lumped, non-linear x-axis dynamics for the Arrow 500 milling machine is presented in Whalley et al. [4]. The arrangement of the system, with a separately excited D.C. motor drive, is as shown in Fig. 1. A block diagram representation of the drive is as Fig. 2, where the transfer functions, non-linear elements and gain parameters are indicated. The transfer function representations in Fig. 2 are: R0 ðsÞ ¼
ðz þ RÞe2GðsÞl þ ðz RÞ , zfðz þ RÞe2GðsÞl þ ðR zÞg
(2.1)
R1 ðsÞ ¼
2zeGðsÞl , ðz þ RÞe2GðsÞl þ ðz RÞ
(2.2)
F 0 ðsÞ ¼
¯ ¯ 2GðsÞl ¯ ðz¯ þ RÞe þ ðz¯ RÞ , ¯ ¯ z¯ þ RÞe ¯ ¯ 2GðsÞl þ ðR ¯ zÞg zfð
(2.3)
Va (s)
+
1
-
ω0 (s)
Back emf
τs + 1
ω1(s) k3
ω0(s)
T0(s)
Ra
Input torque to lead-screw
+
Ra(s)
ω1(s)
R1(s)
Lf s + Rf
− Tb Load torque
σ0 (s)
(s)
1
E
Stress
F0 (s)
Strain
Longitudinal velocity owing to lead-screw drive 0(s) end tensile stress
F1(s)
+
kp
Lead-screw total Force change on + lead-screw Ball nut displacement lateral stiffness Fx (s) Cutting force
Saddle velocity
1
+ Force for acceleration
Saddle transfer function (s) Coulomb friction
ks
x direction velocity of ball nut
Nut end longitudinal velocity
kL
Saddle velocity
1
ω1(s)
x (s)
1(s)
1
x (s)
Traverse drive force − Series stiffness of Lead-screw and Ball nut (s)
+
1(s) +
Saddle and ball nut displacement difference
−
1 Ms + B
Velocity difference
Twist difference
Ball nut thread gain -
s
s
k
Length shortening
+
(s)
1
(s)
(s)
0(s)
f (s)
+
Ls
Backlash
Force to torque constant
−
Lead-screw angular velocity at nut end
+
k1
Lead-screw input end stress
Vf (s)
Lead-screw angular velocity at drive end
k2
+
s
x(s)
Lead-screw length change −
Saddle displacement s
Fig. 2. The distributed-lumped simulation model block diagram of the machine tool axis drive system.
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¯
F 1 ðsÞ ¼
¯ GðsÞl 2ze . 2 GðsÞl ¯ ¯ þ ðz¯ RÞ ¯ ðz¯ þ RÞe
(2.4)
The parameters for R, z; GðsÞ etc. in Eqs. (2.1)–(2.4) are shown in the Nomenclature list with the numerical values for the Arrow 500 machine. The open loop response of the model, following input changes on the motor field voltage and then on the armature voltage, can be obtained from the block diagram simulation. These traces have been employed in the construction of the transfer function matrix, presented in Eq. (2.5), which is formulated in terms of percentage input and output variations, where: (2.5)
yðsÞ ¼ GðsÞuðsÞ. In Eq. (2.5): " GðsÞ ¼
g11 ðsÞ
g12 ðsÞ
g21 ðsÞ
g22 ðsÞ
#
yðsÞ ¼ ½%nðsÞ; %fðsÞT ;
,
uðsÞ ¼ ½%nf ðsÞ; %na ðsÞT
with g11 ðsÞ7
34:5 e0:001s ; ðs þ 37:7107Þ
g12 ðsÞ7
85:5 e0:001s ; ðs þ 100:0Þ
g21 7
0:3 ðs þ 30:0Þ ðs þ 37:7107Þ
and
g22 7
0:48 ðs þ 42Þ : ðs þ 100:0Þ
The model responses following 1% step input changes on the %Vf (s), %Va(s), are shown in Figs. 3A and B, respectively. The measured results, for reference purposes, are also included for comparison with the simulated, transfer function responses, for the Arrow 500 machine. In order that simultaneous table velocity nðsÞ and lead-screw wind-up fðsÞ regulation can be applied, both field and armature voltage controls are required. Thereafter, multivariable control improving dynamic response, and disturbance suppression whilst diminishing load offset, elevating machining quality thereby, will be promoted. Implementation of the regulation strategy will also be simple. In this regard the measurement of motor speed via a tachometer or alternator could be easily provided. Equally, by using strain gauges and slip ring connections, at the motor end of the lead-screw, a voltage proportional to the lead-screw wind-up could be determined by subtracting the strain gauge, voltage outputs. Owing to the compliant elements, comprising the x-axis milling machine drive, the measured transients, shown in Figs. 3A and B, are noisy. Moreover, the results shown are for no load, open-loop operations, with the expectation that heightened excitation, owing to cutting force disturbances would occur, during production runs. In this regard milling operations would generate ‘‘noisy’’, periodic, cyclic, load excitation and internally generated dynamic perturbations. To counter this, closed-loop regulation suppressing load and externally imposed disturbances becomes necessary. The theoretical procedures following, introduce regulation strategies which are specifically designed to achieve limited output coupling, disturbance suppression and recovery whilst improving the speed of response of the system. This regulation will be accommodated with the dissipation of minimum control energy, via the provision of conventional pre and feedback compensators, as specified by Rosenbrock [5], in the final feedback configuration, guaranteeing robust, simple, cost effective implementation. 3. Closed-loop regulation strategy The following procedure employs an inner and outer loop structure for design purposes. Once this design has been established a conventional, multivariable feedback realization will be evaluated. The outer loop controller will primarily be employed to secure specified steady state, reference set point and disturbance containment characteristics. The inner loop will be utilized to attain initial, targeted, dynamic, and load disturbance recovery rates, where the system equation, including disturbance inputs is yðsÞ ¼ GðsÞuðsÞ þ dðsÞ.
(3.1)
The control law for the proposed configuration, for analysis purposes, is uðsÞ ¼ kðsÞðrðsÞ hðsÞyðsÞÞ þ PðrðsÞ FyðsÞÞ,
(3.2)
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Fig. 3. (A) Measured and transfer function responses following a 1% step change in the field voltage. (B) Measured and transfer function responses following a 1% step change in the armature voltage.
where in Eq. (3.1) there are m independent inputs, disturbances and outputs and: F ¼ Diagðf 1 ; f 2 ; . . . f m Þ;
0of j o1;
1pjpm.
With r¯ðsÞ ¼ 0 the closed-loop equation becomes: yðsÞ ¼ ðIm þ GðsÞðkðsÞihhðsÞ þ PFÞÞ1 ðGðsÞPrðsÞ þ dðsÞÞ.
(3.3)
If a steady state matrix Ss is now selected such that: yð0Þ ¼ Ss rð0Þ then from Eq (3.3), with dðsÞ null, the required pre-compensator is P ¼ ðGð0Þ1 þ kð0Þihhð0ÞÞSs ðI FSs Þ1 .
(3.4)
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To achieve ‘‘low’’ steady state interaction Ss would have elements jsi;j j5jsjj j; 1pi; jpm; iaj. Consequently, following the specification of closed-loop, low interaction and substituting for P from Eq. (3.4), results in Eq. (3.3) becoming (3.5) yðsÞ ¼ fIm þ GðsÞ½kðsÞihhðsÞ þ Gð0Þ1 þ kðsÞihhðsÞ ðIm FÞ1 Fg1 fGðsÞPrðsÞ þ dðsÞg. At low frequencies Eq. (3.5) may be approximated by yðsÞ ffi f½Im þ GðsÞkðsÞihhðsÞ½Im þ ðIm FÞ1 Fg1 fGðsÞPrðsÞ þ dðsÞg
(3.6)
If now the elements of F are f1 ¼ f2 ¼ fm ¼ f;
0of o1:0
Eq. (3.6) simplifies to yðsÞ ffi ð1 f Þ½Im þ GðsÞkðsÞihhðsÞ1 ½GðsÞP rðsÞ þ dðsÞ.
(3.7)
In Eq. (3.7) since, from Eq. (3.4): GðsÞP ¼ GðsÞ½Gð0Þ1 þ kð0Þihhð0Þ½Im F1 at low frequencies GðsÞP ffi
1 ðIm þ GðsÞkð0Þihhð0ÞÞ. ð1 f Þ
Consequently, Eq. (3.7), on approaching steady state conditions becomes: (3.8)
yðsÞ ¼ IrðsÞ þ SðsÞdðsÞ, where in Eq. (3.8) the low frequency sensitivity matrix is SðsÞ ¼ ð1 f ÞðIm þ GðsÞkðsÞihhðsÞÞ1 ;
0of o1.
Evidently, from Eq. (3.8), low steady state, output interaction, following reference input changes would be achieved. Moreover, as f is increased from zero there would be increasing, steady state, disturbance rejection with the maintenance of ‘‘adequate’’ stability margins. The design strategy to be adopted here will be to adjust the inner loop k(s) and h(s) vectors providing thereby ‘‘well behaved’’ dynamic conditions. Thereafter, with the pre-compensator P, configured to produce acceptable output coupling, the outer loop feedback gain f becomes the final design parameter enabling the achievement of desired, dynamic and disturbance suppression conditions. 4. Inner loop analysis The Laplace transformed open loop system, given by Eq. (3.1), where the system model G(s) is assumed to be an m m linear, proper or strictly proper realization, admits a factorization: GðsÞ ¼ CL ðsÞLðsÞ
AðsÞ RðsÞCR ðsÞ; dðsÞ
(4.1)
where the elements of LðsÞ; RðsÞ; CL ðsÞCR ðsÞ and
AðsÞ 2 H 1 ; s 2 C. dðsÞ
In Eq. (4.1), L(s) contain the left (row) factors, R(s) contains the right (column) factors and CL(s) and CR(s) contain the transformed, system and actuator, finite time delays, respectively, such that the m m matrices comprising (4.1), as in Whalley and Ebrahimi [6], are LðsÞ ¼ Diagðlj ðsÞ=pj ðsÞÞ; RðsÞ ¼ Diagðrj ðsÞ=qj ðsÞÞ; CR ðsÞ ¼ Diag esT j ; CL ðsÞ ¼ Diag esT k , and A(s) is a non-singular matrix of rational functions, such that det A(s)6¼0, with elements: aij ðsÞ ¼ aij sm1 þ bij sm2 þ gij ;
1pi; jpm.
(4.2)
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Since the transformed input–output-disturbance relationship is (4.3)
yðsÞ ¼ GðsÞuðsÞ þ dðsÞ and if the inner loop control law is uðsÞ ¼ kðsÞð¯rðsÞ hðsÞyðsÞÞ
(4.4)
then combining Eqs. (4.3) and (4.4) yields yðsÞ ¼ ðIm þ GðsÞkðsÞihhðsÞÞ1 ðGðsÞkðsÞ¯rðsÞ þ dðsÞÞ.
(4.5)
The finite time delays in CR(s) may be ordered with TiXTj, 1pjpm i6¼j, so that the forward path gain vector can be arranged as T (4.6) kðsÞ ¼ k1 ðsÞesðT i T 1 Þ ; k2 ðsÞesðT i T 2 Þ . . . ki ðsÞ . . . km ðsÞesðT i T m Þ and if T l XT k ;
1pkpm
lak
T hðsÞ ¼ h1 ðsÞesðT l T 1 Þ ; h2 ðsÞesðT l T 2 Þ ; . . . hl ðsÞ . . . hm ðsÞesðT l T m Þ ,
(4.7)
then if kj ðsÞ ¼ kj fj ðsÞ and
hj ðsÞ ¼ hj wj ðsÞ;
1pjpm.
where fj ðsÞ and wj ðsÞ are proper or strictly proper, stable, realizable, minimum phase, functions then they may be selected such that the determinant of Eq. (4.5), as shown in Whalley [7] is AðsÞ AðsÞ sðT i þT¯ l Þ sðT i þT¯ l Þ kihh ¼ 1 þ e k , (4.8) det Im þ e nðsÞ nðsÞ h dðsÞ dðsÞ where in Eq. (4.8): k ¼ ðk1 ; k2 . . . km ÞT ,
(4.9)
h ¼ ðh1 ; h2 . . . hm Þ
(4.10)
dðsÞ ¼ sk þ a1 sk1 þ a0 and degðnðsÞai;j ðsÞÞok;
1pi; jpm.
The inner product in Eq. (4.8) may be 2 g11 g12 6 . ..
6 . . m1 6 . hAðsÞk ¼ 1; s; . . . s 6 6 b11 b12 4 a11 a12
(4.11) expressed as 32 3 . . . gmm k 1 h1 7 .. 76 k2 h1 7 7 . 76 7 76 6 .. 7. 7 . . . bmm 54 . 5 k m hm . . . amm
(4.12)
If in Eq. (4.12) the gain ratios are k 2 ¼ n1 k 1 ;
k3 ¼ n2 k1 ; . . . km ¼ nm1 k1 ;
k1 a0
(4.13)
and:
hAðsÞk ¼ bðsÞ,
(4.14)
then Eq. (4.14) implies that k1 ½Qh ¼ ðbm1 ; bm2 ; . . . ; b0 ÞT ,
(4.15)
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where in Eq. (4.15): 2 6 g11 þ g12 n1 þ g1m nm1 6 .. 6 . 6 Q¼6 6 6 b11 þ b12 n1 þ b1m nm1 4 a11 þ a12 n1 þ a1m nm1
.. . g21 þ g22 n1 þ g2m nm1 .. .. . . .. . b21 þ b22 n1 þ b2m nm1 .. . a21 þ a22 n1 þ a2m nm1
1843
3 .. . gm1 þ gm2 n1 þ gmm nm1 7 7 .. .. 7 . . 7 7 .. 7 . bm1 þ bm2 n1 þ amm nm1 7 5 .. . am1 þ am2 n1 þ amm nm1
and bj ; 0pjpm 1; are the coefficients of b(s), given in Eq. (4.14). Providing the very weak constraint that n1 ; n2 . . . nm1 can be selected, in Eq. (4.15), so that the resulting Q matrix is invertible, then a unique solution for ðh1 ; h2 ; . . . hm Þk1 exists. Following the choice of a suitable bðsÞ function and the gain ratios n1 ; n2 nm1 , the closed-loop dynamics arising from Eq. (4.5) may be easily computed. Upon satisfying Eq. (4.15) the measurement vector h can be evaluated, once an arbitrary value for k1, k16¼0, has been decided upon. It is apparent from Eq. (4.15) that there will be many values of nj ; 1pjpm 1 which could be selected enabling the inversion of Q. However, choosing these gain ratios arbitrarily, affects the regulation efficiency, the robustness of the feedback configuration to parameter changes and the disturbance rejection properties of the system. To avoid this a systematic, scientific optimization procedure will be employed, following Section 5. 5. Disturbance rejection analysis In practice some form of approximate integration over a limited frequency range, such as phase lag compensation, is usually employed to effect disturbance recovery. To confine output excursions, to input disturbances, the maximum proportional feedback gain, commensurate with acceptable transient conditions and modelling/parameter uncertainties, may be employed. In the following, bounds on the disturbance response, arising from changes in the sensitivity matrix, are determined. This enables the formulation of a functional, J, from which variations in the system’s closed-loop disturbance attenuation can be established. Gain ratios for which there are rapid changes in the performance index J and hence in the disturbance containment capacity of the system, also become apparent. These values indicate a lack of robustness to modelling variations and should be avoided. In many process and military systems the only input signals which are continuously perturbed arise from changes in d(s), so that with the outer loop set points rðsÞ ¼ 0, Eq. (3.8) becomes: (5.1)
yðsÞ ¼ SðsÞdðsÞ, where SðsÞ ¼ ðIm þ GðsÞðkðsÞihhðsÞ þ PFÞÞ1 .
The sensitivity function S(s), in Eq. (5.1), should be such that changes in dðsÞ result in acceptable changes in y(s). These output excursions arise from relatively low frequency variations in dðsÞ, as discussed by Skogestad and Postlethwaite [8]. Moreover, since for physically realisable systems, G(s) must be strictly proper, the closed-loop system provides no attenuation to high frequency disturbances, as SðsÞ ! Im . Consequently, the adjustment of the steady state system, disturbance suppression properties, by increasing f in Eq. (3.8), would ensure enhanced attenuation following low frequency perturbations. Moreover, for increasing values of frequency the singular values of S(io) should be smooth, differentiable and almost monotonic curves, providing a prudent policy of regulation is pursued. The disturbance attenuation, secured by the system, lies between the lower and upper singular values of the S(s) matrix, as shown by Zames [9]. Specifically,
yðsÞ
2 lðSðsÞÞp (5.2)
dðsÞ plðSðsÞÞ, 2
where in Eq. (5.2), lðSðsÞÞ and lðSðsÞÞ are the smallest and largest, singular value amplitudes of S(s), respectively, and kk2 denotes the Euclidean norm, s ¼ io; 0pop1 of y(s) and dðsÞ, respectively. It is also apparent from Eq. (5.2) that providing the upper bound
lðSðsÞÞp3dB, s ¼ io; 0pop1, the system will be stable with a maximum ‘‘high’’ frequency resonant peak p1:4; dðsÞ 2 a0, with the closed-loop outputs being substantially decoupled.
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The singular values of the low frequency sensitivity matrix, given in Eq. (3.8), can be computed in the frequency domain, for f ¼ 0, from det½Im l ðI þ GðsÞðkðsÞihhðsÞÞÞ1 s¼io ¼ 0. The steady state singular values may be estimated from: det½Im l ðI þ Gð0Þðkð0Þihhð0ÞÞÞ1 ¼ 0,
(5.3)
since Gð0Þkð0Þihhð0Þ has singular values of g1 ¼ g2 ¼ gm1 ¼ 0
and
gm ¼
m X
k j hj ,
j¼1
where (5.4)
k ¼ Gð0Þkð0Þ and kj is the jth element of k. Theorem 1. The singular value amplitudes li ; 1pipm, of Eq. (5.3) are: l1 ¼ l2 ; ¼ lm1 ¼ 1
and
lm ¼
1þ
1 Pm
j¼1 kj hj ð0Þ
.
(5.5)
The proof is given in Whalley and Ebrahimi [10]. As shown by Eq. (3.8), SðsÞ ¼ ð1 f ÞðI þ GðsÞkðsÞihhðsÞÞ1 so that the low frequency singular value amplitudes of S(s) are reduced to lj where 1pjpm; f o1.
lj ¼ ð1 f Þlj ;
Consequently, the bound on the steady state attenuation properties of the system becomes
yð0Þ
2 lðSð0ÞÞp
dð0Þ plðSð0ÞÞ.
(5.6)
(5.7)
2
The estimate from Eq. (5.7) is conservative. An accurate measure of the low frequency disturbance attenuation properties of the system can be easily computed from the direct evaluation of Eq. (5.1). In this regard if: sðSð0ÞÞ ¼ largest element of Sð0Þ, sðSð0ÞÞ ¼ smallest element of Sð0Þ, then
yð0Þ
2 sðSð0ÞÞp
dð0Þ psðSð0ÞÞ 2
provides an improved, low frequency, attenuation bound. The employment of an absolute, minimum effort controller, whilst attaining a particular inner closed-loop pole pattern, would not, in general achieve specified, steady state disturbance recovery conditions. However, the disturbance recovery transient would of course, be determined by this pole configuration. To achieve the steady state disturbance recovery effect required, the outer loop feedback gain f could be adjusted, as indicated by Eq. (5.6). This would perturb the system’s transient behaviour and recognition of this should be exercised by aiming for safe, modest response improvements arising from the inner loop dynamics. Thereafter, the control effort required would be the relative minimum effort, commensurate with the achievement of particular steady state, disturbance suppression conditions. As shown by Eq. (3.6), increasing f from zero is instrumental in increasing k(s)S /h(s) to kðsÞihhðsÞ=ð1 f Þ. This is equivalent to increasing b0 of Eq. (4.15) to b0/(1f) elongating and elevating thereby, the performance index J curve, so that J min is moved upwards whilst retaining the original optimum, minimum effort gain ratio value of n. The obvious conclusion from this is that in the absence of all other inputs the control effort expended is devoted entirely to disturbance recovery. Consequently the performance index J, being proportional to the control effort, is also a direct measure of this condition.
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In aiming for this minimum J, for a given pole configuration, the disturbance recovery transient is executed with maximum energy efficiency. By closing the outer loop, upon increasing f from zero, the closed-loop disturbance suppression would be enhanced, whilst perturbing the closed-loop pole pattern. However, providing the S(io) curves are not significantly altered the dominant poles and optimum performance, for low frequency disturbances, prevails. As Skogestad and Postlethwaite [8] observe, any intermediate frequency amplification indicated by the singular values of S(io) should be restricted to o3dB’s. This is equivalent to limiting the closed-loop dynamic magnification, when designing single input–output systems, by confining M (peak) to 1.4 (scalar). However, as stated earlier, this is a conservative measure and the 3 dB limit can normally be transgressed, without immediately affecting the dynamic performance. 6. Optimization Now that a route for designing closed-loop systems, using the transfer function matrix and output measurements has been established, via the theory outlined earlier, the possibility of optimizing this process can be considered. An indication that the freedom exists to do this arises from the arbitrary choice of n1, n2 etc., for the gain ratios. Detecting the absolute minimum control effort required for disturbance suppression under inner loop, closed-loop conditions, with the constraint that the controller model generates a particular polynomial, would provide a useful benchmark result. This polynomial influences the migration pattern of the closed-loop poles so that control effort minimization, whilst attaining the desired system response to input and disturbance changes, could be achieved simultaneously. The controller equation for a system having m inputs and m outputs is given by Eq. (4.4). Consequently, the control effort, at time t, with zero set point changes is ðjk1 h1 j þ jk2 h1 j :jkm h1 jÞjy1 ðtÞj þ ðjk1 h2 j þ jkm h2 jÞjy2 ðtÞj þ ðjk1 hm j þ jk2 hm j þ jkm hm jÞjym ðtÞj Hence, the control energy costs, from zero to Tm, are proportional to ! Z t¼T m X m m X 2 2 2 EðtÞ ¼ ki hj yj ðtÞ dt. t¼0
i¼1
(6.1)
j¼1
For arbitrary changes in the output vector y(t), following arbitrary disturbance changes, minimizing: J¼
m X
h2i
i¼1
m X
k2j
(6.2)
i¼1
would minimize the control energy given by expression (6.1). If the relationships: k3 ¼ n2 k1 ; . . . km ¼ nm1 k1 ; k1 a0
k 2 ¼ n1 k 1 ;
are proposed, then Eq. (6.2) can be written as J ¼ ðk1 Þ2 ð1 þ n21 þ n22 . . . n2m1 Þðh21 þ h22 . . . h2m Þ h21
h22
(6.3)
. . . h2m
and þ ¼ hh; hi. The closed-loop determinant given by Eq. (4.8) with the inner product equated to bðsÞ, as in Eq. (4.14), enables the measurement gains to be determined from Eq. (4.15), where 1 h ¼ k1 1 Q b.
(6.4)
Upon substituting for h from Eq. (6.4), Eq. (6.3) becomes J ¼ ð1 þ n21 þ n22 þ n2m1 ÞbT ðQ1 ÞT Q1 b.
(6.5)
To find the minimum value for J, assuming, for example, that m ¼ 3 gives J ¼ ð1 þ n21 þ n22 ÞbT ðQ1 ÞT Q1 b, where J is minimized when qJ ¼ 0; qn1
qJ ¼0 qn2
and
2 2 q2 J q2 J qJ 40, 2 2 qn1 qn2 qn1 qn2
if ðq2 J=qn21 Þ40: For m43 a numerical minimisation routine would have to be employed to establish the values of n1, n2 y nm1 which minimize J. There are many procedures available for this task, as indicated by Bunday [11], with rapid convergence and high accuracy characteristics. This completes the optimization analysis.
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7. System stability and implementation The input–output relationship for the complete closed-loop system is given by Eq. (3.5). Consequently, the input–output stability condition is dependent on the denominator of this equation. If, for the sake of simplicity, the outer loop feedback gain matrix F is given by F ¼ Diagðf 1 ; f 2 ; . . . f m Þ and if, once again: f1 ¼ f2 ¼ ¼ fm ¼ f;
0of o1;
then the denominator of Eq. (3.5) may be calculated from: kðsÞihhðsÞ Gð0Þ1 f þ det Im þ GðsÞ ð1 f Þ ð1 f Þ
(7.1)
from Eq. (7.1) it is evident that the elements of the feedback–compensator matrix: kðsÞihhðsÞ Gð0Þ1 f þ ð1 f Þ ð1 f Þ become infinite as f ! 1. In practice this would always result in closed-loop system instability and the restriction 0ofo1 applies. For values of f p0:5 the inner loop feedback gain is amplified whilst the effect of the outer loop, arising from G(0)1f/ (1f), is attenuated. Once 0:5of o1:0 both the inner and outer loop gains f/(1f) and 1/(If) become increasingly large. Essentially, amplifying k(s)S/h(s) by 1/(1f), fo1, is equivalent, as stated earlier, to multiplying b0 in Eq. (4.15) by this amount. Consequently, the value of J achieved initially, from the optimization process, is increased further by the use of the outer loop feedback elevating it to J/(1f), whilst the optimum, least effort, gain ratio remains unchanged at n with k becoming, for m ¼ 2, (1, n)T/(1f), for stability assessment purposes. Owing to these factors the system’s disturbance suppression properties are enhanced. Hence, by setting the inner loop frequency response or characteristic equation, such that the amplification of k(s)S/h(s) to k(s)S/h(s)/(1f) can be safely accommodated, dynamic recovery and steady state disturbance rejection can be improved, using the outer loop feedback gain f. Once this exercise has been completed a conventional closed-loop, multivariable feedback structure can be formulated, for implementation purposes. If the forward and feedback path matrices are K(s) and H(s), respectively, then the closedloop relationship becomes: yðsÞ ¼ ðIm þ GðsÞKðsÞHðsÞÞ1 ðGðsÞKðsÞrðsÞ þ dðsÞÞ.
(7.2)
Comparing Eqs. (7.2) and (3.3) it is clear that KðsÞ ¼ P
(7.3)
HðsÞ ¼ P-1 kðsÞihhðsÞ þ F
(7.4)
and
enabling the adoption of the simple feedback configuration, given in Dutton et al. [12], for the least energy, optimum, regulator. 8. Application study To apply the design procedures, the machine tool model presented earlier will be considered, demonstrating effectiveness thereby. For this system model, for the Arrow 500 milling machine: yðsÞ ¼ GðsÞuðsÞ þ dðsÞ; where in Eq. (8.1), from Eq. (2.5): yðsÞ ¼ ðy1 ðsÞ; y2 ðsÞÞT are the Laplace transformed output signals: uðsÞ ¼ ðu1 ðsÞ; u2 ðsÞÞT
(8.1)
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are the Laplace transformed, input signals and: " # g11 ðsÞ g12 ðsÞ GðsÞ ¼ , g21 ðsÞ g22 ðsÞ where gi,j(s), 1pi, jp2, are the transfer functions shown in Eqs. (2.5) and (d1 ðsÞ; d2 ðsÞ)T are the disturbance signals. In this application, the closed-loop system is required to reduce the steady state interaction to a maximum of 25%. Moreover, the velocity response time should be such that steady state conditions can be achieved in approximately 0.1 s, without overshoot problems. Minimization of the steady state output variations to unit step changes, on d1 ðtÞ and d2 ðtÞ, using proportional-passive network compensation alone should be investigated. Enhancement to achieve sðSð0ÞÞp0:4 should be attained by increasing the outer loop feedback with quiescence and disturbance recovery in less than 0.1 s. 9. Inner loop design In order to shape the frequency response characteristics of the inner loop suitable compensators must be selected. First, the system model given in Eq. (8.1) can be arranged as AðsÞ GðsÞ ¼ CL ðsÞLðsÞ RðsÞCR ðsÞ, (9.1) dðsÞ where in Eq. (9.1) a structure similar to Eq. (4.1) can be achieved with dðsÞ ¼ ðs þ 37:7107Þ CL ðsÞ ¼ Diagðe0:001s ; 1Þ; " AðsÞ ¼
RðsÞ ¼ Diag 1; ðs þ 37:7107Þ=ðs þ 100:0Þ ,
34:5
85:5
0:3ðs þ 30:0Þ
0:48ðs þ 42Þ
#
and LðsÞ ¼ CR ðsÞ ¼ I2 . In accordance with Eq. (4.6), for this study kðsÞ ¼ KðsÞðk1 ; k2 ÞT ,
(9.2)
where in Eq. (9.2): KðsÞ ¼ Diagð1; ðs þ 100:0Þ=ðs þ 37:7107ÞÞ and hðsÞ ¼ ðh1 ; h2 ÞHðsÞ,
(9.3)
where in Eq. (9.3): HðsÞ ¼ Diagð1:0; e0:001s Þ. Consequently the equivalent of Eq. (4.12) here is " # 34:5 85:5 AðsÞ 0:3ðs þ 30:0Þ 0:48ðs þ 42Þ k ¼h ke0:001s , hðsÞ dðsÞ ðs þ 37:7107Þ
(9.4)
where Eq. (9.4) has the form of the inner product of Eq. (4.14), where h ¼ ðh1 ; h2 Þ and k ¼ ðk1 ; k2 ÞT . Then an appropriate function b(s) may be selected for the machine tool model. 10. Determination of the controller generated zero If Eq. (9.4) is now considered, then as in Eq. (4.14), but including the denominator d(s): AðsÞ bðsÞ 0:001s hðsÞ k ¼ e . dðsÞ dðsÞ
(10.1)
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Hence, Eq. (10.1) becomes the characteristic equation: 1 ¼ bðsÞe0:001s dðsÞ
(10.2)
and a suitable b(s) function could be selected from consideration of the root locus plot of Eq. (10.2) neglecting the small finite time delay. It is evident that if bðsÞ ¼ b0 ðs þ 5000Þ
(10.3)
and with d(s) from Eq. (8.5) the root locus would be in respect of 1 ¼
ðs þ 5000Þb0 ðs þ 37:7107Þ
and with b0 ¼ 0:02.
(10.4)
The characteristic equation would be 1 ¼
ð0:02s þ 100Þe0:001s , ðs þ 37:7107Þ
with adequate stability margins. The choice of a ‘‘small’’ value for b0 also improves the controller’s robustness, by restricting the control energy dissipation and hence performance index J, as shown in Section 11. 11. Optimization and simulation Following the attainment of the modest improvements in the system dynamics, from the closure of the inner loop, the corresponding minimum effort performance index, since m ¼ 2, becomes: J ¼ ð1 þ n2 ÞbT ðQ1 ÞT Q1 b.
(11.1)
From A(s), given by Eq. (9.1), the Q matrix for this system can now be formulated as " # 37:7107 þ 85:5n 9 þ 20:16n Q¼ 0 0:3 þ 0:48n and from Eqs. (10.3) and (10.4): 5000 b¼ b0 ; b0 ¼ 0:02: 1
(11.3)
Evaluation, using Eqs. (11.2) and (11.3) results in Eq. (11.1) becoming: J¼
(11.2)
2268:3795n4 þ 2841:033n3 þ 3158:088n2 þ 2841:033n þ 889:7085 ð41:04n2 þ 42:21n þ 10:35Þ2
Fig. 4. Performance index J against gain ratio.
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and dJ 74900:6n4 45384:95n3 261573:9n2 200602:13n 45704:5 ¼ . 3 dn ð41:04n2 þ 42:21n þ 10:35Þ
(11.4)
Equating dJ=dn, from Eq. (11.4), to zero: reveals that the extremum values of J arise when: n ¼ 2:4849; 0:59905
and
0:63994 0:019959i:
Fig. 5. Inner loop response following a unit input change on r¯ðtÞ.
f δ1(s) 1
− r1(s)
2
+
20 s+1
−
s + 37.7107
+
P22
+
+
y1(s)
+
0.3(s + 30) s + 37.7101
r(s) + −
P12 r2(s)
e-0.001s
+
P21
+
34.5 +
P11
0.04s + 1
+
+
85.5 s + 100
+
2.4849 (s+100)
+
e-0.001s
+
+ 0.48(s + 42) s + 100
s + 37.7107
+ + δ2(s)
+
0.40172
+ 0.013398
f Fig. 6. Closed-loop system block diagram.
y2(s)
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The graph of the performance index J against gain ratio n, presented in Fig. 4, shows that the absolute minimum value of J is attained, for bðsÞ ¼ 0:02s þ 100, when n ¼ 2:4849, with the constraint that the inner, closed-loop pole is located at, s ¼ 135:011. The feedback measurement gains can now be computed, as in Eq. (4.15), with Q and b from Eqs. (11.2) and (11.3) yielding, with b0 ¼ 0:02 and n ¼ 2:4849. ðh1 ; h2 Þ ¼ ð0:40172; 0:013398Þ,
(11.5)
where k1, in Eq. (11.5) can be selected arbitrarily, initially as unity. This value can be employed subsequently to ‘‘balance’’ the levels of disturbance suppression achieved in each of the output channels, if so desired. The responses from the inner loop following a unit change on rðtÞare shown in Fig. 5. These outputs exhibit a faster transient response reaction time though substantial steady state output interaction is evident. All that remains now is the computation, in accordance with Eq. (3.4), of the pre-compensator P. If here, the F matrix values selected are F ¼ Diagð0:25; 0:25Þ; Diagð0:50; 0:50Þ and Diagð0:75; 0:75Þ
(11.6)
and since steady state interaction between outputs must be limited to o725% then: 0:5 0:0125 Ss ¼ 0:1 0:05 could be specified. This would restrict the amplitude of the pre-compensator elements improving the tolerance of the feedback system to non-linear variations in the actual machine tool system application. Inserting these values of G(0) Ss, F, k(0) and h(0) into Eq. (3.4) results in: 0:66873 2:13 0:60079 2:0805 0:62994 2:1051 P¼ ; and (11.7) 3:0831 2:1595 3:5239 2:1745 4:1404 2:1853 δ1(s) r1(s)
+
2 0.04s + 1
K11
+
34.5 (s + 37.7107)
+
−
s+1
+ K22
+
y1(s)
+
(s + 37.7101)
85.5
K12
20
+
0.3(s) + 30
K21
r2(s)
e-0.001s
(s + 100)
+
+
0.48(s +42) (s +100)
−
e-0.001s
+
+ +
δ2(s) + +
H11(s)
H21(s)
H12(s) +
+
H22(s)
Fig. 7. Conventional closed-loop system block diagram.
y2(s)
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y1(t)
f = 0.25
1
1851
f = 0.5
Output
0.8
f = 0.75
0.6 0.4 f = 0.25 y2(t)
f = 0.5
0.2
f = 0.75
0 0
0.05
0.1
0.15
(A)
0.2 0.25 0.3 Time (sec)
0.35
0.4
0.45
0.5
1.2 f = 0.25, f = 0.5 and f = 0.75
1
y2(t)
Output
0.8 0.6 0.4 y1(t) f = 0.75
0.2
f = 0.5
0
f = 0.25
-0.2 0 (B)
1
2
3
4
5 6 Time (sec)
7
8
9
10
Fig. 8. (A) Closed-loop output response following a unit input step change on r1 ðtÞ. (B) Closed-loop output response following a unit input step change on r2 ðtÞ.
for the selected F matrices, respectively, fully defining the feedback configuration. This results in the block diagram for the close loop system shown in Fig. 6. Implementation in terms of conventional forward and feedback path compensators may now be proposed. In this case, in accordance with Eqs. (7.3) and (7.4), for the F matrices from Eq. (11.6) with P from Eq. (11.7): K¼P and with k ¼ [1,2.4849]T and h(s) from and (11.5) yielding, respectively, 2 0:57541ðsþ81:65Þ 3 0:019191ðsþ81:65Þeð0:001sÞ þ 0:25 ðsþ37:7107Þ ðsþ37:7107Þ 5; HðsÞ ¼ 4 0:35925ðsþ58:03Þ 0:011982ðsþ58:03Þeð0:001sÞ þ 0:25 ðsþ37:7107Þ ðsþ37:7107Þ 2 0:49185ðsþ81:71Þ 3 0:016404ðsþ81:71Þeð0:001sÞ ðsþ37:7107Þ þ 0:50 ðsþ37:7107Þ 5; HðsÞ ¼ 4 0:33802ðsþ56:87Þ 0:011274ðsþ56:87Þeð0:001sÞ þ 0:50 ðsþ37:7107Þ ðsþ37:7107Þ 2 0:40829ðsþ81:8Þ 3 0:013617ðsþ81:8Þeð0:001sÞ ðsþ37:7107Þ þ 0:75 ðsþ37:7107Þ 5; HðsÞ ¼ 4 0:31679ðsþ55:55Þ 0:010565ðsþ55:55Þeð0:001sÞ þ 0:75 ðsþ37:7107Þ ðsþ37:7107Þ
(11.8)
(11.9)
where in Eq. (11.9) the small finite time delay can be ignored in view of the robustness of the regulation strategy employed.
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1 0.8 0.6 y1(t) , f = 0.25
Output
0.4
y1(t) , f = 0.5
0.2 0
y1(t) , f = 0.75
y2(t) , f = 0.75
-0.2
y2(t) , f = 0.5
-0.4
y2(t) , f = 0.25
-0.6 -0.8 -1 0
0.05
0.1
(A)
1.2
0.2
0.25
0.2
0.25
y2(t) , f = 0.25 y2(t) , f = 0.5
1 0.8 Output
0.15 Time (sec)
y2(t) , f = 0.75
0.6 0.4 y1(t) , f = 0.75
0.2
y1(t) , f = 0.5
0 -0.2 (B)
y1(t) , f = 0.25
0
0.05
0.1 0.15 Time (sec)
Fig. 9. (A) Closed-loop output response following a unit input step change on d1 ðtÞ. (B) Closed-loop output response following a unit input step change on d2 ðtÞ.
As Eqs. (11.8) and (11.9) indicate the forward and feedback compensators for this system are full rank matrices. This enables the minimal block diagram configuration, shown in Fig. 7, to be constructed, for implementation purposes where the elements of K and H(s) are Ki,j and Hi,j(s), 1pi; jp2, respectively. From the simulation of this, the output responses following input step changes of unity on r1(t) and r2(t), respectively, can be computed, as shown in Fig. 8A and B. These traces, depict acceptable response variations with no overshoot problems on the principal outputs with steady state interaction restricted to the limits specified. Thereafter, the disturbances offset graphs for inputs steps on d1 ðtÞ and then d2 ðtÞ are presented in Figs. 9A and B. The important table velocity, disturbance rejection characteristics of Fig. 9A, provides excellent recovery transients limiting the loading problem to 720%. 12. Conclusion The regulation issue addressed in this contribution was focused on the control of a machine tool, x-axis, traverse drive. Specifically, the simultaneous control of the work-piece table velocity, whilst confining the lead-screw twist angle was the stated objective. With the achievement of these specification requirements it was shown that: (i) (ii) (iii) (iv) (v) (vi)
Machining variations originating from lead-screw effective torsion-tension length changes would be reduced. Disturbances arising from arbitrary cutting forces would be diminished. Improved transient response and profile following would be available Machining quality, repeatability and accuracy would be enhanced. A simple, output feedback regulation strategy and structure alone, would be required. Implementation costs would be minimized.
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(vii) (viii) (ix) (x)
1853
Rapid cutting load offset recovery would be guaranteed. A robust, low gain, passive feedback configuration would be derived. Low steady state output cross-coupling would be achieved. Minimum maintenance, refit and operating costs and maximum machine tool availability would be attained.
In the initial modelling study both the measured and analytical results were employed in the determination of the axis drive, milling machine, multivariable system model, for control system design purposes. Owing to the robustness of the minimum effort, minimum energy design procedure a simple linear approximation adequately replicated the open-loop, measured response characteristics, for the initial viability study. Reflecting this, the feedback controller, when derived from the inner and outer loop regulators, was uncomplicated comprising little more than a pre compensator of constants. A feedback realization requiring only phase retard elements was also defined. With this optimum regulator the disturbance, suppression properties of the system could also be advanced by increasing f, the outer-loop feedback gain 0ofo1.0. Consequently, the system response for three values of f was evaluated allowing the establishment of the dynamic performance following disturbances and input, set point variations. In accordance with the implementation requirements the pre and feedback compensators for the system were computed, as indicated in Section 11, for f ¼ 0:25; 0:5 and 0:75. Owing to the robustness of the regulation strategy adopted, the feedback controller, finite time delays, were neglected, without significantly affecting the overall system performance, the final configuration of the closed-loop system comprising the conventional structure, shown in Fig. 7. The effectiveness of the controller is evident from the closed-loop response transients from Figs. 8A and B. These show well behaved dynamic characteristics with quiescence being achieved in approximately 0.1 s following input reference changes on r1(t). Equally, twist angle control can be imposed following changes on r2 ðtÞ, as shown in Fig. 9B, although this input would normally remain at zero. The disturbance suppression properties of the system are also pleasing. Fig. 9A shows there is increasing offset rejection as f is increased 0.25ofo0.75. The disturbance recovery times for y1(t) and y2(t) are also rapid and well within the specified limit, for table velocity disturbances d1 ðtÞ. Under normal operating conditions disturbances could not directly affect the lead-screw twist d2 ðtÞ. Consequently, this loading effect would normally be maintained at zero. The response transients shown in Fig. 9B are included for completeness. Equally, the application of step velocity, table disturbances, would not arise in practice as the cutting disturbances would impose additional load torque increases on Tb(s). Therefore, the severity of these loadings would be attenuated by the system elements, as shown in Fig. 2. As a consequence the disturbance response characteristics are in respect of standard benchmark validations, rather than operational conditions. References [1] I.K. Landerborough, A.G. Ulsay, Dynamic modelling and control of a milling process, ASME Journal on Industry 110 (1988) 367–375. [2] R. Whalley, H. Bartlett, M. Ebrahimi, Analytical solution of distributed-lumped parameter network models, Proceedings of IMechE, Part I 212 (13) (1997) 203–218. [3] R. Whalley, M. Ebrahimi, Z. Jamil, The torsional response of spatially dispensed transmission systems, Proceedings of IMechE, Part C 214 (2005) 100–120. [4] R. Whalley, M. Ebrahimi, A. Abdul-Ameer, Hybrid modelling of machine tool axis drives, International Journal of Machine Tools and Manufacture 1646 (2005) 1–17. [5] H.H. Rosenbrock, Computer Aided Control System Design, Academic Press, London, 1974. [6] R. Whalley, M. Ebrahimi, Control of a lamination process, Proceedings of IMechE, Part E 219 (2005) 69–81. [7] R. Whalley, The computation of 2D transfer functions, Proceedings of IMechE, Part I 205 (I1) (1991) 59–67. [8] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control Analysis and Design, Wiley, New York, 1996. [9] G. Zames, Feedback and optimal sensitivity: model reference transformation, multiplicative semi-norms and approximate inverses, IEEE Transactions on Automatic Control 26 (2) (1981) 301–320. [10] R. Whalley, M. Ebrahimi, Closed-loop system disturbance recovery, Proceedings of IMechE, Part C 217 (2003) 631–649. [11] B. Bunday, Basic Optimisation Methods, Edward Arnold, London, 1984. [12] K. Dutton, S. Thompson, R. Burraclough, The Art of Control Engineering, Addison Wesley, Harlow, 1997.
International Journal of Machine Tools & Manufacture 46 (2006) 1835–1853 www.elsevier.com/locate/ijmactool
Optimum, machine tool axis traverse regulation R. Whalley, M. Ebrahimi, A. Abdul-Ameer, S. El-Shalabi School of Engineering, Design and Technology, University of Bradford, UK Received 31 March 2005; received in revised form 11 November 2005; accepted 15 November 2005 Available online 30 January 2006
Abstract A hybrid, distributed parameter model for a milling machine axis drive system is derived from measured results. Spatial variation effects arising from the machine system geometrical changes are incorporated enabling multivariable regulation via feedback control. Design procedures enabling the realization of an optimum, minimum energy, multivariable regulator for this system are presented. Table velocity and lead-screw twist interaction are controlled via armature and field voltage variations, restricting the effect of cutting loads and lead-screw length changes, respectively. The feedback compensators for this system are evaluated and the open and closed-loop model response transients, following reference input variations, are presented. Disturbance suppression and the surface finish capabilities of the closed-loop, minimum energy, system, were investigated. The performance advantages promoted by multivariable minimum effort control are emphasized with the confinement of lead-screw wind-up and cutting load disturbance effects whilst limiting output cross coupling with improving transient behaviour and steady state accuracy. r 2005 Elsevier Ltd. All rights reserved. Keywords: Optimum; Energy; Milling; Machine; Regulative
1. Introduction In order to improve the disturbance rejection capabilities, speed of response and steady state accuracy of machine tools, automatic, feedback control is usually incorporated. This provision also enhances the repeatability of these systems countering wear, environmental and load variations whilst improving surface finish, tolerance limits, integrity and manufacturing efficiency, as in Landerborough and Ulsay [1]. The effectiveness of the control strategy imposed depends upon the fidelity of the machine tool model employed. This realization must replicate machining performance, following input changes, cutting force disturbances, shock and internally generated vibrations. With the advent of broadband, multivariable, distributed parameter modelling methods, as in Whalley et al. [2], the opportunity to include well known dynamic effects, which have been difficult to incorporate hitherto, arises. These perturbations are promoted by spatial dispersion, lead-screw wind-up, interactive torsional and longitudinal loading, deformation and deflection, as shown in Whalley et al. [3]. These detailed realizations enable the formulation of elevated controller design techniques, improving efficiency, accuracy and effectiveness, thereby. Axis regulation strategies, focused on the suppression of disturbances, arising from geometrical, machine tool variations, as well as simultaneously controlling transverse velocity, are now invited. Thereafter, many of the mandatory, precision manufacturing restrictions on temperature, material variations, vibration isolation etc. may be relaxed without compromising machining tolerances, performance or surface finish quality. Equally, regulation robustness, stability, consistency of operation, irrespective of machine configuration, together with the restriction of the effects of arbitrary cutting forces, significantly advantage manufacturing competitiveness. Corresponding author.
E-mail address: [email protected] (M. Ebrahimi). 0890-6955/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.11.011
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Nomenclature A(s) numerator of G(s), matrix ai;j ðsÞ elements of A(s) 1pi, jpm, function ai;j ; bi;j . . . gi;j , coefficients of ai;j ðsÞ, scalar b(s) polynomial function b0 ; b1 bm1 coefficients of bðsÞ, scalar d(s) denominator of G(s), function F outer loop feedback array, matrix f ; f 1 ; f 2 f m outer loop feedback gains, scalar G(s) transfer function array, matrix ¯ GðsÞ system matrix, matrix P(s) pre-compensator array, matrix h feedback path gain, vector h(s) feedback path function, vector Im identity array (m m), matrix J performance index, functional kS/h outer product of k and h, matrix /k,hS inner product of k and h, scalar k forward path gain, vector k(s) forward path function, vector L(s) left (row) factors, matrix n; n1 ; n2 . . . nm1 gain ratios, scalar Q coefficient array, matrix r¯ðsÞ transformed reference input, scalar r(s) transformed reference input, vector R(s) right (column) factors, matrix Ss steady state array, matrix S(s) sensitivity array, matrix u(s) transformed input, vector y(s) transformed output, vector CL(s), CR(s) finite time array, matrix dðsÞ transformed disturbance input, vector T 0 ðsÞ input torque on the lead screw, scalar o1 ðsÞ output lead screw angular velocity, scalar o0 ðsÞ motor speed and input drive end angular velocity of the lead screw, scalar z characteristic impedance of the lead-screw (torsion) ¼ (1.9864 m/s), scalar z¯ characteristic impedance of the lead-screw (tension) ¼ (3.949 107 m/s), scalar GðsÞ propagation function of lead-screw (torsion) ¼ (3.1224 104 s), function GðsÞ propagation function of lead-screw (tension) ¼ (1.9748 104 s), function R termination frictional coefficient (torsion) ¼ (14.65 103 Nm/(rad/s)), scalar ¯ R termination frictional coefficient (tension) ¼ (5.4 106 N/(m/s)), scalar R0 ðsÞ lead-screw input end transfer function (torsion), function F 0 ðsÞ lead-screw input end transfer function (tension), function R1 ðsÞ lead-screw output end transfer function (torsion), function F 1 ðsÞ lead-screw output end transfer function (tension), function s0 ðsÞ input end stress in lead screw, function Fx(s) cutting force in x direction, function f ðsÞ force change on ball nut, function B saddle viscous friction coefficient (1 N/m/s), scalar M saddle mass (307 kg), scalar V a ðsÞ armature voltage, function V f ðsÞ Field voltage, function Tb motor load torque ¼ mf ðsÞ , function E Young’s modulus (200 109 N/m2), scalar fðsÞ angular change in lead-screw at ball-nut end, scalar
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Kp ball nut lateral stiffness (985 106 N/m), scalar k2 back emf constant (0.7639 volt/(rad /s) scalar m ball nut rolling resistance ¼ fiction coefficient ball nut radius ¼ (26 104 m), scalar kL lead-screw and ball nut series stiffness (1.3586 108 N/m), scalar kd angular to linear movement constant (25 104 m/rad), scalar dðsÞ lead-screw length change, function eðsÞ lead-screw strain due to twisting, function vðsÞ saddle velocity, function Lf D.C. motor field inductance, scalar Rf D.C. motor field resistance, scalar t current limiter time constant, scalar k3 armature current-torque constant, scalar Ra D.C. motor armature resistance, scalar fj ðsÞ; wj ðsÞ compensator models, 1pjpm, function
To achieve these objectives, comprehensive modelling studies and multivariable distributed parameter, dynamic analysis, form a necessary prelude to controller design. Additional measurements and hence transducers are also required and the important choice of an appropriate, multiple input–output regulation strategy, must be exercised. In this regard, this submission presents a controller design study for the x-axis, Arrow 500 milling machine system, the configuration of which is shown in Fig. 1. The multivariable, hybrid model for this machine has been validated via measured results and manufacturer’s data enabling the construction of an accurate input–output realization, for analysis purposes.
Spindle motor
Spindle
Fx Cutting force Cutter Lead-screw
v (s)
Work piece
Forward bearing
Saddle velocity
M, B
Slide D.C Motor
Ball nut
d T
Ti Output end torque
To Input end torque
lt Rear bearing
Coupling Va (t) Vf (t) Motor field Voltage
Motor armature Voltage
a0 Angular velocity
0 v0
Output end stress
l
ai i
Input end stress
vi
Longitudinal velocity
Longitudinal velocity Fig. 1. Machine tool x-axis traverse and drive.
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Initially, a simple representation of this system model was employed in the design of a suitable minimum effort, multivariable controller. This regulator is relatively insensitive to parameter variations whilst generating a cooperative feedback strategy, requiring least energy, commensurate with the disturbance suppression, recovery and dynamic response characteristics specified. Evaluation of the controller for a range of feedback gains and pre-compensators was undertaken. Thereafter, the effect of input and load disturbances will be investigated and the performance of this system under closed-loop conditions will be presented prior to incorporation and integration with the manufacturing system. 2. Axis dynamic model A detailed derivation of the distributed-lumped, non-linear x-axis dynamics for the Arrow 500 milling machine is presented in Whalley et al. [4]. The arrangement of the system, with a separately excited D.C. motor drive, is as shown in Fig. 1. A block diagram representation of the drive is as Fig. 2, where the transfer functions, non-linear elements and gain parameters are indicated. The transfer function representations in Fig. 2 are: R0 ðsÞ ¼
ðz þ RÞe2GðsÞl þ ðz RÞ , zfðz þ RÞe2GðsÞl þ ðR zÞg
(2.1)
R1 ðsÞ ¼
2zeGðsÞl , ðz þ RÞe2GðsÞl þ ðz RÞ
(2.2)
F 0 ðsÞ ¼
¯ ¯ 2GðsÞl ¯ ðz¯ þ RÞe þ ðz¯ RÞ , ¯ ¯ z¯ þ RÞe ¯ ¯ 2GðsÞl þ ðR ¯ zÞg zfð
(2.3)
Va (s)
+
1
-
ω0 (s)
Back emf
τs + 1
ω1(s) k3
ω0(s)
T0(s)
Ra
Input torque to lead-screw
+
Ra(s)
ω1(s)
R1(s)
Lf s + Rf
− Tb Load torque
σ0 (s)
(s)
1
E
Stress
F0 (s)
Strain
Longitudinal velocity owing to lead-screw drive 0(s) end tensile stress
F1(s)
+
kp
Lead-screw total Force change on + lead-screw Ball nut displacement lateral stiffness Fx (s) Cutting force
Saddle velocity
1
+ Force for acceleration
Saddle transfer function (s) Coulomb friction
ks
x direction velocity of ball nut
Nut end longitudinal velocity
kL
Saddle velocity
1
ω1(s)
x (s)
1(s)
1
x (s)
Traverse drive force − Series stiffness of Lead-screw and Ball nut (s)
+
1(s) +
Saddle and ball nut displacement difference
−
1 Ms + B
Velocity difference
Twist difference
Ball nut thread gain -
s
s
k
Length shortening
+
(s)
1
(s)
(s)
0(s)
f (s)
+
Ls
Backlash
Force to torque constant
−
Lead-screw angular velocity at nut end
+
k1
Lead-screw input end stress
Vf (s)
Lead-screw angular velocity at drive end
k2
+
s
x(s)
Lead-screw length change −
Saddle displacement s
Fig. 2. The distributed-lumped simulation model block diagram of the machine tool axis drive system.
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¯
F 1 ðsÞ ¼
¯ GðsÞl 2ze . 2 GðsÞl ¯ ¯ þ ðz¯ RÞ ¯ ðz¯ þ RÞe
(2.4)
The parameters for R, z; GðsÞ etc. in Eqs. (2.1)–(2.4) are shown in the Nomenclature list with the numerical values for the Arrow 500 machine. The open loop response of the model, following input changes on the motor field voltage and then on the armature voltage, can be obtained from the block diagram simulation. These traces have been employed in the construction of the transfer function matrix, presented in Eq. (2.5), which is formulated in terms of percentage input and output variations, where: (2.5)
yðsÞ ¼ GðsÞuðsÞ. In Eq. (2.5): " GðsÞ ¼
g11 ðsÞ
g12 ðsÞ
g21 ðsÞ
g22 ðsÞ
#
yðsÞ ¼ ½%nðsÞ; %fðsÞT ;
,
uðsÞ ¼ ½%nf ðsÞ; %na ðsÞT
with g11 ðsÞ7
34:5 e0:001s ; ðs þ 37:7107Þ
g12 ðsÞ7
85:5 e0:001s ; ðs þ 100:0Þ
g21 7
0:3 ðs þ 30:0Þ ðs þ 37:7107Þ
and
g22 7
0:48 ðs þ 42Þ : ðs þ 100:0Þ
The model responses following 1% step input changes on the %Vf (s), %Va(s), are shown in Figs. 3A and B, respectively. The measured results, for reference purposes, are also included for comparison with the simulated, transfer function responses, for the Arrow 500 machine. In order that simultaneous table velocity nðsÞ and lead-screw wind-up fðsÞ regulation can be applied, both field and armature voltage controls are required. Thereafter, multivariable control improving dynamic response, and disturbance suppression whilst diminishing load offset, elevating machining quality thereby, will be promoted. Implementation of the regulation strategy will also be simple. In this regard the measurement of motor speed via a tachometer or alternator could be easily provided. Equally, by using strain gauges and slip ring connections, at the motor end of the lead-screw, a voltage proportional to the lead-screw wind-up could be determined by subtracting the strain gauge, voltage outputs. Owing to the compliant elements, comprising the x-axis milling machine drive, the measured transients, shown in Figs. 3A and B, are noisy. Moreover, the results shown are for no load, open-loop operations, with the expectation that heightened excitation, owing to cutting force disturbances would occur, during production runs. In this regard milling operations would generate ‘‘noisy’’, periodic, cyclic, load excitation and internally generated dynamic perturbations. To counter this, closed-loop regulation suppressing load and externally imposed disturbances becomes necessary. The theoretical procedures following, introduce regulation strategies which are specifically designed to achieve limited output coupling, disturbance suppression and recovery whilst improving the speed of response of the system. This regulation will be accommodated with the dissipation of minimum control energy, via the provision of conventional pre and feedback compensators, as specified by Rosenbrock [5], in the final feedback configuration, guaranteeing robust, simple, cost effective implementation. 3. Closed-loop regulation strategy The following procedure employs an inner and outer loop structure for design purposes. Once this design has been established a conventional, multivariable feedback realization will be evaluated. The outer loop controller will primarily be employed to secure specified steady state, reference set point and disturbance containment characteristics. The inner loop will be utilized to attain initial, targeted, dynamic, and load disturbance recovery rates, where the system equation, including disturbance inputs is yðsÞ ¼ GðsÞuðsÞ þ dðsÞ.
(3.1)
The control law for the proposed configuration, for analysis purposes, is uðsÞ ¼ kðsÞðrðsÞ hðsÞyðsÞÞ þ PðrðsÞ FyðsÞÞ,
(3.2)
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Fig. 3. (A) Measured and transfer function responses following a 1% step change in the field voltage. (B) Measured and transfer function responses following a 1% step change in the armature voltage.
where in Eq. (3.1) there are m independent inputs, disturbances and outputs and: F ¼ Diagðf 1 ; f 2 ; . . . f m Þ;
0of j o1;
1pjpm.
With r¯ðsÞ ¼ 0 the closed-loop equation becomes: yðsÞ ¼ ðIm þ GðsÞðkðsÞihhðsÞ þ PFÞÞ1 ðGðsÞPrðsÞ þ dðsÞÞ.
(3.3)
If a steady state matrix Ss is now selected such that: yð0Þ ¼ Ss rð0Þ then from Eq (3.3), with dðsÞ null, the required pre-compensator is P ¼ ðGð0Þ1 þ kð0Þihhð0ÞÞSs ðI FSs Þ1 .
(3.4)
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To achieve ‘‘low’’ steady state interaction Ss would have elements jsi;j j5jsjj j; 1pi; jpm; iaj. Consequently, following the specification of closed-loop, low interaction and substituting for P from Eq. (3.4), results in Eq. (3.3) becoming (3.5) yðsÞ ¼ fIm þ GðsÞ½kðsÞihhðsÞ þ Gð0Þ1 þ kðsÞihhðsÞ ðIm FÞ1 Fg1 fGðsÞPrðsÞ þ dðsÞg. At low frequencies Eq. (3.5) may be approximated by yðsÞ ffi f½Im þ GðsÞkðsÞihhðsÞ½Im þ ðIm FÞ1 Fg1 fGðsÞPrðsÞ þ dðsÞg
(3.6)
If now the elements of F are f1 ¼ f2 ¼ fm ¼ f;
0of o1:0
Eq. (3.6) simplifies to yðsÞ ffi ð1 f Þ½Im þ GðsÞkðsÞihhðsÞ1 ½GðsÞP rðsÞ þ dðsÞ.
(3.7)
In Eq. (3.7) since, from Eq. (3.4): GðsÞP ¼ GðsÞ½Gð0Þ1 þ kð0Þihhð0Þ½Im F1 at low frequencies GðsÞP ffi
1 ðIm þ GðsÞkð0Þihhð0ÞÞ. ð1 f Þ
Consequently, Eq. (3.7), on approaching steady state conditions becomes: (3.8)
yðsÞ ¼ IrðsÞ þ SðsÞdðsÞ, where in Eq. (3.8) the low frequency sensitivity matrix is SðsÞ ¼ ð1 f ÞðIm þ GðsÞkðsÞihhðsÞÞ1 ;
0of o1.
Evidently, from Eq. (3.8), low steady state, output interaction, following reference input changes would be achieved. Moreover, as f is increased from zero there would be increasing, steady state, disturbance rejection with the maintenance of ‘‘adequate’’ stability margins. The design strategy to be adopted here will be to adjust the inner loop k(s) and h(s) vectors providing thereby ‘‘well behaved’’ dynamic conditions. Thereafter, with the pre-compensator P, configured to produce acceptable output coupling, the outer loop feedback gain f becomes the final design parameter enabling the achievement of desired, dynamic and disturbance suppression conditions. 4. Inner loop analysis The Laplace transformed open loop system, given by Eq. (3.1), where the system model G(s) is assumed to be an m m linear, proper or strictly proper realization, admits a factorization: GðsÞ ¼ CL ðsÞLðsÞ
AðsÞ RðsÞCR ðsÞ; dðsÞ
(4.1)
where the elements of LðsÞ; RðsÞ; CL ðsÞCR ðsÞ and
AðsÞ 2 H 1 ; s 2 C. dðsÞ
In Eq. (4.1), L(s) contain the left (row) factors, R(s) contains the right (column) factors and CL(s) and CR(s) contain the transformed, system and actuator, finite time delays, respectively, such that the m m matrices comprising (4.1), as in Whalley and Ebrahimi [6], are LðsÞ ¼ Diagðlj ðsÞ=pj ðsÞÞ; RðsÞ ¼ Diagðrj ðsÞ=qj ðsÞÞ; CR ðsÞ ¼ Diag esT j ; CL ðsÞ ¼ Diag esT k , and A(s) is a non-singular matrix of rational functions, such that det A(s)6¼0, with elements: aij ðsÞ ¼ aij sm1 þ bij sm2 þ gij ;
1pi; jpm.
(4.2)
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Since the transformed input–output-disturbance relationship is (4.3)
yðsÞ ¼ GðsÞuðsÞ þ dðsÞ and if the inner loop control law is uðsÞ ¼ kðsÞð¯rðsÞ hðsÞyðsÞÞ
(4.4)
then combining Eqs. (4.3) and (4.4) yields yðsÞ ¼ ðIm þ GðsÞkðsÞihhðsÞÞ1 ðGðsÞkðsÞ¯rðsÞ þ dðsÞÞ.
(4.5)
The finite time delays in CR(s) may be ordered with TiXTj, 1pjpm i6¼j, so that the forward path gain vector can be arranged as T (4.6) kðsÞ ¼ k1 ðsÞesðT i T 1 Þ ; k2 ðsÞesðT i T 2 Þ . . . ki ðsÞ . . . km ðsÞesðT i T m Þ and if T l XT k ;
1pkpm
lak
T hðsÞ ¼ h1 ðsÞesðT l T 1 Þ ; h2 ðsÞesðT l T 2 Þ ; . . . hl ðsÞ . . . hm ðsÞesðT l T m Þ ,
(4.7)
then if kj ðsÞ ¼ kj fj ðsÞ and
hj ðsÞ ¼ hj wj ðsÞ;
1pjpm.
where fj ðsÞ and wj ðsÞ are proper or strictly proper, stable, realizable, minimum phase, functions then they may be selected such that the determinant of Eq. (4.5), as shown in Whalley [7] is AðsÞ AðsÞ sðT i þT¯ l Þ sðT i þT¯ l Þ kihh ¼ 1 þ e k , (4.8) det Im þ e nðsÞ nðsÞ h dðsÞ dðsÞ where in Eq. (4.8): k ¼ ðk1 ; k2 . . . km ÞT ,
(4.9)
h ¼ ðh1 ; h2 . . . hm Þ
(4.10)
dðsÞ ¼ sk þ a1 sk1 þ a0 and degðnðsÞai;j ðsÞÞok;
1pi; jpm.
The inner product in Eq. (4.8) may be 2 g11 g12 6 . ..
6 . . m1 6 . hAðsÞk ¼ 1; s; . . . s 6 6 b11 b12 4 a11 a12
(4.11) expressed as 32 3 . . . gmm k 1 h1 7 .. 76 k2 h1 7 7 . 76 7 76 6 .. 7. 7 . . . bmm 54 . 5 k m hm . . . amm
(4.12)
If in Eq. (4.12) the gain ratios are k 2 ¼ n1 k 1 ;
k3 ¼ n2 k1 ; . . . km ¼ nm1 k1 ;
k1 a0
(4.13)
and:
hAðsÞk ¼ bðsÞ,
(4.14)
then Eq. (4.14) implies that k1 ½Qh ¼ ðbm1 ; bm2 ; . . . ; b0 ÞT ,
(4.15)
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where in Eq. (4.15): 2 6 g11 þ g12 n1 þ g1m nm1 6 .. 6 . 6 Q¼6 6 6 b11 þ b12 n1 þ b1m nm1 4 a11 þ a12 n1 þ a1m nm1
.. . g21 þ g22 n1 þ g2m nm1 .. .. . . .. . b21 þ b22 n1 þ b2m nm1 .. . a21 þ a22 n1 þ a2m nm1
1843
3 .. . gm1 þ gm2 n1 þ gmm nm1 7 7 .. .. 7 . . 7 7 .. 7 . bm1 þ bm2 n1 þ amm nm1 7 5 .. . am1 þ am2 n1 þ amm nm1
and bj ; 0pjpm 1; are the coefficients of b(s), given in Eq. (4.14). Providing the very weak constraint that n1 ; n2 . . . nm1 can be selected, in Eq. (4.15), so that the resulting Q matrix is invertible, then a unique solution for ðh1 ; h2 ; . . . hm Þk1 exists. Following the choice of a suitable bðsÞ function and the gain ratios n1 ; n2 nm1 , the closed-loop dynamics arising from Eq. (4.5) may be easily computed. Upon satisfying Eq. (4.15) the measurement vector h can be evaluated, once an arbitrary value for k1, k16¼0, has been decided upon. It is apparent from Eq. (4.15) that there will be many values of nj ; 1pjpm 1 which could be selected enabling the inversion of Q. However, choosing these gain ratios arbitrarily, affects the regulation efficiency, the robustness of the feedback configuration to parameter changes and the disturbance rejection properties of the system. To avoid this a systematic, scientific optimization procedure will be employed, following Section 5. 5. Disturbance rejection analysis In practice some form of approximate integration over a limited frequency range, such as phase lag compensation, is usually employed to effect disturbance recovery. To confine output excursions, to input disturbances, the maximum proportional feedback gain, commensurate with acceptable transient conditions and modelling/parameter uncertainties, may be employed. In the following, bounds on the disturbance response, arising from changes in the sensitivity matrix, are determined. This enables the formulation of a functional, J, from which variations in the system’s closed-loop disturbance attenuation can be established. Gain ratios for which there are rapid changes in the performance index J and hence in the disturbance containment capacity of the system, also become apparent. These values indicate a lack of robustness to modelling variations and should be avoided. In many process and military systems the only input signals which are continuously perturbed arise from changes in d(s), so that with the outer loop set points rðsÞ ¼ 0, Eq. (3.8) becomes: (5.1)
yðsÞ ¼ SðsÞdðsÞ, where SðsÞ ¼ ðIm þ GðsÞðkðsÞihhðsÞ þ PFÞÞ1 .
The sensitivity function S(s), in Eq. (5.1), should be such that changes in dðsÞ result in acceptable changes in y(s). These output excursions arise from relatively low frequency variations in dðsÞ, as discussed by Skogestad and Postlethwaite [8]. Moreover, since for physically realisable systems, G(s) must be strictly proper, the closed-loop system provides no attenuation to high frequency disturbances, as SðsÞ ! Im . Consequently, the adjustment of the steady state system, disturbance suppression properties, by increasing f in Eq. (3.8), would ensure enhanced attenuation following low frequency perturbations. Moreover, for increasing values of frequency the singular values of S(io) should be smooth, differentiable and almost monotonic curves, providing a prudent policy of regulation is pursued. The disturbance attenuation, secured by the system, lies between the lower and upper singular values of the S(s) matrix, as shown by Zames [9]. Specifically,
yðsÞ
2 lðSðsÞÞp (5.2)
dðsÞ plðSðsÞÞ, 2
where in Eq. (5.2), lðSðsÞÞ and lðSðsÞÞ are the smallest and largest, singular value amplitudes of S(s), respectively, and kk2 denotes the Euclidean norm, s ¼ io; 0pop1 of y(s) and dðsÞ, respectively. It is also apparent from Eq. (5.2) that providing the upper bound
lðSðsÞÞp3dB, s ¼ io; 0pop1, the system will be stable with a maximum ‘‘high’’ frequency resonant peak p1:4; dðsÞ 2 a0, with the closed-loop outputs being substantially decoupled.
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The singular values of the low frequency sensitivity matrix, given in Eq. (3.8), can be computed in the frequency domain, for f ¼ 0, from det½Im l ðI þ GðsÞðkðsÞihhðsÞÞÞ1 s¼io ¼ 0. The steady state singular values may be estimated from: det½Im l ðI þ Gð0Þðkð0Þihhð0ÞÞÞ1 ¼ 0,
(5.3)
since Gð0Þkð0Þihhð0Þ has singular values of g1 ¼ g2 ¼ gm1 ¼ 0
and
gm ¼
m X
k j hj ,
j¼1
where (5.4)
k ¼ Gð0Þkð0Þ and kj is the jth element of k. Theorem 1. The singular value amplitudes li ; 1pipm, of Eq. (5.3) are: l1 ¼ l2 ; ¼ lm1 ¼ 1
and
lm ¼
1þ
1 Pm
j¼1 kj hj ð0Þ
.
(5.5)
The proof is given in Whalley and Ebrahimi [10]. As shown by Eq. (3.8), SðsÞ ¼ ð1 f ÞðI þ GðsÞkðsÞihhðsÞÞ1 so that the low frequency singular value amplitudes of S(s) are reduced to lj where 1pjpm; f o1.
lj ¼ ð1 f Þlj ;
Consequently, the bound on the steady state attenuation properties of the system becomes
yð0Þ
2 lðSð0ÞÞp
dð0Þ plðSð0ÞÞ.
(5.6)
(5.7)
2
The estimate from Eq. (5.7) is conservative. An accurate measure of the low frequency disturbance attenuation properties of the system can be easily computed from the direct evaluation of Eq. (5.1). In this regard if: sðSð0ÞÞ ¼ largest element of Sð0Þ, sðSð0ÞÞ ¼ smallest element of Sð0Þ, then
yð0Þ
2 sðSð0ÞÞp
dð0Þ psðSð0ÞÞ 2
provides an improved, low frequency, attenuation bound. The employment of an absolute, minimum effort controller, whilst attaining a particular inner closed-loop pole pattern, would not, in general achieve specified, steady state disturbance recovery conditions. However, the disturbance recovery transient would of course, be determined by this pole configuration. To achieve the steady state disturbance recovery effect required, the outer loop feedback gain f could be adjusted, as indicated by Eq. (5.6). This would perturb the system’s transient behaviour and recognition of this should be exercised by aiming for safe, modest response improvements arising from the inner loop dynamics. Thereafter, the control effort required would be the relative minimum effort, commensurate with the achievement of particular steady state, disturbance suppression conditions. As shown by Eq. (3.6), increasing f from zero is instrumental in increasing k(s)S /h(s) to kðsÞihhðsÞ=ð1 f Þ. This is equivalent to increasing b0 of Eq. (4.15) to b0/(1f) elongating and elevating thereby, the performance index J curve, so that J min is moved upwards whilst retaining the original optimum, minimum effort gain ratio value of n. The obvious conclusion from this is that in the absence of all other inputs the control effort expended is devoted entirely to disturbance recovery. Consequently the performance index J, being proportional to the control effort, is also a direct measure of this condition.
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In aiming for this minimum J, for a given pole configuration, the disturbance recovery transient is executed with maximum energy efficiency. By closing the outer loop, upon increasing f from zero, the closed-loop disturbance suppression would be enhanced, whilst perturbing the closed-loop pole pattern. However, providing the S(io) curves are not significantly altered the dominant poles and optimum performance, for low frequency disturbances, prevails. As Skogestad and Postlethwaite [8] observe, any intermediate frequency amplification indicated by the singular values of S(io) should be restricted to o3dB’s. This is equivalent to limiting the closed-loop dynamic magnification, when designing single input–output systems, by confining M (peak) to 1.4 (scalar). However, as stated earlier, this is a conservative measure and the 3 dB limit can normally be transgressed, without immediately affecting the dynamic performance. 6. Optimization Now that a route for designing closed-loop systems, using the transfer function matrix and output measurements has been established, via the theory outlined earlier, the possibility of optimizing this process can be considered. An indication that the freedom exists to do this arises from the arbitrary choice of n1, n2 etc., for the gain ratios. Detecting the absolute minimum control effort required for disturbance suppression under inner loop, closed-loop conditions, with the constraint that the controller model generates a particular polynomial, would provide a useful benchmark result. This polynomial influences the migration pattern of the closed-loop poles so that control effort minimization, whilst attaining the desired system response to input and disturbance changes, could be achieved simultaneously. The controller equation for a system having m inputs and m outputs is given by Eq. (4.4). Consequently, the control effort, at time t, with zero set point changes is ðjk1 h1 j þ jk2 h1 j :jkm h1 jÞjy1 ðtÞj þ ðjk1 h2 j þ jkm h2 jÞjy2 ðtÞj þ ðjk1 hm j þ jk2 hm j þ jkm hm jÞjym ðtÞj Hence, the control energy costs, from zero to Tm, are proportional to ! Z t¼T m X m m X 2 2 2 EðtÞ ¼ ki hj yj ðtÞ dt. t¼0
i¼1
(6.1)
j¼1
For arbitrary changes in the output vector y(t), following arbitrary disturbance changes, minimizing: J¼
m X
h2i
i¼1
m X
k2j
(6.2)
i¼1
would minimize the control energy given by expression (6.1). If the relationships: k3 ¼ n2 k1 ; . . . km ¼ nm1 k1 ; k1 a0
k 2 ¼ n1 k 1 ;
are proposed, then Eq. (6.2) can be written as J ¼ ðk1 Þ2 ð1 þ n21 þ n22 . . . n2m1 Þðh21 þ h22 . . . h2m Þ h21
h22
(6.3)
. . . h2m
and þ ¼ hh; hi. The closed-loop determinant given by Eq. (4.8) with the inner product equated to bðsÞ, as in Eq. (4.14), enables the measurement gains to be determined from Eq. (4.15), where 1 h ¼ k1 1 Q b.
(6.4)
Upon substituting for h from Eq. (6.4), Eq. (6.3) becomes J ¼ ð1 þ n21 þ n22 þ n2m1 ÞbT ðQ1 ÞT Q1 b.
(6.5)
To find the minimum value for J, assuming, for example, that m ¼ 3 gives J ¼ ð1 þ n21 þ n22 ÞbT ðQ1 ÞT Q1 b, where J is minimized when qJ ¼ 0; qn1
qJ ¼0 qn2
and
2 2 q2 J q2 J qJ 40, 2 2 qn1 qn2 qn1 qn2
if ðq2 J=qn21 Þ40: For m43 a numerical minimisation routine would have to be employed to establish the values of n1, n2 y nm1 which minimize J. There are many procedures available for this task, as indicated by Bunday [11], with rapid convergence and high accuracy characteristics. This completes the optimization analysis.
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7. System stability and implementation The input–output relationship for the complete closed-loop system is given by Eq. (3.5). Consequently, the input–output stability condition is dependent on the denominator of this equation. If, for the sake of simplicity, the outer loop feedback gain matrix F is given by F ¼ Diagðf 1 ; f 2 ; . . . f m Þ and if, once again: f1 ¼ f2 ¼ ¼ fm ¼ f;
0of o1;
then the denominator of Eq. (3.5) may be calculated from: kðsÞihhðsÞ Gð0Þ1 f þ det Im þ GðsÞ ð1 f Þ ð1 f Þ
(7.1)
from Eq. (7.1) it is evident that the elements of the feedback–compensator matrix: kðsÞihhðsÞ Gð0Þ1 f þ ð1 f Þ ð1 f Þ become infinite as f ! 1. In practice this would always result in closed-loop system instability and the restriction 0ofo1 applies. For values of f p0:5 the inner loop feedback gain is amplified whilst the effect of the outer loop, arising from G(0)1f/ (1f), is attenuated. Once 0:5of o1:0 both the inner and outer loop gains f/(1f) and 1/(If) become increasingly large. Essentially, amplifying k(s)S/h(s) by 1/(1f), fo1, is equivalent, as stated earlier, to multiplying b0 in Eq. (4.15) by this amount. Consequently, the value of J achieved initially, from the optimization process, is increased further by the use of the outer loop feedback elevating it to J/(1f), whilst the optimum, least effort, gain ratio remains unchanged at n with k becoming, for m ¼ 2, (1, n)T/(1f), for stability assessment purposes. Owing to these factors the system’s disturbance suppression properties are enhanced. Hence, by setting the inner loop frequency response or characteristic equation, such that the amplification of k(s)S/h(s) to k(s)S/h(s)/(1f) can be safely accommodated, dynamic recovery and steady state disturbance rejection can be improved, using the outer loop feedback gain f. Once this exercise has been completed a conventional closed-loop, multivariable feedback structure can be formulated, for implementation purposes. If the forward and feedback path matrices are K(s) and H(s), respectively, then the closedloop relationship becomes: yðsÞ ¼ ðIm þ GðsÞKðsÞHðsÞÞ1 ðGðsÞKðsÞrðsÞ þ dðsÞÞ.
(7.2)
Comparing Eqs. (7.2) and (3.3) it is clear that KðsÞ ¼ P
(7.3)
HðsÞ ¼ P-1 kðsÞihhðsÞ þ F
(7.4)
and
enabling the adoption of the simple feedback configuration, given in Dutton et al. [12], for the least energy, optimum, regulator. 8. Application study To apply the design procedures, the machine tool model presented earlier will be considered, demonstrating effectiveness thereby. For this system model, for the Arrow 500 milling machine: yðsÞ ¼ GðsÞuðsÞ þ dðsÞ; where in Eq. (8.1), from Eq. (2.5): yðsÞ ¼ ðy1 ðsÞ; y2 ðsÞÞT are the Laplace transformed output signals: uðsÞ ¼ ðu1 ðsÞ; u2 ðsÞÞT
(8.1)
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are the Laplace transformed, input signals and: " # g11 ðsÞ g12 ðsÞ GðsÞ ¼ , g21 ðsÞ g22 ðsÞ where gi,j(s), 1pi, jp2, are the transfer functions shown in Eqs. (2.5) and (d1 ðsÞ; d2 ðsÞ)T are the disturbance signals. In this application, the closed-loop system is required to reduce the steady state interaction to a maximum of 25%. Moreover, the velocity response time should be such that steady state conditions can be achieved in approximately 0.1 s, without overshoot problems. Minimization of the steady state output variations to unit step changes, on d1 ðtÞ and d2 ðtÞ, using proportional-passive network compensation alone should be investigated. Enhancement to achieve sðSð0ÞÞp0:4 should be attained by increasing the outer loop feedback with quiescence and disturbance recovery in less than 0.1 s. 9. Inner loop design In order to shape the frequency response characteristics of the inner loop suitable compensators must be selected. First, the system model given in Eq. (8.1) can be arranged as AðsÞ GðsÞ ¼ CL ðsÞLðsÞ RðsÞCR ðsÞ, (9.1) dðsÞ where in Eq. (9.1) a structure similar to Eq. (4.1) can be achieved with dðsÞ ¼ ðs þ 37:7107Þ CL ðsÞ ¼ Diagðe0:001s ; 1Þ; " AðsÞ ¼
RðsÞ ¼ Diag 1; ðs þ 37:7107Þ=ðs þ 100:0Þ ,
34:5
85:5
0:3ðs þ 30:0Þ
0:48ðs þ 42Þ
#
and LðsÞ ¼ CR ðsÞ ¼ I2 . In accordance with Eq. (4.6), for this study kðsÞ ¼ KðsÞðk1 ; k2 ÞT ,
(9.2)
where in Eq. (9.2): KðsÞ ¼ Diagð1; ðs þ 100:0Þ=ðs þ 37:7107ÞÞ and hðsÞ ¼ ðh1 ; h2 ÞHðsÞ,
(9.3)
where in Eq. (9.3): HðsÞ ¼ Diagð1:0; e0:001s Þ. Consequently the equivalent of Eq. (4.12) here is " # 34:5 85:5 AðsÞ 0:3ðs þ 30:0Þ 0:48ðs þ 42Þ k ¼h ke0:001s , hðsÞ dðsÞ ðs þ 37:7107Þ
(9.4)
where Eq. (9.4) has the form of the inner product of Eq. (4.14), where h ¼ ðh1 ; h2 Þ and k ¼ ðk1 ; k2 ÞT . Then an appropriate function b(s) may be selected for the machine tool model. 10. Determination of the controller generated zero If Eq. (9.4) is now considered, then as in Eq. (4.14), but including the denominator d(s): AðsÞ bðsÞ 0:001s hðsÞ k ¼ e . dðsÞ dðsÞ
(10.1)
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Hence, Eq. (10.1) becomes the characteristic equation: 1 ¼ bðsÞe0:001s dðsÞ
(10.2)
and a suitable b(s) function could be selected from consideration of the root locus plot of Eq. (10.2) neglecting the small finite time delay. It is evident that if bðsÞ ¼ b0 ðs þ 5000Þ
(10.3)
and with d(s) from Eq. (8.5) the root locus would be in respect of 1 ¼
ðs þ 5000Þb0 ðs þ 37:7107Þ
and with b0 ¼ 0:02.
(10.4)
The characteristic equation would be 1 ¼
ð0:02s þ 100Þe0:001s , ðs þ 37:7107Þ
with adequate stability margins. The choice of a ‘‘small’’ value for b0 also improves the controller’s robustness, by restricting the control energy dissipation and hence performance index J, as shown in Section 11. 11. Optimization and simulation Following the attainment of the modest improvements in the system dynamics, from the closure of the inner loop, the corresponding minimum effort performance index, since m ¼ 2, becomes: J ¼ ð1 þ n2 ÞbT ðQ1 ÞT Q1 b.
(11.1)
From A(s), given by Eq. (9.1), the Q matrix for this system can now be formulated as " # 37:7107 þ 85:5n 9 þ 20:16n Q¼ 0 0:3 þ 0:48n and from Eqs. (10.3) and (10.4): 5000 b¼ b0 ; b0 ¼ 0:02: 1
(11.3)
Evaluation, using Eqs. (11.2) and (11.3) results in Eq. (11.1) becoming: J¼
(11.2)
2268:3795n4 þ 2841:033n3 þ 3158:088n2 þ 2841:033n þ 889:7085 ð41:04n2 þ 42:21n þ 10:35Þ2
Fig. 4. Performance index J against gain ratio.
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and dJ 74900:6n4 45384:95n3 261573:9n2 200602:13n 45704:5 ¼ . 3 dn ð41:04n2 þ 42:21n þ 10:35Þ
(11.4)
Equating dJ=dn, from Eq. (11.4), to zero: reveals that the extremum values of J arise when: n ¼ 2:4849; 0:59905
and
0:63994 0:019959i:
Fig. 5. Inner loop response following a unit input change on r¯ðtÞ.
f δ1(s) 1
− r1(s)
2
+
20 s+1
−
s + 37.7107
+
P22
+
+
y1(s)
+
0.3(s + 30) s + 37.7101
r(s) + −
P12 r2(s)
e-0.001s
+
P21
+
34.5 +
P11
0.04s + 1
+
+
85.5 s + 100
+
2.4849 (s+100)
+
e-0.001s
+
+ 0.48(s + 42) s + 100
s + 37.7107
+ + δ2(s)
+
0.40172
+ 0.013398
f Fig. 6. Closed-loop system block diagram.
y2(s)
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The graph of the performance index J against gain ratio n, presented in Fig. 4, shows that the absolute minimum value of J is attained, for bðsÞ ¼ 0:02s þ 100, when n ¼ 2:4849, with the constraint that the inner, closed-loop pole is located at, s ¼ 135:011. The feedback measurement gains can now be computed, as in Eq. (4.15), with Q and b from Eqs. (11.2) and (11.3) yielding, with b0 ¼ 0:02 and n ¼ 2:4849. ðh1 ; h2 Þ ¼ ð0:40172; 0:013398Þ,
(11.5)
where k1, in Eq. (11.5) can be selected arbitrarily, initially as unity. This value can be employed subsequently to ‘‘balance’’ the levels of disturbance suppression achieved in each of the output channels, if so desired. The responses from the inner loop following a unit change on rðtÞare shown in Fig. 5. These outputs exhibit a faster transient response reaction time though substantial steady state output interaction is evident. All that remains now is the computation, in accordance with Eq. (3.4), of the pre-compensator P. If here, the F matrix values selected are F ¼ Diagð0:25; 0:25Þ; Diagð0:50; 0:50Þ and Diagð0:75; 0:75Þ
(11.6)
and since steady state interaction between outputs must be limited to o725% then: 0:5 0:0125 Ss ¼ 0:1 0:05 could be specified. This would restrict the amplitude of the pre-compensator elements improving the tolerance of the feedback system to non-linear variations in the actual machine tool system application. Inserting these values of G(0) Ss, F, k(0) and h(0) into Eq. (3.4) results in: 0:66873 2:13 0:60079 2:0805 0:62994 2:1051 P¼ ; and (11.7) 3:0831 2:1595 3:5239 2:1745 4:1404 2:1853 δ1(s) r1(s)
+
2 0.04s + 1
K11
+
34.5 (s + 37.7107)
+
−
s+1
+ K22
+
y1(s)
+
(s + 37.7101)
85.5
K12
20
+
0.3(s) + 30
K21
r2(s)
e-0.001s
(s + 100)
+
+
0.48(s +42) (s +100)
−
e-0.001s
+
+ +
δ2(s) + +
H11(s)
H21(s)
H12(s) +
+
H22(s)
Fig. 7. Conventional closed-loop system block diagram.
y2(s)
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y1(t)
f = 0.25
1
1851
f = 0.5
Output
0.8
f = 0.75
0.6 0.4 f = 0.25 y2(t)
f = 0.5
0.2
f = 0.75
0 0
0.05
0.1
0.15
(A)
0.2 0.25 0.3 Time (sec)
0.35
0.4
0.45
0.5
1.2 f = 0.25, f = 0.5 and f = 0.75
1
y2(t)
Output
0.8 0.6 0.4 y1(t) f = 0.75
0.2
f = 0.5
0
f = 0.25
-0.2 0 (B)
1
2
3
4
5 6 Time (sec)
7
8
9
10
Fig. 8. (A) Closed-loop output response following a unit input step change on r1 ðtÞ. (B) Closed-loop output response following a unit input step change on r2 ðtÞ.
for the selected F matrices, respectively, fully defining the feedback configuration. This results in the block diagram for the close loop system shown in Fig. 6. Implementation in terms of conventional forward and feedback path compensators may now be proposed. In this case, in accordance with Eqs. (7.3) and (7.4), for the F matrices from Eq. (11.6) with P from Eq. (11.7): K¼P and with k ¼ [1,2.4849]T and h(s) from and (11.5) yielding, respectively, 2 0:57541ðsþ81:65Þ 3 0:019191ðsþ81:65Þeð0:001sÞ þ 0:25 ðsþ37:7107Þ ðsþ37:7107Þ 5; HðsÞ ¼ 4 0:35925ðsþ58:03Þ 0:011982ðsþ58:03Þeð0:001sÞ þ 0:25 ðsþ37:7107Þ ðsþ37:7107Þ 2 0:49185ðsþ81:71Þ 3 0:016404ðsþ81:71Þeð0:001sÞ ðsþ37:7107Þ þ 0:50 ðsþ37:7107Þ 5; HðsÞ ¼ 4 0:33802ðsþ56:87Þ 0:011274ðsþ56:87Þeð0:001sÞ þ 0:50 ðsþ37:7107Þ ðsþ37:7107Þ 2 0:40829ðsþ81:8Þ 3 0:013617ðsþ81:8Þeð0:001sÞ ðsþ37:7107Þ þ 0:75 ðsþ37:7107Þ 5; HðsÞ ¼ 4 0:31679ðsþ55:55Þ 0:010565ðsþ55:55Þeð0:001sÞ þ 0:75 ðsþ37:7107Þ ðsþ37:7107Þ
(11.8)
(11.9)
where in Eq. (11.9) the small finite time delay can be ignored in view of the robustness of the regulation strategy employed.
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1 0.8 0.6 y1(t) , f = 0.25
Output
0.4
y1(t) , f = 0.5
0.2 0
y1(t) , f = 0.75
y2(t) , f = 0.75
-0.2
y2(t) , f = 0.5
-0.4
y2(t) , f = 0.25
-0.6 -0.8 -1 0
0.05
0.1
(A)
1.2
0.2
0.25
0.2
0.25
y2(t) , f = 0.25 y2(t) , f = 0.5
1 0.8 Output
0.15 Time (sec)
y2(t) , f = 0.75
0.6 0.4 y1(t) , f = 0.75
0.2
y1(t) , f = 0.5
0 -0.2 (B)
y1(t) , f = 0.25
0
0.05
0.1 0.15 Time (sec)
Fig. 9. (A) Closed-loop output response following a unit input step change on d1 ðtÞ. (B) Closed-loop output response following a unit input step change on d2 ðtÞ.
As Eqs. (11.8) and (11.9) indicate the forward and feedback compensators for this system are full rank matrices. This enables the minimal block diagram configuration, shown in Fig. 7, to be constructed, for implementation purposes where the elements of K and H(s) are Ki,j and Hi,j(s), 1pi; jp2, respectively. From the simulation of this, the output responses following input step changes of unity on r1(t) and r2(t), respectively, can be computed, as shown in Fig. 8A and B. These traces, depict acceptable response variations with no overshoot problems on the principal outputs with steady state interaction restricted to the limits specified. Thereafter, the disturbances offset graphs for inputs steps on d1 ðtÞ and then d2 ðtÞ are presented in Figs. 9A and B. The important table velocity, disturbance rejection characteristics of Fig. 9A, provides excellent recovery transients limiting the loading problem to 720%. 12. Conclusion The regulation issue addressed in this contribution was focused on the control of a machine tool, x-axis, traverse drive. Specifically, the simultaneous control of the work-piece table velocity, whilst confining the lead-screw twist angle was the stated objective. With the achievement of these specification requirements it was shown that: (i) (ii) (iii) (iv) (v) (vi)
Machining variations originating from lead-screw effective torsion-tension length changes would be reduced. Disturbances arising from arbitrary cutting forces would be diminished. Improved transient response and profile following would be available Machining quality, repeatability and accuracy would be enhanced. A simple, output feedback regulation strategy and structure alone, would be required. Implementation costs would be minimized.
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(vii) (viii) (ix) (x)
1853
Rapid cutting load offset recovery would be guaranteed. A robust, low gain, passive feedback configuration would be derived. Low steady state output cross-coupling would be achieved. Minimum maintenance, refit and operating costs and maximum machine tool availability would be attained.
In the initial modelling study both the measured and analytical results were employed in the determination of the axis drive, milling machine, multivariable system model, for control system design purposes. Owing to the robustness of the minimum effort, minimum energy design procedure a simple linear approximation adequately replicated the open-loop, measured response characteristics, for the initial viability study. Reflecting this, the feedback controller, when derived from the inner and outer loop regulators, was uncomplicated comprising little more than a pre compensator of constants. A feedback realization requiring only phase retard elements was also defined. With this optimum regulator the disturbance, suppression properties of the system could also be advanced by increasing f, the outer-loop feedback gain 0ofo1.0. Consequently, the system response for three values of f was evaluated allowing the establishment of the dynamic performance following disturbances and input, set point variations. In accordance with the implementation requirements the pre and feedback compensators for the system were computed, as indicated in Section 11, for f ¼ 0:25; 0:5 and 0:75. Owing to the robustness of the regulation strategy adopted, the feedback controller, finite time delays, were neglected, without significantly affecting the overall system performance, the final configuration of the closed-loop system comprising the conventional structure, shown in Fig. 7. The effectiveness of the controller is evident from the closed-loop response transients from Figs. 8A and B. These show well behaved dynamic characteristics with quiescence being achieved in approximately 0.1 s following input reference changes on r1(t). Equally, twist angle control can be imposed following changes on r2 ðtÞ, as shown in Fig. 9B, although this input would normally remain at zero. The disturbance suppression properties of the system are also pleasing. Fig. 9A shows there is increasing offset rejection as f is increased 0.25ofo0.75. The disturbance recovery times for y1(t) and y2(t) are also rapid and well within the specified limit, for table velocity disturbances d1 ðtÞ. Under normal operating conditions disturbances could not directly affect the lead-screw twist d2 ðtÞ. Consequently, this loading effect would normally be maintained at zero. The response transients shown in Fig. 9B are included for completeness. Equally, the application of step velocity, table disturbances, would not arise in practice as the cutting disturbances would impose additional load torque increases on Tb(s). Therefore, the severity of these loadings would be attenuated by the system elements, as shown in Fig. 2. As a consequence the disturbance response characteristics are in respect of standard benchmark validations, rather than operational conditions. References [1] I.K. Landerborough, A.G. Ulsay, Dynamic modelling and control of a milling process, ASME Journal on Industry 110 (1988) 367–375. [2] R. Whalley, H. Bartlett, M. Ebrahimi, Analytical solution of distributed-lumped parameter network models, Proceedings of IMechE, Part I 212 (13) (1997) 203–218. [3] R. Whalley, M. Ebrahimi, Z. Jamil, The torsional response of spatially dispensed transmission systems, Proceedings of IMechE, Part C 214 (2005) 100–120. [4] R. Whalley, M. Ebrahimi, A. Abdul-Ameer, Hybrid modelling of machine tool axis drives, International Journal of Machine Tools and Manufacture 1646 (2005) 1–17. [5] H.H. Rosenbrock, Computer Aided Control System Design, Academic Press, London, 1974. [6] R. Whalley, M. Ebrahimi, Control of a lamination process, Proceedings of IMechE, Part E 219 (2005) 69–81. [7] R. Whalley, The computation of 2D transfer functions, Proceedings of IMechE, Part I 205 (I1) (1991) 59–67. [8] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control Analysis and Design, Wiley, New York, 1996. [9] G. Zames, Feedback and optimal sensitivity: model reference transformation, multiplicative semi-norms and approximate inverses, IEEE Transactions on Automatic Control 26 (2) (1981) 301–320. [10] R. Whalley, M. Ebrahimi, Closed-loop system disturbance recovery, Proceedings of IMechE, Part C 217 (2003) 631–649. [11] B. Bunday, Basic Optimisation Methods, Edward Arnold, London, 1984. [12] K. Dutton, S. Thompson, R. Burraclough, The Art of Control Engineering, Addison Wesley, Harlow, 1997.